Complex Analytic Geometry: From the Localization Viewpoint 9789814374705, 9789814374712, 9789814704298

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World Scientific

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COMPLEX ANALYTIC GEOMETRY From the Localization Viewpoint Copyright © 2024 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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ISBN 978-981-4374-70-5 (hardcover) ISBN 978-981-4374-71-2 (ebook for institutions) ISBN 978-981-4704-29-8 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/8324#t=suppl Desk Editors: Nimal Koliyat/Lai Fun Kwong Typeset by Stallion Press Email: [email protected] Printed in Singapore

Preface

Complex analytic geometry is a subject that could be termed, in short, as the study of the sets of common zeros of complex analytic functions. It has a long history — it started with the discovery of complex numbers in the 16th century and then bloomed as the theory of analytic functions of one complex variable, including the Cauchy integral formula, in the 19th century. While the foundation for the case of several variables was laid by H. Cartan, K. Oka et al., the transcendental method was brought in algebraic geometry by G. de Rham, W.V.D. Hodge and K. Kodaira and the modern form of the subject was established in the mid-20th. At about the same time it was enriched by the works of M.F. Atiyah, R. Bott, S.-S. Chern, H. Grauert, P. Griffiths, A. Grothendieck, H. Hironaka, F. Hirzebruch, B. Malgrange, J.W. Milnor, L. Schwartz, J.-P. Serre, I.M. Singer, R. Thom, H. Whitney et al. in various related areas. This trend has been kept growing to the present time and will go on, involving many new ideas and themes. As such, the subject is closely related to many other fields of mathematics and other sciences, where numerous applications have been and will be found. The subjects taken up in this book are rather classical, however we look at them from the localization viewpoint. This means that we are concerned with, among others, local invariants that arise naturally in complex analytic geometry and their relation with global invariants of the manifold or variety, the Poincar´e-Hopf index theorem being a prototype of this. The idea is to look at them as residues associated with the localization of some characteristic classes. Two approaches are taken for this — topological and differential geometric — and the combination of the two brings out further fruitful results. For this, on the one hand we give detailed description of the Alexander duality in combinatorial topology. On the

v

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Complex Analytic Geometry

ˇ other hand we give thorough presentation of the Cech-de Rham cohomology and integration theory on it. This viewpoint provides us with a way for clearer and more precise presentations of the central concepts as well as fundamental and important results that have been treated only globally so far. It also brings new perspectives into the subject and leads to further results and applications. Here are some key features of the book: 1. Detailed description of the dualities and the intersection product in combinatorial topology The Poincar´e, Alexander and Lefschetz dualities on C ∞ manifolds are proved using tiangulations and the dual cellular decompositions. The Thom isomorphism and the Thom class are treated in this context. The intersection product is also defined combinatorially (cf. Chapter 4). These are extended to the case of singular varieties as well (cf. Chapters 13 and 14). ˇ 2. Effective use of Cech-de Rham cohomology This cohomology naturally combines the de Rham cohomology and ˇ the Cech cohomology of locally constant functions. In the global situation, it gives a canonical correspondence between the two cohomologies, which leads to the canonical de Rham theorem. The relative version is conveniently used to describe the Alexander duality, the Thom class and various localizations in terms of differential forms (cf. Chapter 7). ˇ We may also consider Cech-Dolbeault cohomology in a similar manner. In the global situation, this leads to the canonical Dolbeault theorem (cf. Section 11.4). Although it is not treated here, the relative version has many applications including the localization theory for Atiyah classes and explicit representation of Sato hyperfunctions and related operations. 3. Topological and differential geometric definitions of characteristic classes The Chern classes of a complex vector bundle and the Euler class of an oriented real vector bundle are defined topologically, via obsrtuction theory. For general Chern polynomials the corresponding classes of a complex vector bundle are defined differential geometrically, via the Chern-Weil thoery. Modifying this theory, the classes are naturally ˇ defined in the Cech-de Rham cohomology using difference forms for several connections. The relative (localized) version is also discussed in each framework (cf. Chapters 5 and 8). A direct proof is given, including the relative case, of the fact that the differential geometric Chern

Preface

vii

class is the image of the topological one by the canonical morphism from the cohomology with integral coefficient to that with complex coefficient (cf. Section 10.5). 4. Combination of algebra, geometry and analysis The theory of analytic functions of one variable is a model of mathematical theory, where algebra, geometry and analysis are organically combined. We attempted to do similar things in the higher-dimensional case, see in particular Chapters 12 and 13. The book starts off with basic materials on analytic functions of one or several complex variables in Chapter 1. These functions are also referred to as holomorphic functions and are the ones we are mainly concerned with. We introduce the notion of the germ of a holomorphic function and study the structure of the ring of germs of holomorphic functions. The Weierstrass preparation theorem allows us to treat the ring almost as a polynomial ring. It continues by introducing, in Chapter 2, complex manifolds and analytic varieties, the spaces we work on. We study the relation between germs of varieties and ideals in the ring of germs of holomorphic functions. Particularly noteworthy is the Nullstellensatz (zero locus theorem), which we prove in the case of principal ideals. As for the general case, we quote some structual results and give a proof based on these. We also discuss dimensions of varieties. This chapter includes some remarks on the underlying real structures as well, because of their importance. In Chapter 3, fiber bundles, in particular vector bundles, are discussed. The homotopical structure of Stiefel manifold is studied for the obstruction theoretical definition of characteristic classes of vector bundles. The cellular structure of Grassmann manifold, which carries a unversal vector bundle, is analyzed. Some notions concerning differentiable structures are also made precise. We deal with, in Chapter 4, the topics mentioned in the item 1 above, i.e., the Poincar´e, Alexander and Lefschetz dualities, the Thom isomorphism and the intersection product in combinatorial topology. In Chapter 5, we define characteristic classes of vector bundles topologically via the obstruction theory. Their localizations by frames and the associated topological residues are also introduced. In Chapter 6, we review differential forms and their integrations. Such topics as integration along fibers and non-singular foliations are also presented. ˇ In Chapter 7, we discuss the Cech-de Rham cohomology in detail (cf. the ˇ item 2 above). After the de Rham and Cech cohomologies are reviewed,

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ˇ they are combined to form the Cech-de Rham cohomology. The de Rham ˇ cocycles and the Cech cocycles of locally constant functions are naturally ˇ thought of as being Cech-de Rham cocycles and this leads to the canonical ˇ isomorphism between the de Rham cohomology and the Cech cohomology of locally constant functions. We then present the integration theory on ˇ Cech-de Rham cohomology introducing the notion of a system of honeycomb cells. Combining with combinatorial topology, we establish various canonical isomorphisms (Theorem 7.4). The canonical de Rham theorem ˇ immediately follows from this. We discuss the relative Cech-de Rham coˇ homology and prove the relative Cech-de Rham theorem (Corollary 7.8). It is expressed explicitly in terms of integration on topological chains and provides a bridge between topological localizations and differential geometric localizations. It also allows us to describe the dualities in terms of differential forms. The Thom class is expressed explicitly in the relative ˇ Cech-de Rham cohomology. As an application, we prove the Lefschetz fixed point formula in this context. In Chapter 8, characteristic classes of vector bundles are defined differential geometrically via the Chern-Weil theory. The Bott difference forms are then conveniently used to modify the theory to have the classes in ˇ Cech-de Rham cohomology. This way of representing characteristic classes is particularly effective in dealing with the localization problem (cf. item 3 above). In Chapter 9, we discuss some topics related to vector bundles with metrics. Among them are harmonic forms (real and complex), Dolbeault cohomology, Kodaira-Serre duality, Hodge structures, K¨ahler manifolds, Atiyah classes of holomorphic vector bundles and Bott-Chern classes of Hermitian vector bundles. We explain the fundamental idea of localizing characteristic classes in Chapter 10. We then present the residue theorem and give explicit expressions of the residues. Various types of localizations are exhibited. In particular, the differential geometric localizations of Chern classes of vector bundles by frames is discussed in detail. In Chapter 11, we present various tools and notions that are common in complex analytic geometry and are used in later chapters. In particular, we discuss complex analytic spaces, generalizations of varieties, in some detail. In Chapter 12, we study the residues associated with localizations of Chern classes by families of sections of holomorphic vector bundles on complex manifolds. The residue is expressed explicitly (Theorem 12.2) in the case the singular set has an expected dimension, called the “proper case”.

Preface

ix

It shows that, in order to find the residue in the proper case, it suffices to know the residue at an isolated singularity. We give topological, analytic and algebraic expresions of the residue at an isolated singularity. The localized duality (Theorem 12.8) between the Chern class localized by a family of sections of a holomorphic vector bundle and the class of the complex space defined by the family of sections also follows from Theorem 12.2. It is effectively used in the localized intersection theory later in Chapter 14. In Chapter 13, we attempt to do similar things for residues of vector bundles on singular varieties. We discuss Poincar´e, Alexander, Lefschetz and Thom morphisms for singular varieties in combinatorial topology. We then ˇ present de Rham and Cech-de Rham theories for singular varieties and express the above morphisms in terms of integration of differential forms. We discuss Chern classes and their localizations both from topological and differential geometric viewpoints and show that they are essentially the same. We express the residue explicitly (Theorem 13.9), as in the case of manifolds. It again shows that, in order to find the residue in the proper case, it suffices to know the residue at an isolated singularity. We give topological, analytic and algebraic expresions of the residue at an isolated singularity. We also give examples and discuss related topics. The localized duality (Theorem 13.18) between the Chern class localized by a section of a holomorphic vector bundle and the class of the complex space defined by the section also follows from Theorem 13.9. We discuss, in Chapter 14, intersection products of the homology classes of complex subspaces in a complex manifold or in a singular variety. They are treated from the residue theoretical viewpoint and the intersection products are localized at the set of intersection. In the case of manifolds, the intersection product in general is defined combinatorially in Chapter 4, as mentioned above. It is an operation dual to the cup product in cohomology. In general, expected formulas do not hold for the intersection product of the classes of subspaces. The intersection products we consider involve, in some form or another, spaces defined by a family of sections of a vector bundle so that localizations of Chern classes and the associated residues are involved. For the residues, expected formulas always hold, thus in the proper case, for subspaces defined by a family of sections, they also hold. As to the intersection product in sigular varieties, we do not have general combinatorial ways of defining it. Thus we consider intersection products of homology classes that come naturally as the images of the Poincar´e or Alexander morphism and use the cup product in cohomology. For this, as in the case of manifolds, we consider intersection product involving subspaces

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Complex Analytic Geometry

defined by a family of sections and develop the theory in parallel with that case. We then specialize the above considerations to the case of intersection products in singular surfaces. Finally we study the case where a variety and a complex space defined by a section intersect in a dimension greater than expected. In Chapter 15, we discuss the Riemann-Roch theorem in the form A. Grothendieck formulated (cf. Theorem 15.11). It consists of the embedding part and the projection part. We examine the former in detail, as an application of our localization process by the exactness of vector bundle sequences, and prove a prototype of the localized Riemann-Roch theoˇ rem for embeddings on the level of Cech-de Rham cocycles (Corollary 15.1). It immediately yields a theorem in the universal situation (Theorem 15.4). We then present three variations of the localized Riemann-Roch theorem for embeddings (Theorems 15.5–15.7). For the projection part, we outline the proof quoting relevant materials. The Grothendieck-Riemann-Roch theorem contains the Hirzebruch-Riemann-Roch theorem as a special case. The latter is also generalized as the Atiyah-Singer index theorem, which we briefly recall in Notes at the end of the chapter. The book is supplemented with two appendixes, one on commutative algebra and the other on algebraic topology. Exercises provided in the text are meant to help better understand the materials and should mostly be done rather straightforward. Here is a word about the use of terminologies “natural” and “canonical”. A canonical morphism is the one that is induced from a natural idetification or correspondence of the objects. What is natural is somewhat a subjective matter. For example, we could embed the ring of integers Z into the complex numbers C by assigning to 1 in Z any non-zero complex number. In this case, we think the assignment 1 7→ 1 is natural and the morphism Hp (X; Z) → Hp (X; C) induced on the homology of a space X by this identification is canonical. As a typical example appearing in the ˇ text, Cech-de Rham cochains naturally combine de Rham cochains (differˇ ential forms) and Cech cochains of locally constant functions, as mentioned above, and the isomorphism between the de Rham cohomology and the ˇ ˇ Cech cohomology given via the Cech-de Rham cohomology is canonical. Some symbols are used to denote different objects. For example, the letter m is used for the order of a convergent power series, the dimension of a C ∞ manifold and so forth, the letter D for a number of purposes and the notation (f1 , . . . , fr ) for the map with component functions fi , i = 1, . . . , r,

Preface

xi

as well as the ideal generated by the germs of the fi ’s. Hopefully it will be clear from the context what it means and there will be no confusion. Besides calculus and linear algebra, basics of the following are desirable prerequisites: (1) Analytic functions of one complex variable (reviewed in the text as needed). (2) Commutative algebra (reviewed in Appendix A). (3) General topology. (4) Algebraic topology (reviewed in Appendix B). (5) Differentiable manifolds (reviewed in the text as needed). There are numerous literatures on these subjects and the citations made in Notes at the end of each chapter should be supplemented with “for example”. Acknowledgements. This work was supported by JSPS Grants-in-Aid for Scientific Research Grant over the years. I am indebted to many people through collaboration, discussion and advice in their field of expertise and it is not possible to acknowledge all of them. I owe much to J.-P. Brasselet for the item 1 above, to D. Lehmann for the item 2 and to J. Seade for the item 3. I would also like to express my appreciation to M. Abate, P. Aluffi, D. Angella, C. Bisi, F. Bracci, M. Corrˆea, M. Hanamura, N. Honda, T. Izawa, K. Kurano, Y. Matsumoto, J. Noguchi, M. Oka, K. Saito, J. Sch¨ urmann, M. Shiota, M. Soares, N. Tardini, A. Tomassini, F. Tovena and S. Yokura. Special thanks are due to T. Ohmoto for his constant interest, encouragement and advice during the preparation of the manuscripts, which took way longer than initially planned. I would like to extend my gratitude to all who were involved in the editorial and production process of the book, especially Kwong Lai Fun of World Scientific and Nimal Koliyat of ACES, for their generous support.

Tatsuo Suwa Fall 2023

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About the Author

Tatsuo Suwa is a Professor Emeritus at Hokkaido University, Japan. He had held academic positions at the University of Tokyo, the University of Michigan, Hokkaido University and Niigata University. He had visiting positions at institutions in a number of countries, including France, Italy, Mexico, Spain and the U.S.A. He has also been invited to give talks at symposiums and schools all over the world. He published a book on residues of singular holomorphic foliations in 1998. He then noticed that the philosophy and technics of localizing characteristic classes used there are applicable to other problems in Complex Analytic Geometry and related areas, some of which have been realized and are presented in this book.

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Contents

Preface

v

About the Author 1.

Analytic Functions of Several Complex Variables 1.1 1.2 1.3 1.4

2.

1

Multiseries . . . . . . . . . . . . . . . . . . . . Analytic functions of one complex variable . . . Analytic functions of several complex variables Germs of holomorphic functions . . . . . . . . .

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Complex Manifolds and Analytic Varieties 2.1 2.2 2.3 2.4 2.5

3.

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Complex manifolds . . . . . . Analytic varieties . . . . . . . Germs of varieties . . . . . . Nullstellensatz and dimension Underlying real structures . .

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Vector Bundles 3.1 3.2 3.3 3.4 3.5 3.6 3.7

2 5 8 14

24 28 34 38 43 51

Group actions . . . . . . . . . . . . . . . Fiber bundles . . . . . . . . . . . . . . . Vector bundles . . . . . . . . . . . . . . Tangent bundle and vector fields . . . . Stiefel manifold . . . . . . . . . . . . . . Grassmann manifold . . . . . . . . . . . Some topics on differentiable manifolds .

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51 55 61 68 74 77 83

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Complex Analytic Geometry

Dualities and Thom Class 4.1 4.2 4.3 4.4

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Chern Classes and Localization via Obstruction Theory 5.1 5.2 5.3 5.4 5.5 5.6

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Algebraic topology on manifolds . . . . . . Poincar´e, Alexander and Lefschetz dualities Thom isomorphism and Thom class . . . . Intersection product . . . . . . . . . . . . .

95

Index of a family of sections . . . . . . . . . . . . Chern classes of a complex vector bundle . . . . Euler class of an oriented real vector bundle . . . Relative classes . . . . . . . . . . . . . . . . . . . Piecewise-linear manifolds and pseudo-manifolds Localization and topological residues . . . . . . .

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Differential Forms 6.1 6.2 6.3 6.4

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Vector fields and differential forms . . . . . . Integration of differential forms . . . . . . . . Integration along fibers . . . . . . . . . . . . Frobenius theorem and non-singular foliations

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ˇ 7. Cech-de Rham Cohomology 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 8.

de Rham cohomology . . . . . . . . . ˇ Cech cohomology . . . . . . . . . . . . ˇ Cech-de Rham cohomology . . . . . . ˇ Integration of Cech-de Rham cochains

. . . . . . . . . . . . Combination with combinatorial topology . de Rham theorem and related topics . . . . ˇ Relative Cech-de Rham cohomology . . . . Description of dualities in differential forms ˇ Thom class in Cech-de Rham cohomology . Angular form and Bochner-Martinelli form Lefschetz fixed point formula . . . . . . . .

161 167 169 171 177

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ˇ Chern-Weil Theory Adapted to Cech-de Rham Cohomology 8.1 8.2 8.3 8.4

128 131 133 135 136 140

Connections . . . . . . . . . . . . . . . . . . . . . . . Characteristic classes of complex vector bundles . . . Further topics on connections . . . . . . . . . . . . . ˇ Characteristic classes in Cech-de Rham cohomology

177 179 183 188 193 199 205 214 225 229 232 239

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240 243 249 253

Contents

9.

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Vector Bundles with Metrics and Related Topics 9.1 9.2 9.3 9.4 9.5 9.6

255

Metrics on vector bundles and harmonic forms Hodge structures . . . . . . . . . . . . . . . . . Vector bundles on projective space . . . . . . . K¨ ahler manifolds . . . . . . . . . . . . . . . . . Atiyah classes . . . . . . . . . . . . . . . . . . . Hermitian connections and Bott-Chern classes .

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10. Localization 10.1 10.2 10.3 10.4 10.5

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Localization and associated residues . . . . . . . . . Vanishing theorems . . . . . . . . . . . . . . . . . . . Differential geometric localization by frames . . . . Thom class of a complex vector bundle . . . . . . . . Coincidence of topological and differential geometric localizations . . . . . . . . . . . . . . . . . . . . . . .

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Sheaves . . . . . . . . . . . Sheaf cohomology . . . . . . Coherent sheaves . . . . . . Dolbeault theorem . . . . . Complex analytic spaces . . Divisors . . . . . . . . . . . Local complete intersections

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12. Residues of Chern Classes on Manifolds 12.1 12.2 12.3 12.4 12.5 12.6

Triangulation of subanalytic sets . . . Residues of Chern classes on manifolds Grothendieck residues . . . . . . . . . Residues at an isolated singularity . . Examples . . . . . . . . . . . . . . . . Dual class of a complex subspace . . .

325 332 341 349 358 368 375 387

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13. Residues of Chern Classes on Singular Varieties 13.1 13.2

290 300 307 312

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11. Further Topics 11.1 11.2 11.3 11.4 11.5 11.6 11.7

256 268 273 276 278 280

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387 389 390 392 402 405 411

Controlled tube systems for Whitney stratifications . . . . 411 Poincar´e, Alexander, Lefschetz and Thom morphisms . . . . . . . . . . . . . . . . . . . . . . . 412

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13.3 13.4 13.5 13.6 13.7

ˇ de Rham and Cech-de Rham theories for singular varieties . . . . . . . . . . . . . . . . Chern classes and localization . . . . . . . . . Residues of Chern classes on singular varieties Residues at an isolated singularity . . . . . . Examples and related topics . . . . . . . . . .

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14. Intersection Product of Complex Subspaces 14.1 14.2 14.3 14.4 14.5

Refined Whitney sum formula . . . . . . . Intersection product in complex manifolds Intersection product in singular varieties . Intersection product in singular surfaces . Excess intersections . . . . . . . . . . . .

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15. Riemann-Roch Theorem 15.1 15.2 15.3 15.4 15.5 15.6 15.7

Riemann-Roch problem for curves . . . . . Characteristic classes of virtual bundles . . Chern-Weil theory for virtual bundles . . . Local Chern classes and characters . . . . . Universal localized Riemann-Roch theorem Riemann-Roch theorem for embeddings . . Grothendieck-Riemann-Roch Theorem . . .

Appendix A Commutative Algebra A.1 A.2

456 459 469 487 492 495

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496 496 499 503 506 509 516 523

Homological algebra . . . . . . . . . . . . . . . . . . . . . 523 Commutative rings . . . . . . . . . . . . . . . . . . . . . . 539

Appendix B B.1 B.2 B.3

418 426 433 434 440

Algebraic Topology

549

Singular homology . . . . . . . . . . . . . . . . . . . . . . 549 Cell complexes . . . . . . . . . . . . . . . . . . . . . . . . 562 Homology of locally finite chains . . . . . . . . . . . . . . 570

Bibliography

573

Index

581

Chapter 1

Analytic Functions of Several Complex Variables

An analytic function is a function that can locally be expressed as a convergent power series. In the case of one complex variable, it is equivalent to saying that the function is holomorphic, meaning that it is differentialble with respect to the complex variable. As a function of the real and imaginary parts of the variable, it can also be rephrased by saying that the function is continuously differentaible and satisfies the Cauchy-Riemann equations (cf. Theorem 1.1 below). We start by discussing convergence of multiseries, which is necessary for the case of several variables. We then review basics on analytic functions of one complex variable, notably the Cauchy integral formula, which has a significant influence throughout the book. The case of several variables follows with relation to the one variable case. The terms “analytic” and “holomorphic” will be treated as synonyms hereafter. Regular and singular points of a holomorphic map are defined and the behavior of a holomorphic map near a regular point is clarified (Theorems 1.4, 1.5 and 1.6). The notion of the germ of a holomorphic function is introduced and the algebraic structure of the ring of germs of holomorphic functions is discussed. The Weierstrass preparation theorem (Theorems 1.10) allows us to treat convergent power series as polynomials to some extent. These are used to analyze the local structure of the set of common zeros of holomorphic functions, i.e., analytic varieties, in the next chapter. In the following, we denote by C the field of complex numbers and Cn the product of n copies of C. An element z in Cn is expressed as z = (z1 , . . . , zn ) with each zi , i = 1, . . . , n, in C and the set Cn has a natural complex vector space structure of dimension n. The norm of z is defined by p kzk = |z1 |2 + · · · + |zn |2 , which makes Cn a topological vector space. 1

2

Complex Analytic Geometry

Recall that, starting from the field R of real numbers, we may form a real vector space Rm of dimension m with the Euclidean norm. For a √ point z = (z1 , . . . , zn ) in Cn , we write zi = xi + −1 yi with xi and yi the real and imaginary parts of zi . Identifying z with (x1 , y1 , . . . , xn , yn ), we may identify Cn with R2n and by this identification, the above norm in Cn coincides with the Euclidean norm in R2n . 1.1

Multiseries

Let N0 denote the set of non-negative integers. Suppose a complex number cν is assigned to each ν = (ν1 , . . . , νn ) in Nn0 . We call the formal infinite sum ∞ X X cν = cν1 ...νn ν

ν1 ,...,νn =0

a multiseries, or a series for short. P Definition 1.1. A series cν converges to a complex number C if, for every positive number ε, there exists N in N0 with the following property: Nn N1 X X cν − C < ε. ··· for every (N1 , . . . , Nn ) in Nn0 with Ni > N, ν1 =0

νn =0

P

If this is the case, we write C = cν and call it the sum of the series. P We say that cν is convergent if it converges to some complex number. P Note that if a series c is convergent, its sum is uniquely determined. Pν P We also say that cν is absolutely convergent if the series |cν | is convergent. From the definition we have: P Proposition 1.1. A series cν is absolutely convergent if and only if the set o nX |cν | Γ ⊂ Nn0 , finite ν∈Γ

P

|cν | is the supremum of this set. P In general, for a convergent series cν and a finite set Γ in Nn0 , we set X X X cν = cν − cν .

is bounded. In this case the sum

ν ∈Γ /

ν

ν∈Γ

Analytic Functions of Several Complex Variables

3

P Thus if a series cν is absolutely convergent, for every positive number ε, P there exists a finite set Γ such that ν ∈Γ / |cν | < ε. P In the case n = 1, i.e., the case of simple series, if a series cν is convergent, the set {cν } is bounded. Also in this case, if it is absolutely convergent, it is convergent by the Cauchy criterion. In the case n > 1, we P may not have the boundedness of {cν }, even if cν is convergent. However we have: P Proposition 1.2. If a series cν is absolutely convergent, it is convergent. P Moreover the sum C = cν does not depend on the order of summation, P∞ i.e., for every bijection ϕ : N0 → Nn0 , the simple series µ=0 cϕ(µ) converges to C. Proof. Take a bijection ϕ : N0 → Nn0 . By Proposition 1.1, the simple P∞ series µ=0 cϕ(µ) is absolutely covergent, and thus convergent. We denote P its sum by Cϕ and show that cν converges to Cϕ so that Cϕ = C and it does not depend on ϕ. For this, let ε be an arbitrary positive number. Then there exists N0 in N0 such that N0 ε X cϕ(µ) − Cϕ < 2 µ=0

and

∞ X µ=N0 +1

|cϕ(µ) |
N , we set Γ1 = { ν | 0 ≤ νi ≤ Ni }. Then we have X X X cν − Cϕ ≤ cν − Cϕ + |cν | < ε. ν∈Γ1 ν∈Γ0 ν∈Γ1 rΓ0  Example 1.1. Let z = (z1 , . . . , zn ) be a point in Cn with 0 6= |zi | < 1. For each ν in Nn0 , we set z ν = z1ν1 · · · znνn . By Propositions 1.1 and 1.2, the P series ν z ν is convergent and X 1 1 ··· . zν = 1 − z1 1 − zn ν By the convention 00 = 1, the above holds for every z with |zi | < 1. Next we consider series whose terms are functions. Thus let R be a subset in Cn . Suppose we have, for each ν, a complex valued function fν defined on R. We denote by kfν kR the supremum of |fν (z)| on R. P Definition 1.2. A series fν is normally convergent on R if the series P kfν kR is convergent.

4

Complex Analytic Geometry

P P Note that if a series fν converges normally on R, the series fν (z) converges absolutely for each z in R and it defines a function F on R: X F (z) = fν (z). ν

P

Proposition 1.3. If a series fν converges normally on R, it converges uniformly on R, i.e., we may take N in Definition 1.1 independently of z in R, with cν and C replaced by fν (z) and F (z), respectively. Proof. Given a positive number ε. Take ε0 so that 0 < ε0 < ε. By the P normal convergence of fν , there exists a finite set Γ0 in Nn0 such that P 0 n ν ∈Γ / 0 kf kR < ε . Take N in N0 so that Γ0 ⊂ [0, N ] . For (N1 , . . . , Nn ) with Ni > N , we set Γ1 = { ν | 0 ≤ νi ≤ Ni }. Then for an arbitrary finite set Γ in N0n , we have X X fν (z) ≤ kf kR < ε0 . ν∈Γ rΓ1

ν∈Γ rΓ1

Hence we have X X fν (z) − F (z) = fν (z) ≤ ε0 < ε. ν∈Γ1

ν ∈Γ / 1



From the above we see that, for instance, if fν is continuous for each ν, P the sum F = fν is also continuous. Series we are particularly interested in here are “power series”. In the following, for z = (z1 , . . . , zn ) in Cn and ν = (ν1 , . . . , νn ) in Nn0 , we set z ν = z1ν1 · · · znνn ,

|ν| = ν1 + · · · + νn

and ν! = ν1 ! · · · νn !.

Also, for a point a = (a1 , . . . , an ) in Cn and an n-tuple ρ = (ρ1 , . . . , ρn ) of positive real numbers, we set ∆(a, ρ) = { z ∈ Cn | |zi − ai | < ρi } and call it the polydisk with center a and polyradius ρ. We may express it as a product of one-dimensional disks: ∆(a, ρ) = ∆(a1 , ρ1 ) × · · · × ∆(an , ρn ). For a fixed a, they form a fundamental system of neighborhoods of a. P Definition 1.3. A power series with center a is a series fν such that each fν is of the form fν (z) = cν (z − a)ν ,

cν ∈ C.

Analytic Functions of Several Complex Variables

5

In the above, we adopt the convention as in Example 1.1. For power series, we have the following fundamental lemma: P Lemma 1.1. If a series cν (z − a)ν converges absolutely at z = w, the series is normally convergent on the closure ∆(a, ρ) of every polydisk ∆(a, ρ) with ρi < |wi − ai |. Proof. Without loss of generality we may assume that a = (0, . . . , 0), the origin in Cn . For ρ as above, we may find an n-tuple t = (t1 , . . . , tn ) of P real numbers with 0 < ti < 1 and ρi < ti |wi |. Since the series |cν wν | is convergent, there is a constant K such that |cν wν | ≤ K for all ν. If z is a point in ∆(0, ρ), |cν z ν | ≤ |cν |ρν ≤ Ktν and the lemma follows from Example 1.1.



Remark 1.1. As is seen in the proof, the same result holds under the assumption that {|cν wν |} is bounded. Thus if n = 1, only the convergence P of the series cν (z − a)ν at z = w is sufficient to have the same result.

1.2

Analytic functions of one complex variable

We recall some basics on analytic functions of one complex variable. Let U be an open set in the complex plane C and f a complex valued function on U . Definition 1.4. We say that f is analytic at a point a in U if there is a P power series ν≥0 cν (z − a)ν which converges at each point z in a neighborhood of a and satisfies X f (z) = cν (z − a)ν ν≥0

in a neighborhood of a. We say that f is analytic in U if it is analytic at every point in U . In general, let r be a non-negative integer. Recall that a function of real variables is of class C r , C r for short, if its partial derivatives exist up to order r and are continuous. By convention, C 0 means continuous. If all the partial derivatives exist, we say it is C ∞ . √ If we write z = x + −1 y with x and y the real and imaginary parts, we may think of f as above as a function of (x, y). If f is analytic in U , it is C ∞ in U .

6

Complex Analytic Geometry

Definition 1.5. We say that f is holomorphic at a point a in U if the limit f (a + h) − f (a) lim h→0 h exists. We say that f is holomorphic in U if it is holomorphic at every point in U . df (a) and is called the derivaThe above limit, if it exists, is denoted by dz df as a function tive of f at a. If f is holomorphic in U , we may think of dz on U . If f is analytic in U , it is holomorphic in U and the derivative is also analytic in U .

Cauchy integral formula: Let f be a holomorphic function on a neighborhood of the closure of a disk ∆ in C and γ = ∂∆ the boundary of ∆ oriented counterclockwise. Then for z in ∆, Z 1 f (ζ) dζ √ f (z) = . (1.1) 2π −1 γ ζ − z Remark 1.2. The above formula holds more generally if we replace ∆ with a relatively compact open set U in C such that U is a real two-dimensional submanifold of C with piecewise C ∞ boundary ∂U . In this case γ = ∂U consists of a finite number of piecewise C ∞ simple closed curves oriented as the boundary of U (cf. Section 3.7 below). This will be generalized later to the case f is C ∞ (Theorem 11.8 below). √ For a function f on an open set U in C, we write f = u + −1 v with u and v the real and imaginary parts. Theorem 1.1. The following are equivalent: (1) f is analytic in U . (2) f is holomorphic in U . (3) f is C 1 in (x, y) and satisfies the “Cauchy-Riemann equations” in U : ∂v ∂u ∂v ∂u = , =− . ∂x ∂y ∂y ∂x Note that, if we introduce the operators √ √ ∂ 1 ∂ ∂  ∂ 1 ∂ ∂  = − −1 and = + −1 , ∂z 2 ∂x ∂y ∂ z¯ 2 ∂x ∂y we may express the Cauchy-Riemann equations as ∂f = 0. ∂ z¯

Analytic Functions of Several Complex Variables

7

df If this is the case, we have ∂f ∂z = dz . In the following, we use the words “analytic” and “holomorphic” interchangeably. Here are some basic facts that we need. We denote by D a connected open set in C.

Uniqueness of analytic continuation: Let f and g be holomorphic functions on D. If f = g on a neighborhood of a point in D, f = g on D. Open mapping theorem: Let f be a non-constant holomorphic function on D. Then f is open as a map D → C. Maximum principle: Let f be a holomorphic function on D. If there is a point z0 in D such that |f (z0 )| ≥ |f (z)| for all z in a neighborhood of z0 , then f is constant on D. This may be thought of as a special case of the open mapping theorem. We finish this section by reviewing the following: Meromorphic functions Let f and g be holomorphic functions in a neighborhood of a point a in C. Suppose g has no zeros away from a, then f /g is defined and holomorphic away from a. We examin the behavior of f /g near a. If f is not identically equal to zero in a neighborhood of a, we may write f (z) = (z − a)k f1 (z),

0

g(z) = (z − a)k g1 (z)

with k and k 0 non-negative integers and f1 and g1 holomorphic functions without zeros in a neighborhood of a. There are two cases: (1) k ≥ k 0 . In this case, we have 0 f1 (z) f (z) = (z − a)k−k g(z) g1 (z)

so that f /g extends to a holomorphic function in a neighborhood of a. If k > k 0 , the function has a zero at a. We call m = k − k 0 the order of the zero. (2) k < k 0 . In this case, we have 1 f1 (z) f (z) = . g(z) (z − a)k0 −k g1 (z) We say that f /g has a pole at a and call m = k 0 − k the order of the pole.

8

Complex Analytic Geometry

Definition 1.6. A meromorphic function ϕ on an open set U in C is a holomorphic function on the complement of a (possibly empty) set S of isolated points in U such that ϕ has a pole at each point of S. Note that in a neighborhood of each point of U , ϕ may be written ϕ = f /g, where f and g are holomorphic and g is not identically equal to zero. Remark 1.3. 1. In fact, every meromophic function ϕ on an open set U in C may be written ϕ = f /g, where f and g are holomorphic functions on U and g is not identically equal to zero in any connected component of U . This is a property particular to open sets in C. 2. If we denote by OU the set of holomorphic functions on an open set U in C, then the usual addition and multiplication of functions give OU the structure of a ring (in fact a C-algebra). In particular, if U = D is a nonempty connected open set, then OD is an integral domain (cf. Section A.2), by the uniqueness of analytic continuation. Moreover, by 1 above, the set MD of meromorphic functions on D may be thought of as the fraction field of OD . 1.3

Analytic functions of several complex variables

Let f a complex valued function on an open set U in Cn . Definition 1.7. We say that f is analytic at a point a in U if there is a power series X X cν (z − a)ν = cν1 ...νn (z1 − a1 )ν1 · · · (zn − an )νn ν1 ,...,νn ≥0

which converges absolutely at each point z in a neighborhood of a and satisfies X f (z) = cν (z − a)ν in a neighborhood of a. We say that f is analytic in U if it is analytic at every point in U . Note that, in the case n = 1, Definitions 1.4 and 1.7 are equivalent (cf. Remark 1.1). If f is analytic in U , it is continuous in U by Lemma 1.1 and it is analytic in each variable zi separately, since in a neighborhood each point

Analytic Functions of Several Complex Variables

9

a in U we may rearrange the power series as a convergent power series in zi − ai by Proposition 1.2. In fact we have: Theorem 1.2. For a complex valued function f defined on an open set U in Cn , the following conditions are equivalent: (1) f is analytic in U . (2) f is continuous and is analytic in each variable zi in U , i = 1, . . . , n. Proof. It remains to show that (2) implies (1). Let a be a point in U and choose ρ so that ∆(a, ρ) is contained in U . For each i we denote by γi the boundary of ∆(ai , ρi ) oriented counterclockwise. By a repeated use of the Cauchy integral formula (1.1), for z in ∆(a, ρ), Z f (ζ1 , z2 , . . . , zn ) 1 √ dζ1 f (z) = ζ1 − z1 2π −1 γ1 = ··· n Z  Z dζ1 f (ζ1 , . . . , ζn ) 1 √ ··· dζn = ζn − zn 2π −1 γ1 ζ1 − z1 γn  n Z Z 1 f (ζ1 , . . . , ζn ) √ = ··· dζ1 · · · dζn , 2π −1 γ (ζ1 − z1 ) · · · (ζn − zn ) where the last multi-integral is to be performed on γ = γ1 × · · · × γn . Note that we used the continuity of f in the last equality. On the other hand, noting that z − a 1 1 1 i i and = < 1, zi −ai ζi − zi ζi − ai 1 − ζi −ai ζi − ai we have a power series expansion ∞ ν ν X (z1 − a1 ) 1 1 (zn − an ) n 1 ··· = · · · ν1 +1 ν +1 , ζ1 − z1 ζn − zn (ζn − an ) n ν ,...,ν =0 (ζ1 − a1 ) 1

n

which converges normally with respect to ζ on γ. Hence we may integrate term by term and we have a power series expansion, which converges absolutely at z: X f (z) = cν (z − a)ν with  cν =

1 √

2π −1

n Z

Z ... γ

f (ζ1 , . . . , ζn )dζ1 · · · dζn . (ζ1 − z1 )ν1 +1 · · · (ζn − zn )νn +1



10

Complex Analytic Geometry

It is known that we may remove the continuity condition in (2) above (Hartogs’ theorem). In general, let f be a complex valued function defined on an open set √ U in Cn . Writing zi = xi + −1 yi , we may think of f as a function of the real variables (x1 , y1 , . . . , xn , yn ). We recall that (cf. Theorem 1.1) a C 1 function f is analytic in the variable zi if and only if it satisfies the Cauchy-Riemann equation ∂f = 0. ∂ z¯i

(1.2)

If f is analytic in U , by Lemma 1.1, it is C 1 in U , in fact it is C ∞ , since we may differentiate it term by term when expressed as a power series and the resulting series also converges normally in the same domain. Thus from Theorem 1.2, we have: Theorem 1.3. For a function f defined on U , the following are equivalent: (1) f is analytic in U . (2) f is C 1 and satisfies (1.2) in U , for i = 1, . . . , n. In the following, we call an analytic function also a holomorphic function and use the words “analytic” and “holomorphic” interchangeably as in the case of one variable. Note that, for a holomorphic function f , the “partial derivative” ∂f /∂zi coincides with the derivative of f considered as a holomorphic function of the single variable zi and, for every ν, the (higher) partial derivative ∂ν f ∂ |ν| f = ν1 ν ∂z ∂z1 · · · ∂znνn P exists and is holomorphic in U . If f (z) = cν (z − a)ν is a power series expansion of f , then each coefficient cν is given by 1 ∂ν f (a). ν! ∂z ν This series is called the Taylor series of f at a. cν =

Definition 1.8. 1. Let U be an open set in Cn and f : U → Cr a map. We say that f is holomorphic if, writing f componentwise as f = (f1 , . . . , fr ), each fi is holomorphic. 2. Let U and U 0 be two open sets in Cn and f : U → U 0 a map. We say that f is biholomorphic if f is bijective and if both f and f −1 are holomorphic.

Analytic Functions of Several Complex Variables

11

Proposition 1.4. Let U and U 0 be open sets in Cn and Cr , respectively, and let f : U → U 0 and g : U 0 → C be maps. If both f and g are holomorphic, the composition g ◦ f is also holomorphic. Proof. First we note that g ◦ f is C 1 . If we denote coordinates of Cn by (z1 , . . . , zn ) and those of Cr by (w1 , . . . , wr ), using the chain rule we have r r X X ∂g ◦ f ∂fj ∂ f¯j ∂g ∂g (z) = (f (z)) (z) + (f (z)) (z). ∂ z¯i ∂wj ∂ z¯i ∂w ¯j ∂ z¯i j=1 j=1

Then we have the proposition as each term in the right-hand side is zero by the Cauchy-Riemann equations.  For a holomorphic map f from an open set U in Cn to Cr , we write f = (f1 , . . . , fr ) and set  ∂f ∂f1  1 ...  ∂z1 ∂zn   ∂(f1 , . . . , fr )  . .  .. . . ...  = . ∂(z1 , . . . , zn )   ∂f  ∂f r r ... ∂z1 ∂zn We call it the Jacobian matrix of f with respect to z. The rank of f at a point a in U is the rank of the Jacobian matrix at a. Definition 1.9. We say that a point a in U is a regular point of f if the rank of f at a is maximal possible, i.e., min(n, r). Otherwise we say that a is a singular point , or a critical point, of f . Note that, if a is a regular point of f , there is a neighborhood Ua of a in U such that every point in Ua is a regular point of f . In the case n = r, the determinant of the Jacobian matrix is called the Jacobian of f . In this case, a is a regular point of f if and only if det

∂(f1 , . . . , fn ) (a) 6= 0. ∂(z1 , . . . , zn )

If we denote by ui and vi the real and the imaginary parts of fi , we have: 2 ∂(u1 , v1 , . . . , un , vn ) ∂(f1 , . . . , fn ) det = det . (1.3) ∂(x1 , y1 , . . . , xn , yn ) ∂(z1 , . . . , zn ) Exercise 1.1. Verify the identity (1.3).

12

Complex Analytic Geometry

The following three theorems describe the nature of a holomorphic map near its regular point. Thus we consider maps f from a neighborhood of the origin 0 in Cn to Cr . Without loss of generality, we may assume f (0) = 0. We express this situation by writing f : (Cn , 0) → (Cr , 0). Also f : (Cn , 0) → (Cn , 0) being biholomorphic means there exist neighborhoods U and V of 0 such that f is a biholomorphic map of U onto V . First we prove the following using the corresponding theorem in the real variable case: Theorem 1.4 (Inverse mapping theorem). Let f : (Cn , 0) → (Cn , 0) be a holomorphic map. If 0 is a regular point of f , then f is biholomorphic. Proof. By (1.3) and the inverse mapping theorem in the real variable case, there exist neighborhoods U and U 0 of 0 such that f is a bijective map from U onto U 0 with C 1 inverse f −1 . A computation similar to the one in the proof of Proposition 1.4 shows that f −1 is also holomorphic.  From the above, we have the following. Although the proofs are the same as the real case, we give them for the sake of completeness. Theorem 1.5 (Implicit mapping theorem). Let f : (Cn , 0) → (Ck , 0) be a holomorphic map. Suppose n > k and 0 is a regular point of f . Renumbering the functions and the variables if necessary, we may assume det

∂(f1 , . . . , fk ) (0) 6= 0. ∂(z1 , . . . , zk )

Then there exists a holomorphic map ϕ : (Cn−k , 0) → (Ck , 0) such that f (ϕ1 (zk+1 , . . . , zn ), . . . , ϕk (zk+1 , . . . , zn ), zk+1 , . . . , zn ) = 0 for (zk+1 , . . . , zn ) in a neighborhood of 0. Proof. Set z 0 = (z1 , . . . , zk ) and z 00 = (zk+1 , . . . , zn ) and define a holomorphic map g : (Cn , 0) → (Cn , 0) by g(z) = (f (z), z 00 ). Then 0 is a regular point of g and thus g is biholomorphic by Theorem 1.4. We write g −1 = (a, b) with a : (Cn , 0) → (Ck , 0) and b : (Cn , 0) → (Cn−k , 0). Then f (a(z), b(z)) = z 0 ,

b(z) = z 00 .

Thus if we set ϕ(z 00 ) = a(0, z 00 ), ϕ is the desired holomorphic map. Remark 1.4. In the above situation, we may solve the equation f (z1 , . . . , zk , . . . , zn ) = 0

(1.4) 

Analytic Functions of Several Complex Variables

13

for z1 , . . . , zk as functions ϕ1 , . . . , ϕk of (zk+1 , . . . , zn ) in a neighborhood of 0 and the set f −1 (0) is the graph of the map ϕ = (ϕ1 , . . . , ϕk ). The following gives a local normal form of a map near a regular point: Theorem 1.6. 1. Let f : (Cd , 0) → (Cn , 0) be a holomorphic map. Suppose d ≤ n and 0 is a regular point of f . Setting k = n − d and denoting by w = (w1 . . . , wd ) coordinates on Cd , we may assume that ∂(fk+1 , . . . , fn ) (0) 6= 0. det ∂(w1 , . . . , wd ) Then there exists a biholomorphic map h : (Cn , 0) → (Cn , 0) such that (h ◦ f )(w1 , . . . , wd ) = (0, . . . , 0, w1 , . . . , wd ). 2. Let f : (Cn , 0) → (Ck , 0) be a holomorphic map. Suppose n ≥ k and 0 is a regular point of f . We may assume that ∂(f1 , . . . , fk ) det (0) 6= 0. ∂(z1 , . . . , zk ) Then there exists a biholomorphic map h : (Cn , 0) → (Cn , 0) such that (f ◦ h)(z1 , . . . , zk , . . . , zn ) = (z1 , . . . , zk ). Proof. 1. We set z 0 = (z1 , . . . , zk ) and z 00 = (zk+1 , . . . , zn ) and identify Cd with {0} × Cd ⊂ Cn by identifying w with (0, z 00 ), zk+i = wi . Define a map g : (Cn , 0) → (Cn , 0) by g(z) = f (z 00 )+(z 0 , 0). Then it is biholomorphic and h = g −1 is the desired biholomorphic map. 2. If n = k, f is biholomorphic and thus set h = f −1 . If n > k, let g : (Cn , 0) → (Cn , 0) be as in the proof of Theorem 1.5 and set h = g −1 . Then from (1.4), we see that it is the desired map.  Remark 1.5. In the case 2 above, we may think of (f1 , . . . , fk , zk+1 , . . . , zn ) as a coordinate system on a neighborhood of 0 in Cn . The following theorems can be proved as in the one variable case or reducing to that case. We denote by D a connected open set in Cn . Theorem 1.7 (Uniqueness of analytic continuation). Let f and g be holomorphic functions on D. If f = g on a neighborhood of a point in D then f = g on D. Theorem 1.8 (Maximum principle). Let f be a holomorphic function on D. If there is a point z0 in D such that |f (z0 )| ≥ |f (z)| for all z in a neighborhood of z0 , then f is constant on D.

14

1.4

Complex Analytic Geometry

Germs of holomorphic functions

In this section we discuss basic local properties of holomorphic functions. Let H denote the set of pairs (U, f ) of a neighborhood U of 0 and a holomorphic function f on U . We define a relation ∼ in H as follows. For two elements (U, f ) and (V, g) in H, we say (U, f ) ∼ (V, g) if there is a neighborhood W of 0 such that W ⊂ U ∩ V and that the restrictions of f and g to W are the same. Then it is easily checked that ∼ is an equivalence relation in H. The equivalence class of (U, f ) is called the germ of f at 0 and is denoted by f , or simply by f if there is no fear of confusion. We let On denote the quotient set H/ ∼. The set On has the structure of a commutative ring with respect to the operations induced from the addition and multiplication of functions. It has the unity which is the equivalence class of the function constantly equal to 1. We call On the ring of germs of holomorphic functions at 0. In fact it contains the field C as the germs of constant functions and has a natural C-algebra structure. Remark 1.6. In algebraic terms the ring On is expressed as follows (cf. Section A.1). We may introduce an order relation in the set of neighborhoods of 0 by reverse inclusion, i.e., U ≤ V if V ⊂ U . Then it becomes a directed set. For a neighborhood U we denote by OU the ring of holomorphic functions on U and for V ⊂ U let rV U : OU → OV be the restriction. Then (OU , rV U ) is a direct system (cf. Definition A.8) and we may write OU , On = lim −→ U 30

the direct limit on the set of neighborhoods of 0. If we denote by C{z1 , . . . , zn } the set of power series that converge absolutely in some neighborhood of 0, this set also has the structure of a ring. Since (U, f ) ∼ (V, g) if and only if f and g have the same power series expansion, we may identify On with C{z1 , . . . , zn }. We start with some fundamental properties of the ring On . For terminologies and basics of commutative algebra, we refer to Appendix A. In the following, we adopt the convention that C0 = {0} and O0 = C. Proposition 1.5. The ring On is an integral domain. Proof. Suppose f g = 0 for germs f and g in On . Then there is a connected neighborhood U of 0 such that f (z)g(z) = 0 for all z in U . If f 6= 0, there is a point z such that f (z) 6= 0. By the continuity of f , we

Analytic Functions of Several Complex Variables

15

may find a neighborhood Uz of z such that f (z 0 ) 6= 0 for all z 0 in Uz . Since the restriction of g to Uz must be identically equal to 0, we get g = 0 by Theorem 1.7.  By the above proposition, we may form the fraction field of On , which we denote by Mn . Each element in Mn is expressed as f /g and two expressions f /g and f 0 /g 0 stand for the same element if and only if f g 0 = f 0 g. We call an element of Mn a germ of a meromorphic function at 0 in Cn . Proposition 1.6. A germ u in On is a unit if and only if it is the germ of a function u with u(0) 6= 0. Proof. If u is a holomorphic function in a neighborhood of 0 with u(0) 6= 0, then 1/u is a holomorphic function in a neighborhood of 0. Thus the germ of u is a unit in On . The converse is obvious.  If we denote by m the set of non-units in On , it is an ideal in On . Proposition 1.7. The ideal m is the unique maximal ideal in On . Proof. Let I be an arbitrary ideal in On . If I contains a unit, then I = On . If I does not contain units, then I ⊂ m.  Note that the quotient On /m is canonically isomorphic with C. The isomorphism is given by assigning to the class of f the value f (0). Recall that a ring with a unique maximal ideal is called a local ring (cf. Section A.2). We analyze the structure of the ring On by induction on n. For this, we pay attention to a specific variable, which will be zn . First, for a germ P f in On , we write f = |ν|≥0 aν z ν . Definition 1.10. If f 6= 0, the order of f is the integer m such that aν = 0 for all ν with |ν| < m and aν0 6= 0 for some ν0 with |ν0 | = m. The order of the germ 0 is defined to be +∞. Thus the order of f is a positive integer if and only if it is not 0 or a unit. The above notion of order does not depend on the coordinates, while the next one does. Definition 1.11. The order of f in zn is the order of f (0, . . . , 0, zn ) as a power series in zn . We say that f is regular in zn of order m, if the order m of f in zn is a positive integer. Thus f is regular in zn of order m if and only if f (0, . . . , 0, zn ) has a zero of multiplicity m at zn = 0.

16

Complex Analytic Geometry

Lemma 1.2. If the order of f is m, then we may find a suitable coordinate system (ζ1 , . . . , ζn ) of Cn such that the order of f in ζn is m. Proof. As the case f = 0 is obvious, we assume that f 6= 0. If we set P fm = |ν|=m aν z ν , then fm 6= 0. Thus, there is a point c = (c1 , . . . , cn ) 6= 0 such that fm (c) 6= 0. There exists an n × n non-singular matrix C whose n-th row is c. If we set ζ = zC −1 , f (ζC) is of order m in ζ. Since fm ((0, . . . , 0, 1)C) = fm (c) 6= 0, we see that f (ζC) is of order m in ζn .  Remark 1.7. The above arguments work for a finitely many germs in On . Thus, a finite number of germs in On whose orders are positive integers can be simultaneously made to be regular in some variable, by a suitable change of coordinates. We show that the ring On possesses certain properties similar to the ones polynomial rings do. Thus we set z 0 = (z1 , . . . , zn−1 ) and let On−1 denote the ring of germs of holomorphic functions in z 0 . We consider the ring On−1 [zn ] of polynomials in zn with coefficients in On−1 : On−1 [zn ] = { a0 + a1 zn + · · · + ar znr | r ∈ N0 , ai ∈ On−1 }, which is considered as a subring of On and contains On−1 as a subring. Definition 1.12. Let m be a positive integer. A Weierstrass polynomial in zn of degree m is a germ h in On−1 [zn ] of the form h = a0 + a1 zn + · · · + am−1 znm−1 + znm , where a0 , a1 , . . . , am−1 are non-units in On−1 . Thus a Weierstrass polynomial in zn is a non-unit in both On−1 [zn ] and On . Note that in the above, h(0, . . . , 0, zn ) = znm . Hence h is regular in zn of order m. In general, for every germ f in On , we may write in a neighborhood of 0 f (z) = a0 + a1 zn + · · · + am znm + · · · with ai holomorphic functions in z 0 . The germ f is regular in zn of order m if and only if a0 , a1 , . . . , am−1 are non-units in On−1 and am is a unit in m On−1 . In this case, a−1 m (a0 +a1 zn +· · ·+am zn ) is a Weierstrass polynomial in zn of degree m. The Weierstrass preparation theorem below shows that such a germ f is essentially equal to a Weierstrass polynomial of degree m.

Analytic Functions of Several Complex Variables

17

We first prove the following: Theorem 1.9 (Weierstrass division theorem). Let h be a Weierstrass polynomial in zn of degree m. Then, for every germ f in On , there exist uniquely determined germs q in On and r in On−1 [zn ] with deg r < m such that f = qh + r. Moreover, if f is in On−1 [zn ], so is q. Proof. We choose ρ = (ρ1 , . . . , ρn ), ρi > 0, small enough so that the germs f and h have representatives on an open set containing ∆(0, ρ). As h is regular in zn , taking a smaller ρ if necessary, we may assume that h(z) 6= 0 for z = (z1 , . . . , zn ) with |z1 | < ρ1 , . . . , |zn−1 | < ρn−1 , |zn | = ρn . We denote by γ the path in the zn - plane that turns counterclockwise along the circle |zn | = ρn . If we set Z 1 f (z 0 , ζ) dζ √ q(z) = , 2π −1 γ h(z 0 , ζ) ζ − zn then q(z) is holomorphic in ∆(0, ρ). If we further set r(z) = f (z)−q(z)h(z), using the Cauchy integral formula Z f (z 0 , ζ) 1 √ dζ, f (z) = 2π −1 γ ζ − zn we compute r(z) =

1 √

2π −1

Z γ

f (z 0 , ζ) h(z 0 , ζ) − h(z 0 , zn ) dζ. h(z 0 , ζ) ζ − zn

The first function in the integrant does not depend on zn and the second is a polynomial in zn of degree less than m, since h is a Weierstrass polynomial in zn of degree m. Thus we see that r is in On−1 [zn ] and deg r < m. To show the uniqueness of q and r, suppose f = qh + r = q 0 h + r 0 . Choose ρ as above. Then we have r0 (z 0 , zn ) − r(z 0 , zn ) = (q(z 0 , zn ) − q 0 (z 0 , zn ))h(z 0 , zn ) in ∆(ρ, 0). For each fixed a0 in ∆(ρ0 , 0), ρ0 = (ρ1 , . . . , ρn−1 ), h(a0 , zn ) has m zeros in ∆(ρn , 0), counting multiplicities. On the other hand, the left-hand side above is a polynomial in zn of degree at most m − 1 and thus r0 (a0 , zn ) − r(a0 , zn ) = 0 in ∆(ρn , 0). Therefore r 0 = r and also q 0 = q, as On is an integral domain. The last statement is the division algorithm for polynomials. 

18

Complex Analytic Geometry

Theorem 1.10 (Weierstrass preparation theorem). Let f be a germ in On which is regular in zn of order m. Then there is a unique Weierstrass polynomial h in zn of degree m such that f = uh with u a unit in On . Proof. We introduce new variables w = (w1 , . . . , wm ) and we set Om+n = C{w, z} and Om+n−1 = C{w, z 0 }. We set p(w, zn ) = wm + wm−1 zn + · · · + w1 znm−1 + znm and think of it as a Weierstrass polynomial in Om+n−1 [zn ] of degree m. Since On is naturally a subring of Om+n , by the Weierstrass division theorem, there exist unique germs q in Om+n and r in Om+n−1 [zn ] of degree less than m such that f = qp + r.

(1.5)

We write r(w, z 0 , zn ) = bm (w, z 0 ) + bm−1 (w, z 0 )zn + · · · + b1 (w, z 0 )znm−1 with b1 , . . . , bm in Om+n−1 . If we set w = 0 and z 0 = 0 in (1.5), we get f (0, zn ) = q(0, 0, zn )znm + bm (0, 0) + bm−1 (0, 0)zn + · · · + b1 (0, 0)znm−1 . Since f is of order m in zn , q(0, 0, 0) = c 6= 0 and b1 (0, 0) = · · · = bm−1 (0, 0) = bm (0, 0) = 0. If we differentiate the both sides of (1.5) with respect to wj , noting that ∂p/∂wj = znm−j , we get ∂bm ∂bm−1 ∂b1 m−1 ∂q p + qznm−j + + zn + · · · + z . 0= ∂wj ∂wj ∂wj ∂wj n Setting w = 0 and z 0 = 0 again, we see that ( −c for i = j, ∂bi (0, 0) = ∂wj 0 for i > j. Thus ∂(b1 , . . . , bm ) (0, 0) = (−c)m 6= 0 ∂(w1 , . . . , wm ) and by Theorem 1.5, there is a holomorphic map ϕ : (Cn−1 , 0) → (Cm , 0) such that det

b1 (ϕ(z 0 ), z 0 ) = · · · = bm−1 (ϕ(z 0 ), z 0 ) = bm (ϕ(z 0 ), z 0 ) = 0. If we substitute w = ϕ(z 0 ) in (1.5), we get f (z 0 , zn ) = q(ϕ(z 0 ), z 0 , zn )p(ϕ(z 0 ), zn ). If we set u(z 0 , zn ) = q(ϕ(z 0 ), z 0 , zn ) and h(z 0 , zn ) = p(ϕ(z 0 ), zn ), then f = uh with u a unit in On and h a Weierstrass polynomial in zn of degree m. The uniqueness of such u and h is not difficult to see. 

Analytic Functions of Several Complex Variables

19

Remark 1.8. By the above theorem and Remark 1.7, for a finite number of germs, each of which is not 0 or a unit in On , we may find a coordinate system (z1 , . . . , zn ) such that each germ is expressed as the product of a unit and a Weierstrass polynomial in zn . Next we discuss some important properties of the ring On which follow from the above theorems. First we observe that: Lemma 1.3. Let h be a Weierstrass polynomial in On−1 [zn ]. Then h is irreducible in On−1 [zn ] if and only if it is irreducible in On . If it is not irreducible, all of its factors are Weierstrass polynomials in On−1 [zn ]. Proof. Suppose h is not irreducible in On . Then we may write h = f1 f2 with fi , i = 1, 2, non-units in On . Since h is regular in zn , both f1 and f2 are regular in zn and by the preparation theorem, we may write fi = ui hi , i = 1, 2, with ui units in On and hi Weierstrass polynomials in On−1 [zn ]. By the uniqueness, we have h = h1 h2 , and thus h is not irreducible in On−1 [zn ]. Conversely suppose h is not irreducible in On−1 [zn ]. Then h = g1 g2 with gi , i = 1, 2, non-units in On−1 [zn ]. If one of g1 or g2 , say g1 , were a unit in On , we would have ug1 = 1 for some u in On and g2 = uh. From the last part of the division theorem, we would then see that u ∈ On−1 [zn ], which contradicts the fact that g1 is a non-unit in On−1 [zn ]. Thus g1 and g2 are non-units in On and h is not irreducible in On .  Theorem 1.11. The ring On is a unique factorization domain. Proof. We proceed by induction on n. First, we have O0 = C, which is a unique factorization domain, a UFD for short. We assume that On−1 is a UFD. Take a germ f in On which is not 0 or a unit. By changing the coordinate system on Cn if necessary, we may assume that f is regular in zn . By the Weierstrass preparation theorem, there is a unit u in On such that uf is a Weierstrass polynomial in On−1 [zn ], which is a UFD by the induction hypothesis and Gauss’ lemma (Theorem A.15). Since uf is not 0 or a unit, it can be expressed as a product of irreducible elements in On−1 [zn ]. Then the theorem follows from Lemma 1.3.  Since a unit in On has m-th roots for every positive integer m, every germ f , which is not 0 or a unit, has the irreducible decomposition (cf. (A.11)) of the form mr 1 f = pm 1 · · · pr .

20

Complex Analytic Geometry

Recall that, in a UFD, an element is irreducible if and only if it is prime. Theorem 1.12. The ring On is Noetherian. Proof. We proceed by induction on n. We have O0 = C, which is Noetherian. We assume that On−1 is Noetherian. Take an arbitrary ideal I in On . If I = 0, we have nothing to prove. If I 6= 0, we choose a non-zero element h in I. By the Weierstrass preparation theorem, we may assume that h is a Weierstrass polynomial. Thus h is in I ∩On−1 [zn ]. Since I ∩On−1 [zn ] is an ideal in On−1 [zn ], which is Noetherian by the induction hypothesis and the Hilbert basis theorem (Theorem A.17), it has a finite number of generators g1 , . . . , gr . We show that they also generate the ideal I over On . Take an element f in I. Changing the coordinate system of Cn if necessary, we may assume that f is regular in zn . Then by the Weierstrass division theorem, we may write f = qh + r for some q in On and r in On−1 [zn ]. Since the germs f and h are in I, so is r. Thus r is in I ∩ On−1 [zn ]. Since h and r can be written as linear combinations of g1 , . . . , gr over On−1 [zn ], the germ f is a linear combination of g1 , . . . , gr over On .  We give another application of Theorem 1.10. For a point z in Cn , let On,z be the ring of germs of holomorphic functions at z, which is naturally isomorphic with On . For a germ f in On we denote by fz the germ of f at z near 0. Proposition 1.8. If f and g are relatively prime in On , then fz and gz are relatively prime in On,z for all z sufficiently close to 0. Proof. If f or g is a unit, the proposition obviously holds. Thus we assume that f and g are non-units. By the preparation theorem, we may assume that they are Weierstrass polynomials in zn (cf. Remark 1.8) and are relatively prime in On−1 [zn ] (cf. Lemma 1.3). Thus by Theorem A.16, there exist elements a and b in On−1 [zn ] and c 6= 0 in On−1 such that af + bg = c.

(1.6)

Note that the above equality holds in a neighborhood U of 0 in Cn where all the germs involved have representatives. We may assume that U is of the form U = U 0 × U 00 with U 0 a neighborhood of 0 in Cn−1 and U 00 a neighborhood of 0 in C = {zn }. As f is regular in zn , taking a smaller U 0 if necessary, we may assume that for each z 0 in U 0 , f (z 0 , zn ) is not identically zero as a function of zn in U 00 .

Analytic Functions of Several Complex Variables

21

Let a be a point in U . If f (a) 6= 0 or g(a) 6= 0, fa and ga are relatively prime in On,a . Thus assume that f (a) = g(a) = 0. If fa and ga have a common factor ha in On,a with h(a) = 0, ha also divides ca by (1.6) so that the function h(z 0 , zn ) defined near a = (a0 , an ) depends only on z 0 . Thus h(a0 , zn ) vanishes identically in zn . This contradicts the fact that f (a0 , zn ) is not identically zero as a function of zn .  We finish this section by discussing the Riemann extension theorem in the several variable case. Theorem 1.13. Let D be a connected open set in Cn and g a function holomorphic and not identically 0 in D. We set V = { z ∈ D | g(z) = 0 }. If f is a function holomorphic in D r V and is locally bounded, i.e., each point z in D has a neighborhood U such that f is bounded in (D rV ) ∩ U , there is a unique function f˜ holomorphic in D and equal to f on D rV . Proof. The case n = 1 is the classical Riemann extension theorem. Thus we assume that n ≥ 2 and reduce the problem to this case. First we note that, by the assumption and Theorem 1.7, V has no interior points. In particular, for every point z in V , the germ gz is not 0 or a unit. For the existence, it suffices to show that each point z in V has a neighborhood on which there is an extension f˜ of f . Without loss of generality, we may assume z = 0. After a change of coordinates if necessary, we may assume that g is regular in zn , in fact we may assume that g is a Weierstrass polynomial in zn . We choose ρ = (ρ1 , . . . , ρn ), ρi > 0, small enough so that g has a representative on an open set containing ∆(0, ρ) and that g(z) 6= 0 for z = (z 0 , zn ) with z 0 ∈ ∆(0, ρ0 ) and |zn | = ρn , where we set z 0 = (z1 , . . . , zn−1 ) and ρ0 = (ρ1 , . . . , ρn−1 ). Denoting by γ the path in the zn -plane turning counterclockwise along the circle |zn | = ρn , we set Z f (z 0 , ζ) 1 √ dζ. f˜(z) = 2π −1 γ ζ − zn Then f˜ is holomorphic in ∆(0, ρ). For each fixed z 0 in ∆(0, ρ0 ), f (z 0 , zn ) is holomorphic in zn in an open set U containing ∆(0, ρn ) except for a finite number of points (the zeros of g(z 0 , zn )) and is locally bounded. Thus by the Riemann extension theorem in one variable case, f (z 0 , zn ) has a holomorphic extension in U . Therefore by the Cauchy integral formula, f˜ is an extension of f in ∆(0, ρ). The uniqueness follows from the uniqueness of analytic continuation.  Corollary 1.1. If D and V are as in Theorem 1.13, D rV is connected.

22

Complex Analytic Geometry

Proof. If DrV is not connected, there are non-empty open sets U1 and U2 such that DrV = U1 ∪ U2 and U1 ∩ U2 = ∅. The function f , defined by f (z) = 1 for z ∈ U1 and f (z) = 0 for z ∈ U2 , has no extensions.  Notes For Section 1.2, there are a number of textbooks. Here we list [Cartan (1961)]. See 15.12 Theorem in [Rudin (1987)] for Remark 1.3. 1. We list [Fritzsche and Grauert (2002); Gunning and Rossi (1965)] as general references for the theory of analytic functions of several complex variables. The proof of the Weierstrass preparation theorem (Theorem 1.10) given here is based on the one in [Noguchi and Fukuda (1976)].

Chapter 2

Complex Manifolds and Analytic Varieties

A complex manifold is a space that is locally the set of common zeros of holomorphic functions without singularities. It is also described as a spcae obtained by patching open sets in Cn together by biholomorphic maps. These are the spaces on which we consider holomorphic functions, maps and related objects in this book. An analytic variety in a complex manifold is a subset of the manifold that is the common zeros of holomorphic functions possibly with singularities. In this chapter we discuss fundamentals on complex manifolds and analytic varieties. Here is a word about submanifolds and varieties. A submanifold is a variety without singularities. However, when we say that V is a submanifold of a complex manifold M , it is a locally closed set in M , while when we say that V is a variety in M it is a closed set in M . As in the case of functions, we introduce the notion of the germ of a variety. We study the relation between germs of functions and those of varieties. The important point in this is what is called the “Nullstellensatz” (zero locus theorem). We give a proof in the case of principal ideals (Corollary 2.1) and, for the general case, we quote some structural results and give proofs based on these (Theorem 2.8). We also discuss local and global dimensions of varieties and related topics. It is important to keep in mind that a complex manifold M carries an underlying real structure. In particular, as a C ∞ manifold, M has the tangent space at each point. We define the holomorphic tangent space and give the relation with the real tangent space (cf. (2.6)). We also discuss stratification of varieties, which is useful in the study of singular varieties.

23

24

2.1

Complex Analytic Geometry

Complex manifolds

Let M be a Hausdorff topological space with a countable basis. Definition 2.1. A complex chart, or simply a chart, on M is a pair (U, ϕ) of an open set U in M and a homeomorphism ϕ of U onto an open set in Cn for some n ≥ 0. Definition 2.2. A holomorphic atlas on M is a collection of charts A = {(Uα , ϕα )}α∈I such that {Uα }α∈I is a covering of M and that, for each pair (α, β), the map ϕα ◦ ϕβ −1 : ϕβ (Uα ∩ Uβ ) → ϕα (Uα ∩ Uβ ) is holomorphic. Interchanging α and β, we see that the above ϕα ◦ ϕβ −1 is in fact biholomorphic. We also use the convention that a map of the empty set is holomorphic. For two holomorphic atlases A and A0 , we define a relation ∼ saying that A ∼ A0 if their union is a holomorphic atlas. Obviously it is an equivalence relation in the set of holomorphic atlases on M . Definition 2.3. A complex structure on M is an equivalence class of a holomorphic atlas. A complex manifold is a space M together with a complex structure. A complex manifold is often denoted simply by M . The natural number n appearing in Definition 2.1, which is uniquely determined on each connected component of M , is called the (complex) dimension of the component. If all the components have dimension n, we say the dimension of M is n. Remark 2.1. 1. Let X be a topological space and let U = {Uα }α∈I and V = {Vλ }λ∈J be open coverings of X. Recall that V is a refinement of U if there exists a map ι : J → I such that Vλ ⊂ Uι(λ) for all λ. Every open covering of X is equivalent to a covering whose index set is a subset of the power set of X, in the sense that each one of the two is a refinement of the other. Thus we may talk about the “set” of open coverings of X. We may define an order relation in the set of open coverings of X by saying that U ≤ V if V is a refinement of U. In the following we consider the set of open coverings as an ordered set with this ordering. It is a directed set (cf. Section A.1). 2. Likewise we may talk about the set of atlases on a space M as above.

Complex Manifolds and Analytic Varieties

25

3. Recall that a topological space X is paracompact if it is Hausdorff and if every open covering of X admits a locally finite refinement. Since we assumed that the underlying topological space of a complex manifold M to be Hausdorff with countable basis, we see that M is paracompact and the number of connected components is countable. 4. A complex manifold of dimension 0 is a countable set of points with the discrete topology. A complex manifold of dimension one is called a Riemann surface or a (non-singular) complex curve. Also, complex manifold of dimension two is called a (non-singular) complex surface. When these terminologies are used, it is usually understood that the manifold is connected. Let M be a complex manifold. A chart (U, ϕ) on M is said to be holomorphic if it belongs to an atlas representing the complex structure. Let (U, ϕ) be a holomorphic chart on M and suppose M is of dimension n ≥ 1. For a point p in U , we call U a coordinate neighborhood of p and ϕ(p) = (z1 (p), . . . , zn (p)) the local coordinates of p with respect to ϕ. Sometimes we identify U with ϕ(U ) and p with the point (z1 (p), . . . , zn (p)) in ϕ(U ) ⊂ Cn . In this case we call (z1 , . . . , zn ) a coordinate system on U . A complex manifold can be defined by specifying a holomorphic atlas on an appropriate topological space. Example 2.1. 1. A non-empty open subset in Cn is an n-dimensional complex manifold with natural global coordinates. 2. Complex projective space Pn : We denote by ζ = (ζ0 , . . . , ζn ) coordinates on Cn+1 r{0} and introduce a relation ∼ in Cn+1 r{0} by setting, for ζ = (ζ0 , . . . , ζn ) and ζ 0 = (ζ00 , . . . , ζn0 ), ζ ∼ ζ 0 if ζ 0 = tζ for some t in C∗ = Cr{0}. Obviously this is an equivalence relation and we set Pn = (Cn+1 r{0})/ ∼. Endowed with the quotient topology, it is Hausdorff and connected. The equivalence class of ζ = (ζ0 , . . . , ζn ) is denoted by [ζ] = [ζ0 , . . . , ζn ]. The space Pn with the following complex structure is called the n-dimensional complex projective space. First, for each α = 0, . . . , n, the set Uα = { [ζ] ∈ Pn | ζα 6= 0 } is a well-defined open set and Pn is covered by these n + 1 open sets. Also the map ϕα : Uα → Cn given by ϕα ([ζ]) = (ζ0 /ζα , . . . , ζα−1 /ζα , ζα+1 /ζα , . . . , ζn /ζα )

(2.1)

26

Complex Analytic Geometry

is a well-defined homeomorphism. Moreover, it is not difficult to check that for each pair (α, β), the map ϕα ◦ ϕβ −1 : ϕβ (Uα ∩ Uβ ) → ϕα (Uα ∩ Uβ ) is holomorphic. Thus Pn becomes a complex manifold of dimension n. We call [ζ] = [ζ0 , . . . , ζn ] homogeneous coordinates on Pn . We denote the right-hand side of (2.1) by z α = (z1α , . . . , znα ): ( ζi−1 /ζα i = 1, . . . , α, α zi = (2.2) ζi /ζα i = α + 1, . . . , n. It is a holomorphic coordinate system on Uα . We may interpret Pn as the set of complex lines through 0 (onedimensional linear subspaces) in Cn+1 . This viewpoint leads to the construction of a more general class of manifolds, i.e., Grassmann manifolds, which will be discussed in Section 3.6 below. 3. If M and M 0 are complex manifolds of dimensions n and n0 , respectively, the product M × M 0 has naturally the structure of a complex manifold of dimension n + n0 . If (z1 , . . . , zn ) is a coordinate system on U ⊂ M and (z10 , . . . , zn0 0 ) a coordinate system on U 0 ⊂ M 0 , then (z1 , . . . , zn ; z10 , . . . , zn0 0 ) is a coordinate system on U × U 0 ⊂ M × M 0 . See Section 3.1 for some more basic examples. Exercise 2.1. Show that P1 is homeomorphic to the 2-sphere S2 and that it may be interpreted as being obtained from the complex plane C by attaching a “point at ∞”, i.e., the Riemann sphere. Exercise 2.2. Let S2n+1 = { (ζ0 , . . . , ζn ) ∈ Cn+1 | |ζ0 |2 + · · · + |ζn |2 = 1 } be the (2n + 1)-dimensional unit sphere and π the restriction of the canonical surjection Cn+1 r {0} → Pn to S2n+1 . Show that π is surjective (thus Pn is compact) and find the inverse image π −1 (p) for each point p in Pn (cf. Example 3.4 and the expression (3.20) below). Holomorphic maps Let M be a complex manifold with an atlas {(Uα , ϕα )} representing the complex structure. A complex valued function f on an open set U in M is said to be holomorphic if, for every α, the function f ◦ ϕ−1 α is holomorphic on ϕα (U ∩ Uα ). Also, a map f : M → M 0 from M into another complex manifold M 0 with atlas {(Vλ , ψλ )} is said to be holomorphic, if for every −1 pair (α, λ), the map ψλ ◦ f ◦ ϕ−1 (Vλ )). A α is holomorphic on ϕα (Uα ∩ f −1 biholomorphic map is a bijective holomorphic map f such that f is also

Complex Manifolds and Analytic Varieties

27

holomorphic. If there exists a biholomorphic map from M onto M 0 , we say that M is biholomorphic with M 0 and write M ' M 0 . The following is a consequence of Theorem 1.8: Proposition 2.1. Every holomorphic function on a compact connected complex manifold is constant. For a holomorphic map f : M → M 0 , we may define regular and singular points of f as in Definition 1.9, choosing local coordinate systems, and the definition does not depend on the choice of the coordinate systems. Also the behavior of f near a regular point is given as in Theorem 1.6. Definition 2.4. Let f : M → M 0 be a holomorphic map and let n and n0 be dimensions of M and M 0 , respectively. 1. f is an immersion if n ≤ n0 and every point of M is a regular point of f . The map f is an embedding if, furthermore, it is a homeomorphism onto f (M ) with the topology induced from that of M 0 . 2. f is a submersion if n ≥ n0 and every point of M is a regular point of f . Definition 2.5. A holomorphic map π : M → M 0 is a (holomorphic) covering map if it is surjective and if each point of M 0 has a neighborhood U such that each connected component of π −1 (U ) is mapped biholomorphically onto U by π. Thus in this case, dim M = dim M 0 and π is immersive and submersive. Remark 2.2. The set {p} consisting of a point p is a connected complex manifold of dimension 0, with a global chart homeomorphic to C0 = {0}. Any map f : M → {p} of a complex manifold M is holomorphic and every point in M is a regular point of f . Submanifolds Let M be a complex manifold of dimension n. For non-negative integers d and k with d + k = n, we decompose as Cn = Ck × Cd with (z1 , . . . , zk ) and (zk+1 , . . . , zn ) as coordinates on Ck and Cd , respectively. Definition 2.6. A subset V of M is a (complex) submanifold of M if there is an atlas {(Uα , ϕα )} representing the complex structure of M such that for each α, ϕα (V ∩ Uα ) is an open set in Cd = {0} × Cd ⊂ Cn .

28

Complex Analytic Geometry

Thus if V is a submanifold of M , with the topology induced from that of M , it is a complex manifold with the complex structure represented by the atlas {(V ∩ Uα , ϕα |V ∩Uα )}. It is locally closed in M . If V 6= ∅, the integer d in uniquely determined on each connected component of V and is the dimension of the component. The inclusion i : V ,→ M is an embedding. Suppose d ≥ 1 and write ϕα (p) = (z1α (p), · · · , znα (p)) on Uα . Then for α α with V ∩ Uα 6= ∅, ϕα (p) = (0, . . . , 0, zk+1 (p), · · · , znα (p), )) on V ∩ Uα and α α (zk+1 , · · · , zn ) is a coordinate system on V ∩ Uα . Thus, if k ≥ 1, we may think of V as being given by z1α = · · · = zkα = 0 in Uα .

2.2

Analytic varieties

Analytic varieties naturally arise as the set of common zeros of holomorphic functions and include submanifolds as a special class. Let M be a complex manifold. Definition 2.7. An analytic variety in M is a subset V of M such that every point a in M admits a neighborhood U with the property: (*) there exists a holomorphic map f : U → Cr with V ∩ U = f −1 (0). We call f as above a local defining map of V near a. In the following, an analytic variety will simply be called a variety, if there is no fear of confusion. Note that a variety in M is a closed set in M . Remark 2.3. 1. Let V be a subset of M . If every point a in V admits a neighborhood U with the property (*), V is a variety in some open set in M . In particular, if V is a closed set in M as above, it is a variety in M . 2. In the above we allow r to be 0 (cf. Remark 2.2). If r ≥ 1, letting (f1 , . . . , fr ) be the components of f , we may write V ∩ U = { z ∈ U | f1 (z) = · · · = fr (z) = 0 }. We call (f1 , . . . , fr ) as above a system of local defining functions of V and f1 = · · · = fr = 0 local equations for V near a. 3. In particular, M itself and the empty set ∅ are varieties in M . For M , we may take the map M → C0 or the function M → C constantly equal to 0 as a defining map. For ∅, we may take a non-vanishing function, constant function 1 for example, as a defining map. The first part of the following is obvious from the uniqueness of analytic continuation (Theorem 1.7) and the second part follows from Corollary 1.1.

Complex Manifolds and Analytic Varieties

29

Theorem 2.1. Let V be a variety in a connected open set D. If V 6= D, it has no interior points as a subset of D. Moreover, D rV is connected. Remark 2.4. Recall that a subset A of a topological space X is nowhere dense in X, if the closure of A has no interior points in X. Thus, in the above situation, V is nowhere dense in D. In the following, if we simply say a variety, it means a variety in M , thus it is closed in M . We think of a variety as a topological space with the topology induced from that of M . Definition 2.8. Let V be a variety and a a point in V . We say that a is a regular point of V if there is a local defining map f : U → Ck near a such that a is a regular point of f . We say that a is a singular point of V if it is not a regular point. For a variety V , we denote by Vreg and Sing(V ), respectively, the sets of regular and singular points of V . Note that Vreg is an open set in V . Hence Sing(V ) is a closed set in V (and in M ). In fact it is shown that Sing(V ) is again an analytic variety (cf. Theorem 11.5 below). We say that a variety V is non-singular if Sing(V ) = ∅. Example 2.2. Let V be a submanifold of M . Then it is closed in some open set D. Let {(Uα , ϕα )} be an atlas as in Definition 2.6. For every point a of V , take Uα containing a. Then p ◦ ϕα : Uα → Ck is a local defining map on Uα , where p : Cn → Ck is the projection, and V is a variety in D. It is non-singular. Let V be a variety. If a is a regular point of V , by Theorem 1.6, we may assume that k ≤ n in Definition 2.8. Moreover, there is a neighborhood U of a such that V ∩ U is homeomorphic to an open set in Cn−k . In fact, if k ≥ 1, we may assume that U is a coordinate neighborhood with coordinates (z1 , . . . , zn ) such that V ∩ U = {z1 = · · · = zk = 0}. Exercise 2.3. Show that Vreg has a natural structure of complex submanifold of M . From the above, we have: Proposition 2.2. A subset V of M is a complex submanifold of M if and only if it is a non-singular variety in some open set of M .

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Complex Analytic Geometry

Exercise 2.4. Let f : M 0 → M be a holomorphic map of complex manifolds. Show that, if f is an embedding, f (M 0 ) may be endowed with a structure of a submanifold of M so that f is a biholomorphic map of M 0 onto f (M 0 ). Affine algebraic varieties Consider the complex n-space Cn with coordinates (z1 , . . . , zn ) and let, for each i = 1, . . . , r, Pi (z1 , . . . , zn ) be a polynomial in (z1 , . . . , zn ). Then the set V = { (z1 , . . . , zn ) ∈ Cn | Pi (z1 , . . . , zn ) = 0, i = 1, . . . , r } is a variety in Cn , called an affine algebraic variety. Example 2.3. 1. Consider C2 with coordinates (z1 , z2 ). We set P (z1 , z2 ) = z1 z2 and V = { (z1 , z2 ) | P (z1 , z2 ) = 0 }. Thus V consists of two “complex lines” (z1 and z2 “axes”) intersecting in C2 at one point, the origin 0. By definition we see that V r {0} ⊂ Vreg , while by looking at the neighborhood structure of 0, we see that 0 is a singular point of V (cf. Exercise 2.5. 1 below). This can be also checked by studying the behavior of the tangent spaces of the regular part. See also Proposition 11.26. Thus Vreg = V r {0}, which has two connected components each being a one-dimensional complex manifold biholomorphic with C∗ = Cr{0}. 2. Again we consider C2 . We set P (z1 , z2 ) = z13 − z22 and let V be the variety defined by P . By definition we see that V r{0} ⊂ Vreg , while 0 is a singular point of V (cf. Exercise 2.5. 2). Thus Vreg = V r{0}, which has one component biholomorphic with C∗ . Note that V is homeomorphic with C. 3. Consider C3 with coordinates (z1 , z2 , z3 ). We set f (z1 , z2 , z3 ) = z1 z22 −z32 and let V be the variety defined by f . Then Vreg is a two-dimensional complex manifold and Sing(V ) is the z1 -axis. The set V is called the Whitney umbrella. 4. Again consider C3 . We set f (z1 , z2 , z3 ) = z22 − z12 z32 − z13 and let V be the variety defined by f . Then Vreg is a two-dimensional complex manifold and Sing(V ) is the z3 -axis. Exercise 2.5. 1. Let S3 = { (z1 , z2 ) | |z1 |2 + |z2 |2 = 1 } be the threedimensional unit sphere in C2 = R4 . Show that, in Example 2.3. 1, the intersection K = V ∩ S3 consists of two circles that are unknotted but link with each other. 2. Show that, in Example 2.3. 2, K = V ∩ S3 is the “(2, 3)-torus knot”.

Complex Manifolds and Analytic Varieties

31

3. In Example 2.3. 2, find an explicit (holomorphic) homeomorphism from C onto V . 4. Let V be the variety in C4 = {(z1 , z2 , z3 , z4 )} defined by the three equations: z1 z4 − z2 z3 = 0,

z22 − z1 z3 = 0 and z32 − z2 z4 = 0.

Find Vreg and Sing(V ) (cf. Theorem 11.5, also Example 11.10. 1 below). What is the dimension of Vreg ? Projective algebraic varieties Consider the n-dimensional complex projective space Pn with homogeneous coordinates [ζ] = [ζ0 , . . . , ζn ] and let, for each i = 1, . . . , r, Pi (ζ0 , . . . , ζn ) be a homogeneous polynomial in (ζ0 , . . . , ζn ) of degree pi . Then the set V = { [ζ] ∈ Pn | Pi (ζ0 , . . . , ζn ) = 0, i = 1, . . . , r } is a well-defined subset of Pn and is, moreover, a variety in Pn . In fact, in each open set Uα = { ζα 6= 0 }, V is defined by the holomorphic functions fi = Pi (ζ0 , . . . , ζn )/ζαpi , i = 1, . . . , r. Such a variety is called a projective algebraic variety. Note that, as each Uα may naturally be identified with Cn , V ∩Uα is an affine albegraic variety. A non-singular projective algebraic variety is called a projective algebraic manifold. Example 2.4. 1. Let $ : Cn+1 r {0} → Pn be the canonical surjection (cf. Example 2.1. 2). For a (d + 1)-dimensional subspace V of Cn+1 , $(V r {0}) is a projective algebraic manifold of dimension d, which is biholomorphic with Pd and is called a d-plane in Pn . In the case d = n − 1, it is called a hyperplane. 2. A projective algebraic variety in Pn defined by a single (not identically zero) homogeneous polynomial of degree p is called a hypersurface of degree p (cf. Section 2.4 below). A hypersurface of degree one is defined by a linear function `(ζ) = a0 ζ0 + · · · + an ζn with (a0 , . . . , an ) 6= (0, . . . , 0). Thus it is non-singular and is nothing but a hyperplane. If H is a hyperplane, by a linear change of coordinates, we may assume that it is defined by ζ0 = 0. Identifying the open set U0 = {ζ0 6= 0} with Cn = {(ζ1 /ζ0 , . . . , ζn /ζ0 )} and H with Pn−1 = {[ζ1 , . . . , ζn ]}, we may express Pn as a disjoint union Pn = Cn t Pn−1 . This procedure leads to a cellular decomposition of Pn as Pn = Cn t Cn−1 t · · · t C0 (cf. Section B.2). Using this we may compute

32

Complex Analytic Geometry

the homology of Pn : ( n

Hq (P ; Z) =

Z

for q = 0, 2, . . . , 2n,

0

otherwise.

(2.3)

For each i = 0, . . . , n, the group H2i (Pn ; Z) has a canonical generator u2i which is represented by the cycle Pi . For a hypersurface V of degree p, its homology class is given by [V ] = p · u2n−2 (cf. Example 12.5 below). This is a special case of the decomposition of Grassmann manifolds by Schubert cells (cf. Section 3.6). Exercise 2.6. For complex numbers α, β and γ, let Vα,β,γ be the variety in P2 defined by Vα,β,γ = { [ζ0 , ζ1 , ζ2 ] ∈ P2 | ζ0 ζ22 − (ζ1 − αζ0 )(ζ1 − βζ0 )(ζ1 − γζ0 ) = 0 }. 1. Show that, if α, β and γ are mutually distinct, then Vα,β,γ has no singular points. 2. Show that, if γ 6= 0, then V0,0,γ has only one singular point at a = [1, 0, 0], which is equivalent to the one in Example 2.3. 1. 3. Show that V0,0,0 has only one singular point at a = [1, 0, 0], which is equivalent to the one in Example 2.3. 2. Quasi-projective varieties: A set of the form V r V 0 in Pn is called a quasi-projective variety, where V and V 0 are projective algebraic varieties. A quasi-projective manifold is a non-singular quasi-projective variety. Zariski topology: Instead of the usual topology of Cn , we may take as the open sets the complements of the affine algebraic varieties to define a topology on Cn , which is called the Zariski toplogy. Likewise we may define the Zariski topology of Pn by taking as the open sets the complements of the projective algebraic varieties. For a quasi-projective variety V in Pn , it has also the topology induced from the Zariski topology. We denote by VZ the variety V with Zariski topology. Note that the identity map V → VZ is continuous. Algebraic functions: Let V be a quasi-projective variety. A function f : VZ → C is said to be algebraically regular at p in VZ if there exist a neighborhood UZ of p and homogeneous polynomials P and Q of the same degree such that Q is nowhere zero on UZ and f = P/Q on UZ . It is simply called regular, if there is no fear of confusion. A regular function on an open set in VZ is a function regular at every point of the open set.

Complex Manifolds and Analytic Varieties

33

An algebraic function on V is the pull-back of a regular function by the identity V → VZ . Remark 2.5. It is known that every analytic variety in Pn is algebraic (Chow’s theorem). Also, every compact complex manifold of dimension one (compact Riemann surface) can be embedded in Pn for some n. Thus it is biholomorphic with a projective algebraic manifold. Blowing-up This can be defined in generality, however here we discuss the simplest case to illustrate the idea. See Section 11.6 below for the higher-dimensional case and more discussions on this subject. ˜ the variety in Let U be a neighborhood of 0 in C2 = {(z1 , z2 )} and U 1 U × P defined by ˜ = { (z, [ζ]) ∈ U × P1 | z1 ζ1 − z2 ζ0 = 0 }. U If we set Wα = { [ζ] ∈ P1 | ζα 6= 0 }, α = 0, 1, the manifold U ×P1 is covered by two coordinate neighborhoods U × W0 and U × W1 with coordinates (z1 , z2 , ξ) = (z1 , z2 , ζ1 /ζ0 ) and (z1 , z2 , η) = (z1 , z2 , ζ0 /ζ1 ), respectively. In ˜ is given by z1 ξ − z2 = 0 and in U × W1 , by z1 − z2 η = 0. Thus U ˜ U × W0 , U does not have singular points and is a two-dimensional complex manifold, ˜0 = U ˜ ∩ (U × W0 ) and which is covered by two coordinate neighborhoods U ˜ ˜ U1 = U ∩ (U × W1 ) with coordinates (z1 , ξ) and (z2 , η), respectively. If we ˜ → U the restriction of the projection U × P1 → U to U ˜, denote by π : U −1 1 it is a holomorphic map such that π (0) ' P and that its restriction to ˜ −1 (0) is a biholomorphic map onto Ur{0}. This procedure is called the Urπ blowing-up of U at 0. It replaces 0 with the projective line P1 to separate the lines through 0 in U . ˜ , which If V is the variety in Example 2.3. 1, π −1 (V ) is a variety in U ˜0 and by z 2 η = 0 in U ˜1 . Thus we may write is defined by z12 ξ = 0 in U 2 π −1 (V ) = V˜ ∪ π −1 (0), where V˜ consists of two (disjoint) lines, one defined ˜0 and the other by η = 0 in U ˜1 . Notice that π −1 (0) appears by ξ = 0 in U “in double”. Also, if V is the variety in Example 2.3. 2, π −1 (V ) is a variety ˜0 and by z 2 (z2 η 3 − 1) = 0 in ˜ , which is defined by z 2 (z1 − ξ 2 ) = 0 in U in U 1 2 ˜1 . Thus we may write π −1 (V ) = V˜ ∪ π −1 (0), where V˜ is the “parabola” U ˜0 and by z2 η 3 − 1 = 0 in U ˜1 . Again π −1 (0) appears given by z1 − ξ 2 = 0 in U in double. In general, applying this procedure successively, we may “resolve” the singularity of a plane curve (cf. Section 11.6).

34

2.3

Complex Analytic Geometry

Germs of varieties

In this section, we consider the germs of varieties and discuss the relation between these and the ideals in On . In the following, we denote the germ of a function f at 0 also by f . We first introduce a relation ∼ in the set of subsets of Cn . Let A and B be two subsets of Cn . We define A ∼ B if there is a neighborhood U of 0 such that A ∩ U = B ∩ U . It is easily checked that this is an equivalence relation. We call the equivalence class of A the germ of A at 0 and denote it also by A, unless it is necessary to distinguish the two. Usual operations of sets induce those of germs. Thus for two germs A and B at 0, A ∩ B, A ∪ B and ArB are well-defined. The relation A ⊂ B is also well-defined. In Section 1.4, we introduced the ring On of germs of holomorphic functions at 0. Let f1 , . . . , fr be germs in On . We choose a neighborhood U of 0 such that these germs are represented by holomorphic functions on U , which we also denote by f1 , . . . , fr . We set V (f1 , . . . , fr ) = the germ at 0 of { z ∈ U | f1 (z) = · · · = fr (z) = 0 } and call it the germ of the variety defined by f1 , . . . , fr . More generally, let I be an ideal in On . By the Noetherian property of On (cf. Theorem 1.12), there exist a finite number of germs f1 , . . . , fr such that I = (f1 , . . . , fr ), the ideal generated by f1 , . . . , fr . We set V (I) = V (f1 , . . . , fr ) and call it the germ of the variety defined by I. It is easily checked that it does not depend on the choice of generators of I. Thus each ideal in On defines a germ of a variety at 0. Conversely suppose we are given a germ V of a variety at 0. We choose a neighborhood U of 0 such that the germ is represented by a variety in U , which we denote also by V . We set I(V ) = { f ∈ On | f (z) = 0 for all z in V and near 0 }. It is easily checked that this is an ideal in O√n . Recall that (cf. Section A.2) for an ideal I in On , we have the radical I of I. This is again an ideal in On and contains I. See also Section A.2 for the notions of prime and primary ideals . Exercise 2.7. 1. For k = 1, . . . , n, we consider the “coordinate functions” z1 , . . . , zk as germs in On . Show that the ideal p = (z1 , . . . , zk ) is prime and that I(V (p)) = p. √ 2. Show that in the ring O2 , q = (z1 , z22 ) is primary and that q = (z1 , z2 ).

Complex Manifolds and Analytic Varieties

35

We have the following various relations between germs of varieties at 0 and ideals in On : Proposition 2.3. Let V , V1 and V2 be germs of varieties at 0 and I, I1 and I2 ideals in On . 1. If I1 ⊂ I2 , then V (I1 ) ⊃ V (I2 ). 2. If V1 ⊂ V2 , then I(V1 ) ⊃ I(V2 ). 3. V (I1 ) ∩ V (I2 ) = V (I1 + I2 ). 4. V (I1 ) ∪ V (I2 ) = V (I1 I2 ) = V (I1 ∩ I2 ). 5. I(V1 ∪ V2 ) = I(V1 ) ∩ I(V2 ). p 6. I(V ) = I(V ). √ 7. V ( I) = V (I). 8. V (I(V )) = V . √ 9. I(V (I)) ⊃ I. Exercise 2.8. Prove Proposition 2.3. In fact we have the equality in 9 above, which is the content of what is called the “Nullstellensatz” (zero-locus theorem). It will be discussed in the next section. Later in this section we prove this in the case of principal ideals. Definition 2.9. Let V be a germ of a variety at 0. We say that V is irreducible if V 6= ∅ and if V = V1 ∪ V2 implies V1 = V or V2 = V , where V1 and V2 are germs of varieties at 0. The following gives an algebraic characterization of the irreducibility: Theorem 2.2. A germ V of a variety is irreducible if and only if the ideal I(V ) is prime. Proof. First we note that V = ∅ if and only if I(V ) = On . Thus we exclude this case. Suppose I(V ) is not prime. Then there exist germs fi in On , i = 1, 2, such that fi ∈ / I(V ) and f1 f2 ∈ I(V ). We set Vi = V (fi ) ∩ V . Since fi ∈ / I(V ), We have Vi $ V , while V1 ∪ V2 = V ∩ (V (f1 ) ∪ V (f2 )) = V ∩ V (f1 f2 ) = V . Thus V is not irreducible. Conversely, suppose V = V1 ∪ V2 with Vi $ V , i = 1, 2. Then, since I(Vi ) % I(V ), there exists fi which is in I(Vi ) but not in I(V ) for each i. We have f1 f2 ∈ I(V1 ) ∩ I(V2 ) = I(V ). Hence I(V ) is not prime. 

36

Complex Analytic Geometry

Theorem 2.3. Every non-empty germ V of a variety can be expressed as V = V1 ∪ · · · ∪ Vr , where V1 , . . . , Vr are germs of varieties such that each Vi is irreducible and that Vi 6⊂ Vj , if i 6= j. Moreover, V1 , . . . , Vr are uniquely determined by V up to order. Proof. Let V denote the set of germs of non-empty varieties that cannot be expressed as a finite union of irreducible germs of varieties. We show that V = ∅ by contradiction. Suppose V = 6 ∅. Then the set of ideals { I(V ) | V ∈ V } would be non-empty and thus would have a maximal element I(V0 ) by the Noetherian property of On (cf. Theorem 1.12 and Proposition A.19). Then V0 would be a minimal element in V. As V0 is not irreducible, we may write V0 = V1 ∪ V2 with Vi $ V0 , i = 1, 2. Since V0 was minimal in V, both V1 and V2 can be expressed as finite unions of irreducible germs, which is a contradiction. The uniqueness of the expression is not difficult to see.  The above expression of the variety V is referred to as the irreducible decomposition of V . Remark 2.6. Let V be a germ of a variety at 0 in Cn . Take a representative, denoted also by V , of V in a neighborhood U of 0. For z ∈ U , we denote by Vz the germ of V at z. Note that, if z ∈ / V , then Vz = ∅. We remark that even if V is irreducible at 0, Vz may not be irreducible for z ∈ V arbitrarily close to 0. For example, consider the Whitney umbrella (Example 2.3. 3). Now we prove the Nullstellensatz for principal ideals and discuss related topics. We say that an ideal I in On is principal if it is generated by a single germ, i.e., I = (f ) for some f in On . Exercise 2.9. Show that f is irreducible in On if and only if f 6= 0 and (f ) is prime. Theorem 2.4. If p is an irreducible germ in On , then I(V (p)) = (p). Proof. By the preparation theorem, we may assume that p is a Weierstrass polynomial in zn of degree m. Since p is irreducible, p and ∂p ∂zn are relatively prime and we may write (cf. Theorem A.16) ap + b

∂p = c, ∂zn

Complex Manifolds and Analytic Varieties

37

where a and b are in On−1 [zn ] and c is a non-zero germ in On−1 (the discriminant of p up to sign). If p(z 0 , zn ) has a multiple root at zn = a ∂p (z 0 , zn ) = 0 so that c(z 0 ) = 0. Hence p(z 0 , zn ) has m for some z 0 , then ∂z n 0 distinct roots for z with c(z 0 ) 6= 0. Let h be a germ in I(V (p)). Then its representative h(z 0 , zn ) has also m distinct roots in zn for z 0 with c(z 0 ) 6= 0. By the division theorem, we may write r ∈ On−1 [zn ], deg r < m.

h = qp + r, 0

As deg r < m, we have r(z , zn ) = 0 away from the zero set of c. Thus r = 0 as a germ and h is in the ideal (p).  Corollary 2.1 (Nullstellensatz for principal ideals). For a germ f in On , we have: p I(V (f )) = (f ). Proof. The above holds if f is 0 or a unit. Thus we assume that f is mr 1 be the irreducible decomposition. not 0 or a unit andplet f = pm 1 · · · pr On the one hand, (f ) = (p1 · · · pr ) and on the other hand, I(V (f )) = I(V (p1 )) ∩ · · · ∩ I(V (pr )), which is equal to (p1 · · · pr ) by Theorem 2.4.  Proposition 2.4. For a germ f which is not 0 or a unit in On , the following are equivalent: (1) V (f ) is irreducible. (2) There is an irreducible germ p in On such that f = pm for some positive integer m. mr 1 Proof. Let f = pm be the irreducible of f . If 1 · · · pr p p decomposition p r ≥ 2, then, since p1 · · · pr ∈ (f ) but pi ∈ / (f ), (f ) is not prime. Thus by Corollary 2.1 and Theorem 2.2, V (f ) is not irreducible. p We now suppose that f = pm with p irreducible. If f1 f2 ∈ (f ), then k (f1 f2 ) ∈ (f ) for some positve integer k. Thus we may find a germ a in On k such that (f1 f2p ) = apm . This p implies that either f1 or f2 is divisible by  p. Hence f1 ∈ (f ) or f2 ∈ (f ).

The proof of the following is not difficult. Theorem 2.5. Let f be a germ, which is not 0 or a unit, in On . If f = mr 1 pm is the irreducible decomposition of f , then 1 · · · pr V (f ) = V (p1 ) ∪ · · · ∪ V (pr ) is the irreducible decomposition of V (f ).

38

2.4

Complex Analytic Geometry

Nullstellensatz and dimension

In this section we first quote some fundamental results on algebraic structures concerning prime ideals in On and state their geometric consequences on the variety defined by the ideal. We then discuss the Nullstellensatz and dimensions of varieties based on them. For d = 0, . . . , n, we naturally identify the ring Od = C{z1 , . . . , zd } with a subring of On = C{z1 , . . . , zn }. For a prime ideal p in On , we consider the canonical morphism η : Od −→ On /p. We denote by K and L the fraction fields of Od and On /p, respectively. For a germ f in On , its class in On /p is denoted by [f ]. We quote: Theorem 2.6. Let p be a non-zero prime ideal. Then, after a linear change of coordinates of Cn , there is a uniquely determined integer d, 0 ≤ d < n, such that (1) η is injective and On /p is finitely generated over Od as an Od -module, (2) L is generated over K by the single element [zd+1 ] as a field. The statement (1) is proved by a repeated use of the Weierstrass preparation theorem (Theorem 1.10) and (2) follows from the theorem of primitive element in the field theory. Before we state the following theorem, we recall: Definition 2.10. A continuous map f : X → Y of topological spaces is finite if it is proper (cf. Section B.1) and if, for every point y in Y , f −1 (y) is a (possibly empty) finite set. From Theorem 2.6, we see that On /p is integral over Od and that the minimal polynomial P (X) of [zd+1 ] over K is in Od [X] (cf. Proposition A.17). It can be further shown that P (zd+1 ), which is in p, is a Weierstrass polynomial in zd+1 . Let D(P ) denote the discriminant of P (X), which is a non-zero element in Od (cf. (A.12)). We denote by D the germ at 0 of the variety in Cd defined by D(P ). If we set V = V (p), the germ of the variety defined by p, we have the following, which is a consequence of Theorem 2.6: Theorem 2.7. For a suitable decomposition Cn = Cd × Cn−d , there exist a connected neighborhood U 0 of 0 in Cd and a neighborhood U 00 of 0 in Cn−d with the following properties:

Complex Manifolds and Analytic Varieties

39

(1) The germ V has a representative in U = U 0 × U 00 and the map π : V ∩ U −→ U 0 , which is the restriction of the projection Cn → Cd , is a surjective finite map with π −1 (0) = {0}. (2) The germ D has a representative in U 0 and if we set V ∗ = V ∩ U rπ −1 (D), then V ∗ is a connected complex submanifold of dimension d of U . Moreover, the closure of V ∗ in U is V . (3) The restriction of π to V ∗ gives a covering map V ∗ → U 0 rD. With these, we have the following: Theorem 2.8 (Nullstellensatz). For every ideal I in On , we have: √ I(V (I)) = I. Proof. The above holds if I is 0 or On . Thus we assume that I is not 0 or On . By the primary decomposition theorem (Theorem A.19), we see that it suffices to prove the above in the case I = p is a prime ideal. √ We already know that I(V (p)) ⊃ p = p. Thus let f be a germ in I(V (p)) and P (X) = a0 + a1 X + · · · + am−1 X m−1 + X m the minimal polynomial of [f ] over K. Then P (X) is in fact in Od [X] (cf. Proposition A.17) and P (f ) is in p. If m = 1, we set Q(X) = 1 and if m > 1, we set Q(X) = a1 + · · · + am−1 X m−2 + X m−1 . Then P (X) = a0 + Q(X)X and P (f ) = a0 + Q(f )f . Let U = U 0 × U 00 be as in Theorem 2.7. Then, by the surjectivity of π, a0 = 0 so that Q(f )f is in p. If m = 1, then f is in p. If m > 1, by the minimality of P (X), Q(f ) is not in p, thus f is in p.  Corollary 2.2. 1. Let I be an ideal in √ On . If I is prime, then V (I) is irreducible. If V (I) is irreducible, then I is prime. 2. Let I be an ideal in On with I 6= On . If I = q1 ∩ · · · ∩ qr is the primary decomposition of I, then V (I) = V (p1 ) ∪ · · · ∪ V (pr ) is the irreducible decomposition of V (I), where pi =

√

qi .

Proof. The first statement follows from Theorems 2.2 and 2.8 and the second from Proposition 2.3 and Theorem 2.8. 

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Complex Analytic Geometry

Remark 2.7. From the above, we see that there is a one-to-one correspondence between the set of irreducible germs of varieties at 0 in Cn and that of prime ideals in On . Exercise 2.10. Show that, for an ideal I (6= On ) in On , the complex vector space On /I is finite dimensional if and only if V (I) = {0}. Dimension From the item (2) in Theorem 2.7, we have: Theorem 2.9. Let V be an irreducible germ of a variety. Then there is a representative V such that Vreg is connected and dense in V . Dimension at a point: For the germ V of a non-empty variety at 0 in Cn , we define its dimension, denoted by dim V , as follows. If V is irreducible, dim V is the integer d appearing in Theorems 2.6 and 2.7. It is the dimension of the complex manifold Vreg . In general, if V = V1 ∪ · · · ∪ Vr is the irreducible decomposition of V , we set dim V = max1≤i≤r dim Vi . We say that V is pure dimensional if all the components Vi have the same dimension. We also define the codimension by codim V = n − dim V . Note that, if V = V1 ∪· · ·∪Vr is the irreducible decomposition of V , then there is the corresponding decomposition Vreg = C1 ∪ · · · ∪ Cr of Vreg into its connected components Ci . Each Ci is a complex manifold whose closure coincides with Vi . If V is irreducible, i.e., r = 1, then C1 = V1,reg = Vreg . In general, Ci is contained in Vi,reg , however may not coincide with Vi,reg . For example, consider the germ at 0 of the variety in Example 2.3. 1. Remark 2.8. The above definition of dim V coincides with the Krull dimension of On /I(V ) (cf. Section A.2). If V is a germ of a variety at 0, taking a representative V of V , we denote by Vz the germ of V at z near 0, as before. From definition we have the following two propositions: Proposition 2.5. Let V be the germ of a non-empty variety at 0. There is a neighborhood U of 0 and a representative V of V in U such that dim Vz ≤ dim V

for z ∈ V ∩ U.

The equality holds if V is pure dimensional.

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Proposition 2.6. For two germs V and V 0 of non-empty varieties at 0 with V ⊂ V 0 , we have dim V ≤ dim V 0 . Theorem 2.10. Let f be a germ which is not 0 or a unit in On . Then V (f ) is of pure dimension n − 1. Proof. In view of Theorem 2.5, it suffices to show that dim V (p) = n − 1 for every irreducible germ p. From Theorem 2.9, we see that p and V (p) have representatives in a neighborhood U of 0 in Cn such that V (p)reg is a connected complex submanifold of U and is dense in V (p). We show that it is of dimension n − 1. First note that dim V (p)reg ≤ n − 1, since otherwise p would be 0. We may assume that p is a Weierstrass polynomial in zn ∂p ∂p so that p and ∂z are relatively prime, i.e., ∂z ∈ / (p). By Theorem 2.4, n n ∂p { z ∈ V (p) | ∂zn (z) 6= 0 } is non-empty and is an (n − 1)-dimensional complex manifold in V (p)reg . Thus dim V (p)reg = n − 1.  More generally, we quote the following: Theorem 2.11. Let V be an irreducible germ of a variety of dimension d and f a non-unit in On . If f ∈ / I(V ), then V ∩ V (f ) is of pure dimension d − 1. Using this we have: Theorem 2.12. Let V be the germ of a variety of pure dimension n − 1. Then there is a germ f which is not 0 or a unit in On such that I(V ) = (f ). Proof. First we suppose that V is irreducible so that I(V ) is prime (cf. Theorem 2.2). As dim V = n − 1, I(V ) 6= 0. Thus there is a germ f in I(V ) which is not 0 or a unit. Since I(V ) is prime, it contains an irreducible component p of f . We claim that I(V ) = (p). Take an arbitrary germ g in I(V ). Suppose g ∈ / (p) which is equal to I(V (p)) by Theorem 2.4. Then dim V (p) ∩ V (g) = n − 2 by Theorem 2.11, which is a contradiction as it contains V . In general, let V = V1 ∪ · · · ∪ Vr be the irreducible decomposition of V . Then I(Vi ) = (pi ) for some irreducible germ pi , i = 1, . . . , r, so that I(V ) = (p1 · · · pr ).  Remark 2.9. From the proofs of Theorems 2.10 and 2.12, we see that there is a one-to-one correspondence between the set of irreducible germs of varieties of codimension one at 0 in Cn and that of prime principal ideals in On .

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Let f : M → M 0 be a holomorphic map of complex manifolds and V a variety in M . For a point z 0 in M 0 , V ∩ f −1 (z 0 ) is a variety in M . In this situation, we quote the following two theorems: Theorem 2.13 (Upper semi-continuity). For any point a in V , there is a neighborhood U of a in M such that dim(V ∩ f −1 f (z))z ≤ dim(V ∩ f −1 f (a))a

for z ∈ V ∩ U.

Theorem 2.14 (Proper mapping theorem). If f |V is proper, f (V ) is a variety in M 0 and, for z 0 ∈ f (V ), dim f (V )z0 = max{ dim Vz − dim(V ∩ f −1 f (z))z | z ∈ V, f (z) near z 0 }. Global dimension: A variety V in a complex manifold M is said to be globally irreducible, or simply irreducible if there is no fear of confusin, if it is non-empty and cannot be expressed as a union of two varieties V1 and V2 in M with V1 , V2 6= V . This notion should be distinguished from the irreducibility at a point (Definition 2.9). For example the variety V0,0,γ of Exercise 2.6. 2 is globally irreducible, but not irreducible at a. Note that every non-empty variety is expressed as a locally finite union of irreducible varieties. Note also that V is irreducible if and only if the regular part Vreg is connected. Hence, for an irreducible variety V and a point z in V , the dimension of V at z remains constant. We call it the dimension of V . In general, we say that V is pure dimensional, if all the irreducible components of V have the same dimension. A variety V of pure dimension n − 1 in a complex manifold M of dimension n is called a hypersurface in M . By Theorems 2.10 and 2.12, it is equivalent to saying that V is locally defined by a single (not identically zero) holomorphic function with zeros. We finish this section by quoting a generalization of Theorem 1.8 and discussing some of its consequences and related topics. Theorem 2.15. Let V be a connected variety in a complex manifold M and f a holomorphic function on a neighborhood of V . If there is a point z0 in V such that |f (z0 )| ≥ |f (z)| for all z in a neighborhood of z0 in V , then f is constant on V . Corollary 2.3. A compact variety V in a coordinate neighborhood of a complex manifold is a finite set of points.

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Proof. The absolute value of a coordinate function attains its maximum on each connected component of V . Thus the function is constant on each component.  From Corollary 2.3, we have: Proposition 2.7. Let V be a variety in a coordinate neighborhood U of a complex manifold M and f : U → M 0 a holomorphic map into another complex manifold M 0 . If f |V is proper, then f |V is finite. As a related topic, from Theorem 2.14, we have: Proposition 2.8. Let f : M → M 0 be a holomorphic map of complex manifolds and V a variety in M . If f |V is finite, then f (V ) is a variety in M 0 and, for z 0 ∈ f (V ), dim f (V )z0 = max{ dim Vz | z ∈ V, f (z) near z 0 }. In particular, if V is pure d-dimensional, so is f (V ). 2.5

Underlying real structures

C r manifolds Let r denote a non-negative integer or ∞. Recall that a C r structure and a C r manifold of dimension m are defined by replacing, in Definitions 2.1, 2.2 and 2.3, Cn with Rm and “holomorphic maps” with “C r maps”. Also, a C r function on a C r manifold or a C r map between C r manifolds is defined by replacing “holomorphic” with “C r ” in the above. A C 0 manifold is also called a topological manifold and a C r manifold, 1 ≤ r ≤ ∞, a differentiable manifold (of class C r ). If r ≥ 1, we may consider the Jacobian matrix of a C r map f and define regular and singular points of f as in Definition 1.9. We also have the inverse mapping theorem (cf. Theorem 1.4), in fact the proof given there goes back to the real case via (1.3), and the implicit mapping theorem (cf. Theorems 1.5 and 1.6). The notions of C r immersion, embedding and submersion are defined as in Definition 2.4. A C r submanifold is also defined as in Definition 2.6. Thus it is what is called a “regular submanifold”. By restricting to real variables in the definition of analytic functions above, we may consider real analytic functions, which is also referred to as C ω functions. Accordingly, we may define real analytic manifolds, or C ω manifolds for short.

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Identifying Cn with R2n , a complex manifold of dimension n may be naturally thought of as a C r manifold of dimension 2n for every r with 0 ≤ r ≤ ∞ or r = ω. For real manifolds we mostly consider the C ∞ case, since on the one hand it is known that every C r structure, r ≥ 1, contains a unique C ∞ , in fact C ω , structure and on the other hand some tools such as partition of unity are available in the C ∞ case, but not in the C ω case. Real tangent space Let M be a C ∞ manifold of dimension m and {(Uα , ϕα )} a C ∞ atlas representing the C ∞ structure. For a ponit x in Uα , we write ϕα (x) = α α α (xα 1 , . . . , xm ) and call (x1 , . . . , xm ) local coordinates of x relative to ϕα , as in the case of complex manifolds. We recall the following: Definition 2.11. Let x be a point in M . A tangent vector on M at x is an R-linear functional v on the set of C ∞ functions near x satisfying the derivation law: v(f g) = v(f )g(x) + f (x)v(g) for two C ∞ functions f and g near x. We denote by TR,x M the set of tangent vectors at x and call it the (real) tangent space of M at x. It has a natural R-vector space structure. Suppose that (x1 , . . . , xm ) is a coordinate system near x. Then for each
∂ at x, which is defined by i = 1, . . . , m, we have the tangent vector ∂x i x
∂f ∂ ∂ ∂xi x (f ) = ∂xi (x). It will be simply written ∂xi , if there is no fear of ∂ confusion. Then we may take ( ∂x , . . . , ∂x∂m ) as a basis of TR,x M . 1 Differential: Let f : M → M 0 be a C ∞ map of C ∞ manifolds. For each point x in M , we have an R-linear map, called the differential of f at x: f∗ : TR,x M −→ TR,f (x) M 0 . It is defined by f∗ (v)(h) = v(h ◦ f ), for v in TR,x M and a C ∞ function h in a neighborhood of f (x). It will be denoted by f∗,x if it is necessary to make the point explicit. Let m and m0 be dimensions of M and M 0 , respectively. Choose local coordinate systems (x1 . . . , xm ) around x and (y1 , . . . , ym0 ) around f (x) ∂ and set fj = yj ◦ f . Then in terms of the bases ( ∂x , . . . , ∂x∂m ) of TR,x M 1

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and ( ∂y∂ 1 , . . . , ∂y∂ 0 ) of TR,f (x) M 0 , the differential f∗ is represented by the m Jacobian matrix evaluated at x: ∂(f1 , . . . , fm0 ) (x). ∂(x1 , . . . , xm ) Thus f is an immersion if and only if f∗,x is injective for every x in M and f is a submersion if and only if f∗,x is surjective for every x in M (cf. Definition 2.4 with “holomorphic” replaced by “C ∞ ”). f

f0

If M → M 0 → M 00 is a sequence of C ∞ maps, we have 0 (f 0 ◦ f )∗,x = f∗,f (x) ◦ f∗,x .

(2.4)

Remark 2.10. As one of the fundamental facts about C ∞ maps, there is Sard’s theorem: if we set S = { x ∈ M | f∗,x is not surjective } for a C ∞ map f : M → M 0 , then f (S) has measure zero in M 0 . Normal space: Let V be a C ∞ submanifold of M with i : V ,→ M the inclusion. Then for each point x in V , i∗ : TR,x , V → TR,x M is injective so that we may identify TR,x V with a subspace of TR,x M . The quotient space NR,V,x = TR,x M/TR,x V is called the (real) normal space of V in M at x. Exercise 2.11. Let V be as above and x a point of V . Suppose V is of codimension k 0 and is defined by a system of C ∞ functions (f1 , . . . , fk0 ) near x. Let (x1 , . . . , xm ) be a coordinate system in a neighborhood of x in ∂ , . . . , ∂x∂m ) M . We identify TR,x M with Rm = {(ξ1 , . . . , ξm )} by taking ( ∂x 1 m as its basis. Show that, in R , TR,x V is given by m X ∂fi (x) · ξj = 0, ∂x j j=1

i = 1, . . . , k 0 .

Product manifold: Let M and M 0 be C ∞ manifolds of dimensions m and m0 , respectively. If (x1 , . . . , xm ) is a coordinate system near x in M and (x01 , . . . , x0m0 ) a coordinate system near x0 in M 0 , then (x1 , . . . , xm ; x01 , . . . , x0m0 ) is a coordinate system near (x, x0 ) in M × M 0 . From this we see that there is a natural isomorphism TR,(x,x0 ) (M × M 0 ) ' TR,x M ⊕ TR,x0 M 0 .

(2.5)

Holomorphic tangent space Let M be a complex manifold of dimension n. If (z1 , . . . , zn ) is a holomorphic coordinate system on a neighborhood U of a point z in M , then

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√ writing zi = xi + −1 yi with xi and yi the real and the imaginary parts of zi , (x1 , y1 , . . . , xn , yn ) is a C ∞ coordinate system on U . We may think of the vectors √ √ ∂ ∂  ∂  ∂ 1 ∂ 1 ∂ and = − −1 = + −1 ∂zi 2 ∂xi ∂yi ∂ z¯i 2 ∂xi ∂yi c as being in the complexification TR,z M = C ⊗R TR,z M of the real tangent space TR,z M of M at z.

Definition 2.12. The holomorphic tangent space Tz M and the antiholomorphic tangent space T z M of M at z are the subspaces of the complex c vector space TR,z M spanned by ( ∂z∂ 1 , . . . , ∂z∂n ) and ( ∂∂z¯1 , . . . , ∂∂z¯n ), respectively. Obviously we have a decomposition of complex vector spaces: c TR,z M = Tz M ⊕ T z M.

(2.6)

Exercise 2.12. 1. Show that Tz M and T z M do not depend on the choice of the holomorphic coordinates (z1 , . . . , zn ). 2. Show that if f : M → M 0 is a holomorphic map of complex manifolds, its differential f∗ (extended to the complexifications) leaves the holomorphic and antiholomorphic components invariant, i.e., f∗ (Tz M ) ⊂ Tf (z) M 0 and f∗ (T z M ) ⊂ T f (z) M 0 . Let V be a variety in M and i : V ,→ M the inclusion. If z is a regular point of V , we may define Tz V and the differential i∗ : Tz V → Tz M is injective (cf. Exercise 2.3). Thus we may identify Tz V with a subspace of Tz M . We call the quotient space Tz M/Tz V the holomorphic normal space of V in M at z. Remark 2.11. Let V be a complex vector space. Its complex conjugate V is a complex vector space such that the underlying set and the additive structure are the same as those of V but that the scalar multiplication, denoted by ∗, is defined by c ∗ v = c¯v for c ∈ C and v ∈ V. Let R be a subring of R and M an R-module. Also let V be a complex subspace of the complexification Mc = C ⊗R M of M. For an element P P v= ci ⊗ xi , ci ∈ C, xi ∈ M, of V , we set v¯ = c¯i ⊗ xi . Then the set { v¯ | v ∈ V } is a complex subspace of Mc which is canonically isomorphic with V by the correspondence v¯ ↔ v. In particular, in the case M = TR,z M with R = R, we may identify the space T z M with the complex conjugate

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of Tz M via the correspondence n X i=1

n

ci

n

X X ∂ ∂ ∂ ←→ = . ci ∗ c¯i ∂ z¯i ∂zi ∂zi i=1 i=1

Stratifications Stratification is a way to study singular varieties by decomposing them into a union of submanifolds. Definition 2.13. Let S be a closed set in a C ∞ manifold M . A stratification of S is a family {Xα }α∈I of C ∞ submanifolds Xα of M , called strata, satisfying the following conditions: F (1) S = α∈I Xα (disjoint union), (2) each point of S has a neighborhood which intersects with only a finite number of Xα , (3) if Xα ∩ Xβ 6= ∅, then Xα ⊂ Xβ . Let {Xα }α∈I be a stratification of S. Definition 2.14. 1. We say that a pair (Xα , Xβ ) of strata satisfies the Whitney condition at x ∈ Xα ∩ Xβ if the following holds: (*) let {xn } and {yn } be sequences of points in Xα and Xβ , respectively, such that (1) both converge to x, (2) the line joining xn and yn converges to a line L and (3) the tangent space TR,yn Xβ converges to a space T , then L ⊂ T . 2. We say that {Xα }α∈I is a Whitney stratification if every pair (Xα , Xβ ) satisfies the Whitney condition at each point in Xα ∩ Xβ . In (*) above, we consider everything in a local coordinate system around x and think of lines and spaces as being in real Grassmann manifolds (cf. Section 3.6 below). The following is known: Theorem 2.16. Every analytic variety in a complex manifold M admits a Whitney stratification whose strata are complex submanifolds of M . Fundamental properties of Whitney stratifications include the existence of controlled tube systems (cf. Theorem 13.1 below) and the Thom isotopy

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lemma, which states that, if {Xα } is a Whitney stratification of S, then S is “locally topologically trivial” along each stratum Xα . Exercise 2.13. Find a Whitney stratification for each of the varieties V in Example 2.3. Topological dimension Let X be a topological space. An open covering U of X is said to have order m + 1 if there exist m + 1 sets in U whose intersection is non-empty and the intersection of more than m + 1 sets in U is always empty. Definition 2.15. A space X is finite-dimensional if there is an integer m such that every open covering of X admits a refinement having order at most m + 1. The topological dimension of X is the smallest value of such an m. It is known that a topological manifold of dimension m is of topological dimension m. Notes We list [Griffiths and Harris (1978); Kodaira (1986)] as general references on complex manifolds. For analytic varieties, particularly for the details of Section 2.4, we refer to [Gunning and Rossi (1965); Narasimhan (1966); Noguchi (2016)]. As to knots and links that appear in Exercise 2.5, we refer to [Rolfsen (2003)], see also [Milnor (1968)]. See [Hartshorne (1977)] for the corresponding theory in algebraic geometry of the materials in Section 2.3. The Nullstellensatz in algebraic category is referred to as Hilbert Nullstellensatz and that in analytic category as R¨ uckert Nullstellensatz. As to Remark 2.8 and related subjects, we refer to [de Jong and Pfister (2000)]. For fundamental materials on C ∞ manifolds in Section 2.5 and the subsequent chapters, we refer to [Lee (2013); Matsushima (1972)]. In particular, for Remark 2.1. 3 on the topology of manifolds, see Ch. II, §15, Theorem 1 in [Matsushima (1972)]. A C ∞ manifold is often called a smooth manifold. However we avoid the use of the terminology “smooth” here, as it sometimes means “non-singular” in other areas such as algebraic geometry and complex analytic geometry. The fact that every C r structure, r ≥ 1, contains a unique C ∞ structure is proved in [Whitney (1936)]. There it is also proved that every C r structure, r ≥ 1, contains a C ω structure. The uniqueness follows from a result in [Grauert (1958)]. See [Shiga (1976)] for details on this matter.

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For stratifications, we refer to [Gibson, Wirthm¨ uller, du Plessis and Looijenga (1976); Mather (2012); Shiota (1997)]. Theorem 2.16 in the text is due to [Whitney (1965)]. As to the topological dimension, we refer to [Munkres (1975)] and references therein. It is also called the (Lebesgue) covering dimension.

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Chapter 3

Vector Bundles

The tangent space of a manifold “linearly approximates” the manifold at each point. The notion of the tangent bundle naturally comes up as the collection of these spaces. In fact, such objects arise here and there. In this chapter, we review general theory of fiber bundles. After recalling group actions and related topics, we introduce fiber bundles. Among them is a special class of bundles whose fibers are vector spaces, as in the case of tangent bunles. These are called vector bundles and are studied in some detail. We introduce and discuss the structure of the Stiefel manifold W (N, r), the set of r linearly independent vectors in CN . It is used for the obstruction theoretical definition of chracteristic classes of vector bundles in Chapter 5. We then discuss the Grassmann manifold G(N, r), the set of linear subspaces of dimension r in CN . It is a generalization of the projective space and is the orbit space of the natural action of the general linear group GL(r, C) on W (N, r). It admits the tautological vector bundle, of which W (N, r) is the associated principal GL(r, C)-bundle. We study the cellular structure of G(r, N ). The real case is also discussed. We also make precise such notions as orientability, C ∞ maps of subsets, manifolds with boundary, tubular neighborhoods and transversality.

3.1

Group actions

Recall that a topological group is a group G with a topological structure such that the group operations G → G given by g 7→ g −1 and G × G → G given by (g, h) 7→ gh are continuous. Let X be a topological space and G a topological group. We denote by e the unity in G. 51

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Definition 3.1. A continuous left action of G on X is a continuous map α : G × X −→ X such that, denoting α(g, x) by gx, (1) ex = x for all x ∈ X, (2) g(hx) = (gh)x for all g, h ∈ G and x ∈ X. In this case, we say that G acts continuously on X from the left. A left action is also denoted by (G, X). We may also define a right action (X, G) symmetrically to the left one. Let (G, X) be a left action. For a point x in X, the subset Gx = { gx | g ∈ G } of X is called the G-orbit of x. If we introduce a relation ∼ in X by saying that x0 ∼ x if x0 = gx for some g in G, this is an equivalence relation and the equivalence class of x coincides with Gx. The quotient set with the quotient topology is called the orbit space of the action (G, X) and is denoted by G\X. For a right action (X, G), the orbit and the orbit space will be denoted by xG and X/G. A fixed point of an element g of G is a point x such that gx = x. For a point x in X, the set of elements in G that leaves x fixed: Gx = { g ∈ G | gx = x } is a subgroup of G, called the stabilizer of x in G. Note that, if X is Hausdorff, the subgroup Gx is closed for every x. We say that the action is free if Gx = {e} for all x. We also say that the action is effective if T x∈X Gx = {e}. Let (G, X) and (G, X 0 ) be two actions. We say that a map f : X → X 0 is G-equivariant if f (gx) = gf (x) for all (g, x) in G × X. Homogeneous spaces We say that an action (G, X) is transitive if it has only a single orbit. In this case, we say that X is a homogeneous space of G. For example, for a subgroup G0 of G, the right coset space G/G0 = { gG0 | g ∈ G }, which is the orbit space of the natural right action of G0 on G, is a homogeneous space of G with the natural left action of G. We recall:

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Theorem 3.1. Let X be a locally compact Hausdorff space and G a locally compact topological group with countable basis. If X is a homogeneous space of G, then for every x in X, the map G/Gx −→ X

given by gGx 7→ gx

is a G-equivariant homeomorphism. For r with 1 ≤ r ≤ ∞ or r = ω, we may also define C r actions, taking a real Lie group, a C r manifold and a C r map as G, X and α above. Note that a real Lie group is always C ω . Likewise we may consider holomorphic actions, taking a complex Lie group, a complex manifold and a holomorphic map. Here a complex Lie group is a group with a complex manifold structure such that the group operations are holomorphic. As standard real Lie groups, we have the real general linear group GL(p, R), the subgroup GL+ (p, R) of GL(p, R) consisting of matrices with positive determinant, the orthogonal group O(p), the special orthogonal group SO(p) and the unitary group U (q) (cf. Section 9.1 below). The complex general linear group GL(q, C) is a typical complex Lie group. In particular, GL(1, C) is the multiplicative group C∗ of non-zero complex numbers. The following are proved similarly as the corresponding statements for the real case: Theorem 3.2. Let G be a complex Lie group and G0 a closed complex Lie subgroup of G. Then G/G0 admits a complex structure such that the canonical projection G → G/G0 and the natural action G × G/G0 → G/G0 are holomorphic. Theorem 3.3. Let M be a complex manifold and G a complex Lie group acting holomorphically on M . If M is a homogeneous space of G, then for every x in M , Gx is a closed complex Lie subgroup of G and the map G/Gx −→ M

given by

gGx 7→ gx

is a G-equivariant biholomorphic map. Example 3.1. The usual multiplication of matrices on the column vectors defines a left action of GL(p, R) on Rp . This action induces actions of O(p) on the unit ball Bp = { t(x1 , . . . , xp ) ∈ Rp | |x1 |2 + · · · + |xp |2 ≤ 1 } and on its boundary ∂Bp , which is the unit shere Sp−1 (cf. Example 3.9 below). The action (O(p), Sp−1 ) is transitive and the stabilizer of t(1, 0, . . . , 0) may be identified with O(p − 1) so that we have a C ω diffeomorphism

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Sp−1 ' O(p)/O(p − 1). If p > 1, SO(p) also acts transitively on Sp−1 and we have Sp−1 ' SO(p)/SO(p − 1). Likewise GL(q, C) acts on Cq from the left. This action induces actions of U (q) on B2q = { t(z1 , . . . , zq ) ∈ Cq | |z1 |2 + · · · + |zq |2 ≤ 1 } and on its boundary ∂B2q = S2q−1 . The action (U (q), S2q−1 ) is transitive and the stabilizer of t(1, 0, . . . , 0) may be identified with U (q − 1) so that we have a C ω diffeomorphism S2q−1 ' U (q)/U (q − 1). See Section 3.5 below for a more general case.

Properly discontinuous actions We say that an action (G, X) is proper if the map G × X −→ X × X given by (g, x) 7→ (gx, x) is proper, i.e., the inverse image of an arbitrary compact set is compact. A properly discontinuous action is a proper action of a discrete group. If an action (G, X) is properly discontinuous, then the orbit space G\X is Hausdorff. We also have: Lemma 3.1. Let X be a locally compact Hausdorff space and G a discrete group. Then an action (G, X) is properly discontinuous if and only if for every compact set K of X, the set { g ∈ G | gK ∩ K 6= ∅ } is finite. Using the above, the following is proved as in the real case: Theorem 3.4. Suppose G acts on a complex manifold W holomorphically. If the action is properly discontinuously and free, the orbit space M = G\W has the structure of a complex manifold such that the canonical surjection π : W → M is a holomorphic covering map (cf. Definition 2.5). Example 3.2. Complex tori: Let (ω1 , . . . , ω2n ) be a 2n-tuple of elements in Cn linearly independent over R and G = hω1 , . . . , ω2n i the free Abelian group of rank 2n generated by the ωi ’s. Then G acts on Cn by translations and the action is properly discontinuous and free. The quotient T = G\Cn is called a complex torus. It is diffeomorphic to the product S1 × · · · × S1 of 2n copies of a circle S1 . Exercise 3.1. Show that a one-dimensional complex torus is biholomorphic with the one given by G = h1, ωi with = ω > 0 and that it is also biholomorphic with the quotient of C∗ = Cr{0} by the infinite cyclic group hηi generated by ζ 7→ ηζ for some complex number η with 0 < |η| < 1 (in fact, √ η = e2π −1 ω ).

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55

The last viewpoint can be generalized to the higher dimensional case: Example 3.3. Hopf manifolds: Suppose n ≥ 2. Let ηi be complex numbers with 0 < |ηi | < 1, i = 1, . . . , n, and G the infinite cyclic group generated by an element g which acts on W = Cn r{0} by g(z1 , . . . , zn ) = (η1 z1 , . . . , ηn zn ). Then the action is properly discontinuous and free so that M = G\W is a complex manifold of dimension n, called a Hopf manifold. Exercise 3.2. Show that a Hopf manifold is diffeomorphic with S1 × S2n−1 . Thus a Hopf manifold M admits a “C ∞ fiber bundle structure” over Pn−1 with fiber the real torus S1 × S1 (cf. Exercise 2.2 and the subsequent sections). In particular, if η1 = · · · = ηn = η, there is a holomorphic map M → Pn−1 , which gives a “holomorphic fiber bundle structure” with fiber the complex torus hηi\C∗ (cf. Exercise 3.5 below).

3.2

Fiber bundles

Let G be a topological group and F a topological space with a continuous effective left action of G. Also let π : E → X be a continuous map of topological spaces. Definition 3.2. An F -trivialization of π is a pair (U, ψ) of an open set U in X and a homeomorphism ∼

ψ : π −1 (U ) −→ F × U with p ◦ ψ = π, where p : F × U → U is the projection. In this case we also say that ψ is a trivialization of E on U . Definition 3.3. A system of F -trivializations with group G of π is a collection T = {(Uα , ψα )}α∈I of F -trivializations with the following properties: (1) {Uα }α∈I is a covering of X, (2) for each pair (α, β), there exists a continuous map g αβ : Uα ∩ Uβ → G such that ψα ◦ ψβ −1 : F × (Uα ∩ Uβ ) → F × (Uα ∩ Uβ ) is of the form ψα ◦ ψβ −1 (ξ β , x) = (g αβ (x)ξ β , x),

(ξ β , x) ∈ F × (Uα ∩ Uβ ),

where we denote elements of F × Uα by (ξ α , x).

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Note that the maps g αβ are uniquely determined by T as we assumed the action of G on F to be effective. For two systems T and T 0 as above, we write T ∼ T 0 if their union is a system of trivializations. Clearly it is an equivalence relation in the set of systems of F -trivializations. Definition 3.4. A fiber bundle structure with fiber F and group G on the map π : E → X is an equivalence class of systems of F -trivializations of π with group G. A fiber bundle is a map π : E → X together with a fiber bundle structure. A fiber bundle will simply be denoted by π : E → X. We also say that E is a fiber bundle on X. We call E the total space, X the base space and, for x ∈ X, Ex = π −1 (x) the fiber at x, which is homeomorphic with F . Sometimes G is referred to as the structure group and F the typical fiber. If the bundle is defined by a system T = {(Uα , ψα )} as in Definition 3.3, we call the collection {g αβ } the system of transition maps associated with T . In this case we simply say that {g αβ } is a system of transition maps of E. From the effectiveness of the action of G, we have g αβ (x)g βγ (x) = g αγ (x),

x ∈ Uα ∩ Uβ ∩ Uγ .

(3.1)

In particular, g αα (x) = e, the unity in G, and g βα (x) = (g αβ (x))−1 . Conversely, if we are given an open covering {Uα } of X and a collection {g αβ } of continuous maps g αβ : Uα ∩ Uβ → G satisfying (3.1), we may construct a fiber bundle as follows. We consider the disjoint union X = F α α α (F × Uα ) and define a relation ∼ as follows: for (ξ , x ) in F × Uα and β β α α β β (ξ , x ) in F × Uβ , (ξ , x ) ∼ (ξ , x ) if xα = xβ (= x)

and ξ α = g αβ (x)ξ β .

Then it is easy to see that this is an equivalence relation in X. Let E = X/ ∼ be the quotient space with the quotient topology and denote the class of (ξ α , xα ) by [ξ α , xα ]. The map π : E → X given by [ξ α , xα ] 7→ xα is well-defined and continuous. Since (F × Uα )/ ∼ = F × Uα , the map ψα : π −1 (Uα ) → F × Uα given by [ξ α , xα ] 7→ (ξ α , xα ) is an F trivialization on Uα and {(Uα , ψα )} defines a fiber bundle structure on E with {g αβ } as a system of transition maps. We say that a fiber bundle π : E → X is trivial, if the fiber bundle structure defining it contains the “global F -trivialization” (X, ψ), i.e.,

Vector Bundles

57

∼

a homeomorphism ψ : E → F × X commuting with the projections. We refer to F × X as a product bundle. Let π : E → X and π 0 : E 0 → X be two fiber bundles with fiber F and group G and let T = {(Uα , ψα )}α∈I and T 0 = {(Vλ , χλ )}λ∈J be systems of trivializations representing the bundle structures of E and E 0 , respectively. Definition 3.5. An isomorphism ϕ : E → E 0 is a continuous map with: (1) π = π 0 ◦ ϕ, (2) for each pair (α, λ), there exists a continuous map g λα : Uα ∩ Vλ → G such that χλ ◦ ϕ ◦ ψα −1 : F × (Uα ∩ Vλ ) → F × (Uα ∩ Vλ ) is of the form χλ ◦ ϕ ◦ ψα −1 (ξ α , x) = (g λα (x)ξ α , x),

(ξ α , x) ∈ F × (Uα ∩ Vλ ).

Note that the above definition does not depend on the choices of trivializations T and T 0 . Also, an isomorphism is bijective and ϕ−1 is an isomorphism. The composition of two isomorphisms is an isomorphism. If the total spaces E and E 0 are the same, the identity map 1E is an isomorphism if and only if the bundle structures are the same. We say that two bundles E and E 0 are isomorphic and write E ' E 0 if there exists an isomorphism between them. Exercise 3.3. Let E and E 0 be fiber bundles on X as above. 1. Suppose E and E 0 are defined by systems of trivializations T and T 0 αβ on the same coveing U = {Uα }. Let {g αβ } and {g 0 } be the systems of transition maps associated with T and T 0 , respectively. Show that E ' E 0 if and only if, for each α, there exists a continuous map g α : Uα → G such that g αβ (x) = g α (x)−1 g 0

αβ

(x)g β (x).

In particular, the bundle E defined by a system {g αβ } is trivial if and only if, for each α, there exists a continuous map g α : Uα → G such that g αβ (x) = g α (x)−1 g β (x). 2. Let V = {Vλ }λ∈J be a refinement of U, i.e., there is a map ι : J → I such that Vλ ⊂ Uι(λ) for all λ (cf. Remark 2.1. 1). Show that {g ι(λ)ι(µ) } and {g αβ } define isomorphic bundle structures. λµ

3. Let {g αβ } and {g 0 } be two systems on U and V, respectively. Show that they define isomorphic structures if and only if there exists a common refinement W of U and V such that the restrictions are as in 1.

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Definition 3.6. A section of a fiber bundle π : E → X on a subset A of X is a continuous map s : A → E such that π ◦ s = 1A . A section s on A can be described as follows. Let {(Uα , ψα )} be a system of trivializations defining the bundle. Then we may write ψα (s(x)) = (sα (x), x) for x ∈ A ∩ Uα α with s : A ∩ Uα → F a continuous map. For each point x in A ∩ Uα ∩ Uβ , we have sα (x) = g αβ (x)sβ (x). (3.2) α Conversely suppose we have a collection {s } of continuous maps satisfying (3.2). Then setting s(x) = ψα−1 (sα (x), x) for x in A ∩ Uα , we have a section s on A. Note that a “global section” s : X → E defines a homeomorphism of X onto its image s(X), the inverse being π|s(X) . Definition 3.7. A principal G-bundle is a fiber bundle with group G and fiber F = G with the natural left action of G. Exercise 3.4. Show that a principal bundle is trivial if and only if it admits a global section. Let G0 be a subgroup of a topological group G. Then it is a topological group with the topology as a subspace of G. Let π : E → X be a fiber bundle with group G. Definition 3.8. We say that the structure group of the bundle may be reduced to G0 if the bundle structure contains a trivialization such that, in the associated system of transition maps {g αβ }, each g αβ maps Uα ∩ Uβ into G0 . If E → X is a fiber bundle with fiber F and group G and if F 0 is a space with a left G action, using a system {g αβ } for E, we may construct a new bundle E 0 → X with fiber F 0 , as in the way described above. We call such a bundle a bundle associated with E. Remark 3.1. For r with 1 ≤ r ≤ ∞ or r = ω, we have the notion of a C r or holomorphic fiber bundle by assuming that G be a real or complex Lie group, F and X C r or complex manifolds, the action of G on F C r or holomorphic and the transition maps C r or holomorphic. In this case E admits naturally the structure of a C r or complex manifold so that π is a C r or holomorphic submersion. We may also talk about C r or holomorphic sections.

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59

Example 3.4. Let $ : Cn+1r{0} → Pn be the canonical surjection (cf. Example 2.1. 2). We show that it has the structure of a holomorphic principal C∗ -bundle. We denote by ζ = (ζ0 , . . . , ζn ) coordinates on Cn+1 r {0} so that $(ζ) = [ζ] = [ζ0 , . . . , ζn ] are homogeneous coordinates on Pn . For each α = 0, . . . , n, let Uα be the open set in Pn given by ζα 6= 0. Then we have a trivialization ∼

ψα : $−1 (Uα ) −→ C∗ × Uα ,

ζ 7→ (ζα , [ζ]).

ψβ−1 (ζβ , [ζ])

We have ψα ◦ = (ζα /ζβ · ζβ , [ζ]). Thus Cn+1 r {0} is a prin∗ cipal C -bundle with a system of transition functions {g αβ } given by g αβ = ζα /ζβ . The associated line bundle is the tautological bundle on Pn (cf. Section 9.3 below). Exercise 3.5. Show that a Hopf manifold M = hηi\(Cn r {0}) (cf. Example 3.3) admits the structure of a holomorphic fiber bundle on Pn−1 with fiber a one-dimensional complex torus hηi\C∗ . The following is proved similarly as the corresponding statement in the real case: Theorem 3.5. Let G be a complex Lie group and G0 a closed complex Lie subgroup of G. Then π : G → G/G0 admits a local holomorphic section so that it becomes a holomorphic G0 -principal bundle. Homotopy groups Here we briefly review homotopy groups of topological spaces and state some fundamental theorems. Let X be a topological space and x0 a point in X. We denote by π0 (X, x0 ) the set of path components of X with the component of x0 as distinguished element. For a positive integer i, we consider the i-cube I i , I = [0, 1], and denote by πi (X, x0 ) the set of homotopy classes of maps (I i , ∂I i ) → (X, x0 ) relative to ∂I i , or equivalently, the set of homotopy classes of maps (Si , s0 ) → (X, x0 ) relative to a point s0 in Si , a sphere of dimension i. For two maps f, g : (I i , ∂I i ) → (X, x0 ) we define a map f ∗ g : (I i , ∂I i ) → (X, x0 ) by ( f (2t1 , t2 , . . . , ti ) 0 ≤ t1 ≤ 21 , f ∗ g(t1 , . . . , ti ) = 1 g(2t1 − 1, t2 , . . . , ti ) 2 ≤ t1 ≤ 1. Then it induces a group structure on πi (X, x0 ) with the class of the constant map as the identity. We call it the i-th homotopy group of X with

60

Complex Analytic Geometry

base point x0 . It is Abelian if i ≥ 2. Thus we use the additive notation for πi (X, x0 ), i ≥ 2, while we use the multiplicative notation for π1 (X, x0 ), unless it is Abelian. By convention, π0 (X, x0 ) = 0 means that X is path connected, in which case πi (X, x0 ) is determined up to isomorphisms, independent of x0 , and πi (X, x0 ) will simply be denoted by πi (X). If ϕ : (X, x0 ) → (Y, y0 ) is a map of topological spaces, it induces a morphism ϕ∗ : πi (X, x0 ) → πi (Y, y0 ), which assigns to the class of a map f : (I i , ∂I i ) → (X, x0 ) the class of ϕ ◦ f . Example 3.5. For a p-sphere Sp , we have ( 0 i = 0, . . . , p − 1, p πi (S ) = Z i = p. Note that πp (Sp ) has a canonical generator which can be thought of either as the homotopy class of the map (I p , ∂I p ) → (Sp , s0 ) collapsing ∂I p to a point s0 or as the homotopy class of the identity map of Sp . Also, as Cq r{0} deformation retracts to S2q−1 , ( 0 i = 0, . . . , 2q − 2, q πi (C r{0}) = Z i = 2q − 1. The group π2q−1 (Cq r {0}) has a canonical generator, i.e., the homotopy class of the inclusion S2q−1 ,→ Cq r{0}. Homotopy exact sequence: Let π : E → X be a fiber bundle with fiber F . Choose a point x0 in X and identify F with π −1 (x0 ) so that we have an inclusion ι : F ,→ E. Then choose a point e0 in F and consider the homotopy groups of F , E and X with base points e0 , e0 and x0 , respectively. Here we quote the fact that a fiber bundle has the homotopy lifting property for cubes, i.e., every homotopy of maps f : (I i , ∂I i ) → (X, x0 ) can be “lifted” to a homotopy of maps f˜ : (I i , ∂I i , 0) → (E, F, e0 ), 0 = (0, . . . , 0). From this we see that there is a morphism ∂∗ : πi (X, x0 ) → πi−1 (F, e0 ). If i ≥ 2, it assigns to the class of f the class of f˜|∂I i : (∂I i , 0) → (F, e0 ), noting that ∂I i is homeomorphic to Si−1 . In the case i = 1, ∂I = {0, 1} and ∂∗ assigns to the class of f the path component of f˜(1). With these we have the following, where base points are suppressed: Theorem 3.6. For a fiber bundle π : E → X with fiber F , we have an exact sequence of homotopy groups: ι

π

∂

∗ ∗ ∗ · · · −→ πi (F ) −→ πi (E) −→ πi (X) −→ πi−1 (F ) −→ · · · −→ π0 (X).

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61

In the above, the exactness of a sequence of maps involving π0 means that the image equals the inverse image of the distinguished element. Hurewicz theorem: Since (I i , ∂I i ) is homeomorphic with (Bi , Si−1 ), we have Hi (I i , ∂I i ; Z) ' Z for i ≥ 1, and it has a canonical generator µi (cf. Example B.5. 2 and Remark B.13. 2). We have a natural morphism hi : πi (X) −→ Hi (X; Z), which assigns to the homotopy class of f : (I i , ∂I i ) → (X, x0 ), the homology class f∗ µi in Hi (X, x0 ; Z) = Hi (X; Z). Now let k ≥ 0. We say that a space X is k-connected if πi (X) = 0 for i = 0, . . . , k. Theorem 3.7. Suppose X is k-connected. 1. If k = 0, then h1 is an epimorphism with the commutator [π1 (X), π1 (X)] as its kernel. 2. If k ≥ 1, then hk+1 is an isomorphism. 3.3

Vector bundles

In the following, we denote by K either R or C and express the elements of Kl by column vectors, unless otherwise stated. Thus we may represent the group of linear transformations of Kl by GL(l, K) acting from the left. Recall that it has the structure of a real or complex Lie group according as K is R or C. Definition 3.9. A vector bundle is a fiber bundle π : E → X with fiber Kl and group GL(l, K) (cf. Definition 3.4). We say that it is real or complex according as K is R or C. Thus a vector bundle structure is represented by a system of trivializations T = {(Uα , ψα )} as follows: (1) for each α, ψα is a homeomorphism ∼

ψα : π −1 (Uα ) −→ Kl × Uα

with p ◦ ψα = π,

l

where p : K × Uα → Uα is the projection, (2) for each pair (α, β), there is a continuous map g αβ : Uα ∩ Uβ −→ GL(l, K) ψα ◦

ψβ−1 (ξ β , x)

= (g

αβ

β

(x)ξ , x)

with

for (ξ β , x) ∈ Kl × (Uα ∩ Uβ ).

62

Complex Analytic Geometry

Note that each fiber Ex = π −1 (x), x ∈ X, has the structure of a vector space of dimension l over K so that every trivialization restricted to Ex is a K-linear isomorphism. We call l the rank of E. A vector bundle of rank one is called a line bundle. We call {g αβ } the system of transition matrices associated with T . For ξ ∈ Ex , we may write ψα (ξ) = (ξ α , x) with ξ α ∈ Kl . We call ξ α the fiber coordinate of ξ relative to ψα . Thus if x ∈ Uα ∩ Uβ , ξ α = g αβ (x)ξ β . Morphisms:

Let E and F be vector bundles on X.

Definition 3.10. A morphism ϕ : E → F is a continuous map commuting with the projections such that the induced map ϕx : Ex → Fx on each fiber is K-linear. Let l and k be the ranks of E and F . Also let {g αβ } and {hαβ } be systems of transition matrices for E and F , respectively. Then a morphism ϕ : E → F is represented by a collection {ϕα } of maps ϕα : Kl × Uα −→ Kk × Uα of the form ϕα (ξ α , x) = (hα (x)ξ α , x), where hα is a continuous map from Uα into Hom(Kl , Kk ), the space of linear maps of Kl into Kk , satisfying hα (x)g αβ (x) = hαβ (x)hβ (x). In the above each hα is represented by a k × l matrix valued function. We say that the morphism ϕ is an isomorphism if it is furthermore a homeomorphism. In this case ϕ induces a K-isomorphism on each fiber. We also say that E and F are isomorphic (or E is isomorphic with F ), and write E ' F , if there is an isomorphism of E onto F . Remark 3.2. This definition of isomorphism is equivalent to the one given in Definition 3.5 in the case of vector bundles. From the above local description of ϕ, we have the following: Proposition 3.1. Let E and F be two vector bundles on X of rank l and systems of transition matrices {g αβ } and {hαβ }, respectively, on an open covering {Uα }. Then E and F are isomorphic if and only if there exists a continuous map g α : Uα → GL(l, K) for each α, such that −1 αβ

g αβ (x) = g α (x)

h

(x)g β (x)

for x ∈ Uα ∩ Uβ .

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63

In the case the total spaces of E and F are the same, the vector bundle structures defined by two systems {g αβ } and {hαβ } are isomorphic if and only if the corresponding systems of trivializations define the same bundle structure (see Exercise 3.3). Thus a vector bundle is trivial if and only if it is isomorphic to the product Kl × X. We say that a sequence of morphisms of vector bundles ϕ

ψ

E −→ F −→ G ϕx ψx is exact if, for each x in X, the induced sequence Ex → Fx → Gx is exact, i.e., Ker ψx = Im ϕx . Sub and quotient bundles: Let π : E → X be a vector bundle of rank l. 0 For l0 ≤ l, we think of Kl as a subspace of Kl by identifying t(a1 , . . . , al0 ) with t(0, . . . , 0, a1 , . . . , al0 ). Definition 3.11. A subbundle of E is a subset E 0 of E with the following property: there exists a system {(Uα , ψα )} of trivialization for E such that 0 −1 each ψα maps π 0 (Uα ) onto Kl × Uα , π 0 = π|E 0 . In this case, each transition matrix g αβ is of the form αβ

g αβ

αβ

=

g 00 ∗

0 g0

!

αβ

,

(3.3)

αβ

where g 0 : Uα ∩ Uβ → GL(l0 , K) and g 00 : Uα ∩ Uβ → GL(l − l0 , K). αβ 0 0 The map π : E → X has a vector bundle structure of rank l0 with {g 0 } a system of transition matrices. 0 αβ The bundle with fiber Kl−l and system {g 00 } is called the quotient 0 0 bundle of E by E and is denoted by E/E . Note that there is a surjective morphism ϕ : E → E/E 0 so that the sequence ϕ ι 0 −→ E 0 −→ E −→ E/E 0 −→ 0 is exact, where ι denotes the inclusion. In general, if we may choose a system {g αβ } of transition matrices of a vector bundle E with fiber Kl so that each g αβ is of the form (3.3), then E αβ admits a subbundle with {g 0 } as a system of transition matrices. Exercise 3.6. Let ϕ : E → F be a morphism of vector bundles. Show that, if the rank of the restriction ϕx of ϕ to each fiber Ex , x ∈ X, is constant, F F then the kernel Ker ϕ = x∈M Ker ϕx and the image Im ϕ = x∈M Im ϕx of ϕ are subbundles of E and F , respectively. Show also that the quotient bundle E/Ker ϕ is isomorphic with Im ϕ. The quotient F/Im ϕ is called the cokernel of ϕ and is denoted by Coker ϕ.

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Complex Analytic Geometry

Pull-back: Let f : Y → X be a continuous map of topological spaces and π : E → X a vector bundle of rank l. We define the pull-back f ∗ E of E by f by f ∗ E = { (ξ, y) ∈ E × Y | π(ξ) = f (y) }.

(3.4) ˜ Denoting by f and $ the restrictions of the projections onto the first and second factors, we have the commutative diagram: f ∗E $

f˜

/E π

  f / X. Y We see that $ : f ∗ E → X has a vector bundle structure of rank l. Indeed, if {(Uα , ψα )} is a system of trivializations of E, we write ψα (ξ) = (ξ α (ξ), x) and define χα : $−1 (f −1 (Uα )) → Kl × f −1 (Uα ) by χα (ξ, y) = (ξ α (ξ), y). Then {(f −1 (Uα ), χα )} is a system of trivializations of $. Note that (f ∗ E)y = Ef (y) . From the definition we have the following: Proposition 3.2. 1. If 1X is the identity map of X and if E is a vector bundle on X, there is a canonical isomorphism 1∗X E ' E g

given by (ξ, x) ↔ ξ.

f

2. If Z → Y → X is a sequence of continuous maps and if E is a vector bundle on X, there is a canonical isomorphism (f ◦ g)∗ E ' g ∗ (f ∗ E)

given by (ξ, z) ↔ (ξ, y, z), y = g(z).

If A is a subspace of X with the inclusion i : A ,→ X and if π : E → X is a vector bundle on X, i∗ E is called the restriction of E to A and is denoted by E|A . Note that its total space coincides with π −1 A. This notion of pull-back can be defined for other fiber bundles as well. Sections and frames: We recall Definition 3.6 for the notion of a section. A vector bundle π : E → X always admits the zero section, i.e., the map s0 : X → E that assigns to each point x in X the zero of the vector space Ex . The zero section s0 is a homeomorphism of X onto its image Σ, with the restriction of π to Σ its inverse. Let l be the rank of E. Definition 3.12. A frame of E on an open set U in X is an ordered family e(l) = (e1 , . . . , el ) of l sections ei of E on U linearly independent at each point of U .

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65

Let (v1 , . . . , vl ) be a basis of Kl . If ψ : π −1 (U ) → Kl × U is a trivialization of E, it defines a frame e(l) = (e1 , . . . , el ) on U by ei (x) = ψ −1 (vi , x). Conversely a frame e(l) = (e1 , . . . , el ) defines a trivialization ψ by l l X  X ψ(ξ) = ai vi , x for ξ = ai ei (x). i=1

i=1

Suppose (v1 , . . . , vl ) is the standard basis of K l . If {(Uα , ψα )} is a system of trivializations and if {g αβ } is the associated system of transition (l) matrices, for the corresponding system of frames {eα }, we may write: (l)

αβ eβ = e(l) . α g

(3.5)

Let f : Y → X be a continuous map and π : E → X a vector bundle. If s is a section of E on an open set U in X, we have the pull-back f ∗ s, which is a section of f ∗ E (cf. (3.4)) on f −1 (U ) defined by (f ∗ s)(y) = (s(f (y)), y)

for y ∈ f −1 (U ).

(3.6)

Vector bundles with other structures: A vector bundle E is C r or holomorphic if it is so as a fiber bundle (see Remark 3.1), in the latter K being assumed to be C. We may also talk about C r or holomorphic sections. The set of C r sections of E on U is denoted by C r (U ; E). This has a natural structure of vector space by the operations defined by (s1 +s2 )(x) = s1 (x) + s2 (x) and (cs)(x) = cs(x) for s1 , s2 and s in C r (U ; E), c in K and x in U . The set of holomorphic sections of E on U is denoted by Γ (U ; E). This has the structure of a complex vector space. Furthermore, if M is a quasi-projective manifold (cf. Section 2.2), we may consider a regular vector bundle on MZ . An algebraic vector bundle on M is the pull-back of a regular vector bundle by the identity M → MZ . In particular, it is a holomorphic vector bundle. Remark 3.3. The following is known: (V1 ) Let E be a vector bundle on X and f, g : Y → X two continuous maps with Y paracompact. If f and g are homotopic, then f ∗ E ' g ∗ E. (V2 ) The “topological classification” and the “C ∞ classification” of the vector bundles on a C ∞ manifold M are the same. Namely, if E is a continuous vector bundle on M , there exists a C ∞ vector bundle that is isomorphic with E as a continuous bundle and two C ∞ vector bundles are isomorphic as continuous bundles if and only if they are as C ∞ bundles.

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Complex Analytic Geometry

Algebraic operations If we are given some vector bundles, we may construct new ones by algebraic operations. Thus let E and F be vector bundles on X with fibers Kl and Kk , respectively. We choose a suitable covering {Uα } of X and let {g αβ } and {hαβ } be systems of transition matrices for E and F , respectively. F Direct sum E ⊕ F : We set E ⊕ F = x∈X Ex ⊕ Fx with the obviously defined map onto X. It is not difficult to see that it has the structure of a vector bundle with Kl ⊕ Kk ' Kl+k as fiber and {g αβ ⊕ hαβ } as a system of transition maps. By the natural isomorphism Kl ⊕ Kk ' Kl+k , g αβ ⊕ hαβ corrresponds to the matrix  αβ  g 0 . 0 hαβ The direct sum is sometimes called the Whitney sum. In general, let ι

ϕ

0 −→ E 0 −→ E −→ E 00 −→ 0

(3.7)

be an exact sequence of vector bundle morphisms. We say that the sequence splits if there exists a morphism η : E 00 → E such that ϕ ◦ η = 1E 00 , or equivalently, ζ : E → E 0 such that ζ ◦ ι = 1E 0 . A morphism η or ζ as above is called a splitting of (3.7). If this is the case, we have an isomorphism E ' E 0 ⊕ E 00 , the correspondence being given by assigning to each ξ in E the element (ζ(ξ), ϕ(ξ)) in E 0 ⊕ E 00 . Proposition 3.3. If (3.7) is an exact sequence of vector bundles on a paracompact space X, there exists a splitting. Proof. Let U = {Uα } be a locally finite covering of X such that the bundles are trivial on each Uα . Choosing suitable frames, we see that there exists a splitting ηα on each Uα . Let {ρα } be a partition of unity P subordinate to U. Then η = ρα ηα gives a splitting of (3.7).  Remark 3.4. If (3.7) is an exact sequence of C ∞ vector bundles on a C ∞ manifold, there exists a C ∞ splitting. Product E × F : The product E × F of the total spaces has a natural vector bundle structure on X × X. Indeed, we may write E × F = F (x1 ,x2 )∈X×X (Ex1 × Fx2 ). As a vector space, Ex1 × Fx2 = Ex1 ⊕ Fx2 and the local trivial structures on E and F define that on E × F .

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67

If we let d : X ,→ X × X denote the diagonal embedding x 7→ (x, x), then there is a canonical isomorphism d∗ (E × F ) ' E ⊕ F.

(3.8)

We may as well define the product of vector bundles on distinct spaces. Namely, if E and F are vector bundles on X and Y , respectively, E × F is F the vector bundle on X × Y given by E × F = (x,y)∈X×Y (Ex × Fy ). F Morphism Hom(E, F ): Set Hom(E, F ) = x∈X Hom(Ex , Fx ) with the obviously defined map onto X. Then it has the structure of a vector bundle with Hom(Kl , Kk ) ' Kkl as fiber and {Hom(g αβ , hαβ )} as a system of transition maps. Exercise 3.7. Show that, if we represent elements in Hom(Kl , Kk ) by k × l matrices, the bundle Hom(E, F ) is defined by the trivialization {(Uα , ψα )} such that, if (Aα , x) = ψα ◦ ψβ−1 (Aβ , x), Aα is given by Aα = Hom(g αβ , hαβ )(x)Aβ = hαβ (x) · Aβ · (g αβ (x))−1 . In particular if F = K × X, we denote Hom(E, F ) by E ∗ and call it the dual bundle of E. This is the vector bundle with fiber (Kl )∗ ' Kl . If we express the elements in (Kl )∗ by row vectors, then a system of transition −1 matrices is given by {(g αβ ) } with multiplication from the right and if −1 we express them by column vectors, then it is given by {t(g αβ ) } with multiplication from the left. There is a canonical isomorphism (E ∗ )∗ ' E. F Tensor product E ⊗ F : We set E ⊗ F = x∈X Ex ⊗ Fx . Then it has the structure of a vector bundle with Kl ⊗ Kk ' Klk as fiber and {g αβ ⊗ hαβ } as a system of transition maps. The natural isomorphisms Hom(Ex , Fx ) ' Ex∗ ⊗ Fx on the fibers induces a natural isomorphism Hom(E, F ) ' E ∗ ⊗ F. Vr Vr F Vr Exterior power E: We set E = x∈X Ex . Then it has the l Vr l Vr ( ) K ' K r as fiber and { g αβ } as a structure of a vector bundle with Vl system of transition maps. In particular, we denote E by det E and call it the determinant (bundle) of E. It is a vector bundle of rank one defined by the system {det g αβ }. In general, we have a natural isomorphism M Vp Vr Vq (E ⊕ F ) ' E⊗ F. p+q=r

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F Complex conjugate E: We assume that K = C and set E = x∈X E x , where E x denotes the complex conjugate of Ex (cf. Remark 2.11). Then it has the structure of a complex vector bundle with Cl as fiber and {g αβ } as a system of transition matrices. F Complexification E c : We assume that K = R and set E c = x∈X Exc , Exc = C ⊗R Ex . Then it has the structure of a complex vector bundle with Cl as fiber and {g αβ } as a system of transition matrices. Complex bundle as real bundle: The complex vector space Cl is naturally considered as a real vector space of dimension 2l and this defines a natural morphism ρ : GL(l, C) −→ GL(2l, R).

(3.9)

Thus if E is a complex vector bundle of rank l, it has the structure of a real vector bundle of rank 2l. If {g αβ } is a system of transition matrices for E, then {ρ ◦ g αβ } is a system of transition matrices of E as a real bundle. 3.4

Tangent bundle and vector fields

Real tangent bundle Let M be a C ∞ manifold of dimension m. We may give a natural vecF tor bundle structure on the (disjoint) union TR M = x∈M TR,x M of the tangent spaces of M as follows. First, define π : TR M → M by assigning to each tangent vector its base point. Then let {Uα } be a covering of M α by coordinate neighborhoods Uα with coordinates (xα 1 , . . . , xm ). Taking ∂ ∂ ( ∂xα , . . . , ∂xα ) as a basis of TR,x M for each x in Uα , we have a bijection m 1 ψα : π −1 (Uα ) → Rm × Uα . Since we have the relation ∂ ∂xβj

=

m X ∂xα i β ∂x j i=1

(x)

∂ , ∂xα i

j = 1, . . . , m,

(3.10)

for x in Uα ∩ Uβ , we see that ψα ◦ ψβ−1 (ξ, x) = (tαβ (x)ξ, x) for (ξ, x) in Rm × (Uα ∩ Uβ ), where  α  ∂xα ∂x1 1 . . .  ∂xβ ∂xβm   1  α α ∂(x , . . . , x )  . ..  . m 1 .. . = tαβ =   . . .  ∂(xβ1 , . . . , xβm )  α   ∂xα ∂x m m ... ∂xβ1 ∂xβm

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69

Therefore TR M admits the structure of a real vector bundle of rank m with {tαβ } as a system of transition matrices. We call it the (real) tangent bundle of M . The bundle dual to the tangent bundle TR M is called the (real) cotangent bundle and is denoted by TR∗ M . Exercise 3.8. Let M and M 0 be C ∞ manifolds. Consider the product manifold M × M 0 and let p and p0 denote the projections onto the first and the second factors. Show that there is a natural isomorphism TR (M × M 0 ) ' p∗ TR M ⊕ (p0 )∗ TR M 0 , which gives the isomorphism (2.5) on each fiber. Vector fields: A vector field v on an open set U in M is a section of TR M on U . Thus it is expressed as, on each U ∩ Uα , m X ∂ v= fiα (x) α , ∂xi i=1 where the fiα ’s are functions on Uα ∩U . In U ∩Uα ∩Uβ , we have f α = tαβ f β , α ). The vector field v is C ∞ if each fiα is. f α = t(f1α , . . . , fm ∞ For a C vector field v and a C ∞ function f on U , v(f ) is a C ∞ function on U . The bracket operation assigns to a pair (u, v) of C ∞ vector fields on U a C ∞ vector field [u, v] on U satisfying [u, v](f ) = u(v(f )) − v(u(f )). Note that [u, v] is C-bilinear and alternating in (u, v) and that [f u, gv] = f g[u, v] + f u(g)v − gv(f )u for C ∞ functions f and g. We also have the Jacobi identity [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0 for every triple (u, v, w) of vector fields. Real normal bundle: If V is a submanifold of codimension k 0 of M , we may cover V with coordinate neighborhoods Uα on M with coordinates α (xα 1 , . . . , xm ) such that α V ∩ Uα = { x ∈ Uα | xα 1 = · · · = xk0 = 0 }.

Then the restriction tαβ |V of tαβ to V ∩ Uα ∩ Uβ is of the form ! αβ t00 0 αβ t |V = αβ , ∗ t0

(3.11)

70

where t0

Complex Analytic Geometry αβ

αβ

and t00 denote the Jacobian matrices α α ∂(xα ∂(xα k0 +1 , . . . , xm ) 1 , . . . , xk 0 ) and , ∂(xβk0 +1 , . . . , xβm ) ∂(xβ1 , . . . , xβk0 )

α respectively, both restricted to V . Since the restriction of (xα k0 +1 , . . . , xm ) to V form a coordinate system on V ∩ Uα , we see that TR V is a subbundle of TR M |V . We call the quotient bundle the (real) normal bundle of V in M and denote it by NR,V . It is defined by the system of transition matrices αβ {t00 } and we have an exact sequence $

0 −→ TR V −→ TR M |V −→ NR,V −→ 0.

(3.12)

Exercise 3.9. Let f : M → M 0 be a submersion of C ∞ manifolds (cf. Definition 2.4 with “holomorphic” replaced by “C ∞ ”). Show that TR M admits as a subbundle the bundle TR f of vectors that are “tangent to the fibers of f ” so that there is an exact sequence of vector bundles on M : 0 −→ TR f −→ TR M −→ f ∗ TR M 0 −→ 0. Exercise 3.10. Let π : E → M be a C ∞ vector bundle and Σ the image of the zero section s0 : M → E. 1. Show that Σ is a submanifold of E, s0 is a diffeomorphism of M onto Σ and that there is a canonical isomorphism s∗0 NR,Σ ' E. Thus pulling back the sequence (3.12) for (M, V ) = (E, Σ) by s0 , we have the exact sequence of vector bundles on M : 0 −→ TR M −→ s∗0 TR E −→ E −→ 0. 2. Applying the sequence in Exercise 3.9, we have the exact sequence 0 −→ TR π −→ TR E −→ π ∗ TR M −→ 0. Show that there is a canonical isomorphism s∗0 TR π ' E. Thus pulling back the above sequence by s0 , we have the exact sequence 0 −→ E −→ s∗0 TR E −→ TR M −→ 0. 3. Show that there is in fact a canonical isomorphism s∗0 TR E ' E ⊕ TR M. Exercise 3.11. Let M be a C ∞ manifold. Show that the diagonal ∆ = { (x, x) ∈ M × M | x ∈ M } is a closed submanifold of M × M and the map ι : M → M × M given by ι(x) = (x, x) is a diffeomorphism onto ∆. Show moreover that there is an isomorphism ι∗ NR,∆ ' TR M .

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Holomorphic tangent bundle Let M be a complex manifold of dimension n and {Uα } a covering of M by coordinate neighborhoods Uα with holomorphic coordinates (z1α , . . . , znα ). F Then, as in the real case, the union T M = z∈M Tz M of the holomorphic parts of the complexified tangent spaces of M admits the structure of a complex vector bundle of rank n with {τ αβ } given by τ αβ =

∂(z1α , . . . , znα ) ∂(z1β , . . . , znβ )

,

as a system of transition matrices. Since, for each pair (α, β), τ αβ is a holomorphic map from Uα ∩Uβ into GL(n, C), T M is a holomorphic bundle. We F call it the holomorphic tangent bundle of M . Note that T M = z∈M T z M may be identified with the complex conjugate of T M (cf. Remark 2.11 and Section 3.3). The bundle complex dual to T M is called the holomorphic cotangent bundle of M and is denoted by T ∗ M . Definition 3.13. The line bundle det T ∗ M is called the canonical bundle of M and is denoted by KM . We give some fundamental facts on the tangent bundles of a complex manifold. Let TRc M = C ⊗R TR M be the complexification of TR M . Proposition 3.4. There is a natural isomorphism TRc M ' T M ⊕ T M, which induces the decomposition (2.6) on each fiber. Proof. We cover M by coordinate neighborhoods Uα with coordinates √ −1 yiα , then the transition ma(z1α , . . . , znα ). If we write ziα = xα i + αβ c trix t of the vector bundle TR M on Uα ∩ Uβ is the Jacobian matrix of β β β α α α β (xα 1 , y1 , . . . , xn , yn ) with respect to (x1 , y1 , . . . , xn , yn ). It is not difficult to show that this bundle is isomorphic with the one defined by the transition matrix which is equal to the Jacobian matrix of (z1α , . . . , znα , z¯1α , . . . , z¯nα ) of the Cauchy-Riemann with respect to (z1β , . . . , znβ , z¯1β , . . . , z¯nβ ). Because ! αβ τ 0 equations, this matrix is equal to . 0 τ αβ  Proposition 3.5. We have T M ' TR M as real bundles.

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Proof. We cover M by coordinate neighborhoods Uα as in Proposition 3.4. Then again by the Cauchy-Riemann equations, we see that ρ ◦ τ αβ α α α (cf. (3.9)) is identical with the Jacobian matrix of (xα 1 , y1 , . . . , xn , yn ) with β β β β respect to (x1 , y1 , . . . , xn , yn ).  Let v be a section of T M . On a coordinate neighborhood U with coordinates (z1 , . . . , zn ), v is expressed as n X ∂ v= fi (z) ∂zi i=1 with fi complex valued functions on U . The following is not difficult to see: √ Proposition 3.6. Let v be as above. If we write fi = ui + −1 vi with ui and vi real valued functions, then under the isomorphism of Proposition 3.5, v corresponds to the real vector field given by n n X X ∂ ∂ + vi (x, y) . ui (x, y) ∂x ∂y i i i=1 i=1 Definition 3.14. A holomorphic vector field is a holomorphic section of T M . Thus v as above being holomorphic means that each fi is holomorphic. We denote by Γ (U ; T M ) the space of holomorphic vector fields on U . Exercise 3.12. 1. Give detailed proofs of Propositions 3.4–3.6. 2. We may naturally extend the bracket operation [ , ] to the space C ∞ (U ; TRc M ). Show that, if we identify T M with a subbundle of TRc M by Propositions 3.4, the subspaces C ∞ (U ; T M ) and Γ (U ; T M ) of C ∞ (U ; TRc M ) are both stable by [ , ]. 3. Show that the isomorphism of Proposition 3.5 is compatible with the bracket operations on T M and on TR M . Holomorphic normal bundle If V is a complex submanifold of M , T V may naturally be thought of as a subbundle of T M |V , as in the real case. We call the quotient the holomorphic normal bundle of V in M and denote it by NV so that we have the exact sequence $ 0 −→ T V −→ T M |V −→ NV −→ 0. (3.13) Note that, by Proposition 3.5, NV is isomorphic with NR ,V as a real bundle. Note also that the statements in Exercises 3.8–3.11 are valid if we replace “C ∞ ” and various real bundles with “holomorphic” and the corresponding holomorphic bundles.

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Line bundle associated with a hypersurface Let M be a complex manifold and V a hypersurface (possibly with singularities) in M . We may take an open covering {Uα } of M so that in each Uα , V is defined by a “reduced equation” f α = 0, i.e., the germ of f α at each point in V ∩ Uα is reduced (cf. Section 2.3). Note that if V ∩ Uα = ∅, then we may take a non-zero constant as f α . Then, for each pair (α, β), g αβ = f α /f β is a non-vanishing holomorphic function, i.e., a holomorphic function without zeros, on Uα ∩ Uβ and the system {g αβ } satisfies the cocycle condition. Definition 3.15. The line bundle associated with V , denoted by LV , is the line bundle defined by the cocycle {g αβ }. Note that LV is a holomorphic bundle and admits a natural holomorphic section whose zero set is exactly V , i.e., the section represented by the collection {f α } (cf. the paragraph after Definition 3.6). Exercise 3.13. Show that, if V is a non-singular hypersurface in M , there is a natural isomorphism LV |V ' NV . Example 3.6. Hyperplane bundle: Let Pn be the n-dimensional projective space with homogeneous coordinates [ζ0 , . . . , ζn ] and U = {Uα } the covering of Pn given by Uα = {ζα 6= 0}. If H is a hyperplane (cf. Example 2.4) in Pn defined by a linear function `(ζ), it is defined by the holomorphic function `(ζ)/ζα on Uα . The associated line bundle LH is then defined by the system of transition functions {g αβ } with g αβ = ζβ /ζα on the covering U. Thus it is uniquely determined modulo isomorphisms, independently of H. It is called the hyperplane bundle and is denoted by Hn . If V is a hypersurface of degree p, we see that LV is defined by the system of transition functions {(ζβ /ζα )p } so that LV = Hn⊗p , p times tensor product of Hn . Exercise 3.14. Let H be the hyperplane in Pn defined by ζ0 = 0 and p0 the point [1, 0, . . . , 0]. We identify H with Pn−1 = {[ζ1 , . . . , ζn ]} and define a map π : Pn r {p0 } → Pn−1 by [ζ] 7→ [ζ1 , . . . , ζn ]. Show that it has the structure of a holomorphic line bundle isomorphic with Hn−1 . We discuss in Section 11.6 below more generally the line bundle associated with a “divisor”.

74

3.5

Complex Analytic Geometry

Stiefel manifold

In this and the subsequent sections, the elements of CN are denoted by i

column vectors. Also we let ei = t(0, . . . , 0, 1, 0 . . . , 0), i = 1, . . . , N . An r-frame in CN is an ordered family of linearly independent r vectors in CN , 1 ≤ r ≤ N. Definition 3.16. The Stiefel manifold of r-frames in CN , denoted by W (N, r), is the set of r-frames in CN . Let M (N, r) denote the set of N × r complex matrices, which may be identified with CN r . Since an r-frame is expressed by an N × r matrix of rank r, W (N, r) may be thought of as the set of N × r matrices of rank r, which is an open set in M (N, r). Thus W (N, r) is naturally a complex manifold of dimension N r. The complex Lie group GL(N, C) acts transitively on W (N, r) from the left with the stabilizer at (e1 , . . . , er ) the subgroup GN,N −r of GL(N, C) consisting of matrices of the form   Ir ∗ , P 0 ∈ GL(N − r, C) 0 P0 so that W (N, r) is a homogeneous space (cf. Theorem 3.3): W (N, r) ' GL(N, C)/GN,N −r .

(3.14)

Let W0 (N, r) denote the subset of W (N, r) consisting of orthonormal r-frames in CN with respect to the standard Hermitian metric. The unitary group U (N ) acts on W0 (N, r) transitively from the left with the stabilizer at (e1 , . . . , er ) the set of unitary matrices of the form   Ir 0 , U 0 ∈ U (N − r) 0 U0 so that W0 (N, r) is a homogeneous space: W0 (N, r) ' U (N )/U (N − r). We may think of the above as describing the C ∞ manifold structure of W0 (N, r). Thus it is compact, connected and of dimension (2N − r)r. Note that the Gram-Schmidt process gives a deformation retraction W (N, r) −→ W0 (N, r). In particular, W (N, 1) = CN r{0} and W0 (N, 1) = S2N −1 so that S2N −1 ' U (N )/U (N − 1), which is the situation considered in Example 3.1.

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Fiber bundle structure: Suppose r > 1. Taking the first (r −1) vectors of each r-frame we have a map π : W (N, r) −→ W (N, r − 1).

(3.15)

Proposition 3.7. The map (3.15) has a fiber bundle structure with fiber Cr−1 × (Cq r{0}), q = N − r + 1. Proof. We think of W (N, r − 1) as the set of N × (r − 1) matrices of rank r − 1. Set I = { (i1 , . . . , ir−1 ) | 1 ≤ i1 < · · · < ir−1 ≤ N } 0

and for A ∈ W (N, r − 1) and I ∈ I, denote by A0I the (r − 1) × (r − 1) matrix consisting of r − 1 rows of A0 corresponding to I. Then W (N, r − 1) is covered by the open sets UI given by UI = { A0 ∈ W (N, r − 1) | det A0I 6= 0 }. If we represent an element of W (N, r) by an N × r matrix A = (aij ) of rank r, the map π assigns to A the N × (r − 1) matrix A0 consisting of the first r − 1 columns of A. For I ∈ I, we denote by I ∗ the complement of I in (1, . . . , N ). For each I = (i1 , . . . , ir−1 ) ∈ I and i ∈ I ∗ , let AI,i denote the r × r matrix consisting of the i1 , . . . , ir−1 and i-th rows of A. Note that, for a fixed I, at least one of the det AI,i ’s is non-zero, as the rank of A is r. With these, we have a trivialization ∼

ψI : π −1 (UI ) −→ Cr−1 × (Cq r{0}) × UI given by ψI (A) = (t(ai1 r , . . . , air−1 r ), (det AI,i )i∈I ∗ , A0 ) and the system {(UI , ψI )}I∈I defines a fiber bundle structure on π.  Remark 3.5. 1. The inverse image of (e1 , . . . , er−1 ) by π consists of matrices of the form (e1 , . . . , er−1 , a), where a = t(a1 , . . . , ar−1 , ar , . . . , aN ) with t (a1 , . . . , ar−1 ) ∈ Cr−1 and (ar , . . . , aN ) ∈ Cq r{0}. 2. If we use (3.14), we see that the fiber of π is given by GN,N −r+1 /GN,N −r , which is biholomorphic with Cr−1 × (GL(q, C)/Gq,q−1 ) ' Cr−1 × W (q, 1) = Cr−1 × (Cq r{0}). Consider the composition of inclusions ι : S2q−1 ,→ Cq r{0} ,→ W (N, r), where the last one is the inclusion of Cq r {0} as a fiber of the bundle π : W (N, r) → W (N, r − 1). Proposition 3.8. The inclusion ι induces an isomorphism ∼

ι∗ : πi (S2q−1 ) −→ πi (W (N, r))

for i = 0, . . . , 2q − 1.

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Proof.

This can be seen by considering the sequence of maps W (N, r) −→ W (N, r − 1) −→ · · · −→ W (N, 1) = CN r{0}

and applying Proposition 3.7 and Theorem 3.6.



Exercise 3.15. Give a detailed proof of Proposition 3.8. From Proposition 3.8 we have: ( 0 πi (W (N, r)) ' Z

i = 0, . . . , 2q − 2, i = 2q − 1.

(3.16)

Note that π2q−1 (W (N, r)) has a canonical generator, i.e., the homotopy class of ι (cf. Example 3.5). These can be seen also by retracting everything from the beginning to W0 (N, r), as W0 (N, r) → W0 (N, r − 1) is a fiber bundle with fiber S2q−1 . By (3.16) and Theorem 3.7 we have: ( 0 i = 1, . . . , 2q − 2, Hi (W (N, r); Z) ' (3.17) Z i = 0, 2q − 1. Note that H2q−1 (W (N, r); Z) has a canonical generator, i.e., ι∗ ν2q−1 , where ν2q−1 is the canonical generator of H2q−1 (S2q−1 ; Z) (cf. Remark B.13. 2). Real Stiefel manifold We may also consider the real Stielfel manifold V (N 0 , r) of r-frames in 0 RN . As in the complex case, V (N 0 , r) has naturally the structure of a C ∞ manifold of dimension N 0 r. The real Lie group GL(N 0 , R) acts transitively on V (N 0 , r) and we have: V (N 0 , r) ' GL(N 0 , R)/GN 0 ,N 0 −r , where GN 0 ,N 0 −r is the subgroup of GL(N 0 , R) consisting of matrices of the form   Ir ∗ , P 0 ∈ GL(N 0 − r, R). 0 P0 Let V0 (N 0 , r) denote the subset of V (N 0 , r) consisting of orthonormal 0 r-frames in RN with respect to the standard Euclidian metric. The orthogonal group O(N 0 ) acts on V0 (N 0 , r) transitively from left so that V0 (N 0 , r) is a homogeneous space: V0 (N 0 , r) ' O(N 0 )/O(N 0 − r).

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If r < N 0 , SO(N 0 ) also acts on V0 (N 0 , r) transitively so that V0 (N 0 , r) ' SO(N 0 )/SO(N 0 − r). We may think of the two representation of V0 (N 0 , r) as describing its C ∞ manifold structure. Note that the Gram-Schmidt process gives a deformation retraction V (N 0 , r) → V0 (N 0 , r). 0 0 In particular, V (N 0 , 1) = RN r{0} and V0 (N 0 , 1) = SN −1 so that SN

0

−1

' O(N 0 )/O(N 0 − 1).

0

Moreover if N 0 > 1, SN −1 ' SO(N 0 )/SO(N 0 − 1). Suppose r < N 0 and set q 0 = N 0 − r + 1. By arguments somewhat more complicated than the complex case, we have πi (V (N 0 , r)) = 0 for i ≤ q 0 − 2 and ( Z if q 0 − 1 is even or r = 1, 0 πq0 −1 (V (N , r)) = (3.18) Z2 if q 0 − 1 is odd and r > 1. Note that it has a canonical generator. 3.6

Grassmann manifold

An r-plane in CN is an r-dimensional linear subspace of CN . Definition 3.17. The Grassmann manifold of r-planes in CN , denoted by G(N, r), is the set of r-planes in CN . We will see that G(N, r) has a natural complex structure of dimension (N − r)r. An r-plane in CN is determined by an N × r matrix of rank r, whose column vectors forming its basis, i.e., by an element of the Stiefel manifold W (N, r). Two matrices A and A0 in W (N, r) define the same space if and only if there exists P in GL(r, C) such that A0 = AP . Thus G(N, r) is the orbit space of the right action of GL(r, C) on W (N, r): G(N, r) = W (N, r)/GL(r, C). The orbit of A is denoted by [A]. Let I be the set of r-tuples of integers given by I = { (i1 , . . . , ir ) | 1 ≤ i1 < · · · < ir ≤ N }.

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For an element I = (i1 , . . . , ir ) of I, we denote by AI the r × r matrix consisting of the i1 , . . . , ir -th rows of A. Note that the property that det AI 6= 0 does not depend on the
choice of the representative of [A]. The space G(N, r) is covered by Nr open sets UI given by UI = { [A] | det AI 6= 0 }. Each class [A] has a unique representative A◦ with A◦I = Ir , the identity matrix. Denoting by I ∗ the complement of I in (1, . . . , N ), the entries of A◦I ∗ gives a chart ∼

ϕI : UI −→ M (N − r, r) = C(N −r)r . More precisely, for the class of A, A◦ = A(AI )−1 so that ϕI ([A]) = (A(AI )−1 )I ∗ = AI ∗ (AI )−1 . To see that ϕI ◦ϕ−1 I 0 is holomorphic wherever it is defined, take an arbitrary matrix A˜ in M (N − r, r) and let A be the N × r matrix with AI 0 = Ir −1 ˜ If [A] ∈ UI , then ϕI ◦ ϕ−1 ˜ and A(I 0 )∗ = A. , which is I 0 (A) = AI ∗ (AI ) ˜ Thus these charts define a complex structure on G(N, r) holomorphic in A. of dimension (N − r)r. In particular if r = 1, we have A = t(a1 , . . . , aN ) and the correspondence [A] 7→ [ζ0 , . . . , ζN −1 ] = [a1 , . . . , aN ] gives a biholomorphic map of G(N, 1) onto PN −1 . The Grassmann manifolds can be viewed also in the following manner. i

Let ei = t(0, . . . , 0, 1, 0 . . . , 0), i = 1, . . . , N , as in Section 3.5. The complex Lie group GL(N, C) acts on G(N, r) holomorphically and transitively from the left and the stabilizer at the space [e1 , . . . , er ] spanned by (e1 , . . . , er ) is GL(r, N − r; C), which is the set of matrices of the form   P ∗ , P ∈ GL(r, C), Q ∈ GL(N − r, C) 0 Q and is a closed complex Lie subgroup of GL(N, C). Thus we have a biholomorphic map ∼

GL(N, C)/GL(r, N − r; C) −→ G(N, r) (cf. Theorem 3.3). We could alternatively define the complex structure of G(N, r) from this viewpoint. By the Gram-Schmidt process, the unitary group U (N ) also acts transitively on G(N, r) with stabilizer at [e1 , . . . , er ] being U (r) × U (N − r). Thus we have a C ∞ diffeomorphism ∼

G(N, r) −→ U (N )/U (r) × U (N − r), which shows that G(N, r) is connected and compact.

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Schubert cells Recall that every N × r matrix A of rank r can be made to the following “reduced form” Aˆ by elementary transformations on the columns, i.e., by multiplying a suitable matrix in GL(r, C) from the right: (*) For each j, 1 ≤ j ≤ r, we look the j-th column of Aˆ from the bottom and let a ˆij j be the first non-zero entry. Then i1 < · · · < ir and AˆI = Ir , thus a ˆij j = 1. Note that Aˆ is uniquely determined by A. Also A and A0 determine the same element in G(N, r) if and only if Aˆ = Aˆ0 . We say that Aˆ is of type I if AˆI = Ir . If we denote by eI the set of N × r reduced matrices of type I, Pr it is naturally identified with CdI , dI = j=1 (ij − j). We have a bijection G ∼ ψ : G(N, r) −→ eI I∈I

by assigning Aˆ to [A], which may be thought of as giving a cellular decomposition of G(N, r) (cf. Section B.2). The eI ’s are called Schubert cells. The largest cell is for I = (N − r + 1, . . . , N ) and is of complex dimension (N − r)r and the smallest cell is for I = (1, . . . , r) and is of complex dimension 0. We may also describe the Schubert cells as follows. For i = 1, . . . , N , let Vi denote the subspace of CN spanned by e1 , . . . , ei . We set V0 = {0}. For I = (i1 , . . . , ir ) in I, consider the set EI = { L ∈ G(N, r) | dim L ∩ Vi = j, ij ≤ i < ij+1 , 0 ≤ j ≤ r }, where we use the convention that i0 = 0 and ir+1 = N + 1. Then we have EI = ψ −1 (eI ). For an N × r matrix A of rank r and i = 1, . . . , N , we denote by A(i) the (N − i) × r matrix obtained from A by removing the first i rows. We set A(0) = A. Then from the relation dim(L + Vi ) = r + i − dim L ∩ Vi , we see that each element in EI is represented by a matrix A such that rank A(i) = r − j, where we set rank A

(N )

ij ≤ i < ij+1 , 0 ≤ j ≤ r,

= 0. Thus the closure of EI is given by

EI = { L ∈ G(N, r) | dim L ∩ Vij ≥ j, 1 ≤ j ≤ r } and each element is represented by a matrix A such that rank A(ij ) ≤ r − j,

1 ≤ j ≤ r.

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This shows that EI is a subvariety of G(N, r). These are called Schubert varieties, or Schubert cycles. If we define an relation in I by saying that I ≥ I 0 if ij ≥ i0j for j = 1, . . . , r, we may write [ EI = EI 0 . I 0 ≤I

Since there are no cells in real odd dimensions, we have the following: Theorem 3.8. The homology H∗ (G(N, r); Z) is the free Z-module with basis the classes of the Schubert cycles. Dual Grassmann manifold: We set W = CN and W∗ = Hom(CN , C). Then W∗ ' CN . We denote vectors in W∗ by row vectors, which acts on column vectors by multiplication from the left. We may consider the Grassmann manifold G(W∗ , N − r) of (N − r)-planes in W∗ . An element in W∗ is represented by an (N − r) × N matrix B of rank N − r, whose rows forming its basis. Such matrices B and B 0 represents the same element if and only if there is Q ∈ GL(N − r, C) such that B 0 = QB. We have a natural biholomorphic map ∼

D : G(W, r) −→ G(W∗ , N − r),

(3.19)

which assigns to an r-plane L, the (N − r)-plane (W/L)∗ . In terms of matrices, [A] is assigned to [B] with BA = 0. ˆ by interchanging rows and For B, we may define its reduced form B columns in (*): Every (N − r) × N matrix B of rank N − r can be made ˆ by elementary transformations on the to the following “reduced form” B rows, i.e., by multiplying a suitable matrix in GL(N − r, C) from the left: ˆ from the left and (**) For each i, 1 ≤ i ≤ N − r, we look the i-th row of B ˆJ = IN −r , let ˆbji i be the first non-zero entry. Then j1 < · · · < jN −r and B ˆ thus bj i = 1. i

For J = (j1 , . . . , jN −r ) we denote by fJ the cell in G(W∗ , N − r) corresponding to J. By linear algebra we see the following: Proposition 3.9. The cell fJ corresponds to eI by D in (3.19) if and only if J = I ∗ . Example 3.7. For G(N, 1) = PN −1 we have the cellular decomposition G(N, 1) = e(N ) t · · · t e(1) , where e(i) ' Ci−1 .

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Example 3.8. For G(4, 2) we have the cellular decomposition G(4, 2) = e(3,4) t e(2,4) t e(1,4) t e(2,3) t e(1,3) t e(1,2) , where e(3,4) ' C4 , e(2,4) ' C3 , e(1,4) ' e(2,3) ' C2 , e(1,3) ' C and e(1,2) ' {0}. We also have e(2,4) = G(4, 2)re(3,4) e(1,4) = e(1,4) t e(1,3) t e(1,2) ' G(3, 1) ' P2 , e(2,3) = e(2,3) t e(1,3) t e(1,2) ' G(3, 2) ' P2 , and e(1,4) ∩ e(2,3) = e(1,3) = e(1,3) t e(1,2) ' G(2, 1) ' P1 . The cycle e(2,4) is a subvariety of dimension 3 given as follows: e(2,4) ∩ U(3,4) = ∅, e(2,4) ∩ U(i1 ,i2 ) ' C3

for (i1 , i2 ) = (1, 3), (1, 4), (2, 3) and (2, 4).

e(2,4) ∩ U(1,2) is given by, in the canonical coordinate system   a a t 1 0 a31 a41 , by 31 32 = 0, 0 1 a32 a42 a41 a42 which has an isolated singularity at   t 1 0 0 0 . 0100 Exercise 3.16. Verify the statements in Examples 3.7 and 3.8. Universal bundle There are some natural complex vector bundles on the Grassmann manifold. First, there is the tautological bundle π : S → G(N, r), i.e., the bundle whose fiber over a point L in G(N, r) is L itself as a vector space. It is of rank r and can be described as S = { (z, L) ∈ CN × G(N, r) | z ∈ L } with π restriction of the projection to the first factor (see Section 9.3 below for more explicit description as a vector bundle in the case r = 1). Thus S is a subbundle of the trivial bundle IN = CN × G(N, r). The quotient Q = IN /S is called the universal bundle.

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As noted above, the Lie group GL(r, C) acts on the Stiefel manifold W (N, r) from right and the orbit space is the Grassmann manifold: G(N, r) = W (N, r)/GL(r, C) ' GL(N, C)/GL(r, N − r; C). In fact W (N, r) is a principal GL(r, C)-bundle associated with the tautological bundle on G(N, r). The group U (r) acts on W0 (N, r) from the right and there is a natural injection W0 (N, r)/U (r) → G(N, r), which is in fact surjection by the Gram-Schmidt process. Thus we may express G(N, r) = W0 (N, r)/U (r) = U (N )/U (r) × U (N − r). In particular, as G(N, 1) = PN −1 , we have PN −1 = S2N −1 /S1 .

(3.20)

Thus the canonical projection S2N −1 → PN −1 is a principal S1 -bundle. By (3.19), the tautological bundle Sr on G(W, r) corresponds to the universal quotient bundle Qr on G(W∗ , N − r). Proposition 3.10. Let E → X be a complex vector bundle of rank l on a topological space X. If there exists N global sections (s1 , . . . , sN ) of E that span the fiber over every point of X, then there is a map f : X → G(N, r), r = N − l, such that E = f ∗ Q. Proof. Let IN = CN × X be the product bundle and ϕ : IN → E the PN bundle morphism defined by ϕ(t (a1 , . . . , aN ), x) = i=1 ai si (x). Then by assumption, it is surjective and Ker ϕ is a subbundle of IN of rank r = N −l. Let f : X → G(N, r) be defined by f (x) = Ker ϕx . Then we see that E = f ∗ Q.  Note that if X admits a finite covering {Uα } such that E is trivial on each Uα , then E has global sections with the property as above. We discuss more about the universal bundle in Section 5.6 below (cf. Theorem 5.4 and the subsequent paragraph). Real Grassmann manifold We may also consider the real Grassmann manifold G(N 0 , r) of r-planes in 0 RN . As in the complex case, G(N 0 , r) is the orbit space of the right action of GL(r, R) on the real Stiefel manifold V (N 0 , r): G(N 0 , r) = V (N 0 , r)/GL(r, R) and we may define a C ω structure on G(N 0 , r) of dimension (N 0 − r)r.

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The Grassmann manifolds can be viewed also in the following manner. The real Lie group GL(N 0 , R) acts on G(N 0 , r) real analytically and transitively from the left and the stabilizer at the space [e1 , . . . , er ] spanned by (e1 , . . . , er ) is GL(r, N − r; R). Thus we have a C ω diffeomorphism ∼

GL(N 0 , R)/GL(r, N 0 − r; R) −→ G(N 0 , r). By the Gram-Schmidt process, the orthogonal group O(N 0 ) also acts transitively on G(N 0 , r) with stabilizer at [e1 , . . . , er ] being O(r)×O(N 0 −r). Thus we have a C ω diffeomorphism ∼

G(N 0 , r) −→ O(N 0 )/O(r) × O(N 0 − r), which shows that G(N 0 , r) is compact. Also SO(N 0 ) acts transitively on G(N 0 , r) and we have a C ω diffeomorphism ∼

G(N 0 , r) −→ SO(N 0 )/H, where H is the subgroup of SO(N 0 ) of matrices of the form   P 0 , P ∈ O(N 0 ), Q ∈ O(N 0 − r), det P · det Q = 1. 0 Q Thus G(N 0 , r) is connected.

3.7

Some topics on differentiable manifolds

Orientability Orientations of a real vector space: Let V be a real vector space of dimension l0 . Assume that l0 ≥ 1 for the moment. We introduce a relation ∼ in the set of ordered bases of V by saying that (e1 , . . . , el0 ) ∼ (e01 , . . . , e0l0 ), if there exists a matrix P in GL+ (l0 , R) such that (e01 , . . . , e0l0 ) = (e1 , . . . , el0 )P . Then it is an equivalence relation and there are exactly two equivalence classes, each of which is called an orientation of V. An oriented vector space is a vector space together with a prescribed orientation. If V is an oriented vector space, we say a basis (e1 , . . . , el0 ) of V positive or negative according as it belongs or not to the orientation. We adopt the following: 0

Convention 3.1. We orient Rl so that the standard basis (e1 , . . . , el0 ), i

ei = t(0, . . . , 0, 1, 0 . . . , 0), is positive. If V = 0, by convention, we think of it as having two orientations.

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Orientability of C ∞ manifolds: Let M be a C ∞ manifold of dimension m. We say that M is orientable if it is possible to specify an orientation of the tangent space at each point of M so that the specified orientations are compatible with the C ∞ structure in the following sense. For the moment we assume m > 0 and let {(Uα , ϕα )} be a C ∞ atlas representing the C ∞ α structure of M . For a point x in Uα , we write ϕα (x) = (xα 1 , . . . , xm ). Then ( ∂x∂α , . . . , ∂x∂α ) form a basis of TR,x M . If x ∈ Uα ∩ Uβ , ( ∂x∂ β , . . . , ∂x∂β ) 1

m

1

m

also form a basis of TR,x M and the matrix of base change is given by β β α tαβ (x) = ∂(xα 1 , . . . , xm )/∂(x1 , . . . , xm )(x) (cf. (3.10)). Definition 3.18. A C ∞ manifold M is orientable if M admits a C ∞ atlas A = {(Uα , ϕα )} with the following property: (*) for every pair (α, β) and x in Uα ∩ Uβ , tαβ (x) is in GL+ (m, R). For two atlases A and A0 with the property (*) above, we write A ∼ A0 , if their union has the property (*). Then it is an equivalence relation in the set of such atlases. We call an equivalence class an orientation of M . Note that if M is connected and is orientable, there are exactly two orientations of M . We say that M is oriented, if it is orientable and an orientation is specified. Suppose M is oriented. A coordinate system on a connected open set is said to be positive or negative according as it belongs or not to an atlas representing the orientation. Also, an m-form ω is said to be positive or negative according as, when we write ω = f (x) dx1 ∧ · · · ∧ dxm with positive coordinate system (x1 , . . . , xm ), f (x) is a (real) positive or negative valued function. For each point x in M , let (x1 , . . . , xm ) be a positive ∂ , . . . , ∂x∂m ) is a coordinate system near x. We orient TR,x M so that ( ∂x 1 positive basis. Then the orientation does not depend on the choice of the positive coordinate system. In the case m = 0, we think of M as being orientable with each connected component, which is a point, having two orientations. A diffeomorphism f : M → M of an oriented manifold M is said to be orientation preserving if, for each x in M , f∗ : TR,x M → TR,f (x) M transforms a positive basis to a positive basis. Note that Rm is orientable as a C ∞ manifold. In the following, we always orient Rm so that the canonical coordinate system (x1 , . . . , xm ) is positive. This is consistent with the orientation of Rm as a vector space (cf. Convention 3.1).

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If M is a complex manifold, it is always orientable (cf. (1.3)). We orient M so that, if (z1 , . . . , zn ) is a complex coordinate system on M , √ (x1 , y1 , . . . , xn , yn ) is a positive coordinate system, where zi = xi + −1 yi , i = 1, . . . , n. C ∞ maps of subsets We give the following: Definition 3.19. Let M and M 0 be C ∞ manifolds and A a subset of M . A map f : A → M 0 is C ∞ if, for every point a of A, there is a neighborhood U of a in M such that f |A∩U may be extended to a C ∞ map U → M 0 . The following is proved by the partition of unity argument: Proposition 3.11. Let A be a subset of a C ∞ manifold M . If f : A → Rk is C ∞ , f may be extended to a C ∞ map in a neighborhood of A. From the continuity of partial derivatives we have: Proposition 3.12. Let A be a subset of Rm and f : A → Rk a C ∞ map. Suppose there exists an open set D in Rm such that D ⊂ A ⊂ D. Then the values at each point of A of the partial derivatives of the components of an extension of f do not depend on the chosen extension. Manifolds with boundary We set Hm = { (x1 , . . . , xm ) ∈ Rm | x1 ≤ 0 }. Note that C ∞ maps of open sets in Hm are defined as in Definition 3.19. Let R be a Hausdorff topological space with a countable basis. Definition 3.20. A C ∞ manifold with boundary is a space R as above together with a C ∞ structure represented by an C ∞ atlas {(Uα , ϕα )} on R, as in the case of C ∞ manifolds, except each ϕα is a homeomorphism onto an open set in Rm or in Hm . The integer m as above is called the dimension of R. The union of the inverse images of {x1 = 0} in Hm by the ϕα ’s is called the boundary of R and is denoted by ∂R. The interior Int R of R is the set R r∂R, which is a usual C ∞ manifold of dimension m. We do not exclude the case ∂R = ∅.

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Sometimes we use the terminology “a manifold possibly with boundary” to emphasize this. For R as above, we readily see that there is a C ∞ manifold M of dimension m containing R with the following property: for every point p of ∂R there is a coordinate neighborhood U in M with coordinates (x1 , . . . , xm ) such that R ∩ U = { q ∈ U | x1 (q) ≤ x1 (p) }. If ∂R 6= ∅, it is an (m − 1)dimensional C ∞ submanifold of M . In fact if (x1 , . . . , xm ) is a coordinate system as above, then (x2 , . . . , xm ) is a coordinate system on ∂R ∩ U . In this case we say that R is a manifold with boundary in M . Note that such a manifold M is uniquely determined in a neighborhood of R. We say that R is orientable if we may choose M as above so that it is orientable. In this case ∂R is also orientable. We adopt the following: Convention 3.2. If M is oriented so that a coordinate system (x1 , . . . , xm ) as above is positive, we orient ∂R so that (x2 , . . . , xm ) is positive. Example 3.9. The closed unit m-ball Bm = { x ∈ Rm | kxk2 = |x1 |2 + · · · + |xm |2 ≤ 1 } is an m-dimensioanl manifold with boundary. It inherits the orientation of Rm . The unit (m − 1)-sphere Sm−1 = { x ∈ Rm | kxk2 = 1 } is its boundary ∂Bm . It is an (m − 1)-dimensional manifold and is oriented according to Convention 3.2. This is consistent with the orientation of Bm as a closed cell (cf. Remark B.13. 2). Remark 3.6. For each i = 1, . . . , m, we may choose the coordinate system (x1 , . . . , xm ) on M so that R ∩ U = { q ∈ U | xi (q) ≤ xi (p) }. In this case ∂R is given by xi = 0 and (x1 , . . . , xi−1 , xi+1 , . . . , xm ) is a coordinate system on ∂R. Moreover, if M is oriented and if (x1 , . . . , xm ) is positive, ∂R is oriented so that (x1 , . . . , xi−1 , xi+1 , . . . , xm ) is positive or negative according as i is odd or even. Let R be a C ∞ manifold with boundary in M and let M 0 be another C manifold. A C ∞ map f : R → M 0 is defined as in Definition 3.19. Thus it induces a C ∞ map ∂R → M 0 , which will be denoted by ∂f . We quote the following: ∞

Theorem 3.9 (Collar neighborhood theorem). Let R be a C ∞ manifold with boundary. Then ∂R has a neighborhood which is diffeomorphic with ∂R × [0, 1).

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One of the consequences of this is that the inclusion of Int R into R is a homotopy equivalence. Let M be a C ∞ manifold of dimension m. For non-negative integers d0 0 0 and k 0 with d0 +k 0 = m, we decompose as Rm = Rk ×Rd with (x1 , . . . , xk0 ) 0 0 and (xk0 +1 , . . . , xm ) as coordinates on Rk and Rd , respectively. We identify 0 0 Rd with {0} × Rd ⊂ Rm . We also set 0

0

Hd = { (xk0 +1 , . . . , xm ) ∈ Rd | xk0 +1 ≤ 0 }. Definition 3.21. A subset R of M is a C ∞ submanifold of M with boundary, if there is an atlas {(Uα , ϕα )} representing the C ∞ structure of M such 0 0 that for each α, ϕα (R ∩ Uα ) is an open set in Rd or in Hd . If R is as above, it is a d0 -dimensional C ∞ manifold with boundary ∂R, 0 which is the union of the inverse images of {xk0 +1 = 0} in Hd by the ϕα ’s. For such an R, there exists a d0 -dimensional C ∞ submanifold (without boundary) V of M containing R with the following property: for every point p of ∂R there is a coordinate neighborhood U in M with coordinates (x1 , . . . , xm ) such that V ∩ U = { q ∈ U | x1 (q) = · · · = xk0 (q) = 0 } and that R ∩ U = { q ∈ V ∩ U | xk0 +1 (q) ≤ xk0 +1 (p) }. If ∂R 6= ∅, it is a (d0 − 1)-dimensional C ∞ submanifold of V . In fact if (x1 , . . . , xm ) is a coordinate system as above, then (xk0 +2 , . . . , xm ) is a coordinate system on ∂R ∩ U . In this case we say that R is a C ∞ manifold with boundary in V , or in M . The manifold R is orientable if we may choose V as above so that it is orientable. In this case ∂R is also orientable. If R is oriented, ∂R is oriented according to Convention 3.2. Note that ∂R is not equal to the boundary of R in M , in generel. Remark 3.7. In some literature, a “closed manifold” means a compact manifold without boundary. Here if we say that R is a closed submanifold (with boundary) in M , it means that it is a closed subset of M . Orientability of fiber bundles Let (G, F ) be an effective left action of a Lie group G on a C ∞ manifold F possibly with boundary (cf. Definition 3.20). If F has a boundary, we assume that each element of G preserves the boundary. Let π : T → M be a C ∞ fiber bundle on a C ∞ manifold M with fiber F and group G. Note that, if the dimensions of M and F are m and l0 , respectively, then T is a C ∞ manifold of dimension m + l0 . If F has a boundary ∂F , but not M , T is a manifold with boundary ∂T which has the structure of a fiber bundle

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on M with fiber ∂F and group G. We set ∂π = π|∂T . We now suppose that F is oriented (cf. Definition 3.18). Definition 3.22. The bundle π : T → M is orientable, if the structure group may be reduced to a sub-Lie group G0 of G such that each element in G0 preserves the orientation of F . We call a system of trivialization T of π admissible, if it gives a reduction as above. For two admissible systems T and T 0 , we write T ∼ T 0 if their union is also admissible. Then it is an equivalence relation in the set of admissible system of trivializations. An orientation of the bundle π : T → M is defined to be an equivalence class of admissible systems. We say that T is oriented, if it is orientable and an orientation is specified. Suppose the bundle T is oriented. We say a trivialization (U, ψ) of T , with U a connected open set in M , positive or negative according as it belongs or not to an admissible system representing the orientation. In this case, each fiber is oriented so that a positive trivialization induces an orientation preserving diffeomorphism of the fiber onto F . Note that a trivialization (U, ψ) of T induces an isomorphism ∼

TR,t T −→ TR,(y,x) (F × U ) ' TR,y F ⊕ TR,x M, ψ∗

ψ(t) = (y, x).

(3.21)

If π : T → M is an oriented bundle and if M is oriented, then the total space T is orientable. Convention 3.3. We orient the total space T so that the orientation of the fiber followed by that of M gives the orientation of T , i.e., so that the isomorphism (3.21) is orientation preserving for a positive trivialization ψ. Exercise 3.17. Show that, by the above convention, the orientation of ∂T as the boundary of T coincides with the one as the total space of the fiber bundle ∂π : ∂T → M . Let π : E → M be a real C ∞ vector bundle of rank l0 . We may think of 0 the fiber V as being oriented once we fix an isomorphism V ' Rl . Thus we may talk about the orientability of the vector bundle E (cf. Definition 3.22). 0 In particular, if the fiber of E is Rl , E is orientable if and only if the structure group may be reduced to GL+ (l0 , R) (cf. Remark 9.1. 2 below). Since the morphism ρ in (3.9) below maps GL(l, C) into GL+ (2l, R), a complex vector bundle is always orientable. Note that the manifold M is orientable if and only if its tangent bundle TR M is orientable as a bundle (cf. Definitions 3.18 and 3.22). In this case,

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we take the orientation of the bundle TR M so that, if (x1 , . . . , xm ) is a ∂ , . . . , ∂x∂m ) determines a positive coordinate system on M , the frame ( ∂x 1 positive trivialization of TR M . Let V be a submanifold of codimension k 0 of M . Letting NR,V be the normal bundle of V in M , we have the exact sequence (3.12). Proposition 3.13. If M and V are orientable, the bundle NR,V is orientable. α Proof. There is a C ∞ atlas {(Uα , (xα 1 , . . . , xm ))} on M such that V ∩ Uα α α is given by x1 = · · · = xk0 = 0. We specify orientations of M and V . Changing the signs of some variables, if necessary, we may assume that α (xα 1 , . . . , xm ) is a positive coordinate system on M for every α and that the α restriction of (xα k0 +1 , . . . , xm ) to V is a positive coordinate system on V for every α with V ∩ Uα 6= ∅. Thus NR,V is orientable (cf. (3.11)). 

Convention 3.4. If M and V are oriented and if (x1 , . . . , xm ) and (xk0 +1 , . . . , xm ) are positive coordinate systems on M and V , we orient the ∂ ), . . . , $( ∂x∂ 0 )) determines a positive bundle NR,V so that the frame ($( ∂x 1 k trivialization. The total space NR ,V is then oriented according to Convention 3.3. Tubular neighborhoods Let M be a C ∞ manifold and V a submanifold of M with the normal bundle p : NR,V → V . We quote the following: Theorem 3.10 (Tubular neighborhood theorem). There exist a neighborhood U of V in M , a neighborhood W of the image Z of the zero section in NR,V and a diffeomorphism τ of U onto W such that τ (V ) = Z and that (p ◦ τ )|V = 1V . If we take an open ball bundle (cf. Remark 9.1. 1 below) as W , then r = p ◦ τ : U → V is a C ∞ deformation retraction. Such a neighborhood U is referred to as a tubular neighborhood of V in M . Remark 3.8. 1. If M and V are oriented, we orient NR ,V as described after Proposition 3.13. Then may we take τ so that it is orientation preserving.

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2. In the case V is a complex submanifold of a complex manifold M , we can say more about τ (cf. Proposition 15.8 below). Transversality Let f : M → M 0 be a C ∞ map of C ∞ manifolds and V 0 a submanifold of codimension k 0 of M 0 . We look for a condition for f −1 V 0 to be a submanifold of M . Noting that the problem is local, let x be a point in f −1 V 0 . There 0 is a neighborhood U 0 of f (x) in M 0 and a C ∞ submersion g : U 0 → Rk such that V 0 ∩ U 0 = g −1 (0). We then have f −1 V 0 ∩ f −1 U 0 = (g ◦ f )−1 (0). Thus f −1 V 0 is a submanifold in a neighborhood of x if and only if (g ◦ f )∗,x is surjective. Since (g ◦ f )∗,x = g∗,f (x) ◦ f∗,x (cf. (2.4)) and g∗,f (x) is surjective and Ker g∗,f (x) = TR,f (x) V 0 , (g ◦ f )∗,x is surjective if and only if Im f∗,x and TR,f (x) V 0 span TR,f (x) M 0 . Thus we introduce the following: Definition 3.23. We say that f is transverse to V 0 if Im f∗,x + TR,f (x) V 0 = TR,f (x) M 0

for all x ∈ f −1 V 0 .

In this case, we write f t V 0 . f

As we observed, if f t V 0 , then f −1 V 0 is a submanifold of M and, if V 0 6= ∅,

−1

codim f −1 V 0 = codim V 0 . In particular, if M is a submanifold of M 0 and if i : M ,→ M 0 is the inclusion, i−1 V 0 = M ∩ V 0 and i t V 0 if and only if TR,y M + TR,y V 0 = TR,y M 0

for all y ∈ M ∩ V 0 .

In this case we also say M is transverse to V 0 and write M t V 0 . If this is the case, then M ∩ V 0 is a submanifold of M 0 and, if M ∩ V 0 6= ∅, codim(M ∩ V 0 ) = codim M + codim V 0 . Exercise 3.18. Let M be a C ∞ manifold and V a submanifold of codimension k 0 of M . Show that, for every point x in V , there is a k 0 -dimensional submanifold of M which intersects V transversally at x. Let R be a C ∞ manifold with boundary and f : R → M 0 a C ∞ map. Also let V 0 be a submanifold of M 0 . If both f and ∂f : ∂R → M 0 are transverse to V 0 , then f −1 V 0 is a manifold with boundary, (∂f )−1 (V 0 ) is a submanifold of ∂R and ∂(f −1 V 0 ) = f −1 V 0 ∩ ∂R = (∂f )−1 (V 0 ) as sets (cf. Proposition 3.15 below). Here we quote the following:

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Theorem 3.11 (Transversality homotopy theorem). For every C ∞ map f : R → M 0 and a submanifold V 0 of M 0 , there exists a C ∞ map f 0 : R → M 0 homotopic to f such that f 0 t V 0 and ∂f 0 t V 0 . Orientation: Let f : M → M 0 and V 0 be as above. Let m and m0 be dimensions of M and M 0 and k 0 the codimension of V 0 . Suppose f t V 0 and f −1 V 0 6= ∅ so that f −1 V 0 is a submanifold of codimension k 0 of M . Proposition 3.14. In the above situation, if M , M 0 and V 0 are orientable, f −1 V 0 is also orientable. Proof. We specify orientations of M , M 0 and V 0 and let {(Uλ0 , ϕλ )} be a C ∞ atlas representing the orientation of M 0 such that, writing ϕλ (y) = λ 0 λ λ (y1λ , . . . , ym 0 ), V ∩Vλ is given by y1 = · · · = yk 0 = 0 and that the restriction λ λ 0 of (yk0 +1 , . . . , ym0 ) to V is a positive coordinate system on V 0 . Set xλi = yiλ ◦ f , i = 1, . . . , k 0 . We may choose a coordinate system (xλ1 , . . . , xλm ) so that it is positive. Thus f −1 V 0 is orientable.  Convention 3.5. We orient the manifold f −1 V 0 so that the restriction of (xλk0 +1 , . . . , xλm ) is a positive coordinate system. Note that this convention is consistent with Convention 3.4. With this convention, we have Proposition 3.15. Let f : R → M 0 be a C ∞ map. If f t V 0 and ∂f t V 0 , 0

∂(f −1 V 0 ) = (−1)k (∂f )−1 V 0 , as oriented manifolds. Proof. We take (x1 , . . . , xm ) as in the proof of Proposition 3.14 so that R is given by xk0 +1 ≤ 0. If we take the orientation of f −1 V 0 as above, the restriction of (xk0 +2 , . . . , xm ) is a positive coordinate system on ∂(f −1 V 0 ) (cf. Convention 3.2). On the other hand, (x1 , . . . , xk0 , xk0 +2 , . . . , xm ) is a coordinate system on ∂R which is positive or negative according as k 0 is even or odd (cf. Remark 3.6). Thus (xk0 +2 , . . . , xm ) is a coordinate system on (∂f )−1 V 0 positive or negative according as k 0 is even or odd.  In the above situation, let R0 be an (m0 − k 0 )-dimensional C ∞ manifold with boundary in V 0 .

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Proposition 3.16. If f t R0 and f t ∂R0 , then f −1 R0 is a manifold with boundary in M and ∂(f −1 R0 ) = f −1 (∂R0 ), as oriented manifolds. Proof. If V 0 is locally defined by y1 = · · · = yk0 = 0, f −1 V 0 is defined by y1 ◦ f = · · · = yk0 ◦ f = 0. Let (x1 , . . . , xm ) be as before so that the restriction of (xk0 +1 , . . . , xm ) is a positive coordinate system on f −1 V 0 . Let R0 be defined by yk0 +1 ≤ 0. From the condition f t ∂R0 , we may set xk0 +1 = yk0 +1 ◦f and ∂(f −1 R0 ) is given by xk0 +1 ≤ 0. Then (xk0 +2 , . . . , xm ) is a positive coordinate system on ∂(f −1 R0 ) (cf. Convention 3.2). On the other hand, f −1 (∂R0 ) is oriented so that it is a positive coordinate system (cf. Convention 3.5).  Slices: Let M be a C ∞ manifold of dimension m and V a submanifold of codimension k 0 . Definition 3.24. A slice of V in M at x ∈ V is a k 0 -dimensional submanifold D of M containing x, transverse to V at x and diffemorphic with an open k 0 -ball. Taking a coordinate system (x1 , . . . , xm ) on M around x so that V is given by x1 = · · · = xk0 = 0, we see that a slice always exists. If M and V are oriented, we always orient a slice D so that its orientation followed by that of V gives the orientation of M . Likewise, if M is a complex manifold of dimension n and V a complex submanifold of codimension k, we may define a complex slice to be a k-dimensional submanifold D of M , containing x, transverse to V at x and diffemorphic with an open 2k-ball. Ehresmann fibration theorem Definition 3.25. Let f : M → M 0 be a C ∞ map. 1. We say that f is a C ∞ trivial fibration if it admits a trivial C ∞ fiber bundle structure (cf. Section 3.2). 2. The map f is a C ∞ locally trivial fibration, if each point of M 0 has a neighborhood U such that f |f −1 (U ) : f −1 (U ) → U is a C ∞ trivial fibration. Thus f being trivial fibration means that there exist a C ∞ manifold ∼ F and a diffeomorphism ψ : M → F × M 0 such that f = p ◦ ψ, where

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p : F × M 0 → M 0 is the projection. In this case, each fiber f −1 (x), x ∈ M 0 , is diffeomorphic to F . We quote the following: Theorem 3.12. Let f : M → M 0 be a C ∞ map. If it is a proper surjective submersion, it is a C ∞ locally trivial fibration. Remark 3.9. 1. Suppose f : M → M 0 is a locally trivial fibration. If each fiber f −1 (x), x ∈ M 0 , is diffeomorphic to a fixed manifold F , for example this is always the case if M 0 is connected, we may think of f : M → M 0 as a fiber bundle with fiber F and group the group of diffeomorphisms of F . 2. The above theorem holds if we replace M with a C ∞ manifold R with boundary, provided that f restricted to ∂R is also a submersion. In this case each fiber is a C ∞ manifold with boundary. Notes For the proofs of the theorems stated in Section 3.1, see [Kobayashi and Nomizu (1963); Matsushima (1972)]. We list [Steenrod (1951)] as a fundamental reference for fiber bundles. For detailed discussions of Theorem 3.5, see §7 in there. As to the homotopy exact sequence and the Hurewicz theorem, we refer to §17 and §15 in there, see also Ch. 7 of [Spanier (1966)]. For Remark 3.3, we refer to §11 in there, see also §4 of [Hirzebruch (1966)]. For Stiefel and Grassmann manifolds, we also refer to [Steenrod (1951)]. See [Fulton (1984); Ikeda (2018)] and the references therein for the so-called Schubert calculus in Grassmann manifolds. We refer to [Guillemin and Pollack (1974); Lee (2013)] as to Theorems 3.9–3.11. Theorem 3.12 is due to [Ehresmann (1950)].

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Chapter 4

Dualities and Thom Class

In the case of manifolds, the theory of combinatorial topology becomes particularly rich, as there is a deep relation between the (co)homology defined by a triangulation and that defined by its dual cellular decomposition. We investigate such relations and prove the Poincar´e, Alexander and Lefschetz dualities. The Alexander dualty is a localized version of the Poincar´e duality and particularly important to describe localizations in various settings. The Lefschetz duality is for manifolds with boundary and may be thought of as a special case of the Alexander duality. We then prove the Thom isomorphism for a submanifold whose normal bundle is oriented and define the Thom class for such a submanifold. If the manifolds are oriented, the Thom isomorphism is directly related to the Poincar´e and Alexander isomorphisms. As a special case, we define the Thom class of an oriented real vector bundle. This turns out to be the localized version of the Euler class we discuss in the next chapter. We explicitly define the intersection product in homology in terms of combinatorial topology. This is an operation dual to the cup product in cohomology and will particularly be exploited in Chapter 14 below. We refer to Appendix B for basic materials on algebraic topology. Throughout this chapter, we take Z as the coefficient ring of homology and cohomology, unless otherwise stated. Also, we let M denote a C ∞ manifold of dimension m.

95

96

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Complex Analytic Geometry

Algebraic topology on manifolds

C ∞ triangulations Let X be a topological space. A triangulation of X is a pair (K, h) of a simplicial complex K and a homeomorphism h : |K| → X. We abbreviate this by saying that h : |K| → X is a triangulation, or K is a triangulation of X. Let Y be a subspace of X. We say that the triangulation is compatible with Y if there is a subcomplex L of K such that the restriction of h to |L| is a triangulation of Y . Definition 4.1. A triangulation h : |K| → M of a C ∞ manifold M is C ∞ if, for every simplex s of K, h|s is C ∞ and its rank at each point of s is equal to dim s. In the above, we think of s as being in the affine space spanned by s and h|s being C ∞ means C ∞ in the sense of Definition 3.19. Its rank is well-defined by Proposition 3.12. Note that the above definitions also make sense if we replace M with a C ∞ manifold with boundary. The following is known: (T1 ) Every C ∞ manifold M admits a C ∞ triangulation. In fact, if R is a closed C ∞ submanifold of M possibly with boundary, there is a C ∞ triangulation of M compatible with R and ∂R. (T2 ) If K1 and K2 are C ∞ triangulations of M , there exist subdivisions of K1 and K2 that are simplicially isomorphic. In this chapter we do not explicitly use the fact that the triangulations we consider are C ∞ , although it is essential to have a differentiable structure on M for the above facts. Homology via triangulation and dual cellular decomposition We take a triangulation (K0 , h) of M and let K denote the barycentric subdivision of K0 . We further let K 0 be the barycentric subdivision of K, i.e., the second barycentric subdivision of K0 . We take the second barycentric subdivision so that the star of a K0 -subcomplex L of K0 relative to K 0 has the same homotopy type as the polyhedron |L| of L (cf. Proposition B.26). In the following, sometimes a simplex s of K is identified with h(s) and |K| is identified with M .

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Dual cellular decomposition: For a p-simplex s of K, we denote by s∗ the union of (m − p)-simplices of K 0 intersecting with s at its barycenter bs . It is a regular closed (m − p)-cell in |K|, called the cell dual to s. The intersection of s and s∗ consists of the one point bs . The cells dual to simplices in K form a cellular decomposition of |K| = M , which will be denoted by K ∗ . Orientations of simplices and cells: In order to describe the homology and cohomology of M via triangulation or dual cellular decomposition, we fix orientations of simplices of K and cells of K ∗ . As to the orientations of simplices of K 0 , we impose the following conditions. Thus let t be a p-simplex of K 0 . (1) If t ⊂ s, a p-simplex of K, the orientation of t is the same as that of s. ∗ ∗ (2) If t ⊂ s0 , a p-cell of K ∗ , the orientation of t is the same as that of s0 . Note that for t not satisfying either of the above assumptions, there is still freedom of choice of the orientation. Homology and cohomology of M : We denote by Hp (M ) the p-th singular homology of M (cf. Section B.1). An important feature in the case of a manifold is that it can be computed using either the triangulation K or the cellular decomposition K ∗ in the following sense. Thus let (C•K (M ), ∂) be the chain complex with CpK (M ) the free Abelian group generated by K (M ) the boundary the oriented p-simplices in K and ∂ : CpK (M ) → Cp−1 operator defined by p X ∂(v0 , . . . , vp ) = (−1)i (v0 , . . . , vbi , . . . , vp ), i=0

for an oriented simplex s = (v0 , . . . , vp ) with vertices v0 , . . . , vp , and extended linearly (cf. (B.29)). We denote by HpK (M ) the p-th homology of C•K (M ). Denoting by Sp (M ) the group of singular p-chains of M , there is a natural chain morphism η•K : C•K (M ) −→ S• (M ),

(4.1)

which is defined as follows. For an oriented p-simplex s = (v0 , . . . , vp ), let ϕs : ∆p → s be the affine map with ϕs (Pi ) = vi , i = 0, . . . , p. Then ηpK assigns to s the singular simplex σ = h ◦ ϕs : ∆p → s → M . Note that η•K above is injective. It induces an isomorphism on the homology level (cf. (B.30)): ∼

η∗K : HpK (M ) −→ Hp (M ).

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Complex Analytic Geometry ∗

∗

Also if we denote by (C•K (M ), ∂) the chain complex with CpK (M ) the free Abelian group generated by the oriented p-cells in K ∗ , we have a natural isomorphism: ∗

∼

∗

η∗K : HpK (M ) −→ Hp (M ), ∗

(4.2)

∗

where HpK (M ) is the p-th homology of C•K (M ) (cf. Theorem B.22). In our case we have injective chain morphisms ∗

ι

ηK

0

0

• • C•K (M ) −→ S• (M ), C•K (M ) −→

(4.3)

where ι• is the morphism that regards a K ∗ -chain as a K 0 -chain, and the composition induces the isomorphism (4.2). Also the singular cohomology H p (M ) of M can be computed either p • from the cochain complex (CK (M ), δ) with CK (M ) = Hom(CpK (M ), Z) or p K∗ • (M ), Z). from the cochain complex (CK ∗ (M ), δ) with CK ∗ (M ) = Hom(Cp ∗ That is to say that the transposes of the chain morphisms η•K and η•K induce isomorphisms ∼

p ∗ ηK : H p (M ) −→ HK (M )

Denoting chains of K have a chain isomorphism

∼

p ∗ p and ηK ∗ : H (M ) −→ HK ∗ (M ).

(4.4)

by C˘•K (M ) and S˘• (M ) the chain complexes of locally finite and of locally finite singular chains of M , respectively, we morphism η˘•K : C˘•K (M ) → S˘• (M ) as in (4.1). It induces an on the homology level (cf. (B.38)): ∼

˘ pK (M ) −→ H ˘ p (M ). η˘∗K : H

(4.5)

∗ Likewise, considering the complex C˘•K (M ) of locally finite chains of K ∗ , ∗ ∼ ˘ ˘ pK ∗ (M ) → Hp (M ) (cf. (B.37)). we have a canonical isomorphism η˘∗K : H In the following, h , i denotes the paring of chains and cochains, i.e., the Kronecker product (cf. Section A.1).

4.2

Poincar´ e, Alexander and Lefschetz dualities

We prove the dualities in the case of manifolds, which will be generalized to the case of singular varieties in Section 13.2 below. Let M , K0 , K, K 0 and K ∗ be as in Section 4.1. In this section we assume that M is oriented and take orientations of the simplices and cells so that they satisfy the conditions (1) and (2) in Section 4.1 and that they are furthermore compatible with that of M in the following sense:

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(3) The orientation of each m-simplex is the same as that of M . (4) For every p-simplex s of K, 0 < p < m, the orientation of s∗ followed by the orientation of s gives the orientation of M . As a consequence we have: (*) Let s be a p-simplex in K and let t and t0 be an (m − p)-simplex and a p-simplex of K 0 . If t ⊂ s∗ and t0 ⊂ s and if t and t0 span an m-simplex t0 of K 0 , the orientation of t followed by that of t0 gives the orientation of t0 , which is the same as that of M . Poincar´ e duality We define a morphism p ˘K P : CK ∗ (M ) −→ Cm−p (M )

by P (u) =

X

hs∗ , ui s

(4.6)

s

for a p-cochain u of K ∗ , where the sum is taken over all (m − p)-simplices p s of M . For a p-cell s∗ of K ∗ we denote by ϑ(s∗ ) the cochain in CK ∗ (M ) ∗ ∗ 0 ∗ dual to s , i.e., for every p-cell s of K , ( ∗ 1 if s0 = s∗ , 0∗ ∗ hs , ϑ(s )i = 0 otherwise. p ∗ Then CK ∗ (M ) is the free Abelian group generated by the ϑ(s )’s and the ∗ morphism P sends ϑ(s ) to s. Thus P is in fact an isomorphism. We now prove that it is compatible with boundary and coboundary operators so that it induces an isomorphism between the corresponding cohomology and homology. For an oriented p-simplex t of K 0 , we denote by ϑ(t) the cochain in p CK 0 (M ) dual to t, defined similarly as above. Let MK 0 denote the sum of all m-simplices of K 0 , which is an m-cycle.

Ordering of vertices of K 0 : Let V denote the set of vertices of K 0 . Recall that every element v of V is the barycenter of a simplex of K, which is uniquely determined by v. We denote it by sv and introduce an order relation in V by saying that v ≤ v 0 if sv  sv0 . Then it is a simplicial ordering (cf. Definition B.14). We may express each p-simplex t of K 0 as t = ε(v0 , . . . , vp ),

v0 < · · · < vp ,

where ε = ±1 and the sign is to be determined according to the prescribed orientation of t. The cup and cap products are then defined by (B.31) and (B.32).

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Remark 4.1. Let t = ε(v0 , . . . , vp ), v0 < · · · < vp , be a p-simplex of K 0 . For a p-simplex s1 of K, t ⊂ s1 if and only if v0 = bs1 and in this case vi is the barycenter of a (p − i)-simlpex of K which is a face of s1 . Also for an (m − p)-simplex s2 of K, t ⊂ s∗2 if and only if vp = bs2 and in this case there is an m-simplex s of K such that s2 ≺ s and that vi is the barycenter of an (m − i)-simplex of K which is a face of s. Lemma 4.1 (Key lemma). For every p-simplex t of K 0 , we have: ( s if t ⊂ s∗ for some (m − p)-simplex s of K, MK 0 a ϑ(t) = 0 otherwise, as K 0 -chains. Proof. Let t0 be an arbitrary m-simplex of K 0 . Note that, for each p, there exists a unique (m − p)-simplex s of K such that t0 is spanned by a (1) (2) p-simplex t0 of K 0 in s∗ and an (m − p)-simplex t0 of K 0 in s. By our orientation convention (*), we arrange the vertices as t0 = ε(v0 , . . . , vm ), (1) (2) v0 < · · · < vm , so that t0 = ε1 (v0 , . . . , vp ), t0 = ε2 (vp , . . . , vm ) and ε = ε1 ε2 . Note that vp = bs (cf. Remark 4.1). Let t be a p-simplex of K 0 . If it is not in the dual cell of any (m − p)simplex of K, then t0 a ϑ(t) = 0 for every m-simplex t0 of K 0 so that MK 0 a ϑ(t) = 0. Thus suppose t ⊂ s∗ for some (m − p)-simplex s of K. Then we have X (2) X MK 0 a ϑ(t) = t0 a ϑ(t) = t0 = s. (1)

(1)

t0 =t

t0 =t



Recall that, as each cell in K ∗ may be thought of as a K 0 -chain, there ∗ 0 is a natural monomorphism ι : CpK (M ) → CpK (M ), whose transpose is K (M ) → denoted by ι∗ . Likewise there is a natural monomorphism C˘m−p 0 K ˘ Cm−p (M ), which is denoted by κ. Corollary 4.1. The following diagram is commutative: p CK ∗ (M ) O

P

K / C˘m−p (M )

MK 0 a

K / C˘m−p (M ).

ι∗ p CK 0 (M )

0



κ

Lemma 4.2. We have P δu = (−1)p+1 ∂P (u)

for

p u ∈ CK ∗ (M ).

(4.7)

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Proof. Noting that, in (4.7), ι∗ is surjective and compatible with coboundary operators and that κ is injective and compatible with boundary operators, the lemma follows from the property of cap product (B.17).  ∗ Combining with the isomorphisms ηK ˘∗K in (4.5), we have ∗ in (4.4) and η proved:

Theorem 4.1 (Poincar´ e duality). For an oriented C ∞ manifold M of dimension m, the isomorphism P of (4.6) induces an isomorphism ∼ ˘ PM : H p (M ) −→ H m−p (M ).

˘ m (M ) corresponding to [MK 0 ] by We denote by [M ] the class in H K0 ˘ ˘ the isomorphism Hm (M ) ' Hm (M ). Note that it does not depend on the choice of the triangulation by the property (T2 ) of C ∞ triangulations (cf. Section 4.1). By Corollary 4.1, we may write PM (α) = [M ] a α

for α ∈ H p (M ).

Remark 4.2. 1. If M is connected, the cycle MK 0 is called the fundamental cycle of M in K 0 and the class [M ] the fundamental class of M . In this case, ˘ m (M ) ' Z, the class [M ] corresponding there is a canonical isomorphism H to 1. Pm−p 2. Let s be an (m − p)-simplex of K. We may write ∂s = i=0 εi s(i) , where the s(i) ’s are the (m − p − 1)-faces of s and εi = ±1, the sign being determined according to the prescribed orientations. Then, if we set Pm−p ∗ ϑ((∂s)∗ ) = i=0 εi ϑ(s(i) ), by Lemma 4.2, we have δϑ(s∗ ) = (−1)p+1 ϑ((∂s)∗ ). Dual description: We define a morphism p K∗ P 0 : CK (M ) −→ C˘m−p (M )

by P 0 (u) =

X hs, ui s∗

(4.8)

s

for a p-cochain u, where the sum is taken over all p-simplices s of M . For p a p-simplex s of K, we denote by ϑ(s) the cochain in CK (M ) dual to s. 0 ∗ Then P sends ϑ(s) to s and is an isomorphism. We now prove that it is compatible with boundary and coboundary operators so that it induces an isomorphism between the corresponding cohomology and homology. We keep the orientation conventions and ordering of vertices of K 0 . The following is proved as Lemma 4.1, except the cap product becomes right. We give a proof for the sake of completeness.

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Lemma 4.3. For every p-simplex t of K 0 , we have: ( s∗ if t ⊂ s for some p-simplex s of K, ϑ(t) a MK 0 = 0 otherwise as K 0 -chains. Proof. Let t0 be an arbitrary m-simplex of K 0 . Note that, for each p, there exists a unique p-simplex s of K such that t0 is spanned by an (1) (2) (m − p)-simplex t0 of K 0 in s∗ and a p-simplex t0 of K 0 in s. By our orientation convention (*), we arrange the vertices as t0 = ε(v0 , . . . , vm ) so (1) (2) that t0 = ε1 (v0 , . . . , vm−p ), t0 = ε2 (vm−p , . . . , vm ) and ε = ε1 ε2 . Let t be a p-simplex of K 0 . If it is not in any p-simplex of K, then ϑ(t) a t0 = 0 for every m-simplex t0 of K 0 so that ϑ(t) a MK 0 = 0. Thus suppose t ⊂ s for some p-simplex s of K. Then we have X X (1) ϑ(t) a MK 0 = ϑ(t) a t0 = t0 = s∗ . (2)

(2)

t0 =t

t0 =t



Recall that, as each simplex in K may be thought of as a K 0 -chain, 0 there is a natural monomorphism ι0 : CpK (M ) → CpK (M ), whose transpose ∗ K∗ is denoted by ι0 . Likewise there is a natural monomorphism C˘m−p (M ) → 0 K 0 ˘ Cm−p (M ), which is denoted by κ . From the above lemma, we have the following commutative diagram: p CK (M ) O

P0

ι0∗ p CK 0 (M )

aMK 0

K∗ / C˘m−p (M )



κ0

(4.9)

K0 / C˘m−p (M ).

Thus we have (cf. Lemma 4.2, except we use (B.24) instead of (B.17)) P 0 δu = (−1)m−p ∂P 0 (u)

p for u ∈ CK (M ).

and an isomorphism ∼ ˘ P 0 : H p (M ) −→ H m−p (M ).

Remark 4.3. The above isomorphism is given by the right cap product with [M ] and differs from P by a sign of (−1)p(m−p) .

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The case M is compact: In the case M is compact, using the homology and cohomology with C-coefficient (in fact we may use Q as the coefficient), the Poincar´e duality is described as follows. If M is compact, ˘ m−p (M ; C) = Hm−p (M ; C) ' H m−p (M ; C)∗ H and from (B.20) we have: Proposition 4.1. If M is compact, PM sends a class α in H p (M ; C) to the class a in Hm−p (M ; C) such that for every β in H m−p (M ; C).

ha, βi = h[M ], α ` βi

Letting ε∗ : H0 (M ; C) → C be the augmentation, we have (cf. (B.19)): Corollary 4.2. If M is compact, the pairing [M ]a

`

ε

∗ H p (M ; C) × H m−p (M ; C) −→ H m (M ; C) −→ H0 (M ; C) −→ C

is non-degenerate and induces the Poincar´e duality with C-coefficient. Alexander duality Let S be a closed set in M . Suppose that there is a triangulation K0 of M such that S is a K0 -subcomplex of M , i.e., K0 is compatible with S. Recall that the star SK 0 (S) of S in K 0 is the union of simplices of K 0 intersecting with S, i.e., the union of cells of K ∗ intersecting with S. Let OK 0 (S) = SK 0 (S) r ∂SK 0 (S) denote the open star. Note that there is a proper deformation retraction SK 0 (S) → S and a deformation retraction OK 0 (S) → S (cf. Proposition B.26). We have: Proposition 4.2. For a simplex s of K, the following three conditions are equivalent: (1) s ⊂ S,

(2) s∗ ∩ S 6= ∅,

(3) s∗ ∩ OK 0 (S) 6= ∅.

Exercise 4.1. Verify the above. Noting that M rOK 0 (S) is a K ∗ -subcomplex of M , we may write p p ∗ CK ∗ (M, M rOK 0 (S)) = { u ∈ CK ∗ (M ) | hs , ui = 0 for s 6⊂ S }. p • They form a subcomplex of CK ∗ (M ). Denoting by HK ∗ (M, M r OK 0 (S)) its cohomology, we have a natural isomorphism: ∼

p ∗ p ηK ∗ : H (M, M rOK 0 (S)) −→ HK ∗ (M, M rOK 0 (S)).

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Note that there is a natural isomorphism H p (M, M rOK 0 (S)) ' H p (M, M rS). p Now in the sum in (4.6), if u is in CK ∗ (M, M rOK 0 (S)), only (m − p)simplices in S appear. Thus P in (4.6) induces an isomorphism ∼ ˘K p A : CK ∗ (M, M rOK 0 (S)) −→ Cm−p (S).

(4.10)

• • Since (CK ∗ (M, M r OK 0 (S)), δ) is a subcomplex of (CK ∗ (M ), δ) and K K ˘ ˘ (C• (S), ∂) is a subcomplex of (C• (M ), ∂), from Lemma 4.2, we see that A is also compatible with boundary and coboundary operators. Thus we have:

Theorem 4.2 (Alexander duality). For a closed set S which is a subcomplex of M with respect to some triangulation, the isomorphism (4.10) induces an isomorphism ∼

˘ m−p (S). AM,S : H p (M, M rS) −→ H Remark 4.4. Noting that SK 0 (S) r OK 0 (S) = ∂SK 0 (S) and that SK 0 (S) and ∂SK 0 (S) are K ∗ -subcomplexes of M , we may identify the cochain group p p CK Then the Alexander ∗ (M, M r OK 0 (S)) with CK ∗ (SK 0 (S), ∂SK 0 (S)). isomorphism is induced from the isomorphism ∼

p ˘K CK ∗ (SK 0 (S), ∂SK 0 (S)) −→ Cm−p (S),

which is the restriction of P in (4.6). We now try to express A in terms of cap product. We set p p CK 0 (M, M rOK 0 (S)) = { u ∈ CK 0 (M ) | ht, ui = 0 for t with t ∩ S = ∅ }.

Then from Lemma 4.1, we have the commutative diagram: p CK ∗ (M, M rOK 0 (S)) O

A

K / C˘m−p (S)

MK 0 a

K / C˘m−p (S),

ι∗ p CK 0 (M, M rOK 0 (S))

0



κ

(4.11)

where ι∗ and κ are defined as in (4.7). Thus A is again represented by the cap product with the cycle M on the level of chains and cochains. To pass on to the level of homology and cohomology, we proceed as follows.

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105

p p First, we may identify CK 0 (M, MrOK 0 (S)) with CK 0 (SK 0 (S), ∂SK 0 (S)) as in the case of K ∗ -cochain groups in Remark 4.4. We have the cap product (cf. (B.39)) 0 a p ˘K0 C˘rK (SK 0 (S), ∂SK 0 (S))×CK 0 (SK 0 (S), ∂SK 0 (S)) −→ Cr−p (SK 0 (S)). (4.12)

With this, the second horizontal morphism in (4.11) is the cap product with the relative cycle SK 0 (S), the sum of m-simplices of K 0 in SK 0 (S), in K0 C˘m (SK 0 (S), ∂SK 0 (S)). Here we note that there is a commutative diagram K0 C˘m (SK 0 (S))

∂

K0 / C˘m−1 (SK 0 (S))

∂

 K0 / C˘m−1 (SK 0 (S), ∂SK 0 (S)).

=

K C˘m

0

 (SK 0 (S), ∂SK 0 (S))

In the above, ∂ denotes the boundary morphism for relative cochain groups and the vertical arrows are the canonical surjections, of which the first one K0 (∂SK 0 (S)) = 0. For the chain SK 0 (S), ∂SK 0 (S) = 0, is the identity, as C˘m 0 while ∂SK (S) 6= 0 so that it is a cycle only as a relative chain. For this reason we denote SK 0 (S) by SK 0 (S) when considered as a relative chain. Noting that p p HK 0 (SK 0 (S), ∂SK 0 (S)) ' H (M, MrS)

˘ r−p (SK 0 (S)) ' H ˘ r−p (S), and H

(4.12) represents the cap product a ˘ r (SK 0 (S), ∂SK 0 (S)) × H p (M, M rS) −→ ˘ r−p (S). H H

Thus we may write A(α) = [SK 0 (S)] a α

for α ∈ H p (M, M rS).

(4.13)

From this we see that the isomorphism A is independent of the triangulation for which S is a subcomplex (cf. (T2 ) in Section 4.1). From the properties of cup and cap products (cf. (B.18), (B.22)), we have: Proposition 4.3. Let M and S be as above and i : S ,→ M the inclusion. For any class γ ∈ H q (M ), the following diagram is commutative: H p (M, M rS) ( )`γ



H r (M, M rS) where r = p + q.

∼ A

∼ A

˘ m−p (S) /H 

( )ai∗ γ

˘ m−r (S), /H

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Complex Analytic Geometry

Remark 4.5. 1. On the level of chains and cochains, the product (4.12) depends on the ordering of the vertices. For the cycle SK 0 (S) and our specific choice of ordering as in Lemma 4.1, the receiving chain group can K0 be made to be C˘m−p (S). 2. The reason we use the relative homology class of SK 0 (S) instead of the homology class of M is as follows. Observe that we have the cap product 0 a p ˘K0 C˘rK (M ) × CK 0 (M, M rOK 0 (S)) −→ Cr−p (SK 0 (S)).

To see this, let t be an r-simplex of K 0 . Then for every ordering (v0 , . . . , vr ) of the vertices of t, if (vp , . . . , vr ) 6⊂ SK 0 (S), then (v0 , . . . , vp ) ⊂ MrOK 0 (S) p so that t a u = 0 for u in CK 0 (M, M rOK 0 (S)). It induces a ˘ r (M ) × H p 0 (M, M rOK 0 (S)) −→ ˘ r−p (S). H H K

The problem is that there is no corresponding cap product in the singular theory. 3. In the case S is compact, SK 0 (S) is also compact. We have ˘ r (SK 0 (S), ∂SK 0 (S)) = Hr (SK 0 (S), ∂SK 0 (S)) ' Hr (M, M rS). H

(4.14)

For the isomorphism above, let X1 = SK 0 (S) and X2 = M rS in (B.9) and notice that SK 0 (S)rS deformation retracts to ∂SK 0 (S). Thus we have the cap product a

Hr (M, M rS) × H p (M, M rS) −→ Hr−p (S) and we may write A(α) = [M ] a α

for α ∈ H p (M, M rS),

where [M ] denotes the class in Hm (M, M rS) that corresponds to the class [SK 0 (S)] in Hm (SK 0 (S), ∂SK 0 (S)) by the isomorphism (4.14). ˘ r (SK 0 (S), ∂SK 0 (S)) makes sense as ∂SK 0 (S) is a 4. Note that, while H ˘ r (M, M rS) is not defined in general, as M rS is closed set on SK 0 (S), H an open set. Dual description: Let K0 , K, K 0 and K ∗ be as before, except here we assume K0 is a barycentric subdivision of a triangulation of M so that OK (S) deformation retracts to S. Noting that M r OK (S) is a sub-K complex of M , the restriction of P 0 in (4.8) gives a morphism p K∗ A0 : CK (M, M rOK (S)) −→ C˘m−p (SK 0 (S)).

(4.15)

Dualities and Thom Class

107

From Lemma 4.3, we have the commutative diagram: p CK (M, M rOK (S)) O

A0

K∗ / C˘m−p (SK 0 (S)) κ0

ι0 ∗

 aMK 0 p K0 ˘ / CK 0 (M, M rOK (S)) Cm−p (SK 0 (S)).

(4.16)

Thus we have an isomorphism ∼ ˘ A0 : H p (M, M rS) −→ H m−p (S).

Remark 4.6. The above isomorphism differs from A by a sign of (−1)p(m−p) (cf. Remark 4.3). Lefschetz duality Let R be a C ∞ manifold of dimension m with boundary ∂R in M . We may assume that R and ∂R are K0 -subcomplexes of M . We apply the above by setting S = R. Since we have canonical isomorphisms H p (M, M rOK 0 (R)) ' H p (SK 0 (R), ∂SK 0 (R)) ' H p (R, ∂R),

(4.17)

we have: Theorem 4.3 (Lefschetz duality). For a manifold R of dimension m with boundary ∂R in M , we have an isomorphism ∼

˘ m−p (R). L : H p (R, ∂R) −→ H Note that the above isomorphism is given by the cap product with ˘ m (R, ∂R). It is the class corresponding to the relative class [R] of R in H ˘ ˘ m (SK 0 (R), ∂SK 0 (R)). [SK 0 (R)] in the isomorphism Hm (R, ∂R) ' H Remark 4.7. The above theorem holds if R is only piecewise C ∞ (cf. Definition 5.10 below). From the construction we have: Proposition 4.4. For a subcomplex S in the interior of R, we have the commutative diagram: H p (M, M rS)

j∗

o A



˘ m−p (S) H

/ H p (R, ∂R)

j 0∗

o L ı∗



˘ m−p (R) /H

/ H p (M ) o P

ı0∗

 ˘ m−p (M ), /H

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Complex Analytic Geometry

where j : (R, ∂R) ,→ (M, M r S), ı : S ,→ R and ı0 : R ,→ M denote ∗ the inclusions and j 0 denotes the isomorphism in (4.17) followed by the canonical morphism H p (M, M rOK 0 (R)) → H p (M ). Suppose S has a finite number of connected components (Sλ )λ . Then L ˘ p ˘ Hm−p (S) = λ H m−p (Sλ ) and each class α in H (M, M r S) determines ˘ a class aλ in Hm−p (Sλ ) via the Alexander isomorphism. Corollary 4.3. In the above situation, X (ıλ )∗ aλ = [R] a j ∗ α

˘ m−p (R), in H

λ

where ıλ : Sλ ,→ R denotes the inclusion. In many cases we do not consider the auxiliary manifold R and take M itself as R. In this situation we have the commutative diagram j∗

H p (M, M rS)

/ H p (M ) o P

o A



ı∗

˘ m−p (S) H



(4.18)

˘ m−p (M ) /H

and the right-hand side of the identity in Corollary 4.3 is written [M ] a j ∗ α ˘ m−p (M ). in H The above is the basis of the “residue theorem” we discuss in later sections (cf. Theorems 5.2, 10.1 and 10.6). Remark 4.8. Let M and S be as above. The isomorphism (4.6) induces an isomorphism ∼ ˘K p CK ∗ (M rOK 0 (S)) −→ Cm−p (M, S),

which in turn induces another Alexander duality ∼

˘ m−p (M, S). H p (M rS) −→ H The exact sequence δ∗

j∗

· · · → H p−1 (M rS) → H p (M, M rS) → H p (M ) → H p (M rS) → · · · is dual to ∂∗ ˘ ı∗ ˘ ˘ m−p+1 (M, S) → ˘ m−p (M, S) → · · · . ··· → H Hm−p (S) → Hm−p (M ) → H

Dualities and Thom Class

109

Push-forward in cohomology: Let f : M → M 0 be a map of oriented manifolds M and M 0 of dimensions m and m0 , respectively. We define a 0 morphism f∗ : H m−p (M ) → H m −p (M 0 ) so that the following diagram is commutative: H m−p (M ) f∗

H

4.3

P

˘ p (M ) /H f∗



m0 −p

∼

0

(M )

∼ P

 ˘ p (M 0 ). /H

(4.19)

Thom isomorphism and Thom class

Thom class of a submanifold Let M be a C ∞ manifold of dimension m and V a closed submanifold of dimension d0 . We set k 0 = m − d0 . We take a triangulation K0 of M compatible with V and let K and K 0 be as in the previous section. We denote by KV the triangulation of V induced from K and by KV0 its barycentric subdivision. Then KV0 is the set of simplices of K 0 that are in V . We denote by K ∗ and KV∗ the cellular decompositions of M and V dual to K and KV , respectively (cf. Section 4.1). Note that, for a p-simplex s of K in V , its dual s∗ in K ∗ is an (m − p)-cell and that its dual s∗V in KV∗ is a (d0 − p)-cell. They are related by s∗V = s∗ ∩ V as sets. In the case M and V are orientable, there is a more precise formula (cf. Lemma 4.5 below). The simplices and cells of K, K 0 and K ∗ are oriented so that the conditions (1) and (2) in Section 4.1 are satisfied. The simplices of KV and KV0 are oriented as simplices of K and K 0 , respectively. In order to describe the homology and cohomology of V , we impose similar conditions for simplices and cells of KV , KV0 and KV∗ . The condition corresponding to (1) is automatically satisfied. Thus we impose: ∗

(2)V Let t be a p-simplex of KV0 . If t ⊂ s0 V , a p-cell of KV∗ , the orientation ∗ of t is the same as that of s0 V . For a simplex s in KV we would like to relate the orientation of s∗ and that of s∗V . For this we further impose the following condition. To state it, let s be an (m − p)-simplex of K in V . We take a d0 -simplex s1 of KV so that s ≺ s1 . Then there exist a p-simplex t of K 0 in s∗ , a k 0 -simplex t1 of K 0 in s∗1 and a (p − k 0 )-simplex tV of KV0 in s∗V such that t1 and tV span

110

Complex Analytic Geometry

t. Note that the simplices t, t1 and tV have the same orientations as s∗ , s∗1 and s∗V , respectively (cf. (2) and (2)V ). (5) The simplices and cells are oriented so that the orientation of s∗1 followed by the orientation of s∗V gives the orientation of s∗ . Note that, by the tubular neighborhood theorem (Theorem 3.10), the above can be done consistently with other conditions, in particular independently of the choice of s1 , if the normal bundle NR,V of V in M is orientable (cf. Sections 3.4 and 3.7). In this case we may think of s∗1 as being in the fiber direction and we orient the normal bundle NR,V accordingly. Note that the above condition (5) is consistent with Convention 3.3. Now we consider the morphism 0

p k +p TV : CK (M, M rOK 0 (V )), ∗ (V ) −→ CK ∗ V

uV 7→ u,

where u is given by, for each (d0 − p)-simplex s of K (cf. Proposition 4.2), ( if s ⊂ V, hs∗V , uV i (4.20) hs∗ , ui = 0 otherwise. It is clearly an isomorphism. Moreover, with the assumption that the bundle NR,V is oriented as the condition (5), we have 0

δ ◦ TV = (−1)k TV ◦ δ

(4.21)

and thus the following: Theorem 4.4. If the normal bundle of V in M is oriented, the above TV induces an isomorphism ∼

0

TV : H p (V ) −→ H k +p (M, M rV ). We call TV the Thom isomorphism for V . In the following, we assume that the normal bundle NR,V is oriented as above. Definition 4.2. The Thom class of V in M , denoted by ΨM,V or simply by ΨV , is the image of [1] in H 0 (V ) by TV : 0

ΨV = TV ([1]) ∈ H k (M, M rV ), where [1] denotes the class of the cocycle that assigns 1 to each 0-cell (cf. Remark B.9).

Dualities and Thom Class

111

From the definition we have: Proposition 4.5. The Thom class ΨV is represented by a cocycle that as∗ signs 1 or 0 to each oriented k 0 -cell s∗ forming a basis of CkK0 (M ) according as s∗ intersects with V or not. 0

Remark 4.9. 1. The classes in H k (M, MrV ) may also be represented by k0 cocycles in CK 0 (M, M rOK 0 (V )). If ψ is a cocycle representing the Thom class, we have ( 1 if s ⊂ V, ∗ ∗ ∗ hιs , ψi = hs , ι ψi = 0 otherwise, ∗

0

where ι : CkK0 (M ) → CkK0 (M ) denotes a natural monomorphism and ι∗ : k0 k0 CK 0 (M, M rOK 0 (V )) → CK ∗ (M, M rOK 0 (V )) a natural epimorphism. 2. In the above, we do not assume that M or V to be orientable. In the case they are, we have the dulalities on M and V and we have the commutative diagrams as in Propositions 4.9 below. Thom class of an oriented vector bundle Let π : E → M be a C ∞ oriented real vector bundle of rank l0 . We denote by Σ the image of the zero section s0 : M → E. Note that s0 is a diffeomorphism of M onto Σ and that s∗0 NR,Σ = E (cf. Exercise 3.10). We apply the above considerations by letting M , V and k 0 be E, Σ and l0 , respectively. We also replace K with KE . Thus KE is a triangulation of E compatible with Σ. The Thom isomorphism ∼

0

TE : H p (M ) −→ H l +p (E, E rΣ) for E is defined by TE = TΣ ◦ (s∗0 )−1 so that the diagram H p (M ) O o s∗ 0

/ H l0 +p (E, E rΣ) 6

∼ TE

∼

TΣ

H p (Σ). is commutative. The Thom class ΨE of E is defined by ΨE = TE ([1]), which is in fact the Thom class ΨΣ of Σ. The Thom isomorphism may also be written (cf. (4.25), Remark 4.11 and (7.69) below) TE (α) = ΨE ` π ∗ (α)

for α ∈ H p (M ).

(4.22)

112

Complex Analytic Geometry

Rephrasing Proposition 4.5, we have: 0

Proposition 4.6. The Thom class ΨE ∈ H l (E, ErΣ) is represented by a cocycle that assigns 1 or 0 to the dual l0 -cell s∗ in E of each m-simplex s in KE according as s is in Σ or not. 0

Example 4.1. If M is a point, then E = Rl and we have ∼

0

0

0

TRl0 : H 0 ({0}) −→ H l (Rl , Rl r0), the both sides being canonically isomorphic with Z. The Thom class ΨRl0 0 0 0 is the canonical generator of H l (Rl , Rl r 0) (cf. Example B.6. 2 and Remark B.13. 2) and is represented by the cocycle assigning the value 1 to the 0 dual cell in Rl of the vertex 0 and the value 0 to the other l0 -cells. The relation between the Thom class of E and that of each fiber is given as follows. For each point x in M , let ix : (Ex , Ex r0) ,→ (E, ErΣ) be the inclusion. We have the commutative diagram: 0

H l −1 (Ex r0) O

δ∗

/ H l0 (Ex , Ex r0) O

i∗ x

/ H l0 −1 (E rΣ)

i∗ x δ

∗

/ H l0 (E, E rΣ)

j∗

/ H l0 (E) O

/

(4.23)

o π∗ 0

H l (M ). Note that δ ∗ in the first row is an epimorphism for l0 = 1 and an isomorphism for l0 > 1. From the above description, we have: 0

Proposition 4.7. A class Ψ in H l (E, ErΣ) coincides with ΨE if and only if i∗x Ψ = ΨEx for all x in M . We see below (cf. (5.21)) that (π ∗ )−1 j ∗ ΨE = e(E), the Euler class of E. Pull-back: Let f ∗ : M 0 → M be a C ∞ map of C ∞ manifolds. Then the pull-back bundle f ∗ E (cf. (3.4)) is orientable and has a natural orientation. The image of its zero section is given by (f˜)−1 Σ. We have the Thom class 0 Ψf ∗ E in H l (f ∗ E, f ∗ E r(f˜)−1 Σ). There is a canonical morphism 0 0 (f˜)∗ : H l (E, E rΣ) −→ H l (f ∗ E, f ∗ E r(f˜)−1 Σ). The following is a consequence of Proposition 4.7: Proposition 4.8. In the above situation, we have: Ψf ∗ E = (f˜)∗ ΨE .

Dualities and Thom Class

113

Remark 4.10. 1. Let M be a C ∞ manifold and V a closed submanifold of M with the normal bundle p : NR,V → V . By the tubular neighborhood theorem (Theorem 3.10), there exist a neighborhood U of V in M , a neighborhood W of the image Z of the zero section in NR,V and a diffeomorphism τ : (U, V ) → (W, Z) with (p ◦ τ )|V = 1V . We have a composition of isomorphisms ∼

H p (NR,V , NR,V rZ) ' H p (W, W rZ) −→ H p (U, U rV ) ' H p (M, M rV ). ∗ τ

If NR,V is oriented, the Thom class ΨNR,V corresponds to ΨV . 2. We discuss the Thom isomorphism and the Thom class in terms of differential forms in Section 7.9 below, where they are treated in cohomology with C-coefficients. Product formula Suppose we have two pairs of topological spaces (X, A) and (Y, B) with A and B open sets in X and Y , respectively. We denote by (X, A) × (Y, B) the pair (X × Y, (A × Y ) ∪ (X × B)). For example, 0

0

0

0

0

0

l0 = l10 + l20 .

(Rl1 , Rl1 r0) × (Rl2 , Rl2 r0) = (Rl , Rl r0), We have the cup product (cf. Appendix B) `

H p (X × Y, A × Y ) × H q (X × Y, X × B) −→ H p+q ((X, A) × (Y, B)). Letting p1 : (X × Y, A × Y ) −→ (X, A)

and p2 : (X × Y, X × B) −→ (Y, B)

be the projections, we define the cross product ×

H p (X, A)×H q (Y, B) −→ H p+q ((X, A)×(Y, B)) by α×β = p∗1 (α) ` p∗2 (β). Recall that H 1 (R, R r 0) has a canonical generator e. We denote by 0 0 0 e ∈ H l (Rl , Rl r0) the l0 -fold cross product e × · · · × e. We quote: l0

Theorem 4.5. For any pair (X, A) with A open in X, the assignment 0 α 7→ α × el defines an isomorphism ∼

0

0

0

H p (X, A) −→ H p+l ((X, A) × (Rl , Rl r0)). From this, we have (cf. Example 4.1): 0

0

Corollary 4.4. 1. el = ΨRl0 , the Thom class of Rl . 2. ΨRl01 × ΨRl02 = ΨRl0 , l0 = l10 + l20 .

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Complex Analytic Geometry

For i = 1, 2, let Ei be a C ∞ real oriented vector bundle of rank li0 on M . Letting Σi be (the image of) the zero section of Ei , we have the Thom class 0 ΨEi in H li (Ei , Ei rΣi ). Noting that the zero section of the product bundle E1 × E2 (cf. Section 3.3) is ((E1 r Σ1 ) × E2 ) ∪ (E1 × (E2 r Σ2 )), we have 0 the Thom class ΨE1 ×E2 in H l ((E1 , E1 rΣ1 ) × (E2 , E2 rΣ2 )), l0 = l10 + l20 . Theorem 4.6. In the above situation, ΨE1 ×E2 = ΨE1 × ΨE2 . Proof.

This follows from Proposition 4.7 and Corollary 4.4. 2.



Let Ei and Σi be as above, i = 1, 2. We set E = E1 ⊕ E2 and let Σ be the zero section of E. Letting d : M ,→ M × M be the diagonal embedding, we identify E with d∗ (E1 × E2 ) by (3.8). Then we have the commutative diagram E  M

d˜

d

/ E1 × E2  / M × M.

We have a canonical morphism 0

0

˜ ∗ : H l ((E1 , E1 rΣ1 ) × (E2 , E2 rΣ2 )) −→ H l (E, E rΣ). (d) From Theorem 4.6 and Proposition 4.8, we have Corollary 4.5. We have ˜ ∗ (ΨE × ΨE ). ΨE = (d) 1 2 We discuss more on this subject in Sections 5.6, 8.3, 10.3, 10.4 and 14.1 below. Poincar´ e, Alexander and Thom isomorphisms Let M be a C ∞ manifold of dimension m and V a closed submanifold of dimension d0 , as above. We assume that M and V are oriented. In order to describe the dualities for M we impose the conditions (1), (2) in Section 4.1 and (3), (4) in Section 4.2. We also impose the corresponding conditions for the simplices and cells of KV , KV0 and KV∗ . Note that the one corresponding to (1) is already satisfied. Besides (2)V we further impose: (3)V The orientation of each d0 -simplex of KV is the same as that of V .

Dualities and Thom Class

115

(4)V For every p-simplex s, 0 < p < d0 , of KV , the orientation of s∗V followed by the orientation of s gives the orientation of V . Since we assumed that M and V to be oriented, the normal bundle NR,V is orientable (cf. Proposition 3.13) and is oriented according to Convention 3.4. The total space of NR,V is then oriented according to Convention 3.3. By identifying neighborhoods of V in M and NR,V by the tubular neighborhood theorem, we may rephrase the above conventions as (cf. Remark 3.8. 1): Convention 4.1. We orient the bundle NR,V so that the orientation of the fiber of NR,V followed by that of V gives the orientation of M . We also impose the condition (5), which is consistent with the above convention. Then the following diagram is commutative: 0 ∼ / C k ∗+p (M, M rOK 0 (V )) C p ∗ (V ) KV

K

T

o P



C˘dK0 −p (V ).

∼

v

A

Thus we have: Proposition 4.9. In the case M and V are oriented, we have the commutative diagram: ∼ / H k0 +p (M, M rV ) H p (V ) T

o P



˘ d0 −p (V ). H

v

∼

A

From the above, we see that A(ΨV ) = [V ], the fundamental class of V . Note that there is the cup product `

0

(4.24)

0

H k (M, M rV ) × H p (M ) −→ H k +p (M, M rV ). Denoting by i : V ,→ M the inclusion, we have: Proposition 4.10. The following diagram is commutative: H p (M )

ΨV `( )

/ H k0 +p (M, M rV ) o AM,V

i∗

 H p (V )

∼ PV

 ˘ d0 −p (V ), /H

116

Complex Analytic Geometry

Proof.

For a class α ∈ H p (M ), by (4.13), (B.18) and (B.22),

AM,V (ΨV ` α) = [SK 0 (V )] a (ΨV ` α) = ([SK 0 (V )] a ΨV ) a i∗ α = [V ] a i∗ α = PV (i∗ α).



Combining with Proposition 4.9, we have: Corollary 4.6. For a class α in H p (M ), we have TV (i∗ α) = ΨV ` α. Suppose U is a neighborhood of V in M with a deformation retraction r : U → V , for example, the open star OK 0 (V ) of V in K 0 or a tubular neighborhood of V (cf. Theorem 3.10). Then we have isomorphisms 0 0 ∼ H k +p (M, M rV ) ' H k +p (U, U rV ) (excision) and r∗ : H p (V ) → H p (U ). From Corollary 4.6, we have: TV (αV ) = ΨV ` r∗ (αV )

for αV ∈ H p (V ).

(4.25)

Remark 4.11. In fact, the identity in (4.25) holds without the assumption that M and V are orientable. If S is a K0 -subcomplex of M , setting Z = S ∩ V , we have the cup product `

0

0

H k (M, M rV ) × H p (M, M rS) −→ H k +p (M, M rZ). Proposition 4.11. The following diagram is commutative: H p (M, M rS)

ΨV `( )

/ H k0 +p (M, M rZ) o AM,Z

i∗

 H p (V, V rZ) Proof.

∼ AV,Z

 ˘ d0 −p (Z). /H

For a class α ∈ H p (M, M rS), by (4.13), (B.18) and (B.22),

AM,Z (ΨV ` α) = [SK 0 (Z)] a (ΨV ` α) = ([SK 0 (Z)] a ΨV ) a i∗ α. Thus the proposition follows if we show [SK 0 (Z)] a ΨV = [SKV0 (Z)]. For 0

k this, if we let ψ be a cocycle in CK 0 (M, MrOK 0 (V )) representing the Thom class, the class [SK 0 (Z)] a ΨV is represented by the cycle X t a ψ, (4.26) t⊂SK 0 (Z)

Dualities and Thom Class

117

where the sum is taken over m-simplices t of K 0 in SK 0 (Z). For such a simplex t, we write t = ε(v0 , . . . , vk0 , . . . , vm ) and set t1 = ε1 (v0 , . . . , vk0 ) and t2 = ε2 (vk0 , . . . , vm ), where ε, ε1 and ε2 are ±1 and the sign is to be determined accoding to the prescribed orientation. In the above, t2 is in a d0 -simplex s of K, t1 is in its dual k 0 -cell s∗ and vm ∈ Z. Then the cycle in (4.26) is written X hιs∗ , ψi t2 , s

where the sum is taken over d0 -simplices s of K that intersect with Z and d0 -simplices t2 of K 0 in s that intersect with Z. By Remark 4.9. 1, it is P  equal to s⊂V t2 and represents the class [SKV0 (Z)]. Gysin morphism: In the above situation, let i : V ,→ M denote the inclusion. The composition j∗

0

T

0

V H p (V ) −→ H k +p (M, M rV ) −→ H k +p (M )

is referred to as the Gysin morphism and denoted by i∗ so that we have the commutative diagram (cf. (4.18)) H p (V )

i∗

o PV



˘ d0 −p (V ) H 4.4

/ H k0 +p (M ) o PM

i∗



(4.27)

˘ d0 −p (M ). /H

Intersection product

Let M be an oriented and connected C ∞ manifold of dimension m. We take a triangulation K0 of M and let K, K 0 and K ∗ be as in Section 4.1. Intersection product of a dual cell and a simplex Let s1 be an oriented (m − r)-simplex and s2 an oriented s-simplex of K. We denote by ar and al the right and left cap products of chains and cochains of K 0 (cf. Appendix B).

·

Definition 4.3. The intersection product s∗1 s2 is an (r + s − m)-chain of K 0 defined by

·

s∗1 s2 = (ϑ(t1 ) ar MK 0 ) al ϑ(t2 ), where t1 and t2 are an (m − r)-simplex and an (m − s)-simplex of K 0 , respectively, such that t1 ⊂ s1 and t2 ⊂ s∗2 .

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Complex Analytic Geometry

Note that we may also write (cf. (B.26))

·

s∗1 s2 = ϑ(t1 ) ar (MK 0 al ϑ(t2 )). By Lemmas 4.3 and 4.1, ϑ(t1 ) ar MK 0 = s∗1

and MK 0 al ϑ(t2 ) = s2

(4.28)

as K 0 -chains. Thus we see that the definition does not depend on the choice of t1 or t2 . We examine the above definition more closely. First note that we may write t2 = ε2 (v0 , . . . , vm−s ), vm−s = bs2 (cf. Remark 4.1). We have X t al ϑ(t2 ), (4.29) s∗1 s2 = s∗1 al ϑ(t2 ) =

·

t2 ≺t⊂s∗ 1

where the sum is taken over the r-simplices t of K 0 in s∗1 containing t2 as a face. An r-simplex t of K 0 in s∗1 may be written t = ε(v00 , . . . , vr0 ), vr0 = bs1 . 0 ) = (v0 , . . . , vm−s ) and in The condition t2 ≺ t is given by (v00 , . . . , vm−s 0 this case, t al ϑ(t2 ) = εε2 (vm−s , . . . , vr ) = εε2 (bs2 , . . . , bs1 ). Thus X 0 0 0 0 s∗1 s2 = εε2 (vm−s , vm−s+1 , . . . , vr−1 , vr0 ), vm−s = bs2 , vr0 = bs1 .

·

t2 ≺t⊂s∗ 1

·

This shows that s∗1 s2 is non-zero if and only if s1 ≺ s2 .

·

Remark 4.12. 1. The support of s∗1 s2 is the set theoretical intersection of s∗1 and s2 . ∗

2. We may write P 0 α0 ϑ(t1 ) = s∗1 and P α∗ ϑ(t2 ) = s2 (cf. (4.9), (4.7) and (4.28)). Exercise 4.2. Show that in the case s1 = s2 = s, we have

·

s∗ s = bs . Note that ∂s∗1 is a linear combination of cells of K ∗ and that ∂s2 is a linear combination of simplices of K. Lemma 4.4. We have

·

·

·

∂(s∗1 s2 ) = (−1)m−s (∂s∗1 ) s2 + s∗1 ∂s2 . Proof.

By the first identity in (4.29) and (B.17),

·

∂(s∗1 s2 ) = (−1)m−s ((∂s∗1 ) al ϑ(t2 ) − s∗1 al δϑ(t2 ))

·

= (−1)m−s ((∂s∗1 ) s2 − s∗1 al δϑ(t2 )).

Dualities and Thom Class

By (B.33), we have X s∗1 al δϑ(t2 ) =

119

t al δϑ(t2 )

t2 ≺t⊂s∗ 1

= (−1)m−s+1

X

0 0 εε2 (vm−s+1 , . . . , vr−1 , vr0 ),

vr0 = bs1 ,

t2 ≺t⊂s∗ 1

(4.30) where we use the expressions t2 = ε2 (v0 , . . . , vm−s ), vm−s = bs2 , and t = ε(v00 , . . . , vr0 ), vr0 = bs1 , as above. We may write s X (i) ∂s2 = ε(i) s2 , i=0 (i)

where, for each i, s2 is an (s − 1)-simplex of K and ε(i) is to be determined (i) (i) ∗ accoding to the prescribed orientation. We may take t2 ⊂ s2 so that (i) (i) (i) (i) (i) t2 ≺ t2 . Thus we may write t2 = ε2 (v0 , . . . , vm−s , vm−s+1 ), vm−s+1 = bs(i) . 2 0 0 Note that, since vm−s = vm−s = bs2 , in (4.30), vm−s+1 = bs(i) for some 2

(i)

i with t2 ≺ t. On the other hand we have s s X X (i) ∗ ε(i) (s∗1 s2 ) = ε(i) s1 ∂s2 =

·

·

i=0

=

s X i=0

i=0

ε(i)

X

(i)

X

t al ϑ(t2 )

(i)

t2 ≺t⊂s∗ 1

(i)

0 0 εε2 (vm−s+1 , . . . , vr0 ), vm−s+1 = bs(i) , vr0 = bs1 . 2

(i)

t2 ≺t⊂s∗ 1 (i)

Note that the first sum is taken for i such that s1 ≺ s2 . (i) Now we claim that εi ε2 = ε2 for each i, which will prove the lemma. We take an s-simplex t0 of K 0 in s of the form t0 = 0 ε (vm−s , vm−s+1 , wm−s+2 , . . . , wm ). Then ε0 ε(i) (vm−s+1 , wm−s+2 , . . . , wm ) (i) is an (s − 1)-simplex of K 0 in s2 . By our orientation convention (*), (i) ε2 ε0 = ε2 ε0 ε(i) .  Corollary 4.7. For a locally finite r-chain c∗ of K ∗ and a locally finite s-chain c of K, ∂(c∗ c) = (−1)m−s (∂c∗ ) c + c∗ ∂c.

·

·

·

∗

Noting that the intersection of an r-chain of K and an s-chain of K is an (r + s − m)-chain of K 0 , we have a bilinear map

·

∗ K0 (M ). C˘rK (M ) × C˘sK (M ) −→ C˘r+s−m

120

Complex Analytic Geometry

By Corollary 4.7, it induces the intersection product in homology:

·

˘ r+s−m (M ). ˘ r (M ) × H ˘ s (M ) −→ H H Recall that we have the Poincar´e duality (Theorem 4.1): ∼ ˘ PM : H p (M ) −→ H m−p (M ).

In the following, we denote PM by P if there is no fear of confusion. By Definition 4.3 and Remark 4.12. 2, we have the following: Theorem 4.7. The following diagram is commutative: H p (M ) × H q (M ) P ×P

o

 ˘ ˘ s (M ) Hr (M ) × H

`

/ H p+q (M ) o P

 ˘ r+s−m (M ), /H

·

where p + r = m and q + s = m. Thus the intersection product may also be written

·

a b = P (P −1 a ` P −1 b),

˘ r (M ), b ∈ H ˘ s (M ). a∈H

From the properties of the cup product, we see that

·

·

a b = (−1)(m−r)(m−s) b a.

(4.31)

Since P is given by the cap product with the fundamental calss [M ], we may also write

·

a b = [M ] a (P −1 a ` P −1 b) = a a P −1 b.

(4.32)

˘ 0 (M ) = H0 (M ) ' Z, the In the case M is compact and connected, H isomorphism given by the augmentation. Thus if r + s = m, we may think of a b as an integer. We also have, for three homology classes a, b and c,

·

· ·

· ·

(a b) c = a (b c)

(4.33)

Localized intersection product Let S1 and S2 be subcomplexes of M and set S = S1 ∩ S2 . Then we have the intersection product

·

∗ K0 (SK 0 (S)). C˘rK (SK 0 (S1 )) × C˘sK (S2 ) −→ C˘r+s−m

Dualities and Thom Class

121

By Corollary 4.7, it induces the localized intersection product in homology:

·

˘ r+s−m (S). ˘ r (S1 ) × H ˘ s (S2 ) ( −→)S H H

(4.34)

In general, for a K0 -subcomplex S of M , we have the Alexander duality (Theorem 4.2): ∼ ˘ AM,S : H p (M, M rS) −→ H m−p (S). Let S1 and S2 be subcomplexes of M and set S = S1 ∩ S2 as before. Then we have the cup product `

H p (M, M rS1 ) × H q (M, M rS2 ) −→ H p+q (M, M rS). As Theorem 4.7, we have: Theorem 4.8. The following diagram is commutative: H p (M, M rS1 ) × H q (M, M rS2 ) A1 ×A2 o

 ˘ r (S1 ) × H ˘ s (S2 ) H

(

·

`

/ H p+q (M, M rS) o A

)S

 ˘ r+s−m (S), /H

where p + r = m, q + s = m and A, A1 and A2 denote the Alexander isomorphisms for S, S1 and S2 . Thus the localized intersection product may also be written ˘ r (S1 ), b ∈ H ˘ s (S2 ). a∈H (a b)S = A(A−1 a ` A−1 b),

·

1

2

˘ 0 (S) = H0 (S) ' Z. Thus if In the case S is compact and connected, H r + s = m we may think of (a b)S as an integer. We also have formulas corresponding to (4.31) and (4.33). In general from definition we have:

·

Proposition 4.12. Denoting by ı1 : S1 ,→ M , ı2 : S2 ,→ M and ı : S ,→ M the inclusions, ˘ r+s−m (M ). (ı1 )∗ a (ı2 )∗ b = ı∗ ((a b)S ) in H

·

·

Remark 4.13. The diagrams in Theorems 4.7 and 4.8 are compatible via the diagram (4.18). Suppose S has a finite number of connected components (Sλ )λ . Then ˘ r+s−m (S) = L H ˘ H λ r+s−m (Sλ ) and the class (a b)S determines a class ˘ r+s−m (Sλ ) for each λ. We have (a b)Sλ in H X ˘ r+s−m (S), (a b)S = (iλ )∗ (a b)Sλ in H (4.35)

·

·

·

·

λ

where iλ : Sλ ,→ S denotes the inclusion.

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Complex Analytic Geometry

Intersection product with a submanifold Let V be a closed submanifold of dimension d0 of M with i : V ,→ M the inclusion. We assume that V is oriented and that the normal bundle of V is orientable and is oriented as in Convention 4.1. We set k 0 = m − d0 . We may assume that V is a K0 -subcomplex of M . ˘ r (M ) is Convention 4.2. Let c be an r-cycle in M . The class of c in H ˘ r (S) is simply denoted by c, where denote by [c], while the class of c in H S = |c| is the support of c. ˘ d0 (M ) is denoted by [V ], while that in H ˘ d0 (V ) Thus the class of V in H by V . We consider the intersection of V and a homology class in M or in a subcomplex S of M . From the definition of the intersection product and various orientation conventions, we have (cf. the first paragraph of Section 4.3): Lemma 4.5. If we think of V as a d0 -chain of K, for every simplex s of KV , we have

·

s∗V = s∗ V as KV0 -chains. Exercise 4.3. Verify the above. First localization: Letting S1 and S2 be V and M , respectively, in (4.34), we have the intersection product localized at V :

·

˘ r+s−m (V ). ˘ r (V ) × H ˘ s (M ) ( −→)V H H ˘ m−p (M ), we have the product (V Thus, for a class a in H ˘ d0 −p (V ) → H ˘ d0 −p (M ). It is sent to [V ] a by i∗ : H

·

· a)

V

˘ d0 −p (V ). in H

Second localization: Let S be a subcomplex of M with respect to K0 and set Z = V ∩ S. Letting S1 and S2 be V and S, respectively, in (4.34), we have the intersection product localized at Z:

·

˘ r+s−m (Z). ˘ r (V ) × H ˘ s (S) ( −→)Z H H

·

˘ m−p (S), we have the product (V a)Z in H ˘ d0 −p (Z). Thus, for a class a in H If Z has a finite number of connected components, we have the product componentwise and a formula as (4.35).

Dualities and Thom Class

123

The following proposition follows from Propositions 4.10, 4.11, Theorems 4.7 and 4.8: Proposition 4.13. The following diagrams are commutative: H p (M )

∼ PM

˘ m−p (M ) /H

i∗

 H p (V )

∼ PV



(V

·

H p (M, M rS) )V

˘ d0 −p (V ), /H

∼ AM,S

˘ m−p (S) /H

i∗

 H p (V, V rZ)

∼ AV,Z

(V



·

)Z

˘ d0 −p (Z). /H

Remark 4.14. The two diagrams above are compatible via the diagram (4.18) applied for the pairs (M, S) and (V, Z). Intersection product with a map In view of the above, we define intersection products in a more general situation where V is not necessarily a submanifold of M . Thus let V be an oriented C ∞ manifold of dimension d0 and f : V → M a C ∞ map.

·

Definition 4.4. We define the intersection product V f so that the first diagram below is commutative. Also, for a subcomplex S in M , we set Z = f −1 (S). Suppose Z is a subcomplex with respect to some triangulation of V . We then define the localized intersection product (V f )Z so that the second diagram is commutative:

·

H p (M )

∼ PM

˘ m−p (M ) /H

f∗

 H p (V )

∼ PV



(V

·

f

˘ d0 −p (V ), /H

H p (M, M rS) )V

∼ AM,S

f∗

 H p (V, V rZ)

∼ AV,Z

˘ m−p (S) /H 

(V

·

f

)Z

˘ d0 −p (Z). /H

·

˘ m−p (S), we have the product (V f a)Z in Thus, for a class a in H ˘ Hd0 −p (Z). If Z has a finite number of connected components, we have the product componentwise and a formula as (4.35). Remark 4.15. 1. The two diagrams above are compatible via the diagram (4.18) applied for the pairs (M, S) and (V, Z). 2. If V is a submanifold of M and if f = i : V ,→ M is the inclusion, (V i )V and (V i )Z coincide with (V )V and (V )Z , respectively, defined before.

·

·

·

·

124

Complex Analytic Geometry

3. The above products, in the case V or Z is compact, are expressed using differential forms in Section 7.8 below (cf. Propositions 7.13 and 7.20). 4. See Section 7.11 below for an application. Notes For C ∞ triangulations, we refer to [Munkres (1966)]. The statements (T1 ) and (T2 ) in Section 4.1 are due to [Whitehead (1940)]. As to the dualities, we followed the descriptions in [Brasselet (1981)], except for the orientation convention, see also [Suwa (2008)]. In fact there are four ways to present the duality combinatorially. Thus let M be an oriented C ∞ manifold and let S, K and K ∗ be as above. We also denote p p ∗ and CK by CK ∗ the groups of p-cochains of K and K , respectively, and K∗ K ˘ ˘ by Cm−p and Cm−p the groups of locally finite (m − p)-chains of K and K ∗ , respectively. For the duality morphisms (in fact isomorphisms) there are two possibilities: (i) (ii)

∗

p K P : CK (M ) −→ C˘m−p (M ), p K ˘ P : CK ∗ (M ) −→ Cm−p (M ),

∗

p K (SK 0 (S)), A : CK (M, M rS) −→ C˘m−p p K ˘ A : CK ∗ (M, M rS) −→ Cm−p (S).

The morphisms in (ii) are the ones in Section 4.2 above and the ones in (i) are their “duals”. For orientation conventions, denoting by s a simplex in K, there are two possibilities: (a) orientation of s followed by that of s∗ gives the orientation of M , (b) orientation of s∗ followed by that of s gives the orientation of M . The convention (b) is the one we adopted (cf. (4) in Section 4.2). Thus there are four possible combinations (ia), (ib), (iia) and (iib). The convention (ia) is the one adopted in most of the literature, where usually only P is defined. For the definition of A in (i) see Section 7.2 of [Suwa (2008)]. The one in [Brasselet (1981)] is (iia) and ours is (iib). The convention (ii) for the morphism is particularly relevant for A, since the chains involved are in S and this is essential in the localization problem we later deal with. While in (i), the chains are in a neighborhood of S, which deformation retracts to S, and are a little obscured. The reason we use (b) instead of (a) is that if s is in S, s∗ usualy appears in a direction normal to S and we wished to make the convention compatible with Convention 4.1.

Dualities and Thom Class

125

On the level of (co)homology, the duality is given by the left cap product with [M ] in the conventions (ia) and (iib), while it is given by the right cap product with [M ] in (ib) and (iia) (cf. Remark B.11). The difference between them is a sign of (−1)p(m−p) . For the paragraph on the product formula of Thom classes, in particular, Theorem 4.5, we refer to [Milnor and Stasheff (1974)]. We also refer to this for Remark 4.11.

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Chapter 5

Chern Classes and Localization via Obstruction Theory

Characteristic classes of a fiber bundle are invariants of the bundle that measure to what extent the bundle differs from the trivial bundle. There are several ways to define them. In this chapter, we review the Chern classes of complex vector bundles and the Euler class of oriented real vector bundles on cell complexes from the obstruction theoretical viewpoint. In this framework, characteristic classes are obstructions to constructing certain frames of the bundle, the obstruction being given as the mapping degree of the frame on the boundary of each cell. We start with discussing the index of a family of sections at its singularity. We then explain how the Chern classes of a complex vector bundle are defined. A similar construction applies to the Euler class of an oriented real vector bundle. We also discuss the case of relative classes, where a frame is already given on a certain subset. This leads to the localization of characteristic classes by frames. After introducing piecewise C ∞ manifolds and pseudo-manifolds, we discuss the topological localization of Chern classes by a frame and the associated residues that arise via the Alexander duality. The reidue theorem in this setting is stated (Theorem 5.2) and the residues are expressed explicitly (Theorem 5.3). We make similar considerations for the Euler class. The Poincar´e-Hopf theorem (Corollary 5.4) naturally comes up in this setting. In fact, it is a prototype of the residue theorem. We prove that the Thom class is a localized Euler class (Theorem 5.7). For basic materials on algebraic topology, we refer to Appendix B and the references therein. Throughout this chapter we take Z as the coefficient ring of homology and cohomology. Also maps, vector bundles and sections are all continuous, unless otherwise stated.

127

128

5.1

Complex Analytic Geometry

Index of a family of sections

Let X be a topological space and E a complex vector bundle of rank l on X. Definition 5.1. An r-section on a subset A of X is an ordered family s(r) = (s1 , . . . , sr ) of r sections of E on A. A singular point of s(r) is a point where the si ’s fail to be linearly independent over C. An r-frame is an r-section without singularities. A 1-section is nothing but a section. In this case the singular points are the zeros of the section. Recall that an l-frame is simply called a frame (cf. Definition 3.12). In the next section, we define the q-th Chern class cq (E) of E to be the primary obstruction to constructing an r-frame of E, r = l − q + 1. For this purpose, we introduce the notion of the index of an r-section at a point where it is singular or is not defined. Mapping degree Let W (l, r) be the Stiefel manifold of r-frames in Cl (cf. Section 3.5). Recall that H2q−1 (W (l, r)) ' Z, q = l − r + 1 (cf. (3.17)). It has a canonical generator w2q−1 , i.e., there is a natural inclusion ι : S2q−1 ,→ W (l, r) and w2q−1 = ι∗ ν2q−1 with ν2q−1 the canonical generator of H2q−1 (S2q−1 ) ' Z. In particular, if r = 1, then q = l and W (l, 1) = Clr{0}, which deformation retracts to S2l−1 . Suppose we have a map ϕ : S2q−1 → W (l, r). Then it induces a morphism ϕ∗ : H2q−1 (S2q−1 ) −→ H2q−1 (W (l, r)) and we may write ϕ∗ (ν2q−1 ) = d · w2q−1 with d an integer uniquely determined by the homotopy class of ϕ. Definition 5.2. The above integer d is called the mapping degree of ϕ and is denoted by deg ϕ. Remark 5.1. 1. Note that π2q−1 (W (l, r)) ' Z, which has the canonical generator [ι], ι : S2q−1 ,→ W (l, r) being as above. A map ϕ as above defines an element in π2q−1 (W (l, r)), which is of the form d · [ι] with d = deg ϕ.

Chern Classes and Localization via Obstruction Theory

129

Let B2q denote a closed 2q-ball whose boundary is S2q−1 . Then deg ϕ = 0 if and only if ϕ can be extended to a map B2q → W (l, r) and we may think of deg ϕ as the obstruction to extending ϕ to B2q . 2. The mapping degree is also expressed in terms of cohomology. For this, note that both H 2q−1 (S2q−1 ) and H 2q−1 (W (l, r)) are isomorphic with ∗ ∗ Z. Let ν2q−1 and w2q−1 denote their generators dual to ν2q−1 and w2q−1 , ∗ ∗ respectively, i.e., the ones with hν2q−1 , ν2q−1 i = 1 and hw2q−1 , w2q−1 i = 1. 2q−1 For a map ϕ : S → W (l, r), we have the induced morphism ϕ∗ : H 2q−1 (W (l, r)) −→ H 2q−1 (S2q−1 ) and we may write (cf. B.8): ∗ ∗ ϕ∗ (w2q−1 ) = deg ϕ · ν2q−1 .

3. Here is a differential geometric interpretation of the mapping degree (cf. Section 10.5 below for details). We have H 2q−1 ((W (l, r); C) ' C with a canonical generator (cf. Theorem B.20). By the de Rham theorem (Theorem 7.5 below), the generator is represented by a differential form ω2q−1 on W (l, r) and, for a C ∞ map ϕ : S2q−1 → W (l, r), we may express its degree as (cf. Proposition 10.13): Z deg ϕ = ϕ∗ ω2q−1 . S2q−1

An explicit expression of ω2q−1 is given in Proposition 7.28 below. In particular, if r = 1, then q = l and W (l, 1) = Cl r{0}. In this case we may take as ω2q−1 the Bochner-Martinelli form βl (cf. Section 7.10). Note that β1 =

dz 1 √ , 2π −1 z

the Cauchy form. Index of an r-frame We now consider the bundle W (E, r) of r-frames of E on X. This is a bundle associated with E whose fiber W (E, r)x at each point x of X is diffeomorphic with W (l, r). An r-frame of E is nothing but a section of W (E, r). Suppose we have a regular oriented 2q-cell e in X (cf. Section B.2). Letting B2q be the unit closed ball with canonical orientation (cf. Example 3.9 and Remark B.13. 2), we take a characteristic map χ : B2q → e

130

Complex Analytic Geometry

so that it is a homeomorphism determining the orientation of e. We may assume that E is trivial on e. Suppose we are given an r-frame s(r) of E on e r e. Although it is not necessary, in order to fix the idea, we extend s(r) to an r-frame on er{a}, where a is a point in e. It is always possible as er{a} deformation retracts to ere. Denoting by S2q−1 the boundary of B2q , which is a canonically oriented (2q − 1)-sphere, we have a composition of maps: ∂χ

s(r)

p

ϕ : S2q−1 −→ ere −→ W (E, r)|e ' (W (l, r) × e) −→ W (l, r),

(5.1)

where ∂χ = χ|S2q−1 and p is the projection onto the first factor. Definition 5.3. The index I(s(r) , a) of s(r) at a is defined by I(s(r) , a) = deg ϕ. Note that the definition does not depend on the order of the members of s(r) , the choice of a in e or the trivialization of E. Example 5.1. Let U be a neighborhood of 0 in C = {z} and E = C×U the product bundle on U . Also let B2 be a closed 2-ball in U with center 0. For an integer m, let s be the frame of E on S1 = ∂B2 given by s(z) = (z m , z). Then, noting that W (1, 1) = C∗ and the map in (5.1) is given by z 7→ z m , we see that I(s, 0) = m. More generally, if the frame s on S1 is given by s(z) = (f (z), z) with f a non-vanishing C ∞ function, we have (cf. Remark 5.1. 3): Z 1 df √ I(s, 0) = . 2π −1 S1 f The index in Definition 5.3 may also be described as follows. Let s(r) be an r-frame of E on e r e, with r > 1. We write as s(r) = (s(r−1) , sr ), individualizing the last section. The (r − 1)-frame s(r−1) determines a map ϕ1 : S2q−1 −→ W (l, r − 1) similarly as (5.1). Since π2q−1 (W (l, r − 1)) = 0, the map ϕ1 extends to a map B2q → W (l, r − 1). Thus s(r−1) can be extended to an (r − 1)-frame on e and generates a trivial complex subbundle Ir−1 of rank r − 1 of E|e . This way we have an exact sequence ρ

0 −→ Ir−1 −→ E|e −→ E 0 −→ 0,

Chern Classes and Localization via Obstruction Theory

131

where E 0 is a complex vector bundle of rank q on e. We denote by s0r the section of E 0 given by ρ ◦ sr , which is non-vanishing on ere. Consider the composition (cf. (5.1)): ∂χ

s0

p

r ϕ0 : S2q−1 −→ ere −→ W (E 0 , 1) ' W (q, 1) × e −→ W (q, 1).

Then by definition, I(s0r , a) = deg ϕ0 . Proposition 5.1. In the above situation, I(s(r) , a) = I(s0r , a). Proof. As we noted, the map ϕ1 extends to a map B2q → W (l, r − 1), which is also denoted by ϕ1 . We may assume that for every point x in B2q , i

ϕ1 (x) is given by the matrix (e1 , . . . , er−1 ), where ei = t(0, . . . , 1, . . . , 0). This means that we take the trivialization W (E, r)|e ' W (l, r) × e in (5.1) so that for each point x in S2q−1 , ϕ(x) is given by a matrix of the form (e1 , . . . , er−1 , a), where a = t(a1 , . . . , ar−1 , ar . . . , al ) with t(a1 , . . . , ar−1 ) in Cr−1 and (ar . . . , al ) in Cq r{0} = W (q, 1). This also means that we take the trivialization of W (E 0 , 1) so that ϕ0 (x) = (ar . . . , al ). Recall that the map π : W (l, r) → W (l, r − 1) given by taking the first (r − 1) vectors has a fiber bundle structure with fiber Cr−1 × W (q, 1) and that the inclusion of W (q, 1) into a fiber of π induces an isomorphism of the (2q −1)-st homotopy groups (cf. Propositions 3.7 and 3.8). The proposition then follows from Remark 3.5. 1.  5.2

Chern classes of a complex vector bundle

We review the obstruction theoretic construction of the Chern classes of complex vector bundles on cell complexes. Let (X, {eλ }) be a regular cell complex (cf. Section B.2). Recall that the singular homology Hp (X) of X can be computed from the chain complex (C• (X), ∂) with Cp (X) the free Abelian group generated by the oriented p-cells in X. Also the singular cohomology H p (X) can be computed from the cochain complex (C • (X), δ) with C p (X) = Hom(Cp (X), Z) and δ the transpose of ∂. This means that, if we denote by Hp (C• (X)) the p-th homology of (C• (X), ∂) and by H p (C • (X)) the p-th cohomology of (C • (X), δ), there are canonical isomorphisms (cf. Theorem B.22): Hp (C• (X)) ' Hp (X),

H p (C • (X)) ' H p (X).

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Let E be a (continuous) complex vector bundle of rank l on X and r an integer with 1 ≤ r ≤ l. We set q = l − r + 1 as before. We try to construct an r-frame, i.e., a section of W (E, r), on each skeleton X p of X starting from the 0-skeleton and then extending it inductively to larger skeletons. On the way we get a 2q-cochain γ as the obstruction to the construction. It is shown that γ is in fact a cocycle and defines a class in H 2q (X), which will be the q-th Chern class cq (E) of E. First, it is always possible to construct a section s(r) of W (E, r) on X 0 . Let e be a p-cell. If a section s(r) of W (E, r) is given on e r e, it defines a map as (5.1), replacing S2q−1 with Sp−1 ' e r e. Thus s(r) defines an element in πp−1 (W (l, r)). If p ≤ 2l − 2r + 1 = 2q − 1, the section s(r) can be extended to e, since the homotopy group vanishes. This way we may construct a section s(r) of W (E, r) on X 2q−1 . Then we reach to an “obstruction” when p = 2q. Namely, for each 2q-cell e, the r-frame s(r) on ere is extended to an r-frame on e possibly except for a point a in e and the obstruction is given by the index I(s(r) , a). We define a cochain γ by he, γi = I(s(r) , a)

(5.2)

for each 2q-cell e and then extending it linearly. Proposition 5.2. The cochain γ is a cocycle. Proof.

It suffices to show that, for every (2q + 1)-cell e0 , he0 , δγi = 0.

(5.3) e0 .

Let Y denote the subcomplex of X consisting of the cells in Since Y 2q−1 is (2q − 2)-connected, by Theorem 3.7, we have an isomorphism ∼ h : π2q−1 (Y 2q−1 ) → H2q−1 (Y 2q−1 ). We also have H2q−1 (Y 2q−1 ) = Z2q−1 (Y 2q−1 ) = Z2q−1 (Y ). On the other hand, using the fact that the bundle E|e0 is trivial, the given r-section s(r) on X 2q−1 defines a map (r) s∗ : π2q−1 (Y 2q−1 ) → π2q−1 (W (l, r)) ' Z. We denote by α the composition of the maps ∂

h−1

s(r)

∗ C2q (Y ) −→ Z2q−1 (Y ) −→ π2q−1 (Y 2q−1 ) −→ π2q−1 (W (l, r)) ' Z.

By definition, for a 2q-chain c in C2q (Y ), hc, γi = α(c). Since C2q−2 (Y ) is a free Abelian group, Z2q−1 (Y ) is a direct summand of C2q−1 (Y ). Thus (r) s∗ ◦ h−1 extends to a map u : C2q−1 (Y ) → π2q−1 (W (l, r)) ' Z, which may be thought of as a cochain in C 2q−1 (Y ). Then we have, for every chain c in C2q (Y ), hc, γi = α(c) = h∂c, ui = hc, δui so that γ = δu in C 2q (Y ) and (5.3) follows. 

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Remark 5.2. 1. It is also shown that different choices of extensions in each step leads to cocycles that are cohomologous. 2. Precisely speaking, γ is a cochain with coefficients in the “local system formed by π2q−1 (W (Ex , r))”. Since the group GL(l, C) is connected, this system is in fact trivial and we may naturally think of γ as being in C 2q (X). Definition 5.4. The q-th topological Chern class cqtop (E) of E is the class of γ in H 2q (X). Throughout this chapter, we denote cqtop (E) simply by cq (E) and call it the q-th Chern class of E, unless otherwise is stated. The total Chern class of E is defined by c∗ (E) = 1 + c1 (E) + · · · + cl (E), which is an element in the cohomology ring H ∗ (X) and is invertible. Remark 5.3. 1. If E admits an r-frame on X, then clearly cq (E) = 0. Conversely if cq (E) = 0, then it is possible to construct an r-frame of E on X 2q , but not on X in general. Thus cq (E) is referred to as the primary obstruction to constructing an r-frame of E. 2. The classes constructed above are in fact “functorial”, i.e., for a continuous map f : X 0 → X of cell complexes and a complex vector bundle E on X, cq (f ∗ E) = f ∗ cq (E)

in H 2q (X 0 ).

This is proved by taking a “cellular approximation” of f . In particular, the above construction of cq (E) does not depend on the choice of cellular decomposition of X. 3. In Chapter 8 below, we define differential geometric Chern classes for a C ∞ complex vector bundle on a C ∞ manifold via the Chern-Weil theory and discuss, in Chapter 10, the relation between the classes defined from two different approaches. We prove the above funtoriality in the framework of Chern-Weil theory. 5.3

Euler class of an oriented real vector bundle

Let E be an oriented real vector bundle of rank l0 on a regular cell complex (X, {eλ }). We assume again that each cell is oriented. If l0 = 1, the orientability of E implies that E is a trivial bundle. Thus we assume

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that l0 > 1 in the following. In view of (3.18), we may perform a similar construction to obtain the Euler class of an oriented real vector bundle. 0 We denote by V (l0 , 1) the real Stiefel manifold of 1-frames in Rl , which 0 0 is in fact Rl r{0} so that it has the homotopy type of Sl −1 . Thus V (l0 , 1) is (l0 − 2)-connected and πl0 −1 (V (l0 , 1)) ' Z, which has a canonical generator. Hence we have Hl0 −1 (V (l0 , 1)) ' Z with a canonical generator and, for a 0 map ϕ : Sl −1 → V (l0 , 1), we my define its degree, denoted by deg ϕ, as in Definition 5.2. We denote by V (E, 1) the bundle of 1-frames of E on X. Then the Euler class e(E) of E is defined to be the obstruction to constructing a 0 section of V (E, 1), i.e., a non-vanishing section of E, on X l . It will be defined a priori in the cohomology with coefficients in the local system formed by πl0 −1 (V (Ex , 1)). Since GL+ (l0 , R) is connected, this system is in fact trivial and we may naturally think of the obstruction as being in the 0 integral cohomology H l (X; Z), as in the case of Chern classes. We may define the index of a section at its singularity as before. Thus let e be an l0 -cell in X, on which E is trivial. Suppose we have a non-vanishing section s of E on e r {a}, a being a point in e. We have a composition (cf. (5.1)): 0

∂χ

p

s

ϕ : Sl −1 −→ ere −→ V (E, 1)|e ' (V (l0 , 1) × e) −→ V (l0 , 1).

(5.4)

The index of s at a, denoted by I(s, a), is the mapping degree of ϕ: I(s, a) = deg ϕ.

(5.5) p

We try to construct a section s of V (E, 1) on each skeleton X of X inductively from the 0-skeleton, as in the case of Chern classes. Continuing this process, we reach to an “obstruction” when p = l0 . Namely, for each l0 -cell e, the 1-frame s on ere is extended to a 1-frame on e possibly except for a point a in e and the obstruction is given by the index I(s, a). We may 0 define a cochain γ in C l (X) by setting γ(e) = I(s, a) for each l0 -cell e and then extending it linearly. This cochain is in fact a cocycle as in the case of Chern classes. 0

Definition 5.5. The Euler class e(E) of E is the class of γ in H l (X). This class is independent of the choices of various objects used in the definition. Remark 5.4. 1. As in the case of Chern classes, e(E) = 0 if and only if 0 E admits a non-vanishing section on X l . The Euler class is also functorial (cf. Remark 5.3).

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2. If E is a complex vector bundle of rank l, we may think of it as a naturally oriented real bundle of rank 2l and W (l, 1) may be identified with V (2l, 1). In this case, we have e(E) = cl (E), the top Chern class of E. 3. For an oriented real vector bundle E of even rank l0 = 2k 0 , it is known that 0

0

(−1)k cl (E c ) = e(E) ` e(E).

(5.6)

4. The use of (3.18) in general case of frames leads to the obstruction theoretic definition of Stiefel-Whitney classes of real vector bundles.

5.4

Relative classes

Let (X, {eλ }) be a regular cell complex and A a subcomplex of X (cf. Section B.2). Relative Chern classes Let E be a complex vector bundle of rank l on X and suppose we are already given an r-frame s(r) of E on the 2q-skeleton A2q of A, q = l − r + 1. We follow the procedure described in Section 5.2 starting with this frame to obtain an r-frame s˜(r) on X 2q r{isolated points}. To be more precise, the above frame s˜(r) has the following property for a 2q-cell e in X: (

if e ⊂ A, then s˜(r) is defined and equals s(r) on e,

if e 6⊂ A, then s˜(r) is defined on er{a}, where a is a point in e. For every 2q-cell e in A, we have I(˜ s(r) , a) = I(s(r) , a) = 0, a being a point in e. Thus the cocycle γ defined by (5.2) for s˜(r) is in C 2q (X, A). It represents a class in H 2q (X, A), which is denoted by cq (E, s(r) ) and is called the relative Chen class defined by s(r) . The class cq (E, s(r) ) depends on s(r) , but not on the choice of the extension s˜(r) of s(r) (cf. Remark 5.2. 1). Its image by the canonical morphism H 2q (X, A) → H 2q (X) is the usual Chern class cq (E).

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Relative Euler class Let E be an oriented real vector bundle of rank l0 on X and suppose we 0 are already given a non-vanishing section s of E on Al . We follow the procedure described in Section 5.2 for the Euler class starting with this 0 section to obtain a non-vanishing section s˜ on X l r{isolated points}, as in the case of relative Chern classes. Then, for every l0 -cell e in A, we have I(˜ s, a) = I(s, a) = 0. Thus the cocycle γ for s˜ defining the Euler class e(E) 0 0 is in fact in C l (X, A). It represents a class in H l (X, A), which we denote by e(E, s) and call the relative Euler class defined by s. Its image by the 0 0 canonical morphism H l (X, A) → H l (X) is the Euler class e(E). 5.5

Piecewise-linear manifolds and pseudo-manifolds

PL manifolds and smoothings Polyhedra: Let v be a point and B a subset of RN . The cone with vertex v and base B is the set v ∗ B = { x ∈ RN | x = (1 − t)v + tb, b ∈ B, 0 ≤ t ≤ 1 }. Definition 5.6. A polyhedron is a subset P of RN such that each point a in P has a neighborhood S in P of the form S = a ∗ L with L compact. A neighborhood S as above is called a star of a in P and L a link. For example, Rm and Hm = { (x1 , . . . , xm ) ∈ Rm | x1 ≤ 0 } are polyhedra. An open set of a polyhedron is a polyhedron. The polyhedron |K| of a simplicial complex K is a polyhedron. For a point a in |K|, SK (a) is a star and LK (a) a link of a. In fact we have: Proposition 5.3. A subset P of RN is a polyhedron if and only if it is the polyhedron of some simplicial complex K in RN , i.e., P = |K|. We call K as above a simplicial decomposition of P . In this case the identity map |K| → P is a triangulation of P . 0

Definition 5.7. A map f : P → P 0 between polyhedra in RN and RN , respectively, is piecewise-linear, PL for short, if P admits a simplicial decomposition K such that for each simplex s of K, f |s is affine. It is equivalent to saying that the graph Γf = { (x, f (x)) | x ∈ P } of f is 0 a polyhedron in RN +N . Thus if a PL map f : P → P 0 is a homeomorphism,

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f −1 is also PL. A simplicial map f : |K| → |K 0 | of simplicial complexes is PL. If f : P → P 0 is PL and proper, there exist simplicial decompositions K and K 0 of P and P 0 , respectively, such that f : |K| → |K 0 | is simplicial. PL manifolds: A PL manifold with boundary is defined by replacing “C ∞ ” with “PL” in Definition 3.20. Thus let Q be a Hausdorff space with countable basis. Definition 5.8. A PL structure on Q is the equivalence class of an atlas {(Uα , ϕα )} such that (1) for each α, ϕα is a homeomorphism onto an open set in Rm or in Hm , (2) for each pair (α, β), ϕα ◦ ϕ−1 β : ϕβ (Uα ∩ Uβ ) → ϕα (Uα ∩ Uβ ) is PL. The integer m as above is called the dimension of Q. The boundary ∂Q of Q is the set of points in Q that correspond to x1 = 0 in Hm . If ∂Q 6= ∅, it is an (m − 1)-dimensional PL manifold without boundary. A PL manifold Q has a triangulation h : |K| → Q such that each map h−1 ◦ ϕ−1 α : ϕα (Uα ) → |K| is PL. Let M be a PL manifold without boundary. A PL submanifold with boundary in M is defined as in Definition 3.21, replacing “C ∞ ” with “PL”. Thus if Q is a PL submanifold with boundary, it is a PL manifold with boundary. Remark 5.5. A polyhedron P is a PL manifold of dimension m if and only if, for each point a ∈ P , there is a triangulation K of P such that a is a vertex of K and that the link LK (a) of a is a PL (m − 1)-sphere ∂∆m . Piecewise C ∞ homeomorphism: manifold with boundary.

Let P be a polyhedron and R a C ∞

Definition 5.9. A homeomorphism h : P → R is piecewise C ∞ , PD for short, if P has a simplicial decomposition K such that h is a C ∞ triangulation (cf. Definition 4.1). In this case, the atlas on P = |K| given by the open stars of vertices in K makes P into a PL manifold. Let Q be a PL manifold with boundary and R a C ∞ manifold with boundary. A homeomorphism h : Q → R is PD if Q has a triangulation h0 : |K| → Q such that h ◦ h0 : |K| → R is PD.

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Remark 5.6. We may rephrase the statements (T1 ) and (T2 ) in Section 4.1 by saying that every C ∞ manifold M admits a PD homeomorphism Q → M and the PL manifold Q is unique up to PL isomorphism. Smoothings of PL manifolds: Let Q be a PL manifold with boundary. If σ is a C ∞ structure on (the underling topological space of) Q, we denote by Qσ the C ∞ manifold defined by σ. A smoothing of Q is a C ∞ structure σ on Q that makes the identity 1Q : Q → Qσ PD. Thus if h : Q → R is a PD homeomorphism onto a C ∞ manifold R, the C ∞ structure induced from that of R by h is a smoothing of Q. We also call h : Q → R a smoothing. Let Q be a PL manifold and Q1 ⊂ Q r ∂Q a PL submanifold of codimension k. We say that Q1 is flat if it has a neighborhood U in Q such that (U, Q1 ) is PL homeomorphic to (Q1 × Rk , Q1 × {0}). We quote: Theorem 5.1. Let Q be a PL manifold and Q1 ⊂ Q r ∂Q a flat PL submanifold. Also let R be a C ∞ manifold and h : Q → R a smoothing. Then there is a PD isotopy ht : Q → R such that h0 = h and that h1 (Q1 ) is a C ∞ submanifold of R. Moreover, if Q1 is compact, for every neighborhood N of Q1 and every ε, ht can be chosen to be an ε-isotopy relative to QrN . Here we recall some terminologies in the above (cf. Section B.1). In general, if X and Y are topological spaces, an isotopy ht : X → Y is a homotopy {ht }0≤t≤1 such that H : X × I → Y × I, H(x, t) = (ht (x), t), is a homeomorphism, I = [0, 1]. If Q is a PL manifold and R a C ∞ manifold, a PD isotopy is an isotopy ht : Q → R such that the extension H 0 : Q × R → R × R of H given by H 0 (x, t) = (h0 (x), t), for t ≤ 0, and H 0 (x, t) = (h1 (x), t), for t ≥ 1, is a PD homeomorphism. Let ε : Q → R+ = { a ∈ R | a > 0 } be a continuous function. An isotopy ht is an ε-isotopy if d(ht (x), h0 (x)) < ε(x), for all x ∈ Q and t ∈ I, where d( , ) denotes a metric on R. Orientability: Let Q be a PL manifold of dimension m and h : |K| → Q a triangulation. Then, for each (m − 1)-simplex s of K, there exist either one or two m-simlices which contain s as a face. The PL manifold Q is said to be orientable if it admits a triangulation K with the following property: the m-simplices in K can be oriented so that, if s is an (m − 1)-simplex in K and if s1 and s2 are two simplices containing s as a face, then the prescribed orientations of s1 and s2 induce opposite orientations of s. If Q is orientable, it has two orientations, i.e., there are two ways to orient the m-simplices in the above manner. Once we fix an orientation, we say that Q is oriented. In this case ∂Q is naturally oriented.

Chern Classes and Localization via Obstruction Theory

Piecewise C ∞ manifolds:

139

Let M be a C ∞ manifold.

Definition 5.10. A subset R of M is a piecewise C ∞ submanifold of M with boundary, if there exist a C ∞ triangulation (K, h) of M and a subcomplex L of K such that h(|L|) = R and that |L| is a PL submanifold of |K| with boundary. In the above, we also say that R is a piecewise C ∞ manifold with boundary in M . Note that a closed C ∞ submanifold with boundary is a piecewise C ∞ submanifold with boundary. Let R be a piecewise C ∞ submanifold of M with boundary. The dimension of R is defined to be that of |L|, L being as in Definition 5.10. Let d0 = dim R. If ∂R 6= ∅, it is a (d0 − 1)-dimensional piecewise C ∞ manifold without boundary. Definition 5.11. A point a in R is said to be general, if there exist (K, h) and L as above such that a is the image by h of an interior point of a d0 -simplex of L. The set of general points in R is a d0 -dimensional C ∞ submanifold of M without boundary. Let R be as above. We say that R is orientable if |L| is orientable as a PL-manifold. This is equivalent to saying that the set of general points in R is orientable as a C ∞ manifold. If this is the case, ∂R is also orientable. We may think of the set of general points of ∂R as the boundary of a C ∞ manifold with boundary whose interior is the set of general points of R. If R is oriented, ∂R is oriented according to Convention 3.2. Pseudo-manifolds Definition 5.12. A pseudo-manifold of dimension d0 in M is a subcomplex X of M , with respect to some triangulation of M , satisfying the following conditions: (1) Every simplex in X is a face of some d0 -simplex in X. (2) Every (d0 − 1)-simplex is the face of exactly two d0 -simplices. The pseudo-manifold X is orientable if (3) The d0 -simplices in X can be oriented so that, if s is a (d0 − 1)-simplex in X and if s1 and s2 are the two simplices that contain s in their

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boundary, then the prescribed orientations of s1 and s2 induce opposite orientations of s. An orientable pseudo-manifold X is said to be oriented, once orientations of d0 -simplices in X satisfying (3) above are fixed. We say that X 0 0 is irreducible if X rX d −2 is connected, where X d −2 denotes the (d0 − 2)0 skeleton of X. A general point of X is a point in X r X d −2 . If X is an oriented pseudo-manifold, the union of all the d0 -simplices in X is a cycle. In particular, if X is irreducible, it is called the fundamental cycle of X ˘ d0 (X) the fundamental class of X. In this case, there is a and its class in H ˘ d0 (X) ' Z, the fundamental class corresponding canonical isomorphism H to 1. In general, we have a decomposition into irreducible components: [ X= Xi . i

˘ d0 (X) is generated by the fundamental classes of If X is oriented, then H the Xi ’s. Example 5.2. 1. A closed submanifold of a C ∞ manifold is a non-singular pseudo-manifold. 2. An analytic variety V in a complex manifold is an oriented pseudomanifold. The irreducible decomposition of V as a variety gives that as a pseudo-manifold. A general point of V is nothing but a non-singular point of V .

5.6

Localization and topological residues

In this section, we let M be a C ∞ manifold of dimension m and let K0 , K, K 0 and K ∗ be as in Section 4.1. Duals of characteristic classes Suppose M is oriented and connected so that we have the Poincar´e duality (Theorem 4.1). Chern classes: Let E be a complex vector bundle of rank l on M . We use the dual cellular decomposition K ∗ to define the Chern classes of E (cf. Section 5.2). Denoting by s(r) an r-frame already constructed on the

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(2q − 1)-skeleton (K ∗ )2q−1 of K ∗ , the Poincar´e dual of cq (E) is represented ˘ m−2q (M ) by the cycle in H X C= I(s(r) , bs ) s, (5.7) s

where the sum is taken over all the (m − 2q)-simplices s of K and I(s(r) , bs ) denotes the index of s(r) on s∗ ∩ (K ∗ )2q−1 at the barycenter bs (cf. Definition 5.3, we may take bs as a there). Euler class: Let E be an oriented real vector bundle of rank l0 on M . Denoting by s a non-singular section already constructed on the (l0 − 1)˘ m−l0 (M ) by skeleton of K ∗ , the Poincar´e dual of e(E) is represented in H the cycle X C= I(s, bs ) s, (5.8) s

where the sum is taken over all the (m − l0 )-simplices s of K. Localized Chern classes Let E be a complex vector bundle of rank l on M and S a K0 -subcomplex of M . We apply the considerations of Section 5.4 by letting A be the subK ∗ -complex M r OK 0 (S). Thus suppose we are already given an r-frame s(r) of E on the 2q-skeleton of M rOK 0 (S), q = l − r + 1. Then we have the relative class in H 2q (M, M rOK 0 (S)) ' H 2q (M, M rS), which is denoted by cqS,top (E, s(r) ) and is called the topological localization of cq (E) by s(r) . It will also be denoted by cqS (E, s(r) ), cqtop (E, s(r) ) or cq (E, s(r) ), if there is no fear of confusion. The class cqtop (E, s(r) ) depends on s(r) , but not on the choice of the extension s˜(r) of s(r) . Its image by the canonical morphism H 2q (M, M rS) → H 2q (M ) is the usual Chern class cq (E). Recall that, in the above construction, for an (m − 2q)-simplex s of K, the following are equivalent (cf. Proposition 4.2): (1) s 6⊂ S,

(2) s∗ ∩ S = ∅,

(3) s∗ ⊂ M rOK 0 (S).

Note that the point a may be assumed to be the barycenter bs of s. Suppose M is oriented so that we have the Alexander isomorphism (Theorem 4.2): ∼ ˘ A : H 2q (M, M rS) −→ H m−2q (S). Definition 5.13. The topological residue TRescq (s(r) , E; S) of s(r) for cq (E) at S is the image of cqtop (E, s(r) ) by A.

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Remark 5.7. In order to have the above localization and residue, it suffices to have s(r) on ∂SK 0 (S) ∩ (K ∗ )2q (cf. Remark 4.4). Suppose that S has only a finite number of connected components (Sλ )λ . Then we have a decomposition ˘ m−2q (S) = L H ˘ H λ m−2q (Sλ ) ˘ m−2q (Sλ ) for and accordingly we have the residue TRescq (s(r) , E; Sλ ) in H each λ. We have (cf. (5.7)): Proposition 5.4. In the above situation, the residue TRescq (s(r) , E; Sλ ) is represented by the cycle X Cλ = I(˜ s(r) , bs ) s, (5.9) s

where the sum is taken over the (m − 2q)-simplices of K in Sλ . Note that the class of the cycle does not depend on the choice of s˜(r) . ˘ 0 (Sλ ) = H0 (Sλ ) ' Z, In particular, if 2q = m and if Sλ is compact, H the isomorphism given by the augmentation (cf. Appendix B). Thus we may regard TRescq (s(r) , E; Sλ ) as an integer and we have X TRescq (s(r) , E; Sλ ) = I(˜ s(r) , bs ), (5.10) s

where the sum is taken over all the 0-simplices s of K in Sλ , in fact bs = s. Let R be an m-dimensional manifold possibly with boundary in M . We may assume that R and ∂R are K0 -subcomplexes of M . In the above considerations, we let S be R. Thus suppose we are given an r-frame s(r) of E on the 2q-skeleton of M rOK 0 (R). Then we have the relative class in H 2q (M, MrOK 0 (R)) ' H 2q (R, ∂R), which is denoted by cqR,top (E, s(r) ), or simply by cqR (E, s(r) ). Its image by the canonical morphism H 2q (R, ∂R) → H 2q (M ) is the usual Chern class cq (E). Remark 5.8. To have the above relative class cqR (E, s(r) ), it suffices to have s(r) on ∂SK 0 (R) ∩ (K ∗ )2q (cf. Remark 5.7). Suppose M is oriented so that we have the Lefschetz isomorphism (Theorem 4.3): ∼ ˘ L : H 2q (R, ∂R) −→ H m−2q (R).

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143

Recall that it is given by the left cap product with the class [R]. The class [R] a cqR (E, s(r) ) is represented by a cycle as in the right-hand side of (5.9), where the sum is taken over the (m − 2q)-simplices of K in R. ˘ 0 (R) = H0 (R) and there In particular, if 2q = m and if R is compact, H is the augmentation ε∗ : H0 (R) → Z. In this case, ε∗ ([R] a cqR (E, s(r) )) is equal to the Kronecker product h[R], cqR (E, s(r) )i (cf. (B.19)). It is an integer given by X I(˜ s(r) , bs ), s

where the sum is taken over all the 0-simplices s of K in R, in fact bs = s. Coming back to the previous situation, let S be a K0 -subcomplex of M and s(r) an r-frame of E on the 2q-skeleton of M r OK 0 (S). Let R be as above and suppose S ⊂ Int R. The r-frame s(r) restricts to an r-frame, denoted also by s(r) , on the 2q-skeleton of M rOK 0 (R). Thus we have the relative class cqR (E, s(r) ) in H 2q (R, ∂R) as above. From Corollary 4.3, we have the following “residue theorem”: Theorem 5.2. Let M be an oriented C ∞ manifold of dimension m and E a complex vector bundle of rank l on M . Also let S be a subcomplex of M with a finite number of connected components (Sλ )λ and s(r) an r-frame of E on 2q-skeleton of MrOK 0 (S), q = l − r + 1. In this situation, it holds: ˘ m−2q (Sλ ), which 1. For each λ, we have the residue TRescq (s(r) , E; Sλ ) in H is represented by the cycle (5.9). 2. If S is in the interior of a manifold R of dimension m possibly with boundary in M , X ˘ m−2q (R), in H (ıλ )∗ TRescq (s(r) , E; Sλ ) = [R] a cqR (E, s(r) ) λ

where ıλ : Sλ ,→ R denotes the inclusion. Remark 5.9. 1. If we take M as R in 2 above, the right-hand side is ˘ m−2q (M ). expressed as [M ] a cq (E) in H ˘ 0 (R) = H0 (R). In this case each 2. In the case 2q = m, if R is compact, H Sλ is compact and, appling the augmentation ε∗ : H0 (R) → Z to the both sides of the equality in 2 in Theorem 5.2, we have (cf. (B.2)) X TRescq (s(r) , E; Sλ ) = h[R], cqR (E, s(r) )i λ

as integers. Suppose M is compact. If we take M as R, the right-hand side is expressed as h[M ], cq (E)i.

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Complex Analytic Geometry

Transverse residues: Let M , K0 , K, K ∗ and S be as above. Suppose the maximum dimension of the simplices of K in S is m − 2q and let S 0 be an oriented submanifold of M of dimension m − 2q which is contained in S. We may assume that the orientations of simplices in K are compatible with that of S 0 . Let x be a point in S 0 and D a slice of S 0 in M at x (cf. Definition 3.24). We may assume that x is the barycenter bs of some (m − 2q)-simplex s of K in S 0 and that s∗ is in D (cf. Theorem 5.1). We may also extend the triangulation K 0 on s∗ throughout D. Let s(r) be an r-frame of E on the 2q-skeleton of M r OK 0 (S), as before. Restricting E and s(r) to D, we have the localization cqtop (E|D , s(r) |D ) and the residue TRescq (s(r) |D , E|D ; x), which correspond to each other by the Alexander isomorphism H 2q (D, Dr{x}) ' H0 ({x}). As H0 ({x}) ' Z, we may think of TRescq (s(r) |D , E|D ; x) as an integer, which is referred to as the transverse residue at x. In fact it is given by TRescq (s(r) |D , E|D ; x) = I(s(r) , bs ). Remark 5.10. 1. If we denote by ι : D ,→ M the inclusion, we have the commutative diagram (cf. Proposition 4.13): H 2q (M, M rS)

∼ AM,S

ι∗

 H 2q (D, Dr{x})

˘ m−2q (S) /H (D

∼ AD,x

·

 ˘ 0 ({x}) /H

)x

so that we may write

·


TRescq (s(r) |D , E|D ; x) = D TRescq (s(r) , E; S) x . 2. By Remarks 4.4 and 5.7, in the above notation, TRescq (s(r) |s∗ , E|s∗ ; bs ) makes sense and is equal to TRescq (s(r) |D , E|D ; x). As a function of x, TRescq (s(r) |D , E|D ; x) is locally constant. Thus, if S is connected, it is constant. From Proposition 5.4, we have (cf. Convention 4.2): 0

Proposition 5.5. If Sλ is an (m − 2q)-dimensional submanifold of M , TRescq (s(r) , E; Sλ ) = TRescq (s(r) |D , E|D ; x) · Sλ where x is a point in Sλ and D a slice of Sλ at x.

˘ m−2q (Sλ ), in H

Chern Classes and Localization via Obstruction Theory

145

The above expression of the residue is generalized to the case Sλ is a pseudo-manifold. Thus let Sλ be a connected component of S as above and suppose it is an oriented pseudo-manifold of dimension m − 2q (cf. Definition 5.12). If s is an (m − 2q)-simplex of K in Sλ , s∗ is a 2q-cell such that s∗ ∩ Sλ = {bs }. Thus we have the transverse residue TRescq (s(r) |s∗ , E|s∗ ; bs ), which is equal to I(s(r) , bs ) (cf. Remark 5.10. 2). From Proposition 5.4, we have: Theorem 5.3. Suppose Sλ is an oriented pseudo-manifold of dimension ˘ m−2q (Sλ ) is represented m − 2q. Then the residue TRescq (s(r) , E; Sλ ) in H by the cycle X TRescq (s(r) |s∗ , E|s∗ ; bs ) · s, s

where s runs through all the (m − 2q)-simplices of K in Sλ . Corollary 5.1. In the above situation, suppose that the set of general points of Sλ has the structure of a C ∞ submanifold of dimension m − 2q of M S and that, in the irreducible decomposition Sλ = i Sλ,i , the set {Sλ,i }i is locally finite. Then we have: X ˘ m−2q (Sλ ), TRescq (s(r) , E; Sλ ) = TRescq (s(r) |D , E|D ; xλ,i )·[Sλ,i ] in H i

where xλ,i is a general point of Sλ,i and D a slice of Sλ,i at xλ,i . ˘ m−2q (Sλ ). In fact In the above, [Sλ,i ] denotes the class of Sλ,i in H ˘ m−2q (Sλ ) is a free Abelian group generated by these classes. H In the case Sλ is a submanifold, the above reduces to Proposition 5.5. Remark 5.11. In Chapter 10 below we discuss localization problems in various settings mainly from the differential geometric viewpoint and give a general residue theorem (Theorem 10.1). The localization of Chern classes by frames in this context is treated in Section 10.3, where the differential geometric counterpart of Theorem 5.2 is given (cf. Theorem 10.6). We then prove that these two are essentially the same (cf. Theorem 10.13). In Chapters 12 and 13, we discuss these residues in detail. We now give some fundamental examples. Hyperplane bundle: Let Pn be the n-dimensional complex projective space with homogeneous coordinates [ζ0 , . . . , ζn ]. We denote by Uα the open set in Pn defined by ζα 6= 0, α = 0, . . . , n. Let H be the hyperplane defined

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Complex Analytic Geometry

by ζ0 = 0. The hyperplane bundle LH is the line bundle defined by the transition functions {ζβ /ζα } (cf. Example 3.6, where it is denoted by Hn ). It admits a section s represented by the collection (sα ) given by sα = ζ0 /ζα . The zero set of s is exactly H so that we have the localization c1 (LH , s) in H 2 (Pn , Pn rH) of c1 (LH ) and the associated residue TResc1 (s, LH ; H), which is the image of c1 (LH , s) by the Alexander isomorphism ∼

H 2 (Pn , Pn rH) −→ H2n−2 (H). Proposition 5.6. We have: TResc1 (s, LH ; H) = [H]

in H2n−2 (H).

Proof. By Proposition 5.5, it suffices to show that the transverse residue is equal to 1. Thus let x be a point in H and assume that it is in Uα , α 6= 0. We have a trivialization LH |Uα ' C × Uα and s is expressed as [ζ] 7→ (z, [ζ]), z = ζ0 /ζα . Let D be a complex 1-dimensional disk transverse to H at x. We may think of z as a coordinate on D and we see that the transverse residue is 1 (cf. Example 5.1).  The above is a special case of Theorem 12.2 below. Corollary 5.2. The first Chern class c1 (LH ) of the hyperplane bundle is the Poincar´e dual of [H]. Universal bundle: More generally we consider the universal bundle Q on the Grassmann manifold G(N, r) of r-planes in CN , which is of rank l = N − r (cf. Section 3.6). For q = 1, . . . , l, we have the q-th Chern class cq (Q) of Q. Let I q denote the r-tuple of integers (l+1−q, l+2, l+3, . . . , l+r = N ). Note that the Schubert variety EI q has complex dimension lr − q and each element is represented by a matrix A such that the rank of A(l+1−q) , which is an (r + q − 1) × r matrix, is less than or equal to r − 1. We have the exact sequence of vector bundles on G(N, r): π

0 −→ S −→ IN −→ Q −→ 0.

(5.11)

Let si be the section of Q defined by si = π(ei ), i = 1, . . . , N , and set s(k) = (s1 , . . . , sk ), k = l − q + 1. Theorem 5.4. For q = 1, . . . , l, the singular set of s(k) is EI q and TRescq (s(k) , Q; EI q ) = [EI q ]

in H2(lr−q) (EI q ).

Proof. If we think of G(N, r) as the union of the eI ’s, then (s1 , . . . , sk ) fails to be linearly independent exactly on eI q . Moreover, in the coordinate neighborhood UI q , EI q is given by al+2−q,1 = · · · = al+1,1 = 0. 

Chern Classes and Localization via Obstruction Theory

147

Corollary 5.3. The Schubert class [EI q ] is the Poincar´e dual of cq (Q). Remark 5.12. More generally, let I = (i1 , . . . , ir ) be an arbitrary element in I. For each j = 1, . . . , r, let λj be defined by λj = N − r + j − ij and set ∆I (Q) = det(cλj +i−j (Q))1≤i,j≤r . In the above, the product is the cup product. Since the Chern classes are of even degree, we may define the determinant as in the case of usual numbers. Then we have (Giambelli’s formula): [G(N, r)] a ∆I (Q) = [EI ], i.e., the Schubert class [EI ] is the Poincar´e dual of ∆I (Q). In view of Theorem 3.8, we see that the cohomology H ∗ (G(N, r); Z) is the graded Z-algebra Z[c1 (Q), . . . , cl (Q)], the polynomial ring over Z freely generated by the Chern classes of the universal bundle. Localized Euler class Let E be an oriented real vector bundle of rank l0 on M and S a K0 subcomplex of M . Suppose we are already given a non-vanishing section s of E on the l0 -skeleton of M r OK 0 (S). Then we have the relative class 0 0 in H l (M, M rOK 0 (S)) ' H l (M, M rS), which we denote by e(E, s) and call the localization of e(E) by s. Its image by the canonical morphism 0 0 H l (M, M rS) → H l (M ) is the Euler class e(E). Suppose M is oriented so that we have the Alexander isomorphism ∼

0

˘ m−l0 (S). A : H l (M, M rS) −→ H Definition 5.14. The topological residue TRese (s, E; S) of s for e(E) at S is the image of e(E, s) by A. Remark 5.13. As in the case of Chern classes, it suffices to have s on 0 ∂SK 0 (S) ∩ (K ∗ )l to have the localization and residue (cf. Remark 5.7). If S has a finite number of connected components (Sλ )λ we have the ˘ m−l0 (Sλ ) for each λ. It is represented by the residue TRese (s, E; Sλ ) in H cycle (cf. (5.8)) X Cλ = I(˜ s, bs ) s, (5.12) s

where the sum is taken over the (m − l0 )-simplices of K in Sλ .

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Complex Analytic Geometry

˘ 0 (Sλ ) = H0 (Sλ ) ' Z and In particular, if l0 = m and if Sλ is compact, H we may regard TRese (s, E; Sλ ) as an integer and we have X TRese (s, E; Sλ ) = I(˜ s, bs ), (5.13) s

where the sum is taken over all the 0-simplices s of K in Sλ , in fact bs = s. Let R be an m-dimensional manifold possibly with boundary in M . In the above considerations, we let S be R. Thus suppose we are given a non-vanishing section s of E on the l0 -skeleton of M r OK 0 (R). Then we 0 have the relative class eR (E, s) in H l (R, ∂R). Remark 5.14. Similar remark as in Remark 5.8 apply to the case of Euler class. If M is oriented, we have the Lefschetz isomorphism: 0 ∼ ˘ L : H l (R, ∂R) −→ H m−l0 (R). The class [R] a eR (E, s) is represented by a cycle as in the right-hand side of (5.12), where the sum is taken over the (m − l0 )-simplices of K in R. ˘ 0 (R) = H0 (R) and there In particular, if l0 = m and if R is compact, H is the augmentation ε∗ : H0 (R) → Z. We have ε∗ ([R] a eR (E, s)) = h[R], eR (E, s)i. It is an integer given by X I(˜ s, bs ), s

where the sum is taken over all the 0-simplices s of K in R, in fact bs = s. Coming back to the previous situation, let S be a K0 -subcomplex of M and s a non-vanishing section of E on the l0 -skeleton of M r OK 0 (S). Let R be as above and suppose S ⊂ Int R. The section s restricts to a non-vanishing section on the l0 -skeleton of M rOK 0 (R). Thus we have the 0 relative class eR (E, s) in H l (R, ∂R) as above. We have the residue theorem as Theorem 5.2 for the Euler class: Theorem 5.5. In the above situation, it holds: ˘ m−l0 (Sλ ), which is 1. For each λ, we have the residue TRese (s, E; Sλ ) in H represented by the cycle (5.12). 2. We X have (ıλ )∗ TRese (s, E; Sλ ) = [R] a eR (E, s)

˘ m−l0 (R). in H

λ

Remark 5.15. Similar remarks as in Remark 5.9 apply to this case, replacing cq , 2q and s(r) with e, l0 and s, respectively.

Chern Classes and Localization via Obstruction Theory

149

Transverse residues: As in the case of Chern classes, we may consider the transverse residue. For simplicity we consider the case Sλ is a submanifold of dimension m − l0 . Let x be a point in Sλ and D a slice of Sλ in M at x. Then restricting E and s to D, we have the transverse residue TRese (s|D , E|D , x), which is an integer independent of x. We have (cf. Proposition 5.5): Proposition 5.7. If Sλ is an (m − l0 )-dimensional submanifold of M , TRese (s, E; Sλ ) = TRese (s|D , E|D ; x) · Sλ

˘ m−l0 (Sλ ). in H

In the case Sλ is a pseudo-manifold, we have expressions of the residues similar to the ones in Theorem 5.3 and Corollary 5.1. Poincar´ e-Hopf theorem Euler-Poincar´ e characteristic: Let X be a topological space with finite homology type and bp the p-th Betti number (cf. (B.13). Here we set A = ∅). Suppose bp = 0 for p > p0 . Then the alternating sum p0 X χ(X) = (−1)p bp p=0

is called the Euler-Poincar´e characteristic of X. If X = |K| is the polyhedron of a finite simplicial complex K, then X satisfies the above conditions and it is shown that its Euler-Poincar´e characteristic is also given as X χ(X) = (−1)p kp , p≥0

where kp is the number of p-simplices in K. If M is a compact C ∞ manifold, it admits a triangulation by a finite simplicial complex and we have χ(M ) =

m X p=0

(−1)p bp =

m X (−1)p kp ,

(5.14)

p=0

where m is the dimension of M . Note that, if m is odd, then χ(M ) = 0 by the Poincar´e duality. Likewise we may consider the Euler-Poincar´e characteristic χ(R) of a compact manifold R with boundary.

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Complex Analytic Geometry

Euler class of the tangent bundle: Let M be a C ∞ manifold of dimension m. If E = TR M is the tangent bundle of M , then l0 = m 0 so that (K ∗ )l = M . The sections of TR M are vector fields. Definition 5.15. Let a be a point in M and U a neighborhood of a. For a non-vanishing vector field v on U r{a}, its Poincar´e-Hopf index PH(v, a) at a is the index I(v, a) as defined in (5.5). Now suppose that M is oriented so that TR M is also oriented. In this case, we define the Euler class e(M ) of M to be the Euler class of TR M . In the above, we may think of a as a vertex of K and may write (cf. (5.13)) PH(v, a) = TRese (v, TR M ; a). More generally, let S be a K0 -subcomplex of M . For a compact connected component Sλ of S and a non-vanishing vector field v on U r Sλ , where U is a neighborhood of Sλ , we define (cf. (5.13)) PH(v, Sλ ) = TRese (v, TR M ; Sλ ).

(5.15)

Let R be a compact manifold of dimension m with boundary in M . For a non-vanishing vector field v on a neighborhood of ∂R, we have the relative class eR (TR M, v) in H m (R, ∂R) (cf. Remark 5.14). We then define PH(v, R) = h[R], eR (TR M, v)i.

(5.16)

Then for a compact subcomplex S of M in the interior of R and a nonvanishing vector field v on M rS, by Theorem 5.5 (see also Remark 5.15), we have X PH(v, Sλ ) = PH(v, R). λ

In the above situation, if M is compact, we have X PH(v, Sλ ) = h[M ], e(M )i.

(5.17)

λ

On the other hand, there exists a vector field v0 having a singularity of index 1 at the barycenter of each even-dimensional simplex and a singularity of index −1 at the barycenter of each odd-dimensional simplex, and for v0 we have (cf. (5.14)) X PH(v0 , bs ) = χ(M ), s

where s runs through all the simplices in K. From (5.17), we have χ(M ) = h[M ], e(M )i.

(5.18)

Chern Classes and Localization via Obstruction Theory

151

Thus we have: Theorem 5.6. Let M be a compact oriented C ∞ manifold. For a vector field v defined and non-vanishing away from a compact subsomplex S with a finite number of connected components Sλ , X PH(v, Sλ ) = χ(M ). λ

Corollary 5.4 (Poincar´ e-Hopf theorem). Let M be as above. For a vector field v defined and non-vanishing on M , except for a finite number of points a1 , . . . , ar , r X

PH(v, ai ) = χ(M ).

i=1

Remark 5.16. Let R be a C ∞ manifold of dimension m with boundary in M . For a point p in ∂R, we take a coordinate system (U, (x1 , . . . , xm )) around p in M so that R ∩ U is given by x1 ≤ 0. We say that a tangent Pm ∂ at p is pointing outward, or inward, if a1 > 0 or vector v = i=1 ai ∂x i a1 < 0, respectively. Suppose R as above is compact and oriented. For a vector field v on a neighborhood of ∂R in M pointing outward at every point of ∂R, we have PH(v, R) = χ(R). On the other hand, if v is pointing inward at every point of ∂R, we have PH(v, R) = χ(R) − χ(∂R). Thus the above results are generalized for such vector fields. Case of complex vector bundles: From the construction we have the following (cf. Remark 5.4. 2): Proposition 5.8. If E is a complex vector bundle of rank l, we may think of it as an oriented real vector bundle of rank 2l and we have e(E, s) = cl (E, s)

in H 2l (M, M rS).

If M is oriented, we also have TRese (s, E; Sλ ) = TRescl (s, E; Sλ )

˘ m−2l (Sλ ). in H

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Complex Analytic Geometry

Suppose M is a complex manifold of dimension n. Then the q-th Chern class cq (M ) of M is defined to be the q-th Chern class cq (T M ) of the holomorphic tangent bundle T M . Recall that (cf. Proposition 3.5) T M can be naturally identified with the real tangent bundle TR M . Thus a section of T M can be considered as either a complex vector field or a real vector field. As a special case of Proposition 5.8, we have: Proposition 5.9. Let v be a section of T M defined and non-vanishing on a neighborhood of a, possibly except for at a. Then its Poincar´e-Hopf index PH(v, a) as defined in Definition 5.15 coinsides with the index as defined in Definition 5.3 (with r = 1). We also see that, since the top Chern class cn (M ) is the primary obstruction to constructing a (non-vanishing) vector field, it coincides with the Euler class e(M ) of TR M : cn (M ) = e(M ). In particular, if M is compact, (5.18) reads χ(M ) = h[M ], cn (M )i.

(5.19)

Example 5.3. Let M = P1 be the Riemann sphere with homogeneous coordinates [ζ0 , ζ1 ] (cf. Exercise 2.1). For i = 0, 1, let Ui be the open set given by ζi 6= 0. On U0 , z = ζ1 /ζ0 is a coordinate and on U1 , z 0 = ζ0 /ζ1 is a coordinate. Let a0 = [1, 0] and a1 = [0, 1]. For every integer d, we consider the holomorphic vector field v on P1 r {a0 , a1 } given by v = ∂ ∂ = −(z 0 )2−d ∂z z d ∂z 0 . We have (cf. Example 5.1) PH(v, a0 ) + PH(v, a1 ) = d + 2 − d = 2, confirming that χ(P1 ) = 2. Later in Section 8.2, we represent the Chern classes of a complex vector bundle by differential forms using connections for the bundle. If M is compact, (5.19) may be written (cf. Proposition 7.11 below) Z χ(M ) = cn (M ), (5.20) M

which is referred to as the “Gauss-Bonnet formula”.

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153

Thom class as a localized Euler class Let M be a C ∞ manifold of dimension m and π : E → M an oriented real vector bundle of rank l0 . We apply the above consideration to the “diagonal section” of the pull-back π ∗ E. Recall that (cf. (3.4)) the pull-back of E to itself is a vector bundle on E given by π ∗ E = { (ξ1 , ξ2 ) ∈ E × E | π(ξ1 ) = π(ξ2 ) }. We think of it as a vector bundle $ : π ∗ E → E on the second factor with $ the restriction of the projection. We denote by Σ the image of the zero section of π : E → M , which is narurally diffeomorphic with M . The bundle π ∗ E admits the diagonal section s∆ defined by s∆ (ξ) = (ξ, ξ) for ξ in E, whose zero set is Σ. Thus we have the localization e(π ∗ E, s∆ ) in 0 H l (E, E rΣ) of e(π ∗ E) by s∆ . Suppose M is oriented so that Σ is also oriented. We orient the total space E so that the orientation of the fiber followed by that of Σ gives the orientation of E (cf. Convention 3.3). Then we have the corresponding ˘ m (Σ) (note that E is (m+l0 )-dimensional). residue TRese (s∆ , π ∗ E; Σ) in H 0 Recall that we have the Thom class ΨE of E in H l (E, E rΣ). Theorem 5.7. In the above situation, 1. e(π ∗ E, s∆ ) = ΨE

0

in H l (E, E rΣ).

2. If M is oriented, TRese (s∆ , π ∗ E; Σ) = Σ

˘ m (Σ). in H

Proof. It suffices to prove the statement 1 as the statement 2 follows directly from 1. This time we take a triangulation K0 of E compatible with Σ. Let γ be a cocycle representing e(π ∗ E, s∆ ). For an m-simplex s not in Σ, we have hs∗ , γi = 0, as the dual l0 -cell s∗ does not intersect Σ. Suppose s is in Σ and let U be a neighborhood of bs such that we have a trivialization 0 E|U ' Rl × U . Then we have a trivialization 0

0

0

π ∗ E|π−1 (U ) ' Rl × π −1 (U ) ' Rl × Rl × U and the section s∆ is expressed as (ξ, x) 7→ (ξ, ξ, x). As the l0 -cell s∗ may be thought of as being in the fiber of E at bs , we have hs∗ , γi = 1 (cf. (5.5)). By Proposition 4.6 we have the theorem.  Remark 5.17. 1. By the “functoriality” of the obstruction cocycles (cf. Remark 5.4. 1) and Theorem 5.7, we have π ∗ e(E) = e(π ∗ E) = j ∗ ΨE . As

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Complex Analytic Geometry

the map π : E → M is a deformation retraction, it induces an isomorphism 0 0 ∼ π ∗ : H l (M ) → H l (E) and we have e(E) = (π ∗ )−1 j ∗ ΨE .

(5.21)

2. The Thom class is a universal lozalization of the Euler class in the following sense. Given a section s : M → E of E with the zero set S, which is a subcomplex of M . We have the induced morphism 0

0

s∗ : H l (E, E rΣ) −→ H l (M, M rS). By the functoriality of relative obstruction cocycles, we have e(E, s) = s∗ e(π ∗ E, s∆ ) = s∗ ΨE . If M is oriented, we have the commutative diagram: 0

H l (E, E rΣ) l0



˘ m (Σ) /H

∼ A

s∗

H (M, M rS)

∼ A



(M

·

s

)S

(5.22)

˘ m−l0 (S), /H

·

where (M s )S denotes the intersection product with s localized at S (cf. Definition 4.4). Thus we may express the residue as TRese (s, E; S) = (M

· Σ) s

S.

If E is a complex vector bundle of rank l, we have the topological localization cltop (π ∗ E, s∆ ) in H 2l (E, E r M ), which coincides with e(π ∗ E, s∆ ) (cf. Proposition 5.8) so that we have: Corollary 5.5. For a complex vector bundle E of rank l, 1. cltop (π ∗ E, s∆ ) = ΨE

in H 2l (E, E rΣ).

2. If M is oriented, TRescl (s∆ , π ∗ E; Σ) = Σ

˘ m (Σ). in H

Remark 5.18. 1. Remark 5.17. 2 applies with e and l0 replaced by cl and 2l, i.e., the Thom class of a complex vector bundle is a universal localization of the top Chern class. Namely, If E → M is a complex vector bundle of rank l and s a section of E on M with the zero set S, which is a subcomplex of M , then cltop (E, s) = s∗ ΨE .

(5.23)

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155

We come back to this point and review this from differential geometric viewpoint in Section 10.4 below (cf. also Section 7.9). 2. The previous example of hyperplane bundle (Proposition 5.6) may be thought of as a special case of Corollary 5.5. Let Pn be the projective space with homogeneous coordinates [ζ0 , . . . , ζn ]. Also let H be the hyperplane defined by ζ0 = 0 and p the point [1, 0, . . . , 0]. Recall that (cf. Exercise 3.14) the map π : Pn r{p} → H, [ζ] 7→ [0, ζ1 , . . . , ζn ], may be identified with the hyperplane bundle π : Hn−1 → Pn−1 . With this identification, π ∗ Hn−1 is the restriction of Hn → Pn to Hn−1 and the diagonal section s∆ is the restriction of the canonical section to Hn−1 : ? _ π ∗ Hn−1 / Hn−1 Hn o  Pn o



? _ Hn−1

π



π

/ Pn−1 .

Moreover, by excision we have H 2 (Pn , Pn rH) = H 2 (Hn−1 , Hn−1 rH). Whitney sum formula for localized classes Let E be an oriented real vector bundle of rank l0 on M . Also let s be a section of E on M whose zero set S is a K0 -subcomplex of M . In this 0 situation, we have the localization e(E, s) in H l (M, MrS). If M is oriented, ˘ m−l0 (S) as the image of e(E, s) by we have the residue TRese (s, E; S) in H the Alexander duality 0 ∼ ˘ H l (M, M rS) −→ H m−l0 (S). Note that, if s is the zero section, then S = M and e(E, s) is the Euler class e(E). For i = 1, 2, let Ei be an oriented real vector bundle of rank li0 on M . Also let si be a section of Ei on M whose zero set Si is a K0 -subcomplex of M . We set E = E1 ⊕ E2 , l0 = l10 + l20 , and s = s1 ⊕ s2 . Then the zero set S of s is given by S = S1 ∩ S2 . In this situation, we have the localizations 0 0 e(Ei , si ) in H li (M, M rSi ), i = 1, 2, and e(E, s) in H l (M, M rS). We also have the cup product 0

`

0

0

H l1 (M, M rS1 ) × H l2 (M, M rS2 ) −→ H l (M, M rS). Theorem 5.8. In the above situation, 1. e(E, s) = e(E1 , s1 ) ` e(E2 , s2 ). 2. If M is oriented,

·

TRese (s, E; S) = (TRese (s1 , E1 ; S1 ) TRese (s2 , E2 ; S2 ))S

˘ m−l0 (S). in H

156

Complex Analytic Geometry

Proof. The statement 1 follows from Corillary 4.5 and Remark 5.17. 2 and the statement 2 from Theorem 4.8.  In the case of complex vector bundles, we have: Corollary 5.6. In the above situation, suppose Ei is a complex vector bundle of rank li , i = 1, 2, and set l = l1 + l2 . 1 2 1. cltop (E, s) = cltop (E1 , s1 ) ` cltop (E2 , s2 ).

2. If M is oriented,

·

TRescl (s, E; S) = (TRescl1 (s1 , E1 ; S1 ) TRescl2 (s2 , E2 ; S2 ))S ˘ m−2l (S). in H Following is an application of the above, which will be stated for complex vector bundles and Chern classes, however, it holds for real oriented vector bundles and Euler classes as well. Suppose we have an exact sequence of complex vector bundles on M : 0 −→ E 0 −→ E −→ E 00 −→ 0. We denote by l0 , l and l00 the (complex) ranks of E 0 , E and E 00 , respectively. Let s be a section of E 0 whose zero set S is a K0 -subcomplex of M . We may think of s also as a section of E so that we have the localizations 0 cl (E 0 , s) and cl (E, s). Moreover, if M is oriented, we have the residues ˘ m−2l (S). ˘ m−2l0 (S) and TRescl (s, E; S) in H TRescl0 (s, E 0 ; S) in H Proposition 5.10. In the above situation, we have: 00

TRescl (s, E; S) = TRescl0 (s, E 0 ; S) a i∗ cl (E 00 )

˘ m−2l (S), in H

where i : S ,→ M denotes the inclusion. Proof. By Proposition 3.3, E is isomorphic with E 0 ⊕ E 00 and s corre0 00 sponds to s ⊕ 0. Thus by Corollary 5.6, cl (E, s) = cl (E 0 , s) ` cl (E). The proposition then follows from Proposition 4.3, letting p = 2l0 , r = 2l and 00 γ = cl (E 00 ).  Example 5.4. Let P2 be the projective plane with homogeneous coordinates [ζ0 , ζ1 , ζ2 ] and set M = P2 r {[0, 0, 1]}, i.e., the total space of the hyperplane bundle on P1 = {[ζ0 , ζ1 ]}. The manifold M is covered by two open sets U (i) given by ζi 6= 0, i = 0, 1. On U (0) , we set z1 = ζ1 /ζ0 and z2 = ζ2 /ζ0 , and on U (1) , we set z10 = ζ0 /ζ1 and z20 = ζ2 /ζ1 . Consider the vector field v (section of T M ) given by ∂ ∂ = z20 0 . v = z2 ∂z2 ∂z2

Chern Classes and Localization via Obstruction Theory

157

The singular set S of v is the projective line ζ2 = 0. We wish to find Resc2 (v, T M ; S), which is in H0 (S) and is equal to the Poincar´e-Hopf index of v at S. Let π : M → S be the bundle map so that we have the exact sequence 0 −→ T π −→ T M −→ π ∗ T S −→ 0. We may think of v as a section of T π and its transverse residue is 1. By Proposition 5.5, we have TResc1 (v, T π; S) = S. Using Proposition 5.10, we have TResc2 (v, T M ; S) = S a i∗ c1 (π ∗ T S) = S a c1 (T S) = χ(S) = 2. Note that v can be extended to a vector field on P2 , which is given by v = −z100

∂ ∂ − z200 00 00 ∂z1 ∂z2

in the coordinates z100 = ζ0 /ζ2 and z200 = ζ1 /ζ2 . Thus v has a singularity of index 1 at [0, 0, 1] and the above result is consistent with the Poincar´e-Hopf theorem, as χ(P2 ) = 3. Remark 5.19. Later we show the Whitney sum formula for the total Chern classes (cf. (8.15)) as well as the formula corresponding to the one in Proposition 5.10 (cf. Theorem 10.7 and Corollary 10.2) in the framework of Chern-Weil theory. See also Corollary 10.5 for the differential geometric counterpart of Corollary 5.6. The Whitney sum formula is further refined in Section 14.1 below. Intermediate Thom classes The identity in Corollary 5.5. 1 suggests the possibility of defining the “intermediate Thom classes” for a complex vector bundle by localizing appropriate Chern classes. Let M be a C ∞ manifold of dimension m and π : E → M a complex vector bundle of rank l. Let E r = E ⊕ · · · ⊕ E be the direct sum of r copies of E with the projection ρ : E r → M . Consider the “fiber product”: /E

ρ∗ E

π

$

 Er

ρ

 / M.

Thus ρ∗ E is a vector bundle of rank l on E r given by ρ∗ E = { (ξ, (ξ1 , . . . , ξr )) ∈ E × E r | π(ξ) = ρ(ξ1 , . . . , ξr ) }.

(5.24)

158

Complex Analytic Geometry

Note that the condition π(ξ) = ρ(ξ1 , . . . , ξr ) means that ξ1 , . . . , ξr and ξ are all in the same fiber of π. The bundle ρ∗ E has a natural r-section, i.e., (r) the “diagonal r-section” s∆ = (s1 , . . . , sr ) given by si (ξ1 , . . . , ξr ) = (ξi , (ξ1 , . . . , ξr )). Let W (l, r) denote the Stiefel manifold of ordered r frames in Cl (cf. Section 3.5). It is an open subset in (Cl )r = Clr and is given by W (l, r) = (Cl )r rΣ0 , where Σ0 = { (v1 , . . . , vr ) ∈ (Cl )r | v1 ∧ · · · ∧ vr = 0 }.

(5.25)

lr

If we represent elements in C by l × r matrices, Σ0 is the common zero set of all the r × r minors and has codimension l − r + 1. Let W (E, r) denote the Stiefel bundle of r frames in E, which is a fiber bundle with fiber W (l, r) associated with E and is an open subset in E r . We have W (E, r) = E r rΣ, where Σ = { (ξ1 , . . . , ξr ) ∈ E r | ξ1 ∧ · · · ∧ ξr = 0 }. (r) s∆ : r

It coincides with the singular set of Σ = (r) localization cq (ρ∗ E, s∆ ) ∈ H 2q (E r , E rΣ) by q = l − r + 1.

(5.26)

(r) S(s∆ ). Thus we have the (r) s∆ of cq (ρ∗ E) ∈ H 2q (E r ),

Definition 5.16. The topological q-th Thom class ΨEq of E is defined by (r)

ΨEq = cqtop (ρ∗ E, s∆ )

in H 2q (E r , E r rΣ).

Note that ΨEl = ΨE is the Thom class defined previously. We have a diagram as (4.23) with l0 , Ex , Ex r0, E, E rM and π replaced by 2q, Exr , Exr rΣx , E r , E r rΣ and ρ, respectively. Noting that δ ∗ is an isomorphism and Exr rΣx ' W (l, r), we have H 2q (Exr , Exr rΣx ) ' H 2q−1 (W (l, r)) ' Z. We also have: cq (E) = (ρ∗ )−1 j ∗ ΨEq . As in the case of Thom class, the intermediate Thom class ΨEq may be thought of as a universal localization of cq (E). In general, if we have a collection s(r) = (s1 , . . . , sr ) of r sections of E, we may think of it as a section of ρ : E r → M , which induces a morphism ∗

s(r) : H 2q (E r , E r rΣ) −→ H 2q (M, M rS), where S = S(s(r) ). We have ∗

cq (E, s(r) ) = s(r) ΨEq .

Chern Classes and Localization via Obstruction Theory

159

If M is oriented, we have a commutative diagram: H 2q (E r , E r rΣ) s(r)

∼ A

˘ m+2lr−2q (Σ) /H

∗

 H 2q (M, M rS)

∼ A



(M

·

s(r)

)S

˘ m−2q (S). /H

(r)

We have TRescq (ρ∗ E, s∆ ; Σ) = A(ΨEq ) = Σ and TRescq (E, s(r) ; S) = (M

·

s(r) Σ)S .

Remark 5.20. In Section 10.4 below we define the classes corresponding to the above via differential geometric localization. Thus we refer the above classes as topological intermediate Thom classes. Notes Obstruction theory for characteristic classes of fiber bundles is thoroughly explained in [Steenrod (1951)], to which we refer for details of the materials in this chapter. The descriptions given here are modifications of those in [Brasselet, Lehmann, Seade and Suwa (2001)]. As a general reference for characteristic classes, we list [Milnor and Stasheff (1974)], to which we refer for the equality (5.6), see also [Hirzebruch (1966)]. For piecewise-linear manifolds, we refer to [Hirsch and Mazur (1974); Rourke and Sanderson (1972)]. In particular, Theorem 5.1 is due to [Hirsch and Mazur (1974)]. See [Brasselet, Seade and Suwa (2009)] and references therein for Remark 5.16. Let E be a complex vector bundle of rank l on an oriented C ∞ manifold M . Also let s be a section of E with zero set S. Then TRescl (s, E; S) corresponds to what is called the “localized top Chern class” defined in the algebraic category in [Fulton (1984)]. The formula in Proposition 5.10 above corresponds to the one in Example 14.1.3 in there and the formulas in Corollary 5.6 to the one in Example 17.4.8.

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Chapter 6

Differential Forms

Differential forms are objects that represent cochains explicitly in the cohomology theory of C ∞ manifolds, the exterior derivative corresponding to the coboundary operator and the dual operation being given by integration. In fact, if we pass on to the cohomology, the cohomology defined by differential forms is canonically isomorphic with the (singular or simplicial) cohomology of the manifold. That is the content of the de Rham theorem, which we prove in the next chapter. In this setting, the exterior product corresponds to the cup product. In this chapter, we discuss some topics related to differential forms on ∞ C and complex manifolds. Integration and integration along fibers are discussed. We also recall the Frobenius theorem and non-singular holomorphic foliations.

6.1

Vector fields and differential forms

Let M be a C ∞ manifold and U an open set in M . Recall that a vector field on U is a section on U of the tangent bundle TR M of M (cf. Section 3.4). Differential forms are alternating forms on the space of vector fields: Definition 6.1. A complex valued differential p-form of class C ∞ , a C ∞ Vp c p-form for short, on U is a C ∞ section of the bundle (TR M )∗ on U . We denote by Ap (U ) the set of C ∞ p-forms on U , which is by definition Vp C ∞ (U ; (TRc M )∗ ) . The set A0 (U ) is thought of as the set of C ∞ functions on U and it has the structure of a C-algebra. For each p, Ap (U ) has the structure of an A0 (U )-module.

161

162

Complex Analytic Geometry

Let {Uα } be a covering of M by coordinate neighborhoods Uα with α α α coordinates (xα 1 , . . . , xm ). For a point x in Uα , we denote by (dx1 , . . . , dxm ) ∂ ∂ ∗ the basis of TR ,x M dual to the basis ( ∂xα , . . . , ∂xα ) of TR ,x M . A p-form m 1 ω on U is expressed as, on U ∩ Uα , X α ω= fiα1 ,...,ip (x) dxα (6.1) i1 ∧ · · · ∧ dxip , 1≤i1 k, we prove the lemma by induction on k. If k = 0, then ω = 0 and the lemma holds by setting η = 0. Suppose it holds for k = k0 and we prove it for k = k0 + 1. We may write ω = ω1 + ω2 ∧ d¯ zk0 +1 , where ω1 and ω2 are forms not involving d¯ zk0 , k 0 > k0 . We may write X ω2 = fJ d¯ zJr{k0 +1} . J3k0 +1

¯ 1 does not have terms involving d¯ Since ∂ω zk0 +1 ∧ d¯ zj for j > k0 + 1 and ¯ ¯ ¯ 0 = ∂ω = ∂ω1 + ∂ω2 ∧ d¯ zk0 +1 , we see that ∂fJ =0 ∂ z¯j for J containing k0 + 1 and j > k0 + 1. Thus if we set Z 1 dζk0 +1 ∧ dζ¯k0 +1 √ gJ (z) = , fJ (z1 , . . . , ζk0 +1 , . . . , zn ) ζk0 +1 − zk0 +1 2π −1 U k0 +1 by Lemma 11.3, we have ∂gJ = fJ ∂ z¯k0 +1 If we set

and

η0 =

∂gJ = 0, ∂ z¯j

X

for j > k0 + 1.

gJ d¯ zJr{k0 +1} ,

J3k

¯ 0 does not involve d¯ ¯ the form ω − (−1)q−1 ∂η zk0 , k 0 > k0 and ∂-closed. By induction hypothesis, there exists a (0, q − 1)-form η1 such that ω − q−1 ¯ 0 = ∂η ¯ 1 and ω = ∂((−1) ¯ (−1)q−1 ∂η η0 + η1 ).  Corollary 11.3. The Dolbeault complex (11.7) is acyclic and gives a resolution 0 → Ω p → A p,• of the sheaf Ω p of holomorphic p-forms. Remark 11.5. The above is a weak form in the sense that the domain of definition of η as given in the proof is smaller than that of ω. This is sufficient to have Corollary 11.3 and the following theorem. For the full version we need further assumption on U . For example, if ¯ each Ui is convex, we may proceed further to prove that if ω is ∂-closed on ¯ U , it is ∂-exact on U . This is one of the essential ingredients in the proof of Theorem 11.7. Theorem 11.9 (Dolbeault theorem). There is an isomorphism q p H∂p,q ¯ (M ) ' H (M ; Ω ).

Further Topics

353

Proof. We set A p,n+1 = 0 and, for r = 0, . . . , n, we denote by K p,r the kernel of ∂¯p,r : A p,r → A p,r+1 . Then we have K p,0 = Ω p and K p,n = A p,n . We have the short exact sequences: 0 −→ K

p,r

∂¯

−→ A p,r −→ K

p,r+1

−→ 0,

0 ≤ r ≤ n,

where we set K p,n+1 = 0. We have the associated long exact sequence (cf. Theorem 11.1): · · · −→ H q (M ; K δ

∗

−→ H

p,r

q+1

∂¯∗

) −→ H q (M ; A p,r ) −→ H q (M ; K

(M ; K

p,r

) −→ H

q+1

(M ; A

p,r

p,r+1

)

) −→ · · · .

(11.14)

First, letting q = r = 0, we have the exact sequence ∂¯∗

0 −→ Γ (M ; Ω p ) −→ Γ (M ; A p,0 ) −→ Γ (M ; K

p,1

),

which gives the theorem for q = 0. Suppose q ≥ 1. Since the sheaf A p,r is fine, by Proposition 11.4, δ ∗ : H q (M ; K

p,r+1

∼

) −→ H q+1 (M ; K

p,r

).

Thus by a repeated use of the above H q (M ; Ω p ) = H q (M ; K

p,0

) ' H 1 (M ; K

p,q−1

δ∗

p,q−1

).

From (11.14), ∂¯∗

Γ (M ; A p,q−1 ) −→ Γ (M ; K

p,q

) −→ H 1 (M ; K

Thus we have the theorem for q ≥ 1.

) −→ 0. 

From Theorems 11.9 and 11.7, we have: Corollary 11.4. For a holomorphically convex domain D, the Dolbeault ¯ is acyclic: complex (Γ (D; A p,• ), ∂) H∂p,q ¯ (D) = 0

for q ≥ 1.

Remark 11.6. 1. We use the above theorem for M holomorphically convex domains, i.e., the case the right-hand side vanishes for q ≥ 1, to obtain a canonical Dolbeault theorem (cf. Corollary 11.6 below). The difference between the above isomorphism and the canonical one is a sign of q(q+1) (−1) 2 . ˇ r (M ; C) is isomorphic 2. We could also prove, for a C ∞ manifold M , that H with Hdr (M ) in a similar manner, using the resolution 0 → C → A • . However the isomorphism obtained this way differs from the canonical one r(r+1) (cf. Corollary 7.4, also (11.8) and (11.9)) by a sign of (−1) 2 .

354

Complex Analytic Geometry

ˇ Cech-Dolbeault cohomology Let M be a complex manifold of dimension n. We have the Dolbeault com¯ Let U = {Uα }α∈I be an open covering of M . Then we have plex (A p,• , ∂). ¯ the double complex (C • (U; A p,• ), δ, (−1)• ∂). We denote the associated p,• ¯ single complex by (A (U), ϑ). Thus M ¯ ϑ¯ = δ + (−1)q1 ∂. Ap,q (U) = C q1 (U; A p,q2 ), q1 +q2 =q

ˇ ˇ We could define the Cech-Dolbeault cohomology as in the case of Cechde Rham cohomology. Here we define simply as ˇ Definition 11.16. The Cech-Dolbeault cohomology of type (p, q) of U is the q-th hypercohomology of A p,• on U: q p,• ). Hϑp,q ¯ (U) = H (U; A

From Proposition 11.6 we have the following, which can also be proved as Theorem 7.1: Theorem 11.10. If U is locally finite, then the inclusion Ap,q (M ) ,→ C 0 (U, Ap,q ) ⊂ Ap,q (U) induces an isomorphism ∼

p,q H∂p,q ¯ (M ) −→ Hϑ ¯ (U).

We consider only locally finite coverings hereafter. Definition 11.17. A covering U = {Uα } of M is analytically good, if every non-empty intersection Uα0 ...αr is biholomorphic to a holomorphically convex domain in Cn . By Corollary 11.4, if U is analytically good, it is good for A p,• (cf. Definition 11.9). From Proposition 11.7, we have the following, which can also be proved as Theorem 7.2: Theorem 11.11. If U is analytically good, the inclusion C q (U; Ω p ) ,→ C q (U; Ap,0 ) ⊂ Ap,q (U) induces an isomorphism ∼

H q (U; Ω p ) −→ Hϑp,q ¯ (U). From Theorems 11.10 and 11.11, we have: Corollary 11.5. If U is analytically good, there is a canonical isomorphism q p H∂p,q ¯ (M ) ' H (U; Ω ).

Further Topics

355

Note that every complex manifold admits an analytically good covering and the set of analytically good coverings is cofinal in the set of coverings of M . Since H q (U; Ω p ) is independent of the analytically good covering U by Corollary 11.5, we have: Theorem 11.12. If U is an analytically good covering of a complex manifold M , the canonical morphism is an isomorphism: ∼

H q (U; Ω p ) −→ H q (M ; Ω p ). Combining with Corollary 11.5, we have: Corollary 11.6 (Canonical Dolbeault theorem). There is a canonical isomorphism q p H∂p,q ¯ (M ) ' H (M ; Ω ).

¯ An explicit correspondence is given as (7.10) with D replaced by ϑ. In fact we used the Dolbeault theorem (Theorem 11.9) for Corollary 11.6, however the advantage of this is that it gives a canonical correspondence. In the following, we consider the covering U = {Ui }ni=1 of Cn r{0} given by Ui = {zi 6= 0}. It is an analytically good covering. The n-form, the Cauchy form in n-variables,  1 n dz ∧ · · · ∧ dz 1 n √ κn = z · · · z 2π −1 1 n may be thought of as a cocycle c in C n−1 (U; Ω n ) given by c1...n = κn . We ¯ also have the Bochner-Martinelli form βn , which is a ∂-closed (n, n−1)-form n on C r{0} (cf. Section 7.10). Proposition 11.16. In the isomorphism (Cn r0) ' H n−1 (U; Ω n ) H∂n,n−1 ¯ of Corollary 11.6, the class of βn corresponds to the class of (−1)

n(n−1) 2

κn .

Proof. If n = 1, the cohomologies are the same and β1 = κ1 . Thus we assume n ≥ 2. We may think of βn as being in C 0 (U; A n,n−1 ) ⊂ A n,n−1 (U) and κn in C n−1 (U; A n,0 ) ⊂ A n,n−1 (U). We construct a cochain χ in Ln−2 A n,n−2 (U) = p=0 C p (U; A n,q ), q = n − p − 2, so that βn − (−1)

n(n−1) 2

¯ κn = ϑχ

in A n,n−1 (U).

356

Complex Analytic Geometry

Pn−2 Writing χ = p=0 χp , χp ∈ C p (U; A n,q ), this is expressed as  ¯ 0,  βn = ∂χ  ¯ p, 0 = δχp−1 + (−1)p ∂χ 1 ≤ p ≤ n − 2,   −(−1) n(n−1) n−2 2 κ = δχ .

(11.15)

n

Note that the condition in the middle is vacuous if n = 2. Let 0 ≤ p ≤ n − 2 so that 0 ≤ q ≤ n − 2. For a (p + 1)-tuple of integers I = (i0 , . . . , ip ) with 1 ≤ i0 < · · · < ip ≤ n, let I ∗ = (j0 , . . . , jq ) denote the complement of {i0 , . . . , ip } in {1, . . . , n} with 1 ≤ j0 < · · · < jq ≤ n. Setting Φ(z) = dz1 ∧ · · · ∧ dzn , zI = zi0 · · · zip , |I ∗ | = j0 + · · · + jq and q X d (−1)µ z¯jµ d¯ zj0 ∧ · · · ∧ d¯ zjµ ∧ · · · ∧ d¯ zjq , Φ¯I ∗ (z) = µ=0

we define a cochain χp by q! Cn Φ¯I ∗ (z) ∧ Φ(z) q(n + p − 1) χpI = (−1)εI . , εI = |I ∗ | + (n − 1)! zI kzk2(q+1) 2 Then it can be shown that it satisfies (11.15).



Remark 11.7. Let W be a holomorphically convex neighborhood of 0 in Cn and W = {Wi }ni=1 the covering of Wr0 given by Wi = W ∩Ui . Then we have a canonical isomorphism H∂n,n−1 (Wr0) ' H n−1 (W; Ω n ), under which ¯ n(n−1)

the class of βn (restricted to Wr0) corresponds to the class of (−1) 2 κn (restricted to W). Suppose the class of θ corresponds to the class of γ under ¯ = 0, the above isomorphism. If h is a holomorphic function on V , since ∂h we see that the class of hθ corresponds to the class of hγ (cf. the relation ˇ similar to (7.10) for Cech-Dolbeault cocycles and (11.15)). In the above situation set R1 = { z ∈ Cn | kzk2 ≤ nε2 }. The boundary ∂R1 is a usually oriented (2n − 1)-sphere S2n−1 . We also set Γ = { z ∈ Cn | |zi | = ε, i = 1, . . . , n }, which is an n-cycle oriented so that arg z1 ∧ · · · ∧ arg zn is positive. ¯ Proposition 11.17. Let θ be a ∂-closed (n, n − 1)-form on Cn r 0 and γ n−1 n a cocycle in C (U; Ω ). If the class of θ corresponds to the class of γ by the canonical isomorphism H∂n,n−1 (Cn r0) ' H n−1 (U; Ω n ), ¯ then Z Z θ = (−1) S2n−1

n(n−1) 2

γ. Γ

Further Topics

Proof.

357

Recall that we have canonical isomorphisms ∼

∼

H∂n,n−1 (Cn r0) −→ Hϑn,n−1 (U) ←− H n−1 (U; Ω n ). ¯ ¯ The assumption implies that there exists a cochain χ in A n,n−2 (U) such ¯ Consider the commutative diagram that θ − γ = ϑχ. A n,n−2 (U) A



¯ ϑ

n,n−1

/ A 2n−2 (U) D

(U)

 / A 2n−1 (U)

/ A 2n−2 (U ∩ S2n−1 ) D

R  / A 2n−1 (U ∩ S2n−1 ) S2n−1/ C,

where U ∩S2n−1 denotes the covering of S2n−1 consisting of the Ui ∩S2n−1 ’s. For each i = 1, . . . , n, we set Qi = { z ∈ S2n−1 | |zi | ≥ |zj | for all j 6= i }. Then {Qi } is a honeycomb system adapted to U ∩ S2n−1 and, by the Stokes ˇ formula for Cech-de Rham cochains, Z Z n Z n−1 X X X Z γ. θ− (Dχ)i0 ...ip = 0= Dχ = S2n−1

Qi0 ...ip

p=0 i0 0} for some real valued C ω function fij on U . 387

388

Complex Analytic Geometry

For example, a variety in a complex manifold is semianalytic. This can be seen by taking the real and imaginary parts of defining functions at each point in M (cf. Definition 2.7). Definition 12.2. A subset X in M is subanalytic if every point a in M has a neighborhood U such that X ∩ U = π(A), where A is a relatively compact semianalytic set in M ×M 0 , for some C ω manifold M 0 , and π : M ×M 0 → M the projection. Semianalytic sets are subanalytic. Also, a polyhedron P in RN (cf. Definition 5.6) is subanalytic in an open set in RN containing P as a closed set. Let X ⊂ M and X 0 ⊂ M 0 be subanalytic sets. A continuous map f : X → X 0 is said to be subanalytic if the graph of f is subanalytic in M × M 0. Let X be a subanalytic set in M and K a simplicial complex in RN (cf. Definition B.13). Thus |K| is subanalytic in some open set in RN . Definition 12.3. A subanalytic triangulation of X is a triangulation (K, h) of X such that h : |K| → X is a subanalytic homeomorphism. We quote the following: Theorem 12.1. Let X be a subanalytic set in a C ω manifold M . Then: 1. There exists a subanalytic triangulation (K, h) of M compatible with X such that h is C 1 . We call such a triangulation a C 1 triangulation of (M, X) for short. 2. If (K1 , h1 ) and (K2 , h2 ) are C 1 triangulations of (M, X), there exists a common refinement (K, h), i.e., (K, h) is a C 1 triangulation of (M, X) and there is a commutative diagram 9 |K1 |

g1

h1

h

|K| g2

%

% /M 9

h2

|K2 |

,

where, for each i = 1, 2, (1) (K, gi ) is a C 1 triangulation of (|Ki |, |Li |), Li being the subcomplex of Ki with hi (|Li |) = X,

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(2) every simplex of Ki is the union of the images by gi of a finite number of simplices of K. In the statement 1 above, h being C 1 means the following. For each simplex s of K, we think of it as being in the affine space spanned by s and require that h|s be C 1 in the sense of Definition 3.19 with C ∞ replaced by C 1 . Contrary to the case of C ∞ triangulations as in Definition 4.1, we do not require that the rank of h|s at each point of s be equal to dim s. In the theorem above, the set of points in s where the rank of h|s is smaller than dim s has measure zero in s. 12.2

Residues of Chern classes on manifolds

Let M be a complex manifold of dimension n and E a holomorphic vector bundle of rank l on M . Let s(r) = (s1 , . . . , sr ) be a holomorphic r-section of E. Then it defines a complex space S = (S, OS ) in M (cf. Section 11.5). It is given as follows. Let F be the ideal sheaf in OM locally generated by the (r × r)-minors of the matrix of the local components of the si ’s. The support S of S is the support of OM /F , which is the singular set S(s(r) ) of s(r) and OS = ι−1 (OM /F ), where ι : S ,→ M is the inclusion. By (A.16), We have dim S ≥ n − q, q = l − r + 1 (cf. (A.16), in the case r = 1 this also follows from Theorem 2.11). We take a C 1 triangulation K0 of M compatible with S as in Theorem 12.1 and let K be its barycentric subdivision. Also, let K ∗ denote the cellular decomposition dual to K, which is constructed from the barycentric subdivision K 0 of K (cf. Section 4.1). In this situation, we have the topological localization cqtop (E, s(r) ) in 2q H (M, MrS; Z) (cf. Section 5.6) and the differential geometric localization cqdiff (E, s(r) ) in H 2q (M, M r S; C) (cf. Section 10.3). We also have the ˘ 2(n−q) (S; Z) and Rescq (s(r) , E; S) associated residues TRescq (s(r) , E; S) in H ˘ in H2(n−q) (S; C). We have seen (cf. Corollary 10.6) that Rescq (s(r) , E; S) is the image of TRescq (s(r) , E; S) by the canonical morphism ˘ 2(n−q) (S; Z) −→ H ˘ 2(n−q) (S; C). H

(12.1)

Now we consider the case where S is of pure dimension n − q. In the following, we refer this situation as the proper case. Let (Si )i be the (global) irreducible components of S. Then the set {Si }i is locally finite. ˘ 2(n−q) (S; Z) and it is the free Abelian group Each Si defines a class [Si ] in H generated by the [Si ]’s. Thus the morphism (12.1) is injective so that we

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may identify the two residues: Rescq (s(r) , E; S) = TRescq (s(r) , E; S). (12.2) S Let pi be a non-singular point of Si◦ = Si r j6=i Sj and Di a complex slice of Si in M at pi (cf. Definition 3.24). Recall that Di is a locally closed complex sudmanifold of dimension q in M through pi and transverse to Si (r) at pi . The r-section si = s(r) |Di of Ei = E|Di has an isolated singularity (r) at pi so that we have the residue Rescq (si , Ei ; pi ), which is an integer (cf. Theorem 10.11). Considering a triangulation as above of a neighborhood of S that contains Di as a closed set and is compatible with S ∪ Di , we may assume that pi is the barycenter bs of a 2(n − q)-simplex s in the non-singular part of Si◦ and that the 2q-cell s∗ dual to s is in Di . Then by Corollary 10.6, or as a special case of (12.2), (r)

(r)

Rescq (si , Ei ; pi ) = TRescq (si , Ei ; pi ),

pi = bs .

Note that this number does not depend on the choice of pi on the nonsingular part of Si◦ , since the residue is locally constant in pi (topologically, this follows from the homotopy invariance of the indices and differential geometrically by continuity of the residue with respect to pi ) and the nonsingular part of Si◦ is connected. We restate Corollary 5.1 (see also Corollary 10.1) in the above situation: Theorem 12.2. Suppose S is pure (n − q)-dimensional. Then we have: X (r) ˘ 2(n−q) (S; Z). TRescq (s(r) , E; S) = TRescq (si , Ei ; pi ) · [Si ] in H i

Thus we see that, in order to find the residue in the proper case, it suffices to know the residue at an isolated singularity. We discuss this situation in Section 12.4 below. 12.3

Grothendieck residues

In this section, we briefly review Grothendieck residues, which are used for analytic expressions of the residues. Let U be a neighborhood of the origin 0 in Cn and f1 , . . . , fn holomorphic functions on U such that their common set of zeros V (f1 , . . . , fn ) consists only of 0. For small positive numbers εi , i = 1, . . . , n, we set Γ = { z ∈ U | |fi (z)| = εi , i = 1, . . . , n },

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which is an n-cycle in U . It is oriented so that the form dθ1 ∧ · · · ∧ dθn is positive, θi = arg fi . For a holomorphic n-form ω on U , we set    Z ω 1 n ω √ (12.3) Res0 = f1 , . . . , fn f · · · fn 2π −1 1 Γ and call it the Grothendieck residue of ω/f1 · · · fn at 0. Note that this residue is alternating in (f1 , . . . , fn ). In general, this is computed as follows. From the condition V (f1 , . . . , fn ) = {0}, we see p that, for each i, zi is in the radical (f1 , . . . , fn ) of the ideal (f1 , . . . , fn ) (cf. Theorem 2.8 (Nullstellensatz)). Hence there is a positive integer ki Pn such that ziki is in (f1 , . . . , fn ) and we may write ziki = j=1 cij fj with cij ∈ On . Then     det(cij )ω ω Res0 = Res0 k1 . f1 , . . . , fn z1 , . . . , znkn If we write ω = f dz1 ∧ · · · ∧ dzn with f in On , the right-hand side of the above is, by the Cauchy integral formula, the coefficient of z1k1 −1 · · · znkn −1 in the power series expansion of f det(cij ). In particular, if f = (f1 , . . . , fn ) is non-degenerate, i.e., if the Jacobian Jf = det (∂fi /∂zj ) is non-zero at 0, then we have   f (0) ω Res0 = . f1 , . . . , f n Jf (0) Example 12.1. In the case n = 1, the residue (12.3) is the usual Cauchy residue at 0 of the meromorphic 1-form ω/f1 . Example 12.2. If ω = df1 ∧ · · · ∧ dfn , then   df1 ∧ · · · ∧ dfn Res0 f1 , . . . , fn is a positive integer which is simultaneously equal to (cf. Sections 12.4 and 12.5 below): (1) the mapping degree of f = (f1 , . . . , fn ), thus the Poincar´e-Hopf index Pn at 0 in Cn of the vector field v = i=1 fi · ∂/∂zi , (2) dimC O/(f1 , . . . , fn ), where O denotes the ring of germs at 0 of holomorphic functions on Cn and (f1 , . . . , fn ) the ideal generated by the germs of f1 , . . . , fn at 0. It can also be interpreted as the intersection number (D1 at 0 of the divisors Di defined by fi (cf. (14.9) below).

· ··· ·D

n )0

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Example 12.3. In particular, if fi = ∂f /∂zi for some holomorphic function f on U , then it is the Milnor number µ(V, 0) of the hypersurface V defined by f at 0: " ∂f

# ∂f d ∂z1 ∧ · · · ∧ d ∂z n = µ(V, 0), Res0 ∂f ∂f ∂z1 , . . . , ∂zn which is also the multiplicity m(f, 0) of f at 0 (cf. Section 11.7 and Section 12.5 below).

12.4

Residues at an isolated singularity

Let M be a complex manifold of dimension n and E a holomorphic vector bundle of rank l on M , l ≥ n. Let r = l − n + 1 and suppose we have a holomorphic r-section s(r) with an isolated singularity at p in M . Thus this is a proper case. In this situation, we have TRescn (s(r) , E; p) and Rescn (s(r) , E; p), which are identified. It is an integer, in fact we will see that it is positive in the holomorphic case. Recall that Rescn (s(r) , E; p) is expressed as follows (cf. (10.11)). Let U be a neighborhood of p where the bundle E is trivial. We may assume that U is a coordinate neighborhood and sometimes we identify p with 0 in Cn and U with a neighborhood of 0. Letting U0 = U r{p} and U1 = U , we consider the covering {U0 , U1 } of U . We take an s(r) -trivial connection ∇0 for E on U0 and a connection ∇1 for E on U1 trivial with respect to some holomorphic frame e(l) = (e1 , . . . , el ) of E. Thus cn (∇0 ) = 0 and cn (∇1 ) = 0. Let R be a compact real 2n-dimensional manifold with C ∞ boundary in U containing p in its interior. Then we have Z Rescn (s(r) , E; p) = − cn (∇0 , ∇1 ). (12.4) ∂R

We give various expressions of this number. Topological expression We have already seen that Rescn (s(r) , E; p) may be expressed as a mapping degree, even if E and s(r) are not holomorphic, in fact s(r) may not be defined at p (cf. Definition 5.3, (5.10) and Theorem 10.11). Let S2n−1 denote a small (2n − 1)-sphere in U with center p. Then we have the mapping as given in (5.1): ϕ : S2n−1 −→ W (l, r).

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Theorem 12.3. In the above situation, we have: Rescn (s(r) , E; p) = deg ϕ. Remark 12.1. If E and s(r) are holomorphic, then Rescn (s(r) , E; p) is a positive integer, by our orientation convention. Residue at a non-degenerate zero: Let us assume that l = n. Thus Pn r = 1 and we have only one section s. We write s = i=1 fi ei with fi holomorphic functions on U . We have W (l, r) = W (n, 1) = Cn r{0}, which deformation retracts to S2n−1 . We say that p is a non-degenerate zero of s if det

∂(f1 , . . . , fn ) (p) 6= 0. ∂(z1 , . . . , zn )

In this case, (f1 , . . . , fn ) form a coordinate system around p. Hence from Theorem 12.3 we have: Proposition 12.1. If p is a non-degenerate zero of s, Rescn (s, E; p) = 1. Analytic expression First we prepare some preliminary materials. Lemma 12.1. Let V be a germ of a variety of dimension d at 0 in Cn and g1 , . . . , gN germs in On . Suppose V (g1 . . . , gN ) ∩ V = {0}. Then there exists an N × d matrix C = (cij ) of complex numbers such that, for germs PN fj = i=1 cij gi , j = 1, . . . , d, we have V (f1 , . . . , fd ) ∩ V = {0}. Proof.

It suffices to show the following for ν = 1, . . . , d:

(*) If there is an N ×(ν−1) matrix (cij ) such that dim V (f1 , . . . , fν−1 )∩V = PN d − ν + 1, for fj = i=1 cij gj , j = 1, . . . , ν − 1, then there exist complex numbers ciν , i = 1, . . . , N , such that dim V (f1 , . . . , fν ) ∩ V = d − ν, for PN fν = i=1 ciν gi . In the above, if ν = 1, V (f1 , . . . , fν−1 ) is understood to be (the germ S at 0 of) Cn . To show (*), let V (f1 , . . . , fν−1 ) ∩ V = q Vq be the irreducible decomposition (cf. Theorem 2.3). Since V (g1 . . . , gN ) ∩ V = {0}, for each q, there exist a point xq in a representative of Vq and gi with gi (xq ) 6= 0. Let Hq denote the hyperplane in CN = {(ξ1 , . . . , ξN )} defined

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PN

fν =

i=1 gi (xq )ξi P N i=1 ciν gi .

= 0. Let (c1ν , . . . , cN ν ) be a point in CN r

S

q

Hq and set

Then Vq 6⊂ V (fν ) for each q. We have S V (f1 , . . . , fν ) ∩ V = q (Vq ∩ V (fν )).

Since each Vq is irreducible and Vq 6⊂ V (fν ), dim Vq ∩ V (fν ) = dim Vq − 1 (cf. Theorem 2.11). Therefore, we have (*) and the lemma.  Note that the above is also valid if d = n, in which case V is the germ of Cn . In fact it is this case that we consider in this chapter and the general case of d < n will be used in the next chapter.
Let r and l be integers with 1 ≤ r ≤ l and set N = rl . We denote by M (l, C) the space of l × l complex matrices, which is naturally identified Vr 2 with Cl . For a matrix A in M (l, C), its r-th exterior power A is an N × N matrix defined as in Section 8.2. The following is proved by linear algebra: Lemma 12.2. For every matrix A in M (l, C) and every neighborhood W of A, there exists a matrix A0 in W such that the matrix consisting of the Vr 0 first d columns of A satisfies the condition of Lemma 12.1. Let E, U , e(l) and s(r) be as in the beginning of this section. We write Pl si = j=1 fji ej , i = 1, . . . , r, with fji holomorphic functions on U . Let F be the l × r matrix whose (i, j)-entry is fij . We set I = { (i1 , . . . , ir ) | 1 ≤ i1 < · · · < ir ≤ l }.

(12.5)

For an element I = (i1 , . . . , ir ) in I, let FI denote the r × r matrix whose k-th row is the ik -th row of F , k = 1, . . . , r, and set ϕI = det FI . Note that Fp is the ideal generated by the germs of the ϕI ’s and S(s(r) ) is the set of common zeros of the ϕI ’s. From the assumption S(s(r) ) = {p}, we have the following as a consequence of Lemmas 12.1 and 12.2 with d = n: Lemma 12.3. We may choose a holomorphic frame e(l) = (e1 , . . . , el ) of E so that there exist n elements I (1), . . . , I (n) in I with the property V (ϕI (1) , . . . , ϕI (n) ) = {p}, where V (ϕI (1) , . . . , ϕI (n) ) denotes the common zero set of ϕI (1) , . . . , ϕI (n) . Let e(l) be a frame of E as in Lemma 12.3. We write I (α) = α = 1, . . . , n, and let F (α) be the l × l matrix obtained (α) from the l × l identity matrix by replacing the ij -th column with the j-th (α) column of F , j = 1, . . . , r. Note that det F = ϕI (α) . Let Fˇ (α) denote the adjoint matrix of F (α) and set Θ(α) = dF (α) · Fˇ (α) ,

(α) (α) (i1 , . . . , ir ),

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which is an l × l-matrix with entries holomorphic 1-forms. The significance of this matrix will be clarified in the proof of the following theorem. Denoting by σ ˜n the polarization of the n-th elementary invariant polynomial σn (cf. Section 8.2), we have the holomorphic n-form σ ˜n (Θ(1) , . . . , Θ(n) ) on U . Theorem 12.4. In the above notation,   σ ˜ (Θ(1) , . . . , Θ(n) ) Rescn (s(r) , E; p) = Resp n . ϕI (1) , . . . , ϕI (n) Proof. Recall that the residue is given by (12.4), where ∇0 is an s(r) trivial connection for E on U0 = U r{p} and ∇1 the e(l) -trivial connection for E on U1 = U . We compute this by setting R = { q ∈ U | |ϕI (1) (q)|2 + · · · + |ϕI (n) (q)|2 ≤ nε2 } ˇ with ε a small positive number. The idea is to introduce a suitable CdRcochain that constitute a bridge between the (2n − 1)-form cn (∇0 , ∇1 ) and the n-form appearing in the Grothendieck residue in the statement. We consider the covering U = {U (α) }nα=1 of U0 given by U (α) = { q ∈ U0 | ϕI (α) (q) 6= 0 } ˇ and work on the Cech-de Rham cohomology of U. On U (α) , we may replace, (l) in the frame e(l) , (ei(α) , . . . , ei(α) ) with (s1 , . . . , sr ) to obtain a frame eα for r

1

(l)

E. The matrix of frame change from e(l) to eα is given by F (α) . We (l) denote by ∇(α) the connection for E on U (α) trivial with respect to eα . (α) (α) (l) The connection matrix θ of ∇ with respect to the frame e is obtained by setting P = (F (α) )−1 and θ = 0 in (8.4): θ(α) = F (α) · d(F (α) )−1 = −dF (α) · (F (α) )−1 = −

1 ϕI (α)

Θ(α) .

We define an alternating cochain τ in A2n−2 (U) by τα1 ...αq = cn (∇0 , ∇1 , ∇(α1 ) , . . . , ∇(αq ) ), which is a (2n − q − 1)-form on U (α1 ...αq ) = U (α1 ) ∩ · · · ∩ U (αq ) . We claim that its coboundary is given, for (α1 , . . . , αq ) with 1 ≤ α1 < · · · < αq ≤ n, as follows:   q = 1, −cn (∇0 , ∇1 )  (Dτ )α1 ...αq = 0 q = 2, . . . , n − 1, (12.6)   −cn (∇ , ∇(1) , . . . , ∇(n) ) q = n. 1

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To prove this, first notice that, since ∇0 and ∇(α) are all s(r) -trivial, cn (∇0 , ∇(α1 ) , . . . , ∇(αq ) ) = 0

for q ≥ 1.

(12.7)

Then we compute cn (∇1 , ∇(α1 ) , . . . , ∇(αq ) ) via Proposition 8.4. Thus let ∇t denote the connection for E on U (α1 ...αq ) given by q q   X X ∇t = 1 − t ν ∇1 + tν ∇(αν ) . ν=1

ν=1

Since the connection matrix θ1 of ∇1 with respect to e(l) is zero, the connection matrix θt of ∇t with respect to e(l) is given by q q q   X X X θt = 1 − tν θ1 + tν θ(αν ) = tν θ(αν ) . ν=1

ν=1

ν=1

The curvature matrix κt of ∇t is given by κt = dθt + θt ∧ θt , which is holomorphic for each fixed t. The form cn (∇1 , ∇(α1 ) , . . . , ∇(αq ) ) is a constant multiple of the integral on ∆n of the sum of the forms α1

αq

σ ˜n (κt , . . . , θ1 , . . . , θq , . . . , κt ), which are holomorphic (2n − q)-forms on U (α1 ...αq ) . Thus we have cn (∇1 , ∇(α1 ) , . . . , ∇(αq ) ) = 0

for q = 1, . . . , n − 1.

(12.8)

Now we compute Dτ . For q = 1, we have, by (12.7) and (12.8), (Dτ )α = dcn (∇0 , ∇1 , ∇(α) ) = −cn (∇0 , ∇1 ). For q = 2, . . . , n, we have, by (12.7), (Dτ )α1 ...αq =

q X

\ (αν ) , . . . , ∇(αq ) ) (−1)ν cn (∇0 , ∇1 , ∇(α1 ) , . . . , ∇

ν=1

+ (−1)q−1 dcn (∇0 , ∇1 , ∇(α1 ) , . . . , ∇(αq ) ) = −cn (∇1 , ∇(α1 ) , . . . , ∇(αq ) ) + cn (∇0 , ∇(α1 ) , . . . , ∇(αq ) ) = −cn (∇1 , ∇(α1 ) , . . . , ∇(αq ) ), which proves (12.6) in view of (12.8). Denoting by ι the inclusion map ∂R ,→ U0 , we let ι∗ U be the covering of ∂R by the open sets ∂R∩U (α) . Then, as a honeycomb system {R(α) }nα=1 adapted to ι∗ U, we take R(α) = { q ∈ ∂R | |ϕI (α) (q)| ≥ |ϕI (β) (q)|

for all β }

and, for a q-tuple (α1 , . . . , αq ) with 1 ≤ α1 < · · · < αq ≤ n, we set R(α1 ...αq ) = R(α1 ) ∩ · · · ∩ R(αq ) , which is a (2n − q)-dimensional manifold

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with boundary oriented as an intersection of honeycomb cells. Considering the integration Z : A2n−1 (ι∗ U) −→ C, ∂R R from the identity ∂R Dτ = 0, we obtain Z Rescn (s(r) , E; p) = cn (∇1 , ∇(1) , . . . , ∇(n) ). R(1...n)

By Proposition 8.4, setting Φ = (−1) we have cn (∇1 , ∇(1) , . . . , ∇(n) ) =

n(n−1) 2

n! σ ˜n (θ(1) , . . . , θ(n) ),

 √−1 n Z

Φ dt1 · · · dtn  √−1 n n(n−1) σ ˜n (θ(1) , . . . , θ(n) ) = (−1) 2 2π  1 n σ n(n−1) ˜n (Θ(1) , . . . , Θ(n) ) √ . = (−1) 2 ϕI (1) · · · ϕI (n) 2π −1 Noting that the n-cycle Γ appearing in the Grothendieck residue with n(n−1) respect to the functions ϕI (1) , . . . , ϕI (n) is given by Γ = (−1) 2 R(1...n) , we obtain the formula.  2π

∆n

Special cases: 1. The case l = n and r = 1. We may write s(1) = (s), Pn s = i=1 fi ei . Then we may set ϕI (i) = fi , i = 1, . . . , n. We see that Θ(i) is the l × l matrix whose i-th column is t (df1 , . . . , dfn ) with all the other entries equal to 0. Thus we have σ ˜n (Θ(1) , . . . , Θ(n) ) = df1 ∧ · · · ∧ dfn and   df ∧ · · · ∧ dfn Rescn (s, E; p) = Resp 1 . f1 , . . . , f n In the case r = 1 and n = 1, the proof is directly done by computing the difference form c1 (∇0 , ∇1 ), which turns out to be (cf. Example 10.3) df 1 . c1 (∇0 , ∇1 ) = − √ 2π −1 f For general n, the above can be also obtained from Theorem 10.8, Propositions 11.16 and 11.17. 2. The case n = 1 and r = l. Let e(l) = (e1 , . . . , el ) be an arbitrary frame of Pl E and write si = j=1 fij ej , i = 1, . . . , l. Let F = (fij ) and set ϕ = det F . Then we may set ϕI (1) = ϕ and we have σ1 (Θ) = dϕ so that we have   dϕ Resc1 (s(r) , E; p) = Resp . ϕ Note that Resc1 (s(r) , E; p) in this case coincides with the residue Resc1 (s, det E; p) of the section s = s1 ∧ · · · ∧ sl of the line bundle Vl det E = E at p.

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Fundamental properties of the residues We consider the situation in the beginning of this section. Letting T be a neighborhood of 0 in C = {t}, we consider the bundle ˜ = E × T on U ˜ = U × T . Suppose we have an r-tuple of holomorphic E (r) ˜ ˜ such that s˜(r) (z, 0) = s(r) (z). For t in T , we set sections s˜ of E on U (r) (r) ˜ Et = E|U ×{t} and st (z) = s˜(r) (z, t). We call such an s˜(r) (or st ) a perturbation of s(r) . Sometimes we identify U × {t} with U and E|t with E. (r)

Lemma 12.4. We have dim S(˜ s(r) ) = 1 and S(st ) is a non-empty finite set. (r)

Proof. By the upper semicontinuity of the dimension of S(st ) (cf. Theorem 2.13), we have dim S(˜ s(r) ) ≤ 1. On the other hand, we have (r) codim S(˜ s ) ≤ l − r + 1 = n by (A.16).  P (r) Lemma 12.5. The sum q∈S(s(r) ) Rescn (st , Et ; q) is continuous in t. t

˜ 0 be an s˜(r) -trivial connection for E ˜ on U ˜0 = U ˜ r S(˜ Proof. Let ∇ s(r) ) ˜ ˜ ˜ and ∇1 a connection for E on U . The statement follows from Theorem 10.6 ˜ 0 and ∇ ˜ 1 and using by computing the residues taking the restrictions of ∇ (10.11).  Since the residues are integers, from the above, we have: P (r) Proposition 12.2. The sum q∈S(s(r) ) Rescn (st , Et ; q) is constant. In t particular, X (r) Rescn (s(r) , E; p) = Rescn (st , Et ; q). (r)

q∈S(st )

Note that this also follows from Theorems 5.2 and the homotopy invariance of topological residues. We prepare a proposition for later use. Suppose one of the sections in s(r) is non-zero at p. For example we assume that s1 (p) 6= 0 so that we have an exact sequence of vector bundles on a neighborhood of p: 0 −→ I −→ E −→ E 0 −→ 0,

(12.9)

where I denotes the trivial line bundle determined by s1 and E 0 is a vector (r−1) bundle (still trivial) of rank l − 1. Let s0 = (s02 , . . . , s0r ) denote the 0 (r − 1)-sections of E determined by (s2 , . . . , sr ). The following is proved as Lemma 10.4:

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Proposition 12.3. In the above situation, we have Rescn (s(r) , E; p) = Rescn (s0

(r−1)

, E 0 ; p).

We also need the following: Lemma 12.6. If r = 1, there exists always a “good perturbation” of s, i.e., ˜ near 0 such that s˜(z, 0) = s(z) and that st a holomorphic sections s˜ of E has only non-degenerate zeros, for t 6= 0. Pn Proof. We may write s = i=1 fi ei and think of f = (f1 , . . . , fn ) as a holomorphic map of U into Cn . We assume that 0 is a degenerate zero of s and consider the hypersurface C in U defined by det

∂(f1 , . . . , fn ) = 0. ∂(z1 , . . . , zn )

By Theorem 2.14, D = f (C) is a hypersurface in a neighborhood of 0 in Cn . Let T be a one-dimensional disk intersection with D only at 0. Then Pn st = i=1 (fi − t)ei is a desired perturbation.  Algebraic expression We refer to Section A.2 for relevant materials in algebra. Let E, U , e(l) and s(r) be as in the beginning of this section. Also let F , FI and ϕI = det FI be as before. We denote by OU the sheaf of germs of holomorphic functions on U and by F the ideal sheaf in OU generated by the (germs of) ϕI ’s. Note that F does not depend on the choice of the frame e(l) of E. We have the complex space S = (S, OS ) in U with support S = S(sr ) = {p} and structure sheaf OS = ι−1 (OU /F ), ι : S ,→ U being the inclusion, as above. First we consider the case l = n and r = 1. Thus we have one section Pn s = i=1 fi ei and Fp = (f1 , . . . , fn ). Obviously we have: Proposition 12.4. If p is a non-degenerate zero of s, dimC On /(f1 , . . . , fn ) = 1. Now we come back to the general case and let s˜(r) = (˜ s1 , . . . , s˜r ) be a (r) ˜ perturbation of s as before. We define FI and ϕ˜I as above, using the s˜i ’s. Let T be a small neighborhood of 0 in C and F˜ the ideal sheaf generated by ˜ = U × T . Let S˜ be the complex space in U ˜ with support the ϕ˜I ’s in OU˜ , U (r) −1 ˜ is S˜ = S(˜ s ) and structure sheaf OS˜ = (˜ι) (OU˜ /F˜ ), where ˜ι : S˜ ,→ U ˜ the inclusion. We denote by p : U = U × T → T the projection onto the

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second factor and by π the restriction of p to S˜ so that π = p ◦ ˜ι. We have a morphism (π, $) : S˜ −→ (T, OT )

(12.10)

˜ , O ˜ ) and (p, p∗ ) : as the composition of the canonical morphism S˜ → (U U ˜ ˜ (U , OU˜ ) → (T, OT ). For each x in S, $x : OT,π(x) → OS,x ˜ is the composition of p∗x : OT,π(x) → OU˜ ,x and the canonical epimorphism OU˜ ,x → OU˜ ,x /F˜x = OS,x ˜ . Let Ft be the ideal sheaf generated by the ϕI,t ’s in OUt , Ut = U × {t}. The fiber of (π, $) over t ∈ T is the complex space St in S˜ with support (r) St = S(st ) and structure sheaf OSt = ι−1 t (OUt /Ft ), where ιt : St ,→ Ut is the inclusion. By Lemma 12.4, π : S˜ → T is a finite map. Thus each point x in ˜ S is isolated in St , t = π(x), and by the Nullstellensatz, dimC OSt ,x = dimC OUt ,x /Ft,x is finite (cf. (A.18)). Proposition 12.5. In the above situation, X dimC OUt ,q /Ft,q . dimC OU,p /Fp = (r)

q∈S(st )

Proof. We claim that the morphism (π, $) in (12.10) is flat (cf. Section A.2). Then the proposition follows from (A.21). Let x be a point in S˜ and set t = π(x). In the following, we set Ox0 = OU˜ ,x , Ox = OS,x and Ot = OT,t . Note that Ox0 and Ot are regular local ˜ rings of dimensions n+1 and 1, respectively. We have ht F˜x = n = l−r+1. Hence by (A.17), the ring Ox is CM. Since the morphism $x : Ot → Ox is finite (cf. (A.20)), Ox is a CM Ot -module by (A.14). By (A.15), denoting by mt the maximal ideal in Ot , depth(mt ; Ox ) + pdOt Ox = depth mt . We have depth(mt ; Ox ) = dimOt Ox = dimOx Ox = 1 and depth mt = dim Ot = 1. Therefore, pdOt Ox = 0 and (π, $) is flat.  From Propositions 12.1, 12.2, 12.4 and 12.5 and Lemma 12.6, we have: Corollary 12.1. In the case l = n and r = 1, Rescn (s, E; p) = dimC OU,p /Fp = dimC On /(f1 , . . . , fn ).

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401

We continue with the general situation where l ≥ n. We assume that s1 (p) 6= 0 as in the situation of Proposition 12.3. Then we may write Pl 0 0 0 holomorphic functions on U and ej , i = 2, . . . , r, with fij s0i = j=2 fij (l−1)

e0 = (e02 , . . . , e0l ) a frame of E 0 (cf. (12.9)). Let F 0 be the (r −1)×(l −1) 0 matrix whose (i, j)-entry is fij . We set I 0 = { (i2 , . . . , ir ) | 2 ≤ i2 < · · · < ir ≤ l }. For an element I 0 = (i2 , . . . , ir ) in I 0 , let FI0 0 denote the (r − 1) × (r − 1) matrix consisting of the columns of F 0 corresponding to I 0 and set ϕ0I 0 = det FI0 0 . Note that the set of common zeros of the ϕ0I 0 ’s consists only of p. Let 0 Fp denote the ideal of OU,p generated by the ϕ0I 0 ’s. Proposition 12.6. In the above situation, we have Fp = Fp0

and thus

dimC OU,p /Fp = dimC OU,p /Fp0 .

Proof. We may assume, without loss of generality, that f11 (p) 6= 0. Then, we may take as (e02 , . . . , e0l ) the sections determined by (e2 , . . . , el ). For i ≥ 2, we have   l X f11 f1j 1 fi1 s1 + si = fi1 fij ej . f11 j=2

Hence 0 fij

1 = f11

f11 f1j fi1 fij .

For I 0 = (i2 , . . . , il ), we compute ϕ(1,I 0 ) = f11 · ϕ0I 0 . Thus the ideal Fp0 is generated by { ϕ(1,I 0 ) | I 0 ∈ I 0 }. On the other hand, for every I = (i1 , . . . , ir ), considering the determinant of the (r + 1) × (r + 1) matrix whose first and second rows are (f11 , f1i1 , . . . , f1ir ) and whose k-th row is (fk−1,1 , fk−1,i1 , . . . , fk−1,ir ), k ≥ 3, we have f11 · ϕI =

r X (−1)j−1 f1ij · ϕ(1,i1 ,...,ibj ,...,ir ) . j=1

Hence we have Fp0 = Fp . Theorem 12.5. We have Rescn (s(r) , E; p) = dimC OU,p /Fp .



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Proof. We prove this by induction on r. The case r = 1 is Corollary 12.1. Suppose that the statement is true for (r − 1)-sections (with isolated singularity). If there is a section in s(r) which does not vanish at p, we are done by Propositions 12.3 and 12.6. Otherwise, we take a perturbation s1,t of (r) s1 so that s1,t (p) 6= 0, for t 6= 0, and set st = (s1,t , s2 , . . . , sr ). Recalling that none of the si ’s vanishes on U r{0}, we see that, for t 6= 0 and every (r) (r) point q ∈ S(st ), there is a section in st which does not vanish at q. The theorem follows from Propositions 12.2, 12.3, 12.5 and 12.6 and the induction hypothesis.  Special cases: 1. The case l = n and r = 1. In this case, we restate Pn Corollary 12.1. We may write s(1) = (s), s = i=1 fi ei . Then we have Fp = (f1 , . . . , fn ) and Rescn (s, E; p) = dimC On /(f1 , . . . , fn ). 2. The case n = 1 and r = l. Let e(l) = (e1 , . . . , el ) be an arbitrary frame of Pl E and write si = j=1 fij ej , i = 1, . . . , r. Let F = (fij ) and set ϕ = det F . Then we have Fp = (ϕ) and Resc1 (s, E; p) = dimC O1 /(ϕ). . 12.5

Examples

In the following, we denote by M a complex manifold of dimension n. Poincar´ e-Hopf index of a vector field We take as E the holomorphic tangent bundle T M . A section of T M may be thought of either as a complex vector field or as a real vector field (cf. Proposition 3.6). Let S be a compact set in M and Sλ a connected component of S. For a non-vanishing C ∞ vector field v on U rSλ , where U is a neighborhood of Sλ , we may define the Poincar´e-Hopf index of v at Sλ by (cf. Remark 10.6) PH(v, Sλ ) = Rescn (v, T M ; Sλ ). If Sλ is a subcomplex with respect to some triangulation of M , this definition coincides with the one in (5.15) by Proposition 5.8 and Corollary 10.6. If Sλ consists of a point p and if v is defined and holomorphic

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in a neighborhood of p, it has algebraic and analytic expressions (cf. Theorems 12.5 and 12.4, also Example 12.2). Let R be a compact manifold of dimension 2n with boundary in M . For a non-vanishing vector field v on a neighborhood of ∂R, we may also define (cf. (10.3)) Z cnR (T M, v), PH(v, R) = R

which coincides with the one in (5.16) (cf. Proposition 7.18 and Remark 7.18). Suppose S has only a finite number of connected components (Sλ )λ . Then if v is a non-vanishing vector field on M rS and if S is in the interior of R, we have (cf. Proposition 10.2 and Theorem 10.6) X PH(v, Sλ ) = PH(v, R). λ

If M is compact, taking M as R, we have Z X PH(v, Sλ ) = cn (M ), λ

M

where cn (M ) = cn (T M ) and the right-hand side equals the Euler-Poincar´e characteristic χ(M ) of M (cf. (5.20), Gauss-Bonnet formula). Index of a (1, 0)-form We take as E the holomorphic cotangent bundle T ∗ M of M . Then a section of T ∗ M is a (1, 0)-form. Let θ be a C ∞ (1, 0)-form. For a compact connected component Sλ of the zero set or the set of points where it is not defined S(θ) of θ having a neighborhood disjoint from the other components, we define the index Ind(θ, Sλ ) of θ at Sλ by Ind(θ, Sλ ) = Rescn (θ, T ∗ M ; Sλ ). If M is compact and if S(θ) admits only a finite number of connected components (Sλ ), we have (cf. Proposition 10.2, Theorem 10.6, (5.20)) X Ind(θ, Sλ ) = (−1)n χ(M ). λ

If Sλ consists of a point p and if θ is defined and holomorphic in a neighborhood of p, Ind(θ, p) has the topological, algebraic and analytic expressions as given above.

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Multiplicity of a function For a holomorphic function f on M , its differential df is a section of T ∗ M . The zero set S(df ) of df coincides with the critical set C(f ) of f . Definition 12.4. The multiplicity m(f, Sλ ) of f at a compact connected component Sλ of C(f ) as above is defined by m(f, Sλ ) = Rescn (df, T ∗ M ; Sλ ). Note that, if Sλ consists of a point p, it coincides with the multiplicity m(f, p) of f at p described in Example 12.3. Now we consider the global situation. Let f : M → C be a holomorphic map of M onto a complex curve (Riemann surface) C. The differential df : T M → f ∗ T C of f determines a section of T ∗ M ⊗f ∗ T C, which is also denoted by df . The set of zeros of df is the critical set C(f ) of f . Suppose C(f ) is a compact set with a finite number of connected components (Sλ )λ . Then we have the residue Rescn (df, T ∗ M ⊗ f ∗ T C; Sλ ) for each λ. If M is compact, we have (cf. Proposition 10.2, Theorem 10.6) Z X Rescn (df, T ∗ M ⊗ f ∗ T C; Sλ ) = cn (T ∗ M ⊗ f ∗ T C). M

λ

We look at the both sides of the above more closely. In the following, we set D(f ) = f (C(f )), the set of critical values. Then, if M is compact, f defines the structure of a C ∞ locally trivial fibration on MrC(f ) → CrD(f ) (cf. Theorem 3.12). Denoting by F a general fiber of f , we have: Lemma 12.7. If M is compact and if D(f ) consists of isolated points, Z cn (T ∗ M ⊗ f ∗ T C) = (−1)n (χ(M ) − χ(F ) · χ(C)). M

Suppose that f (Sλ ) consists of a point. Taking a coordinate on C around f (Sλ ), we think of f as a holomorphic function near Sλ . Then we may write Rescn (df, T ∗ M ⊗ f ∗ T C; Sλ ) = Rescn (df, T ∗ M ; Sλ ) = m(f, Sλ ), the multiplicity of f at Sλ . Thus we have Theorem 12.6. Let f : M → C be a holomorphic map of a compact complex manifold M of dimension n onto a complex curve C. If D(f ) consists of isolated points, then we have X m(f, Sλ ) = (−1)n (χ(M ) − χ(F ) · χ(C)). λ

Residues of Chern Classes on Manifolds

405

In particular, we have: Corollary 12.2. In the above situation, if the critical set C(f ) of f consists of only isolated points, X m(f, p) = (−1)n (χ(M ) − χ(F ) · χ(C)). p∈C(f )

The above notion of multiplicity and the formula for these multiplicities will be generalized to the case of functions on singular varieties in Section 13.7 below. 12.6

Dual class of a complex subspace

We come back to the situation of the beginning of Section 12.2 and let M , E, s(r) , S, S and so forth be as given there. We adopt Convention 4.2 for notation of the class of a cycle. We assume that dim S = n − q so that we may identify the topological residue TRescq (s(r) , E; S) and the differential geometric residue Rescq (s(r) , E; S) (cf. (12.2)). In this case, we may also identify cqtop (E, s(r) ) and cqdiff (E, s(r) ), which we denote by cq (E, s(r) ). Let (Si ) be the irreducible (r) components of S. Also let pi , Di , Ei and si be as before. Definition 12.5. The multiplicity mi of Si in S is defined by (r)

mi = Rescq (si , Ei ; pi ). Note that mi has topological, analytic and algebraic expressions as given in Theorems 12.3–12.5. P Definition 12.6. The homology class of S is defined by [S] = mi [Si ] P ˘ ˘ in H2(n−q) (M ; Z) or by S = mi [Si ] in H2(n−q) (S; Z). In this situation, we have: Theorem 12.7. The class cq (E) corresponds to [S] under the Poincar´e ∼ ˘ duality H 2q (M ; Z) → H 2(n−q) (M ; Z). In fact, we have the following more “precise” theorem, which is a direct consequence of Theorem 12.2: Theorem 12.8. Let S be the complex space of dimension n − q in M as ˘ 2(n−q) (S; Z) is equal to TRescq (s(r) , E; S), above. Then the class of S in H

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which may be identified with Rescq (s(r) , E; S). Thus the class cq (E, s(r) ) corresponds to the class S under the Alexander duality ∼ ˘ H 2q (M, M rS; Z) → H 2(n−q) (S; Z).

We specialize the above to the case r = 1. In this case, we have q = l and we are to consider the top Chern class. Let M and E be as above. Also let s be a holomorphic section of E and S the complex space defined by (the ideal generated by the local components of) s. Thus the support of S is the zero set S of s. In this situation, we have the localization cl (E, s) in H 2l (M, M rS) and the associated residue ˘ 2(n−l) (S). In general, dim S ≥ n − l and the equality TRescl (s, E; S) in H holds if and only if s is a regular section (cf. Definition 11.31). The following is a special case of Theorem 12.8: Theorem 12.9. Let S be the complex space of dimension n − l in M as ˘ 2(n−l) (S; Z) is equal to TRescl (s, E; S), above. Then the class of S in H which may be identified with Rescl (s, E; S). Thus the class cl (E, s) corresponds to the class S under the Alexander duality ∼ ˘ H 2l (M, M rS; Z) −→ H 2(n−l) (S; Z).

Remark 12.2. 1. Corollary 5.5 (see also Theorem 10.9) may be understood in this context. 2. In the above situation, we call cl (E, s) the Thom class of S in M and denote it by ΨS (cf. Definition 15.4 and the subsequent paragraph). This is used to express the localized Riemann-Roch theorem for embeddings in a certain case in Section 15.6 below (cf. Theorem 15.5). Localization by meromorphic sections Let M be a complex manifold of dimension n and D a divisor on M (cf. Section 11.6). Recall that D is represented by a system {(Uα , ϕα )}, where {Uα } is a covering of M , ϕα is in Γ (Uα ; M ∗ ), for each α, and f αβ = ϕα /ϕβ is in Γ (Uα ∩ Uβ ; O ∗ ), for each pair (α, β). The system {f αβ } defines the associated line bundle LD and the collection {ϕα } the natural meromorphic section sD of LD . Conversely, let L be a holomorphic line bundle on M and s a meromorphic section of L. Let {Uα } be a covering of M trivializing L and {f αβ } a system of transition functions. Then s is represented by a collection {ϕα } of meromorphic functions such that ϕα = f αβ ϕβ on Uα ∩ Uβ . If ϕα is in

Residues of Chern Classes on Manifolds

407

Γ (Uα ; M ∗ ) for each α, then s defines a divisor D. Moreover L is the line bundle LD associated with D and s is the natural section sD of LD . In the above situation, we have the topological localization c1top (LD , sD ) in H 2 (M, M r |D|; Z) (cf. Section 5.6) and the differential geometric localization c1diff (LD , sD ) in H 2 (M, M r|D|; C) (cf. Section 10.3). We also ˘ 2(n−1) (|D|; Z) and have the associated residues TResc1 (sD , LD ; |D|) in H ˘ Resc1 (sD , LD ; |D|) in H2(n−1) (|D|; C). We have seen (cf. Corollary 10.6) that Resc1 (sD , LD ; |D|) is the image of TResc1 (sD , LD ; |D|) by the canonical morphism ˘ 2(n−1) (|D|; Z) −→ H ˘ 2(n−1) (|D|; C). H

(12.11)

Suppose D has only a finite number of connected components (Sλ )λ . ˘ 2(n−1) (|D|; Z) = L H ˘ Then we have a decomposition H λ 2(n−1) (Sλ ; Z) and ˘ 2(n−1) (Sλ ; Z) for accordingly we have the residue TResc1 (sD , LD ; Sλ ) in H each λ. We have (cf. Theorem 5.2): X ˘ 2(n−1) (M ; Z), (ıλ )∗ TResc1 (sD , LD ; Sλ ) = M a c1 (LD ) in H λ

where ıλ : Sλ ,→ M denotes the inclusion. Passing to the homology with C-coefficient, we have a similar statement for Resc1 (sD , Sλ ; |D|) (cf. Theorem 10.6). Example 12.4. Let C be a compact Riemann surface and L a holomorphic line bundle on C. Suppose we have a non-zero meromorphic section s of L. Then the support S of the divisor defined by s is the set of zeros and poles of s. The previous construction gives us the localization c1 (L, s) in H 2 (C, C rS) of c1 (L) in H 2 (C). Note that S consists of a finite number of points and, for each p in S, TResc1 (s, L; p) and Resc1 (s, L; p) are the same integer. We have X Resc1 (s, L; p) = hC, c1 (L)i. p∈S

The residue Resc1 (s, L; p) is given as follows. We choose an open neighborhood U of p not containing any other points of S and trivializing L. Let e be a holomorphic frame of L on U , and write s = f e with f a meromorphic function on U . Let S1 be a circle around p in U oriented counterclockwise. Then we have (cf. Example 10.3) Z df 1 √ 1 , Resc (s, L; p) = 2π −1 S1 f which is the order of zero or pole according as f has a zero or a pole at p.

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P Let D a non-zero divisor on M and write D = mi Vi , mi 6= 0. Then ˘ each Vi defines a class [Vi ] in H2(n−1) (|D|; Z) and it is the free Abelian group generated by the [Vi ]’s. Thus the morphism (12.11) is injective so that we may identify the two residues: Resc1 (sD , LD ; |D|) = TResc1 (sD , LD ; |D|). (12.12) S Let pi be a non-singular point of Vi r j6=i Vj and Di a complex slice of Vi in M at pi , similarly as in the beginning of Section 12.2. Note that in this case Di is complex one dimensional. The section si = sD |Di of Li = LD |Di is non-vanishing and holomorphic away from pi so that we have the residues TResc1 (si , Li ; pi ) and Resc1 (si , Li ; pi ), both of which are the same integer ordVi (sD ) (cf. Example 10.3). It is the coefficient mi of Vi in D (cf. (11.19)). P Definition 12.7. For a divisor D = mi Vi , we define its homology class P ˘ 2n−2 (M ; Z) or by D = P mi [Vi ] in H ˘ 2n−2 (|D|, Z). by [D] = mi [Vi ] in H If D is non-zero and effective, we have seen that it may be thought of as a complex space of pure dimension n − 1 in M (cf. Section 11.6). In this case, the natural section sD of the associated line bundle LD is holomorphic and D is the complex space defined by sD . The multiplicity of Vi in D is mi so that the above definition coincides with the one in Definition 12.6. Note that, if D = 0, then |D| = ∅ and the both sides of (12.12) are zero. The statement of Theorem 12.2 holds in the case of divisors as well: TResc1 (sD , LD ; |D|) = TResc1 (si , Li ; pi ) · [Vi ]

˘ 2(n−1) (|D|; Z) in H

so that we have: Theorem 12.10. Let D be a divisor on M . Then the class of D in ˘ 2(n−1) (|D|; Z) is equal to TResc1 (sD , LD ; |D|), which may be identified H with Resc1 (sD , LD ; |D|). Thus the class c1 (LD , sD ) corresponds to the class D under the Alexander duality ∼ ˘ H 2 (M, M r|D|; Z) −→ H 2(n−1) (|D|; Z).

˘ 2(n−1) (M ; Z) is the Poincar´e dual of In particular, the class [D] in H 2 c1 (LD ) in H (M ; Z). Thus if D1 and D2 are two linearly equivalent divisors, then [D1 ] = [D2 ]. Also, if D1 and D2 are two divisors, the intersection product [D1 ] [D2 ] ˘ 2(n−2) (M ; Z) is the Poincar´e dual of c1 (LD ) ` c1 (LD ) in H 4 (M ; Z) in H 1 2 (cf. Theorem 4.7). If M is compact, for n divisors D1 , . . . , Dn on M , the “global intersection number” ]([D1 ] · · · [Dn ]) is defined and is given by

·

·

·

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hM, c1 (LD1 ) · · · c1 (LDn )i, where the product is the cup product (cf (14.8) Tn below). If the intersection i=1 |Di | consists of isolated points, then this number is the sum of the intersection numbers at the points of intersection (cf. Theorem 14.2 below). See Example 12.2 for this “local intersection number” when the divisors are effective. We descuss these more in detail in Section 14 below. In the case D is a non-zero effective divisor, Theorem 12.10 is a special case of Theorem 12.9, where l = 1. Example 12.5. Let V be the projective algebraic variety in Pn = {[ζ0 , . . . , ζn ]} defined by a homogeneous polynomial P of degree p (cf. Section 2.2). The function ϕ = P/ζ0p is a well-defined meromorphic function on Pn , which is given as the quotient of P/ζαp by ζ0p /ζαp on each affine open set Uα = {ζα 6= 0}. Thus, if we denote by D∞ the hyperplane defined by ζ0 = 0, then V is linearly equivalent to p D∞ and [V ] = p [D∞ ] in H2n−2 (Pn ; Z). Recall that [D∞ ] = [Pn−1 ] is a generator of H2n−2 (Pn ; Z) ' Z. Also, the intersection product of k copies of [Pn−1 ] generates H2n−2k (Pn ; Z) ' Z, for k = 1, . . . , n (cf. Example 2.4. 2). Notes The presentation of this chapter is mainly based on Section 4 of [Suwa (2008)] and the references therein. For subanalytic sets, we refer to [Bierstone and Milman (1988); Shiota (1997)] and the references therein. Theorem 12.1 is due to [Ohmoto and Shiota (2017)]. We list [Griffiths and Harris (1978); Hartshorne (1966)] as references for Section 12.3. In Section 12.4, see [Suwa (2005)] for algebraic expression and [Suwa (2003a)] for analytic expression with more examples. The formula in Corollary 12.2 is proved in [Iversen (1971)], see also Example 14.1.5 in [Fulton (1984)]. See [Izawa and Suwa (2003)] for a precise proof of Lemma 12.7. ˇ See [Suwa (2000a)] for a proof of Theorem 12.9 via Cech-de Rham cohomology and some more materials in this direction, including the above type of duality for non-compact varieties, which is effectively used in [Brasselet, Lehmann, Seade and Suwa (2001)].

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Chapter 13

Residues of Chern Classes on Singular Varieties

We describe explicitly the residues determined by holomorphic r-sections of vector bundles on singular varieties. For this, we first look into the neighborhood structure of a Whitney stratification, introducing the notion of a controlled tube system. We then discuss the Poincar´e, Alexander, Lefschetz and Thom morphisms for singular varieties, along the line of Sections 4.2 and 4.3. In order to represent ˇ these in terms of differential forms, we discuss the de Rham and Cechde Rham theories for varieties. We define the Chern classes of complex vector bundles on varieties and their localizations by r-frames from both topological and differential geometric viewpoints. The classes defined by two ways are shown to be essentially the same. As in the case of r-sections on complex manifolds, we have a fundamental theorem (Theorem 13.9) saying that the residue in the proper case is determined by the residues at isolated singularities of the r-section that appear on the transverse slices of irreducible components of the singular set. We give various expressions for the residue at an isolated singularity as well as some fundamental examples. We also discuss the local duality concerning complex spaces defined by an r-section, as in the case of complex manifolds. It will be used in Chapter 14 below for the localized intersection theory.

13.1

Controlled tube systems for Whitney stratifications

Let M be a C ∞ manifold and X a submanifold of M . A tube around X is a triple T = (U, r, ρ), where U is an open neighborhood of X in M , r : U → X a C ∞ submersive retraction and ρ a non-negative C ∞ function on U such that ρ−1 (0) = X and that each point x ∈ X is a unique 411

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non-degenerate critical point of ρ|r−1 (x) . For example, if r : U → X is a tubular neighborhood (cf. Theorem 3.10) and ρ the function corresponding to the square of distance from the zero section of the normal bundle with respect to some Riemannian metric, then (U, r, ρ) is a tube. It can be shown that every tube is essentially given as above. Let S be a closed set in M and {Xα } a stratification of S (cf. Definition 2.13). A tube system for {Xα } is a collection {Tα }, where, for each α, Tα = (Uα , rα , ρα ) is a tube around Xα . The tube system {Tα } is controlled if, for every pair (α, β) with Xα ∩ (Xβ rXβ ) 6= ∅, the following holds: rα ◦ rβ (x) = rα (x)

and ρα ◦ rβ (x) = ρα (x)

for all x ∈ Uα ∩ Uβ with rβ (x) ∈ Uα . We quote: Theorem 13.1. Every Whitney stratification admits a controlled tube system. Here are some aspects of controlled tube systems. Let {Tα } be a tube system for {Xα }. For each α, we take a positive function εα on Xα so that Tα = { x ∈ Uα | ρα (x) ≤ εα (rα (x)) } is a closed ball bundle on Xα . Thus Oα = { x ∈ Uα | ρα (x) < εα (rα (x)) } is an open ball bundle on Xα . Then rα |Tα : Tα → Xα is a proper deformation retraction and rα |Oα : Oα → Xα a deformation retraction. If {Tα } is controlled, then: (1) For every pair (α, β) with Xα ∩ (Xβ rXβ ) 6= ∅, ∂Tα intersects both Xβ and ∂Tβ transversally. S (2) The inclusion S ,→ α Oα is a homotopy equivalence. 13.2

Poincar´ e, Alexander, Lefschetz and Thom morphisms

Let M be a complex manifold of dimension n and V an analytic variety of pure dimension d in M . We denote by Sing V the singular set of V and by Vreg = V rSing V the regular part, which is naturally oriented as a complex manifold. We take a C 1 triangulation (K0 , h) of M compatible with V and Sing V as in Theorem 12.1. Let K be the barycentric subdivision of K0 and K ∗ the cellular decomposition dual to K, which is constructed from the barycentric subdivision K 0 of K (cf. Section 4.1). We denote by KV the triangulation of V induced from K and by KV0 its barycentric subdivision. Then KV0 is the set of simplices of K 0 that are in V (cf. Section 4.3). Note that if s is a p-simplex of K in V , then s∗ is a (2n−p)-cell of K ∗ and s∗ ∩V

Residues of Chern Classes on Singular Varieties

413

is a (2d − p)-chain of KV0 . The chain s∗ ∩ V is a sum of KV0 -simplices each of which is oriented so that its orientation followed by that of s gives the orientation of Vreg . In this section, homology and cohomology are with Z-coefficient. We also use the notation in Section 4.1 for various chain and cochain groups. Thus, denoting by C•KV (V ) and S• (V ) the complexes of (finite) chains of KV and of singular chains of V , there is a natural chain morphism η•KV : C•KV (V ) −→ S• (V ), which is defined as (4.1). It induces isomorphisms ∼

η∗KV : HpKV (V ) −→ Hp (V )

∼

p ∗ and ηK : H p (V ) −→ HK (V ). V V

(13.1)

Also, denoting by C˘•KV (V ) and S˘• (V ) the complexes of locally finite chains of KV and of locally finite singular chains of V , respectively, there is a natural chain morphism η˘•KV : C˘•KV (V ) → S˘• (V ), which induces an isomorphism ∼ ˘ ˘ pKV (V ) −→ η˘∗KV : H Hp (V ).

(13.2)

In the following, the suffix V in KV or in KV0 will be omitted if there is no fear of confusion. Poincar´ e morphism We define a morphism p ˘K PV : CK 0 (V ) −→ C2d−p (V )

by P (uV ) =

X

hs∗ ∩ V, uV i s,

(13.3)

s

where the sum is taken over all (2d − p)-simplices s of KV . We may prove that it is compatible with boundary and coboundary operators along the same line as in the case of manifolds (cf. Section 4.1). Thus let VK 0 denote the fundamental cycle of V in K 0 , i.e., the sum of 2d-simplices of K 0 in V . The following is proved as Lemma 4.1: Lemma 13.1. For every p-simplex t of K 0 in V , we have: ( s if t ⊂ s∗ ∩ V for some (2d − p)-simplex s of KV , VK 0 a ϑ(t) = 0 otherwise, as KV0 -chains.

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Complex Analytic Geometry

From this we have the commutative diagram P

/ C˘ K (V ) 2d−p

VK 0 a

%  K0 C˘2d−p (V ),

p CK 0 (V )

κ

where κ is the natural monomorphism, and we have a relation as the one ∗ in Lemma 4.2. Thus, combining with the isomorphisms ηK in (13.1) for 0 V KV0 and η˘∗KV in (13.2), we have: Theorem 13.2. The morphism P in (13.3) induces a morphism ˘ 2d−p (V ). PV : H p (V ) −→ H We call PV the Poincar´e morphism. It is given by the cap product with the fundamental class [V ] and is independent of the triangulation. If V is non-singular, this is the Poincar´e isomorphism in Theorem 4.1. Alexander morphism Let Z be a K0 -subcomplex of V . We denote by SK 0 (Z) and OK 0 (Z) the star and the open star of Z in K 0 . We may also consider SKV0 (Z) = SK 0 (Z) ∩ V and OKV0 (Z) = OK 0 (Z) ∩ V . Both SK 0 (Z) and SKV0 (Z) proper deformation retract to Z and both OK 0 (Z) and OKV0 (Z) deformation retract to Z. Note that V rOKV0 (Z) = V rOK 0 (Z). We may write p p CK 0 (V, V rOK 0 (Z)) = { uV ∈ CK 0 (V ) | ht, uV i = 0 for t with t ∩ Z = ∅ }, p where t denotes a p-simplex of KV0 . Denoting by HK 0 (V, V rOK 0 (Z)) the ∗ cohomology of CK 0 (V, V rOK 0 (Z)), we have a natural isomorphism ∼

p ∗ p ηK 0 : H (V, V rOK 0 (Z)) −→ HK 0 (V, V rOK 0 (Z)).

There is also a natural isomorphism H p (V, V rOK 0 (Z)) ' H p (V, V rZ). p If uV is in CK 0 (V, V rOK 0 (Z)), in the sum in (13.3), only the simplices of K in Z appear. Hence the morphism PV induces a morphism p ˘K AV : CK 0 (V, V rOK 0 (Z)) −→ C2d−p (Z).

(13.4)

It is compatible with the boundary and coboundary operators so that we have:

Residues of Chern Classes on Singular Varieties

415

Theorem 13.3. For a subcomplex Z of V , the morphism AV in (13.4) induces a morphism ˘ 2d−p (Z). AV,Z : H p (V, V rZ) −→ H It is called the Alexander morphism. If V is non-singular, this is the Alexander isomorphism in Theorem 4.2. Remark 13.1. As in the case of manifolds, we may write (cf. (4.13)) for α ∈ H p (V, V rZ), A(α) = [SKV0 (Z)] a α ˘ 2d (SK 0 (Z), ∂SK 0 (Z)) of the relative where [SKV0 (Z)] denotes the class in H V V 0 0 cycle SK 0 (Z) in C˘ K (SK 0 (Z), ∂SK 0 (Z))(= C˘ K (SK 0 (Z)). V

2d

V

2d

V

V

Also, if Z is compact, we may write (cf. Remark 4.5. 3) for α ∈ H p (V, V rZ). A(α) = [V ] a α Lefschetz morphism Let R be a real 2n-dimensional manifold with boundary in M such that R contains Sing V in its interior and that ∂R is transverse to Vreg . We may assume that the triangulation K0 is also compatible with R and ∂R. We set Q = R ∩ V and apply the above considerations for Z = Q. Since we have natural isomorphisms H p (V, V rOK 0 (Q)) ' H p (SKV0 (Q), ∂SKV0 (Q)) ' H p (Q, ∂Q), (13.5) where the boundaries are taken in V , we obtain a morphism ˘ 2d−p (Q), L : H p (Q, ∂Q) −→ H (13.6) which we call the Lefschetz morphism. Note that the above morphism is ˘ 2d (Q, ∂Q). It given by the cap product with the relative class [Q] of Q in H ˘ 2d (Q, ∂Q) ' is the class corresponding to [SKV0 (Q)] in the isomorphism H ˘ 2d (SK 0 (Q), ∂SK 0 (Q)). H V

V

If V is non-singular, (13.6) is the Lefschetz isomorphism in Theorem 4.3. From the construction, we have (cf. Proposition 4.4): Proposition 13.1. Let R be a real 2n-dimensional manifold possibly with boundary in M as above. For a subcomplex Z in the interior of Q = R ∩ V , we have the commutative diagram: H p (V, V rZ) 

j∗

A

˘ 2d−p (Z) H

∗

/ H p (Q, ∂Q) 

j 0∗

L

˘ 2d−p (Q) /H

0∗

/ H p (V ) 

P

˘ 2d−p (V ), /H

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Complex Analytic Geometry

where j : (Q, ∂Q) ,→ (V, V r Z),  : Z ,→ Q and 0 : Q ,→ V denote ∗ the inclusions and j 0 denotes the isomorphism in (13.5) followed by the canonical morphism H p (V, V rOK 0 (Q)) → H p (V ). Suppose Z has a finite number of connected components (Zλ )λ . Then p ˘ 2d−p (Z) = L H ˘ H λ 2d−p (Zλ ) and each class α in H (V, V rZ) determines a ˘ 2d−p (Zλ ) via the Alexander morphism. class aλ in H Corollary 13.1. In the above situation, X (λ )∗ aλ = [Q] a j ∗ α

˘ 2d−p (Q), in H

λ

where λ : Zλ ,→ Q denotes the inclusion. In some situations, we do not consider the auxiliary manifold R and take M itself as R. In this case, we have the commutative diagram H p (V, V rZ) A



˘ 2d−p (Z) H

j∗

∗

/ H p (V ) 

(13.7)

P

˘ 2d−p (V ) /H

and the right-hand side of the identity in Corollary 13.1 is written V a j ∗ α ˘ 2d−p (V ). in H The above is the basis of the residue theorem we discuss in Section 13.4 below (cf. Theorems 13.5 and 13.6). Thom morphism We consider the morphism p 2k+p TV : CK (M, M rOK 0 (V )), 0 (V ) −→ CK ∗

uV 7→ u,

where u is given by, for each (2d − p)-simplex s of K, ( hs∗ ∩ V, uV i if s ⊂ V, ∗ hs , ui = 0 otherwise. Then it is compatible with coboundary operators and induces a morphism TV : H p (V ) −→ H 2k+p (M, M rV ),

(13.8)

called the Thom morphism. If V is non-singular, it is the isomorphism introduced in Section 4.3. Definition 13.1. The Thom class of V , denoted by ΨV , is the image of [1] by TV : H 0 (V ) → H 2k (M, M rV ).

Residues of Chern Classes on Singular Varieties

417

If V is non-singular, it coincides with the one in Definition 4.2. Remark 13.2. A remark similar to Remark 4.9. 1 applies to this case. 2k Namely, If ψ is a cocycle in CK 0 (M, M r OK 0 (V )) representing the Thom class, we have ( 1 if s ⊂ V, ∗ ∗ ∗ hιs , ψi = hs , ι ψi = 0 otherwise, where ι and ι∗ are defined similarly as in Remark 4.9. 1 with k 0 = 2k. From definition we see that the following diagram is commutative:

P



/ H 2k+p (M, M rV )

T

H p (V )

˘ 2d−p (V ). H

v

∼

(13.9)

A

Thus the Thom class ΨV in H 2k (M, M r V ) correspons to the funda˘ 2d (V ) by the Alexander isomorphism AM,V . mental class V in H Note that there is the cup product `

H 2k (M, M rV ) × H p (M ) −→ H 2k+p (M, M rV ). Denoting by i : V ,→ M the inclusion, the following is proved as Proposition 4.10: Proposition 13.2. The following diagram is commutative: H p (M )

ΨV `( )

/ H 2k+p (M, M rV ) o AM,V

i∗

 H p (V )

PV

 ˘ 2d−p (V ). /H

In particular, combined with (13.9), we have: TV (i∗ α) = ΨV ` α

for α ∈ H p (M ). ∼

As in (4.25), if we use the isomorphisms r∗ : H p (V ) → H p (OK 0 (V )) ∼ and H 2k+p (M, M rV ) → H 2k+p (OK 0 (V ), OK 0 (V )rV ), the above identity may be expressed as TV (αV ) = ΨV ` r∗ (αV )

for αV ∈ H p (V ).

(13.10)

If S is a K0 -subcomplex of M , setting Z = S ∩ V , we have the cup product `

H 2k (M, M rV ) × H p (M, M rS) −→ H 2k+p (M, M rZ).

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Complex Analytic Geometry

The following is proved as Proposition 4.11 (cf. Remark 13.2): Proposition 13.3. The following diagram is commutative: H p (M, M rS)

ΨV `( )

/ H 2k+p (M, M rZ) o AM,Z

i∗

 H p (V, V rZ)

AV,Z

 ˘ 2d−p (Z). /H

Gysin morphism: In the above situation, let i : V ,→ M denote the inclusion. As in Section 4.3, the composition j∗

T

V H 2k+p (M, M rV ) −→ H 2k+p (M ) H p (V ) −→

is referred to as the Gysin morphism and denoted by i∗ . We have the commutative diagram (cf. (4.18)) H p (V ) 

/ H 2k+p (M )

i∗

o PM

PV

˘ 2d−p (V ) H

(13.11)

i∗

 ˘ 2d−p (M ). /H

If V is non-singular, this coincides with (4.27). 13.3

ˇ de Rham and Cech-de Rham theories for singular varieties

Let M be a complex manifold of dimension n and V a variety of dimension d in M . Also, let K0 , K, K 0 and K ∗ be as in the previous section. Integration Since the triangulation h : |K0 | → M is C 1 , for a p-simplex s of K, or of K 0 , and a C ∞ p-form ω on a neighborhood of h(s) in M , we may define the integration as (6.15): Z Z ω = (h|s )∗ ω, s

s

which extends to integrations on finite chains. Suppose Sing V is compact. Let R be a real 2n-dimensional compact manifold possibly with boundary in M containing Sing V in its interior such that ∂R is transverse to Vreg . We set Q = R ∩ V , which may be assumed

Residues of Chern Classes on Singular Varieties

419

to be a finite K-chain. For a neighborhood U of Q in M , we have the integration Z : A2d (U ) −→ C. Q

We have the Stokes theorem (cf. Theorem 6.2): Z Z dω = ω for ω ∈ A2d−1 (U ). Q

∂Q

If V is compact, we have the integration Z : Hd2d (U ) −→ C,

(13.12)

V

where U is a neighborhood of V in U . Let Z be a compact set in V (which may not be compact) containing Sing V . Let U0 be a neighborhood of V rZ in M and U1 a neighborhood of Z in M and consider the covering U = {U0 , U1 } of U = U0 ∪ U1 . Let R1 be a compact 2n-dimensional manifold with boundary in U1 containing Z in its interior such that ∂R1 is transverse to Vreg . We set Q1 = R1 ∩ V and Q01 = −∂Q1 . Then we have the integration Z Z Z : A2d (U, U0 ) −→ C given by ξ 7→ ξ1 + ξ01 , (13.13) V

Q1

Q01

which induces Z

2d : HD (U, U0 ) −→ C.

Let R be a real 2n-dimensional compact manifold with boundary in M such that R contains Sing V in its interior and that ∂R is transverse to Vreg . We set Q = R∩V and let O be a neighborhood of ∂Q in M . Letting U0 = O and U1 a neighborhood of Q in M , we consider the covering U = {U0 , U1 } of U = U0 ∪ U1 . We have the integration Z Z Z 2d : HD (U, U0 ) −→ C given by [ξ] 7→ ξ1 − ξ01 . (13.14) Q

∂Q

de Rham isomorphism Let U be a neighborhood of V in M such that the inclusion i : V ,→ U is a homotopy equivalence. For example, letting {Xα } be a Whitney stratifiS cation of V , we may take as U the union of open ball bundles Oα with respect to a controlled tube system for {Xα }. Then we have an isomorphism ∼

i∗ ◦ χd : Hdp (U ) −→ H p (V ; C)

(13.15)

420

Complex Analytic Geometry ∼

as the composition of χd : Hdp (U ) → H p (U ; C) (cf. Theorem 7.5) and ∼ i∗ : H p (U ; C) → H p (V ; C). To give a simplicial expression of the above, we assume that U contains SK (V ) and modify K away from a neighborhood of V to obtain a triangulation KU of U as in Theorem 12.1, which is always possible. Then the ∗ isomorphism i∗ ◦ χd corresponds, via the isomorphism ηK of (13.1), to the isomorphism ∼

p i∗ ◦ ψd : Hdp (U ) −→ HK (V ; C),

which is the composition of ψd : ∼ p p i∗ : HK (U ; C) → HK (V ; C). The U morphism

(13.16)

∼ Hdp (U ) →

p HK (U ; C) (cf. (7.30)) and U ∗ isomorphism i ◦ ψd is induced from the

p i∗ ◦ ψd : Ap (U ) −→ CK (V ; C)

(13.17)

that assigns to a p-form ω the cochain u given by Z hs, ui = ω for every p-simplex s of K in V . s

In the following, we identify the singular homology and the simplicial homology, via the isomorphism η∗ in (13.1) or η˘∗ in (13.2), and the singular cohomology and the simplicial cohomology, via the isomorphism η ∗ in (13.1). Also, we use the integration of forms on simplicial chains, rather than that on singular chains, and use ψ instead of χ to denote the morphism from the space of forms to that of topologicalR cochains (cf. Section 7.5). If V is compact, we have the integration V : Hd2d (U ) → C (cf. 13.12). The following is proved as Proposition 7.11: Proposition 13.4. Suppose V is compact and let U be a neighborhood of V in M such that the inclusion i : V ,→ U is a homotopy equivalence. Then, for a class α in H 2d (V ; C) and the class [ω] in Hd2d (U ) corresponding to α under the isomorphism (13.15), we have Z hV, αi = [ω]. V

Poincar´ e morphism: Let U be a neighborhood of V such that the inclusion i : V ,→ U is a homotopy equivalence, as above. In terms of differential forms, the Poincar´e morphism (cf. Theorem 13.2) with C-coefficient ∼ ˘ 2d−p (V ; C) P˜ : H p (U ) −→ H p (V ; C) −→ H (13.18) d

is induced from the morphism i∗ ◦ψ 0

P

p ˘K Ap (U ) −→d CK 0 (V ; C) −→ C2d−p (V ; C),

ω 7→

XZ s

s∗ ∩V

 ω s,

Residues of Chern Classes on Singular Varieties

421

where i∗ ◦ ψd0 denotes the morphism (13.17) for K 0 and the sum is taken over the (2d − p)-simplices s of K in V . ˘ 0 (V ; C) = In particular consider the case p = 2d. If V is compact, H H0 (V ; C) and there is the augmentation ε∗ : H0 (V ; C) → C. In this case, we have Z ε∗ P˜ ([ω]) = ω, V

for any closed 2d-form ω on U , which restates the equality in Proposition 13.4. If, moreover, V is connected, ε∗ is an isomorphism so that (13.18) may be thought of as given by the integration Z : Hd2d (U ) −→ C. V

ˇ Cech-de Rham isomorphism Let U be a neighborhood of V in M such that the inclusion i : V ,→ U is a homotopy equivalence and U a covering of U . Then we have a canonical isomorphism ∼

p HD (U) −→ H p (V ; C) ∼ p (U) → HD

(13.19) ∗

p

p

∼

p

H (U ; C) and i : H (U ; C) → H (V ; C). as the composition of If there is a honeycomb system adapted to U, the former is given by χD as in (7.24) (cf. Theorem 7.6). We may give a simplicial expression of the above isomorphism as in the de Rham case. We do this in the case U consists of two open sets ; U = {U0 , U1 }, for simplicity. Thus let KU be a triangulation of U as before. Suppose there exists a honeycomb system {R0 , R1 } adapted to U that is also adapted to K in V , i.e., for each p-simplex s of K in V , s ∩ Ri is a p-chain, i = 0, 1, and s ∩ R01 is a (p − 1)-chain with respect to some subdivision of K (cf. Definition 7.9). Then the isomorphism (13.19) ∗ corresponds, via the isomorphism ηK of (13.1), to ∼

p p (V ; C), (U) −→ HK i∗ ◦ ψD : HD

(13.20)

which is induced from the morphism p i∗ ◦ ψD : Ap (U) −→ CK (V ; C)

(13.21)

ˇ that assigns to a CdR cochain ξ = (ξ0 , ξ1 , ξ01 ) the cochain u given by Z Z Z hs, ui = ξ0 + ξ1 + ξ01 s∩R0

s∩R1

s∩R01

for every p-simplex s of K in V (cf. Remark 7.10).

422

Complex Analytic Geometry

ˇ Relative Cech-de Rham isomorphism Let Z be a K0 -subcomplex of V containing Sing(V ). Let U be a neighborhood of V in M such that the inclusion i : V ,→ U is a homotopy equivalence. We take a covering U = {U0 , U1 } of U as follows. For U0 , we take a neighborhood of V rZ in U such that the inclusion i0 : V rZ ,→ U0 is a homotopy equivalence. For U1 , we take a neighborhood of Z such that U0 ∪ U1 = U . For example, we may take a covering defined as follows: (C) Let {Xα } be a Whitney stratification of V . We may assume that the stratum Xα0 of the largest dimension is equal to V r Z and that {Xα }α6=α0 is a Whitney stratification of Z. Let {Oα } be a system of open ball bundles with respect to a controlled tube system for {Xα }. S With these, we set U0 = Oα0 , U1 = α6=α0 Oα and U = U0 ∪ U1 . Then there is a C ∞ deformation retraction r0 : U0 → V rZ and the inclusion i : V ,→ U is a homotopy equivalence. In this case, i1 : Z ,→ U1 is also a homotopy equivalence. We may assume that U contains SK (V ) and modify K away from a neighborhood of V to obtain a triangulation KU of U , as before. Suppose {R0 , R1 } is a honeycomb system subordinate to U, i.e., Ri ⊂ Ui , i = 0, 1, (cf. Remark 7.5) satisfying the following: Condition (V, Z): (1) V ⊂ R0 ∪ R1 , (2) adapted to K 0 in V ,

(3) adapted to Z.

Recall that (2) means that, for each p-simplex t of K 0 in V , t ∩ Ri is a p-chain, i = 0, 1, and t ∩ R01 is a (p − 1)-chain with respect to some subdivision of K 0 (cf. Definition 7.9) and (3) means that R1 ⊂ OK 0 (Z) (cf. Definition 7.12). For example, take the barycentric subdivision K 00 of K 0 and set R0 = SK 00 (V r OK 0 (Z)) and R1 = SK 00 (Z), the stars in K 00 . Then {R0 , R1 } is a desired honeycomb system. We have a morphism p 0 i∗ ◦ ψD : Ap (U) −→ CK 0 (V ; C),

(13.22)

0

which is defined as (13.21) for K . It induces a morphism p 0 i∗ ◦ ψD : Ap (U, U0 ) −→ CK (13.23) 0 (V, V rOK 0 (Z); C) ˇ that assigns to a relative p-CdR cochain ξ = (ξ1 , ξ01 ) the cocycle u given by Z Z ht, ui = ξ1 + ξ01 t∩R1

for a p-simplex t of K 0 in V .

t∩R01

Residues of Chern Classes on Singular Varieties

423

0 Theorem 13.4. The morphism i∗ ◦ ψD in (13.23) induces an isomorphism ∼

p 0 i∗ ◦ ψD : HD (U, U0 ) −→ H p (V, V rZ; C).

Proof.

The morphism in (13.22) also induces a morphism Z  p p A (U0 ) −→ CK 0 (V rOK 0 (Z); C), given by ξ0 7→ t 7→

 ξ0 ,

t∩R0

(13.24) and we have a commutative diagram of complexes with exact rows, omitting the coefficient C in the bottom row: 0

/ A• (U, U0 )

/ A• (U)

/ A• (U0 )

/0

0

 / C • 0 (V, V rOK 0 (Z)) K

 / C • 0 (V ) K

 / C • 0 (V rOK 0 (Z)) K

/ 0.

We denote by j0 the composition of the inclusions that are homotopy equivalences i0

V rOK 0 (Z) ,→ V rZ ,→ U0 . Then the morphism (13.24) may be thought of as the composition ψ0

j∗

p p d 0 Ap (U0 ) −→ CK 0 (U0 ; C) −→ CK 0 (V rOK 0 (Z); C), U0

where KU0 0 denotes a triangulation of U0 obtained by extending the trian∼ gulation K 0 on V r OK 0 (Z). Thus it induces an isomorphism Hdp (U0 ) → p HK 0 (V rOK 0 (Z); C) (cf. (13.16)). In conclusion, we have the following commutative diagram with exact rows: ···

/ H p−1 (U0 ) d o

···

/H

p−1

 (V rZ)

/ H p (U, U0 )

/ H p (U) D

 / H (V, V rZ)

 / H (V )

D

p

o

p

We then have the theorem by the five lemma.

/ H p (U0 ) d

/ ···

o

 / H (V rZ) p

/ ··· . 

Remark 13.3. In the absolute case, we assumed the existence of a honeycomb system that is also adapted with the triangulation to have the isomorphism in (13.20). In the relative case, we need a honeycomb system that is adapted to the triangulation and the set Z. As noted above, such a honeycomb system exists so that we always have the isomorphism in Theorem 13.4.

424

Complex Analytic Geometry

Alexander morphism: Let V , Z, U = {U0 , U1 } and {R0 , R1 } be as ˇ above. In terms of CdR cochains, the Alexander morphism (cf. Theorem 13.3) with C-coefficient ∼ p ˘ 2d−p (Z, C) A˜ : HD (U, U0 ) −→ H p (V, V rZ; C) −→ H

(13.25)

is induced from the morphism ∗

0

i ◦ψ p ˘K Ap (U, U0 ) −→D CK 0 (V, V rOK 0 (Z); C) −→ C2d−p (Z; C),

ξ 7→

XZ

Z

s∗ ∩Q1

s

 ξ01 s,

ξ1 +

(13.26)

s∗ ∩Q01

where Q1 = R1 ∩ V , Q01 = R01 ∩ V and the sum is taken over the (2d − p)simplices of K in Z. ˘ 0 (Z; C) = In particular consider the case p = 2d. If Z is compact, H H0 (Z; C) and there is the augmentation ε∗ : H0 (Z; C) → C. In this case, since the 2n-cells s∗ appearing in (13.26) cover SK 0 (Z), we have Z Z ˜ ε∗ A([ξ]) = ξ1 + ξ01 , (13.27) Q1

Q01

R

which may as well be written V ξ (cf. (13.13)). If, moreover, Z is connected, ε∗ is an isomorphism and we have: Proposition 13.5. If Z is compact and connected, in the Alexander morphism 2d HD (U, U0 ) −→ H0 (Z; C) ' C,

the class of ξ = (ξ1 , ξ01 ) is sent to the complex number

R V

ξ.

The following is proved as Proposition 7.17: Proposition 13.6. Suppose Z is compact. Then in (13.25), a class [ξ] in p HD (U, U0 ) is sent to the class of a (2d − p)-cycle c in Z such that Z Z Z η1 = ξ1 ∧ η1 + ξ01 ∧ η1 c

Q1

Q01

for every closed (2d − p)-form η1 on U1 . In the case p = 2d, we recover (13.27). Before we proceed to the Lefschetz morphism, we consider the following situation. Let R be a real 2n-dimensional compact manifold with boundary in M such that R contains Sing V in its interior and that ∂R is transverse

Residues of Chern Classes on Singular Varieties

425

to Vreg . We set Q = R ∩ V . By Rsetting Z = Q in the above, we have the integration, which we denote by Q : Z 2d : HD (U, U0 ) −→ C. Q

As noted above, it coincides with the integration (13.13) for Z = Q. We also have a canonical isomorphism (cf. Theorem 13.4) : p HD (U, U0 ) ' H p (Q, ∂Q; C). In this situation, we have a generalization of Proposition 13.4: Proposition 13.7. In the above situation, for a class α in H 2d (Q, ∂Q; C) 2d and the class [ξ] in HD (U, U0 ) corresponding to α under the above isomorphism, we have Z h[Q], αi =

[ξ]. Q

R RRemarkR 13.4. In the equality above, the right-hand side is given by Q ξ = ξ + Q01 ξ01 with Q1 and Q01 as in (13.13) for Z = Q. If ξ01 is defined Q1 1 on a neighborhood of ∂Q,Z we mayZ write (cf. Z (13.14) and Remark 7.18) ξ1 −

[ξ] = Q

Q

ξ01 . ∂Q

Lefschetz morphism: Let R be a real 2n-dimensional manifold with boundary in M containing Sing V in its interior such that ∂R is transverse ˇ to Vreg . We set Q = R ∩ V . In terms of CdR cochains, the Lefschetz morphism (cf. (13.6)) with C-coefficient ∼ ˜ : H p (U, U0 ) −→ ˘ 2d−p (Q; C) L H p (Q, ∂Q; C) −→ H (13.28) D is obtained by setting Z = Q in (13.25). Thus it is induced from the morphism assigning to ξ = (ξ1 , ξ01 ), the cycle as given in (13.26). In particular, in the case p = 2d and Q is compact, we have an expression as (13.27): Z ε∗ L([ξ]) =

ξ Q

ˇ for any 2d-CdR cocycle ξ, which restates the equality in Proposition 13.7. If, moreover, Q is connected, we have a statement as in Proposition 13.5 for the Lefschetz duality: Proposition 13.8. If Q is compact and connected, in the Lefschetz morphism 2d HD (U, U0 ) −→ H0 (Q; C) ' C, R the class of ξ = (ξ1 , ξ01 ) is sent to Q ξ.

426

Complex Analytic Geometry

If Q is compact, we have a statement corresponding to Proposition 13.6, setting Z = Q. Remark 13.5. With the expressions as above of the duality morphisms, we have a diagram similar to the one in Proposition 13.1 with C-coefficient.

13.4

Chern classes and localization

Let M , V , K0 , K and K 0 be as before. Chern classes from topological viewpoint If we have a continuous complex vector bundle EV of rank l on V , we may define the Chern class cq (EV ) in H 2q (V ; Z) by obstruction theory using the triangulation K or K 0 as in Section 5.2. If we use K 0 , it is the primary obstruction to constructing an r-frame on the 2q-skeleton (K 0 )2q of K 0 , r = l − q + 1. The image of cq (EV ) by the Poincar´e morphism is ˘ 2(d−q) (V ; Z) by the cycle represented in H XX I(s(r) , bt ) s, (13.29) s

t

where the first sum is taken over all the 2(d − q)-simplices s of K in V , the second sum is taken over all the 2q-simplices t of K 0 in s∗ ∩ V and bt denotes the barycenter of t. Recall that we may construct an r-frame s(r) on the (2q − 1)-skeleton of K 0 and I(s(r) , bt ) is the index of s(r) on ∂t at bt . Topological localization Let Z be a K0 -subcomplex of V containing Sing(V ). Suppose we are already (r) given an r-frame sV of EV on the 2q-skeleton of V rOK 0 (Z). Then, for a (r) 2q-simplex t of K 0 not intersecting with Z, we have I(sV , bt ) = 0. Thus the 2q cocycle representing the Chern class cq (EV ) is in fact in CK 0 (V, V rOK 0 (Z)). (r) 2q It represents a class in H (V, V rZ; Z), which is denoted by cqZ,top (EV , sV ) (r)

and is called the topological localization of cq (EV ) by sV at Z. It will also (r) (r) (r) be denoted by cqZ (EV , sV ), cqtop (EV , sV ) or cq (EV , sV ). 2q Its image by the canonical morphism H (V, V r Z; Z) → H 2q (V ; Z) is the Chern class cq (EV ), however as a relative class it depends on the (r) frame sV .

Residues of Chern Classes on Singular Varieties

(r)

427

(r)

Definition 13.2. The topological residue TRescq (sV , EV ; Z) of sV for (r) cq (EV ) at Z is the image of cqtop (EV , sV ) by the Alexander morphism ˘ 2(d−q) (Z; Z). AV,Z : H 2q (V, V rZ; Z) −→ H (r)

Note that TRescq (sV , EV ; Z) is represented by a cycle as (13.29), where the first sum is taken over the 2(d − q)-simplices of K in Z. Note also that the possible singular point bt of s(r) in t may not be in Z. Suppose now that Z has only a finite number of connected components (Zλ )λ . Then we have a decomposition M ˘ 2(d−q) (Z; Z) = ˘ 2(d−q) (Zλ ; Z) H H λ (r)

˘ 2(d−q) (Zλ ; Z) and accordingly, we have the residue TRescq (sV , EV ; Zλ ) in H for each λ. It is represented by a cycle as (13.29), where the sum is taken over the 2(d − q)-simplices in Zλ . (r) In the case d = q and Zλ is compact, the residue TRescq (sV , EV ; Zλ ) is in H0 (Zλ ; Z) ' Z and may be thought of as an integer. It is given by X (r) TRescq (sV , EV ; Zλ ) = I(s(r) , bs ), (13.30) s

where the sum is taken over all the 0-simplices s of K in Zλ , in fact bs = s. Let R be a real 2n-dimensional manifold with boundary in M containing Sing V in its interior such that ∂R is transverse to Vreg . We set Q = R ∩ V . In the above considerations, we let Z be Q. Thus suppose we are given an (r) r-frame sV of EV on the 2q-skeleton of V r OK 0 (Q). Then we have the (r) relative class cqQ (EV , sV ) in H 2q (V, V rOK 0 (Q); Z) ' H 2q (Q, ∂Q; Z). (r)

(r)

To have the above relative class cqQ (EV , sV ), it suffices to have sV on the 2q-skeleton of ∂SKV0 (Q) (cf. Remark 5.8). Coming back to the previous situation, let Z be a K0 -subcomplex of (r) V containing Sing V and suppose we have an r-frame sV of EV on the 2q-skeleton of V rOK 0 (Z). Let R be as above and suppose that Z ⊂ Int R. (r) The r-frame sV restricts to an r-frame on the 2q-skeleton of V rOK 0 (Q). (r) Thus we have the relative class cqQ (EV , sV ) in H 2q (Q, ∂Q; Z) as above. From Proposition 13.1, we have the following “residue theorem”: Theorem 13.5. In the above situation it holds: (r) ˘ 2(d−q) (Zλ ; Z). 1. For each λ, we have the residue TRescq (sV , EV ; Zλ ) in H

428

Complex Analytic Geometry

2. We have X (r) (r) (λ )∗ TRescq (sV , EV ; Zλ ) = [Q] a cqQ (EV , sV )

˘ 2(d−q) (Q; Z), in H

λ

where λ : Zλ ,→ Q denotes the inclusion. Note that if we take M as R, then Q = V in the statement 2 above and ˘ 2(d−q) (V ; Z). the right-hand side is expressed as V a cq (EV ) in H We now treat the same problem from the differential geometric viewpoint. In fact these two are essentially the same (cf. Theorem 13.8 below). Chern classes from differential geometric viewpoint Let V be a variety in a complex manifold M as before. Also let U be a neighborhood of V in M such that the inclusion i : V ,→ U is a homotopy equivalence. Suppose we have a continuous complex vector bundle EV on V . Then we have a vector bundle E on U by pulling back EV by a homotopy inverse ϕ of i ; E = ϕ∗ EV . Note that E admits a C ∞ structure, which is essentially unique (cf. Remark 3.3). Thus choosing a connection for E, we may define the Chern classes cq (E) of E in Hd2q (U ). Recall the ∼ isomorphism i∗ ◦ ψd : Hd2q (U ) → H 2q (V ; C) (cf. (13.16)). Definition 13.3. The differential geometric Chern classes cq (EV ) of EV is defined to be the image in H 2q (V ; C) of cq (E) by i∗ ◦ ψd . Remark 13.6. Suppose we have a C ∞ vector bundle E on U and set EV = E|V (= i∗ E). Then ϕ∗ EV ' E for a homotopy inverse ϕ of i, as i ◦ ϕ ' 1U . We may as well define the Chern classes of EV using i∗ ◦ ψD in (13.20). Consider in particular the case the covering U of U consists of two open sets ; U = {U0 , U1 }, as before. If E is as above, its q-th Chern class cq (E) 2q in HD (U) is represented by the cocycle cq (∇∗ ) in A2q (U) given by cq (∇∗ ) = (cq (∇0 ), cq (∇1 ), cq (∇0 , ∇1 )),

(13.31)

where ∇0 and ∇1 are connections for E on U0 and U1 , respectively (cf. (10.10)). If U admits a suitable honeycomb system so that the iso∼ 2q morphism i∗ ◦ ψD : HD (U) → H 2q (V ; C) in (13.20) is defined, we may define the Chern class cq (EV ) as the image of cq (E) by i∗ ◦ ψD . Remark 13.7. We may also define the Chern class of a virtual bundle on V using (8.16) and the above arguments.

Residues of Chern Classes on Singular Varieties

429

Differential geometric localization For the relative case, let Z be a K0 -subcomplex of V containing Sing(V ). We take a covering U as defined in (C) above. We may take a homotopy inverse ϕ of i : V ,→ U so that we have the commutative diagram: U0

ı

r0

 V rZ



/U 

/U ,

/V >

ϕ

ϕ0

0

where U 0 is the space obtained from U by retracting U0 in U to V rZ, ϕ0 is a homotopy inverse of the inclusion i0 : V ,→ U 0 such that ϕ0 ◦ i0 = ϕ ◦ i, ı and  are the inclusions. Let EV be a continuous vector bundle on V and E the pullback ϕ∗ E, as above. Recall that E has a C ∞ structure. Note that EV |VrZ also has (r) a C ∞ structure (cf. Remark 3.3). Suppose we have a C ∞ r-frame sV of (r) EV on V r Z. Then we will see that there is a “localization” cq (EV , sV ) in H 2q (V, V rZ; C) of the Chern class cq (EV ) in H 2q (V ; C), q = l − r + 1. (r) In fact we have the localization ci (EV , sV ) for i ≥ l − r + 1, however, the case i = l − r + 1 is of essential interest (cf. Remark 10.5). First we show that E|U0 is isomorphic with r0∗ (EV |VrZ ) as a C ∞ bundle. To see this, noting that ϕ0 ◦ i0 ' 1V , we have E|U0 = ı∗ ϕ∗ EV = r0∗ (ϕ0 ◦ )∗ EV = r0∗ ((ϕ0 ◦ i0 )∗ EV )|VrZ ) ' r0∗ (EV |VrZ ). (r)

Let ∇ be an sV -trivial connection for EV on V rZ. We take as ∇0 in (13.31) the connection corresponding to the pull-back r0∗ ∇ (cf. Section 8.3). Then cq (∇0 ) = r0∗ cq (∇) = 0 by Proposition 10.5. Thus the cocycle is 2q in A2q (U, U0 ) and it defines a class in HD (U, U0 ), which we denote by (r) (r) q c (E, sV ). It does not depend on the choice of the sV -trivial connection ∇ or on the choice of the connection ∇1 (cf. Proposition 10.1). Recall that in this situation there is always an isomorphism (cf. Theorem 13.4) ∼

2q 0 i∗ ◦ ψD : HD (U, U0 ) −→ H 2q (V, V rZ; C). (r)

(r)

We denote the image of cq (E, sV ) by cqdiff (EV , sV ) and call it the differ(r) ential geometric localization of cq (EV ) at Z by sV . It is sent to the class q 2q c (EV ) by the canonical morphism H (V, V rZ; C) → H 2q (V ; C). Suppose now that Z has only a finite number of connected components (Zλ )λ . We have the Alexander morphism M ˘ 2(d−q) (Z; C) = ˘ 2(d−q) (Zλ ; C). A : H 2q (V, V rZ; C) −→ H H λ

430

Complex Analytic Geometry

(r) ˘ 2(d−q) (Zλ ; C), which Thus, for each λ, cqdiff (EV , sV ) defines a class in H (r)

(r)

we denote by Rescq (sV , EV ; Zλ ) and call the residue of sV at Zλ with respect to cq . The residues are expressed in the following way (cf. (13.25)). For each λ, we choose a neighborhood Uλ of Zλ in U1 , so that the Uλ ’s are mutually disjoint. We may assume that OK 0 (Zλ ) ⊂ Uλ . Let {R0 , R1 } be a honeycomb system subordinate to U and satisfying Condition (V, Z) above. We set Qλ = R1 ∩ V ∩ Uλ and Q0λ = −∂Qλ . Then the residue (r) Rescq (sV , EV ; Zλ ) is represented by the 2(d − q)-cycle Z  XZ Cλ = cq (∇1 ) + cq (∇0 , ∇1 ) s, (13.32) s

s∗ ∩Qλ

s∗ ∩Q0λ

where the sum is taken over the 2(d − q)-simplices of K in Zλ . ˘ 0 (Zλ ; C) = H0 (Zλ ; C) ' C. In particular, if d = q and if Zλ is compact, H (r) Thus we may regard Rescq (sV , EV ; Zλ ) as a complex number and we have Z Z (r) q Rescq (sV , EV ; Zλ ) = c (∇1 ) + cq (∇0 , ∇1 ). (13.33) Qλ

Q0λ

Let R be a real 2n-dimensional manifold with boundary in M containing Sing V in its interior such that ∂R is transverse to Vreg . We set Q = R ∩ V . (r) If we have an r-frame sV on V rQ, or on a neighborhood of ∂Q, we have (r) the relative class cqQ (E, sV ) in H 2q (Q, ∂Q; C). Coming back to the previous situation, suppose Z is contained in the (r) interior of a 2n-dimensional manifold R as above. An r-frame sV as above restricts to an r-frame on V rQ, or on a neighborhood of ∂Q, and we have (r) the relative class cqQ (EV , sV ) in H 2q (Q, ∂Q; C). We have the differential geometric counterpart of Theorem 13.5: Theorem 13.6. In the above situation it holds: (r) ˘ 2(d−q) (Zλ ; C). 1. For each λ, we have the residue Rescq (sV , EV ; Zλ ) in H

2. We have X (r) (r) (λ )∗ Rescq (sV , EV ; Zλ ) = [Q] a cqQ (EV , sV )

˘ 2(d−q) (Q; C). in H

λ

If we take M as R, then Q = V in the statement 2 above and the ˘ 2(d−q) (V ; C). right-hand side is expressed as V a cq (EV ) in H Remark 13.8. 1. In (13.32) and (13.33), the second integration is performed on V rZ and we may replace ∇0 with the connection ∇ originally

Residues of Chern Classes on Singular Varieties

431

given on V rZ (precisely speaking, the connection for E|VrZ corresponding to ∇ by the isomorphism E|VrZ ' EV |VrZ ). 2. Suppose we have a C ∞ vector bundle E on U and an r-section s(r) of E on U such that S(s(r) ) ∩ V is a K0 subcomplex of V containing Sing(V ). (r) Then the above procudure works by setting EV = E|V , sV = s(r) |V and (r) Z = S ∩ V , S = S(s ) (cf. Remark 13.6). In this setting, the localization (r) cq (EV , sV ) is the image of cq (E, s(r) ) by the canonical morphism i∗ : H 2q (U, U rS; C) −→ H 2q (V, V rZ; C). 3. In the case d = q and Z = {p} is an isolated point, the residue (r) Rescd (sV , EV ; {p}) is a complex number given as follows (cf. (13.33) and 1, 2 above). Take a neighborhood O of p in U such that E|O is trivial and let B be a closed 2n-ball with center p in O such that S = ∂B intersects transversely with V (cf. Theorem 11.16). Then letting L = V ∩ S be the link of V oriented as the boundary of V ∩ B, we have Z (r) Rescd (sV , EV ; {p}) = − cd (∇, ∇1 ), L

(r) sV -trivial

where ∇ is an connection for EV on V r{p} and ∇1 a connection for E on O trivial with respect to some frame. Example 13.1. Let V be a one-dimensional variety in a neighborhood U of 0 with an isolated singularity at 0. Let E = C × U be the product bundle on U . Suppose we have a C ∞ section s of E on U such that S(s)∩V = {0}. Set EV = E|V and sV = s|V . Then we have the localization c1 (EV , sV ) in H 2 (V, V r {0}; C) and the residue Resc1 (sV , EV ; 0) in H0 ({0}; C) = C, which we try to find. If we denote by e the frame of E given by e(z) = (1, z), we may write s = f e with f a C ∞ function on U . Let ∇0 be an s-trivial connection for E on U r S(s) and ∇1 a connection for E on U trivial with respect to e. Then, letting L be the link of V at 0, we have (cf. Remark 13.8): Z 1 Resc (sV , EV ; 0) = − c1 (∇0 , ∇1 ). L 1

The difference form c (∇0 , ∇1 ) is computed as in Example 10.3 and we have c1 (∇0 , ∇1 ) = − 2π√1 −1 df f so that Z df 1 √ . (13.34) Resc1 (sV , EV ; 0) = 2π −1 L f

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Complex Analytic Geometry

If f is holomorphic on U , then the residue (13.34) may also be   df expressed as the Grothendieck residue Res0 relative to V (cf. Secf V tion 13.6, Theorem 13.11 below). If V is a complete intersection defined by h1 = · · · = hn−1 = 0 in U , it is equal to   df ∧ dh1 ∧ · · · ∧ dhn−1 Res0 . f, h1 , . . . , hn−1 In this case, it is also equal to dim On /(f, h1 , . . . , hn−1 ) (cf. Example 12.2, Theorem 13.12 below). Coincidence of topological and differential geometric localizations Let M and V be as above and let EV be a continuous complex vector bundle of rank l on V . Then we have the Chern class cqtop (EV ) in H 2q (V ; Z) defined via obstruction theory and the Chern class cqdiff (EV ) in H 2q (V ; C) defined via Chern-Weil theory. As in the case of manifolds, they are essentially the same as stated below. The following can be proved similarly as Theorem 10.12: Theorem 13.7. In the above situation, cqdiff (EV ) is the image of cqtop (EV ) by the canonical morphism H 2q (V ; Z) → H 2q (V ; C). We have the following theorem for localized classes, which can be proved the same way as Theorem 10.13. Thus let Z be a subvariety of V of dimen(r) sion smaller than d = dim V and containing Sing V and sV an r-frame of (r) EV on V rZ, r = l − q + 1. Then we have the localized classes cqtop (EV , sV ) (r)

in H 2q (V, V rZ; Z) and cqdiff (EV , sV ) in H 2q (V, V rZ; C). (r)

Theorem 13.8. In the above situation, cqdiff (EV , sV ) is the image of (r) cqtop (EV , sV ) by the canonical morphism H 2q (V, V rZ; Z) → H 2q (V, V rZ; C). (r)

Corollary 13.2. In the above situation, the residue Rescq (sV , EV ; Z) is (r) the image of the topological residue TRescq (sV , EV ; Z) by the canonical ˘ 2(d−q) (Z; Z) → H ˘ 2(d−q) (Z; C). morphism H In particular, if d = q and if Z is compact and connected, the both may be thought of as an identical integer.

Residues of Chern Classes on Singular Varieties

13.5

433

Residues of Chern classes on singular varieties

Let V be a variety of dimension d in a complex manifold M of dimension n. Also, let EV be a continuous vector bundle of rank l on V . Let Z be a (r) subvariety of V containing Sing V and suppose an r-frame sV of EV is given on V rZ. (r) In this situation, we have the topological localization cqtop (EV , sV ) (r) ˘ 2(d−q) (Z; Z), in H 2q (V, V r Z; Z) and the residue TRescq (s , EV ; Z) in H V

(r)

q = l − r + 1. If sV is C ∞ , we also have the differential geometric localiza(r) (r) tion cqdiff (EV , sV ) in H 2q (V, V rZ; C) and the residue Rescq (sV , EV ; Z) in ˘ 2(d−q) (Z; C). We have seen (cf. Corollary 13.2) that Rescq (s(r) , EV ; Z) is H V (r)

the image of TRescq (sV , EV ; Z) by the canonical morphism ˘ 2(d−q) (Z; Z) −→ H ˘ 2(d−q) (Z; C). H

(13.35)

Now we consider the case where Z is of pure dimension d − q. We refer this situation as the proper case, as in Chapter 12. Let (Zi )i be the ˘ 2(d−q) (Z; Z) irreducible components of Z. Each Zi defines a class [Zi ] in H and it is the free Abelian group generated by the [Zi ]’s. Thus the morphism (13.35) is injective so that we may identify the two residues: (r)

(r)

Rescq (sV , EV ; Z) = TRescq (sV , EV ; Z). (13.36) S ◦ Let pi be a non-singular point of Zi = Zi r j6=i Zj and Di a complex slice of Zi◦ in M at pi (cf. Definition 3.24). Recall that Di is a locally closed complex sudmanifold of dimension q + k in M through pi and transverse to Zi at pi . Let U be a neighborhood of Z in M that contains Di as a closed submanifold and take a triangulation K0 of U as in Theorem 12.1, compatible with V ∩U and Z ∪Di , and let K be its barycentric subdivision. Let K 0 and K ∗ be as before. We may assume that pi is the barycenter bs of a 2(d − q)-simplex s in the non-singular part of Zi◦ and that the 2(q + k)-cell s∗ dual to s is in Di . Thus Di is also transverse to V . The (r) (r) r-section si = sV |Di ∩V of Ei = EV |Di ∩V has an isolated singularity at (r) pi so that we have the residue TRescq (si , Ei ; pi ), which is an integer. If (r) (r) sV is C ∞ , we also have Rescq (si , Ei ; pi ), which may be identified with (r) TRescq (si , Ei ; pi ). Note that this number does not depend on the choice of pi as in the case of residues on complex manifolds. Theorem 13.9. In the above situation, we have X (r) (r) TRescq (sV , EV ; Z) = TRescq (si , Ei ; pi ) · [Zi ] i

in

˘ 2(d−q) (Z; Z). H

434

Complex Analytic Geometry

(r)

Proof. By definition, the residue TRescq (sV , EV ; Z) is represented by the cycle XX I(s(r) , bt ) s, (13.37) s

t

where the first sum is taken over all the 2(d − q)-simplices s of K in Z, the second sum is taken over all the 2q-simplices t of K 0 in s∗ ∩V and I(s(r) , bt ) denotes the index of s(r) at the barycenter bt of t. Since s is a simplex of dimension 2(d − q), its interior is in the nonsingular part of Z and we may express the cycle (13.37) as XXX i

I(s(r) , bt(i) ) s(i) ,

s(i) t(i)

where the second sum is taken over all the 2(d − q)-simplices s(i) of K in Zi and the third sum is taken over all the 2q-simplices t(i) of K 0 in P (r) s∗(i) ∩ V . Moreover, we have t(i) I(s(r) , bt(i) ) = TRescq (si , Ei ; pi ), which is independent of s(i) .  Remark 13.9. If V is Cohen-Macaulay, in particular if it is a local complete intersection, then dim Z ≥ d − q (cf. Remark 11.13, Proposition A.23 and (A.16)).

13.6

Residues at an isolated singularity

Let V be a variety of dimension d in a complex manifold M of dimension n = d + k and let E be a holomorphic vector bundle of rank l (≥ d) on a neighborhood of V in M . Suppose that V has at most an isolated singularity at p. Let r = l − d + 1 and suppose we have a holomorphic r-section s(r) of E with S(s(r) ) ∩ V = {p}. Then, setting EV = E|V and (r) (r) (r) sV = s(r) |V , we have TRescd (sV , EV ; p) and Rescd (sV , EV ; p), which are identified. It is an integer, in fact we will see that it is positive. (r) Recall that Rescd (sV , EV ; p) is expressed as follows (cf. (13.33) and Remark 13.8). Let U be a neighborhood of p in M where the bundle E is trivial and set W = U ∩ V . We may assume that U is a coordinate neighborhood and sometimes we identify p with 0 in Cn and U with a neighborhood of 0. Let ∇0 be an s(r) -trivial connection for E on UrS(s(r) ) and ∇1 a connection for E on U trivial with respect to some holomorphic

Residues of Chern Classes on Singular Varieties

435

frame e(l) = (e1 , . . . , el ) of E. Then, letting L be the link of V at p, we have Z (r) Rescd (sV , EV ; p) = − cd (∇0 , ∇1 ). (13.38) L

(r) sV -trivial

Note that we may replace ∇0 with an W r{p}. We give various expressions of this number.

connection ∇ for EV on

Topological expression We take a small closed 2n-ball B with center p and let S be the boundary ∂B, which is a sphere of dimension 2n − 1. Recall that the link L of V at p is the intersection V ∩ S, which is a naturally oriented C ∞ manifold of dimension 2d − 1 (cf. Section 11.7). Let (Li )i be the connected components (r) of L. Since EV is trivial on W , sV defines, for each i, a map (cf. (5.1)) ϕi : Li −→ W (l, r). Since W (l, r) is 2l − 2r = 2d − 2 connected and H2d−1 (W (l, r), Z) = Z, we may consider the mapping degree deg ϕi of ϕi . Theorem 13.10. In the above situation, (r)

Rescd (sV , EV ; p) =

X

deg ϕi .

i (r)

Proof. We try to compute TRescd (sV , EV ; p), as it is the same as (r) Rescd (sV , EV ; p). We may assume that B = s∗ is the 2n-cell dual to (r) (2d−1) p. We may also extend sV |∂B to K 0 ∩ (s∗ ∩ V ), which includes p. (r) Note that, in this process, we only keep s |∂B∩V unchanged and modify (r) (2d−1) sV in (Int B) ∩ (V r{p}) to obtain a frame on K 0 ∩ (s∗ ∩ V ). If we (r) denote this frame by s0 , by definition, we have (cf. (13.29)) X (r) (r) TRescd (sV , EV ; p) = I(s0 , bt ), t

where the sum is taken over all the 2d-simplices t of K 0 in s∗ ∩ V and bt (r) denotes the barycenter of t. By definition of I(s0 , bt ) and properties of mapping degree, we have the theorem.  (r)

For the above formula, we only need to have a continuous frame sV on W r{p}.

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Analytic expression Grothendieck residues relative to a subvariety: Let U be a neighborhood of 0 in Cn and V a variety of dimension d in U . We assume that V contains 0 and that V r{0} is non-singular. Let f1 , . . . , fd be holomorphic functions on U and V (f1 , . . . , fd ) the variety defined by them. Suppose that V (f1 , . . . , fd ) ∩ V = {0}. For small positive numbers εi , i = 1, . . . , d, we set Γ = { z ∈ U ∩ V | |fi (z)| = εi , i = 1, . . . , d }, which is a d-cycle in V r{0}. It is oriented so that the form dθ1 ∧ · · · ∧ dθd is positive, θi = arg fi . For a holomorphic d-from ω on U , we set    1 d Z ω ω √ Res0 = f1 , . . . , f d V 2π −1 Γ f1 · · · fd and call it the Grothendieck residue of ω/f1 · · · fd at 0 relative to V . In the case d = n, it is the usual Grothendieck residue in Section 12.3. If V is a complete intersection defined by h1 = · · · = hk = 0 in U , we have     ω ω ∧ dh1 ∧ · · · ∧ dhk Res0 = Res0 . f1 , . . . , fd V f1 , . . . , fd , h1 , . . . , hk Exercise 13.1. Verify the above. Analytic expression: We consider the situation in the beginning of this Pl section. We write si = j=1 fji ej , i = 1, . . . , r, with fji holomorphic functions on U . Let F be the l × r matrix whose (i, j)-entry is fij . Let I be the set as given in (12.5) and let FI and ϕI be as defined before. If we write eI = ei1 ∧ · · · ∧ eir , we have X s1 ∧ · · · ∧ sr = ϕI eI . I∈I

and S(s(r) ) is the set of common zeros of the ϕI ’s. From the assumption S(s(r) ) ∩ V = {p}, we have the following by Lemmas 12.1 and 12.2: Lemma 13.2. We may choose a holomorphic frame e(l) = (e1 , . . . , el ) of E so that there exist d elements I (1) , . . . , I (d) in I with V (ϕI (1) , . . . , ϕI (d) ) ∩ V = {p}. Let e(l) be a frame of E as in Lemma 13.2. We write I (α) = α = 1, . . . , d, and let F (α) and Θ(α) be defined as in the

(α) (α) (i1 , . . . , ir ),

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437

paragraph after Lemma 12.3. Thus F (α) is an l × l matrix of holomorphic functions and Θ(α) is an l × l matrix of holomorphic 1-forms. With these we have: Theorem 13.11. In the above notation,   σ ˜d (Θ(1) , . . . , Θ(d) ) (r) Rescd (sV , EV ; p) = Resp . ϕI (1) , . . . , ϕI (d) V The above theorem is proved exactly the same way as Theorem 12.4. The only difference is that, in Theorem 12.4, U is a neighborhood of p in a manifold, while in Theorem13.11, it is replaced by a neighborhood W of p in a possibly singular variety. However everything is performed on W r{p}, which is non-singular. Special cases: 1. The case l = d and r = 1. Let e(l) = (e1 , . . . , ed ) be an Pd arbitrary frame of E and write s = i=1 fi ei . Then we may set ϕI (i) = fi , i = 1, . . . , d, and we have σ ˜d (Θ(1) , . . . , Θ(d) ) = df1 ∧ · · · ∧ dfd . Thus   df ∧ · · · ∧ dfd Rescd (sV , EV ; p) = Resp 1 . f1 , . . . , fd V 2. The case d = 1 and r = l. Let e(l) = (e1 , . . . , el ) be an arbitrary frame of Pr E and write si = j=1 fij ej , i = 1, . . . , l. Let F = (fij ) and set ϕ = det F . Then we may set ϕI (1) = ϕ and we have σ1 (Θ) = dϕ so that we have   dϕ (r) Resc1 (sV , EV ; p) = Resp . ϕ V Fundamental properties of the residues Let p be an isolated singular point in V and suppose that V is a complete intersection defined by h = (h1 , . . . , hk ) : (U, p) → (Ck , 0). Let T be a small neighborhood of 0 in Ck . For a point t in T , we set Vt = h−1 (t). Let C(h) denote the critical set of h and D(h) = h(C(h)) the discriminant, which is a hypersurface in T (cf. Proposition 11.28). We have Sing(Vt ) = C(h) ∩ Vt , which consists of at most a finite number of points. We set Et = E|Vt and (r) (r) st = s(r) |Vt so that S(st ) = S(s(r) )∩Vt . By the assumption S(s(r) )∩V = (r) (r) {p}, we have dim S(s ) ≤ k. Hence S(st ) also consists of at most a finite (r) number of points. Note that even if q is in Sing(Vt ), if q ∈ / S(st ), then (r) Rescd (st , Et ; q) = 0. We set V0 = V .

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(r)

Lemma 13.3. We have dim S(s(r) ) = k and S(st ) is a non-empty finite set. Proof. By assumption, we have dim S(s(r) ) ≤ k. On the other hand, codim S(s(r) ) ≤ l − r + 1 = d, by (A.16).  P (r) Lemma 13.4. The sum q∈S(s(r) ) Rescd (st , Et ; q) is continuous in t. t

Proof. Let ∇0 be an s(r) -trivial connection for E on U rS(s(r) ) and ∇1 a connection for E on U . Then by Theorem 13.6, the above sum is equal to an integral over Qt = R ∩ Vt , which is continuous in t.  Since T rD(h) is dense in T , by Lemma 2.7, we have P (r) Proposition 13.9. The sum q∈S(s(r) ) Rescn (st , Et ; q) is constant. In t particular, X (r) (r) Rescn (sV , EV ; p) = Rescn (st , Et ; q), (r)

q∈S(st )

which is a non-negative integer. (r)

Remark 13.10. By Lemma 13.3, Rescn (sV , EV ; p) is in fact a positive integer. Algebraic expression We refer to Section A.2 for relevant materials in algebra. Let E, U , and s(r) be as in the beginning of this section. Let e(l) be a holomorphic frame of E on U . Also let F , FI and ϕI = det FI be as before. We denote by OU the sheaf of germs of holomorphic functions on U and by F the ideal sheaf in OU generated by the (germs of) ϕI ’s. Note that F does not depend on the choice of the frame e(l) of E. We have the complex space S = (S, OS ) in U with support S = S(sr ) and structure sheaf OS = ι−1 (OU /F ), where ι : S ,→ U is the inclusion (cf. Section 11.5). We suppose that V is a complete intersection defined by h : (U, p) → (Ck , 0) as above. Let T be a small neighborhood of 0 in Ck . Denoting by π the restriction of h to S, we have a morphism (π, $) : S −→ (T, OT )

(13.39)

as the composition of the canonical morphism S → (U, OU ) and (h, h∗ ) : (U, OU ) → (T, OT ). For each x in S, $x : OT,π(x) → OS,x

Residues of Chern Classes on Singular Varieties

439

is the composition of h∗x : OT,π(x) → OU,x and the canonical epimorphism OU,x → OU,x /Fx = OS,x . We suppose that V is a complete intersection defined by h : (U, p) → (Ck , 0) as above. Let T be a small neighborhood of 0 in Ck and, for a point (r) t in T , we set Vt = h−1 (t). Also let S(st ) = S(s(r) ) ∩ Vt , as before. Let IV = (h1 , . . . , hk ) be the ideal sheaf of V in OU and FV the ideal sheaf generated by F and IV in OU . Also, for t = (t1 , . . . , tk ) ∈ T , we denote by FVt the ideal sheaf generated by F and IVt = (h1 − t1 , . . . , hk − tk ). The intersection Z of V and S in U is the complex space with support Z = S ∩ V = {p} and structure sheaf OZ = ι−1 (OU /FV ), where ι : Z ,→ U is the inclusion (cf. Section 11.5). Also, the intersection Zt of Vt and S in U is the complex space with support Zt = S ∩ Vt and structure sheaf OZt = ι−1 t (OU /FVt ), where ιt : Zt ,→ U is the inclusion. It is the fiber of (π, $) over t. By Lemma 13.3, π : S → T is a finite map. Thus each point x in S is isolated in Zt , t = π(x), and dimC OZt ,x = dimC OU,x /FVt ,x is finite. Proposition 13.10. In the above situation, X dimC OU,p /FV,p = dimC OU,q /FVt ,q . (r)

q∈S(st )

Proof. This is proved as Proposition 12.5 with a little modification, re˜ S and St with S, Z and Zt , respectively. placing S, We claim that the morphism (π, $) in (13.39) is flat, which will prove the proposition. Let x be a point in S and set t = π(x). In the following, we set Ox0 = OU,x , Ox = OS,x and Ot = OT,t . Note that Ox0 and Ot are regular local rings of dimensions n and k, respectively. We have ht Fx = n − k = d = l − r + 1. Hence by (A.17), the ring Ox is CM. Since the morphism $x : Ot → Ox is finite, Ox is a CM Ot -module. By (A.15), denoting by mt the maximal ideal in Ot , depth(mt ; Ox ) + pdOt Ox = depth mt . We have depth(mt ; Ox ) = dimOt Ox = dimOx Ox = k and depth mt = dim Ot = k. Therefore, pdOt Ox = 0 and (π, $) is flat.  Since the regular values of h are dense, by Proposition 13.9, Theorem 12.5 and Proposition 13.10, we have the following theorem. Theorem 13.12. We have (r)

Rescd (sV , EV ; p) = dimC OU,p /FV,p .

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Complex Analytic Geometry

Remark 13.11. As we can see from the above proofs, the assumption that V is a complete intersection is necessary only to ensure that V admits a “smoothing” in U .

13.7

Examples and related topics

In the following, we denote by M a complex manifold of dimension n and by V a variety of dimension d in M . Virtual index of a vector field Suppose V is an SLCI, i.e., a local complete intersection defined by a section of a holomorphic vector bundle N of rank k on M (cf. Definition 11.29). We have the virtual tangent bundle τV = (T M − N )|V of V . We also have its Chern class c∗ (τV ) in H ∗ (V ; C) (cf. Remark 13.7). As Vreg is a complex manifold, it has the holomorphic tangent bundle T Vreg , which is canonically isomorphic with the real tangent bundle TR Vreg of Vreg as a real bundle. Let Z be a subvariety of V containing Sing(V ) and v a C ∞ non-vanishing vector field on V rZ. Then we may define the localization cdZ (τV , v) by v of the class cd (τV ) as follows. Let U be a neighborhood of V in M such that the inclusion i : V ,→ U is a homotopy equivalence. Also let U = {U0 , U1 } be a covering of U such that U0 is a neighborhood of V r Z with a C ∞ deformation retraction r0 : U0 → V r Z and U1 is a neighborhood of Z in U . For i = 0, 1, let N ∇M i and ∇i be connections for T M and N on Ui , respectively, and set ? M 2d ∇i = (∇i , ∇N i ). Then the d-th Chern class in HD (U) of the virtual bundle (T M − N )|U is represented by the cocycle (cf. Sections 8.3 and 8.4) cd (∇?∗ ) = (cd (∇?0 ), cd (∇?1 ), cd (∇?0 , ∇?1 )).

(13.40)

Suppose a non-vanishing C ∞ vector field v is given on W0 = V rZ. Let ∇ be a v-trivial connection for T Vreg on W0 . We take connections ∇M and ∇N for T M |W0 and N |W0 so that (∇, ∇M , ∇N ) is compatible with the first row of (11.26) (cf. Proposition 8.5). Noting that T M |U0 ' r0∗ (T M |W0 ) and N N |U0 ' r0∗ (N |W0 ), we take as ∇M 0 and ∇0 the connections for T M |U0 and ∗ N ∗ M N |U0 corresponding to r0 ∇ and r0 ∇ , respectively. Then we have the vanishing cd (∇?0 ) = r0∗ cd (∇) = 0 (cf. Proposition 8.7) and the cocycle in 2d (13.40) defines a class in HD (U, U0 ), whose image in H 2d (V, V rZ; C) by the isomorphism of Theorem 13.4 is the localization cdZ (τV , v). We recall

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441

the Alexander morphism (cf. (13.25)) 2d ˘ 0 (Z; C). HD (U, U0 ) −→ H

If Z has a finite number of connected components (Zλ ), for each λ, we have ˘ 0 (Zλ ; C). If Zλ is a compact component, the residue Rescd (v, τV ; Zλ ) in H ˘ we have H0 (Zλ ; C) = H0 (Zλ ; C) ' C, the isomorphism being given by the augmentation, so that we may think of Rescd (v, τV ; Zλ ) as a number. Definition 13.4. For a compact connected component Zλ , we define the virtual index Vir(v, Zλ ) of v at Zλ as the residue Rescd (v, τV ; Zλ ). The virtual index is expressed as (cf. (13.33)) Z Z cd (∇?1 ) + cd (∇?0 , ∇?1 ). Vir(v, Zλ ) = Qλ

(13.41)

Q0λ

Note that we may replace ∇?0 with ∇? = (∇M , ∇N ) (cf. Remark 13.8. 1). Proposition 13.11. If Zλ is in Vreg , then Vir(v, Zλ ) = PH(v, Zλ ). Proof. Note that, in this case, the integrations in (13.41) are performed on Vreg and it suffices to define various connections involved on a neighborhood W of Zλ in Vreg . Let ∇, ∇M and ∇N be as above and set ∇? = (∇M , ∇N ). We take a connections ∇0 for T Vreg on W and connections M N M N ∇0 and ∇0 for T M and N on W so that (∇0 , ∇0 , ∇0 ) is compatible ? M N with the first row of (11.26) on W and set ∇0 = (∇0 , ∇0 ). Then, by d 0? d 0 d ? 0? Proposition 8.7, c (∇ ) = c (∇ ) on W and c (∇ , ∇ ) = cd (∇, ∇0 ) on W rZ.  Let R be a compact manifold of dimension 2n with boundary in M containing Sing(V ) in its interior such that ∂R is transverse to V . We set Q = R ∩ V . For a non-vanishing vector field v on a neighborhood of ∂Q in V , we may also define Vir(v, Q) in H0 (Q; C) ' C. If Z is contained in the interior of Q, for a non-vanishing vector field v on V rZ (cf. Theorem 13.6), X Vir(v, Zλ ) = Vir(v, Q). λ

In particular, we have (cf. Proposition 13.4): Proposition 13.12. In the above situation, if V is compact, we have Z X Vir(v, Zλ ) = cd (τV ). λ

V

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Complex Analytic Geometry

Now we consider the case Zλ consists of an isolated point p. In this case, we take as U a neighborhood of 0 in Cn , p being identified with 0, and set W = U ∩ V (cf. the beginning of Section 13.6). Suppose W is a CI defined by (h1 , . . . , hk ) : U → Ck . Let v be a non-vanishing C ∞ vector field on W0 = W r{0}. Then the (k + 1)-tuple  ∂  ∂ v (k+1) = v, ,..., ∂h1 W0 ∂hk W0 of sections of T M |W0 form a frame so that we have the residue Rescd (v (k+1) , T M |V ; 0). Theorem 13.13. In the above situation, Vir(v, 0) = Rescd (v (k+1) , T M |V ; 0). Proof. We denote by ∂ (k) the frame ($(∂/∂h1 ), . . . , $(∂/∂hk )) of N |W0 and by ν (k) the frame of NU extending ∂ (k) . We take a v-trivial connection ∇ for T Vreg on W0 , a v (k+1) -trivial connection ∇M for T M |W0 and a ∂ (k) trivial connection ∇N for NW0 so that (∇, ∇M , ∇N ) is compatible with the first row of (11.26) and set ∇? = (∇M , ∇N ). Also, let ∇M 1 be an arbitrary (k) connection for T U and let ∇N be the ν -trivial connection for N and set 1 N ∇?1 = (∇M , ∇ ). Let R be a 2n-dimensional manifold in U containing 0 in 1 1 its interior such that ∂R is transverse to V r{0} and set Q = R ∩ V . Then (cf. (13.33)) Z Z (k+1) d M Rescd (v , T M |V ; 0) = c (∇1 ) − cd (∇M , ∇M 1 ). Q

∂Q

∗ N ∗ N From c∗ (∇?1 ) = c∗ (∇M 1 )/c (∇1 ) and c (∇1 ) = 1, we have

cd (∇?1 ) = cd (∇M 1 ).

(13.42)

d ? ? In fact we may take ∇M 1 so that it is 0. To find c (∇ , ∇1 ), recall that it is ˜ ? ) over the 1-simplex [0, 1], where ∇ ˜ ? = (∇ ˜ M, ∇ ˜ N) given by integrating cd (∇ M M M N N N N ˜ ˜ with ∇ = (1−t)∇ +t∇1 and ∇ = (1−t)∇ +t∇1 . Since ∇ = ∇N 1 ˜ N = ∇N and moreover, c∗ (∇ ˜ N ) = 1, as ∇N is ∂ (k) on W0 , we have ∇ ˜ ? ) = cd (∇ ˜ M ) exactly as above. Therefore we trivial. Thus we have cd (∇ have cd (∇? , ∇?1 ) = cd (∇M , ∇M 1 ), which together with (13.42) implies the equality. 

Remark 13.12. 1. Thus it has a topological expression as in Theorem 13.10. However, even if v is defined and holomorphic in a neighborhood of p, we cannot use the algebraic or analytic expression in Theorems 13.12

Residues of Chern Classes on Singular Varieties

443

or 13.11, as the vector fields ∂/∂hi cannot be extended holomorphically through 0. 2. As is seen in the proof, we could use an arbitrary frame of N on U0 instead of ∂ (k) . In particular, if we take the frame (grad h1 , . . . , grad hk ), we get essentially what is called the GSV-index of v. Now we consider the situation of Theorem 11.18, with f replaced by h. Thus suppose we have a holomorphic map h : U → U 0 from a neighborhood U of 0 in Cn onto a neighborhood U 0 of 0 in Ck with h(0) = 0 such that V = h−1 (0) is a complete intersection of dimension d = n − k with an isolated singularity at 0. In this case we may assume that, for each t ∈ U 0 , Vt = h−1 (t) is a complete intersection of dimension d in U . Denoting by C(h) the set of critical points of h, we have Sing(Vt ) = C(h) ∩ Vt . Let Bε and B0δ be as in Theorem 11.18. Set Sε = ∂Bε and suppose we are given a non-vanishing vector field v˜ on a neighborhood U0 (disjoint from C(h)) of Sε ∩ h−1 (B0δ ) in U , which is tangent to U0 ∩ V . Setting h0 = h|U0 , we have the exact sequence of vector bundles 0 −→ T h0 −→ T U0 −→ h∗0 T U 0 −→ 0,

(13.43)

where T h0 is the bundle of vectors tangent to the fibers of h0 . Considering a C ∞ splitting of the above, we see that, taking a smaller U0 if necessary, v˜ defines a non-vanishing C ∞ section v˜0 of T h0 . We set v = v˜0 |V and vt = v˜0 |Vt . Set Q = Bε ∩ V and, for each t ∈ B0δ , Qt = Bε ∩ Vt . Then we have the virtual index Vir(vt , Qt ). If t ∈ B0δ rD(h), then Vt is non-singular and Qt = Ft , where Ft is a Milnor fiber. Theorem 13.14. In the above situation, Vir(vt , Qt ) does not depend on t. In particular, Vir(v, Q) = PH(vt , F t )

for t ∈ / D(h).

Proof. We compute Vir(vt , Qt ) using (the restriction to Vt of) connections as follows. We take connections ∇000 , ∇0 and ∇00 for T h|U0 , T U |U0 and h∗ T U 0 |U0 , respectively, so that (1) ∇000 is v˜0 -trivial, (2) the triple (∇000 , ∇0 , ∇00 ) is compatible with (13.43). We set ∇?0 = (∇0 , ∇00 ). There is a (trivial) vector bundle N with a frame (ν1 , . . . , νk ) on U and a surjective bundle morphism $

T U −→ N −→ 0

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Complex Analytic Geometry

∂ ) = νi , i = 1, . . . , k. The restriction of the such that, on U r C(h), $( ∂h i above to Vt,reg gives the sequence $

0 −→ T Vt,reg −→ T U |Vt,reg −→ NVt,reg −→ 0,

(13.44)

where NVt,reg is the normal bundle of Vt,reg . Thus τVt = (T U − N )|Vt , the virtual tangent bundle of Vt . The bundle T U 0 is also trivial of rank k and we may identify h∗0 T U 0 with N |U0 so that the restriction of (13.43) to Vt coincides with the restriction of (13.44) to U0 . Let ∇1 and ∇01 be arbitrary connections for T U and N , respectively, on U and set ∇?1 = (∇1 , ∇01 ). Then we have Z Z Vir(vt , Qt ) = cd (∇?1 ) − cd (∇?0 , ∇?1 ), Qt

∂Qt

which varies continuously on t. For t ∈ BεrD(h), this is equal to PH(vt , Ft ), which is an integer. Therefore Vir(vt , Qt ) is constant, as Bε rD(h) is connected and dense in Bε .  Recall that there exists a vector field v on W0 = U0 ∩V pointing outward on Q (cf. Remark 11.12). Corollary 13.3. For a vector field v as above, we have Vir(v, 0) = 1 + (−1)d µ(V, 0). Euler-Poincar´ e characteristic of singular varieties Let V be a local complete intersection defined by a section of a holomorphic vector bundle N of rank k on M and τV = (T M − N )|V as before. If V is non-singular, from (3.13) we have cd (τV ) = cd (V ) and moreover if V is compact, by the “Gauss-Bonnet formula” (cf. (5.20)) Z χ(V ) = cd (τV ). V

More generally, we have: Theorem 13.15. Let V be an SLCI of dimension d in M . If V is compact and has only isolated singularities p1 , . . . , pr , Z r X χ(V ) = cd (τV ) + (−1)d+1 µ(V, pi ). V

i=1

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445

Proof. For each i = 1, . . . , r, we take a closed ball Bi in M around pi so Sr that they are mutually disjoint. Then V ∗ = Vr i=1 Int Bi is a C ∞ manifold with boundary. Since ∂V ∗ is an odd-dimensional manifold, χ(∂V ∗ ) = 0 and we have χ(V ) = χ(V ∗ ) +

r X

χ(Bi ∩ V ).

i=1

The space Bi ∩ V is the cone with vertex pi and base Si ∩ V and is contractible, where Si = ∂Bi . Thus χ(Bi ∩ V ) = 1. For each i, we take a neighborhood Oi of pi containing Bi and a vector field v˜i on Oi r{pi } as in Remark 11.12. We may assume that the Oi ’s are Sr mutually disjoint. Let v be the vector field on i=1 (Oi r {pi }) ∩ V which is the restriction of v˜i on (Oi r{pi }) ∩ V . By Propositions 13.12, Z r X cd (τV ) = Vir(v, V ∗ ) + Vir(v, pi ). V ∗

i=1 ∗

Since V is in Vreg , Vir(v, V ) = PH(v, V ∗ ) by Proposition 13.11. We have PH(v, V ∗ ) = χ(V ∗ ), since the vector field v is pointing inward on ∂V ∗ and χ(∂V ∗ ) = 0 (cf. Remark 5.16). By Corollary 13.3, Vir(v, pi ) = 1 + (−1)d µ(V, pi ).  Note that we need only the compactness of V but not of M . Example 13.2. Let V be a complete intersection in M = Pn (cf. Example 11.11. 3). Then, from (8.16), (9.22) and N = Hn⊗p1 ⊕ · · · ⊕ Hn⊗pk , where Hn denotes the hyperplane bundle, we have c∗ (τV ) = i∗ c∗ (T Pn − N ),

c∗ (T Pn − N ) = (1 + hn )n+1 ·

k Y

1 . 1 + p ν hn ν=1

In the above, i : V ,→ Pn denotes the inclusion, hn = c1 (Hn ) and the product is the cup product. On the other hand, V is the Alexander dual of a localization of ck (N ) (cf. Theorem 12.9), in particular [V ] is the Poincar´e dual of ck (N ) and may be written i∗ V = [V ] = Pn a ck (N ) in H2d (Pn ). Qk We have ck (N ) = ν=1 pν · hkn . Since hn is the Poincar´e dual of [Pn−1 ], we have Pn a hn = [Pn−1 ] = j∗ Pn−1 , where j : Pn−1 ,→ Pn is the inclusion. Thus, by (B.18) and (B.23), Pn a h2n = j∗ Pn−1 a hn = j∗ (Pn−1 a j ∗ hn ). Since j ∗ hn = hn−1 , the first Chern class of the hyperplane bundle on Pn−1 ,

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Complex Analytic Geometry

we have Pn−1 a j ∗ hn = j∗0 Pn−2 , where j 0 : Pn−2 ,→ Pn−1 is the inclusion. Hence Pn a h2n = [Pn−2 ]. Continuing this, we have Pn a ck (N ) =

k Y

pν · [Pd ].

ν=1

Thus we compute, using (B.8) and noting that h[Pd ], hdn i = 1, Z cd (τV ) = hV, i∗ cd (T Pn − N )i = hi∗ V, cd (T Pn − N )i V k h Y = (1 + hn )n+1 ·

i pν , 1 + pν hn d ν=1

(13.45)

where [ ]d denotes the coefficient of hdn in [ ]. Thus we may compute χ(V ) from Theorem 13.15. Let V0 be a non-singular complete intersection in PnR of dimension d with the same multi-degree as V . Then we have χ(V0 ) = V0 cd (V0 ), which is also given by (13.45). Hence we have the following formula, which is readily proved by a direct argument as well: χ(V ) = χ(V0 ) + (−1)d+1

r X

µ(V, pi ).

(13.46)

i=1

See Theorem 14.3 below for an application of Theorem 13.15 to the case V is a curve in a surface. Virtual index of a (1, 0)-form Let V be a LCI defined by a section of a holomorphic vector bundle N of rank k on M as before. This time we consider the virtual cotangent bundle τV∗ = (T ∗ M − N ∗ )|V . Let Z be a closed subset of V containing Sing(V ) and θ a C ∞ non-vanishing (1, 0)-form on V r Z. Then we may define the localization cdZ (τV∗ , θ) of the class cd (τV∗ ) as in the case of vector fields. If Z has a finite number of connected components (Zλ ), for each λ we have the residue Rescd (θ, τV∗ ; Zλ ) and the residue formula. For a compact connected component Zλ , we define the virtual index Vir(θ, Zλ ) of θ at Zλ as the residue Rescd (θ, τV∗ ; Zλ ). Now we consider the case Zλ consists of an isolated point p. In this case, we take as U a neighborhood of 0 in Cn , p being identified with 0, and set W = U ∩V , as in the case of the virtual index of a vector field. Suppose W is a CI defined by (h1 , . . . , hk ) : U → Ck . For a (1, 0)-form θ on W0 = Wr{0},

Residues of Chern Classes on Singular Varieties

447

we consider the (k + 1)-tuple θ(k+1) = (θ, dh1 |W0 , . . . , dhk |W0 ) of sections of T ∗ M |W0 . It should be emphasized that here we take the restrictions as sections and not as differential forms. The following is proved as Theorem 13.13: Theorem 13.16. In the above situation, Vir(θ, 0) = Rescd (θ(k+1) , T ∗ M |V ; 0). Thus it has a topological expression as in Theorem 13.10. Moreover, if θ is defined and holomorphic in a neighborhood of p, we have the analytic and algebraic expressions in Theorems 13.11 and 13.12. Multiplicity of a function on a local complete intersection Let V be an LCI defined by a section of N and consider the virtual cotangent bundle τV∗ = (T ∗ M − N ∗ )|V as above. Let f˜ be a holomorphic function on M and let f and f 0 be its restrictions to V and Vreg , respectively. We define the singular set S(f ) of f by S(f ) = Sing(V ) ∪ C(f 0 ). As in the case of vector bundles, we may define the localization of the d-th Chern class of τV∗ by df , which in turn defines the residue Rescd (df, τV∗ ; Z) at each compact connected component Z of S(f ). For simplicity we consider the case Z consists of a point p. We define the virtual multiplicity m(f, e p) of f at p by m(f, e p) = Vir(df, p) = Rescd (df, τV∗ ; p).

(13.47)

The multiplicity of f at p is then defined by m(f, p) = m(f, e p) − µ(V, p).

(13.48)

Note that, if p is in Vreg , we have Rescd (df, τV∗ ; p) = Rescd (df, T ∗ Vreg ; p). On the other hand, in this case we have µ(V, p) = 0 so that m(f, p) coincides with the one in Definition 12.4, see also Example 12.3. Let f˜ : M → C be a holomorphic map onto a non-singular complex curve C and set f = f˜|V , f 0 = f˜|Vreg and S(f ) = Sing(V ) ∪ C(f 0 ). We further set V0 = V r S(f ) and f0 = f˜|V0 . Thus df0 is a non-vanishing section of the bundle T ∗ V0 ⊗ f0∗ T C, which is of rank d. If we look at cd (ε), ε = τV∗ ⊗ f ∗ T C, we see that there is a canonical localization cdZ (ε, df ) in H 2d (V, V rZ; C) of cd (ε), Z = S(f ). Let (Zλ )λ be the connected components of Z. Then cdZ (ε, df ) defines, for each λ, the residue Rescd (df, τV∗ ⊗ f ∗ T C; Zλ ). If Zλ is compact, it is given by a formula similar to (13.33), see also (13.41). Note that, if Zλ is

448

Complex Analytic Geometry

in the non-singular part Vreg , it coincides with the one in Section 12.5. If V is compact, we have (cf. Proposition 13.12) Z X Rescd (df, τV∗ ⊗ f ∗ T C; Zλ ) = cd (τV∗ ⊗ f ∗ T C). V

λ

Now we assume that S(f ) consists of a finite number of isolated points and examine the both sides more closely. First, we may write, for a point p in S(f ), Rescd (df, τV∗ ⊗ f ∗ T C; p) = m(f, e p) = m(f, p) − µ(V, p). If we set C0 = C r f (S(f )), V00 = V r f −1 C0 and f00 = f |V00 , then : V00 → C0 is a C ∞ fiber bundle (cf. Theorem 3.12). Let F denote the fiber. f00

Lemma 13.5. If V is compact, then we have Z X cd (τV∗ ⊗ f ∗ T C) = (−1)d (χ(V ) − χ(F ) χ(C)) + µ(V, p). V

p∈S(f )

Proof. Using the properties of Chern classes and noting that C is complex one-dimensional, we have cd (τV∗ ⊗ f ∗ T C) = (−1)d (cd (τV ) − cd−1 (τV ) · f ∗ c1 (C)). Thus, in view of Theorem 13.15, it suffices to prove Z cd−1 (τV ) · f ∗ c1 (C)) = χ(F )χ(C).

(13.49)

V

For this, let f (S(f )) = {t1 , . . . , tk } and, for each ti , take an open disk Ci around ti in C so that they are mutually disjoint and that T C|Ci is trivial. Set C0 = C rS(f ) and consider the covering C = {C0 , {Ci }ki=1 } of C. We set V00 = V rf −1 (C0 ) as above. Let Ui = (f˜)−1 Ci and let U0 be a tubular neighborhood of V00 in M with a deformation retraction r0 : U0 → V00 . We may take U0 small enough so that f˜ does not have critical points in U0 . Set S f˜0 = f˜|U0 . Let U denote the covering {U0 , {Ui }ki=1 } of U0 ∪ Ui . In the following, we try to prove (13.49) taking connections for various bundles. As to the bundles on V , it is sufficient to consider connections on V00 and Ui . C We take a connection ∇C 0 for T C on C0 . For each i = 1, . . . , k, let ∇i be a connection for T C on Ci trivial with respect to some frame of T C. 1 2 Noting that c1 (∇C i ) = 0, c (C) is represented, in A (C), by the cocycle 1 C 1 C C c1 (∇C ∗ ) = (c (∇0 ), 0, (c (∇0 , ∇i ))i ).

Residues of Chern Classes on Singular Varieties

449

We have an exact sequence of vector bundles on V00 : ∗

0 −→ T f00 −→ T V00 −→ f00 T C −→ 0. Take connections ∇f0 and ∇V0 for T f00 and T V00 , respectively, so that (∇f0 , ∇V0 , f ∗ ∇C 0 ) is compatible with the above sequence. Then we have c∗ (∇V0 ) = c∗ (∇f0 ) · f ∗ c∗ (∇C 0 ), in particular, cd−1 (∇V0 ) = cd−1 (∇f0 ) + cd−2 (∇f0 ) · f ∗ c1 (∇C 0 ).

(13.50)

Take connections ∇0 and ∇00 for T M |V00 and NV00 so that (∇V0 , ∇0 , ∇00 ) is compatible with 0 −→ T V00 −→ T M |V00 −→ NV00 −→ 0 and set ∇?0 = (∇0 , ∇00 ). Then we have cd−1 (∇?0 ) = cd−1 (∇V0 ).

(13.51)

Also let ∇i and ∇0i be connections for T M and N , respectively, on Ui and set ∇?i = (∇i , ∇0i ). Then cd−1 (τV ) is represented, in A2(d−1) (U), by the cocycle cd−1 (∇?∗ ) = (cd−1 (∇?0 ), (cd−1 (∇?i ))i , (cd−1 (∇?0 , ∇?i ))i ). The class cd−1 (τV ) · f ∗ c1 (C) is then represented by the cup product cd−1 (∇?∗ ) · f ∗ c1 (∇C ∗) d−1 C = (cd−1 (∇?0 ) · f ∗ c1 (∇C (∇?0 ) · f ∗ c1 (∇C 0 ), 0, (c 0 , ∇i ))i ).

From (13.51) and (13.50) and by the dimension reason, we have d−1 cd−1 (∇?0 ) · f ∗ c1 (∇C (∇f0 ) · f ∗ c1 (∇C 0)=c 0 ), C d−1 C cd−1 (∇?0 ) · f ∗ c1 (∇C (∇f0 ) · f ∗ c1 (∇C 0 , ∇i ) = c 0 , ∇i ).

For each i = 1, . . . , k, let Di be a closed disk in Ci centered at ti , and let Sk D0 = Cr i=1 Int Di . Then {D0 , {Di }ki=1 } is a honeycomb system adapted to C. If we set Qi = f −1 Di and Q0i = f −1 D0i , D0i = −∂Di , using the projection formula, the left-hand side of (13.49) becomes Z

(∇f0 )

d−1

c Q0

∗ 1

·f c

(∇C 0)

+

k Z X i=1

Z = D0

C cd−1 (∇f0 ) · f ∗ c1 (∇C 0 , ∇i )

Q0i

f∗ cd−1 (∇f0 ) · c1 (∇C 0)+

k Z X i=1

D0i

C f∗ cd−1 (∇f0 ) · c1 (∇C 0 , ∇i ).

450

Complex Analytic Geometry

In the above, f∗ cd−1 (∇f0 ) denotes the integration along the fibers of f and it is equal to the constant χ(F ). On the other hand, Z Z k Z X 1 C C c1 (∇C ) + c (∇ , ∇ ) = c1 (C) = χ(C), 0 0 i D0

i=1

D0i

C

which proves (13.49).



Thus we have: Theorem 13.17. In the above situation, we have X m(f, p) = (−1)d (χ(V ) − χ(F ) χ(C)) . p∈S(f )

The above generalizes the formula in Corollary 12.2 for singular varieties. The virtual multiplicity m(f, ˜ p) = Rescd (df, τV∗ ; p) is given as follows. Let U be a neighborhood of p in M so that the bundle N admits a frame Pk (ν1 , . . . , νk ) on U . We write σ = i=1 hi νi with hi holomorphic functions on U . Then V is defined by (h1 , . . . , hk ) in U . Consider the (k + 1)-tuple of sections s(k+1) = (df˜, dh1 , . . . , dhk ) of T ∗ U . By assumption we have S(s(k+1) ) ∩ V = {p}. Since the rank of (k+1) (k+1) T ∗ U is n, we have the residue Rescd (sV , T ∗ U |V ; p), sV = s(k+1) |V . The following is proved as Theorem 13.13: (k+1)

m(f, e p) = Rescd (sV

, T ∗ U |V ; p).

(13.52)

The virtual multiplicity m(f, e p) was defined as the residue of df on the ∗ virtual bundle τV (cf. (13.47)) and this definition led us to a global formula as in Theorem 13.17. The identity (13.52) shows that it coincides with the (k+1) residue of sV = (df˜|V , dh1 |V , . . . , dhk |V ) on the vector bundle T ∗ U |V . Thus we have various expressions for m(f, e p) as given in the previous section; by Theorem 13.10, we have a topological expression, by Theorem 13.11 we have a way to compute m(f, e p) explicitly and by Theorem 13.12 we may express m(f, e p) = dimC On /(J(f˜, h1 , . . . , hk ), h1 , . . . , hk ),

(13.53)

where J(f˜, h1 , . . . , hk ) denotes the Jacobian ideal of the map (f˜, h1 , . . . , hk ), i.e., the ideal generated by the (k+1)×(k+1) minors of the Jacobian matrix ∂(f˜,h1 ,...,hk ) ∂(z1 ,...,zn ) .

Residues of Chern Classes on Singular Varieties

451

The multiplicity m(f, p) is also interpreted as follows. Thus let Vf˜ denote the complete intersection defined by (f˜, h1 , . . . , hk ), assuming f˜(p) = 0. Then, from (13.48), (13.53) and the identity µ(V, p) + µ(V ˜, p) = dimC On+k /(J(f˜, h1 , . . . , hk ), h1 , . . . , hk ), f

which follows from (11.23), we get m(f, p) = µ(Vf˜, p). Dual classes We come back to the general situation as in the beginning of Section 13.5. Thus let V be a variety of dimension d in M . Here we let EV be a holomorphic vector bundle of rank l on V . Note that it makes sense to talk about holomorphic vector bundles on varieties as we have the notion of a holomorphic function on a variety (cf. Section 11.5). Suppose we are (r) given a holomorphic r-section sV of EV . Let Z be the analytic space in V (r) defined by sV , i.e., the analytic space locally defined by the ideal generated by the r × r minors of the matrix whose entries are the local components (r) of the sV,i ’s. The support Z of Z is the singular set of sV . We assume that Z contains Sing V . If we set q = l − r + 1, we have the topological (r) and differential geometric localizations of cq (EV ) by sV and the associated (r) (r) residues TRescq (sV , EV ; Z) and Rescq (sV , EV ; Z) as their images by the Alexander morphism. In the following, we assume that dim Z = d − q (proper case) and let (Zi )i be the irreducible components of Z. Note that in this case we may (r) (r) identify TRescq (sV , EV ; Z) and Rescq (sV , EV ; Z) (cf. (13.36)). (r) Let pi , Ei and si be as in the paragraph following (13.36). Then we (r) (r) have the residues TRescq (si , Ei ; pi ) and Rescq (si , Ei ; pi ), which may be identified. It is a positive integer. Definition 13.5. The multiplicity mi of Zi in Z is defined by (r)

mi = Rescq (si , Ei ; pi ). Note that, the multiplicity mi has a topological expression as in Theo(r) rems 13.10. If EV and sV are the restrictions of the ones on M , it has an analytic expression as in Theorem 13.11. If, moreover, V is a local complete intersection, it has an algebraic expression as in Theorem 13.12. P Definition 13.6. The homology class of Z is defined by [Z] = mi [Zi ] P ˘ ˘ in H2(d−q) (V ; Z) or by Z = mi [Zi ] in H2(d−q) (Z; Z).

452

Complex Analytic Geometry

From Theorem 13.9, we have: Theorem 13.18. Let Z be the complex subspace of dimension d−q of V as ˘ 2(d−q) (Z; Z) is equal to TRescq (s(r) , EV ; Z), above. Then the class of Z in H V (r)

(r)

which may be identified with Rescq (sV , EV ; Z). Thus the class cq (EV , sV ) is sent to the class Z by the Alexander morphism ˘ 2(d−q) (Z; Z). H 2q (V, V rZ; Z) −→ H ˘ 2(d−q) (V ; Z) is the image of cq (LD ) in In particular, the class [Z] in H H (V ; Z) by the Poincar´e morphism. 2q

Notes The presentation of this chapter is mainly based on Sections 5 and 6 of [Suwa (2008)]. For controlled tube systems and Theorem 13.1, we refer to [Gibson, Wirthm¨ uller, du Plessis and Looijenga (1976); Mather (2012); Shiota (1997)]. As to the Poincar´e, Alexander and Thom morphisms, we followed the description of [Brasselet (1981)], except for the orientation convention (cf. Notes in Chapter 4). In Section 13.6, see [Suwa (2003a)] for the analytic expression and [Suwa (2005)] for the algebraic expression with some more examples. As to the Grothendieck residue relative to a subvariety, see Ch.IV, 8 in [Suwa (1998)]. In Section 13.7, the virtual index is introduced in [Lehmann, Soares and Suwa (1995)]. The GSV-index of a vector field at an isolated singularity is introduced in [G´ omez-Mont, Seade and Verjovsky (1991)] for hypersurfaces and in [Seade and Suwa (1996)] for complete intersections. As to the material there, we refer to the above literatures as well as [Brasselet, Seade and Suwa (2009); Suwa (2014)] and references therein. In [Brasselet, Lehmann, Seade and Suwa (2001)], a generalized Milnor number µ(V, S) is defined for each compact connected component S of the singular set of an SLCI V and, if V is compact, a formula as in Theorem 13.15 is proved. In the case V is a hypersurface, the number and the formula coincide with the ones in [Parusi´ nski (1988)], see also [Parusi´ nski and Pragacz (1995)]. As to the formula (13.46), see also [Dimca (1992)]. The index of a 1-form is introduced and the algebraic expression is given in [Ebeling and Gusein-Zade (2001)]. For details on the multiplicity of a function, we refer to [Izawa and Suwa (2003)]. It is possible to define the multiplicity of a function at a non-isolated component of S(f ) using the

Residues of Chern Classes on Singular Varieties

453

number µ(V, S) above. See [Suwa (2005)] for more examples of related local invariants. As another topic along the line above, there is the theory of characteristic classes of singular varieties. For the treatment in the framework of this book, we refer to [Brasselet, Lehmann, Seade and Suwa (2001); Brasselet, Seade and Suwa (2009); Suwa (1996)] and a survey paper [Suwa (2003b)]. We list, as related works, [Aluffi (1999); Brasselet, Sch¨ urmann and Yokura (2010); Brasselet and Schwartz (1981); Fulton and Johnson (1980); MacPherson (1974); Parusi´ nski and Pragacz (2001); Schwartz (1965)] and survey papers [Sch¨ urmann and Yokura (2007); Yokura (2020)]. We also list [Brasselet (2022)] as a survey paper of characterictic classes in general.

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Chapter 14

Intersection Product of Complex Subspaces

In this chapter, we discuss intersection products of the homology classes of complex spaces in complex manifolds or in singular varieties. The formulas we prove concerning intersection products are localized at the set of intersection. The corresponding global formulas are consequences of these. We first generalize the Whitney sum formula for localized Chern classes (cf. Corollaries 5.6 and 10.5) so that it fits our purposes (Corollary 14.1). In the case of manifolds, the intersection product in homology is defined combinatorially in Section 4.4. It is an operation dual to the cup product in cohomology. In general, the intersection product of the classes of subspaces is different from the class of the intersection of them as complex subspaces. The intersection products we consider involve, in some form or another, complex spaces defined by a family of sections of a vector bundle. In this situation, we have the localization of the Chern class of the bundle by the family of sections as well as the associated residue. The class of the complex space defined by the family of sections coincides with the residue if it has an expected dimension (proper case). We always have expected formulas for residues. Thus in the proper case, we have expected formulas for spaces as defined above. We start by considering the intersection product of subspaces defined by a holomorphic section of a vector bundle. The aforementioned Whitney sum formula for localized classes and the duality in Theorem 12.9 are key ingredients for this. As a special case, we discuss intersection products of divisors and give B´ezout’s theorem as an example (Example 14.1). Intersection product with the class of a (fixed) subvariety is then studied. It leads to the intersection theory of the classes of complex spaces in complex manifolds in general. In particular, we consider the intersection product of the class of a variety and the class of the complex space defined by a

455

456

Complex Analytic Geometry

family of holomorphic sections. Here we use the duality as in Theorem 12.8. As a related topic, we prove the adjunction formula for singular curves (Theorem 14.3). As to the intersection product in sigular varieties, we do not have general combinatorial ways of defining it. Thus we consider intersection products of homology classes that come naturally as the images of the Poincar´e or Alexander morphism and use the cup product in cohomology. For this, as in the case of manifolds, we consider intersection product involving subspaces defined by a family of sections and develop the theory in parallel with that case. We also discuss intersection products of divisors. Here we need to be a little careful, as unlike the case of complex manifolds, we have to distinguish between Cartier and Weil divisors. We also discuss the multiplicity of a variety at a point, giving its various expressions. We then specialize the above considerations to the case of intersection products in singular surfaces. In particular, we study the behavior of intersection products under blowing-up, which is applied to define intersection product of Weil curves. Finally we study the case where a variety and a complex space defined by a section intersect in a dimension greater than expected.

14.1

Refined Whitney sum formula

We recall the construction of the Euler class of an oriented real vector bundle and its localization in Sections 5.3, 5.4 and 5.6. Here we do it for bundles on simplicial complexes. In the following, a map means a continuous map 0 0 and Rl r{0} is denoted by Rl r0. Let l0 be an integer with l0 > 1. Recall that there is a canonical isomorphism (cf. Example B.6) 0

0

∼

0

0

0

δ ∗ : H l −1 (Sl −1 ) −→ H l (Bl , Sl −1 ) ' Z. 0

0

0

0

0

0

There is also a canonical isomorphism H l (Rl , Rl r0) ' H l (Bl , Sl −1 ). The 0 0 0 group H l (Rl , Rl r 0) has the canonical generator ΨRl0 , the Thom class, 0 which is equal to the l0 -fold cross product el of the canonical generator 0 0 0 e of H 1 (R, R r 0) (cf. Corollary 4.4. 1). The generator of H l (Bl , Sl −1 ) 0 corresponding to ΨRl0 is denoted by ul , which coincides with δ ∗ (νl∗0 −1 ) (cf. Remark 5.1. 2). 0 0 0 0 Thus we may define the degree of a map (Bl , Sl −1 ) → (Rl , Rl r0) as well.

Intersection Product of Complex Subspaces

457 0

0

Proposition 14.1. Giving a homotopy class of a map ϕ : Sl −1 → Rl r0 0 0 0 0 is equivalent to giving that of a map ψ : (Bl , Sl −1 ) → (Rl , Rl r0), in fact 0 0 0 0 that of a map (Bl , Bl r0) → (Rl , Rl r0). 0

Proof. Let ϕ be as above. Consider the product I × Sl −1 with I = [0, 1], 0 0 the unit interval, and define ϕ˜ : I × Sl −1 → Rl by (t, x) 7→ tϕ(x). The 0 0 0 ball Bl is obtained from I × Sl −1 by shrinking {0} × Sl −1 to a point and 0 0 0 0 ϕ˜ induces a map ψ : (Bl , Sl −1 ) → (Rl , Rl r{0}). The homotopy class of ϕ determines that of ψ as the homotopy class of such a map is uniquely determined by its degree. The converse is obvious.  0

0

0

Let E = Rl × Bl be the product bundle of rank l0 on Bl . Suppose we 0 have a non-vanishing section s of E on Sl −1 . We extend it to a section on 0 0 Bl non-vanishing section on Bl r0. We have a composition as in (5.4): 0

0

0

s

0

0

p

0

0

ψ : (Bl , Sl −1 ) −→ (Rl , Rl r0) × Bl −→ (Rl , Rl r0),

(14.1)

where p denotes the projection. Then the index I(s, 0) of s at 0 is the mapping degree of ψ: I(s, 0) = deg ψ. 0

0

Suppose we have two product bundles Ei = Rli × Bli , i = 1, 2. For 0 each i, let si be a non-vanishing section of Ei on Sli −1 . The product 0 0 0 0 0 0 (Bl1 , Sl1 −1 ) × (Bl2 , Sl2 −1 ) is homeomorphic with (Bl , Sl −1 ), l0 = l10 + l20 . If we identify the two pairs of spaces, the product E = E1 × E2 is the product 0 0 0 bundle Rl × Bl on Bl and s = s1 × s2 is a non vanishing section of E on 0 Sl −1 . Lemma 14.1. In the above situation, we have I(s, 0) = I(s1 , 0) · I(s2 , 0). 0

0

0

0

Proof. Let ψi : (Bli , Sli −1 ) → (Rli , Rli r 0) be the map as defined in 0 0 0 0 0 0 (14.1) for si . Then we have ψi∗ : H li (Rli , Rli r0) → H li (Bli , Sli −1 ) and by 0 0 definition, ψi∗ (eli ) = deg ψi · uli . Also if we let ψ denote the map for s, we 0 0 0 0 0 0 have ψ ∗ : H l (Rl , Rl r 0) → H l (Bl , Sl −1 ). The lemma follows from the commutativity of the diagram 0

0

0

0

0

0

H l1 (Rl1 , Rl1 r0) × H l2 (Rl2 , Rl2 r0)

×

ψ1∗ ×ψ2∗

 0 0 0 0 0 0 H l1 (Bl1 , Sl1 −1 ) × H l2 (Bl2 , Sl2 −1 ) and Corollary 4.4. 2.

×

/ H l0 (Rl0 , Rl0 r0) 

ψ∗

/ H l0 (Bl0 , Sl0 −1 ) 

458

Complex Analytic Geometry

Let X be a space with a triangulation K0 . We denote by K the barycentric subdivision of K0 and by K 0 that of K, as before. Also let E → X be an oriented real vector bundle of rank l0 on X. We endow each simplex of K 0 with an orientation and define the Euler class of E and its localization using the triangulation K 0 . We also fix a simplicial ordering on the vertices of K 0 (cf. Definition B.14). Let Z be a K0 -subcomplex of X and suppose we have a non-vanishing section s of E on the l0 -skeleton of K 0 in X r OK 0 (Z). Then we have the 0 Euler class e(E, s) of E localized by s in H l (X, X r Z). Recall that it is represented by the cocycle constructed as follows. First we extend s to a 0 non-vanishing section s˜ of E on X l r{isolated points}, these isolated points being the barycenter of some K 0 -simplices in SK 0 (Z). The class e(E, s) is represented by the cocycle γ that assigns to each l0 -simplex t of K 0 the index I(˜ s, bt ) of s˜ at the barycenter bt of t. For i = 1, 2, let Ei be an oriented real vector bundle of rank li0 on X. Also let Zi be a K0 -subcomplex of X and si a non-vanishing section of Ei on the li0 -skeleton of K 0 in X rOK 0 (Zi ). We set E = E1 ⊕ E2 , l0 = l10 + l20 , Z = Z1 ∩ Z2 and s = s1 ⊕ s2 . Note that OK 0 (Z) = OK 0 (Z1 ) ∩ OK 0 (Z2 ). In this situation, s is a non-vanishing section of E on the l0 -skeleton of K 0 in X r OK 0 (Z). Indeed, let t = ε(v0 , · · · , vl0 ), v0 < · · · < vl0 , be an oriented l0 -simplex of K 0 and set t1 = ε1 (v0 , · · · , vl10 ), t2 = ε2 (vl10 , · · · , vl0 ), where ε, ε1 and ε2 are ±1, to be determined according to the prescribed orientations. If t is in X rOK 0 (Z), then either t1 is in X rOK 0 (Z1 ) or t2 is in X r OK 0 (Z2 ). There is a homeomorphism t ' t1 × t2 , by which s corresponds to s1 × s2 . Hence s is non-vanishing on t. 0 Thus we have the localizations e(Ei , si ) in H li (X, X rZi ), i = 1, 2, and 0 e(E, s) in H l (X, X rZ). We also have the cup product 0

0

`

0

H l1 (X, X rZ1 ) × H l2 (X, X rZ2 ) −→ H l (X, X rZ). Theorem 14.1. In the above situation, e(E, s) = e(E1 , s1 ) ` e(E2 , s2 ). Proof. For i = 1, 2, let s˜i be an extension of si and γi the cocycle representing the class e(Ei , si ) as above. Let t be an oriented l0 -simplex of K 0 and t1 , t2 be as above. Then γ1 ` γ2 is the cocycle that assigns I(˜ s1 , bt1 ) · I(˜ s2 , bt2 ) to t. By Lemma 14.1, I(˜ s1 , bt1 ) · I(˜ s2 , bt2 ) = I(˜ s1 × s˜2 , bt1 × bt2 ). There is a homeomorphism (t1 × t2 , bt1 × bt2 ) ' (t, bt ), by which s˜1 × s˜2 corresponds to an extension s˜ of s. The class e(E, s)

Intersection Product of Complex Subspaces

459

is represented by the cocycle γ that assigns I(˜ s, bt ), which is equal to I(˜ s1 , bt1 ) · I(˜ s2 , bt2 ) by the above argument.  Corollary 14.1. In the above situation, suppose Ei is a complex vector bundle of rank li , i = 1, 2, and set l = l1 + l2 . Then we have cl (E, s) = cl1 (E1 , s1 ) ` cl2 (E2 , s2 ). Remark 14.1. 1. In fact, if we use the cross product 0

0

×

0

H l1 (X, X rZ1 ) × H l2 (X, X rZ2 ) −→ H l ((X, X rZ1 ) × (X, X rZ2 )), we have e(E1 × E2 , s1 × s2 ) = e(E1 , s1 ) × e(E2 , s2 ). The formula in Theorem 14.1 also follows from this by applying the mor0 0 phism d∗ : H l ((X, X rZ1 ) × (X, X rZ2 )) → H l (X, X rZ) induced by the diagonal embedding d : X ,→ X × X. 2. The advantage of the formula in Corollary 14.1 over that in Corollary 5.6 or in Corollary 10.5 is in two ways: (1) it can be applied to non-vanishing sections that are defined only away from the “singular set”, for example, in the case of divisors, and (2) it can be applied to vector bundles on triangulable spaces, for example, singular varieties.

14.2

Intersection product in complex manifolds

Let M be a complex manifold of dimension n. We have the intersection ˘ ∗ (M ; Z), which is denoted by (cf. Section 4.4). product in H Let E be a holomorphic vector bundle of rank l on M . If s(r) is a holomorphic r-section of E with singular set S, we have the topological localization cqtop (E, s(r) ) in H 2q (M, MrS; Z) and the differential geometric localization cqdiff (E, s(r) ) in H 2q (M, MrS; C), q = l−r+1. We also have the ˘ 2(n−q) (S; Z) and Rescq (s(r) , E; S) associated residues TRescq (s(r) , E; S) in H ˘ in H2(n−q) (S; C). Let S be the complex space defined by s(r) . It has S as its support and, in general, dim S ≥ n−q. In the case dim S = n−q (proper case), we may identify TRescq (s(r) , E; S) and Rescq (s(r) , E; S) (cf. (12.2)). Moreover, in this case, we may define the homology class of S; [S] in ˘ 2(n−q) (M ; Z) or S in H ˘ 2(n−q) (S; Z) (cf. Definition 12.6). The class S H (r) coincides with TRescq (s , E; S) = Rescq (s(r) , E; S) (cf. Theorems 12.8 and 12.9).

·

460

Complex Analytic Geometry

In the following, through the end of this chapter, the homology and cohomology are with Z-coefficient, unless otherwise stated, and the coefficient is omitted. Intersection product of subspaces defined by a section For each j = 1, . . . , ν, let Ej be a holomorphic vector bundle of rank lj on M . Let sj be a holomorphic section of Ej and Sj the complex space in M defined by sj . The support Sj of Sj is the zero set of sj . We have the (topological) localization clj (Ej , sj ) in H 2lj (M, M r Sj ) of clj (Ej ) by sj . We also have the residue TResclj (sj , Ej ; Sj ) as the image of clj (Ej , sj ) by the Alexander isomorphism ∼ ˘ H 2lj (M, M rSj ) −→ H 2(n−l ) (Sj ). j

Note that dim Sj ≥ n − lj and that the equality holds if and only if sj is a regular section. If this is the case (proper case), we may define ˘ 2(n−l ) (Sj ). It coincides with the class of Sj , denoted also by Sj , in H j TResclj (sj , Ej ; Sj ), which may be identified with Resclj (sj , Ej ; Sj ) (cf. Theorem 12.9). Let E = E1 ⊕ · · · ⊕ Eν be the direct sum and s the section of E given by s = s1 ⊕ · · · ⊕ sν . Let S be the complex space in M defined by s. As a complex subspace of M , it is the intersection of the Sj ’s. The support S of Tν S is the zero set of s and is given by S = j=1 Sj . We have the localization P ν cl (E, s) in H 2l (M, M r S) by s, l = j=1 lj . We also have the residue l TRescl (s, E; S) as the image of c (E, s) by the Alexander isomorphism ∼ ˘ H 2l (M, M rS) −→ H 2(n−l) (S). Note that dim S ≥ n − l and that the equality holds if and only if s is a regular section. In this case, we may define the class of S, denoted ˘ 2(n−l) (S). It coincides with TRescl (s, E; S), which may be also by S, in H identified with Rescl (s, E; S). We have the cup product H 2l1 (M, M rS1 ) × · · · × H 2lν (M, M rSν ) −→ H 2l (M, M rS) and we have (cf. Corollary 14.1) cl1 (E1 , s1 ) ` · · · ` clν (Eν , sν ) = cl (E, s).

(14.2)

From (14.2) and Theorem 4.8, we have:

·

(TRescl1 (s1 , E1 ; S1 ) · · ·

· TRes

clν (sν , Eν ; Sν ))S

= TRescl (s, E; S)

˘ 2(n−l) (S). in H

(14.3)

Intersection Product of Complex Subspaces

461

Proposition 14.2. In the above situation, we have: 1. If dim Sj = n − lj , for each j, (S1

· ··· ·S )

˘ 2(n−l) (S). in H

= TRescl (s, E; S)

ν S

2. Moreover, if dim S = n − l, (S1 Proof.

· ··· ·S )

ν S

˘ 2(n−l) (S). in H

=S

The proposition follows from Theorem 12.9.



Remark 14.2. If dim Sj = n − lj , for each j, and if S is compact, we may write (cf. Remark 4.5. 3) (S1

· ··· ·S )

ν S

= M a cl (E, s)

in H2(n−l) (S),

where M denotes the class of M in H2n (M, M rS). Note that, if l = n and if S is compact and connected, TRescn (s, E; S) is in H0 (S) ' Z and may be regarded as an integer. Suppose that dim Sj = n − lj , for each j, and that S has only a finite number of connected components (Sλ )λ . Then the λ component of (S1 · · · Sν )S is given by (S1 · · · Sν )S,λ = TRescl (s, E; Sλ ) in ˘ 2(n−l) (Sλ ) and we have H X ˘ 2(n−l) (S), (14.4) (iλ )∗ (S1 · · · Sν )S,λ = (S1 · · · Sν )S in H

·

·

·

·

·

·

·

·

λ

where iλ : Sλ ,→ S denotes the inclusion. Passing from the homologies of the Sj ’s and S to that of M , we have from (14.3) (cf. Remark 4.13),

·

·

[TRescl1 (s1 , E1 ; S1 )] · · · [TResclν (sν , Eν ; Sν )] = [TRescl (s, E; S)]

˘ 2(n−l) (M ). in H

−1 l Since [TRescl (s, E; S)] = PM (c (E)) = M a cl (E), we have:

Proposition 14.3. In the above situation, we have: 1. If dim Sj = n − lj , for each j,

·

·

[S1 ] · · · [Sν ] = M a cl (E)

˘ 2(n−l) (M ). in H

2. Moreover, if dim S = n − l,

·

·

[S1 ] · · · [Sν ] = [S]

˘ 2(n−l) (M ). in H

462

Complex Analytic Geometry

Intersection product of divisors The divisors we consider in the following are assumed to be non-zero. Recall that, for a divisor D on M , there is the associated line bundle LD together with the natural meromorphic section sD (cf. Section 11.6). ˘ 2(n−1) (|D|), which We have the topological residue TResc1 (sD , LD ; |D|) in H may be identified with the differential geometric residue Resc1 (sD , LD ; |D|) ˘ 2(n−1) (|D|), denoted also (cf. (12.12)). We have the homology class of D in H by D, which coincides with TResc1 (sD , LD ; |D|) (cf. Theorem 12.10). The results in the previous paragraph hold in this case with a little modification, in fact they include the case of effective divisors as a special case. Thus, for divisors D1 , . . . , Dν on M , we have the intersection product ˘ 2(n−ν) (S), S = Tν |Dj | with |Dj | the support of (D1 · · · Dν )S in H j=1 Dj . We set E = LD1 ⊕ · · · ⊕ LDν and s = sD1 ⊕ · · · ⊕ sDν . Then we have the localization cν (E, s) in H 2ν (M, M rS) and the associated residue ˘ 2(n−ν) (S). If the Dj ’s are effective, the section s is holoTRescν (s, E; S) in H morphic and defines a complex space S whose support is S. Moreover, if ˘ 2(n−ν) (S) is equal to TRescν (s, E; S). dim S = n−ν, then the class of S in H We have, as Proposition 14.2, the following:

·

·

Proposition 14.4. In the above situation, 1.

(D1

· ··· ·D )

ν S

˘ 2(n−ν) (S). in H

= TRescν (s, E; S)

2. Moreover, if the Dj ’s are effective and if dim S = n − ν, (D1

· ··· ·D )

ν S

=S

˘ 2(n−ν) (S). in H

Note that, if S is compact, we may write (cf. Remark 14.2) (D1

· ··· ·D )

ν S

= M a cν (E, s)

in H2(n−ν) (S).

·

·

In the case ν = n and S is compact, (D1 · · · Dn )S is in H0 (S) and we define the intersection number of the Dj ’s by ](D1

· ··· ·D

where ε∗ : H0 (S) → Z is the We also have a formula finite number of connected of (D1 · · · Dν )S is given ˘ 2(n−ν) (Sλ ) and we have H X (iλ )∗ (D1 · · · Dν )S,λ

·

·

·

λ

·

n)

= ε∗ ((D1

· ··· ·D

n )S ),

(14.5)

augmentation. as (14.4). Thus suppose that S has only a components (Sλ )λ . Then the λ component by (D1 · · · Dν )S,λ = TRescν (s, E; Sλ ) in

·

= (D1

·

· ··· ·D )

ν S

˘ 2(n−ν) (S). in H

(14.6)

Intersection Product of Complex Subspaces

463

In the global situation, we have (cf. Proposition 14.3. 1)

·

·

[D1 ] · · · [Dν ] = M a cν (E)

˘ 2(n−ν) (M ), in H

(14.7)

where cν (E) = c1 (LD1 ) ` · · · ` c1 (LDν ). In particular, if ν = n and if M is compact, we may define the intersection number of the [Dj ]’s by

·

·

·

·

]([D1 ] · · · [Dn ]) = ε∗ ([D1 ] · · · [Dn ]), where ε∗ : H0 (M ) → Z is the augmentation. By (14.7), we have

·

·

]([D1 ] · · · [Dn ]) = hM, c1 (LD1 ) ` · · · ` c1 (LDn )i.

(14.8)

It coincides with the one in (14.5). Tn Let D1 , . . . , Dn be n divisors on M . Suppose that S = j=1 |Dj | consists of isolated points. For each point p in S, we have the localized intersection product (D1 · · · Dn )p in H0 ({p}) ' Z. Thus it may be regarded as an integer. From (14.6), we have the following:

·

·

Theorem 14.2. Let M be a complex manifold of dimension n. For divisors Tn D1 , . . . , Dn on M such that S = j=1 |Dj | consists of a finite number of isolated points, we have X (D1 · · · Dn )p = ](D1 · · · Dn ),

·

·

·

·

p∈S

·

·

which coincides with ]([D1 ] · · · [Dn ]), if M is compact. In the case each Dj is defined by a holomorphic function fj near p, the local intersection number (D1 · · · Dn )p is simultaneously equal to

·

·

(1) the mapping degree of f = (f1 , . . . , fn ), at p, (2) the Grothendieck residue   df ∧ · · · ∧ dfn Resp 1 , f1 , . . . , fn

(14.9)

(3) dimC O/(f1 , . . . , fn ). (cf. Example 12.2, Theorems 12.3–12.5). Example 14.1. Let Pj (ζ0 , . . . , ζn ) be homogeneous polynomials of degree dj , j = 1, . . . , n. For each j, Pj defines an effective divisor Dj in the projective space Pn . If [ζ0 , · · · , ζn ] denote homogeneous coordinates on Pn , d the divisor Dj is defined by the holomorphic function Pj (ζ0 , . . . , ζn )/ζαj on

464

Complex Analytic Geometry ⊗d

Uα = {ζα 6= 0}. Note that LDj = Hn j , where Hn denotes the hyperplane bundle on Pn . Tn In this situation, if S = j=1 |Dj | consists of isolated points, we have (B´ezout’s theorem): X (D1 · · · Dn )p = d1 · · · dn .

·

·

p∈S

This follows from Theorem 14.2 and (14.8). Intersection product with a subvariety Let V be a variety of dimension d in M with the embedding i : V ,→ M . Also let K0 be a triangulation of M as in Section 13.2. We may define the intersection product with V as in the case of submanifolds (cf. Section 4.4). We consider the intersection of V and a homology class in M or in a K0 -subcomplex S of M . First localization: Letting S1 and S2 be V and M , respectively, in (4.34), we have the intersection product localized at V :

·

˘ 2d−p (V ). ˘ 2d (V ) × H ˘ 2n−p (M ) ( −→)V H H ˘ 2n−p (M ), we have the product (V For a class a in H ˘ ˘ 2d−p (M ). [V ] a by i∗ : H2d−p (V ) → H We have the Alexander duality

·

· a)

V

. It is sent to

∼ ˘ AM,V : H 2k (M, M rV ) −→ H 2d (V ),

under which the Thom class ΨV of V corresponds to the class V (cf. Definition 13.1 and (13.9)). Letting S1 = V and S2 = ∅ in Theorem 4.8, we may write (V

· a)

V

−1 = AM,V (ΨV ` PM a).

Second localization: Let S be a K0 -subcomplex of M and set Z = V ∩ S. Letting S1 and S2 be V and S, respectively, in (4.34), we have the intersection product localized at Z:

·

˘ 2d (V ) × H ˘ 2n−p (S) ( −→)Z H ˘ 2d−p (Z). H

(14.10)

·

˘ 2n−p (S), we have the product (V a)Z in H ˘ 2d−p (Z). For a class a in H Letting S1 = V and S2 = S in Theorem 4.8, we may write (V

· a)

Z

= AM,Z (ΨV ` A−1 M,S a).

(14.11)

Intersection Product of Complex Subspaces

465

The following proposition follows from Propositions 13.2, 13.3, Theorems 4.7 and 4.8: Proposition 14.5. The following diagrams are commutative: H p (M )

∼ PM

i∗

 H p (V )

˘ 2n−p (M ) /H 

PV

(V

·

H p (M, M rS)

∼ AM,S

i∗

)V

 H p (V, V rZ)

˘ 2d−p (V ), /H

˘ 2n−p (S) /H 

AV,Z

(V

·

)Z

˘ 2d−p (Z). /H

Thus, for a class α in H p (M ), we have (V

·P

M (α))V

= V a i∗ (α).

(14.12)

Also, if Z is compact, for a class α in H p (M, M rS), we have (cf. Remark 13.1) (V

·A

M,S (α))Z

= V a i∗ (α).

(14.13)

Remark 14.3. 1. The two diagrams in Proposition 14.5 are compatible through the diagrams (4.18) and (13.7). 2. Likewise, for a holomorphic map f : V → M , we may define the intersection product with f as in Definition 4.4 and, if Z = f −1 (S) has a finite number of connected components, we have the product componentwise and a formula as (4.35). Intersection product of varieties Let V and W be varieties of dimensions d and h, respectively, in M . We set Z = V ∩ W , which is a variety in M . Letting S = W in (14.10), we have the intersection product

· We have the intersection product (V · W ) ˘ [V ] · [W ] by the canonical morphism H

˘ 2d (V ) × H ˘ 2h (W ) ( −→)Z H ˘ 2(d+h−n) (Z). H

˘ 2(d+h−n) (Z). It is sent to in H ˘ 2(d+h−n) (Z) → H2(d+h−n) (M ). Now we assume that Z is of pure (d + h − n)-dimensional and let (Zi ) be irreducible components, which is locally finite. In this case, we have an ˘ 2(d+h−n) (Z) is a free explicit expression of the above intersection. Since H Abelian group generated by the classes [Zi ] in this case, we may write X (V W )Z = di [Zi ],

·

i

Z

466

Complex Analytic Geometry

for some integers di . For each i, we take a point pi in the non-singular part S of Zi◦ = Zi r j6=i Zj and let Di be a complex slice of Zi at pi (cf. Definition 3.24). Thus it is a (2n − d − h)-dimensional complex submanifold of M transverse to Zi◦ at pi . Let U be a neighborhood of Z in M that contains Di as a closed submanifold and take a triangulation K0 of U as in Theorem 12.1, compatible with V ∩ U , W ∩ U and Z ∪ Di , and let K be its barycentric subdivision. Let K 0 and K ∗ be as before. We may assume that pi is the barycenter bs of a 2(d + h − n)-simplex s in the non-singular part of Zi◦ and that the 2(2n − d − h)-cell s∗ dual to s is in Di . Thus Di is also transverse to both V and W so that Di ∩ V and Di ∩ W are complex submanifolds of dimensions (n − h) and (n − d), respectively, of M . Note that (Di ∩ V ) ∩ (Di ∩ W ) = Di ∩ V ∩ W = {pi }. The following shows that di is the “transverse intersection number” of V and W at pi : Proposition 14.6. We have

·

di = ((Di ∩ V ) (Di ∩ W ))pi . Proof.

We claim that

·

· ·W)

((Di ∩ V ) (Di ∩ W ))pi = (Di (V

Z )pi .

(14.14)

Since the right-hand side above is di (cf. Exercise 4.2), the proposition then follows. To show (14.14), let ι : Di ,→ M denote the inclusion. Then we have the following diagram (cf. Proposition 4.13): ˘ 2d (V ) /H

∼

H 2(n−d) (M, M rV )

AM,V

ι∗

 H 2(n−d) (Di , Di r(Di ∩ V ))

(Di

∼

·

(14.15) )Di ∩V

 / H2(n−h) (Di ∩ V ).

ADi ,Di ∩V

·

·

Thus (Di V )Di ∩V = Aι∗ A−1 V and likewise (Di W )Di ∩W = Aι∗ A−1 W , with A denoting appropriate Alexander isomorphisms. Considering the diagram in Theorem 4.8 for the case (M, S1 ) = (Di , Di ∩ V ), (M, S2 ) = (Di , Di ∩ W ) and using the functoriality of the cup product, we have

·

((Di ∩ V ) (Di ∩ W ))pi = ADi ,pi ι∗ (A−1 V ` A−1 W ), where ∼

ADi ,pi : H r (Di , Di r{pi }) −→ H0 ({pi }) ' Z denotes the Alexander isomorphism with r = 2(2n − d − h). Considering again the diagram in Theorem 4.8 for the case (M, S1 ) = (M, V ),

Intersection Product of Complex Subspaces

467

(M, S2 ) = (M, W ), we see that the right-hand side above is equal to ADi ,pi ι∗ A−1 (V W )Z , which in turn equals the right-hand side of (14.14) by the commutativity of the diagram obtained by replacing V with Z in (14.15). 

·

Subspaces defined by a family of sections Let E be a holomorphic vector bundle of rank l on M . Also, let s(r) be a holomorphic r-section of E and S the complex space defined by s(r) , as in the beginning of this section. Let V be a variety of dimension d in M and set EV = E|V and (r) (r) sV = s(r) |V . Let Z be the complex space in V defined by sV . It is the intersection of V and S as a complex subspace of M and the support Z of Z is given by Z = V ∩ S. In this situation, we have the (topologi(r) cal) localization cq (EV , sV ) in H 2q (V, V rZ). Note that it is the image of cq (E, s(r) ) by the pull-back i∗ : H 2q (M, M rS) −→ H 2q (V, V rZ). (r)

We also have the residue TRescq (sV , EV ; Z) as its images by the Alexander morphism ˘ 2(d−q) (Z). H 2q (V, V rZ) −→ H (r)

Recall that, if dim Z = d − q, then TRescq (sV , EV ; Z) may be identi(r) fied with the differential geometric residue Rescq (sV , EV ; Z) (cf. (13.36)). Moreover, in this case, we may define the homology class of Z; [Z] in ˘ 2(d−q) (V ) or Z in H ˘ 2(d−q) (Z) (cf. Definition 13.6). The class Z coincides H (r)

(r)

with TRescq (sV , EV ; Z) = Rescq (sV , EV ; Z) (cf. Theorem 13.18). By Proposition 14.5, (r) ˘ 2(d−q) (Z). (14.16) (V TRescq (s(r) , E; S))Z = TRescq (sV , EV ; Z) in H

·

Proposition 14.7. In the above situation, we have: 1. If dim S = n − q, (V

· S)

Z

(r)

= TRescq (sV , EV ; Z)

˘ 2(d−q) (Z). in H

2. Moreover, if dim Z = d − q, (V

· S)

Z

=Z

˘ 2(d−q) (Z). in H

Proof. The first statement follows from (14.16) and Theorem 12.8 and the second from Theorem 13.18. 

468

Complex Analytic Geometry

Remark 14.4. 1. If dim S = n − q and if Z is compact, we may write (cf. (14.13)) (V

· S)

Z

= V a cq (EV , sV )

in H2(d−q) (Z).

2. See Section 14.5 below for the case r = 1 (thus q = l) and dim S > n − l or dim Z > d − l. Passing from the homology of Z to that of V , we have from (14.16) (cf. Remark 14.3. 1), (V

· [TRes

(r) , E; S)])V cq (s

(r)

= [TRescq (sV , EV ; Z)]V

˘ 2(d−q) (V ), in H

where [ ]V denotes the class in the homology of V . Since (r)

[TRescq (sV , EV ; Z)]V = PV (cq (EV )) = V a cq (EV ), we have: Proposition 14.8. In the above situation, we have: 1. If dim S = n − q, (V

· [S])

V

= V a cq (EV )

˘ 2(d−q) (V ). in H

2. Moreover, if dim Z = d − q, (V

· [S])

V

˘ 2(d−q) (V ). in H

= [Z]V

Adjunction formula for singular curves The classical “adjunction formula” says that, if C is a compact non-singular complex curve in a complex surface S (complex manifold of dimension two), then we have

·

χ(C) = −C (KS + C).

(14.17)

In the above, χ(C) denotes the Euler-Poincar´e characteristic of C, which is equal to 2 − 2g with g the genus of C. Also, KS is the canonical divisor of S and the means intersection product. In the classical case, it is usually assumed that S is compact and the intersection product is in S. In this case, the intersection product is in H0 (S) and the right-hand side of (14.17) is understood to be the image by the augmentation ε∗ : H0 (S) → Z, which is an isomorphism, if S is connected. This formula follows from (6.11) using various relations among characteristic classes.

·

Intersection Product of Complex Subspaces

469

We prove a generalization of (14.17) as an application of Theorem 13.15 and the previous considerations. Here S is not assumed to be compact and the intersection product is in C. Theorem 14.3. Let C be a compact singular curve with singularities p1 , . . . , pr in a complex surface S. Then we have

·

χ(C) = −](C (KS + C)) +

r X

µ(C, pi ),

i=1

where µ(C, pi ) is the Milnor number of C at pi . Proof. First note that C ∩ |KM + C| = C and, since C is compact, ](C (KS + C)) makes sense (cf. (14.5)). Noting that C is an SLCI defined by the canonical section sC of LC , it suffices to show that (cf. Theorem 13.15) Z c1 (τC ) = −](C (KS + C)), (14.18)

·

·

C

where τC = i∗ (T S − LC ) with i : C ,→ S the inclusion. We have (C (KS + C))C = (C ([KS ] + [C]))C in H0 (C). The classes V2 [KS ] and [C] are the images of c1 ( T ∗ S) = c1 (T ∗ S) = −c1 (T S) and ∼ ˘ c1 (LC ), respectively, by the Poincar´e isomorphism P : H 2 (S) → H 2 (S). Thus by (14.12),

·

·

·

−(C ([KS ] + [C])C = C a i∗ c1 (T S − LC ) = C a c1 (τV ). Taking the images of the both sides by the augmentation ε∗ : H0 (C) → Z, we have (14.18).  In the case C is connected, the number to as the virtual genus of C.

14.3

1 2

·

](C (KS + C)) + 1 is referred

Intersection product in singular varieties

In this section, we let M denote a complex manifold of dimension n. Intersection product in a variety Let V be a variety of dimension d in M and take a triangulation of M as in Section 13.2. We wish to define the intersection product in V , denoted

470

by

Complex Analytic Geometry

·

V

, so that the following diagram is commutative:

P ×P

 ˘ r (V ) × H ˘ s (V ) H

/ H p+q (V )

`

H p (V ) × H q (V )

·

V



P

˘ r+s−2d (V ), /H

where P denotes the Poincar´e morphism, p + r = 2d and q + s = 2d. However, since P is not an isomorphism in general, we may define the product only for classes that come naturally from cohomology. The situation is similar for localized intersection product. Let Z1 and Z2 be subcomplexes of V and set Z = Z1 ∩ Z2 . We wish to define the intersection product in V localized at Z, denoted by ( V )Z , so that the following diagram is commutative:

·

H p (V, V rZ1 ) × H q (V, V rZ2 ) A1 ×A2

 ˘ r (Z1 ) × H ˘ s (Z2 ) H

(

·

V

`

)Z

/ H p+q (V, V rZ) 

A

˘ r+s−2d (Z), /H

where A, A1 and A2 denote the Alexander morphisms for Z, Z1 and Z2 , p + r = 2d and q + s = 2d. Intersection product of subspaces defined by a section Let V be a variety of dimension d in M . For each j = 1, . . . , ν, let EV,j be a holomorphic vector bundle of rank lj on V . Let sV,j be a holomorphic section of EV,j and Zj the complex space in V defined by sV,j . We assume that the support Zj of Zj contains Sing(V ). Then we have the localization clj (EV,j , sV,j ) in H 2lj (V, V rZj ) by sV,j , as in Section 13.4. We also have the residue TResclj (sV,j , EV,j ; Zj ) as the image of clj (EV,j , sV,j ) by the Alexander morphism ˘ 2(d−l ) (Zj ). Aj : H 2lj (V, V rZj ) −→ H j If dim Zj = d − lj , then TResclj (sV,j , EV,j ; Zj ) may be identified with the differential geometric residue Resclj (sV,j , EV,j ; Zj ) (cf. (13.36)). Moreover, ˘ 2(d−l ) (Zj ) is equal to TRes lj (sV,j , EV,j ; Zj ) (cf. Thethe class of Zj in H j c orem 13.18). Let EV = EV,1 ⊕ · · · ⊕ EV,ν be the direct sum and sV the section of EV given by sV = sV,1 ⊕ · · · ⊕ sV,ν . Let Z be the complex space in V defined

Intersection Product of Complex Subspaces

471

by sV . As a complex subspace of V , it is the intersection of the Zj ’s. The Tν support Z of Z is the zero set of sV and is given by Z = j=1 Zj . We Pν have the localization cl (EV , sV ) in H 2l (V, V r Z) by sV , l = j=1 lj . We also have the residue TRescl (sV , EV ; Z) as the image of cl (EV , sV ) by the Alexander morphism ˘ 2(d−l) (Z). A : H 2l (V, V rZ) −→ H If dim Z = d−l, TRescl (sV , EV ; Z) may be identified with Rescl (sV , EV ; Z). ˘ 2(d−l) (Z) is equal to TRescl (sV , EV ; Z). Moreover, the class of Z in H In view of (14.3), we give the following: Definition 14.1. The intersection product of the TResclj (sV,j , EV,j ; Zj )’s in V localized at Z is defined by (TRescl1 (sV,1 , EV,1 ; Z1 )

·

···

·

TResclν (sV,ν , EV,ν ; Zν ))Z ˘ 2(d−l) (Z). (14.19) = TRescl (sV , EV ; Z) in H V

V

We have the cup product H 2l1 (V, V rZ1 ) × · · · × H 2lν (V, V rZν ) −→ H 2l (V, V rZ) and we have (cf. Corollary 14.1) cl1 (EV,1 , sV,1 ) ` · · · ` clν (EV,ν , sV,ν ) = cl (EV , sV ). Thus (14.19) is compatible with the Alexander morphisms Aj and A. In the case EV,j and sV,j are the restriction of the ones Ej and sj on M , by (14.16), TResclj (sV,j , EV,j ; Zj ) = (V TRescl (sV , EV ; Z) = (V

· TRes

· TRes

clj (sj , Ej ; Sj ))Zj

and

cl (s, E; S))Z

so that (14.19) may be written ((V

· TRes

· · · · · (V · TRes = (V · TRes (s, E; S))

cl1 (s1 , E1 ;S1 ))Z1

V

V

cl

Z

clν (sν , Eν ; Sν ))Zν )Z

˘ 2(d−l) (Z). in H

Here we recall the identity (14.3). See Propositions 14.15 and 14.16 below for related identities. Coming back to the situation of the beginning of this paragraph, we have:

472

Complex Analytic Geometry

Proposition 14.9. In the above situation, we have: 1. If dim Zj = d − lj , for j = 1, . . . , ν, then the intersection product of the Zj ’s in V localized at Z is defined and is given by ˘ 2(d−l) (Z). in H (Z1 V · · · V Zν )Z = TRescl (sV , EV ; Z)

·

·

2. Moreover, if dim Z = d − l, (Z1 Proof.

·

···

V

·

˘ 2(d−l) (Z). in H

Zν )Z = Z

V

This follows from Theorem 13.18.



Remark 14.5. If dim Zj = d − lj , for j = 1, . . . , ν, and if Z is compact, we may write (Z1

·

···

V

·

Zν )Z = V a cl (EV , sV )

V

in H2(d−l) (Z).

Note that, if l = d and if Z is compact and connected, then TRescd (sV , EV ; Z) is in H0 (Z) ' Z and may be regarded as an integer. Suppose that dim Zj = d − lj , for j = 1, . . . , ν, and that Z has only a finite number of connected components (Zλ )λ . Then the λ component of (Z1 V · · · V Zν )Z is given by (Z1 V · · · V Zν )Z,λ = TRescl (sV , EV ; Zλ ) ˘ 2(d−l) (Zλ ) and we have in H X ˘ 2(d−l) (Z), (iλ )∗ (Z1 V · · · V Zν )Z,λ = (Z1 V · · · V Zν )Z in H

·

·

·

·

·

·

·

·

λ

(14.20) where iλ : Zλ ,→ Z denotes the inclusion. Passing from the homologies of the Zj ’s and Z to that of V , we define the intersection product of the [TResclj (sV,j , EV,j ; Zj )]V ’s in V by

·

[TRescl1 (sV,1 , EV,1 ; Z1 )]V

V

···

·

V

[TResclν (sV,ν , EV,ν ; Zν )]V

= [TRescl (sV , EV ; Z)]V

˘ 2(d−l) (V ). in H

Since [TRescl (sV , EV ; Z)]V = PV (cl (EV )) = V a cl (EV ), we have: Proposition 14.10. In the above situation, we have: 1. If dim Zj = d − lj , for j = 1, . . . , ν, then the intersection product of the [Zj ]V ’s in V is defined and is given by ˘ 2(d−l) (V ). in H [Z1 ]V V · · · V [Zν ]V = V a cl (EV )

·

·

2. Moreover, if dim Z = d − l, [Z1 ]V

·

V

···

·

V

[Zν ]V = [Z]V

˘ 2(d−l) (V ). in H

Intersection Product of Complex Subspaces

473

Intersection product with a subvariety Let V 0 be a variety of dimension d0 in V with the embedding ι : V 0 ,→ V . Also let Z be a subcomplex of V and set Z 0 = V 0 ∩ Z. We wish to define the intersection product in V with V 0 , denoted by (V 0 V )V 0 , and the intersection product in V with V 0 localized at Z 0 , denoted by (V 0 V )Z 0 , so that the following diagrams are commutative:

·

H p (V )

PV

˘ 2d−p (V ) /H

ι∗

 H p (V 0 )



PV 0

(V 0

·

V

˘ 2d0 −p (V 0 ), /H

H p (V, V rZ) )V 0

˘ 2d−p (Z) /H

AV,Z

ι∗

 H p (V 0 , V 0 rZ 0 )

·



(V 0

·

V

)Z 0

˘ 2d0 −p (Z 0 ). /H

AV 0 ,Z 0

Subspaces defined by a family of sections Let V be a variety of dimension d in M . Also, let EV be a holomor(r) phic vector bundle of rank l on V , sV a holomorphic r-section of EV and (r) Z the complex space in V defined by sV . We assume that the support (r) Z of Z contains Sing(V ). Then we have the localization cq (EV , sV ) in (r) H 2q (V, V rZ), q = l − r + 1, and the associated residue TRescq (sV , EV ; Z) ˘ 2(d−q) (Z) as its image by the Alexander morphism in H ˘ 2(d−q) (Z). H 2q (V, V rZ) −→ H Let V 0 be a subvariety of V of dimension d0 and set EV 0 = EV |V 0 and (r) (r) = sV |V 0 . Let Z 0 be the complex space in V 0 defined by sV 0 . As a complex subspace of V , it is the intersection of V 0 and Z and the support Z 0 of Z 0 is given by Z 0 = V 0 ∩ Z. We assume that Z 0 contains Sing(V 0 ). Then (r) we have the topological localization cq (EV 0 , sV 0 ) in H 2q (V 0 , V 0 r Z 0 ) and (r) the topological residue TRescq (sV 0 , EV 0 ; Z 0 ) as its image by the Alexander morphism ˘ 2(d0 −q) (Z 0 ). H 2q (V 0 , V 0 rZ 0 ) −→ H

(r) sV 0

In view of (14.16), we give the following: (r)

Definition 14.2. The intersection pruduct of V 0 and TRescq (sV , EV ; Z) in V localized at Z 0 is defined by (r) (r) ˘ 2(d0 −q) (Z 0 ). in H (V 0 V TRescq (s , EV ; Z))Z 0 = TRescq (s 0 , EV 0 ; Z 0 )

·

V

V

(14.21) Proposition 14.11. Definitions 14.1 and 14.2 are compatible in the case the both make sense.

474

Complex Analytic Geometry

Proof. Suppose there exist a holomorphic vector bundle F of rank k 0 = d − d0 on V and a holomorphic section t of F such that V 0 is defined by t. 0 We assume V 0 contains Sing(V ). Then we have the localization ck (F, t) in 0 H 2k (V, V r V 0 ) and V 0 = TResck0 (t, F ; V 0 ) (cf. Theorem 13.18). Also let EV be a holomorphic vector bundle of rank l on V and sV a holomorphic section of EV . Let Z be the complex space in V defined by sV . We assume that its support Z containes the singular set Sing(V ) of V . We set EV 0 = EV |V 0 and sV 0 = sV |V 0 . Let Z 0 be the complex space in V 0 defined by sV 0 . The support Z 0 of Z 0 is given by Z 0 = Z ∩ V 0 . We assume that Z 0 contains Sing(V 0 ). In this situation, we may define (V 0 V TRescl (sV , EV ; Z))Z 0 in two ways, i.e., by Definition 14.1 and by Definition 14.2, and we are to show that they are the same. On the one hand, by Definition 14.2,

·

(V 0

·

V

TRescl (sV , EV ; Z))Z 0 = TRescl (sV 0 , EV 0 ; Z 0 ) = AV 0 ,Z 0 (cl (EV 0 , sV 0 )).

On the other hand, by Definition 14.1, the left-hand side is equal to (TResck0 (t, F ; V 0 )

·

V

TRescl (sV , EV ; Z))Z 0 = TResck0 +l (t⊕sV , F ⊕EV ; Z 0 ),

which is equal to, by Corollary 14.1, 0

0

AV,Z 0 (ck +l (F ⊕ EV , t ⊕ sV ) = AV,Z 0 ((ck (F, t) ` cl (EV , sV )). Thus the proposition follows from the commutativity of the diagram 0

ck (F,t)`( )

/ H 2k0 +p (V, V rZ 0 )

H p (V, V rZ) ι∗

 H p (V 0 , V 0 rZ 0 )

AV 0 ,Z 0



AV,Z 0

˘ 2d0 −p (Z 0 ), /H

where ι : V 0 ,→ V is the embedding. For this, let γ be a cocycle in 2k0 0 k0 CK (F, t). Then, for a 2d0 -simplex s of K, 0 (V, V r V ) representing c V ∗ 0 0 s ∩ V is a 2k -chain of KV and we have (cf. (13.37)): ( 1 if s ⊂ V 0 , ∗ hs ∩ V, γi = 0 otherwise. The commutativity is then proved as Proposition 4.10, see also Proposition 13.2.  (r)

If EV and sV (14.16),

are the restrictions of E and s(r) on M , we have, by (r)

· TRes (s , E; S)) ; Z ) = (V · TRes (s , E; S))

TRescq (sV , EV ; Z) = (V (r)

TRescq (sV 0 , EV 0

0

(r)

cq

0

cq

(r)

and

Z

Z0

Intersection Product of Complex Subspaces

475

so that (14.21) is written (contraction of V ) (V 0

·

V

(V

· TRes

(r) , E; S))Z )Z 0 cq (s

= (V 0

· TRes (r)

Proposition 14.12. Suppose that EV and sV and s(r) on M . Then, if dim S = n − q, (V 0

·

(V

V

· S)

Z )Z 0

·

(r) , E; S))Z 0 . cq (s

are the restrictions of E

˘ 2(d0 −q) (Z 0 ). in H

= (V 0 S)Z 0

Coming back to the situation of the beginning of this paragraph, we have: Proposition 14.13. In the above situation, we have: 1. If dim Z = d−q, then the intersection product (V 0 at Z 0 is defined and is given by (V 0

·

(r)

Z)Z 0 = TRescq (sV 0 , EV 0 ; Z 0 )

V

·

V

Z)Z 0 in V localized

˘ 2(d0 −q) (Z 0 ). in H

2. Moreover, if dim Z 0 = d0 − q, (V 0 Proof.

·

V

Z)Z 0 = Z 0

˘ 2(d0 −q) (Z 0 ). in H

The statements follow from Theorem 13.18.



Remark 14.6. If Z 0 is compact and if dim Z = d − q, we may write (V 0

·

Z)Z 0 = V 0 a cq (EV 0 , sV 0 )

V

in H2(d0 −q) (Z 0 ).

Passing from the homology of Z 0 to that of V 0 , we define the intersection (r) pruduct of V 0 and [TRescq (sV , EV ; Z)]V in V localized at V 0 by (V 0

·

V

(r)

[TRescq (sV , EV ; Z)]V )V 0 (r)

= [TRescq (sV 0 , EV 0 ; Z 0 )]V 0

˘ 2(d0 −q) (V 0 ). in H

(14.22)

Since (r)

[TRescq (sV 0 , EV 0 ; Z 0 )]V 0 = PV 0 (cq (EV 0 )) = V 0 a cq (EV 0 ),

(14.23)

we have: Proposition 14.14. In the above situation, we have:

·

1. If dim Z = d − q, then the intersection product (V 0 V [Z]V )V 0 in V localized at V 0 is defined and is given by ˘ 2(d0 −q) (V 0 ). (V 0 V [Z]V )V 0 = V 0 a cq (EV 0 ) in H

·

2. Moreover, if dim Z 0 = d0 − q, (V 0

·

V

[Z]V )V 0 = [Z 0 ]V 0

˘ 2(d0 −q) (V 0 ). in H

476

Complex Analytic Geometry

Distributive property For each j = 1, . . . , ν, let Ej be a holomorphic vector bundle of rank lj on M and sj a holomorphic section of Ej , as in Section 14.2. Also, let Pν E = E1 ⊕ · · · ⊕ Eν , l = j=1 lj and s = s1 ⊕ · · · ⊕ sν . We denote by Sj and S the complex spaces in M defined by sj and s, as before. Let V be a variety of dimension d in M and set EV,j = Ej |V , sV,j = sj |V , EV = E|V and sV = s|V . We denote by Zj and Z the complex spaces in V defined by sV,j and sV . The support Zj of Zj is given by Zj = V ∩ Sj and that of Z by Z = V ∩ S. We assume that each Zj contains Sing(V ). Proposition 14.15. In the above situation, we have: 1. If each sj , j = 1, . . . , ν, and s are regular sections, we have (V

· (S · · · · · S ) ) = ((V · S ) · 1

ν S Z

·

·

˘ 2(d−l) (Z), · · · V (V Sν )Zν )Z in H where the both sides are well-defined under the condition. 1 Z1

V

2. Moreover, if dim Zj = d − lj , for each j, then (V (V

· (S · · · · · S )

ν S )Z

1

= (Z1

·

V

···

·

V

·S )

Zν )Z

= Zj so that ˘ in H2(d−l) (Z). j Zj

·

Proof. If sj is a regular section, the product (V Sj )Zj is equal to TResclj (sV,j , EV,j ; Zj ) by Proposition 14.7. 1. If, moreover, s is a regular section, then (S1 · · · Sν )S = S by Proposition 14.2. Thus (V (S1 · · · Sν )S )Z = (V S)Z , which is equal to TRescl (sV , EV ; Z) by Proposition 14.7. 1. Hence, by Definition 14.1, we have the identity in the first statement. The second statement follows from Proposition 14.7. 2 and the first. 

·

·

·

·

· ·

·

·

As noted in the proof, (S1 · · · Sν )S = S. Also, if dim Z = d − l, (Z1 V · · · V Zν )Z = Z (cf. Proposition 14.9. 2). Let furthermore V 0 be a subvariety of V of dimension d0 and set EV 0 ,j = Ej |V 0 , sV 0 ,j = sj |V 0 , EV 0 = E|V 0 and sV 0 = s|V 0 . We denote by Zj0 and Z 0 the complex spaces in V 0 defined by sV 0 ,j and sV 0 . The support Zj0 of Zj0 is given by Zj0 = V 0 ∩ Zj and that of Z 0 by Z 0 = V 0 ∩ Z. We assume that each Zj0 contains Sing(V 0 ).

·

·

Proposition 14.16. In the above situation we have: 1. If dim Zj = d − lj , for each j, and dim Z = d − l, (V 0

·

· = ((V · V

(Z1

·

V

···

V

Z1 )Z10

0

V

Zν )Z )Z 0

·

V0

···

·

V0

(V 0

·

V

Zν )Zν0 )Z 0

˘ 2(d0 −l) (Z 0 ), in H

Intersection Product of Complex Subspaces

477

where the both sides are well-defined under the condition. 2. Moreover, if dim Zj0 = d0 − lj , for each j, then (V 0 that (V 0

·

V

(Z1

·

V

···

·

V

Zν )Z )Z 0 = (Z10

·

V0

···

·

V0

·

V

Zj )Zj0 = Zj0 so

˘ 2(d0 −l) (Z 0 ). in H

Zν0 )Z 0

·

Proof. If dim Zj = d − lj , for each j, the product (V 0 V Zj )Zj0 makes sense and is equal to TResclj (sV 0 ,j , EV 0 ,j ; Zj0 ) by Proposition 14.13. 1. If, moreover, dim Z = d − l, then (Z1 V · · · V Zν )Z = Z by Proposition 14.9. Thus the product (V 0 V (Z1 V · · · V Zν )Z )Z 0 = (V 0 V Z)Z 0 makes sense and is equal to TRescl (sV 0 , EV 0 ; Z 0 ) by Proposition 14.13. 1. Hence, by Definition 14.1 for the case V is V 0 , we have the identity in the first statement. The second statement follows from Proposition 14.13. 2 and the first. 

·

·

·

·

·

·

·

·

As noted in the proof, (Z1 V · · · V Zν )Z = Z. Also, if dim Z 0 = d0 − l, · · · V 0 Zν0 )Z 0 = Z 0 (cf. Proposition 14.9. 2).

·

(Z10 V 0

·

Intersection product of divisors Meromorphic functions and divisors on singular varieties are defined as in the case of complex manifolds with a little modification (cf. Section 11.6). First, let V be a non-empty germ of a variety at 0 in Cn and V = Sr decomposition with pi prime ideals in O = On . i=1 V (pi ) the irreducible Tr Thus V = V (I), I = i=1 pi (cf. Section 2.3). The quotient OV = O/I is the ring of germs of holomorphic functions on V (cf. Section 11.5). The Tr Sr set S = i=1 (O r pi ) = O r i=1 pi is mutiplicative in O and its image S 0 in OV is multiplicative in OV (cf. Section A.2). The fraction ring MV = (S 0 )−1 OV is the ring of germs of meromorphic functions on V . It is canonically isomorphic with S −1 O/I · S −1 O (cf. Proposition A.21). Thus every meromorphic function germ on V is represented by a germ of the form f /g with f ∈ O and g ∈ S. Note that S 0 does not contain any zero divisor in OV so that MV naturally contains OV . Next, let V be a variety of dimension d in a complex manifold M . If we let IV be the ideal sheaf of V , OV = i−1 (O/IV ) is the sheaf of germs of holomorphic functions on V , where O = OM and i : V ,→ M is the inclusion (cf. Section 11.5). Note that, for an open set W in V and an open set U in M with W ⊂ U , there is a canonical morphism Γ (U ; O) → Γ (W ; OV ). For f in Γ (U ; O), we denote by [f ] its image in Γ (W ; OV ). We define the sheaf MV of germs of meromorphic functions on V as in the case of complex manifolds, relpacing O with OV . For each point z in V , MV,z

478

Complex Analytic Geometry

is isomorphic with the ring of germs of meromorphic functions defined as above, i.e., if we define Sz in Oz and Sz0 in OV,z = Oz /IV,z as above, then MV,z ' (Sz0 )−1 OV,z ' Sz−1 Oz /(IV,z · Sz−1 Oz ). A meromorphic function ψ on an open set W in V is a section of MV on W . Taking a suitable open set U in M with W = U ∩ V , such a function is represented by a system {(Uα , f α , g α )}, where {Uα } is a covering of U and f α , g α are holomorphic functions on Uα such that the germ gzα at z is in Sz for all z in Wα = Uα ∩ V , the germs fzα and gzα are relatively prime for all z in Uα and that [f α ] · [g β ] = [f β ] · [g α ] in Wα ∩ Wβ . We write ψ = [f α ]/[g α ] on Wα . A pole of ψ is a zero of [g α ] for some α. Denoting by OV∗ and MV∗ the sets of invertible elements in OV and MV , respectively, we define the sheaf DivV of Cartier divisors on V by a sequence as (11.17). A Cartier divisor on V is then defined to be a section of DivV on V . A Cartier divisor D is represented by a system {(Wα , ψ α )}, where {Wα } is an open covering of V , ψ α is in Γ (Wα ; MV∗ ), for each α, and ψ α /ψ β is in Γ (Wα ∩ Wβ ; OV∗ ), for each pair (α, β). Associated with D is a line bundle LD on V together with a natural meromorphic section sD . The bundle LD is defined by the system of transition functions {ψ α /ψ β } and the section sD is represented by the collection {ψ α }. P A Weil divisor on V is a locally finite formal sum mi Vi0 of varieties 0 Vi of codimension one in V with integral coefficients mi . For a Weil divisor P S DW = mi Vi0 , we define its support by |DW | = Vi0 . P Definition 14.3. For a non-zero Weil divisor DW = mi Vi0 , we define its P P W 0 ˘ homology class by [D ] = mi [Vi ] in H2d−2 (V ; Z) or by DW = mi [Vi0 ] ˘ 2d−2 (|DW |, Z). in H We would like to show that a Cartier divisor naturally defines a Weil divisor. Since the notion of the order of a meromorphic function along a subvariety of codimension one is not defined as easily as in the case of complex manifolds, we proceed as follows. Thus let D be a non-zero Cartier divisor. The set of zeros and poles of sD is a variety of codimension one in V (cf. Theorem 2.11), which we denote by |D| and call the support of D. In the following we always asssume that |D| contains Sing(V ). Then we have the topological and differential geometric localizations of c1 (LD ) ˘ 2(d−1) (|D|; Z) by sD and the associated residues TResc1 (sD , LD ; |D|) in H ˘ 2(d−1) (|D|; C) as their images by the Alexander and Resc1 (sD , LD ; |D|) in H morphism ˘ 2(d−1) (|D|). H 2 (V, V r|D|) −→ H

Intersection Product of Complex Subspaces

479

S If we let |D| = Vi0 be the irreducible decomposition, each Vi0 defines a ˘ 2(d−1) (|D|) and it is the free Abelian group generated by class [Vi0 ] in H 0 the [Vi ]’s. Thus we may identify the two residues ; TResc1 (sD , LD ; |D|) = Resc1 (sD , LD ; |D|). S Let pi be a non-singular point of Vi0 r j6=i Vj0 and Di a complex slice of Vi0 in M at pi , as in Section 13.5. The intersection Di ∩ V is a one-dimensional variety in Di . The section si = sD |Di ∩V of Li = LD |Di ∩V is non-vanishing away from pi so that we have the residues TResc1 (si , Li ; pi ) and Resc1 (si , Li ; pi ), both of which may be thought of as the same integer. Definition 14.4. The order of sD along Vi0 is defined by ordVi0 (sD ) = TResc1 (si , Li ; pi ). Note that the definition does not depend on the choice of the point pi as above. We then make the following: Definition 14.5. We define the Weil divisor associated with D to be the P locally finite formal sum ordVi0 (sD ) · Vi0 . The support of the Weil divisor defined by a Cartier divisor D coincides with |D|. Unlike the case of complex manifolds, a Weil divisor does not define a Cartier divisor in general (cf. Example 14.4 below). For a non-zero Cartier divisor D, the statement as in Theorem 13.9 holds as well: X ˘ 2(d−1) (|D|). TResc1 (sD , LD ; |D|) = TResc1 (si , Li ; pi ) · [Vi0 ] in H i

Thus we have: Theorem 14.4. Let D be a Cartier divisor on V . Then the class in ˘ 2(d−1) (|D|; Z) of the Weil divisor DW associated with D is equal to H TResc1 (sD , LD ; |D|), which may be identified with Resc1 (sD , LD ; |D|). Thus the class c1 (LD , sD ) is sent to the class DW by the Alexander morphism ˘ 2(d−1) (|D|; Z). H 2 (V, V r|D|; Z) −→ H ˘ 2(d−1) (V ; Z) is the image of c1 (LD ) in In particular, the class [DW ] in H H 2 (V ; Z) by the Poincar´e morphism. P We say that a Weil divisor DW = mi Vi0 is effective if mi ≥ 0 for all i. Also, a Cartier divisor D is effective if its associated Weil divisor is effective. This is equivalent to saying that the natural section sD of the associated bundle LD is holomorphic.

480

Complex Analytic Geometry

If D is a non-zero effective Cartier divisor, sD defines a complex space of dimension d − 1 in V whose support is |D|. In this case, the order ordVi0 (sD ) in Definition 14.4 of sD along Vi0 coincides with the multipicity mi in Definiton 13.5 of Vi0 in the complex space. Thus the homology class of the associated Weil divisor DW coincides with that of the complex space (cf. Definitions 13.6, 14.3 and 14.5) so that Theorem 14.4 is a special case of Theorem 13.18, where r = 1 and q = l = 1. The divisors we consider in the following are assumed to be non-zero. For a Cartier divisor D, we denote the associated Weil divisor also by D to avoid heavy notation. Let D1 , . . . , Dν be Cartier divisors on V and set EV = LD1 ⊕ · · · ⊕ LDν , Tν sV = sD1 ⊕ · · · ⊕ sDν and Z = j=1 |Dj |. We assume that each |Dj | contains Sing(V ). As sV is a non-vanishing holomorphic section of EV away from Z, we have the topological localization cν (EV , sV ) in H 2ν (V, V r Z) ˘ 2(d−ν) (Z). In view of and the associated residue TRescν (sV , EV ; Z) in H Definition 14.1 and Theorem 14.4, we make the following Definition 14.6. The intersection product of (the classes of the Weil divisors associated with) the Dj ’s in V localized at Z is defined by (D1

·

···

V

·

˘ 2(d−ν) (Z). in H

Dν )Z = TRescν (sV , EV ; Z)

V

In the case the Dj ’s are effective, the above is compatible with Proposition 14.9. 1. If dim Z = d − ν, then TRescν (sV , EV ; Z) is equal to the class of the complex space Z defined by sV in V (cf. Theorem 13.18). If Z is compact, we may write (cf. Remark 14.5) (D1

·

V

···

·

V

Dν )Z = V a cν (EV , sV )

in H2(d−ν) (Z).

·

In particular, if ν = d and if Z is compact, (D1 V · · · H0 (Z) and we define the intersection number of the Dj ’s by ](D1

·

···

V

·

V

Dd ) = ε∗ ((D1

·

V

···

·

V

·

V

Dν )Z is in

Dd )Z ),

(14.24)

where ε∗ : H0 (Z) → Z is the augmentation. In the global situation, we have (cf. Proposition 14.10. 1) [D1 ]V

·

V

···

·

V

[Dν ]V = V a cν (EV )

˘ 2(d−ν) (V ), in H

where cν (EV ) = c1 (LD1 ) ` · · · ` c1 (LDν ). In particular, if ν = d and if V is compact, we may define the intersection number ]([D1 ]V V · · · V [Dd ]V ), which coincides with the one in (14.24).

·

·

Intersection Product of Complex Subspaces

481

Note that the above product makes sense, if one of the divisors is only P Weil. Indeed, suppose D1 = mi Vi0 is Weil and the others are Cartier. Tν ∗ ∗ We set EV = LD2 ⊕ · · · ⊕ LDν , sV = sD2 ⊕ · · · ⊕ sDν and Z ∗ = j=2 |Dj |. Then the product (D2 V · · · V Dν )Z ∗ is defined and is given by (cf. Definition 14.6)

·

·

(D2

·

···

V

V

·

·

Dν )Z ∗ = TRescν−1 (s∗V , EV∗ ; Z ∗ )

·

˘ 2(d−ν+1) (Z ∗ ). in H

·

˘ 2(d−ν) (V 0 ∩ Z ∗ ) and Thus (Vi0 V (D2 V · · · V Dν )Z ∗ )Vi0 ∩Z ∗ is defined in H i ∗ ∗ ∗ 0 is equal to TRescν−1 (sV |Vi0 , EV |Vi0 ; Vi ∩ Z ) by Definition 14.2, for each i. P Definition 14.7. Let D1 = mi Vi0 be a Weil divisor and D2 , . . . , Dν Cartier divisors on V . The intersection product of D1 , . . . , Dν in V localized Tν at Z = j=1 |Dj | is defined by (D1

·

V

·

D2 X

V

···

·

V

Dν )Z

·

·

·

˘ 2(d−ν) (Z), in H mi · (i )∗ (Vi0 V (D2 V · · · V Dν )Z ∗ )Vi0 ∩Z ∗ Tν where Z ∗ = j=2 |Dj | and i : Vi0 ∩ Z ∗ ,→ Z is the inclusion. =

Remark 14.7. In the case D1 is Cartier and mi Vi0 is the associated Weil divisor, the above definition coincides with the one in Definition 14.6 (cf. Proposition 14.11). In particular, if ν = d and if Z is compact, we may define the intersection number ](D1 V · · · V Dd ) as in (14.24). Letting D1 , D2 , . . . , Dν , Z, Z ∗ , EV∗ and s∗V be as above, we have:

·

·

Proposition 14.17. If Z is compact and if dim Z ∗ = d − ν + 1, we may write (D1

·

V

D2

·

V

···

·

V

Dν )Z = D1 a cν−1 (EV∗ ||D1 | , s∗V ||D1 | )

in H2(d−ν) (Z),

where D1 denotes the relative class of D1 in H2(d−1) (|D1 |, |D1 |rZ). Proof. Since Vi0 ∩ Z ∗ is compact and dim Z ∗ = d − ν + 1, we may write (cf. Remark 14.6) (Vi0

·

V

(D2

·

V

···

·

V

Dν )Z ∗ )Vi0 ∩Z ∗ = Vi0 a cν−1 (EV∗ |Vi0 , s∗V |Vi0 ).

Denoting by ıi : Vi0 ,→ |D1 | the inclusion, we have cν−1 (EV∗ |Vi0 , s∗V |Vi0 ) = (ıi )∗ cν−1 (EV∗ ||D1 | , s∗V ||D1 | ). Thus (i )∗ (Vi0

·

V

(D2

·

V

···

·

V

Dν )Z ∗ )Vi0 ∩Z ∗ = (ıi )∗ Vi0 a cν−1 (EV∗ |Vi0 , s∗V |Vi0 ). 

482

Complex Analytic Geometry

Passing from the homology of Z to that of |D1 |, from (14.22) with Z ∗ as Z and (14.23), (D1

·

V

[D2

·

···

V

·

Dν ]V )|D1 |

V

˘ 2(d−ν) (|D1 |). in H

= D1 a cν−1 (EV∗ ||D1 | )

(14.25)

In the above situation, we have a formula as (14.20) and we have: Theorem 14.5. Let V be a variety of dimension d and let D1 , . . . , Dd be Td divisors, that are Cartier (possibly except for one) on V . If Z = j=1 |Dj | consists of a finite number of isolated points, we have X (D1 V · · · V Dd )p = ](D1 V · · · V Dd ),

·

·

·

·

p∈Z

where (D1

·

V

···

·

V

Dd )p is the intersection number at p.

In some cases we may express the intersection number at a point explicitly. Thus let V be a variety of pure dimension d in a neighborhood U of 0 in Cn possibly with an isolated singularity at 0. Let D1 , . . . , Dd be efTd fective Cartier divisors on V . Suppose j=1 |Dj | = {0} so that we have the intersection product (D1 V · · · V Dd )0 in H0 ({0}) ' Z. If U is sufficiently small, each Dj is defined by the restriction of a holomorphic function fj on U . Note that V (f1 , . . . , fd ) ∩ V = {0}. By Definition 14.6, (D1 V · · · V Dd )0 has various expressions as given in Section 13.6. Namely, we have a topological expression as in Theorem 13.10. Also, by Theorem 13.11, we have   df1 ∧ · · · ∧ dfd . (14.26) (D1 V · · · V Dd )0 = Res0 f1 , . . . , f d V

·

·

·

·

·

·

Moreover, if V is a complete intersection defined by h1 , . . . , hk in U , by Theorem 13.12, we have (D1

·

V

···

·

V

Dd )0 = dim On /(f1 , . . . , fd , h1 , . . . , hk ).

P In the case D1 = mi Vi0 is a Weil divisor, we have expressions as above on the Vi0 ’s. For example,   X df2 ∧ · · · ∧ dfd (D1 V · · · V Dd )0 = mi Res0 . (14.27) f2 , . . . , f d V0

·

·

i

Intersection Product of Complex Subspaces

483

Multiplicity at a point Let U be a neighborhood of 0 in Cn and V a variety of dimension d in U which contains 0. The tangent cone C0 (V ) of V at 0 is the complex space in U defined by the ideal sheaf generated by the leading homogeneous polynomials of germs in the ideal of V at 0. It is known that its support C0 (V ) has the same dimension as V . Example 14.2. 1. Let V be the variety in C2 = {(z1 , z2 )} defined by z13 − z22 = 0 (cf. Example 2.3. 2). Then C0 (V ) is the complex space defined by the ideal (z22 ) (a “double line”). 2. Let V be the variety in C3 = {(z1 , z2 , z3 )} defined by z1 z2 − z32 = 0 (cf. Exercise 11.11). Then C0 (V ) is V itself. 3. Let V be the variety in C3 = {(z1 , z2 , z3 )} defined by z1 z22 − z32 = 0 (cf. Example 2.3. 3). Then C0 (V ) is the complex space defined by the ideal (z32 ) (a “double plane”). A hyperplane H through 0 is a variety of dimension n − 1 in U defined by a linear function `. We may think of H as the complex space defined by a section sH of the (trivial) line bundle LH and ` the component of sH with respect to some frame of LH . Note that sH is a regular section. Let {H1 , . . . , Hi } be a collection of hyperplanes, 1 ≤ i ≤ d. We say that it is excellent for V if dim C0 (V ) ∩ H1 ∩ · · · ∩ Hi = d − i. In this case, sH1 ⊕ · · · ⊕ sHi is a regular section of LH1 ⊕ · · · ⊕ LHi . If {H1 , . . . , Hd } is excellent for V , then V ∩H1 ∩· · ·∩Hd = {0}. Thus we have the localized intersection product (V H1 · · · Hd )0 in H0 ({0}; Z) = Z, the identification being made by the augmentation, and we may think of it as a positive integer.

· ·

·

Definition 14.8. The multiplicity of V at 0 is defined by m(V, 0) = (V

·H · ··· ·H ) , 1

d 0

where {H1 , . . . , Hd } is a collection of d hyperplanes excellent for V . In the above situation, we set E = LH1 ⊕ · · · ⊕ LHd , s = sH1 ⊕ · · · ⊕ sHd and S = H1 ∩ · · · ∩ Hd . Then s is a regular section of E with zero set S. We set EV = E|V and sV = s|V . Let Z be the complex space in V defined by sV . The support of Z is {0} and Z defines a class (a number) ˘ 0 ({0}; Z) = H0 ({0}; Z) = Z, which is also denoted by Z. in H

484

Complex Analytic Geometry

Proposition 14.18. We have m(V, 0) = TRescd (sV , EV ; 0), which is equal to Z. Proof.

We have m(V, 0) = (V

· (H · · · · · H )

d S )0 .

1

Noting that the complex space defined by s is reduced, we have, from Proposition 14.2. 2, (H1 · · · Hd )S = S. The proposition follows from Proposition 14.7. 

·

·

From Theorems 13.11 and 13.12, we have: Corollary 14.2. In the above situation, suppose 0 is an isolated singularity of V . Let `i be a defining linear function of Hi , i = 1, . . . , d. Then we have   d`1 ∧ · · · ∧ d`d m(V, 0) = Res0 . `1 , . . . , `d V Moreover, if V is a CI defined by h1 , . . . , hk , k = n − d, it may also be expressed as m(V, 0) = dimC On /(h1 , . . . , hk , `1 , . . . , `d ). Example 14.3. Let V be a hypersurface in U defined by f . Let m be the order of f at 0 (cf. Definition 1.10). By a suitable change of coordinates, we may assume that the order of f in zn is m (cf. Lemma 1.2). If we let Hi be the hyperplane defined by zi , i = 1, . . . , n − 1, then {H1 , . . . , Hn−1 } is excellent for V and we see that (V H1 · · · Hn−1 )0 is the m-fold point in the zn -plane (cf. Example 11.7). Thus m(V, 0) = m. For example, if V is as in 1, 2 or 3 of Example 14.2, m(V, 0) = 2.

· ·

·

Here is another expression of the multiplicity at a point. The projective cone P C0 (V ) of V at 0 is the complex space in Pn−1 ˜ → U be defined by the homogeneous polynomials defining C0 (V ). Let π : U the blowing up of U at 0 and D the exceptional divisor, which is biholomorphic with Pn−1 . Let V˜ denote the proper transform of V (cf. Section 11.6). ˜ Lemma 14.2. The intersection E of V˜ and D as a complex subspace of U is the projective cone P C 0 (V ).

Intersection Product of Complex Subspaces

485

˜α be the coordinate neighborhood in U ˜ with coordinates Proof. Let U α α (zα+1 , ξ1 , . . . , ξn−1 ) as in the construction of the blowing-up in Section 11.6. Let f be a defining function of V . Let p be the order of f at 0 and P write f (z) = ν≥p fν (z), where fν (z) is homogeneous of degree ν in z = (z1 , . . . , zn ). The function f˜ appearing in (11.20) is written X ν−p α α α fν (ξ1α , . . . , ξαα , 1, ξα+1 . f˜(zα+1 , ξ1α , . . . , ξn−1 )= , . . . , ξn−1 ) · zα+1 ν≥p

The intersection of V˜ and D is defined by the ideal generated by the f˜’s α α and zα+1 . Thus it is generated by the fp (ξ1α , . . . , ξαα , 1, ξα+1 , . . . , ξn−1 )’s and zα+1 .  In general, let X be a complex space of dimension d in Pn . If we denote by LH the hyperplane bundle on Pn , the class [X] a c1 (LH )d is in H0 (Pn ). Definition 14.9. The degree of X is defined by deg X = ε∗ ([X] a c1 (LH )d ), where ε∗ : H0 (Pn ) → Z denotes the augmentation. Let {H1 , . . . , Hd } be a collection of d hyperplanes in Pn . We say that it is in general position, or general for short, with respect to X if Z = X ∩ H1 ∩ · · · ∩ Hd consits of (a finite number of) isolated points. In this case, we have the intersection product (X H1 · · · Hd )Z in H0 (Z).

· ·

·

Proposition 14.19. If (H1 , . . . , Hd ) is general with respect to X,

· · ··· ·H )

deg X = ε∗ ((X H1

d Z ),

where ε∗ : H0 (Z) → Z denotes the augmentation. Proof.

By a repeated use of (4.32), we have

· ·

·

[X] a c1 (LH )d = [X] [H] · · · [H],

·

·

·

which is equal to [X] [H1 ] · · · [Hd ] in H0 (Pn ). If we let ι : Z ,→ Pn be the inclusion, we have ι∗ ((X H1 · · · Hd )Z ) = [X] [H1 ] · · · [Hd ] (cf. Proposition 4.12). The proposition follows from the fact that tha augmentations are compatible with ι∗ . 

· ·

·

Proposition 14.20. We have: m(V, 0) = deg P C 0 (V ).

·

·

·

486

Complex Analytic Geometry

˜ → U be the blowing up of U at 0 and D the excepProof. Let π : U tional divisor, which is biholomorphic with Pn−1 (cf. Section 11.6). Let {H1 , . . . , Hd } be a collection of d hyperplanes through 0 excellent for V . ˜ i that of Hi for each i. The Let V˜ denote the proper transform of V and H ˜ ˜ is the projective intersection E of V and D as a complex subspace of U cone P C 0 (V ) (cf. Lemma 14.2). By Proposition 14.7, (V˜ D)E = E. Also ˜ i and D as a complex subspace of U ˜ is a hyperplane the intersection Hi of H ˜ i )H = Hi . in D = Pn−1 and (D H i ˜1 ∩ · · · ∩ H ˜ d−1 ∩ (H ˜ d ∪ D) = E ∩ H ˜1 ∩ · · · ∩ H ˜ d−1 Noting that Z = V˜ ∩ H

·

·

consists of a finite number of points, we consider the intersection product ˜1 · · · H ˜ d−1 (H ˜ d + D))Z in H0 (Z). (V˜ H ˜1 · · · H ˜ d−1 )Z = On the one hand, it is equal to the product (V˜ D H ˜ ˜ (E H1 · · · Hd−1 )Z , which is equal to (E D H1 D · · · D Hd−1 )Z by a repeated use of Proposition 14.13 or Proposition 14.15. It is mapped to deg E by the augmentation ε∗ : H0 (Z) → Z. ˜1 · · · H ˜ d−1 ) ˜ is the proper trans˜ = (V˜ H On the other hand, W W form of W = (V H1 · · · Hd−1 )W . We have, since Z is compact (cf. Remark 14.4. 1), ˜ (H ˜ d + D))Z = W ˜ a c1 (L ˜ (W | ˜ , π ∗ s| ˜ ))

· · · · · ·

·

·

· ·

·

· ·

·

· · · · · ·

·

Hd +D W

W

˜ a $∗ c1 (LH |W , s|W ), =W d which is sent to W a c1 (LHd |W , s|W ) = (V ˜. where $ is the restriction of π to W

·H · ··· ·H ·H ) 1

d−1

d 0

by $∗ , 

Remark 14.8. The independence of m(V, 0) of the choice of a collection of hyperplanes excellent for V in Definition 14.8 can also be seen from the above proposition. We finish this section by giving some remarks on the multiplicity at a point. Let U and V be as above. Lemma 14.3. Let P be a non-singular hypersurface (through 0) in U . Suppose P intersects transversely with Vreg near 0 and set V 0 = V ∩ P , then m(V 0 , 0) = m(V, 0). Proof. Since P is non-singular, by a suitable change of coordinates, we may assume that P is a hyperplane. Note that dim V 0 = d − 1 (cf. Theorem 2.11). If {H2 , . . . , Hd } is a collection of hyperplanes excellent for V 0 , then {P, H2 , . . . , Hd } is a collection of hyperplanes excellent for V . We have

· · ··· ·H ) .

m(V 0 , 0) = (V 0 H2

d 0

Intersection Product of Complex Subspaces

On the other hand, from the assuptions we have V 0 = (V sition 14.7).

487

·P)

V0

(cf. Propo

Let V be a variety of dimension d in a neighborhood U of 0 in Cn containing 0. Let {H1 , . . . , Hd } be a collection of hyperplanes in U excellent for V . Let E = LH1 ⊕ · · · ⊕ LHd and s = sH1 ⊕ · · · ⊕ sHd . For each i, let Gi denote the intersection of V and Hi as a complex space in U . We may think of it as the complex space in V defined by the section sHi |V of LHi |V . If we denote by Gi the support of Gi , by Proposition 14.15. 1, m(V, 0) = ((V Since dim Gi = d − 1, (V

·H ) · 1 G1

·H )

i Gi

V

···

·

V

(V

·H )

d Gd )0 .

= Gi so that (cf. Proposition 14.15. 2)

m(V, 0) = (G1

·

V

···

·

V

Gd ) 0 .

Let V 0 be a subvariety of V containing 0. Let {H2 , . . . , Hd } be a collection of hyperplanes excellent for V 0 and V and let G0i be the intersection of V 0 and Hi as a complex space. Denoting by G0i the support of G0i , we have

·

m(V 0 , 0) = ((V 0 H2 )G02 We have (V

0

·H )

i G0i

=

G0i

= (V

0

·

m(V 0 , 0) = (V 0

·

V0

···

·

V0

·

(V 0 Hd )G0d )0 .

Gi )G0i . Thus by Proposition 14.16,

V

·

V

G2

·

V

···

·

V

Gd )0 .

Lemma 14.3 can also be seen from the above arguments. 14.4

Intersection product in singular surfaces

In this section, we apply the considerations in the previous sections to the case of singular surfaces. A curve or a surface means a variety of pure dimension one or two, respectively. Let V be a (possibly singular) surface in a complex manifold M . Let D be a Cartier divisor on V and denote by LD the associated line bundle on V . The bundle LD admits a natural meromorphic section sD and its first Chern class c1 (LD ) is localized at the support |D| of D by sD . The localized class c1 (LD , sD ) is in H 2 (V, V r |D|). We have the associated residue TResc1 (sD , LD ; |D|) as the image of c1 (LD , sD ) by the Alexander morphism ˘ 2 (|D|). H 2 (V, V r|D|) −→ H By Theorem 14.4, the class of the Weil divisor associated with D is equal to TResc1 (sD , LD ; |D|), which may be identified with Resc1 (sD , LD ; |D|).

488

Complex Analytic Geometry

Let D1 and D2 be divisors on V . As we have seen in the previous ˘ 0 (Z), section, we may define the intersection product (D1 V D2 )Z in H Z = |D1 | ∩ |D2 |, if at least one of them is Cartier. If D2 is Cartier, the ˘ 0 (Z) → H ˘ 0 (|D1 |) is image of (D1 V D2 )Z by the canonical morphism H given by (cf. (14.25))

·

·

(D1

·

[D2 ]V )|D1 | = D1 a c1 (LD2 ||D1 | ).

V

(14.28)

·

˘ 0 (|D1 |) → It is further sent to [D1 ]V V [D2 ]V = [D1 ]V a c1 (LD2 ) by H ˘ H0 (V ). If Z is compact, we have (cf. Proposition 14.17) (D1

·

V

D2 )Z = D1 a c1 (LD2 ||D1 | , sD2 ||D1 |)

in H0 (Z),

(14.29)

where D1 is the class of D1 in H2 (|D1 |, |D1 |rZ). In this case, we have the intersection number ](D1 V D2 ) as defined in (14.24). In particular, if Z consists of a finite number of isolated points, we have (Theorem 14.5) X (D1 V D2 )p = ](D1 V D2 ).

·

·

·

p∈Z

If V has an isolated singularity at p and if D1 and D2 are Cartier the intersection number at p is given, for example (cf. (14.26)), by   df1 ∧ df2 . (D1 V D2 )p = Resp f1 , f2 V P If D1 = mi Vi0 is Weil, we have (cf. (14.27))   X df2 (D1 V D2 )p = mi Resp . f2 V 0

·

·

i

If D is a Cartier divisor with compact support in V , then Z = |D| and (cf. (14.28)) ](D

·

V

D) = ε∗ (D a c1 (LD ||D| )),

(14.30)

where ε∗ : H0 (|D|) → Z denotes the augmentation. It is referred to as the self-intersection number of D. Effect of blowing-up Let U be a neighborhood of 0 in Cn and V a surface in U containing 0. ˜ → U be the blowing-up of U at 0, D = π −1 (0) the exceptional Let π : U divisor, which may be thought of as Pn−1 , V˜ the proper transform of V . ˜ is the projective The intersection E of V˜ and D as a complex subspace of U

Intersection Product of Complex Subspaces

489

·

cone P C 0 (V ) (cf. Lemma 14.2). By Proposition 14.7, (V˜ D)E = E. The complex space E is considered as a Cartier divisor on V˜ . Let C be a curve through 0 in V . The proper transform C˜ of C is a curve in V˜ and Z = C˜ ∩ E consists of a finite number of points so that we have (C˜ ˜ E)Z in H0 (Z).

·

V

Lemma 14.4. We have m(C, 0) = ](C˜

·

V˜

E).

˜ Proof. Let H be a hyperplane through 0, excellent for C and V , and H ˜ ∪ D) = C˜ ∩ D = C˜ ∩ E the proper transform of H. Noting that Z = C˜ ∩ (H consists of a finite number of points, we consider the intersection product ˜ + D))Z in H0 (Z). On the one hand, by Proposition 14.12, it is (C˜ (H equal to (C˜ D)Z = (C˜ V˜ E)Z . On the other hand we have, since Z is compact (cf. Remark 14.4. 1), ˜ + D))Z = C˜ a c1 (L ˜ | ˜ , π ∗ s| ˜ )) (C˜ (H H+D C C = C˜ a $∗ c1 (LH |C , s|C ),

·

·

·

·

·

d

which is sent to C a c1 (LH |C , s|C ) = (C H)0 by $∗ , where $ is the ˜ restriction of π to C.  Let ρ : V˜ → V denote the restriction of π. For a curve C in V through 0, the analytic inverse image ρ∗ C may be thought of as a divisor on V˜ . In particular, if C is Cartier, so is ρ∗ C. Lemma 14.5. If C is Cartier, the multiplicity m(C, 0) is divisible by m(V, 0) and if we set m(C, V ; 0) = m(C, 0)/m(V, 0), we have ρ∗ C = C˜ + m(C, V ; 0)E. Proof. Since C is Cartier, we may write ρ∗ C = C˜ + kE for some integer k. Let H be a hyperplane through 0, excellent for C and V . Let G be the intersection of V and H as complex subspace of M . We have (V H)G = G. ˜ be the intersection of V˜ and H ˜ as complex subspace of M ˜ . We Also let G ∗ ˜ ˜ ˜ ˜ have (V H)G˜ = G. We set Z = G∩|ρ C|, which consists of a finite number ˜ ˜ ρ∗ C)Z in two ways. On the one hand, since of points and compute (G V ˜ ∩ C˜ = ∅, we have G ˜ ˜ E)Z . ˜ ˜ ρ∗ C)Z = k(G (G V V On the other hand, ˜ ˜ ρ∗ C)Z = G ˜ a ρ∗ c1 (LC , s), (G V 1 which is sent to G a c (LC , s) = (G V C)0 = m(C, 0) by ρ∗ . As in the ˜ ˜ = ](E ˜ G). proof of Proposition 14.20, m(V, 0) = ](E H)  V

·

·

·

·

·

·

·

·

·

490

Complex Analytic Geometry

Theorem 14.6. Let ρ : V˜ → V be the restriction of π as above. Let C1 and C2 be curves in V . Suppose C1 ∩ C2 = {0} and C2 is Cartier. Then (C1

·

V

X

C2 )0 =

(C˜1

·

V˜

C˜2 )q + m(C1 , 0) · m(C2 , V ; 0).

q∈ρ−1 (0)

Proof. Noting that Z = C˜1 ∩ |ρ∗ C2 | consists of a finite number of points, we consider the product (C˜1 V˜ ρ∗ C2 )Z . On the one hand, by Lemmas 14.4 and 14.5, it is equal to

·

(C˜1

·

V˜

(C˜2 + m(C2 , V ; p)E))Z = (C˜1

·

V˜

C˜2 )Z + m(C2 , V ; 0)(C˜1

·

V˜

E)Z ,

·

where (C˜1 V˜ E)Z is mapped to m(C1 , 0) by ε∗ . On the other hand, it is equal to C˜1 a ρ∗ c1 (LC2 |C1 , sC2 |C1 ) (cf. (14.29)), which is mapped to  C1 a c1 (LC2 |C1 , sC2 |C1 ) = (C1 V C2 )0 by ρ∗ .

·

We give the following expression for the multiplicity: Proposition 14.21. m(V, 0) = −](E

·

V˜

E).

·

Proof. We have ](E V˜ E) = ε∗ (E a c1 (LE |E )) (cf. (14.30)). Consider the commutative diagram of inclusions E

˜ j



 V˜

/D ı

˜i

 ˜. /U

We have LE |E = ∗ (˜i)∗ LD = (˜j)∗ ı∗ LD . Recall ı∗ LD = −LH , negative of the hyperplane bundle on D = Pn−1 . Since we may write LH = ı∗ LH˜ , LE |E = −(˜j)∗ ı∗ LH˜ = −∗ (˜i)∗ LH˜ = −∗ LG˜ , we have E a c1 (LE |E ) =  −E a c1 (LG˜ |E ). If C1 and C2 are two Cartier curves such that C1 ∩ C2 = {0}, then |ρ∗ C1 | ∩ |ρ∗ C2 | = E, which is compact. We compute, using Lemmas 14.4 and 14.5 and Proposition 14.21, (C1

·

V

C2 )0 = ](ρ∗ C1

·

V˜

ρ∗ C2 ).

(14.31)

Intersection Product of Complex Subspaces

491

Intersection product of Weil curves Let V be a surface in a complex manifold M . Let C1 and C2 be two (distinct) curves in V . If at least one of them is Cartier, We have the local and global intersection numbers of C1 and C2 . If C1 and C2 are only Weil curves, we proceed as follows. ˜ → M be the blowing-up at p and Let p be a point in V . Let πp : M ρp : V˜ → V the restriction of πp , where V˜ is the proper transform of V under πp . Let C be a curve on V through p and C˜ the proper transform of C under πp . Definition 14.10. The total transform of C under ρp is defined by m(C, p) E. ρ?p C = C˜ + m(V, p) If C is Cartier, this coincides with the inverse image ρ∗p C by Lemma 14.5. In general m(C, p)/m(V, p) is a rational number and ρ?p C is different from ρ∗p C. Let C1 and C2 be curves on V through p such that they intersect only at p near p. If either C˜1 or C˜2 is Cartier, we may define the intersection number by (C1

·

V

C2 )p =

X

(C˜1

q∈ρ−1 (p)

·

V˜

m(C1 , p) · m(C2 , p) C˜2 )q + , m(V, p)

(14.32)

which reduces to the formula in Theorem 14.6, if either C1 or C2 is Cartier. By Lemma 14.4 and Proposition 14.21, we may write (C1

·

V

C2 )p = ](ρ?p C1

·

V˜

ρ?p C2 ),

which reduces to (14.31), if both C1 and C2 are Cartier. If both C˜1 and C˜2 are not Cartier, we define the intersection number by recursion of (14.32). Note that, if either one of C1 and C2 is not Cartier at p, (C1 V C2 )p is only a rational number, in general, for m(V, p) might not divide m(C1 , p) · m(C2 , p).

·

In the global situation, if C1 ∩ C2 consists of a finite number of isolated points, we define by recursion X ](C1 V C2 ) = (C1 V C2 )p .

·

·

p∈C1 ∩C2

492

Complex Analytic Geometry

More generally, if Z = C1 ∩ C2 is compact, we define ](C1

·

V

C2 ) = ](ρ?p C1

·

V˜

ρ?p C2 ).

In particular, the self-intersection number of a compact curve C in V is given by ](C

·

V

C) = ](ρ?p C

·

V˜

ρ?p C).

Remark 14.9. 1. In the above, we need not to resolve the singularities of V , we only need to take blowing-ups sufficiently many times so that the curve becomes Cartier. 2. Intersections of Weil divisors are defined by extending the above linearly. Example 14.4. Let V be defined by z1 z2 − z32 = 0 in C3 = {(z1 , z2 , z3 )} (cf. Exercise 11.11). Let C1 and C2 be curves in V defined by z1 = z3 = 0 and z2 = z3 = 0, respectively. Then C1 and C2 are Weil curves (only C1 ∪ C2 is Cartier). Since m(V, 0) = 2, m(C1 , 0) = m(C2 , 0) = 1, and C˜1 and C˜2 are disjoint, we compute (C1

14.5

·

V

C2 )0 = ](C˜1

·

V˜

m(C1 , 0) · m(C2 , 0) 1·1 1 C˜2 ) + =0+ = . m(V, 0) 2 2

Excess intersections

Let M be a complex manifold of dimension n and E a holomorphic vector bundle of rank l on M . Also let s be a holomorphic section of E and S the zero set of s. In this situation, we have the residues TRescl (s, E; S) ˘ 2(n−l) (S; Z) and Rescl (s, E; S) in H ˘ 2(n−l) (S; C). The complex space S in H defined by s has S as its support. Recall that dim S ≥ n − l, where the equality holds if and only if s is a regular section. In the case dim S = n − l, ˘ 2(n−l) (S; Z) we may define the homology class of S, denoted also by S, in H (cf. Definition 12.6) and we have (cf. Theorem 12.9) S = TRescl (s, E; S), which may also be identified with Rescl (s, E; S). Now we consider the case dim S > n − l. Suppose there exists a subbundle E 0 of rank l0 = n−dim S of E such that s is a section of E 0 . Then we may ˘ 2(n−l0 ) (S; Z) and we have S = TRes l0 (s, E 0 ; S). define the class of S in H c Let E 00 denote the quotient bundle E/E 0 , which is of rank l00 = l − l0 .

Intersection Product of Complex Subspaces

493

From Proposition 5.10 (see also Corollary 10.2), we have: Theorem 14.7. In the above situation, we have: 00

˘ 2(n−l) (S), in H

TRescl (s, E; S) = S a i∗ cl (E 00 ) where i : S ,→ M denotes the inclusion.

Let E, s and S be as in the beginning of this section. Let V be a variety of dimension d in M and set EV = E|V and sV = s|V . Let Z be the complex space in V defined by sV . Its support Z is given by Z = V ∩ S. Suppose s is a regular section so that dim S = n − l. Then, by Proposition 14.7. 1, (V

· S)

Z

˘ 2(d−l) (Z). in H

= TRescl (sV , EV ; Z)

(14.33)

In general, dim Z ≥ d − l by Theorem 2.11. If dim Z = d − l, we may define the homology class of Z (cf. Definition 13.6) and we have Z = TRescl (sV , EV ; Z) so that (cf. Proposition 14.7. 2) (V

· S)

Z

= Z.

Now we consider the case dim Z > d−l, which means that s vanishes on V “more than expected” so that V and S intersect in a dimension higher than expected. Suppose there exists a subbundle EV0 of rank l0 = d − dim Z of EV such that sV is a section of EV0 . Then we may define the class of Z ˘ 2(d−l0 ) (Z; Z) and we have Z = TRes l0 (sV , E 0 ; Z). Let E 00 denote the in H V V c quotient bundle EV /EV0 , which is of rank l00 = l − l0 . From Corollary 14.1, we have 0

00

cl (EV , sV ) = cl (EV0 , sV ) ` cl (EV00 ). We also have a statement as in Proposition 4.3 in this situation and we have (cf. the proof of Proposition 5.10): 00

TRescl (sV , EV ; Z) = TRescl0 (sV , EV0 ; Z) a j ∗ cl (EV00 )

˘ 2(d−l) (Z), in H

where j : Z ,→ V denotes the inclusion. The left-hand side of the above is equal to (V S)Z by (14.33). Thus we have:

·

Theorem 14.8 (Excess intersection formula). In the above situation, we have: (V

· S)

Z

00

= Z a j ∗ cl (EV00 )

˘ 2(d−l) (Z). in H

494

Complex Analytic Geometry

Notes The adjunction formula (14.17) was generalized in [Kodaira (1960)] for a possibly singular curve C in a complex surface S as:

·

e = −(KS + C) C + χ(C)

r X

c(C, pi ),

i=1

e is a non-singular model of C and c(C, pi ) is an invariant of C at where C the singular point pi , which is related to the Milnor number µ(C, pi ) by c(C, pi ) = µ(C, pi ) + si − 1 with si the number of (local) branches of C at e − Pr (si − 1) = χ(C), the above formula is equivalent to pi . Since χ(C) i=1 the one in Theorem 14.3. For details on tangent cones and the multiplicity of a variety at a point, we refer to Chapter 7 of [Whitney (1972)]. The terminology “excellent” appears there. The multiplicity in Definition 14.8 above coincides with the one in there. It also coincides with the one in p.79 of [Fulton (1984)] (cf. Proposition 14.20). The intersection theory on normal singular surfaces is teated in [Mumford (1961)], see also [Sakai (1984)]. Our approach to the intersection theory on singular surfaces is based on residue theory. It is consistent with the one in the references above and, moreover, allows us to relate the intersection theory to the residue theories of holomorphic foliations and maps on singular varieties. The description follows (and make them more precise) that in [Bracci and Suwa (2004)], for which we also refer to for an application. The intersection theory in algebraic category is comprehensively presented in [Fulton (1984)]. The content of this chapter corresponds to some parts in there, in fact it is inspired by them. For example, the basic intersection product construction there is done intersecting the cormal cone by the zero section. In our framework, it appears as a residue (see, e.g., Propositions 14.7 and 14.13 in the text). The notion of the order there is defined in terms of the length of a module. It is replaced with some residue in our setting (cf. Definition 14.4). The excess intersection formula is also discussed in generality there. The formula in Theorem 14.8 corresponds to the one in Example 6.1.7 in there.

Chapter 15

Riemann-Roch Theorem

The Riemann-Roch problem originates in finding the dimension of the space of meromorphic functions, on a compact Riemann surface, with prescribed order of zeros and poles. After the work of K. Kodaira for compact complex surfaces, it was formulated and proved for projective algebraic manifolds by F. Hirzebruch as the equality between the alternating sum of the dimensions of cohomology groups with coefficients in the sheaf of holomorphic sections of a holomorphic vector bundle and a certain topological invariant. It was then generalized in two directions ; one as a functorial equality by A. Grothendieck and the other as the index theorem of elliptic operators by M.F. Atiyah and I.M. Singer. In this chapter, we mostly discuss the former, i.e., the GrothendieckRiemann-Roch theorem (cf. Theorem 15.11). The proof for this is divided into two parts; for embeddings and for projections. We examin the embedding case in detail, applying our localization approach, and prove a prototype of the localized Riemann-Roch theorem for embeddings on the ˇ level of Cech-de Rham cocycles (Corollary 15.1). It immediately yields a theorem in the universal situation (Theorem 15.4). We then present three variations of the localized Riemann-Roch theorem for embeddings (Theorems 15.5–15.7). For the projection part, we outline the proof quoting relevant materials. The relation with the Atiyah-Singer index theorem is briefly mentioned in Notes at the end of the chapter. In the following, we sometimes omit the symbol ∧ for the exterior product of forms and the symbol ` for the cup product of cohomology classes and simply denote them as products.

495

496

15.1

Complex Analytic Geometry

Riemann-Roch problem for curves

Let C be a compact Riemann surface. The original Riemann-Roch problem consists in finding the dimension of the space of meromorphic functions on C with prescribed orders of zeros and poles. To be more precise, let M (C) P be the space of meromorphic functions on C and D = mi pi a divisor on C. If we set L(D) = { ϕ ∈ M (C) | (ϕ) + D ≥ 0 }, then a meromorphic function ϕ is in L(D) if and only if, for i with mi > 0, it has a pole of order at most mi at pi , and, for i with mi < 0, it has a zero of order at least −mi at pi . In terms of bundles and sheaves, we have an isomorphism L(D) ' H 0 (C; O(LD )). The isomorphism is given by assigning to each meromorphic function in L(D) the section which is locally represented by the holomorphic function obtained by multiplying ϕ by a (local) defining function of D. We denote by M 1 (C) the space of meromorphic 1-forms on C. For a meromorphic 1-form ω, we may define the associated divisor (ω) as in the case of meromorphic functions. We set I(D) = { ω ∈ M 1 (C) | (ω) − D ≥ 0 }. Then we have an isomorphism I(D) ' H 0 (C; Ω 1 (L∗D )). We denote by l(D) and i(D) the dimensions of L(D) and I(D), respecP tively. We also let d = mi , the degree of D, and g the genus of C, i.e., 0 1 dim H (C; Ω ) (cf. the paragraph after Corollary 9.6). Then the classical Riemann-Roch theorem says that l(D) = d − g + i(D) + 1.

(15.1)

We see later that this is a special case of the Hirzebruch-Riemann-Roch theorem (Theorem 15.12).

15.2

Characteristic classes of virtual bundles

K-groups Let X be a paracompact topological space and let At (X) denote the free Abelian group generated by the isomorphism classes of continuous complex

Riemann-Roch Theorem

497

vector bundles on X. For simplicity, we denote the isomorphism class of a vector bundle E also by E. The K-group of continuous vector bundles on X, denoted by K t (X), is the Abelian group obtained as the quotient of At (X) by the subgroup generated by the elements of the form E −(E 0 +E 00 ) for which there is an exact sequence 0 −→ E 0 −→ E −→ E 00 −→ 0. In the following, we consider characteristic classes of elements in K t (X) and, for this purpose, it is not necessary to distinguish the class [E] in K t (X) of an element E in At (X) from E, so that we also denote [E] by E. Thus E + F = E ⊕ F in K t (X), as every short exact sequence of continuous vector bundles on a paracompact space splits (cf. Proposition 3.3). Note that every element in K t (X) is expressed as E − F , which is referred as a virtual bundle (cf. Section 8.3). In fact K t (X) has the structure of a ring with multiplication coming from the tensor products of vector bundles with the identity the class of the product bundle C × X. If f : X 0 → X is a continuous map, there is a natural ring morphism ∗ f : K t (X) → K t (X 0 ) coming from the pull-back of vector bundles. If M is a C ∞ manifold we can also construct the K-group K d (M ) of C ∞ vector bundles on M . There is a caninical morphism K d (M ) → K t (M ), which is in fact an isomorphism (cf. Remark 3.3). Chern character and Todd class Let X be a regular cell complex. For a complex vector bundle E of rank l Pl on X, we have the (total) Chern class c∗ (E) = i=0 ci (E) in H ∗ (X; Z), c0 (E) = 1 (cf. Section 5.2). We consider a formal factorization l X

ci (E) ti =

i=0

l Y

(1 + γj t).

(15.2)

j=1

Thus ci (E) is the i-th elementary symmetric function in the γj ’s. Definition 15.1. The Chern character ch∗ (E) of E is the class in H ∗ (X; Q) given by ch∗ (E) =

l X

e γj .

j=1

Note that the right-hand side above is a symmetric series in the γj ’s so that it is expressed in terms of Chern classes. More explicitly, if we set

498

si (E) =

Complex Analytic Geometry

Pl

j=1

γji , then ch∗ (E) = l +

X si (E) i≥1

i!

.

The classes ci = ci (E) and si = si (E) are related by Newton’s formula: si − c1 si−1 + c2 si−2 − · · · + (−1)i i ci = 0, i ≥ 1. (15.3) ∗ ∗ While c is multiplicative, ch is additive, i.e., if 0 −→ E 0 −→ E −→ E 00 −→ 0 (15.4) is an exact sequence of complex vector bundles, c∗ (E) = c∗ (E 0 ) · c∗ (E 00 ) and ch∗ (E) = ch∗ (E 0 ) + ch∗ (E 00 ). From the latter, we have a morphism ch∗ : K t (X) −→ H ∗ (X; Q). (15.5) If we have two complex vector bundles E and F , ch∗ (E ⊗ F ) = ch∗ (E) · ch∗ (F ) so that ch∗ in (15.5) is in fact a ring morphism, the product in the right-hand side being the cup product. Pr In particular, for a virtual bundle of the form ξ = i=0 (−1)i Ei with Ei complex vector bundles, i = 0, . . . , r, r r X Y (−1)i ch∗ (Ei ), (15.6) c∗ (Ei )(i) and ch∗ (ξ) = c∗ (ξ) = i=0

i=1

where (i) = (−1)i . Definition 15.2. Using the factorization (15.2), we define the Todd class of E in H ∗ (X; Q) by l Y γj . td(E) = 1 − e−γj j=1 Note that it can be also expressed in terms of Chern classes of E. It is multipicative, i.e., for an exact sequence as (15.4), td(E) = td(E 0 ) · td(E 00 ). Note that the series in the right-hand side starts with 1 so that td(E) is invertible in H ∗ (X; Q). Pl Vi For a complex vector bundle E of rank l, we set λE = i=0 (−1)i E P V V i 0 l E = in K t (X) so that ch∗ (λE ) = i=0 (−1)i ch∗ ( E), where we set C × X. With these we have the following fundamental formula: Theorem 15.1. For a complex vector bundle E of rank l, ch∗ (λE ∗ ) = cl (E) · td(E)−1 in H ∗ (X; Q).

Riemann-Roch Theorem

499

Vi This can be proved using the expression of ch∗ ( E ∗ ) in terms of the γj ’s. We prove this on the level of differential forms below (cf. Theorem 15.2). We then prove a “localized version” of this, which is one of the essential ingredients in the proof of the localized Riemann-Roch theorem for embeddings (cf. Corollary 15.1). 15.3

Chern-Weil theory for virtual bundles

In this section, we let M denote a C ∞ manifold. A fundamental relation in linear algebra Let A be an l × l complex matrix. We set λi A =

i X

j

I ∧ · · · ∧ A ∧ · · · ∧ I,

λ0 A = I.

j=1

Note that (I − e−A )/A makes sense, even if A is not invertible. Lemma 15.1. We have l  I − e−A  X i . (−1)i tr(e−λ A ) = det A · det A i=0

Proof.

We have det(I − e−A ) =

l X Vi (−1)i tr( e−A ) i=1

and, if A is in GL(l, C), the identity follows from Vi −A i e = e−λ A . Since GL(l, C) is dense in M (l, C),we have the lemma.



Chern character form and Todd form Let E be a C ∞ complex vector bundle on M . For a connection ∇ for √ E, let K denote its curvature and set A = ( −1/2π)K. We represent K locally by a curvature matrix. Recall that the total Chern form is defined by (cf. Section 8.2) c∗ (∇) = det(I + A).

500

Complex Analytic Geometry

Definition 15.3. Using the following particular invariant series, the Chern character form and the Todd form of ∇ are defined by  A  . ch∗ (∇) = tr(eA ) and td(∇) = det I − e−A Proposition 15.1. The images of ch∗ (E) and td(E) by the canonical monomorphism H ∗ (M ; Q) → H ∗ (M ; C) ' Hd∗ (M ) are represented by ch∗ (∇) and td(∇), respectively. The properties of determinant and trace imply that c∗ and td are multiplicative, while ch∗ is additive on the form level as well, i.e., if (∇0 , ∇, ∇00 ) is a triple of connections compatible with the exact sequence (15.4), we have c∗ (∇) = c∗ (∇0 ) · c∗ (∇00 ),

td(∇) = td(∇0 ) · td(∇00 ),

(15.7)

ch∗ (∇) = ch∗ (∇0 ) + ch∗ (∇00 ).

Note that I − e−A is divisible by A and the result is invertible so that  I − e−A  td(∇)−1 = det A also makes sense. If we set si (∇) = tr(Ai ), then it is a closed 2i-form on M . Denoting by l the rank of E, we have c∗ (∇) = 1 +

l X i=1

ci (∇)

and

ch∗ (∇) = l +

X si (∇) i≥1

i!

.

The forms ci = ci (∇) and si = si (∇) are again related by Newton’s formula (15.3). Note that the constant term in td(∇) is 1 and that td(∇) can be expressed as a series (in fact a polynomial) in ci (∇). We have also the following formula on the form level (cf. Theorem 15.1), which is a direct consequence of Lemma 15.1: Theorem 15.2. Let E be a C ∞ complex vector bundle of rank l and ∇ a connection for E. Then we have l X

Vi (−1)i ch∗ ( ∇∗ ) = cl (∇) · td(∇)−1 ,

i=0

Vi ∗ where ∇ denotes the connection for E ∗ dual to ∇ and ∇ the connection Vi ∗ V V0 ∗ 0 ∗ for E induced by ∇∗ . Here we set E = C × M and ∇ = d, the exterior differential. ∗

Riemann-Roch Theorem

501

Characteristic forms of virtual bundles If we have a C ∞ complex vector bundle Ei on M , for each i = 0, . . . , r, Pr i we may consider the virtual bundle ξ = i=0 (−1) Ei as an element in K d (M ). Letting ∇(i) be a connection for Ei , we denote by ∇? the family of connections (∇(r) , . . . , ∇(0) ) and define its total Chern form c∗ (∇? ) by r Y ∗ ? c (∇ ) = c∗ (∇(i) )(i) , i=0 i

where (i) = (−1) . The q-th Chern form cq (∇? ) is the component of c∗ (∇? ) of degree 2q. More generally, for an invariant series ϕ, we write ϕ = P (c1 , c2 , · · · ) as before and set ϕ(∇? ) = P (c1 (∇? ), c2 (∇? ), . . . ). In particular, we have r X ch∗ (∇? ) = (−1)i ch∗ (∇(i) ). i=0

As in the case of vector bundles we have difference forms for families of connections (cf. Proposition 8.3): Proposition 15.2. Suppose we have p + 1 families of connections ∇?ν = (0) (q) (∇ν , . . . , ∇ν ), ν = 0, . . . , p. Then we have a form ϕ(∇?0 , . . . , ∇?p ) alternating in the p + 1 entries and satisfying p X c?ν , . . . , ∇?p ) + (−1)p dϕ(∇?0 , . . . , ∇?p ) = 0. (−1)ν ϕ(∇?0 , . . . , ∇ ν=0

Proof. This is proved as Proposition 8.3. Thus, consider the product ˜ (i) Rp × M with projection ρ : Rp × M → M . For each i = 0, . . . , r, let ∇ ∗ be the connection for ρ Ei given by p p   X X (i) ˜ (i) = 1 − ∇ tν ρ∗ ∇0 + tν ρ∗ ∇ν(i) . ν=1

ν=1

˜ ? )), where ˜ ? = (∇ ˜ (r) , . . . , ∇ ˜ (0) ) and define ϕ(∇? , . . . , ∇?p ) = ρ0∗ (ϕ(∇ We set ∇ 0 0 p ρ : ∆ × M → M is the restriction of ρ.  For two families of connections ∇?0 and ∇?1 , we have ϕ(∇?1 ) − ϕ(∇?0 ) = dϕ(∇?0 , ∇?1 ). Hence we reprove that the class [ϕ(∇? )] of the closed form ϕ(∇? ), which is in fact ϕ(ξ), depends only on ξ and not on the choice of ∇? . In particular, the total Chern class c∗ (ξ) is the class of c∗ (∇? ) and is also given by the first identity in (15.6). The q-th Chern class cq (ξ) is the component of c∗ (ξ) in Hd2q (M ) and is the class of cq (∇? ).

502

Complex Analytic Geometry

Virtual bundle associated with a sequence Let ψr

ψ1

0 −→ Er −→ · · · −→ E1 −→ E0 −→ 0

(15.8)

be a sequence of C ∞ complex vector bundles on M and, for each i, let ∇(i) be a connection for Ei . We say that the family (∇(r) , . . . , ∇(0) ) is compatible with the sequence if, for each i, the following diagram is commutative: A0 (M, Ei )

ψi

/ A0 (M, Ei+1 )

∇(i)

 A1 (M, Ei )

1⊗ψi



∇(i+1)

/ A1 (M, Ei+1 ).

The following two propositions generalize Propositions 8.5 and 8.6: Proposition 15.3. Suppose the sequence (15.8) is exact. Given a connection ∇(0) for E0 , then there exist connections ∇(i) for Ei , i = 1, . . . , r, such that the family (∇(r) , . . . , ∇(0) ) is compatible with (15.8). Proof. We proceed by induction on r. It holds obviously if r = 1. Suppose it holds for r = k ≥ 1 and prove that it also holds for r = k + 1. Identifying Ek+1 with a subbundle of Ek by ψk+1 , we have the exact sequences: ϕ

0 −→ Ek+1 −→ Ek −→ Ek /Ek+1 −→ 0

(15.9)

and 0 −→ Ek /Ek+1 −→ Ek−1 −→ · · · −→ E1 −→ E0 −→ 0

(15.10)

By induction hypothesis, there exist connections ∇ for Ek /Ek+1 and ∇(i) for Ei , i = 1, . . . , k − 1, such that (∇, ∇(k−1) , . . . , ∇(0) ) is compatible with (15.10). By Proposition 8.5, there exist connections ∇(k+1) and ∇(k) for Ek+1 and Ek such that (∇(k+1) , ∇(k) , ∇) is compatible with (15.9). Then (∇(k+1) , . . . , ∇(0) ) is compatible with (15.8).  Proposition 15.4. Suppose the sequence (15.8) is exact. For a family (∇(r) , . . . , ∇(0) ) of connections compatible with (15.8), r Y i=0 i

where (i) = (−1) .

c∗ (∇(i) )(i) = 1,

Riemann-Roch Theorem

503

Proof. We prove by induction on r. If r = 1, the statement is true, as c∗ (∇(1) ) = c∗ (∇(0) ). Now suppose the statement is true for r = k and show for the case r = k + 1. Again, the sequence (15.8) splits into (15.9) and (15.10). Since ∇(k+1) and ∇(k) are compatible with ψk+1 , ∇(k) induces a connection ∇ for Ek /Ek+1 so that (∇(k+1) , ∇(k) , ∇) is compatible with (15.9). By Proposition 8.6, we have c∗ (∇(k) ) = c∗ (∇(k+1) ) · c∗ (∇). On the other hand, as (∇, ∇(k−1) , . . . , ∇(0) ) is compatible with (15.10), by the induction hypothesis, we have the proposition.  From this we have the following: Proposition 15.5. Suppose the sequence (15.8) is exact. Let ϕ be an in(r) (0) variant polynomial and ∇?k = (∇k , . . . , ∇k ), k = 0, . . . , p, families of connections compatible with (15.8). Then ˆ ˆ ?0 , . . . , ∇ ˆ ?p ) = ϕ(∇(0) , . . . , ∇(0) ϕ(∇ p ) in particular ϕ(ξ) = ϕ(E0 ), 0

(r)

(1)

ˆ ? denotes the family of connections (∇ , . . . , ∇ ) for the virtual where ∇ k k k Pr i−1 bundle ξˆ = Ei . Similarly for the other “partitions” of the i=1 (−1) virtual bundle ξ.

15.4

Local Chern classes and characters

In this section also, we let M denote a C ∞ manifold. ˇ Characteristic cocycles in the Cech-de Rham complex Using difference forms we may define characteristic classes of virtual bun∗ ˇ (U) for an arbitrary covering U dles in the Cech-de Rham cohomology HD of M (cf. Section 8.4). For simplicity we consider coverings U consisting of two open sets U0 and U1 . In fact this is the case we consider in the following. Pr If ξ = i=0 (−1)i Ei is a virtual bundle, we take a family of connections (r) (0) ∇?ν = (∇ν , . . . , ∇ν ) for ξ on each Uν , ν = 0, 1. For the collection ∇?∗ = (∇?0 , ∇?1 ) and a symmetric series ϕ, we define a cochain ϕ(∇?∗ ) in A∗ (U) by ϕ(∇?∗ ) = (ϕ(∇?0 ), ϕ(∇?1 ), ϕ(∇?0 , ∇?1 )). ∗ It is in fact a cocycle and defines a class [ϕ(∇?∗ )] in HD (U). It does not depend on the choice of the collection of families of connections ∇?∗ and ∗ corresponds to the class ϕ(ξ) under the isomorphism HD (U) ' Hd∗ (M ).

504

Complex Analytic Geometry

Now we prove a formula which describes explicitly the difference between the cocycle for the product of two symmetries series and the product of cocycles for these series. Although it is stated for connections of a single vector bundle, the same formula holds for families of connections of a virtual bundle. Thus let E be a complex vector bundle on M . Note that ϕψ(∇) = ϕ(∇) · ψ(∇), for symmetric series ϕ and ψ and a connection ∇ for E. Let ∇i be a connection for E on Ui , i = 0, 1. We set ˜ = (1 − t)ρ∗ ∇0 + tρ∗ ∇1 with ρ : R × U01 → U01 , as in the proof of ∇ Proposition 15.2. Lemma 15.2. For two symmetric series ϕ and ψ, we have ϕψ(∇0 , ∇1 ) = ϕ(∇0 ) · ψ(∇0 , ∇1 ) + ϕ(∇0 , ∇1 ) · ψ(∇1 ) − d τ01 , where τ01 is a form on U01 given by ˜ · dψ(ρ∗ ∇1 , ∇)). ˜ τ01 = ρ0∗ (ϕ(ρ∗ ∇0 , ∇) ˜ · ψ(∇)) ˜ and Proof. By definition, the left-hand side is equal to ρ0∗ (ϕ(∇) the sum of the first two terms in the right-hand side is equal to ˜ + ϕ(∇) ˜ · ψ(ρ∗ ∇1 )). ρ0∗ (ϕ(ρ∗ ∇0 ) · ψ(∇) We have ˜ · ψ(∇) ˜ − (ϕ(ρ∗ ∇0 ) · ψ(∇) ˜ + ϕ(∇) ˜ · ψ(ρ∗ ∇1 )) ϕ(∇) ˜ · dψ(ρ∗1 ∇, ∇) ˜ − ρ∗ (ϕ(∇0 ) · ψ(∇1 )). = dϕ(ρ∗ ∇0 , ∇) If we denote by i the inclusion of the boundary {0, 1} of [0, 1] into [0, 1] and by ∂ρ0 the restriction of ρ0 to {0, 1}, the lemma follows from the identity ρ0∗ ◦ ρ∗ = 0, Proposition 6.3 and ˜ · dψ(ρ∗ ∇1 , ∇)) ˜ (∂ρ0 )∗ ◦ i∗ (ϕ(ρ∗ ∇0 , ∇) = ϕ(∇0 , ∇1 ) · dψ(∇1 , ∇1 ) − ϕ(∇0 , ∇0 ) · dψ(∇1 , ∇0 ) = 0.



From Lemma 15.2, we have the following: Proposition 15.6. For two symmetric series ϕ and ψ, we have, in A∗ (U), ϕψ(∇∗ ) = ϕ(∇∗ ) ` ψ(∇∗ ) + Dτ, where τ = (0, 0, τ01 ) with τ01 a form on U01 as given in Lemma 15.2. Remark 15.1. In the situation of Proposition 15.6, if ϕ(∇∗ ) is in A∗ (U, U0 ), i.e., if ϕ(∇0 ) = 0, so is ϕψ(∇∗ ), since ϕψ(∇0 ) = ϕ(∇0 ) · ψ(∇0 ). The proposition shows that the class ϕψ(E) coincides with ϕ(E) ` ψ(E) ∗ in HD (U, U0 ), since τ is also in A∗ (U, U0 ).

Riemann-Roch Theorem

505

Localization by exactness We discuss the localization theory of characteristic classes of virtual bundles by exactness. Let S be a closed set in M . Letting U0 = M rS and U1 a neighborhood of S in M , we consider the covering U = {U0 , U1 } of M . Note that the cup product of a cochain in A∗ (U, U0 ) and a cochain in A∗ (U) is in A∗ (U, U0 ) and it induces a right H ∗ (M ; C)-module structure on H ∗ (M, M rS; C). Let 0 −→ Er −→ · · · −→ E1 −→ E0 −→ 0

(15.11)

∞

be a complex of C complex vector bundles on M which is exact on U0 . Then we will see below that, for each q > 0, there is a canonical localization cqS (ξ) in H 2q (M, M r S; C) of the Chern class cq (ξ) in H 2q (M ; C) of the Pr virtual bundle ξ = i=0 (−1)i Ei . The basic fact is the following “vanishing theorem”, which follows from Proposition 15.4: Lemma 15.3. If ∇?0 is a family of connections on U0 compatible with (15.11), then, for each q > 0, cq (∇?0 ) = 0. In fact, the above holds for the difference form of a finite number of families of connections compatible with (15.11) on U0 . For a symmetric series ϕ without constant term, we also have a similar vanishing ϕ(∇?0 ) = 0. Let ∇?0 be a family of connections compatible with (15.11) on U0 and Pr ? ∇1 an arbitrary family of connections for ξ = i=0 (−1)i Ei on U1 . Then the class cq (ξ) is represented by the cocycle cq (∇?∗ ) = (cq (∇?0 ), cq (∇?1 ), cq (∇?0 , ∇?1 )) in A2q (U). By Lemma 15.3, we have cq (∇?0 ) = 0 and thus the cocycle is in A2q (U, U0 ) and it defines a class cqS (ξ) in H 2q (M, M r S; C). It is sent to cq (ξ) by the canonical morphism j ∗ : H 2q (M, M rS; C) → H 2q (M ; C). It is not difficult to see that the class cqS (ξ) does not depend on the choice of the family of connections ∇?0 compatible with (15.11) or on the choice of the family of connections ∇?1 . If ϕ is a symmetric series without constant term, we may also define the localized class ϕS (ξ) of ϕ(ξ). In particular, noting that the alternating sum of the ranks of Ei is zero, if M r S 6= ∅, we have the localized Chern character ch∗S (ξ) in the relative cohomology H ∗ (M, MrS; C), which is sent to ch∗ (ξ) by the morphism j ∗ . It is the class of the cocycle ch∗ (∇?∗ ) = (0, ch∗ (∇?1 ), ch∗ (∇?0 , ∇?1 )) in A∗ (U, U0 ).

506

Complex Analytic Geometry

Let F be a C ∞ complex vector bundle on M and ∇ a connection for F . Then its Chern character ch∗ (F ) is the class of the cocycle ch∗ (∇) = (ch∗ (∇)|U0 , ch∗ (∇)|U1 , 0) in A∗ (U). The complex 0 −→ F ⊗ Er −→ · · · −→ F ⊗ E1 −→ F ⊗ E0 −→ 0 (r)

(0)

is exact on U0 and the family ∇ ⊗ ∇?0 = (∇ ⊗ ∇0 , . . . , ∇ ⊗ ∇0 ) of connections is compatible with the above sequence on U0 . We set F ⊗ ξ = Pr (r) (0) i ? i=0 (−1) F ⊗Ei and let ∇⊗∇1 denote the family (∇⊗∇1 , . . . , ∇⊗∇1 ). Then ch∗ (F ⊗ ξ) is the class of the cocycle ch∗ (∇ ⊗ ∇?∗ ) = (0, ch∗ (∇ ⊗ ∇?1 ), ch∗ (∇ ⊗ ∇?0 , ∇ ⊗ ∇?1 )). We have ch∗ (∇ ⊗ ∇?1 ) = ch∗ (∇) · ch∗ (∇?1 ), ch∗ (∇ ⊗ ∇?0 , ∇ ⊗ ∇?1 ) = ch∗ (∇) · ch∗ (∇?0 , ∇?1 ). Hence, recalling the definition of the cup product, we have ch∗ (∇ ⊗ ∇?∗ ) = ch∗ (∇) ` ch∗ (∇?∗ )

in A∗ (U, U0 ),

thus ch∗S (F ⊗ ξ) = ch∗ (F ) · ch∗S (ξ) 15.5

in H ∗ (M, M rS; C).

(15.12)

Universal localized Riemann-Roch theorem

In this section, we let M be a C ∞ manifold and E a C ∞ complex vector bundle of rank l on M . Koszul complex Let s be a C ∞ section of E. The Koszul complex associated with (E, s) is the sequence Vl ds ds Vi ∗ ds Vi−1 ∗ ds ds V0 ∗ 0 −→ E ∗ −→ · · · −→ E −→ E −→ · · · −→ E −→ 0, (15.13) Vi ∗ Vi−1 ∗ V0 ∗ where E = C × M and ds = dis : E → E is defined by ds (ϕ1 ∧ · · · ∧ ϕi ) =

i X (−1)j−1 ϕj (s(x))ϕ1 ∧ · · · ∧ ϕ cj ∧ · · · ∧ ϕi j=1

Riemann-Roch Theorem

507

for ϕ1 , . . . , ϕi in Ex∗ , x ∈ M . It is not difficult to see that it forms a complex of vector bundles, i.e., dis ◦ di+1 = 0. Let S denote the zero set of s. s Proposition 15.7. The sequence (15.13) is exact on M rS. Proof. Let x be a point in M r S and take a neighborhood U of x in M r S where E is trivial. Since we may take s as a part of a frame of E on U , there exists a frame (e∗1 , . . . , e∗l ) of E ∗ on U such that e∗1 (s) = 1 and e∗j (s) = 0, for j ≥ 2. Then we see that both Im di−1 and Ker dis are spanned s ∗ ∗ by the elements ej1 ∧ · · · ∧ ejr , 2 ≤ j1 < · · · < jr ≤ l.  From definitions of dual and exterior product of connections (Section 8.3), we have: Lemma 15.4. Let ∇ be a connection for E on M r S. If ∇ is s-trivial, Vl V0 ∗ then the family ( ∇∗ , . . . , ∇ ) is compatible with (15.13) on M rS. Exercise 15.1. Show the above lemma. In the above situation, we consider two localized classes. Thus let U0 = MrS and U1 a neighborhood of S and consider the covering U = {U0 , U1 }. Let ∇0 be an s-trivial connection for E on U0 and ∇1 an arbitrary connction for E on U1 . Localized top Chern class of E: Since cl (∇0 ) = 0, we have the localization clS (E, s) in H 2l (M, M rS; C), which is represented by the cocycle cl (∇∗ ) = (0, cl (∇1 ), cl (∇0 , ∇1 ))

in A2l (U, U0 ).

Note that the class clS (E, s) is in the image of H 2l (M, M r S; Z) → H 2l (M, M r S; C), provided that S is a subcomplex of real codimension greater than or equal to 2 with respect to some triangulation of M (cf. Theorem 10.13). Pl Vi ∗ Localized Chern character of λE ∗ : We set λE ∗ = i=0 (−1)i E . V ? Since ch∗ ( ∇∗0 ) = 0 by Lemma 15.4, we have the localization ch∗S (λE ∗ , s) in H ∗ (M, M rS; C), which is represented by the cocycle V? V? V? V? ch∗ ( ∇∗∗ ) = (0, ch∗ ( ∇∗1 ), ch∗ ( ∇∗0 , ∇∗1 )) in A∗ (U, U0 ), V? ∗ Vl V0 ∗ where ∇ν denote the family of connections ( ∇∗ν , . . . , ∇ν ) on Uν , for ν = 0, 1.

508

Complex Analytic Geometry

Inverse Todd class of E: We also have the inverse Todd class td−1 (E), which is represented by the cocycle td−1 (∇∗ ) = (td−1 (∇0 ), td−1 (∇1 ), td−1 (∇0 , ∇1 ))

in A∗ (U).

Note that the class td−1 (E) is in the image of the canonical monomorphism H ∗ (M ; Q) → H ∗ (M ; C). In order to apply Proposition 15.6, let ρ : R × U01 → U01 be the ˜ = (1 − t)ρ∗ ∇0 + tρ∗ ∇1 for projection and we consider the connection ∇ V? ˜ ∗ Vl ˜ ∗ V0 ˜ ∗ ∗ ρ E. Let ∇ denote the family of connections ( ∇ ,..., ∇ ). Let ρ0 : [0, 1] × U01 → U01 be the restriction of ρ. We also have the cup product `

A∗ (U, U0 ) × A∗ (U) −→ A∗ (U, U0 ). Theorem 15.3. In the above situation, we have V? ch∗ ( ∇∗∗ ) = cl (∇∗ ) ` td−1 (∇∗ ) + Dτ

in A∗ (U, U0 ),

˜ · d td−1 (ρ∗ ∇1 , ∇)). ˜ where τ = (0, 0, τ01 ), τ01 = ρ0∗ (cl (ρ∗ ∇0 , ∇) Proof.

We claim that (cl · td−1 )(∇∗ ) is in A∗ (U, U0 ) and that V? ch∗ ( ∇∗∗ ) = (cl · td−1 )(∇∗ ) in A∗ (U, U0 ).

(15.14)

Then the theorem follows from Proposition 15.6 (see also Remark 15.1). First, we have (cl · td−1 )(∇∗ ) = ((cl · td−1 )(∇0 ), (cl · td−1 )(∇1 ), (cl · td−1 )(∇0 , ∇1 )). We have (cl · td−1 )(∇0 ) = cl (∇0 ) · td−1 (∇0 ) = 0. By Theorem 15.2 and definition of difference forms, we have V? (cl · td−1 )(∇1 ) = cl (∇1 ) · td−1 (∇1 ) = ch∗ ( ∇∗1 ), ˜ · td−1 (∇)) ˜ (cl · td−1 )(∇0 , ∇1 ) = ρ0∗ (cl (∇) V ? ˜ ∗ ) = ch∗ (V? ∇∗0 , V? ∇∗1 ). = ρ0∗ ch∗ ( ∇ Hence we have (15.14) and the theorem.



Corollary 15.1 (Prototype of localized RR for embeddings). We have ch∗S (λE ∗ , s) = clS (E, s) · td(E)−1 .

Riemann-Roch Theorem

509

This is an essential ingredient for the localized Riemann-Roch theorem for embeddings. Remark 15.2. 1. The equality in Corollary 15.1 holds a priori in H ∗ (M, MrS; C). However, since the right-hand side is in H ∗ (M, MrS; Q), so is the left-hand side and the equality is in H ∗ (M, M rS; Q). 2. If we take the image of the above equality by the canonical morphism j ∗ : H ∗ (M, MrS; Q) → H ∗ (M ; Q), it becomes the formula of Theorem 15.1. Universal situation Let π : E → M be a C ∞ complex vector bundle of rank l, as above. We denote by Σ the image of the zero section. Then we have the Thom class ΨE in H 2l (E, E rΣ) and the Thom isomorphism ∼

T : H ∗ (M ) −→ H 2l+∗ (E, E rΣ), which may be expressed as T (α) = ΨE ·π ∗ α (cf. Sections 4.3 and 7.9). Since ΨE = cl (π ∗ E, s∆ ) (cf. Theorem 10.9), applying Corollary 15.1 to π ∗ E and s∆ , we have: Theorem 15.4 (Universal localized RR for embeddings). In the above situation, we have ch∗Σ (λπ∗ E ∗ , s∆ ) = ΨE · td(π ∗ E)−1 = T (td(E)−1 )

in H ∗ (E, E rΣ; Q).

Once we have a section s : M → E, we recover the equality in Corollary 15.1 by pulling back the above by s (cf. Theorem 10.10). As mentioned before the equality in Corollary 15.1 is an essential step in the proof of the Riemann-Roch theorem for embeddings. The next step is to identify clS (E, s) with the Thom class of a certain vector bundle. We may interpret Theorem 15.4 as the Riemann-Roch theorem for the embedding Σ ,→ E with td(E) the Todd class of the normal bundle E of Σ in E.

15.6

Riemann-Roch theorem for embeddings

Let M be a C ∞ manifold and N a C ∞ complex vector bundle of rank k on M . Also let s be a C ∞ section of N and V the zero set of s. Then we have

510

Complex Analytic Geometry

the localized top Chern class ckV (N, s) in H 2k (M, M rV ) and the localized Pk Vi ∗ Chern character ch∗V (λN ∗ , s) in H ∗ (M, M rV ), λN ∗ = i=0 (−1)i N . If we apply Theorem 15.3 in this situation, it already gives a prototype ˇ of the Riemann-Roch theorem for embeddings on the level of Cech-de Rham cocycles. On the cohomology level, it reads (cf. Corollary 15.1) ch∗V (λN ∗ , s) = ckV (N, s) · td(N )−1

in H ∗ (M, M rV ).

(15.15)

In the following, we examine the above equality more closely in the following three cases: (a) M is a complex manifold, X = (X, OX ) a complex space in M defined by a regular section (cf. Definition 11.32) and V = X the underlying variety. (b) M is a C ∞ manifold and V a closed submanifold of M such that its normal bundle NV admits the structure of a complex vector bundle. (c) M is a complex manifold and V a closed complex submanifold of M . We will see that, in each of these cases, ckV (N, s) may be interpreted as the Thom class of a subspace or a submanifold and that the right-hand side of (15.15) is the image of the inverse Todd class of some bundle by the Thom morphism. We also give an interpretation of the left-hand side in each case. These constitute a Riemann-Roch theorem for embeddings. I. Case (a) Let M be a complex manifold of dimension n and X = (X, OX ) the complex space in M defined by a regular section s of a holomorphic vector bundle N of rank k on M . Thus X is of dimension d = n−k. In view of Theorem 12.9, we give the following: Definition 15.4. The Thom class ΨX of X is defined by ΨX = ckX (N, s)

in H 2k (M, M rX).

Let U be a neighborhood of X in M with a deformation retraction r : U → X, for example, taking a triangulation K0 of M compatible with X, we may take the open star OK 0 (X) in the second barycentric subdivision K 0 of K0 as U . Note that H 2k+p (M, MrX) ' H 2k+p (U, UrX) by excision. We define the Thom morphism TX : H p (X) −→ H 2k+p (M, M rX)

by α 7→ ΨX ` r∗ (α).

Riemann-Roch Theorem

511

Then we have the following commutative diagram:

PX



/ H 2k+p (M, M rX)

TX

H p (X)

˘ 2d−p (X), H

v

∼

AM,X

where PX is the Poincar´e morphism, given by the cap product with X (cf. Section 13.2). Denoting by i : X ,→ M the embedding, we define the Gysin morphism i∗ : H p (X) → H 2k+p (M ) as the composition (cf. Section 13.2) j∗

T

X H p (X) −→ H 2k+p (M, M rX) −→ H 2k+p (M ).

Remark 15.3. If X is reduced, the above Thom class and Thom isomorphism coincide with those for V = X (cf. Theorem 12.9, Definition 13.1, (13.9) and (13.10)). If we set NX = N |X , the bundle r∗ NX admits a C ∞ structure and is isomorphic with N |U as C ∞ bundle (cf. Remark 3.3). Thus the right-hand side of (15.15) is written TX (td(NX )−1 ). As for the left-hand side, from the Koszul complex for (N, s), we have a sequence Vk Vk−1 ∗ ds ds ds ds 0 −→ OM ( N ∗ ) −→ OM ( N ) −→ · · · −→ OM (N ∗ ) −→ OM . The following is proved using the fact that s is a regular section: Lemma 15.5. The above sequence is exact. Since the image of ds : OM (N ∗ ) → OM is the ideal sheaf I of OM generated by the local components of s and OX = i−1 (OM /I ), where i : X → M is the embedding, we have the exact sequence of OM -modules Vk 0 −→ OM ( N ∗ ) −→ · · · −→ OM (N ∗ ) −→ OM −→ i∗ OX −→ 0. (15.16) In view of the above, we define (cf. (15.21) below) ch∗X (i∗ OX ) = ch∗X (λN ∗ , s).

512

Complex Analytic Geometry

Then from (15.15), we have: Theorem 15.5 (RR for embeddings I). In the case (a), we have: ch∗X (i! OX ) = TX (td(NX )−1 ) ch∗ (i! OX ) = i∗ (td(NX )−1 )

in H ∗ (M, M rX), in H ∗ (M ).

Remark 15.4. 1. For a complex vector bundle F on X, the complex Vk ∗ 0 −→ r∗ F ⊗ N −→ · · · −→ r∗ F ⊗ N ∗ −→ r∗ F −→ 0. is exact on U rX. Thus if we set γ(F ) = r∗ F ⊗ λN ∗ , then we have the localized Chern character ch∗X (γ(F )) and by (15.12), ch∗X (γ(F )) = ch∗ (r∗ F ) · ch∗X (λN ∗ , s). Thus we have: ch∗X (γ(F )) = TX (ch∗ (F ) · td(NX )−1 ) ch∗ (γ(F )) = i∗ (ch∗ (F ) · td(NX )−1 )

in H ∗ (M, M rX), in H ∗ (M ).

2. The above equalities hold a priori in the cohomology with C coefficients, however, since the right-hand sides of the both are in the cohomology with Q coeffitients, they hold in the cohomology with Q coeffitients. II. Case (b) Let M be a C ∞ manifold of dimension m and V a closed C ∞ submanifold of dimension d0 of M . We assume that the normal bundle p : NR,V → V admits the structure of a complex vector bundle of rank k, 2k = m − d0 . Thus we denote it by NV . Recall that (cf. Section 4.3) we have the Thom isomorphism ∼

TV : H p (V ) −→ H 2k+p (M, M rV ).

(15.17)

The Thom class ΨV of V is the image of [1] ∈ H 0 (V ) by TV . Let Z denote the image of the zero section of NV . By the tubular neighborhood theorem (Theorem 3.10), there exist a neighborhood U of V in M , a neighborhood W of Z in NV and a C ∞ diffeomorphism τ : U → W such that τ (V ) = Z and that (p ◦ τ )|V = 1V . We have isomorphisms ∼

H ∗ (M, M rV ) ' H ∗ (U, U rV ) ←− H ∗ (W, W rZ) ' H ∗ (NV , NV rZ). ∗ τ

The Thom class ΨV of V corresponds to the Thom class ΨNV of the bundle NV under the above isomorphism and the Thom isomorphism (15.17) cor∼ responds to the Thom isomorphism TNV : H p (Z) → H 2k+p (NV , NV r Z).

Riemann-Roch Theorem

513

Note that, if we denote by s∆ the diagonal section of the bundle p∗ NV on NV , its zero set is Z and we have ΨNV = ckZ (p∗ NV , s∆ ) (cf. Theorem 10.9). If we set N = r∗ NV , r = p ◦ τ : U → V , it is isomorphic with τ ∗ (p∗ NV |W ) (cf. Proposition 3.2. 2) and we have the commutative diagram N  U

∼

/ p∗ NV |W

∼ τ

(15.18)

 / W.

If we let s be the section of N corresponding to s∆ , its zero set is V and we have ΨV = ckV (N, s). Thus in this cases, (15.15) may be written ch∗V (λN ∗ , s) = TV (td(NV )−1 )

in H ∗ (M, M rV ).

(15.19)

For a complex vector bundle F on V , if we set γ(F ) = r∗ F ⊗ λN ∗ , as in the case (a) (cf. Remark 15.4. 1), we have the localized Chern character ch∗V (γ(F )), which is equal to ch∗ (r∗ F ) · ch∗V (λN ∗ , s). Combined with (15.19), we have: Theorem 15.6 (RR for embeddings II). In the case (b), for a complex vector bundle F on V , we have: ch∗V (γ(F )) = TV (ch∗ (F ) · td(NV )−1 ) ch∗ (γ(F )) = i∗ (ch∗ (F ) · td(NV )−1 )

in H ∗ (M, M rV ), in H ∗ (M ).

Remark 15.5. 1. A remark similar to the one in Remark 15.4. 2 applies to this case as well. 2. Taking a connection of F , we may write down the formula on the cocycle level. Case (c) is a special case of (b) and Theorem 15.6 holds as it is. To say more about the left-hand side, we prepare some materials. Real analytic functions ω Let M be a real analytic manifold. It is known that the sheaf CM of rings of real analytic functions on M is coherent.

514

Complex Analytic Geometry

ω If M is a complex manifold, the sheaf CM contains the sheaf OM of holomorphic functions as a subsheaf of rings. In the following, we omit the subscript M in the sheaf notation, if there is no fear of confusion. It is known that, if M is a coherent O-module, then C ω ⊗O M is a coherent C ω -module and that the assignment M 7→ C ω ⊗O M is an exact functor from the category of coherent O-modules to that of coherent C ω -modules. Let M be a complex manifold and V a closed complex submanifold of M . We denote by IV the ideal sheaf of V in M and by IZ the ideal sheaf of Z in NV , where Z is the image of the zero section of NV . We quote the following “complex tubular neighborhood theorem”, which is a refinement of Theorem 3.10 in this situation:

Proposition 15.8. There exist a neighborhood U of V in M , a neighborhood W of Z in NV and a real analytic isomorphism τ : U → W such that τ (V ) = Z, (p ◦ τ )|V = 1V and that τ −1 |NV,z ∩W is holomorphic for each z in V . Moreover, a map τ as above induces an isomorphism ∼

ω (τ, τ ∗ ) : (U, CUω ⊗OU IV ) −→ (W, CW ⊗OW IZ ).

See Definition 11.5 and the subsequent paragraph for what is meant by (τ, τ ∗ ) being an isomorphism. Characteristic classes of coherent sheaves Let M be a complex manifold and M a coherent OM -module. Suppose there exist, on M , a complex 0 −→ Er −→ · · · −→ E1 −→ E0 −→ 0

(15.20)

of real analytic vector bundles and an epimorphism C (E0 ) → C ⊗O M such that the sequence ω

ω

0 −→ C ω (Er ) −→ · · · −→ C ω (E1 ) −→ C ω (E0 ) −→ C ω ⊗O M −→ 0 is exact. We reffer to a sequence as above a locally free resolution of M by real analytic vector bundles. In this situation, we define the Chern character of M in H ∗ (M ) by r X ch∗ (M ) = ch∗ (ξ), ξ= (−1)i Ei . (15.21) i=0

If S is the support of M , it is an analytic variety in M and the complex (15.20) is exact on M rS. Thus we may define the localized class ch∗S (M ) in H ∗ (M, M rS).

Riemann-Roch Theorem

515

The definition above does not depend on the choice of the locally free resolution. Remark 15.6. If M is a coherent OM -module, M admits a locally free resolution by real analytic vector bundles on any compact set in M . III. Case (c) Let M be a complex manifold and V a closed complex submanifold of codimension k of M . In this case, the normal bundle p : NV → V is holomorphic and there exists a tubular neighborhood U with a deformation retraction r = p ◦ τ : U → V as in Proposition 15.8. Let i : V ,→ U ⊂ M and j : Z ,→ W ⊂ N denote the embeddings. As in the case (b), we have the Thom isomorphism (15.17) and the Thom class is given by ΨV = ckV (N, s). Here N is the bundle on U given by N = r∗ NV . Since r = p ◦ τ , N is a real analytic vector bundle which is isomorphic with p∗ NV |W , and s is the (real analytic) section corresponding to the diagonal section s∆ (cf. (15.18)). Since s∆ is a regular section of p∗ NV , as in the case (a) (let M , N and s be NV , p∗ N and s∆ , respectively, in (15.16)), we have the exact sequence Vk 0 −→ O( p∗ NV∗ ) −→ · · · −→ O(p∗ NV∗ ) −→ O −→ O/IZ −→ 0, where O = OW . By Proposition 15.8, after tensoring the above sequence with C ω , we have a resolution of i! OV by the Kozsul complex for (N, s). Thus (15.15) may be written ch∗V (i! OV ) = TV (td(NV )−1 )

in H ∗ (M, M rV ).

By Proposition 15.8, we may replace λN ∗ with i! OV and for a holomorphic vector bundle F on V , we may replace γ(F ) with i! F . Thus we have: Theorem 15.7 (RR for embeddings III). In the case (c), for a holomorphic vector bundle F on V , we have: ch∗V (i! F ) = TV (ch∗ (F ) · td(NV )−1 ) ch∗ (i! F ) = i∗ (ch∗ (F ) · td(NV )−1 )

in H ∗ (M, M rV ), in H ∗ (M ).

Remark 15.7. 1. A remark similar to the one in Remark 15.4. 2 applies to this case as well. 2. If V is compact, we may take a coherent OV -module as F (cf. Remark 15.6).

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Complex Analytic Geometry

From the exact sequence 0 −→ T V −→ T M |V −→ NV −→ 0, we have td−1 (NV ) = i∗ td−1 (M ) · td(V ) so that we have: Corollary 15.2. In the situation of Theorem 15.7, it holds ch∗ (i! F ) · td(M ) = i∗ (ch∗ (F ) · td(V )).

15.7

Grothendieck-Riemann-Roch Theorem

The original Grothendieck-Riemann-Roch theorem is stated and proved for proper morphisms of non-singular quasi-projective varieties (over an arbitrary algebraically closed field K). Here we make interpretations in our context (in the case K = C). The key point in this is what is called the “GAGA principle”. GAGA principle a Let M be a quasi-projective manifold. It is equipped with the sheaf OM of rings of algebraic functions, which is a subsheaf of the sheaf OM of holomorphic functions. Recall that (cf. Sections 2.2 and 11.1), if we denote by MZ and O r , respectively, the manifold M with Zarisky topology and the sheaf of rings of regular functions on MZ , then O a = 1−1 O r , where 1 : M → MZ is the identity. For a coherent O r -module F , we set F a = 1−1 F and F h = O ⊗O a F a which is a coherent O-module, as taking tensor product is right exact.

Theorem 15.8. Let M be a projective algebraic manifold. 1. For a coherent O r -module F , there is a canonical isomorphism ∼ H p (MZ ; F ) −→ H p (M ; F h ). 2. Let F and G be coherent O r -modules. Then every morphism F h → G h comes from a unique morphism F → G . 3. For every coherent O-module M , there is a coherent O r -module F such that F h is isomorphic with M . Moreover such a sheaf F is determined uniquely by M up to isomorphisms.

Riemann-Roch Theorem

517

K-group of projective algebraic manifolds Let M be a projective algebraic manifold of dimension n. We denote by K h (M ) and Kh (M ) the K-groups of holomorphic vector bundles and of coherent O-modules on M , respectively. There is a natural morphism K h (M ) → Kh (M ), which is induced by F 7→ O(F ). Theorem 15.9. The above morphism is an isomorphism: ∼

K h (M ) −→ Kh (M ). This is proved using the fact that every coherent O r -module on MZ admits a locally free resolution by vector bundles and the GAGA principle. Thus we may define the Chern character ch∗ : Kh (M ) ' K h (M ) −→ K d (M ) ' K t (M ) −→ H ∗ (M ; Q). Let f : M → M 0 be a holomorphic map of projective algebraic manifolds. We have a morphism f ∗ : K h (M 0 ) → K h (M ), which is induced from E 7→ f ∗ E for any holomorphic vector bundle E on M 0 . In view of Theorem 15.9, we also have a morphism f ∗ : Kh (M 0 ) → Kh (M ). Let f : M → M 0 be as above. If F is a coherent OM -module, then p R f∗ F are coherent OM 0 -modules and Rp f∗ F = 0 for p > dim M (cf. Theorem 11.14 and Remark 11.8 (1)). We have a morphism f! : Kh (M ) −→ Kh (M 0 ), P which is induced from F 7→ p (−1)p Rp f∗ F (cf. (11.16)). Note that this can also be described using coherent O r -modules and the GAGA principle. If we have two such maps f : M → M 0 and g : M 0 → M 00 , we have (g ◦ f )! = g! ◦ f! . This is proved by applying the spectral sequence E2p,q = Rp g∗ Rq f∗ F

(15.22) (E2p,q , H r )

with

and H r = Rr (g ◦ f )∗ F

(cf. Proposition 11.22). Suppose we have two maps fi : Mi → Mi0 , i = 1, 2. Then there is a morphism Kh (M1 ) ⊗ Kh (M2 ) −→ Kh (M1 × M2 ), given by ξ1 ⊗ ξ2 7→ ξ1 × ξ2 = p∗1 ξ1 ⊗ p∗2 ξ2 . For the map (f1 , f2 ) : M1 × M2 → M10 × M20 , we have (f1 , f2 )! (ξ1 × ξ2 ) = (f1 )! ξ1 × (f2 )! ξ2 .

(15.23)

518

Complex Analytic Geometry

For the projective space PN , we set O(n) = OPN (HN ⊗n ). We quote: Theorem 15.10. 1. Kh (PN ) is generated by the classes of O(n), 0 ≤ n ≤ N . We also have   N +n χ(PN ; O(n)) = . N 2. For any projective algebraic manifold M , the morphism Kh (PN ) ⊗ Kh (M ) −→ Kh (PN × M ) is surjective. Grothendieck-Riemann-Roch Theorem Theorem 15.11. Let f : M → M 0 be a holomorphic map of projective algebraic manifolds. Then, for every ξ in Kh (M ), ch∗ (f! ξ) · td(M 0 ) = f∗ (ch∗ (ξ) · td(M )) Proof.

in H ∗ (M 0 ; Q).

The proof goes as follows:

Step I. Suppose we have maps f : M → M 0 and g : M 0 → M 00 as above. If the formula holds for f and g, then it also holds for g ◦ f . Indeed, this follows from (g ◦ f )∗ = g∗ ◦ f∗ and (15.22). Step II. Let f : M → M 0 be as in the statement of the theorem. Denoting by i : M ,→ PN an embedding, we may express f as the composition f˜

p2

M −→ PN × M 0 −→ M 0 , where f˜(z) = (i(z), f (z)) and p2 is the projection onto the second factor. Note that f˜ is a closed embedding. We have already proved the formula for f˜ (cf. Corollary 15.2), so we are left to prove it for the projection p2 . Step III. Suppose we have two maps fi : Mi → Mi0 , i = 1, 2. If the formula hold for (fi , ξi ), then it hold for (f1 × f2 , ξ1 × ξ2 ). This follows from (15.23).

Riemann-Roch Theorem

519

Since we may write p2 : PN × M 0 → M 0 as p × 1M 0 : PN × M 0 → {pt} × M 0 , we are left to prove the formula for the case p : PN → {pt} and ξ = O(n), i.e., to prove ch∗ (p! O(n)) · td({pt}) = p∗ (ch∗ (O(n)) · td(PN ))

in H ∗ ({pt}; Q).

∼

Step IV. Note that H ∗ ({pt}; Q) = H 0 ({pt}; Q) → Q, the isomorphism is being given by the composition ε∗ P{pt} of the Poincar´e isomorphism for the point and the augmentation ε∗ : H0 ({pt}; Q) → Q. We compute the images of the both sides by ε∗ P{pt} . In the left-hand side, td({pt}) = 1 and Ri p∗ O(n) is the locally free sheaf on {pt} of rank dim H i (PN ; O(n)). Thus ε∗ P{pt} (LHS) = χ(PN ; O(n)). On the other hand, ε∗ P{pt} (RHS) = ε∗ p∗ PPN ([ch∗ (O(n)) · td(PN )]N ) = ε∗ PPN ([ch∗ (O(n)) · td(PN )]N ) = hPN , [ch∗ (O(n)) · td(PN )]N i, where the first equality is by definition of p∗ , the second by (B.2) and the third by (B.19). Hence it amounts to prove hPN , [ch∗ (O(n)) · td(PN )]N i = χ(PN ; O(n)). From (9.22), we have c∗ (PN ) = (1 + γ)N +1 , γ = c1 (HN ). Thus N +1  γ . td(PN ) = 1 − e−γ On the other hand, ch∗ (O(n)) = enγ and the equality follows from Theorem 15.10. 1.  If we let M 0 = {pt} in Theorem 15.11, by a similar computation as in Step IV above, we have: Theorem 15.12 (Hirzebruch-Riemann-Roch theorem). Let M be a projective algebraic manifold of dimension n. For a holomorphic vector bundle E on M , we have χ(M ; O(E)) = hM, [ch∗ (E) · td(M )]n i, where [ ]n denotes the component in H 2n (M ). In particular, let M = C be a compact Riemann surface and E = LD the line bundle determined by a divisor D on C. Then, in the left-hand side, dim H 0 (C; O(LD )) = l(D) and dim H 1 (C; O(LD )) = dim H 0 (C; Ω 1 (LD ∗ )) = i(D), by the Kodaira-Serre duality (cf. (9.13)).

520

Complex Analytic Geometry

On the other hand, we have ch∗ (LD ) = 1+c1 (LD ) and td(C) = 1+ 12 c1 (C). Thus the right-hand side is given by hC, c1 (LD ) + 21 c1 (C)i, which is equal to d − g + 1 so that we recover the classical formula (15.1). Notes As to Kodaira’s work on the Riemann-Roch theorem for compact complex surfaces, we refer to [Kodaira (1951)]. We list [Atiyah (1967); Hirzebruch (1966)] as references for Section 15.2. Theorem 15.1 is proved in [Hirzebruch (1966)] and its expression on the form level (Theorem 15.2) in [Harvey and Lawson (1993)]. Theorem 15.3 ˇ is a localized version of that in terms of Cech-de Rham cocycles and is an essential ingredient for the localized Riemann-Roch theorem for embeddings in our framework, as noted in the text. The notion of local Chern character and class appears in [Atiyah and Hirzebruch (1961, 1962)] as the authors introduce “generalized difference bundles”. It is then formulated explicitly in [Baum, Fulton and MacPherson (1975); Iversen (1976)]. We deal with the problem, in Section 15.4, using families of connections compatible with sequences of vector bundles and ˇ defining the classes in the relative Cech-de Rham cohomology. The classes we define have all the necessary properties and should coincide with the ones in the literatures above. Hence they are in the cohomology H ∗ (M, MrS; Q) with Q coefficient. Theorem 15.4 is proved in [Iversen (1976)] by different approaches. Our treatment of local Chern character and class as well as the proof of Theorem 15.3 are taken from [Suwa (2000b)]. Also the Chern-Weil theory for virtual bundles that are necessary for these is taken from [Suwa (1998)]. See (18.D) Corollary in [Matsumura (1980)] as to Lemma 15.5 and (15.16). For details of the paragraphs “Real analytic functions” and “Characteristic classes of coherent sheaves”, we refer to [Atiyah and Hirzebruch (1961, 1962)]. Theorems 15.6 and 15.7 are proved in [Atiyah and Hirzebruch (1962)]. Although it is not stated explicitly, they are proved in localized form. In the algebraic category, the formulas are proved for a locally free OV -module on an LCI by analyzing the graph construction in [Baum, Fulton and MacPherson (1975)]. These formulas are also proved at the level of differential forms and currents in [Harvey and Lawson (1993)], See also [Bismut (1998)]. The Riemann-Roch theorem for embeddings in localized form is used to compute residues of singular holomorphic distributions or foliations, see [Suwa (2012)].

Riemann-Roch Theorem

521

For the paragraph “GAGA principle”, we refer to [Serre (1956)] and for “K-group of projective algebraic manifolds” to [Borel et Serre (1958)], where the Grothendieck-Riemann-Roch theorem is proved. The Hirzebruch-Riemann-Roch theorem (Theorem 15.12) is due to [Hirzebruch (1966)]. As to the Riemann-Roch theorem for singular spaces, it is proved for quasi-projective varieties in [Baum, Fulton and MacPherson (1975)] and for complex spaces in [Levy (1987)]. Theorem 15.12 may be proved for an arbitrary compact complex manifold as a special case of the Atiyah-Singer index theorem (cf. [Atiyah and Singer (1963)] and subsequent papers). The theorem asserts that, for an elliptic differential operator on a compact manifold, the “analytic index” is equal to the “toplogical index”. For example, let M be an oriented C ∞ manifold of dimension m. Then we have the operator d : Ar (M ) → Ar+1 (M ). If we endow M with a Riemannian metric, we have the adjoint d∗ : Ar (M ) → Ar−1 (M ) (cf. Section 9.1). The operator M M D = d + d∗ : Ar (M ) −→ Ar (M ) r:even

r:odd

is shown to be elliptic. Suppose that M is compact. Then, on the one hand, the analytic index of D, which is dim Ker D − dim Coker D by definition, is L equal to dim Ker D − dim Ker D∗ , where D∗ = d∗ + d : r:odd Ar (M ) → L r r:even A (M ). Thus it is equal to M M dim Hr (M ) − dim Hr (M ) = χ(M ), r:even

r:odd

the Euler-Poincar´e characteristic of M (cf. Proposition 9.5). On the other hand, if m is even, it is shown that, using (5.6) and Theorem 15.1, the topological index is hM, e(M )i with e(M ) the Euler class of M . Thus we have the equality (5.18) for the case m is even. Note that, if m is odd, the both indices are zero. Likewise let M be a complex manifold of dimension n and E a holomorphic vector bundle on M . Then we have ∂¯ : Ap,q (M ; E) → Ap,q+1 (M ; E). If we endow M and E with Hermitian metrics, we have the adjoint operator ∂¯h∗ : Ap,q (M ; E) → Ap,q−1 (M ; E) with h the metric on E (cf. (9.11)). The operator M M D = ∂¯ + ∂¯∗ : A0,q (M ; E) −→ A0,q (M ; E) h

q:even

q:odd

is shown to be elliptic. If M is compact, the analytic index is the EulerPoincar´e characteristic χ(M ; O(E)) and the topological index is given by

522

Complex Analytic Geometry

hM, [ch∗ (E) · td(M )]n i. Thus the equality in Theorem 15.12 holds for an arbitrary compact complex manifold. We list [Freed (2021)] as a survey article on the Atiyah-Singer index theorem.

Appendix A

Commutative Algebra

In this appendix, we review some materials on commutative algebra necessary for our purpose. In Section A.1, we recall homological algebra and related topics and, in Section A.2, we discuss structures of rings and modules. In the following, we let R denote a commutative ring with unity 1, which may possibly be equal to 0, and deal with R-modules. In particular, if R = Z, the ring of integers, they are Abelian groups and if R = C, they are complex vector spaces. A morphism of R-modules is a map ϕ : M → M0 of R-modules compatible with the R-module structures. An injective morphism is called a monomorphism and a surjective one an epimorphism. An isomorphism is a bijective morphism ϕ. In this case, its inverse ϕ−1 is also an isomorphism. If there is an isomorphism between M and M0 , we write M ' M0 . Sometimes a morphism is called a homomorphism. In general, for a set X, 1X denotes the identity map of X.

A.1

Homological algebra

In the following, i, k, p, q and r will denote integers. Homology Definition A.5. A chain complex of R-modules is a collection {(Kp , dp )}, where for each p, Kp is an R-module and dp : Kp → Kp−1 is an R- morphism with the property dp ◦ dp+1 = 0.

523

524

Complex Analytic Geometry

Thus a chain complex is expressed by a sequence of R-morphisms dp+1

dp

· · · −→ Kp+1 −→ Kp −→ Kp−1 −→ · · · with Im dp+1 ⊂ Ker dp . A chain complex as above is written (K• , d), or simply K• . The morphisms dp are called boundary operators. We set Zp (K• ) = Ker dp and Bp (K• ) = Im dp+1 . An element in Kp , Zp (K• ) or Bp (K• ) is called, respectively, a p-chain, a p-cycle or a p-boundary. The p-th homology of K• is defined by Hp (K• ) = Zp (K• )/Bp (K• ), which inherits naturally an R-module structure. For a p-cycle c, we denote by [c] its class in Hp (K• ). If [c] = [c0 ], we write c ∼ c0 and say that they are homologous. Note that the above notions and namings came originally from topology (cf. Appendix B below). Chain morphisms: Let (K• , d• ) and (L• , d• ) be two chain complexes. A chain morphism ϕ• : K• → L• is a collection {ϕp } of morphisms ϕp : Kp → Lp such that ϕp−1 ◦ dp = dp ◦ ϕp . In this case ϕ• induces a morphism for each p: ϕ∗ : Hp (K• ) −→ Hp (L• ). The composition of two chain morphisms is defined in an obvious manner and its induced morphism on the homology is the composition of the induced morphisms. A sequence of morphisms ψ•

ϕ•

J• −→ K• −→ L• ψp

ϕp

is said to be exact if Jp → Kp → Lp is exact, i.e., Im ψp = Ker ϕp , for all p. The complex (K• , d• ) with Kp = 0 and dp = 0 for all p is called the zero complex and is denoted by 0. Proposition A.9. If the sequence ψ•

ϕ•

0 −→ J• −→ K• −→ L• −→ 0 is exact, it induces a long exact sequence ψ∗

ϕ∗

d

∗ · · · −→ Hp (J• ) −→ Hp (K• ) −→ Hp (L• ) −→ Hp−1 (J• ) −→ · · · .

Commutative Algebra

525

In the above, d∗ , which is referred to as the connecting morphism, assigns to a class [c] in Hp (L• ) the class [a] in Hp−1 (J• ) such that c = ϕp (b) and dp (b) = ψp−1 (a) with b in Kp . Let (K• , d) be a chain complex. A subcomplex of K• is a collection {Kp0 } 0 such that Kp0 is a sub-R-module of Kp and that dp (Kp0 ) ⊂ Kp−1 for all p. 0 00 In this case, {(Kp , dp )} is a chain complex. We set Kp = Kp /Kp0 . Since dp 00 induces a morphism Kp00 → Kp−1 , which we also denote by dp , K•00 is also a chain complex, called the quotient complex of K• by K•0 . In this case we have a short exact sequence of complexes 0 −→ K•0 −→ K• −→ K•00 −→ 0. Note that, in a sequence as in Proposition A.9, J• is isomorphic with a subcomplex of K• and L• with a quotient complex. Chain homotopy: Let (K• , d) and (L• , d) be two complexes and ϕ• and ϕ0• two chain morphisms of K• to L• . A chain homotopy between ϕ• and ϕ0• is a collection {hp } of morphisms hp : Kp → Lp+1 such that dp+1 ◦ hp + hp−1 ◦ dp = ϕp − ϕ0p . If there is a chain homotopy between ϕ• and ϕ0• , we write ϕ• ' ϕ0• . From the definition we have: Proposition A.10. If ϕ• ' ϕ0• , they induce the same morphism on the homology: ϕ∗ = ϕ0∗ : Hp (K• ) −→ Hp (L• ). A chain morphism ϕ• : K• → L• is said to be a chain equivalence if there is a chain morphism ψ• : L• → K• such that ψ• ◦ ϕ• ' 1K• and ϕ• ◦ ψ• ' 1L• . As a consequence of Proposition A.10, we have: Corollary A.3. If ϕ• : K• → L• is a chain equivalence, the induced morphism is an isomorphism: ∼

ϕ∗ : Hp (K• ) −→ Hp (L• ). Cohomology As a notion dual to chain complex, we have a cochain complex of R-modules. It is a collection {(K p , dp )}, where for each p, K p is an R-module and

526

Complex Analytic Geometry

dp : K p → K p+1 is a morphism with the property dp ◦ dp−1 = 0. Thus it is expressed by a sequence of morphisms dp−1

dp

· · · −→ K p−1 −→ K p −→ K p+1 −→ · · · with Im dp−1 ⊂ Ker dp . It is also written (K • , d), or simply K • . The morphisms dp are called coboundary operators. We set Z p (K • ) = Ker dp and B p (K • ) = Im dp−1 . An element in K p , Z p (K • ) or B p (K • ) is called, respectively, a p-cochain, a p-cocycle or a p-coboubdary. The p-th cohomology of K • is defined by H p (K • ) = Z p (K • )/B p (K • ). It will be also denoted by Hdp (K • ), if we wish to make the coboudary operator d explicit. For a p-cocycle u, we denote by [u] its class in H p (K • ). If [u] = [u0 ], we write u ∼ u0 and say that they are cohomologous. Morphisms of cochain complexes are defined similarly as for morphisms of chain complexes and we also have: Proposition A.11. If the sequence ψ•

ϕ•

0 −→ J • −→ K • −→ L• −→ 0 is exact, it induces a long exact sequence ψ∗

ϕ∗

d∗

· · · −→ H p (J • ) −→ H p (K • ) −→ H p (L• ) −→ H p+1 (J • ) −→ · · · . In the above, d∗ assigns to a class [u] in H p (L• ) the class [w] in H p+1 (J • ) such that u = ϕp (v) and dp (v) = ψ p+1 (w) with v in K p . A subcomplex of K • is a collection K 0• = {K 0p } such that K 0p is a sub-R-module of K p and that dp (K 0p ) ⊂ K 0p+1 . In this case, (K 0p , dp ) is a complex. We set K 00p = K p /K 0p . Since dp induces a morphism K 00p → K 00p−1 , which we also denote by dp , K 00• is also a complex, called the quotient complex. In this case we have a short exact sequence of complexes 0 −→ K 0• −→ K • −→ K 00• −→ 0. The notion of homotopy between two cochain morphisms is also defined as in the case of homology. Thus let (K • , d) and (L• , d) be two complexes and ϕ• and (ϕ0 )• two cochain morphisms of K • to L• . A cochain homotopy between ϕ• and (ϕ0 )• is a collection {hp } of morphisms hp : K p → Lp−1 such that dp−1 ◦ hp + hp+1 ◦ dp = ϕp − (ϕ0 )p .

Commutative Algebra

527

If there is a cochain homotopy between ϕ• and (ϕ0 )• , we write ϕ• ' (ϕ0 )• . In this case, they induce the same morphism on the cohomology: ϕ∗ = (ϕ0 )∗ : H p (K • ) −→ H p (L• ). A cochain morphism ϕ• : K • → L• is said to be a cochain equivalence if there is a cochain morphism ψ • : L• → K • such that ψ • ◦ ϕ• ' 1K • and ϕ• ◦ ψ • ' 1L• . In this case, the induced morphism is an isomorphism: ∼

ϕ∗ : H p (K • ) −→ H p (L• ). Dual complex: If K• is a chain complex, we may construct an associated cochain complex. Thus we set K p = Hom(Kp , R), the R-module of morphisms of Kp to R. We define the Kronecker product h , i as an R-bilinear form Kp × K p −→ R

given by (c, u) 7→ hc, ui = u(c).

The coboundary operator dp : K p → K p+1 is defined as the transpose of dp : hc0 , dp ui = hdp c0 , ui

for u ∈ K p and c0 ∈ Kp+1 .

(A.1)

Then we see that dp ◦ dp−1 = 0 so that (K • , d) is a cochain complex, called the complex dual to K• . Note that the Kronecker product descends to homology and cohomology, i.e., the R-bilinear form Hp (K• ) × H p (K • ) −→ R

given by ([c], [u]) 7→ h[c], [u]i = hc, ui

is well-defined. If ϕ• : K• → L• is a chain morphism, we have the transpose cochain morphism ϕ• : L• → K • , which is defined by ϕp (v) = v ◦ ϕp , in other words, hc, ϕp (v)i = hϕp (c), vi

for c ∈ Kp and v ∈ Lp .

Note that the above relation descends to homology and cohomology. For two morphisms ϕ• and ϕ0• , a homotopy between ϕ• and ϕ0• induces a homotopy between ϕ• and (ϕ0 )• . Acyclicity: Let K • be a complexe with K p = 0 for p < 0. We say that K • is acyclic if H p (K • ) = 0 for p ≥ 1. The following is not difficult to see: Proposition A.12. If there is a collection {hp } of morphisms hp : K p → K p−1 such that dp−1 ◦ hp + hp+1 ◦ dp = 1K p •

then K is acyclic.

for p ≥ 1,

528

Complex Analytic Geometry

Spectral sequences Filtration: Let M be an R-module. A (decreasing) filtration on M is a family (F p M)p of sub-R-modules of M with F p M ⊃ F p+1 M. We set M GM = Gp M, Gp M = F p M/F p+1 M p

and call it the graded module associated with (F p M). For a ∈ F p M, we denote by [a]p its class in Gp M. Let k0 be a non-negative integer. Definition A.6. A spectral sequence is a system (Ekp,q , H r ) as follows: = 0 if (1) for each p, q and k ≥ k0 , Ekp,q is an R-module such that Ekp,q 0 p < 0 or q < 0, p,q p+k,q−k+1 (2) there is a morphism dp,q satisfying k : Ek → Ek p−k,q+k−1 dp,q = 0, k ◦ dk

(3) there is an isomorphism p,q p−k,q+k−1 Ek+1 ' Ker dp,q , k / Im dk p,q thus, if k > max{p, q + 1}, then Ekp,q ' Ek+1 ' · · · , which is denoted p,q by E∞ , (4) for each r, H r is a filtered R-module such that H r = 0 for r < 0 and that the filtration satisfies F p H r = H r for p ≤ 0 and F p H r = 0 for p ≥ r + 1, i.e.,

H r = F 0 H r ⊃ F 1 H r ⊃ · · · ⊃ F r+1 H r = 0, p,q (5) there is an isomorphism E∞ ' Gp H p+q . p,q p,q Note that from (3), if Ekp,q = 0, then Ek+1 = · · · = E∞ = 0, in p,q particular if p < 0 or q < 0, Ek = 0 for all k ≥ k0 . Considering the case q = 0, if k1 ≥ max{2, k0 }, there is a sequence of epimorphisms p,0 Ekp,0 −→ Ekp,0 −→ · · · −→ E∞ ' F pH p. 1 1 +1

(A.2)

Also considering the case p = 0, if k1 ≥ max{1, k0 }, there is a sequence of monomorphisms 0,q H q /F 1 H q ' E∞ −→ · · · −→ Ek0,q −→ Ek0,q . 1 +1 1

These are called edge morphisms.

(A.3)

Commutative Algebra

529

Suppose Ekp,q = 0 for q > 0. Then it can be easily checked that the 1 morphisms in (A.2) are isomorphisms and that F p H p = H p . Likewise if Ekp,q = 0 for p > 0, the morphisms in (A.3) are isomorphisms and F 1 H q = 1 0. Thus we have: Proposition A.13. 1. Let k1 ≥ max{2, k0 }. If Ekp,q = 0 for q > 0, then 1 r,0 r Ek1 ' H for every r. 2. Let k1 ≥ max{1, k0 }. If Ekp,q = 0 for p > 0, then Ek0,r ' H r for every r. 1 1 Definition A.7. We say that a spectral sequence (Ekp,q , H r ) degenerates at Ek1 if dp,q k = 0 for every (p, q) and k ≥ k1 . p,q Thus in this case, we have Ekp,q ' · · · ' E∞ ' Gp H p+q . For example, 1 in the situations of Proposition A.13, the spectral sequence degenerates at Ek1 .

Filtered complexes: Let (K • , d) be a complex of R-modules with K r = 0 for r < 0. A filtration on the complex K • is a family (F p K • ) of subcomplexes with F p K • ⊃ F p+1 K • . Thus for each r, (F p K r ) is a filtration on K r and d(F p K r ) ⊂ F p K r+1 . We assume that F p K r = K r , for p ≤ 0 and F p K r = 0, for p ≥ r + 1.

(A.4)

We have the quotient complex Gp K • = F p K • /F p+1 K • . For each r, by (A.4), Gp K r = 0 for p < 0 or p > r. The filtration (F p K • ) induces a filtration on the cohomology H r (K • ): F p H r (K • ) = Z r (K • ) ∩ F p K r /B r (K • ) ∩ F p K r . Proposition A.14. For a filtered complex K • , there is a spectral sequence (Ekp,q , H r ) with k0 = 0, E0p,q = Gp K p+q , Proof.

E1p,q ' H p+q (Gp K • )

and

H r = H r (K • ).

For each triple (p, q, k) of non-negative integers, we set Zkp,q = { a ∈ F p K p+q | da ∈ F p+k K p+q+1 }, Bkp,q = (d(F p−k+1 K p+q−1 ) + F p+1 K p+q ) ∩ Zkp,q Ekp,q

=

and

Zkp,q /Bkp,q .

We verify the conditions in Definition A.6 successively. In the following, the class of a ∈ Zkp,q in Ekp,q will be denoted by [a]k .

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Complex Analytic Geometry

(1) We have Z0p,q = F p K p+q and B0p,q = F p+1 K p+q so that E0p,q = Gp K p+q ,

[a]0 = [a]p ,

(A.5)

and by (A.4), we have the condition (1) in Definition A.6. (2) Since dZkp,q ⊂ Zkp+k,q−k+1 and dBkp,q ⊂ Bkp+k,q−k+1 , the morphism p,q p+k,q−k+1 dp,q , k : Ek → Ek

[a]k 7→ [da]k ,

p−k,q+k−1 is well-defined and satisfies dp,q = 0. k ◦ dk p,q p,q (3) Since Zk+1 ⊂ Zkp,q , we have a morphism Zk+1 → Ekp,q , a 7→ [a]k . p,q p,q As a ∈ Zk+1 , we have dk [a]k = 0 and this is a morphism into Ker dp,q k . p,q p−k,q+k−1 Moreover, if a ∈ Bk+1 , we see that [a]k ∈ Im dk . Thus we have a well-defined morphism p,q p−k,q+k−1 Ek+1 −→ Ker dp,q , k / Im dk

[a]k+1 7→ [[a]k ],

(A.6)

which is an isomorphism. Indeed, for the surjectivity, take [a]k ∈ Ker dp,q k , p,q p+k,q−k+1 i.e., a ∈ Zk and da ∈ Bk . Then we may write da = da1 + b,

a1 ∈ F p+1 K p+q , b ∈ F p+k+1 K p+q+1 .

p,q We have a − a1 ∈ Zk+1 and [a − a1 ]k = [a]k as a1 is in fact in Bkp,q . p,q Thus [a − a1 ]k+1 7→ [[a]k ]. For the injectivity, take [a]k+1 ∈ Ek+1 such that [a]k ∈ Im dkp−k,q+k−1 . Then there exists a0 ∈ Zkp−k,q+k−1 such that p,q a − da0 ∈ Bkp,q . From this we see that a ∈ Bk+1 .

(4) We set H r = H r (K • ) for the condition (4). (5) If k > max{p, q + 1}, then Zkp,q = Z p+q (K • ) ∩ F p K p+q , Bkp,q = (B p+q (K • ) + F p+1 K p+q ) ∩ Zkp,q , Ekp,q = Zkp,q /Bkp,q and dk = 0. Thus p,q p,q and it is straightforward to see that there is an Ekp,q = Ek+1 = · · · = E∞ isomorphism p,q E∞ ' Gp H p+q (K • ),

[a]∞ ↔ [[a]]p ,

(A.7)

p,q = Z p+q (K • ) ∩ F p K p+q in F p H p+q (K • ). where [a] is the class of a ∈ Z∞ p,q r Therefore, (Ek , H ) as defined above is a spectral sequence. may be written Finally, we try to find E1p,q . Any element in Ker dp,q 0 [a]0 , where a ∈ Z0p,q with [da]0 = 0, i.e, da ∈ B0p,q+1 = F p+1 K p+q+1 . The last condition means a ∈ Z1p,q and in the step (3) above, we may set a1 = 0. p p In the identity (A.5), dp,q 0 [a]0 = [da]0 = [da] = d[a] , where the last d is p p+q the one induced on G K . Thus we have

E1p,q ' Ker d0 / Im d0 = H p+q (Gp K • ),

[a]1 ↔ [[a]p ].



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Proposition A.15. Let (Ekp,q , H r ) be the spectral sequence as constructed in Proposition A.14. If there exists k1 such that dp,q k1 = 0 for every (p, q), then it degenerates at k1 , i.e., dp,q = 0 for every (p, q) and k ≥ k1 . k Proof.

From (A.6) and the assumption, we have an isomorphism ∼

Ekp,q −→ Ekp,q , 1 +1 1

[a]k+1 7→ [a]k

for every (p, q). By this isomorphism, for (p + k1 + 1, q − k1 ), dk1 +1 [a]k1 +1 = [da]k1 +1 corresponds to [da]k1 = dk1 [a]k1 , which is 0 by the assumption. Thus dp,q k1 +1 = 0 for every (p, q). Continuing this argument, we have the proposition.  Double complexes: A double complex of R-modules is a collection K •,• = (K p,q , d0 , d00 ) such that (1) for each fixed q, K •,q = (K p,q , d0 ) is a complex, (2) for each fixed p, K p,• = (K p,q , d00 ) is a complex and (3) d0 ◦ d00 + d00 ◦ d0 = 0. L If we set K r = p+q=r K p,q and define d : K r → K r+1 by d = d0 + d00 , we get a complex (K • , d), called the associated single complex. In the following we assume that K p,q = 0 if p < 0 or q < 0. On the complex K • , there are two natural filtrations: M M 0 p • F K = K i,•−i and 00F q K • = K •−j,j . i≥p

j≥q

For these filtrations we have 0 p

G K • ' K p,•−p

and

00 q

G K • ' K •−q,q .

Theorem A.13. For a double complex K •,• , there are two spectral sequences (0Ekp,q , H r ) and (00Ekp,q , H r ) with E0p,q ' K p,q ,

0

E0p,q ' K q,p ,

00

r

r

E1p,q ' Hdq00 (K p,• ),

0

E1p,q ' Hdq0 (K •,p ),

00

E2p,q ' Hdp0 (Hdq00 (K •,• )),

0

E2p,q ' Hdp00 (Hdq0 (K •,• ))

00

and

•

H = H (K ). Moreover, the both sequences degenerate at E2 so that Hdp0 (Hdq00 (K •,• )) ' 0Gp H p+q (K • )

and

Hdp00 (Hdq0 (K •,• )) ' 00Gp H p+q (K • ).

Explicit correspondences in the above isomorphisms are given in the proof.

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Complex Analytic Geometry

Proof. The theorem basically follows from Proposition A.14, except for the last statement. We look at the first sequence, the second one being P i,p+q−i similar. For a ∈ 0F p K p+q , we write a = with ai,p+q−i ∈ i≥p a i,p+q−i K . We have 0Z0p,q = 0F p K p+q , 0B0p,q = 0F p+1 K p+q and E0p,q = 0Gp K p+q ' K p,q ,

0

[a]0 ↔ ap,q .

00 p,q In the last isomorphism, dp,q so that 0 corresponds to d on K p,q−1 = H p+q (0Gp K • ) ' Hdq00 (K p,• ). Ker dp,q 0 / Im d0

We may write (A.8), we have

0 p,q Z1

0 p

= {a ∈ F K

p+q

E1p,q ' Hdq00 (K p,• ),

0

00 p,q

|d a

(A.8)

= 0 } and, by (A.6) and

[a]1 ↔ [ap,q ].

(A.9)

We may also write 0 p,q Z2

= { a ∈ 0F p K p+q | d00 ap,q = 0 and d0 ap,q + d00 ap+1,q−1 = 0 }.

Recall that dp,q : 0E1p,q → 0E1p+1,q is given by [a]1 7→ [da]1 and in the 1 0 p+1,q isomorphism E1 ' Hdq00 (K p+1,• ), [da]1 ↔ [d0 ap,q + d00 ap+1,q−1 ] = 0 p,q 0 p,q [d a ] = d [a ], where the last d0 is the one induced by d0 on Hdq00 (K p,• ); d0 : Hdq00 (K p,• ) → Hdq00 (K p+1,• ). Thus, by (A.6) and (A.9), we have E2p,q ' Hdp0 (Hdq00 (K •,• )),

0

[a]2 7→ [[ap,q ]].

(A.10)

p,q p+2,q−1 As to the last statement, recall that dp,q is given by 2 : E2 → E2 p+2 0 p+2,q−1 •,• [a]2 7→ [da]2 . By the isomorphism E2 ' Hd0 (Hdq−1 (K )), [da]2 00 p+2,q−1 0 p+1,q−1 00 p+2,q−2 0 p+1,q−1 is sent to [[(da) ]] = [[d a +d a ]] = [[d a ]] = [d0 [ap+1,q−1 ]] = 0. Hence dp,q = 0 for every (p, q) and the sequence degen2 p,q erates at E2 by Proposition A.15. Thus 0E∞ ' · · · ' 0E2p,q , [a]∞ 7→ [a]2 , p+q • 0 p p+q 0 p,q (K ) ∩ F K . By (A.7) and (A.10), we have where a ∈ Z∞ = Z

G H p+q (K • ) ' Hdp0 (Hdq00 (K •,• )),

0 p

[[a]]p 7→ [[ap,q ]].



Direct limit Let X be a set. An order relation, or an ordering, in X is a relation ≤ in X with the following properties: (1) For every x in X, x ≤ x. (2) If x ≤ y and y ≤ z, then x ≤ z. (3) If x ≤ y and y ≤ x, then x = y.

Commutative Algebra

533

An ordered set is a set together with an order relation. If Y is a subset of an ordered set X, then Y is naturally an ordered set. We say that Y is cofinal, if for each x in X, there exists y in Y such that x ≤ y. An order relation is total, if it also has the following property: (4) For every x and y in X, either x ≤ y or y ≤ x. A directed set is an ordered set Λ such that for every λ and µ in Λ, there exists ν in Λ with λ ≤ ν and µ ≤ ν. Definition A.8. A direct system of R-modules on Λ is a system (Mλ , fµλ ), where for each λ in Λ, Mλ is an R-module and for each pair (λ, µ) with λ ≤ µ, fµλ : Mλ → Mµ is a morphism such that (1) fλλ = 1Mλ , (2) fνµ ◦ fµλ = fνλ

for λ ≤ µ ≤ ν.

F We introduce a relation ∼ in the disjoint union λ Mλ by saying that xλ ∼ xµ if there exists ν with λ, µ ≤ ν such that fνλ (xλ ) = fνµ (xµ ). Then it is an equivalent relation and the direct limit of the system is defined by: G M = lim Mλ = Mλ / ∼ . −→ λ∈Λ

λ∈Λ

For each xλ , we denote by [xλ ] its equivalence class. We may define the addition in M by [xλ ] + [xµ ] = [fνλ (xλ ) + fνµ (xµ )], taking ν with λ, µ ≤ ν. Multiplication of R on M is naturally defined and M becomes an R-module. For each λ, we have a canonical morphism fλ : Mλ → M, which assigns [xλ ] to xλ . If λ ≤ µ, we have fλ = fµ ◦ fµλ . If each Mλ is an R-algebra and fµλ is an R-algebra morphism, the direct limit M also admits the structure of an R-algebra. If Λ0 is a cofinal subset of a directed set Λ, then Λ0 is a directed set. Moreover, a direct system (Mλ , fµλ ) on Λ defines a direct system on Λ0 . There is a canonical isomorphism M0 = lim Mλ ' M = lim Mλ . −→ −→ λ∈Λ0

λ∈Λ

Taking direct limits is an exact functor (cf. the item “Abelian categories” below). Five lemma The proof of the following is rather straightforward:

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Complex Analytic Geometry

Lemma A.6. Let R be a commutative ring with unity and let M1 f1

 N1

/ M3

/ M2 f2

/ M4

f3

 / N2

f4

 / N3

 / N4

/ M5 f5

 / N5

be a commutative diagram of R-modules with exact rows. 1. If f1 is an epimorphism and f2 and f4 are monomorphisms, then f3 is a monomorphism. 2. If f5 is a monomorphism and f2 and f4 are epimorphisms, then f3 is an epimorphism. Modules Here we collect some basics on R-modules. An R-module M is finitely generated if there exists a finite number of elements x1 , . . . , xp in M that generate M, i.e., every element of M is written Pp i=1 ai xi , ai ∈ R. This is equivalent to saying that there is an epimorphism Rp −→ M −→ 0, where Rp denotes the direct sum of p copies of R. We think of the zero module as being finitely generated. An R-module M is free if it is isomorphic with a direct sum of copies of R. If M is a finitely generated free module, i.e., M ' Rr for some r, it is shown that r is uiquely determined by M. It is called the rank of M over R. In the case R is a field, it is the dimension of the vector space M over R. An R-module P is projective if for every epimorphism ϕ : M → N and every morphism f : P → N of R-modules, there exists a morphism g : P → M such that f = ϕ ◦ g: P g

M

{ ϕ

f

 / / N.

This is equivalent to saying that the functor Hom(P, · ) is exact (cf. Example A.1. 1 below). Free modules are projective. An R-module is projective if and only if it is a direct summand of a free module. For every R-module M, there exist a projective module P and an exact sequence P → M → 0. An R-module I is injective if for every monomorphism ϕ : N → M and every morphism f : N → I of R-modules, there exists a morphism g : M → I

Commutative Algebra

535

such that f = g ◦ ϕ: N/

ϕ

/M

f

 | I

g

.

This is equivalent to saying that the functor Hom( · , I) is exact. It is shown that, for every R-module M, there exist an injective module I and an exact sequence 0 → M → I. We quote: Proposition A.16. A Z-module, i.e., an Abelian group, M is injective if and only if it is divisible in the sense that, for every x ∈ M and n ∈ Z, n 6= 0, there exists x0 ∈ M such that nx0 = x. An R-module F is flat if for every monomorphism M0 → M of R-modules, F ⊗R M0 → F ⊗R M is a monomorphism. This is equivalent to saying that the functor F ⊗R · is exact. Projective modules are flat. An R-module M is torsion-free if ax = 0 (a ∈ R, x ∈ M) implies that either a is a zero divisor in R or x = 0. If R is an integral domain, this is equivalent to saying that ax = 0 implies a = 0 or x = 0. Flat modules are torsion-free. If R is a principal ideal domain (cf. Section A.2), the converse is also true. Abelian categories Here we recall some basics on Abelian categories, without going too much in details. Categories and functors: A category is a non-empty collection C of objects such that, for every pair of objects (A, B) in C, associated is a set Hom(A, B), called the set of morphisms of A to B, and, for every triplet (A, B, C), there is a map Hom(A, B) × Hom(B, C) → Hom(A, C), called the composition of morphisms, with the following properties: (1) The composition of morphisms is associative. (2) For every object A, Hom(A, A) contains an element 1A , called the identity morphism, which is a left and right unity for the composition. An isomorphism is a morphism that admits an inverse. For u ∈ Hom(A, B) and v ∈ Hom(B, C), we denote the composition by v ◦ u ∈ Hom(A, C). Let C and C 0 be categories. A covariant functor F : C → C 0 is a “function” that assigns to each object A ∈ C an object F (A) ∈ C 0 and to each

536

Complex Analytic Geometry

morphism u ∈ Hom(A, B) a morphism F (u) ∈ Hom(F (A), F (B)) so that F (1A ) = 1F (A) and F (v ◦ u) = F (v) ◦ F (u). Likewise a contravariant functor F : C → C 0 is defined by letting F (u) ∈ Hom(F (B), F (A)) and F G F (v ◦ u) = F (u) ◦ F (v). For a sequence C → C 0 → C 00 of functors, the composition G ◦ F is defined in an obvious way. Abelian categories: A category C is additive if, for every pair of objects (A, B), Hom(A, B) has the structure of an Abelian group and the composition is bilinear. We also assume that, for every pair (A, B), the “direct sum” A ⊕ B exists and that there exists an object A (“zero” of C) with 1A = 0. For a morphism u in C, the notions of “kernel” Ker u, “cokernel” Coker u, “image” Im u and “coimage” Coim u may be defined (they may not exist). There is a canonical morphism Coim u → Im u. A functor F : C → C 0 of an additive category to another is additive if F (u + v) = F (u) + F (v) for every u, v ∈ Hom(A, B). An Abelian category is an additive category C with the following properties: (1) For every morphism u in C, Ker u and Coker u exist. (2) For every morphism u in C, the canonical morphism Coim u → Im u is an isomorphism. Note that (1) implies the existence of coimage and image for every morphism. u v Let C be an Abelian category. A sequence of morphisms A → B → C in C is exact if Ker v = Im u. Let F : C → C 0 be an additive covariant functor of an Abelian category to another. The functor F is left exact if, for every exact sequence of the form 0 → A0 → A → A00 , the sequence 0 → F (A0 ) → F (A) → F (A00 ) is exact. It is right exact if, for every exact sequence A0 → A → A00 → 0, the sequence F (A0 ) → F (A) → F (A00 ) → 0 is exact. It is exact if it is both left and right exact. This is equivalent to saying that, for every exact sequence A → B → C, F (A) → F (B) → F (C) is exact. Likewise we may define exactness of contravariant functors. For example a contravariant additive functor F is left exact if, for every exact sequence A0 → A → A00 → 0, the sequence 0 → F (A00 ) → F (A) → F (A0 ) is exact. Example A.1. 1. Let R be a commutative ring with unity. We have the category Mod(R), whose objects are R-modules and whose morphisms are R-morphisms. For any R-module M, HomR (M, · ) : Mod(R) → Mod(R)

Commutative Algebra

537

is a covariant left exact functor and, for any R-module N, HomR ( · , N) : Mod(R) → Mod(R) is a contravariant left exact functor. Also, M ⊗R · : Mod(R) → Mod(R) and · ⊗R N : Mod(R) → Mod(R) are right exact covariant functors. 2. Let X be a topological space. We have the category Sh(X), whose objects are sheaves of Abelian groups on X and whose morphisms are sheaf morphisms. Taking the sections Γ (X; · ) : Sh(X) → Mod(Z) is a left exact covariant functor. See Remark 11.1 for more examples. Derived functors: Let C be an Abelian category. If there is an exact sequence of the form 0 → A0 → A → A00 → 0, we say that A0 is a sub-object of A and A00 a quotient of A. In this case we write A00 = A/A0 . Projective and injective objects in C are defined as in the case of R-modules. We say that C has enough projective or injective objects if every object in C is a quotient of a projective object or a sub-object of an injective object, respectively. A (cochain) complex in C is a sequence of morphisms dp−1

dp

· · · −→ K p−1 −→ K p −→ K p+1 −→ · · · such that dp ◦ dp−1 = 0 for all p. In this case Im dp−1 is a sub-object of Ker dp and H p (K • ) = Ker dp / Im dp−1 is the p-th cohomology of K • . Let F : C → C 0 be a covariant additive functor of an Abelian category to another. Suppose C has enough injective objects. Then, for each nonnegative integer p, we may define the p-th right derived functor Rp F : C → C 0 as follows. For any object A ∈ C, take an injective resolution, i.e., an exact sequence of the form 0 → A → I 0 → I 1 → · · · , with I p injective objects, p ≥ 0, and set Rp F (A) = H p (F (I • )). It is uniquely determined modulo isomorphisms. Moreover, if F is left exact, R0 F (A) = F (A) and an exact sequence 0 → A0 → A → A00 → 0 yields an exact sequence 0 → F (A0 ) → F (A) → F (A00 ) → R1 F (A0 ) → R1 F (A) → R1 F (A00 ) → · · · → Rp F (A0 ) → Rp F (A) → Rp F (A00 ) → Rp+1 F (A0 ) → · · · . If F is a contravariant functor, we may define the left derived functor Lp F . That is, for an object A, take an injective resolution as above. Then F (I • ) is a chain complex and we set Lp F (A) = Hp (F (I • )). The left derived functor of a covariant functor or the right derived functor of a contravariant functor is defined by taking projective resolutions. Namely, let P • → A → 0 be a projective resolution. If F is covariant, Lp F (A) = Hp (F (P • )) and if F is contravariant, Rp F (A) = H p (F (P • )).

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Example A.2. 1. For a commutative ring R with unity, the category Mod(R) of R-modules has enough projective and injective objects. For an R-module M, the p-th right derived functor of HomR (M, · ) is denoted by ExtpR (M, · ) and, for an R-module N, the p-th right derived functor of HomR ( · , N) is denoted by ExtpR ( · , N). For any pair (M, N), there is a canonical isomorphism between the ExtpR (M, N)’s defined in two ways. For example, for a fixed N, ExtpR (M, N) is defined by taking a projective resolution of M. We have (1) Ext0R (M, N) = HomR (M, N). (2) ExtpR (M, N) = 0 for p ≥ 1, if M is projective. (3) A short exact sequence 0 → M0 → M → M00 → 0 yields a long exact sequence 0 −→ HomR (M00 , N) −→ HomR (M, N) −→ HomR (M0 , N) −→ Ext1R (M00 , N) −→ Ext1R (M, N) −→ Ext1R (M0 , N) −→ · · · . Also, the p-th left derived functors of M ⊗R · and · ⊗R N are denoted by TorRp (M, · ) and TorRp ( · , N), respectively. For any pair (M, N), there is a canonical isomorphism between the TorRp (M, N)’s defined in two ways. 2. For a topological spaceX, the category Sh(X) of sheaves of Abelian groups on X has enough injective objects. The p-th derived functor of Γ (X; · ) is denoted by H p (X; · ). This is one way of defining the cohomology H p (X; S ) of a sheaf S on X. Grothendieck spectral sequence: The theory of spectral sequences may also be developed in Abelian categories. In particular, we quote the following: Theorem A.14. Let C, C 0 and C 00 be Abelian categories and let F : C → C 0 and G : C 0 → C 00 be additive covariant functors. Suppose that C and C 0 have enough injective objects, G is left exact and F transforms injective objects to G-acyclic objects, i.e., if I is injective, Rq G(F (I)) = 0 for q > 0. Then, for any object A in C, there is a spectral sequence (Ekp,q , H r ) in C 00 with E2p,q = Rp G(Rq F (A))

and

H r = Rr (G ◦ F )(A).

For an example, see Proposition 11.22.

Commutative Algebra

A.2

539

Commutative rings

In the following a ring means a commutative ring with unity 1, which may possibly be equal to 0. On the other hand, for a field we assume that 1 6= 0. Let a and b elements in a ring R. We say that b divides a and write b | a if a = bc for some c in R. Integral domains: A zero divisor in a ring R is an element a in R such that there is an element b 6= 0 in R with ab = 0. An integral domain is a ring R 6= 0 without non-zero zero divisors, i.e., if ab = 0, for a, b in R, then a = 0 or b = 0. If R is an integral domain, we may form the fraction field F (R) of R as follows. We introduce a relation ∼ in R × (R r {0}) by (a, b) ∼ (a0 , b0 ) if and only if ab0 = a0 b. It is an equivalence relation and the equivalence class of (a, b) is denoted by a/b. Then F (R) = R × (Rr{0})/ ∼ becomes a field with the addition a/b+c/d = (ad+bc)/bd and multiplication a/b · c/d = ac/bd. Identifying a with a/1, we may think of R as a subring of F (R). Unique factorization domains: We say that an element u in a ring R is a unit if there is an element v in R such that uv = 1. Let R be an integral domain. An element a in R is irreducible if a is not a unit and if the identity a = bc for elements b and c in R implies that either b or c is a unit. Note that 0 is not irreducible. We say that R is a unique factorization domain, or a UFD for short, if every element a in R which is not 0 or a unit can be expressed as a product of irreducible elements in R and the expression is unique up to the order and multiplications by units. Thus an element a as above may be expressed as mr 1 a = u pm 1 · · · pr ,

(A.11)

where u is a unit, the mi ’s are positive integers and the pi ’s are irreducible elements such that if i 6= j, pi is not a product of a unit and pj . An expression as above, which is unique up to the order and multiplications by units, is called the irreducible decomposition of a For example, the ring Z of integers is a UFD. Also a field is a UFD, since any of its elements is either 0 or a unit. Then ring On of germs of holomorphic functions at 0 in Cn is a UFD (cf. Theorem 1.11). Let R be an integral domain. An element a in R is prime if a is not 0 or a unit and if a | bc, for elements b and c in R, then either a | b or a | c. If a is prime, it is irreducible. If R is a UFD, every irreducible element is prime.

540

Complex Analytic Geometry

Theorem A.15 (Gauss’ lemma). If R is a UFD, so is the polynomial ring R[X] in the variable X. Let R be a UFD. For non-zero elements a and b there is always the greatest common divisor gcd(a, b), which is unique up to multiplication by units. We say that a and b are relatively prime if gcd(a, b) is a unit. Note that if either a or b is a unit, then a and b are relatively prime. Resultant: Let R be an integral domain. Pm Pn i j i=0 ai X and Q(X) = j=0 bj X , am bn R(P, Q) is an element in R defined by am · · · a0 .. . am R(P, Q) = bn · · · b0 .. . bn

For two polynomials P (X) = 6= 0, in R[X], their resultant

..

. ··· ..

. ···

a0 . b0

The discriminant D(P ) of P (X) is defined by D(P ) = (−1)

m(m−1) 2

1 R(P, P 0 ), am

(A.12)

where P 0 (X) = a1 + · · · + mam−1 X m−1 . Theorem A.16. 1. For P (X) and Q(X) as above, there exist A(X) and B(X) in R[X] such that A(X)P (X) + B(X)Q(X) = R(P, Q). 2. Suppose R is a UFD. Then P (X) and Q(X) are relatively prime if and only if R(P, Q) 6= 0. Integral extensions: Let R be a subring of a ring S. An element α of S is algebraic over R if there is a non-zero polynomial P (X) in R[X] with P (α) = 0. It is integral over R if we may take a monic polynomial as P (X) in the above. The ring S is integral over R if every element of S is integral over R. If S is finitely generated as an R-module, S is integral over R. Let K be a subfield of a field L. In this case an element α in L is algebraic over K if and only if it is integral over K and among the monic polynomials in K[X] as above, there is a unique polynomial with least degree, which is called the minimal polynomial of α.

Commutative Algebra

541

Let R be an integral domain and F (R) its fraction field. We say that R is normal if every element in F (R) which is integral over R is in R. For example, a UFD is normal. Proposition A.17. Let R be a normal integral domain and K = F (R) its fraction field. Let K ⊂ L be a field extension and α an element in L. Suppose α is integral over R, thus in particular α is algebraic over K. In this situation the minimal polynomial of α is in R[X]. Principal ideal domains: A principal ideal domain, a PID for short, is an integral domain R such that every ideal I is principal, i.e., I = (a) for some a in R. For example, Z and an arbitrary field are PID. Also the polynomial ring K[X] of one variable over a field K is a PID. A PID is a UFD. Proposition A.18. If M is a free module over a PID, every submodule of M is free. Noetherian rings: We say that a ring R is Noetherian if every ideal in R is finitely generated, i.e., if I is an ideal in R, there exist a finite number of elements x1 , . . . , xr in I such that every element x in I is written Pr x = i=1 ai xi with ai ∈ R. Proposition A.19. For a ring R, the following are equivalent: (1) R is Noetherian. (2) Every ascending sequence I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · · of ideals in R is stationary, i.e., Ik = Ik+1 = · · · , for some k. (3) Every non-empty set of ideals in R has a maximal element. For example, the ring Z is Noetherian. Also a field is Noetherian, since any of its ideals is either 0 or itself. The ring On is Noetherian (cf. Theorem 1.12). Theorem A.17 (Hilbert basis theorem). If R is Noetherian, so is R[X]. Maximal ideals: proper if I 6= R.

In the following we say that an ideal I in a ring R is

542

Complex Analytic Geometry

We say that an ideal m in R is maximal if m is proper and if there are no ideals I with m $ I $ R. This is equivalent to saying that the quotient R/m is a field. A ring with a unique maximal ideal is called a local ring. For example the ring On is a local ring with the ideal m of non-units in On as its unique maximal ideal (cf. Proposition 1.7). There is a canonical isomorphism On /m ' C. We quote: Theorem A.18 (Nakayama’s lemma). Let (R, m) be a local ring and M a finitely generated R-module. If mM = M, then M = 0. There is a canonical isomorphism M/mM ' R/m ⊗R M and it is an R/m-vector space. As a consequence of the above, we have: Proposition A.20. Let (R, m) and M be as above and let x1 , . . . , xr be elements in M. If the classes of x1 , . . . , xr span the vector space M/mM, then x1 , . . . , xr generate M as an R-module. Prime and primary ideals: For an ideal I in a ring R, its radical is defined by √ I = { a ∈ R | ak ∈ I for some positive integer k }, which is an ideal in R containing I. An ideal p in R is said to be prime if R/p is an integral domain, i.e., p √ is proper and ab ∈ p implies a ∈ p or b ∈ p. If p is prime, then p = p. Every maximal ideal is prime. The ideal {0} is prime if and only if R is an integral domain. If R is an integral domain, for an element p 6= 0 in R, p is prime if and only if (p) is prime. Thus, if R is a UFD, p is irreducible if and only if (p) is prime. We say that an element a in R is nilpotent if ak = 0 for some positive integer k. A proper ideal q in R is said to be primary if every zero divisor in R/q √ is nilpotent, i.e., ab ∈ q implies a ∈ q or b ∈ q. This is also equivalent to √ saying that ab ∈ q implies a ∈ q or b ∈ q or a, b ∈ q. √ If q is primary, then q is prime. We quote the following: Theorem A.19 (Primary decomposition theorem). If R is Noetherian, for every proper ideal I in R, there exist a finite number of primary ideals q1 , . . . , qr such that I = q1 ∩ · · · ∩ qr .

Commutative Algebra

543

Moreover, if I cannot be expressed as the intersection of ideals in a proper √ √ subset of {q1 , . . . , qr }, then the prime ideals q1 , . . . , qr are uniquely determined by I. In the above we call

√

√ q1 , . . . , qr the associated primes of I.

Fraction rings: A subset S of a ring R is said to be multiplicative if 1 ∈ S and if s, s0 ∈ S implies ss0 ∈ S. For such a set S, we construct the fraction ring, denoted by S −1 R, of R with respect to S as follows. We introduce a relation ∼ in R × S by (a, s) ∼ (a0 , s0 ) if and only if s00 (as0 − a0 s) = 0 for some s00 ∈ S. It is an equivalence relation and the equivalence class of (a, s) is denoted by a/s. Then S −1 R = R × S/ ∼ becomes a ring with the usual addition and multiplication of fractions. There is a natural morphism R → S −1 R given by a 7→ a/1. It is injective if and only if S does not contain any zero divisor in R. The above morphism gives S −1 R an R-module structure and it is a flat R-module (cf. Section A.1). If R is an integral domain, S = Rr{0} is multiplicative and S −1 R is the fraction field F (R) of R. In general, for an R-module M and a multiplicative set S in R, we may define S −1 M = { x/s | x ∈ M, s ∈ S } the same way as S −1 R. It is an S −1 R-module and the assignment x/s 7→ (1/s) ⊗ x induces a well-defined ∼ S −1 R-module isomorphism S −1 M → S −1 R ⊗R M. Let ϕ : R → R0 be a ring morphism. If S is a multiplicative set in R, S 0 = ϕ(S) is a multiplicative set in R0 . The morphism ϕ gives R0 an R-module structure and the assignment a0 /s 7→ a0 /ϕ(s) induces a ∼ −1 well-defined ring isomorphism S −1 R0 → S 0 R0 . Proposition A.21. Let I be an ideal in R. For a multiplicative set S in R, we denote by S 0 its image in R/I. Then there is a canonical ring isomorphism S −1 R/I · S −1 R ' S 0 Proof.

−1

(R/I)

given by [a/s] ↔ [a]/[s].

We have S

0 −1

(R/I) ' S −1 (R/I) ' S −1 R ⊗R (R/I) ' S −1 R/I · S −1 R,

where the last isomorphism is a consequence of the fact that S −1 R is a flat R-module.  If p is a prime ideal in R, then S = Rrp is multiplicative. In this case, S −1 R is denoted by Rp . It is a local ring with pRp as its maximal ideal. By Proposition A.21, there is a canonical field isomorphism Rp /pRp ' F (R/p).

544

Complex Analytic Geometry

More generally, let p1 , . . . , pr be prime ideals in R. Then the set S = Sr Tr i=1 (R r pi ) = R r i=1 pi is multiplicative in R. We set I = i=1 pi . In this situation, we have an isomorphism as in Proposition A.21. Note that S 0 does not contain any zero divisor in R/I in this case. In the case R = On , R/I and (S 0 )−1 (R/I) are the rings of germs of holomorphic and meromorphic functions, respectively, on the germ of the Sr varaity V (I) = i=1 V (pi ).

Tr

Height of an ideal: Let R be a ring with R 6= 0. A prime chain of length n is a sequence of n + 1 prime ideals in R: p0 % p1 % · · · % pn . The height of a prime ideal p, denoted by ht p, is the supremum of the lengths of the prime chains with p0 = p. The height ht I of a proper ideal I is the minimum of the heights of the prime ideals containing I. Krull dimension: The dimension of a ring R 6= 0, denoted by dim R, is the supremum of the heights of the prime ideals in R. It is also referred to as the Krull dimension of R. From definition, for every proper ideal I, dim R/I + ht I ≤ dim R. Example A.3. Let On be the ring of germs of holomorphic functions at 0 in Cn = {(z1 , . . . , zn )}. Then (z1 , . . . , zn ) % (z1 , . . . , zn−1 ) % · · · % (z1 ) % 0 is a prime chain of length n. In fact it can be shown that dim On = n. The annihilator of an R-module M is the ideal in R defined by Ann(M) = { a ∈ R | aM = 0 }. If M 6= 0, we define the (Krull) dimension of M by dim M = dim(R/ Ann(M)). If M = 0, we set dim M = −1. It is also denoted by dimR M. Regular sequences: Let R be a ring and M an R-module. An element a in R is a zero divisor on M if there is an element x 6= 0 in M such that ax = 0. A sequence a1 , . . . , ar of elements of R is M-regular if (1) a1 is not a zero divisor on M, (2) for each i = 2, . . . , r, ai is not a zero divisor on M/(a1 , . . . , ai−1 )M, (3) M = 6 (a1 , . . . , ar )M.

Commutative Algebra

545

If the ai ’s belong to an ideal I, we say that a1 , . . . , ar is an M-regular sequence in I. For example, if M = R = On , then z1 , . . . , zn is an On -regular sequence in the maximal ideal m. In fact, for any sequence ν1 , . . . , νn of positive integers, the sequence z1ν1 , . . . , znνn is an On -regular sequence in m. Depth of an ideal: Let R be a ring and M an R-module. Let I be an ideal in R with IM 6= M. The depth of I on M, denoted by depth(I; M), is the length of a maximal M-regular sequence in I. The depth of I on R is simply called the depth of I and is denoted by depth I. Let (R, m) and (S, n) be Noetherian local rings and ϕ : R → S a local morphism, i.e., a morphism with ϕ(m) ⊂ n. The morphism ϕ endows S with an R-module structure. We say that ϕ is finite if S is finitely generated over R. In this case, we have: dimS S = dimR S,

depth(n; S) = depth(m; S).

(A.13)

Cohen-Macaulay modules: Let (R, m) be a Noetherian local ring and M a finitely generated R-module. The R-module M is said to be CohenMacaulay, CM for short, if M = 0, or if M 6= 0 and depth(m; M) = dim M. The ring R is CM if it is CM as an R-module. For example, On is a CM ring. We have: Proposition A.22. Suppose a1 , . . . , ar is an M-regular sequence in m. Then M is CM if and only if M/(a1 , . . . , ar )M is. Proposition A.23. Let (R, m) be a CM local ring. Then: 1. For every proper ideal I in R, ht I = depth(I; R),

ht I + dim R/I = dim R.

2. A sequence a1 . . . , ar in m is R-regular if and only if ht(a1 . . . , ar ) = r. In this case, every associated prime of (a1 . . . , ar ) has height r. Note that, in general, a sequence being regular depends on the order of the elements, however, in the situation of Proposition A.23, it does not. From (A.13), we have: If ϕ : R → S is finite and if S is a CM ring, then S is a CM R-module. (A.14)

546

Complex Analytic Geometry

Projective dimension: Let R be a ring and M an R-module. The projective dimension of M, denoted by pdR M, is the minimum of the lengths of projective resolution of M. We quote the Auslander-Buchsbaum formula, which says that if R is a Noetherian local ring, M is finite over R and pd M is finite, depth(m; M) + pd M = depth m.

(A.15)

Determinantal ideals: Let R be a ring and f : Rm → Rn an Rmorphism, which may be represented by an n × m matrix. We assume that m ≥ n and denote by I(f ) the ideal generated by all the n × n minors of f . We assume I(f ) 6= R. Then we have: ht I(f ) ≤ m − n + 1.

(A.16)

We also have: If R is CM and if ht I(f ) = m − n + 1, then R/I(f ) is CM.

(A.17)

Analytic rings: Let On = C{z1 , . . . , zn } denote the ring of convergent power series in n variables. A ring R is an analytic ring if R ' On /I for some proper ideal I in On (for some n). The ring On /I is a Noetherian local ring with m/I as its unique maximal ideal, where m is the maximal ideal in On . The ring R inherits the structure of a Noetherian local ring via the above isomorphism. If X = (X, OX ) is a complex space, then OX,x is an analytic ring for each point x in X (cf. Section 11.5). For an ideal I in On , we denote by V (I) the germ at 0 (in Cn ) of the variety defined by I (cf. Section 2.3). If R = On /I, then dimR R = dim V (I). We denote by dimC the dimension of a complex vector space. By the Nullstellensatz (Theorem 2.8), dimR R = 0 if and only if dimC R is finite.

(A.18)

Let ϕ : R → S be a local morphism of analytic rings. The morphism ϕ induces C = R ⊗R R/mR → S ⊗R R/m. We say that ϕ is quasi-finite if this morphism makes S ⊗R R/m a finite-dimensional complex vector space. Clearly a finite morphism is quasi-finite. The coverse is also true: ϕ is finite if and only if it is quasi-finite.

(A.19)

Let (f, ϕ) : X = (X, OX ) → T = (T, OT ) be a morphism of complex spaces with T reduced so that we may think of OT as the sheaf of holomorphic functions (cf. Section 11.5). For each point x in X, ϕ induces a local morphism ϕx : OT,f (x) −→ OX,x , which endows OX,x with an OT,f (x) -module structure.

Commutative Algebra

547

We think of each point t in T as a complex subspace defined by the ideal It of holomorphic functions vanishing at t. The fiber Xt of (f, ϕ) over t is the inverse image of t by (f, ϕ). Thus it is the complex space with support Xt = f −1 (t) and structure sheaf OXt = i−1 t (OX /(OX · It )), where it : Xt ,→ X is the inclusion. For each point x in Xt , we have the exact sequence OX,x ⊗OT ,t mt −→ OX,x −→ OX,x ⊗OT ,t OT,t /mt −→ 0, where mt is the maximal ideal in OT,t . Thus noting that (It )t = mt , we have OXt ,x = OX,x /(OX,x · mt ) ' OX,x ⊗OT ,t OT,t /mt ' OX,x ⊗OT ,t C. Hence by (A.18) and (A.19), we see that x is an isolated point in Xt if and only if ϕx is finite.

(A.20)

Suppose now that f is a finite map (cf. Definition 2.10) so that every point x in X is isolated in Xt , t = f (x), and dimC OXt ,x is finite. Let t be a point in T . For a point x in Xt , we set ν(x) = dimC OXt ,x and P ν(t) = x∈Xt ν(x). We say that (f, ϕ) is flat if OX,x is a flat OT ,f (x) module, for every x in X. In this situation, we have: (f, ϕ) is flat if and only if ν(t) is locally constant.

(A.21)

Notes The contents of Section A.1 may be found in most of the text books dealing with homological algebra. For spectral sequences, in particular the ones in Abelian categories, we refer to [Grothendieck (1957)]. For Abelian categories, we refer to [Grothendieck (1957); Kashiwara and Schapira (1990)]. We list [Bourbaki (1961-65); de Jong and Pfister (2000); Eisenbud (1995); Hotta (1987); Matsumura (1980)] as general references for Section A.2. For (A.16), (A.19) and (A.21), see [Macaulay (1916)], [Narasimhan (1966)] and [Douady (1968)], respectively.

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Appendix B

Algebraic Topology

We summarize basic materials in algebraic topology relevant to our purposes. On the way we fix notation and orientation conventions. We adopt the singular homology theory as our reference theory and use cellular decompositions or triangulations of the space for actual computations. In the following, for a subset A in a topological space X, we denote by Int A and A the interior and the closure of A in X, respectively. A map between topological spaces means a continuous map, unless otherwise stated. We also let R denote a commutative ring with unity. In our case it will be mostly Z, Q, R or C.

B.1

Singular homology

We refer to Section A.1 for basics on homological algebra. Affine spaces: An affine space is a subset A of RN of the form A = v + V with v a point in RN and V a linear subspace of RN . In this case V is determined uniquely by A and v is determined modulo elements in V. The dimension of A is the dimension of V. For p + 1 points v0 , . . . , vp in A and Pp Pp p + 1 real numbers t0 , . . . , tp with i=0 ti = 1, the point i=0 ti vi is in A. Let v0 , . . . , vp be points in RN . Then the set p p n o X X A(v0 , . . . , vp ) = v ∈ RN v = t i vi , ti = 1 i=0

i=0

is the minimal affine space containing v0 , . . . , vp . In fact, letting V be the subspace spanned by v1 − v0 , . . . , vp − v0 , we may write A(v0 , . . . , vp ) = v0 + V. We say that v0 , . . . , vp are in general position if dim V = p. 549

550

Complex Analytic Geometry

A map f : A → Rk is affine if f ((1 − t)v + tv 0 ) = (1 − t)f (v) + tf (v 0 ), for any points v and v 0 in A. This is equivalent to saying that the map f˜ : V → Rk defined by f˜(v) = f (v0 + v) − f (v0 ), for a point v0 in A, is linear. It is sometimes called linear in the affine sense, or simply linear. Simplices: Let v0 , . . . , vp be points in general position in RN . The psimplex with vertices v0 , . . . , vp is the set p p o n X X N t i vi , ti = 1, ti ≥ 0 . (B.1) s= v∈R v= i=0

i=0

For a point v in s, the numbers (t0 , . . . , tp ) as in (B.1) are uniquely determined by v and are called the barycentric coordinates of v. For each i, ti Pp 1 is the barycentric coordinate of v relative to vi . The point p+1 i=0 vi is called the barycenter of s and is denoted by bs . Note that in the case p = 0, s = bs = {v0 }. Singular homology In Rp , the points P0 = (0, . . . , 0), P1 = (1, 0, . . . , 0), . . . , Pp = (0, . . . , 0, 1) are in general position. The standard p-simplex ∆p is the p-simplex with vertices P0 , . . . , Pp . Let X be a topological space. A singular p-simplex of X is a continuous map σ : ∆p −→ X. We denote by Sp (X; R) the free R-module generated by all the singular pL simplices of X, i.e., the direct sum Rσ with σ all the singular p-simplices of X. If there is no fear of confusion, we omit the coefficient ring R and denote Sp (X; R) by Sp (X). Thus an element c in Sp (X), called a singular P p-chain, is expressed uniquely as a finite sum c = aσ σ with aσ elements in R and σ singular p-simplices. For p > 0 and i = 0, . . . , p, we denote by εip : ∆p−1 → ∆p the affine map with ( Pj for j < i, εip (Pj ) = Pj+1 for j ≥ i. In short, εip is the operation that leaves Pi out of ∆p . For a singular p-simplex σ, p > 0, and i = 0, . . . , p, the i-th face di σ is the singular (p − 1)-simplex defined by di σ = σ ◦ εip : ∆p−1 → X. We extend di as an R-morphism of Sp (X) to Sp−1 (X) and define the boundary operator p X ∂ = ∂p : Sp (X) −→ Sp−1 (X) by ∂p = (−1)i di . i=0

Algebraic Topology

551

We set Sp (X) = 0 for p < 0 and ∂0 = 0. Then we see that ∂p ◦ ∂p+1 = 0 and (S• (X), ∂) is a chain complex (cf. Section A.1). By convention, we set Sp (∅) = 0 for all p. Definition B.9. The p-th singular homology Hp (X; R) of X with coefficients in R is the p-th homology of (S• (X), ∂). Remark B.8. There is a canonical isomorphism Sp (X; R) ' R⊗Z Sp (X; Z), through which the two R-modules may be identified. However, this relation does not directly descend to the homology level in general (cf. Theorem B.20 below). P P We define the augmentation ε: S0 (X) → R by ε( aσ σ) = aσ . Since ε◦∂ : S1 (X) → R is the zero map, it induces a morphism ε∗ : H0 (X; R) → R, which is also called the augmentation. Noting that we may regard a 0simplex as a point and a 1-simplex as a path in X, we see that, if X(6= ∅) is path connected, ε∗ is an isomorphism. F In general, if X = λ Xλ is the decomposition into path components, L then Hp (X; R) = λ Hp (Xλ ; R). Example B.4. If X = {x}, the space consisting of one point x, there is a unique singular p-simplex (the constant map) for each p and we see that ( R p = 0, Hp ({x}; R) ' 0 p 6= 0. In the following, the coefficients ring R in the homology is also omitted if there is no fear of confusion. Homotopy: Let f, f 0 : X → Y be two maps of topological spaces. We say f is homotopic to f 0 if there is a map F : X × I → Y such that F (x, 0) = f (x) and F (x, 1) = f 0 (x) for all x ∈ X, where I denotes the interval [0, 1]. In this case, we write f ' f 0 . The map F is called a homotopy between f and f 0 . Setting ft = F ( , t), we also say {ft }0≤t≤1 is a homotopy from f0 = f to f1 = f 0 . Let A be a subspace of X. A homotopy F as above is relative to A if F (x, t) = f (x) for all x ∈ A and t ∈ I. A map f : X → Y is a homotopy equivalence if there is a map g : Y → X such that g ◦ f ' 1X and f ◦ g ' 1Y . Such a map g is called a homotopy inverse of f .

552

Complex Analytic Geometry

Recall that a map f : X → Y is said to be proper if the inverse image of each compact set in Y by f is compact. Let f, f 0 : X → Y be two proper maps. A proper homotopy between f and f 0 is a map F : X × I → Y as above that is proper. Induced morphism: Let f : X → Y be a continuous map of topological spaces. It induces a chain morphism f• : S• (X) → S• (Y ) by assigning to σ : ∆p → X the composition f ◦σ : ∆p → Y . It in turn induces a morphism f∗ : Hp (X) −→ Hp (Y ). It is compatible with the augmentations, i.e., the following diagram is commutative: H0 (X)

f∗

/ H0 (Y ) ε

ε∗

∗ '  R.

(B.2)

If two maps f, f 0 : X → Y are homotopic, then f• , f•0 : S• (X) → S• (Y ) are chain homotopic so that f∗ = f∗0 : Hp (X) → Hp (Y ). If f : X → Y is a homotopy equivalence, f• : S• (X) → S• (Y ) is a chain equivalence so that ∼ f∗ : Hp (X) → Hp (Y ) is an isomorphism. Deformation retract: Let A be a subspace of X and i : A ,→ X the inclusion. We say that A is a retract of X if there is a continuous map r : X → A such that r ◦ i = 1A . Such a map r is called a retraction. Also, A is a deformation retract of X if there is a retraction r such that i◦r ' 1X . ∼ Thus in this case, i is a homotopy equivalence so that i∗ : Hp (A) → Hp (X) with r∗ its inverse. A space X is contractible if there is a point x0 in X such that {x0 } is a deformation retract of X. A proper deformation retraction is a deformation retraction that is proper. Relative homology: Let A be a subspace of X and i : A ,→ X the inclusion. The complex (S• (A), ∂) may naturally be thought of as a subcomplex of (S• (X), ∂) by identifying a singular simplex σ : ∆ → A with i ◦ σ : ∆ → X. Let (S• (X, A), ∂) denote the quotient complex, S• (X, A) = S• (X)/S• (A). Proposition B.24. For each p, Sp (X, A) is a free R-module with basis the classes of singular p-simplices σ of X with σ(∆p ) 6⊂ A.

Algebraic Topology

553

This can be seen from the decomposition Sp (X) = Sp (A) ⊕ Sp0 , where Sp0 is the free R-module generated by singular p-simplices σ of X with σ(∆p ) 6⊂ A. Note that S•0 is not naturally a subcomplex of S• (X) in general. We denote by Hp (X, A; R) the p-th homology of (S• (X, A), ∂) and call it the p-th relative homology of X mod A, or the p-th homology of the pair (X, A). Note that S• (X, ∅) = S• (X), thus Hp (X, ∅ ; R) = Hp (X; R). Denoting by j : (X, ∅) ,→ (X, A) the inclusion, we have the exact sequence of complexes: j•

i

• S• (X) −→ S• (X, A) −→ 0. 0 −→ S• (A) −→

(B.3)

From this, we have the long exact sequence (cf. Proposition A.9): j∗

i

∂

∗ ∗ · · · −→ Hp (A) −→ Hp (X) −→ Hp (X, A) −→ Hp−1 (A) −→ · · · .

(B.4)

We may also describe Hp (X, A) as follows. We set Zp (X, A) = { c ∈ Sp (X) | ∂c ∈ Sp−1 (A) }, Bp (X, A) = { c ∈ Sp (X) | c ∼ c0 , c0 ∈ Sp (A) }. Then we have Hp (X, A) ' Zp (X, A)/Bp (X, A). If we have a triple (X, A1 , A2 ) of spaces with A2 ⊂ A1 ⊂ X, we have the exact sequence of complexes j•

i

• 0 −→ S• (A1 , A2 ) −→ S• (X, A2 ) −→ S• (X, A1 ) −→ 0,

which yields a long exact sequence: i

j∗

∂

∗ · · · → Hp (A1 , A2 ) → Hp (X, A2 ) → Hp (X, A1 ) →∗ Hp−1 (A1 , A2 ) → · · · . (B.5) If f : (X, A) → (Y, B) is a continuous map of pairs of topological spaces, i.e., a continuous map f : X → Y with f (A) ⊂ B, it induces a chain morphism f• : S• (X, A) → S• (Y, B) by assigning to the class of σ the class of f ◦ σ. Thus it in turn induces a morphism

f∗ : Hp (X, A) −→ Hp (Y, B).

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Complex Analytic Geometry

Singular cohomology As a notion dual to the singular homology, we have the singular cohomology. Thus, for a topological space X, we set S p (X) = HomR (Sp (X); R) and define the coboundary operator δ = δ p : S p (X) −→ S p+1 (X) as the transpose of ∂ (cf. (A.1)): hc0 , δui = h∂c0 , ui

for u ∈ S p (X) and c0 ∈ Sp+1 (X).

Definition B.10. The p-th singular cohomology H p (X; R) is the p-th cohomology of the complex (S • (X), δ). Remark B.9. Let 1 denote the 0-cochain defined so that it assigns 1 to each singular 0-simplex and is extended R-linearly. Then we see that it is a cocycle and we have its class [1] in H 0 (X; R). If f : X → Y is a continuous map, it induces a cochain morphism f • : S • (Y ) → S • (X) and thus a morphism f ∗ : H p (Y ) −→ H p (X). If two maps f, f 0 : X → Y are homotopic, then f • , (f 0 )• : S • (Y ) → S • (X) are cochain homotopic so that f ∗ = (f 0 )∗ : H p (Y ) → H p (X). If f : X → Y is a homotopy equivalence, f • : S • (Y ) → S • (X) is a cochain equivalence ∼ so that f ∗ : H p (Y ) → H p (X) is an isomorphism. For a subspace A of X, we let S • (X, A) denote the cochain complex dual to S• (X, A). The p-th relative cohomology H p (X, A; R) of X mod A is the p-th cohomology of the complex (S • (X, A), δ). From (B.3), for each p, we have the exact sequence (cf. Example A.2. 1) jp

ip

0 −→ S p (X, A) −→ S p (X) −→ S p (A) −→ Ext1R (Sp (X, A), R). Thus we may identify S p (X, A) with { u ∈ S p (X) | hc, ui = 0 for c ∈ Sp (A) }.

(B.6)

Moreover Ext1R (Sp (X, A), R) = 0 by Proposition B.24 and ip is surjective. Thus we have a long exact sequence: δ∗

j∗

i∗

· · · −→ H p−1 (A) −→ H p (X, A) −→ H p (X) −→ H p (A) −→ · · · .

(B.7)

The Kronecker product on chains and cochains descends to a product on homology and cohomology, which we still call the Kronecher product and denote by the same symbol: Hp (X, A) × H p (X, A) −→ R,

h[c], [u]i = hc, ui.

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If f : (X, A) → (Y, B) is a continuous map of pairs of topological spaces, it induces a morphism f ∗ : H p (Y, B) −→ H p (X, A). In this situation, we have ha, f ∗ (β)i = hf∗ (a), βi

for a ∈ Hp (X, A), β ∈ H p (Y, B).

(B.8)

Excision A pair (X1 , X2 ) of subspaces of X is called an excisive couple if the inclusion S• (X1 ) + S• (X2 ) ,→ S• (X1 ∪ X2 ) is a chain equivalence. It is shown that, if Int X1 ∪ Int X2 = X, then (X1 , X2 ) is an excisive couple. If (X1 , X2 ) is an excisive couple, the morphism S• (X1 )/S• (X1 ∩X2 ) → S• (X1 ∪X2 )/S• (X2 ) induced by the inclusion (X1 , X1 ∩ X2 ) ,→ (X1 ∪ X2 , X2 ) is a chain equivalence so that there exist canonical isomorphisms ∼

Hp (X1 , X1 ∩ X2 ) −→ Hp (X1 ∪ X2 , X2 ),

(B.9)

∼

H p (X1 ∪ X2 , X2 ) −→ H p (X1 , X1 ∩ X2 ).

Let S be a closed set in X and U an open set in X containing S. Then (U, X rS) is an excisive couple and there exist canonical isomorphisms ∼

Hp (U, U rS) −→ Hp (X, X rS),

∼

H p (X, X rS) −→ H p (U, U rS). (B.10)

Mayer-Vietoris sequence Let X1 and X2 be subspaces of X. Then there is an exact sequence of chain complexes j•

i

• 0 −→ S• (X1 ∩ X2 ) −→ S• (X1 ) ⊕ S• (X2 ) −→ S• (X1 ) + S• (X2 ) −→ 0,

where i• (c) = (c, −c) and j• (c1 , c2 ) = c1 +c2 . Thus if (X1 , X2 ) is an excisive couple, we have the following exact sequence: j∗

i

∗ · · · −→ Hp (X1 ∩ X2 ) −→ Hp (X1 ) ⊕ Hp (X2 ) −→

(B.11)

∂

∗ Hp (X1 ∪ X2 ) −→ Hp−1 (X1 ∩ X2 ) −→ · · · ,

where ∂∗ assigns to the class of c1 + c2 the class of c in X1 ∩ X2 such that ∂c1 = c and ∂c2 = −c. Likewise we have the exact sequence in cohomology: j∗

i∗

· · · −→ H p (X1 ∪ X2 ) −→ H p (X1 ) ⊕ H p (X2 ) −→ δ∗

H p (X1 ∩ X2 ) −→ H p+1 (X1 ∪ X2 ) −→ · · · .

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Complex Analytic Geometry

In the above, denoting by ik : X1 ∩ X2 ,→ Xk and jk : Xk ,→ X1 ∪ X2 , k = 1, 2, the inclusions, we may write j ∗ (α) = (j1∗ (α), j2∗ (α)) and i∗ (α1 , α2 ) = i∗1 (α1 ) − i∗2 (α2 ). For every cocycle u ∈ S p (X1 ∩ X2 ) there exist cochains uk ∈ S p (Xk ), k = 1, 2, such that u = ip1 (u1 ) − ip2 (u2 ). The connecting morphism δ ∗ assigns to the class of u the class of v ∈ S p+1 (X1 ∪ X2 ) such that j1p+1 (v) = δu1 and j2p+1 (v) = δu2 . Example B.5. 1. Consider the unit q-sphere in Rq+1 = {(x1 , . . . , xq+1 )}: Sq = { x ∈ Rq+1 | kxk2 = |x1 |2 + · · · + |xq+1 |2 = 1 }. We may write Sq = S + ∪ S − with S + = Sq r {(0, . . . , 0, 1)} and S − = Sq r{(0, . . . , 0, −1)}. Note that both S + and S − are contractible. In the case q = 0, S0 = {−1, 1} and we have: ( R⊕R p = 0, Hp (S0 ; R) ' 0 otherwise. In the case q ≥ 1, Sq ∩ {xq+1 = 0} = Sq−1 is a deformation retract of S ∩ S − . By a repeated use of (B.11), we have: ( R p = 0, q, q Hp (S ; R) ' 0 otherwise. +

2. Let Bq = { x ∈ Rq | kxk ≤ 1 } be the closed unit q-ball in Rq , q ≥ 1. Then Sq−1 = ∂Bq is the unit (q − 1)-sphere. From the contractibility of the ball and (B.4) we have: ( R p = q, q q−1 Hp (B , S ; R) ' 0 otherwise. Note that the above holds also for q = 0, if we set B0 = {0} and S−1 = ∅. In this case, H0 (B0 , S−1 ; R) = H0 ({0}; R) has a canonical generator, i.e., the class of the constant map ∆0 → {0}. Note also that, if q > 1, ∼

∂∗ : Hq (Bq , Sq−1 ; R) −→ Hq−1 (Sq−1 ; R)

(B.12)

is an isomorphism. Since Rq and Rq r{0} deformation retract to Bq and Sq−1 , respectively, there is a canonical isomorphism Hp (Rq , Rq r{0}; R) ' Hp (Bq , Sq−1 ; R).

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For the cohomology we have: Example B.6. 1. For the unit sphere Sq in Rq+1 , q ≥ 1, H p (Sq ; R) '

( R

p = 0, q,

0

otherwise.

2. For the closed unit q-ball Bq in Rq and the unit (q−1)-sphere Sq−1 = ∂Bq , q ≥ 1, H p (Bq , Sq−1 ; R) '

( R 0

p = q, otherwise.

The above holds also for q = 0. In this case, H 0 (B0 , S−1 ; R) = H 0 ({0}; R) has a canonical generator, i.e., the class of the cocycle that assigns 1 to the constant map ∆0 → {0}. ∼ Note that, if q > 1, δ ∗ : H q−1 (Sq−1 ; R) → H q (Bq , Sq−1 ; R) is an isomorphism. There is also a canonical isomorphism H p (Rq , Rq r {0}; R) ' H p (Bq , Sq−1 ; R). Spaces with finite homology type: A pair (X, A) of topological spaces is said to have a finite homology type if, for every p, Hp (X, A; Z) is finitely generated. For such a pair, by the structure theorem for finitely generated Abelian groups, we have Hp (X, A; Z) ' Zbp ⊕ Tp ,

Tp = Z/(t1 ) ⊕ · · · ⊕ Z/(trp ),

(B.13)

where bp is a non-negative integer and t1 , . . . , trp positive integers with t1 6= 1 and t1 | t2 | · · · | trp . Note that rp may possibly be 0. The integers bp and t1 , . . . , trp are uniquely determined by (X, A). We call Zbp the free part of Hp (X, A; Z) and bp the p-th Betti number of (X, A). We also call Tp the torsion part and t1 , . . . , trp the torsion coefficients. The torsion part may also be expressed as Tp = Z/(pk11 ) ⊕ · · · ⊕ Z/(pkl l ), where the pi ’s are prime numbers (not necessarily distinct) and the ki ’s are positive integers, i = 1, . . . , l. The powers pki i are uniquely determined. Similar statements hold if we replace Z with a principal ideal domain.

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Complex Analytic Geometry

Universal coefficient theorem The following is proved using Propositions A.18 and B.24 (cf. Example A.2. 1 for Tor and Ext): Theorem B.20. Let R be a principal ideal domain. Then, for a pair of topological spaces (X, A) and an R-algebra S, there exist canonical exact sequences 0 → S ⊗R Hp (X, A; R) → Hp (X, A; S) → TorR1 (Hp−1 (X, A; R), S) → 0, 0 → Ext1R (Hp−1 (X, A; R), S) → H p (X, A; S) → HomR (Hp (X, A; R), S) → 0, both of which split. Here are some of the consequences of the above: (1) If S is R-torsion-free, thus R-flat, there exists a canonical isomorphism Hp (X, A; S) ' S ⊗R Hp (X, A; R). (2) If Hp−1 (X, A; R) is R-free, thus R-projective and R-flat, then there exist canonical isomorphisms Hp (X, A; S) ' S⊗R Hp (X, A; R), H p (X, A; S) ' HomR (Hp (X, A; R), S). (3) If R is a field, then there exists a canonical isomorphism H p (X, A; R) ' Hp (X, A; R)∗ . Note that the isomorphism is induced by the Kronecker product. (4) Suppose R = Z. If S is divisible, thus Z-injective (cf. Proposition A.16), then there exists a canonical isomorphism H p (X, A; S) ' HomZ (Hp (X, A; Z), S). (5) Suppose (X, A) is of finite homology type. Then, for K = Q, R or C, we have (cf. (B.13)) Hp (X, A; K) ' Kbp ,

H p (X, A; K) ' Kbp .

We also have H p (X, A; Z) ' Zbp ⊕ Tp−1 . For the last isomorphism, let R = S = Z in the second sequence in Theorem B.20. First note that HomZ (Hp (X, A; Z), Z) ' Zbp . To compute Ext1Z (Hp−1 (X, A; Z), Z), take a free resolution of Hp−1 (X, A; Z) of the form τ

0 −→ Zrp−1 −→ Zbp−1 ⊕ Zrp−1 −→ Hp−1 (X, A; Z) −→ 0, where τ is a morphism given by (n1 , . . . , nr ) 7→ (0; n1 t1 , . . . , nr tr ), r = rp−1 . Then we see that Ext1Z (Hp−1 (X, A; Z), Z) ' Z/(t1 )⊕· · ·⊕Z/(trp−1 ) = Tp−1 .

Algebraic Topology

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Products Cup product:

We define the cup product `

S p (X) × S q (X) −→ S p+q (X),

(u, v) 7→ u ` v

by setting hσ, u ` vi = hdp+1 · · · dp+q σ, uihd0 · · · dp−1 σ, vi

(B.14)

for a (p + q)-simplex σ and extending it R-linearly. Remark B.10. For a (p + q)-simplex σ, dp+1 · · · dp+q σ is the “first psimplex” of σ, i.e., the composition of the affine map ε : ∆p → ∆p+q given by ε(Pi ) = Pi , 0 ≤ i ≤ p, and σ. Also, d0 · · · dp−1 σ is the “last q-simplex” of σ, i.e., the composition of the affine map ε0 : ∆q → ∆p+q given by ε0 (Pi ) = Pp+i , 0 ≤ i ≤ q, and σ. If q = 0, the former is σ itself and the latter the 0-simplex that assigns to P0 the point σ(Pp ) in X. The cup product is R-bilinear and we have δ(u ` v) = δu ` v + (−1)p u ` δv.

(B.15)

This shows that the above induces the cup product on the cohomology: `

H p (X) × H q (X) −→ H p+q (X), L which makes H ∗ (X) = p≥0 H p (X) a graded R-algebra with unity. The unity is the class [1] of the cochain that assigns 1 to each 0-simplex. Note that the cup product in H ∗ (X) is anticommutative, i.e., for α ∈ H p (X) and β ∈ H q (X), α ` β = (−1)pq β ` α. Let A1 and A2 be subspaces of X. If the couple (A1 , A2 ) is excisive, we see that we also have the cup product `

H p (X, A1 ) × H q (X, A2 ) −→ H p+q (X, A1 ∪ A2 ). If f : X → Y is a continuous map, f ∗ : H ∗ (Y ) → H ∗ (X) is a morphism of graded R-algebras with unity.

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Complex Analytic Geometry

Cap product:

We define the cap product a

Sr (X) × S p (X) −→ Sr−p (X) by cau=

X

aσ hdp+1 · · · dr σ, ui d0 · · · dp−1 σ

(B.16)

σ

P for an r-chain c = aσ σ and a p-cochain u, r ≥ p (cf. Remark B.10). Then c a u is R-linear in c and u and we have ∂(c a u) = (−1)p (∂c a u − c a δu).

(B.17)

c a (u ` v) = (c a u) a v

(B.18)

We also have

p

q

for c ∈ Sr (X), u ∈ S (X) and v ∈ S (X). From (B.17) we have the cap product a

Hr (X) × H p (X) −→ Hr−p (X). The relation (B.18) holds in homology and cohomology and the cap product L makes H∗ (X) = r≥0 Hr (X) a right H ∗ (X)-module. In the case r = p, we have ε(c a u) = hc, ui,

(B.19)

where ε : S0 (X) → R is the augmentation. Thus from (B.18), if p + q = r, hc, u ` vi = hc a u, vi.

(B.20)

The above identities hold also in homology and cohomology. Let A be a subspace of X and i : A ,→ X the inclusion. Then for c in Sr (A) and u in S p (X, A), c a u = 0. Thus we have a

Sr (X, A) × S p (X, A) −→ Sr−p (X),

(B.21)

which induces a

Hr (X, A) × H p (X, A) −→ Hr−p (X). Also we may define a

Hr (A) × H p (X) −→ Hr−p (A)

by a a α = a a i∗ α.

(B.22)

For a continuous map f : (X, A) → (Y, B), we have f∗ (a a f ∗ (β)) = f∗ (a) a β for a ∈ Hr (X, A) and β ∈ H p (Y, B), which generalizes (B.8).

(B.23)

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561

Remark B.11. The cap product above is what is called the left cap product. There is also the right cap product a

S p (X) × Sr (X) −→ Sr−p (X), which is defined by uac=

X

aσ hd0 · · · dq−1 σ, ui dq+1 · · · dr σ

σ

P for an r-chain c = aσ σ and a p-cochain u, r ≥ p. Then u a c is R-linear in u and c and we have ∂(u a c) = (−1)r−p δu a c + u a ∂c.

(B.24)

(u ` v) a c = u a (v a c)

(B.25)

We also have

for c ∈ Sr (X), u ∈ S p (X) and v ∈ S q (X). Moreover we have (u ar c) al v = u ar (c al v),

(B.26)

where al and ar denote left and right cap products, respectively. From (B.24) we have the right cap product a

H p (X) × Hr (X) −→ Hr−p (X). The relation (B.25) holds in homology and cohomology and the right cap product makes H∗ (X) a left H ∗ (X)-module. Note that, on the homology and cohomology level, the left and right cap products differ by a sign of (−1)p(r−p) . K¨ unneth formula: Let X and Y be topological spaces. Denoting by p1 and p2 the projections X × Y → X and X × Y → Y , we define the cross product ×

H p (X) × H q (Y ) −→ H p+q (X × Y )

by α × β = p∗1 (α) ` p∗2 (β).

Suppose the coefficient ring R is a principal ideal domain. If X is a space of finite homology type and if either H ∗ (X) or H ∗ (Y ) is R-free, then there is an isomorphism H ∗ (X × Y ) ' H ∗ (X) ⊗ H ∗ (Y ),

(B.27)

where α × β in H ∗ (X × Y ) corresponds to α ⊗ β in H ∗ (X) ⊗ H ∗ (Y ).

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B.2

Complex Analytic Geometry

Cell complexes

In this section we review cell complexes and, as a special case, we discuss simplicial complexes. They provide practical ways of representing homology and cohomology of spaces that admit a cellular or simplicial decomposition.

Cell complexes Let X be a Hausdorff topological space. Definition B.11. A cell in X is a subset e of X that admits a surjective continuous map χ : Bp → e such that χ(Sp−1 ) ⊂ ere and that χ|BprSp−1 is a homeomorphism onto e. A map χ as above is called a characteristic map of e. The integer p, which is determined uniquely by e, is called the dimension of e. A cell e of dimension p is called a p-cell and is sometimes denoted by ep . In the above we allow p to be equal to 0 and a 0-cell is a point of X, which has a unique characteristic map. A subset of the form e is referred to as a closed cell. In particular, Bp is a closed cell in Rp with the identity map as a characteristic map. Definition B.12. A cell complex is a Hausdorff topological space X together with a family {eλ }λ∈Λ of cells in X with the following properties: F F (1) X = λ∈Λ eλ (disjoint union). We set X p = pλ ≤p eλ , pλ = dim eλ . (2) For every λ ∈ Λ, eλ reλ ⊂ X pλ −1 . In this case, we say that {eλ }λ∈Λ is a cellular decomposition of X. We call X p the p-skeleton of X. A subcomplex of a cell complex (X, {eλ }λ∈Λ ) is a cell complex (A, {eλ }λ∈Λ0 ) such that A ⊂ X and Λ0 ⊂ Λ. For every p, X p is a subcomplex. The intersection and union of subcomplexes are subcomplexes. A cell complex (X, {eλ }λ∈Λ ) is said to be finite if Λ is a finite set. It is locally finite if, for every x ∈ X, there exists a finite subcomplex A with x ∈ Int A. A subcomplex of a locally finite cell complex X is a closed set in X. A locally finite cell complex satisfies the conditions for what is called a “CW-complex” and in the following we only consider locally finite cell complexes.

Algebraic Topology

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Homology of cell complexes First we prove the following: Lemma B.7. Let e be a p-cell in X. If χ : (Bp , Sp−1 ) → (e, e r e) is a characteristic map of e, it induces an isomorphism ∼

χ∗ : Hq (Bp , Sp−1 ; R) −→ Hq (e, ere; R)

for all q.

Proof. Since Bp r{0} deformation retracts to Sp−1 , there is a canonical isomorphism Hq (Bp , Sp−1 ) ' Hq (Bp , Bp r{0}), which is canonically isomorphic with Hq (BprSp−1 , (BprSp−1 )r{0}) by excision. By similar arguments, we see that there is a canonical isomorphism Hq (e, ere) ' Hq (e, er{x}), x = χ(0) ∈ e. Since χ|BprSp−1 is a homeomorphism, we have the lemma.  Thus, for a p-cell e in X (cf. Example B.5. 2, with p and q interchanged), ( R q = p, (B.28) Hq (e, ere; R) ' 0 otherwise. Let (X, {eλ }λ∈Λ ) be a cell complex. For each p-cell eλ , the inclusion iλ : (eλ , eλ reλ ) ,→ (X p , X p−1 ) induces a morphism (iλ )∗ : Hq (eλ , eλ reλ ; R) −→ Hq (X p , X p−1 ; R). With these, we have: Theorem B.21. In the above situation, we have an isomorphism M M ∼ (iλ )∗ : Hq (eλ , eλ reλ ; R) −→ Hq (X p , X p−1 ; R), where the direct sum is taken over all the p-cells eλ of X. Thus Hp (X p , X p−1 ; R) is a free R-module and Hq (X p , X p−1 ; R) = 0 for q 6= p. Orientation of cells: Let e be a p-cell in X. Setting R = Z in (B.28), we have Hp (e, ere; Z) ' Z. An orientation of e, or of e, is a generator of Hp (e, e r e; Z). Thus each cell has two orientations. An oriented cell is a cell together with a specified orientation. Remark B.12. A 0-cell e0 = {x}, x ∈ X, has a canonical orientation, i.e., the class of the constant map ∆0 → {x} in H0 (e, ere; Z) = H0 ({x}; Z).

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Complex Analytic Geometry

Homology and cohomology of cell complexes: (X, {eλ }), we set

For a cell complex

Cp (X; R) = Hp (X p , X p−1 ; R). For each p-cell eλ , we choose and fix an orientation µλ ∈ Hp (eλ , eλ reλ ; Z) and let iλ : (eλ , eλ reλ ) ,→ (X p , X p−1 ) be the inclusion as before. Then by Theorem B.21, Cp (X; Z) is the free Z-module (Abelian group) generated by the (iλ )∗ (µλ )’s, for all the p-cells eλ in X. By Theorem B.21 and (B.28), we have Hp−1 (X p , X p−1 ; Z) = 0. Hence by Theorem B.20, we have: Proposition B.25. There is a canonical isomorphism Cp (X; R) ' R ⊗Z Cp (X; Z). Thus intuitively we may say that Cp (X; R) is the free R-module generated by all the oriented closed p-cells in X. We define the boundary operator (we omit the coefficient R) ∂p : Cp (X) −→ Cp−1 (X) as the connecting morphism ∂

∗ Hp (X p , X p−1 ) −→ Hp−1 (X p−1 , X p−2 )

in the long exact sequence for the triple (X p , X p−1 , X p−2 ) (cf. (B.5)). Then we have ∂p−1 ◦ ∂p = 0 and (C• (X; R), ∂• ) is a chain complex. If A is a subcomplex of X, we may think of Cp (A; R) as the free Rmodule generated by the oriented closed p-cells in A, by Proposition B.25, and identify with a direct summand of Cp (X; R). The boundary operators are compatible with the inclusion Cp (A; R) ,→ Cp (X; R) and we have the quotient complex C• (X; R)/C• (A; R), which we denote by C• (X, A; R). Note that we may think of Cp (X, A; R) as the free R-module generated by the closures of oriented p-cells in X rA. Denoting by C • (A; R), C • (X; R) and C • (X, A; R) the cochain complexes dual to C• (A; R), C• (X; R) and C• (X, A; R), respectively, we have the exact sequence, as in the case of singular cochains: 0 −→ C • (X, A; R) −→ C • (X; R) −→ C • (A; R) −→ 0. We have: Theorem B.22. For a pair (X, A) of cell complexes, there exist canonical isomorphisms: Hp (C• (X, A; R)) ' Hp (X, A; R),

H p (C • (X, A; R)) ' H p (X, A; R).

Algebraic Topology

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Example B.7. The sphere Sq , q ≥ 1, may be identified with the space obtained from Bq by shrinking Sq−1 = ∂Bq to a point. Let χ : Bq → Sq be the projection in this process. Setting e0 = χ(Sq−1 ) and eq = Sq re0 , we have a cellular decomposition {e0 , eq } of Sq . Suppose q ≥ 2. For Bq , we set eq = Bq r Sq−1 and for Sq−1 , we take the above decomposition {e0 , eq−1 }. Then {e0 , eq−1 , eq } is a cellular decomposition of Bq . Applying Theorem B.22, we come back to the results of Example B.5. A cell is said to be regular if it admits a characteristic map that is a homeomorphism. A cell complex is regular if every cell is regular. As particular regular cell complexes, there are “simplicial complexes”. Simplicial complexes Let s be a p-simplex with vertices v0 , . . . , vp in RN . A face of s is a simplex with vertices a subset of {v0 , . . . , vp }. If t is a face of s, we write t ≺ s. Definition B.13. A simplicial complex is a collection K of simplices in RN satisfying the following conditions: (1) Every face of a simplex in K is in K. (2) If s and t are in K and if s ∩ t 6= ∅, s ∩ t is a face of each of s and t. S (3) Every point x in |K| = s∈K s has a neighborhood intersecting with only a finite number of simplices in K. The set |K| as above is called the polyhedron of K. Let K and K 0 be simplicial complexes. A simplicial map of K to K 0 is a continuous map f : |K| → |K 0 | which carries each simplex of K linearly (in the affine sense) onto a simplex of K 0 . It is a simplicial isomorphism if, in addition, f is a homeomorphism. A subcomplex of K is a subset of K which itself is a simplicial complex. If L is a subcomplex of K, |L| is a closed subset of |K|. Let A be a subset of |K|. The star and the link of A in K are defined, respectively, by SK (A) = { t ∈ K | t ≺ s for some s, A ∩ s 6= ∅ }, LK (A) = { t ∈ SK (A) | A ∩ t = ∅ }. They are subcomplexes of K. We set SK (A) = |SK (A)| and LK (A) = |LK (A)| and call them also the star and the link of A. The open star of A is OK (A) = SK (A)rLK (A). Note that |K|rOK (A) is the polyhedron of the subcomplex consisting of simplices that are disjoint from A.

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S For a simplex s, we set es = sr ts t. Thus es = s. If K is a simplicial complex, (|K|, {es }) is a regular cell complex. If L is a subcomplex of K, |L| is a sub-cell complex of |K|. Orientation of simplices: Let s be a p-simplex with vertices v0 , . . . , vp . Assume that p ≥ 1 for the moment. We introduce a relation ∼ in the set of ordered vertices of s by saying that (vi0 , . . . , vip ) ∼ (vj0 , . . . , vjp ) if (i0 , . . . , ip ) is an even permutation of (j0 , . . . , jp ). Then it is an equivalence relation and there are exactly two equivalence classes, each of which is called an orientation of s. An oriented simplex is a simplex together with a specified orientation. The specified orientation is called positive and the other one negative. A simplex with the orientation represented by an ordered vertices (v0 , . . . , vp ) is also denoted by (v0 , . . . , vp ) and the simplex with the opposite orientation by −(v0 , . . . , vp ). Then for every permutation ρ of (0, . . . , p), (vρ(0) , . . . , vρ(p) ) = sgn ρ · (v0 , . . . , vp ). By convention we think of a 0-simplex s with vertex v as having two orientations and denote s with the “specified orientation” by (v) and s with the “other orientation” by −(v). Remark B.13. 1. Specifying an orientation of the standard p-simplex ∆p in Rp , p ≥ 1, is equivalent to specifying an orientation of Rp as a vector space (cf. Section 3.7), since every orientation of ∆p is represented by an ordered vertices of the form (P0 , Pi1 , . . . , Pip ). The standard orientation of ∆p is the one represented by (P0 , P1 , . . . , Pp ). We always endow ∆p with the standard orientation ; ∆p = (P0 , P1 , . . . , Pp ), which is consistent with Convention 3.1. In the case p = 0, we specify an “orientation” of ∆0 so that we may write ∆0 = (P0 ). The boundary ∂∆p is oriented as ∂(P0 , . . . , Pp ) =

p X

(−1)i (P0 , . . . , Pbi , . . . , Pp ).

i=0

We may think of ∆p as a p-dimensional piecewise C ∞ manifold with boundary in Rp (cf. Definition 5.10). The above orientation of the boundary is consistent with Convention 3.2, modified for piecewise C ∞ manifolds. ∼

2. There is a natural homeomorphism h : (∆p , ∆p rInt ∆p ) → (Bp , Sp−1 ), p ≥ 0. Thus Hp (∆p , ∆prInt ∆p ; Z) ' Z (cf. Example B.5. 2). Moreover the homology has a canonical generator, i.e., the class [1p ] of the identity map 1p of ∆p . This in turn gives a canonical generator h∗ ([1p ]) of Hp (Bp , Sp−1 ; Z).

Algebraic Topology

567

We always think of Bp as an oriented closed cell with this orientation. By (B.12), Hp (Sp ; Z), p ≥ 1, has also a canonical generator. 3. The above definition of orientation of a simplex s is made to be compatible with that of the cell es in the following sense. For an oriented p-simplex s = (v0 , . . . , vp ), p ≥ 1, we define an affine map χ : ∆p → s by χ(Pi ) = vi . We may think of it as a characteristic map for es and, noting that es = s, we endow the cell es with the orientation χ∗ ([1p ]) ∈ Hp (s, s r es ; Z) (cf. Lemma B.7). In the case s is a 0-simplex with vertex v, we specify an orientation of s ; s = (v). Define χ : ∆0 → s by χ(P0 ) = v, then χ∗ ([10 ]) ∈ H0 (s, sres ; Z) = H0 ({v}; Z) is the canonical orientation of es (cf. Remark B.12). Homology and cohomology of simplicial complexes: Let K be a simplicial complex. By Remark B.13, we may think of the chain group Cp (|K|; R) as the free R-module generated by the oriented p-simplices in K with the boundary operator given by ∂(v0 , . . . , vp ) =

p X (−1)i (v0 , . . . , vbi , . . . , vp ).

(B.29)

i=0

If L is a subcomplex of K, we may identify Cp (|L|; R) with a direct summand of Cp (|K|; R) and we have the quotient chain complex C• (|K|, |L|; R) = C• (|K|; R)/C• (|L|; R). Denoting by C • (|K|, |L|; R) the cochain complex dual to C• (|K|, |L|; R), we have canonical isomorphisms (cf. Theorem B.22) Hp (C• (|K|, |L|; R)) ' Hp (|K|, |L|; R), H p (C • (|K|, |L|; R)) ' H p (|K|, |L|; R).

(B.30)

The above isomorphisms may also be interpreted as follows. Let s be an oriented p-simplex of K. Regarding the map χ : ∆p → s ⊂ |K| defined in Remark B.13. 3 as a singular simplex of |K|, we have a morphism ηp : Cp (|K|, |L|; R) −→ Sp (|K|, |L|; R). We also have its transpose η p : S p (|K|, |L|; R) −→ C p (|K|, |L|; R). Then the above morphisms are compatible with the boundary and coboundary operators, respectively, and induce the isomorphisms in (B.30).

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Complex Analytic Geometry

Cup and cap products: of vertices of K.

Let K be a simplicial complex and V the set

Definition B.14. An ordering on V is simplicial if its restriction to the set of vertices of each simplex of K is total. For example, a total ordering on V is simplicial. The cup and cap products are defined on the level of simplicial chains and cochains. For this we fix a simplicial ordering on V. For an oriented simplex s of K we denote by ϑ(s) its dual in C p (|K|; R), i.e., the element defined by ( 1 if s0 = s, 0 hs , ϑ(s)i = 0 otherwise and extended R-linearly. Then C p (|K|; R) is the free R-module generated by the ϑ(s)’s. The cup product `

C p (|K|; R) × C q (|K|; R) −→ C p+q (|K|; R) is defined by setting, for oriented p-simplex s1 , q-simplex s2 and (p + q)simplex s hs, ϑ(s1 ) ` ϑ(s2 )i = ε ε1 ε2

(B.31)

if we may write s1 = ε1 (v0 , . . . , vp ), s2 = ε2 (vp , . . . , vp+q ) and s = ε(v0 , . . . , vp+q ) with v0 < · · · < vp+q , setting 0 otherwise, and extending R-linearly. The cap product a

Cr (|K|; R) × C p (|K|; R) −→ Cr−p (|K|; R) is defined by hc a u, vi = hc, u ` vi for an r-chain c, a p-cochain u and an (r − p)-cochain v. Thus if s = ε(v0 , . . . , vr ) with v0 < · · · < vr and s1 = ε1 (v0 , . . . , vp ), s a ϑ(s1 ) = εε1 (vp , . . . , vr ).

(B.32)

Also, for s and s1 as above, s a δϑ(s1 ) = εh∂(v0 , . . . , vp+1 ), ϑ(s1 )i(vp+1 , . . . , vr ) = (−1)p+1 εε1 (vp+1 , . . . , vr ).

(B.33)

Algebraic Topology

569

If L is a subcomplex of K, for c in Cr (|L|; R) and u in C p (|K|, |L|; R), c a u = 0. Thus we have a

Cr (|K|, |L|; R) × C p (|K|, |L|; R) −→ Cr−p (|K|; R).

(B.34)

Remark B.14. 1. The above products induce the cup and cap products on the level of homology and cohomology and correspond to the ones on singular homology and cohomology under the isomorphisms of (B.30). 2. The above cap product is the left cap product. The right cap product a

C p (|K|; R) × Cr (|K|; R) −→ Cr−p (|K|; R) is defined by hu a c, vi = hc, v ` ui for an r-chain c, a p-cochain u and an (r − p)-cochain v. Thus if s = ε(v0 , . . . , vr ) with v0 < · · · < vr and s1 = ε1 (vr−p , . . . , vr ), ϑ(s1 ) a s = εε1 (v0 , . . . , vr−p ). (B.35) Barycentric subdivision: Let K be a simplicial complex. A subdivision of K is a simplicial complex K 0 such that |K 0 | = |K| and that each simplex of K 0 is contained in a simplex of K. A particular type of subdivisions of K is given as follows. For a simplex s, we denote by bs its barycenter. The barycentric subdivision K 0 of K is the set of simplices obtained in the following way. Take a sequence of simplices s0  · · ·  sp in K. Then the simplices in K 0 consist of {bs0 , . . . , bsp } for all the sequences as above. It is again a simplicial complex. A subcomplex L of K is full if s is a simplex in K whose vertices are all in L, then s is in L. For any subcomplex L of K, L0 is full in K 0 , the barycentric subdivision of K. We give a proof of the following for the sake of completeness: Proposition B.26. If L is a full subcomplex of K, |L| is a deformation retract of SK 0 (|L|). Proof. Let x be a point in OK (|L|). For each vertex v in L, we define a number x(v) as follows. If there is a simplex s in K such that x is in s and v is a vertex of s, x(v) is the barycentric coordinate of x relative to v. Note that it is a positive number not depending on such an s. If there are P no such simplices, we set x(v) = 0. Noting that v∈L x(v) is a finite sum, which is non-zero, we set X 1 x(v)v. r(x) = X x(v) v∈L v∈L

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Complex Analytic Geometry

It is in a face of a simplex of K whose vertices are in L. Since L is full, P the point is in |L|. If x is in |L|, v∈L x(v) = 1 and r(x) = x. Thus it is a retraction of OK (|L|) onto |L|. Moreover, it is a deformation retraction by the homotopy rt (x) = (1 − t)x + t r(x). The restriction of r to SK 0 (|L|) is a desired deformation retraction.  Note that the above retraction r : SK 0 (|L|) → |L| is proper. B.3

Homology of locally finite chains

Let X be a locally compact topological space. A locally finite singular p-chain of X is a formal sum X c= aσ σ, σ

with σ singular p-simplices of X and aσ ∈ R, such that, for every compact set C in X, { σ | aσ 6= 0, σ(∆p ) ∩ C 6= ∅ } is a finite set. If c is a locally finite singular p-chain as above, its support S |c| = aσ 6=0 σ(∆p ) is closed and the boundary ∂c is a well-defined locally finite singular (p − 1)-chain. The set S˘p (X; R) of locally finite chains is an R-module and we have a chain complex (S˘• (X; R), ∂• ). The homology ˘ p (X; R) of this complex is called the homology of locally finite singular H chains. There is a natural inclusion S• (X; R) ,→ S˘• (X; R), which induces a morphism ˘ p (X; R). Hp (X; R) −→ H If X is compact, it is an isomorphism. If f : X → Y is a proper map of locally compact spaces, it induces a morphism ˘ p (X) −→ H ˘ p (Y ), f∗ : H which is determined by the proper homotopy class of f . Remark B.15. The homology of locally finite singular chains is not a homotopy invariant as the following example shows: ( p = q, q ˘ p (R ; Z) = Z H 0 p 6= q.

Algebraic Topology

571

If A is a closed subset of X, then the inclusion A ,→ X is proper and we may identify S˘• (A; R) with a subcomplex of S˘• (X; R). Thus we have the complex S˘• (X, A; R) = S˘• (X; R)/S˘• (A; R) of relative chains, which defines ˘ p (X, A; R) of locally finite singular chains. We have the relative homology H also the exact sequence as (B.4) for this homology theory. For A as above, we may define the cap product a S˘r (X, A) × S p (X, A) −→ S˘r−p (X)

as in (B.16), see also (B.21). It induces the cap product a ˘ r (X, A) × H p (X, A) −→ ˘ r−p (X). H H

(B.36)

Let (X, {eλ }λ∈Λ ) be a (locally finite) cell complex. We denote by ˘ Cp (X; R) the R-module of formal sums of oriented closed p-cells in X with Q coefficients in R, i.e., the direct product Reλ with eλ all the oriented closed p-cells in X. Then we have a chain complex (C˘• (X; R), ∂). If A is a sub-cell complex of X, C˘• (A; R) is a subcomplex of C˘• (X; R) and we may consider the quotient complex C˘• (X, A; R) = C˘• (X; R)/C˘• (A; R). Then there is a canonical isomorphism ˘ p (X, A; R). Hp (C˘• (X, A; R)) ' H

(B.37)

For a simplicial complex K, C˘p (|K|; R) is the R-module of formal sums of oriented p-simplices in K with coefficients in R. If L is a sub-simplicial complex of K, C˘• (|L|; R) is a subcomplex of C˘• (|K|; R) and we have the quotient complex C˘• (|K|, |L|; R) = C˘• (|K|; R)/C˘• (|L|; R). There is a canonical isomorphism ˘ p (|K|, |L|; R). Hp (C˘• (|K|, |L|; R)) ' H

(B.38)

The cap product (B.36) may also be described in terms of locally finite chains of K (cf. (B.34)): a C˘r (|K|, |L|; R) × C p (|K|, |L|; R) −→ C˘r−p (|K|; R).

(B.39)

Notes As a basic reference on general topology, we list [Munkres (1975)]. There are a number of references on algebraic topology. Here we list [Greenberg and Harper (1981); Hattori (1991); Spanier (1966)].

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Bibliography

Abate, M., Bracci, F., Suwa, T. and Tovena, F. (2013). Localization of Atiyah classes, Rev. Mat. Iberoam. 29, pp. 547-578. Abate, M., Bracci, F and Tovena, F. (2008). Index theorems for holomorphic maps and foliations, Indiana Univ. Math. J. 57, pp. 2999-3048. Aluffi, P. (1999). Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc. 351, pp. 3989-4026. Angella, D., Suwa, T., Tardini, N. and Tomassini, A. (2020). Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms, Complex Manifolds 7, pp. 194-214. Atiyah, M.F. (1957). Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 84, pp. 181-207. Atiyah, M.F. (1967). K theory, Benjamin, New York. Atiyah, M.F. and Bott, R. (1967-8). A Lefschetz fixed point formula for elliptic complexes: I, II, Ann. Math. 86, pp. 374-407, 88, pp. 451-491. Atiyah, M.F. and Hirzebruch, F. (1961). Analytic cycles on complex manifolds, Topology 1, pp. 25-45. Atiyah, M.F. and Hirzebruch, F. (1962). The Riemann-Roch for analytic embeddings, Topology 1, pp. 151-166. Atiyah, M.F. and Singer, I.M. (1963). The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69, pp. 422-433. Baum, P. and Bott, R. (1972). Singularities of holomorphic foliations, J. Differential Geom. 7, pp. 279-342. Baum, P., Fulton, W. and MacPherson, R. (1975). Riemann-Roch for singular varieties, Publ. Math. IHES 45, pp. 101-145. Bierstone, E. and Milman, P.D. (1988). Semianalytic and subanalytic sets, Publ. Math. IHES 67, pp. 5-42. Bisi, C., Bracci, F., Izawa, T. and Suwa, T. (2016). Localized intersection of currents and the Lefschetz coincidence point theorem, Annali di Mat. Pura ed Applicata 195, pp. 601-621. Bismut, J.-M. (1998). Local index theory and higher analytic tosion, Documenta Math. Extra Volume ICM 1998, I, pp. 143-162.

573

574

Complex Analytic Geometry

Bismut, J.-M., Gillet, H. and Soul´e, C. (1988). Analytic torsion and holomorphic determinant bundles I: Bott-Chern forms and analytic torsion, Comm. Math. Phys. 115, pp. 49-78. Borel, A. et Serre, J.-P. (1958). Le th´eor`eme de Riemann-Roch, Bull. Soc. Math. France 86, pp. 97-136. Bott, R. (1972). Lectures on characteristic classes and foliations, Lectures on Algebraic and Differential Topology, Lecture Notes in Math. 279, Springer, Berlin, pp. 1-94. Bott, R. and Chern, S.S. (1965). Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114, pp. 71-112. Bott, R. and Tu, L. (1982). Differential forms in algebraic topology, Graduate Texts in Math. 82, Springer, Berlin. ´ ements de Math´ematique, Alg`ebre Commutative, Chap. Bourbaki, N. (1961-65). El´ 1-7, Hermann, Paris. Bracci, F. and Suwa, T. (2004). Residues for singular pairs and dynamics of biholomorphic maps of singular surfaces, Intern. J. Math. 15, pp. 443-466. Brasselet, J.-P. (1981). D´efinition combinatoire des homomorphismes de Poincar´e, Alexander et Thom pour une pseudo-vari´et´e, Caract´eristique d’Euler-Poincar´e, Ast´erisque 82-83, Soc. Math. France, pp. 71-91. Brasselet, J.-P. (1995). Aspects combinatoires des singularit´es, Singularity Theory, World Scientific, Singapore, pp. 74-127. Brasselet, J.-P. (2022). Characteristic classes, Chapter 5 in Handbook of Geometry and Topology of Singularities III, J.L. Cisneros-Molina, D.-T. Lˆe and J. Seade eds., Springer Berlin. Brasselet, J.-P., Lehmann, D., Seade, J. and Suwa, T. (2001). Milnor classes of local complete intersections, Trans. Amer. Math. Soc. 354, pp. 1351-1371. Brasselet, J.-P., Sch¨ urmann, J. and Yokura, S. (2010). Hirzebruch classes and motivic Chern classes for singular spaces, J. Topology and Analysis 2, pp. 1-55. Brasselet, J.-P. et Schwartz, M.-H. (1981). Sur les classes de Chern d’un ensemble analytique complexe, Ast´erisque 82-83, Soci´et´e Math´ematique de France, pp. 93-147. Brasselet, J.-P., Seade, J. and Suwa, T. (2009). Vector fields on singular varieties, Lecture Notes in Math. 1987, Springer, Berlin. Brasselet, J.-P. and Suwa, T. (2021). Local and global coincidence homology classes, J. Fixed Point Theory Appl. 23, 24. Burgos Gil, J.I., Kramer, J. and K¨ uhn, U. (2005). Arithmetic characteristic classes of automorphic vector bundles, Documenta Math. 10, pp. 619-716. Burgos Gil, J.I. and Lit¸kanu, R. (2010). Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions, Documenta Math. 15, pp. 73-176. Cartan, H. (1961). Th´eorie ´el´ementaire des fonctions analytiques d’une ou plusieurs variables complexes, Hermann, Paris. English translation : Elementary Theory of Analytic Functions of One or Several Complex Variables, Dover Publications, New York, 1995. Cartan, H. et Serre, J.-P. (1953). Un th´eor`eme de finitude concernant les vari´et´es analytiques compactes, C. R. Acad. Sci. Paris 237, pp. 128-130.

Bibliography

575

Corrˆea, M. and Suwa, T. (2020). Localization of Bott-Chern classes and Hermitian residues, J. London Math. Soc. 101, pp. 349-372. de Jong, T. and Pfister, G. (2000). Local analytic geometry, Advanced Lectures in Math., Vieweg, Wiesbaden. Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D. (1975). Real homotopy theory of K¨ ahler manifolds, Invent. math. 29, pp. 245-274. Demailly, J.-P. (2012). Complex analytic and differential geometry, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf. Dimca, A. (1992). Singularities and Topology of Hypersurfaces, Springer, Berlin. Douady, A. (1968). Flatness and privilege, Enseign. Math. 14, pp. 47-74. Ebeling, W. and Gusein-Zade, S. (2001). On the index of a holomorphic 1-form on an isolated complete intersection singularity, Doklady Math. 64, pp. 221-224. Ehresmann, C. (1950). Les connexions infinit´esimales dans un espace fibr´e diff´erentiable, Colloque de Topologie, Bruxelles, pp. 29-55. Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Math. 150, Springer, Berlin. Fischer, G. (1976). Complex analytic geometry, Lecture Notes in Math. 538, Springer, Berlin. Freed, D.S. (2021). The Atiyah-Singer index theorem, Bull. Amer. Math. Soc. 58, pp. 517-566. Fritzsche, K. and Grauert, H. (2002). From holomorphic functions to complex manifolds, Graduate Texts in Math. 213, Springer, Berlin. Fr¨ olicher, A. (1955). Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. 41, pp. 641-644. ˇ Fujisawa, K. (2018). Thom form in equivariant Cech-de Rham theory, Intern. J. Math. 29, 1850050. Fulton, W. (1984). Intersection theory, Springer, Berlin. Fulton, W. and Johnson, K. (1980). Canonical classes on singular varieties, Manuscripta Math. 32, pp. 381-389. Gibson, C.G., Wirthm¨ uller, K., du Plessis, A.A. and Looijenga, E.J.N. (1976). Topological stability of smooth mappings, Lecture Notes in Math. 552, Springer, Berlin. Gillet, H. and Soul´e, C. (1990). Characteristic classes for algebraic vector bundles with Hermitian metric, I, Ann. Math. 131, pp. 163-203. Godement, R. (1958). Topologie Alg´ebrique et Th´eorie des Faisceaux, Hermann, Paris. G´ omez-Mont, X., Seade, J. and Verjovsky, A. (1991). The index of a holomorphic flow with an isolated singularity, Math. Ann. 291, pp. 737-751. Goresky, M. and MacPherson, R. (1980). Intersection homology theory, Topology 19, pp. 135-162. Goresky, M. and MacPherson, R. (1985). Lefschetz fixed point theorem for intersection homology, Comment. Math. Helv. 60, pp. 366-391. Grauert, H. (1958). On Levi’s problem and the imbedding of real-analytic manifolds, Ann. Math. 68, pp. 460-472.

576

Complex Analytic Geometry

Grauert, H. (1960). Ein Theorem der analytischen Garbentheorie und die Mod´ ulr¨ aume komplexer Strukturen, Publ. math. I.H.E.S. 5, pp.5-64. Grauert, H. and Remmert, R. (1984). Coherent analytic sheaves, Grundlehren der math. Wissenschaften 265, Springer, Berlin. Greenberg, M. and Harper, J. (1981). Algebraic topology : A first course, AddisonWesley. Greuel, G.-M. (1975). Der Gauß-Manin Zusammenhang isolierter Singularit¨ aten von vollst¨ andigen Durchschnitten. Math. Ann. 214, pp. 235-266. Griffiths, P. and Harris, J. (1978). Principles of Algebraic Geometry, John Wiley & Sons, New York. Grothendieck, A. (1957). Sur quelques points d’alg`ebre homologique, Tˆ ohoku Math. J. 9, pp. 119-227. Grothendieck, A. (1960-61). Techniques de construction en g´eom´etrie analytique. II. G´en´eralit´es sur les espaces annel´es et les espaces analytiques, S´eminaire Henri Cartan 13, no 1, exp. no 9, pp. 1-14. Guillemin, V. and Pollack, A. (1974). Differential topology, Prentice-Hall, Englewood Cliffs, New Jersey. Gunning, R. and Rossi, H. (1965). Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, New Jersey. Hamm, H. (1971). Lokale topologische Eigenschaften komplexer R¨ aume, Math. Ann. 191, pp. 235-252. Hartshorne, R. (1966). Residues and duality, Lecture Notes in Math. 20, Springer, Berlin. Hartshorne, R. (1977). Algebraic geometry, Graduate Texts in Math. 52, Springer, Berlin. Harvey, R. and Lawson, H.B. (1993). A theory of characteristic currents associated with a singular connection, Ast´erisque 213, Soc. Math. France. Hattori, A. (1991). Topology, Iwanami Shoten, Tokyo (in Japanese). Hauser, H. (2003). The Hironaka theorem on resolution of singularities (Or : A proof we always wanted to understand ), Bull. Amer. Math. Soc. 40, pp. 323403. Hironaka, H. (1964). Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. of Math. 79, pp. 109-203, pp. 205-326. Hirsch, M.W. and Mazur, B (1974). Smoothings of piecewise linear manifolds, Ann. of Math. Studies 80, Princeton Univ. Press, Princeton, New Jersey. Hirzebruch, F. (1966). Topological methods in algebraic geometry, Third ed., Grundlehren der math. Wissenschaften 131, Springer, Berlin. Honda, N., Izawa, T. and Suwa, T. (2023). Sato hyperfunctions via relative Dolbeault cohomology, J. Math. Soc. Japan 75, pp. 229-290. Hotta, R. (1987). Introduction to algebra - Groups, rings & modules -, Shokabo, Tokyo (in Japanese). Husemoller, D. (1994). Fibre bundles, Third ed., Graduate Texts in Math. 20, Springer, Berlin. Ikeda, T, (2018), Lectures on enumerative geometry, Univ. of Tokyo Press, Tokyo (in Japanese).

Bibliography

577

Iversen, B. (1971). Critical points of an algebraic function, Inventiones Math. 12, pp. 210-224. ´ Norm. Sup. 9, pp. 155Iversen, B. (1976). Local Chern classes, Ann. Scient. Ec. 169. Izawa, T. and Suwa, T. (2003). Multiplicity of functions on singular varieties, Internat. J. Math. 14, pp. 541-558. Kashiwara, M. and Schapira, P. (1990). Sheaves on manifolds, Grundlehren der math. Wissenschaften 292, Springer, Berlin. Kobayashi, S. and Nomizu, K. (1963). Foundations of differential geometry, Vol. I, John Wiley & Sons, New York. Kodaira, K. (1951). The theorem of Riemann-Roch on compcat analytic surfaces, Amer. J. Math. 73, pp. 813-875. Kodaira, K. (1954). On K¨ ahler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. Math. 60, pp. 28-48. Kodaira, K. (1960). On compact complex analytic surfaces, I, Ann. Math. 71, pp. 111-152. Kodaira, K. (1986). Complex manifolds and deformation of complex structures, Grundlehren der math. Wissenschaften 283, Springer, Berlin. Lˆe, D.-T. (1974). Calculation of Milnor number of isolated singularity of complete intersection, Funct. Anal. Appl. 8, pp. 127-131. Lee, J.M. (2013). Introduction to smooth manifolds, Second ed., Graduate Texts in Math. 218, Springer, Berlin. Lefschetz, S. (1926). Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28, pp. 1-49. Lefschetz, S. (1937). On the fixed point formula, Ann. Math. 38, pp. 819-822. ˇ Lehmann, D. (1991). Syst`emes d’alv´eoles et int´egration sur le complexe de Cechde Rham, Publications de l’IRMA 23 No VI, Universit´e de Lille I. Lehmann, D., Soares, M. and Suwa, T. (1995). On the index of a holomorphic vector field tangent to a singular variety, Bol. Soc. Bras. Mat. 26, pp. 183199. Levy, R.N. (1987). The Riemann-Roch theorem for complex spaces, Acta Math. 158, pp. 149-188. Lojasiewicz, S. (1991). Introduction to complex analytic geometry, Birkh¨ auser. Looijenga, E.J.N. (1984). Isolated singular points on complete intersections, London Math. Soc. Lecture Note Series 77, Cambridge Univ. Press, Cambridge. Macaulay, F.S. (1916). Algebraic theory of modular systems, Cambridge Tracts 16, Cambridge Univ. Press, Cambridge. MacPherson, R. (1974). Chern classes for singular varieties, Ann. Math. 100, pp. 423-432. Malgrange, B. (1957). Faisceaux sur des vari´et´es analytiques r´eelles, Bull. Soc. Math. France 85, pp.231-237. Mather, J. (2012). Notes on topological stability, Bull. Amer. Math. Soc. (New Series) 49, pp. 475-506. Matsumura, H. (1980). Commutative algebra, Second ed., Benjamin/Cummings, Reading, Massachusetts.

578

Complex Analytic Geometry

Matsushima, Y. (1972). Differentiable manifolds, Translated by E.T. Kobayashi, Marcel Dekker, New York. Milnor, J. (1968). Singular points of complex hypersurfaces, Ann. of Math. Studies 61, Princeton Univ. Press, Princeton, New Jersey. Milnor, J. and Stasheff, J. (1974). Characteristic classes, Ann. of Math. Studies 76, Princeton Univ. Press, Princeton, New Jersey. Mumford, D. (1961). The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. I.H.E.S. 9, pp. 5-22. Munkres, J.R. (1966). Elementary differential topology, Ann. of Math. Studies 54, Princeton Univ. Press, Princeton, New Jersey. Munkres, J.R. (1975). Topology, a first course, Prentice-Hall, Englewood Cliffs, New Jersey. Narasimhan, R. (1966). Introduction to the theory of analytic spaces, Lecture Notes in Math. 25, Springer, Berlin. Noguchi, H. and Fukuda, T. (1976). Elementary catastrophes, Kyoritsuzensho 208, Kyoritsu Shuppan, Tokyo (in Japanese). Noguchi, J. (2016). Analytic function theory of several variables : Elements of Oka’s coherence, Springer, Berlin. Ohmoto, T. and Shiota, M. (2017). C 1 -triangulations of semialgebraic sets, J. Topology 10, pp. 765-775. Oka, M. (1997). Non-degenerate complete intersection singularity, Actualit´es Math´ematiques, Hermann, Paris. Parusi´ nski, A. (1988). A generalization of the Milnor number, Math. Ann. 281, pp. 247-254. Parusi´ nski, A. and Pragacz, P. (1995), A formula for the Euler characteristic of singular hypersurfaces, J. Algebraic Geom. 4, pp. 337-351. Parusi´ nski, A. and Pragacz, P. (2001), Characteristic classes of hypersurfaces and characteristic cycles, J. Algebraic Geom. 10, pp. 63-79. Rolfsen, D. (2003). Knots and links, AMS Chelsea Publishing, Amer. Math. Soc., Providence, Rhode Island. Rourke, C. and Sanderson, B. (1972). Introduction to piecewise-linear topology, Springer, Berlin. Rudin, W. (1987). Real and complex analysis, Third ed., McGraw-Hill, New York. Sakai, F. (1984). Weil divisors on normal surfaces, Duke Math. J. 51, pp. 877-887. Sch¨ urmann, J.and Yokura, S (2007). A survey of characteristic classes of singular spaces, Singularity Theory, Dedicated to Jean-Paul Brasselet on his 60th birthday, World Scientific, Singapore, pp. 865-952. Schwartz, M.-H. (1965). Classes caract´eristiques d´efinies par une stratification d’une vari´et´e analytique complexe, C.R. Acad. Sci. Paris 260, pp. 3262-3264, 3535-3537. Seade, J. (2019). On Milnor’s fibration theorem and its offspring after 50 years, Bull. Amer. Math. Soc. 56, pp. 281-348. Seade, J. and Suwa, T. (1996). A residue formula for the index of a holomorphic flow, Math. Ann. 304, pp. 621-634.

Bibliography

579

Seade, J. and Suwa, T. (1998). An adjunction formula for local complete intersections, Internat. J. Math. 9, pp. 759-768. Serre, J.-P. (1955a). Un th´eor`eme de dualit´e, Comment. Math. Helv. 29, pp. 9-26. Serre, J.-P. (1955b). Faisceux alg´ebriques coh´erents, Ann. Math. 61, pp. 197-278. Serre, J.-P. (1956). G´eom´etrie alg´ebrique et g´eom´etrie analytique, Ann. Inst. Fourier 6, pp. 1-42. Serre, J.-P. (1975). Alg`ebre Locale - Multiplicit´es, Troisi`eme ´ed., Lecture Notes in Math. 11, Springer, Berlin. Shiga, K. (1976). Manifold theory I, Geometry i, Iwanamikˆ oza Kisosˆ ugaku, Iwanami Shoten, Tokyo (in Japanese). Shiota, M. (1997). Geometry of subanalytic and semialgebraic sets, Progress in Math. Birkh¨ auser, Boston. Spanier, E.H. (1966). Algebraic topology, McGraw-Hill, New York. Steenrod, N. (1951). The topology of fiber bundles, Princeton Univ. Press, Princeton, New Jersey. Suwa, T. (1996). Classes de Chern des intersections compl`etes locales, C.R. Acad. Sci. Paris 324, pp. 67-70. Suwa, T. (1998). Indices of vector fields and residues of singular holomorphic foliations, Actualit´es Math´ematiques, Hermann, Paris. Suwa, T. (2000a). Dual class of a subvariety, Tokyo J. Math. 23, pp. 51-68. Suwa, T. (2000b). Characteristic classes of coherent sheaves on singular varieties, Singularities – Sapporo 1998, 279-297, Adv. Stud. Pure Math., 29, Kinokuniya, Tokyo, 2000. Suwa, T. (2003a). Residues of Chern classes, J. Math. Soc. Japan 55, pp. 269-287. Suwa, T. (2003b). Characteristic classes of singular varieties, Sugaku Expositions, Amer. Math. Soc. 16, pp. 153-175. Suwa, T. (2005). Residues of Chern classes on singular varieties, Singularit´es Franco-Japonaises, Marseille 2002, S´eminaires et Congr`es 10, Soc. Math. France, pp. 265-285. Suwa, T. (2008). Residue theoretical approach to intersection theory, real and complex singularities, Contemporary Math., Amer. Math. Soc. 459, pp. 207-261. ¯ ˇ Suwa, T. (2009). Cech-Dolbeault cohomology and the ∂-Thom class, Singularities – Niigata-Toyama 2007, Adv. Studies in Pure Math. 56, Math. Soc. Japan, pp. 321-340. Suwa, T. (2012). Residues of singular holomorphic distributions, Singularities in Geometry and Topology, Strasbourg 2009, IRMA Lectures in Mathematics and Theoretical Physics 20, European Math. Soc., pp. 207-247. Suwa, T. (2014). GSV-indices as residues, J. of Singularities 9, pp. 206-218. Suwa, T. (2022a). Relative Dolbeault cohomology, Riv. Mat. Univ. Parma 13, pp. 307-352. Suwa, T. (2022b). Residues and hyperfunctions, Chapter 8 in Handbook of Geometry and Topology of Singularities III, J.L. Cisneros-Molina, D.-T. Lˆe and J. Seade eds., Springer Berlin. Suwa, T. (2023). Representation of relative sheaf cohomology, Forum Math. 35, pp. 1047-1076.

580

Complex Analytic Geometry

Tsikh, A.K. (1988). Weakly holomorphic functions on complete intersections, and thier holomorphic extension, Math. USSR Sbornik 61, pp. 421-436. Vick, J.W. (1994). Homology theory : An introduction to algebraic topology, Second ed., Gaduate Texts in Math., Springer, Berlin. Voisin, C. (2007). Hodge theory and complex algebraic geometry. I, Translated from the French by Leila Schneps, Reprint of the 2002 English edition, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge. Weil, A. (1952). Sur les th´eor`emes de de Rham, Comment. Math. Helv. 26, pp. 119-145. Wells, R.O. Jr. (2008). Differential analysis on complex manifolds, Third ed., Graduate Texts in Math. 65, Springer, New York. Whitehead, J.H.C. (1940). On C 1 -complexes, Ann. Math. 41, pp. 809-824. Whitney, H. (1936). Differentiable manifolds, Ann. Math. 37, pp. 645-680. Whitney, H. (1965). Tangents to an analytic variety, Ann. Math. 81, pp. 496-549. Whitney, H. (1972). Complex analytic vrieties, Addison-Wesley Publishing Co. Yokura, S. (2020). Motivic Milnor-Hirzebruch classes revisited, Singularities – Kagoshima 2017, World Scientific, Singapore, pp. 265-294.

Index

Betti number, 149, 272, 557 B´ezout’s theorem, 464 Bianchi identity, 242 blowing-up, 33, 273, 373, 374, 488 Bochner-Martinelli form, 129, 230, 283, 355 Bott-Chern class, 285 cohomology, 284 boundary, 524 operator, 524, 550 bundle canonical, 71 cotangent holomorphic, 71 real, 69 determinant, 67 hyperplane, 73, 146, 283, 310 line, 62 normal holomorphic, 72 real, 70 principal, 58 quotient, 63 sub, 63 tangent holomorphic, 71 real, 69 tautological, 59, 81, 273 universal, 81, 146, 273 vector, 61 virtual, 250, 497

Abelian category, 536 acyclic complex, 527 adjunction formula, 166, 468 affine algebraic variety, 30 map, 550 space, 549 Alexander duality, 104 isomorphism, 114 morphism, 415 algebraic manifold, 31 vector bundle, 65 alternating cochain, 182, 184 analytic inverse image, 366 restriction, 366 variety, 28 analytically good covering, 354 angular form, 228, 230 global, 228 associated prime, 543 Atiyah class, 280 form, 280 Atiyah-Singer index theorem, 521 atlas, 24 augmentation, 551 barycenter, 550 barycentric subdivision, 569 581

582

Complex Analytic Geometry

C-algebra, 358 C-ringed space, 360 canonical bundle, 71 divisor, 373 cap product, 560, 568 in forms, 214 Cartier divisor, 369, 478 Cauchy form, 129, 230 in n-variables, 355 integral formula, 6 Cauchy-Riemann equations, 6 ˇ Cech cohomology, 179 of a sheaf, 333 of forms, 181 of locally constant functions, 180 complex, 180 theorem, 200 ˇ Cech-de Rham cohomology, 183 complex, 183 Stokes theorem, 194, 196 theorem, 199 ˇ Cech-Dolbeault cohomology, 354 cell, 562 characteristic map of, 562 complex, 562 finite, 562 homology of, 564 sub, 562 dual, 97 regular, 565 cellular decomposition, 31, 97, 562 chain, 524 complex, 523 equivalence, 525 homotopy, 525 morphism, 524 characteristic class, 247, 279 form, 246 chart, 24

Chern character, 497 form, 500 class of a complex manifold, 152 relative, 135 via Chern-Weil theory, 239 via obstruction theory, 133 form, 247 Chern-Weil theory, 239 coboundary, 526 coboundary operator, 526 ˇ in Cech complex, 179, 181 cochain, 526 complex, 525 equivalence, 527 homotopy, 526 cocycle, 526 cofinal, 533 Cohen-Macaulay (CM) module, 545 coherent sheaf, 343 cohomologous, 526 cohomology Bott-Chern, 284 ˇ Cech, 179 ˇ Cech-de Rham, 183 ˇ Cech-Dolbeault, 354 de Rham, 178 Dolbeault, 264 of a cochain complex, 526 of sheaves, 332 ˇ relative Cech-de Rham, 205 relative de Rham, 206 singular, 554 collar neighborhood theorem, 86 complete intersection, 376 global, 378 set-theoretic, 376 complex acyclic, 527 ˇ Cech, 180 ˇ Cech-de Rham, 183 chain, 523 cochain, 525 de Rham, 177, 339 Dolbeault, 264, 339

Index

double, 531 of sheaves, 339 complex analytic space, 361 complex conjugate of vector space, 46 of vector bundle, 68 complex curve, 25 complex manifold, 24 complex slice, 92 complex space, 361 model, 361 complex structure, 24 complex subspace, 365 complex surface, 25 complexification of vector bundle, 68 cone, 136 connecting morphism, 525 connection, 240 dual, 252 Hermitian, 281 matrix, 242 metric, 282 of type (1, 0), 278 coordinate neighborhood, 25 system, 25 coordinates homogeneous, 26 local, 25 cotangent bundle holomorphic, 71 real, 69 sheaf, 381 covering analytically good, 354 good, 186, 340 simplicial, 180 covering map, 27, 39 critical point, 11 set, 376 cross product, 113, 561 cup product, 201, 559, 568 ˇ of Cech-de Rham cochains, 200

583

curvature, 241 matrix, 242 cycle, 524 deformation retract, 552 proper, 552 degree of a map, 128 of a projective algebraic space, 485 determinant bundle, 67 de Rham cohomology, 178 complex, 177, 339 theorem, 199 difference form, 247 differentiable manifold, 43 differential, 44 differential form, 161 of type (p, q), 164 with coefficients in a vector bundle, 163 dimension Krull, 40, 544 of a complex manifold, 24 of a C r manifold, 43 of a variety, 40 topological, 48 direct image sheaf, 329 direct limit, 14, 326, 533 direct sum of connections, 250 of vector bundles, 66 discriminant of a polynomial, 540 divisor canonical, 373 Cartier, 369, 478 effective, 372, 479 principal, 370 Weil, 370, 478 Dolbeault cohomology, 264 complex, 264, 339 Theorem, 352 Dolbeault-Grothendieck lemma, 351

584

Complex Analytic Geometry

domain, 348 holomorphically convex, 349 double complex, 531 dual cell, 97 duality Alexander, 104 Lefschetz, 107 Poincar´e, 101 effective divisor, 372, 479 Ehresmann fibration theorem, 92 elementary invariant polynomial, 243 embedding, 27 epimorphism, 523 equivariant, 52 Euler class, 134, 141, 150 localized, 147 relative, 136 sequence, 274 Euler-Poincar´e characteristic of a holomorphic vector bundle, 358 of a space, 149, 272 exact sequence of sheaves, 328 of vector bundles, 63 excellent, 483 exceptional divisor, 373 excision, 207, 555 exterior derivative, 162 exterior product of connections, 252 of differential forms, 162 family of sections (r-section), 128 fiber bundle, 56 fiber integration, 169 ˇ on CdR cochains, 227 filtration, 528 Hodge, 268 fine sheaf, 335 finite homology type, 557 map, 38 finitely generated module, 534

fixed point, 52 flat module, 535 fraction field, 539 ring, 543 frame, 64, 74, 128 orthonormal, 258, 261 free module, 534 Frobenius theorem, 172 Fr¨ olicher spectral sequence, 271 Fubini-Study form, 275, 283 metric, 276 function algebraic, 33 analytic one variable, 5 several variables, 8 holomorphic, 26 one variable, 6 several variables, 10 meromorphic, 8, 15, 369, 478, 544 of class C r , 5 real analytic, 43 regular, 32 fundamental class, 101, 140 cycle, 101, 140 Gauss’ lemma, 19, 540 Gauss-Bonnet formula, 152, 403, 444 general point of a pseudo-manifold, 140 genus, 277 germ of a complex space, 364 of a holomorphic function, 14 of a variety, 34 good covering, 186, 340 Grassmann manifold, 26, 77, 146 Grothendieck residue, 391 spectral sequence, 538 Grothendieck-Riemann-Roch theorem, 518

Index

585

group action, 51 effective, 52 free, 52 properly discontinuous, 54 transitive, 52 GSV-index, 443 Gysin morphism, 117, 418, 511

Hopf manifold, 55, 59, 275, 276 Hurewicz theorem, 61 hypercohomology, 340 hyperplane, 31, 483 bundle, 73, 146, 283, 310 hypersurface, 42 projective algebraic, 31

harmonic form, 263, 265 height, 544 Hermitian connection, 281 form, 262 inner product, 259 metric, 260 vector bundle, 260 vector space, 259 Hilbert basis theorem, 20, 541 Hodge filtration, 268 metric, 277 structure, 269 holomorphic distribution, 304 foliation, 174 tangent bundle, 71 space, 46 vector field, 72 holomorphically convex domain, 349 homogeneous space, 52 homologous, 524 homology of a chain complex, 524 of locally finite chains, 98, 570 singular, 551 homomorphism, 523 homotopy, 551 chain, 525 cochain, 526 equivalence, 551 exact sequence, 60 group, 59 inverse, 551 proper, 552 honeycomb system, 189

ideal maximal, 15, 542 primary, 34, 542 prime, 34, 542 principal, 36 immersion, 27, 45 implicit mapping theorem, 12 in general position (general), 485 index of a 1-form, 403 of a section, 134, 457 of an r-section, 130 Poincar´e-Hopf, 150, 402 virtual, 441 injective module, 534 inner product, 256 Hermitian, 259 integral domain, 14, 539 integration along fibers, 169 of differential forms, 167 intersection of complex subspaces, 365 intersection product, 117 localized, 121 invariant (symmetric) polynomial, 243 inverse image sheaf, 329 inverse mapping theorem, 12 irreducible decomposition of a variety, 36 of an element, 19, 539 element, 19, 539 globally, 42 variety, 35 isomorphism Alexander, 114

586

Complex Analytic Geometry

of fiber bundles, 57 of modules, 523 of vector bundles, 62 Poincar´e, 114 Thom, 114 isotopy, 138 Jacobian matrix, 11 K-group, 497 K¨ ahler manifold, 276 metric, 276 Kodaira embedding theorem, 278 Kodaira-Serre duality, 266, 267, 358 Koszul complex, 506 Kronecker product, 98, 527 Krull dimension, 40, 544 K¨ unneth formula, 561 Laplacian, 263, 265, 267 Lefschetz duality, 107 fixed point formula, 236 morphism, 415 line bundle, 62 associated with a divisor, 369 associated with a hypersurface, 73 linearly equivalent, 370 link, 136, 565 of singularity, 379 local C-algebra, 358 local complete intersection (LCI), 381 local defining function, 28 map, 28 local ring, 15, 542 localization, 141, 289–291, 426 differential geometric, 307, 389, 429 topological, 141, 389, 426 localized intersection product, 121 locally free sheaf, 342 m-fold point, 364, 484 manifold complex, 24

differentiable, 43 K¨ ahler, 276 piecewise C ∞ , 139 piecewise-linear, 137 projective algebraic, 31 pseudo, 139 quasi-projective, 32 real analytic, 43 topological, 43 with boundary, 85 map biholomorphic, 10, 26 finite, 38 holomorphic, 10, 26 mapping degree, 128 maximum principle, 7, 13 meromorphic function, 8, 15, 369, 478, 544 metric Fubini-Study, 276 Hermitian, 260 Hodge, 277 K¨ ahler, 276 Riemannian, 257 Milnor fiber, 379 number, 380 minimal polynomial, 540 model complex space, 361 monomorphism, 523 morphism of complex spaces, 361 of modules, 523 of sheaves, 327, 330 of vector bundles, 62 multiplicative subset, 543 multiplicity of a function, 380, 392, 404, 447 of a variety at a point, 483 of an irreducible component, 405, 451 Nakayama’s lemma, 542 Newton’s formula, 498 nilpotent element, 362, 542 non-singular variety, 29

Index

normal bundle holomorphic, 72 real, 70 sheaf, 381 space, 45 nowhere dense, 29 Nullstellensatz, 39 for principal ideals, 37 obstruction, 129 open mapping theorem, 7 orbit, 52 order of a meromorphic section, 370, 479 of a pole, 7 of a power series, 15 in a variable, 15 of a zero, 7 order of a covering, 48 order relation (ordering), 532 of vertices, 99 simplicial, 99, 568 total, 533 orientation of a cell, 563 of a fiber bundle, 88 of a manifold, 84 of a simplex, 566 of a vector bundle, 88 paracompact, 25 piecewise C ∞ homeomorphism, 137 manifold, 139 piecewise-linear (PL) manifold, 137 map, 136 Poincar´e duality, 101 in forms, 215, 266 isomorphism, 114 lemma, 178 morphism, 414 residue map, 166

587

Poincar´e-Hopf index, 150, 402 theorem, 151 polarization, 243 pole of a meromorphic function, 7, 369, 478 polydisk, 4 polyhedron, 136 of a simplicial complex, 565 presheaf, 326 primary decomposition, 543 ideal, 34, 542 prime element, 20, 539 ideal, 34, 542 principal divisor, 370 principal ideal, 36 domain (PID), 541 product cap, 560, 568 cup, 201, 559, 568 exterior, 162 Kronecker, 527 ˇ of Cech cochains, 201 product formula for Thom classes, 114 projection formula, 171, 227 projective algebraic manifold, 31 algebraic variety, 31 space, 25 projective dimension, 546 projective module, 534 proper deformation retract, 552 homotopy, 552 map, 552 mapping theorem, 42 proper case, 389, 433 proper ideal, 541 proper transform, 374 properly discontinuous action, 54 properly ordered index set, 182 pseudo-manifold, 139

588

Complex Analytic Geometry

pull-back ˇ in Cech cohomology, 203 ˇ in Cech-de Rham cohomology, 204 of a connection, 249 of a differential form, 163 of a section, 65 of a vector bundle, 64 push-forward in cohomology, 109 quasi-projective manifold, 32 variety, 32 radical of an ideal, 34, 542 rank of a vector bundle, 62 real analytic function, 43 manifold, 43 reduced complex space, 362 germ of a holomorphic function, 346 reduced expression, 182, 184 refinement, 24 regular cell, 565 function, 32 in a variable, 15 point of a holomorphic map, 11, 27 of a variety, 29 section, 384 sequence, 544 relative ˇ Cech-de Rham cohomology, 205 Chern class, 135 de Rham cohomology, 206 Euler class, 136 homology, 553 residue, 290, 293 differential geometric, 307 Grothendieck, 391 theorem, 143, 148, 294, 308, 427 topological, 141, 427 transverse, 144, 149, 294 resolution of a sheaf, 339

restriction of a sheaf, 326 resultant, 540 Ricci identity, 303 Riemann extension theorem, 21 sphere, 26 surface, 25, 407, 496 Riemannian metric, 257 vector bundle, 257 ring local, 15, 542 Noetherian, 20, 541 of convergent power series, 14 of germs of holomorphic functions, 14 Sard’s theorem, 45 Schubert cell, 79 class, 147 cycle, 80 variety, 80, 146 section of a fiber bundle, 58 of a sheaf, 326 self-intersection number, 488 semianalytic set, 387 series, 2 absolutely convergent, 2 convergent, 2 multi, 2 normally convergent, 3 power, 4 Taylor, 10 sheaf, 325 coherent, 343 constant, 326 cotangent, 381 direct image, 329, 367 fine, 335 inverse image, 329 locally free, 342 morphism, 327, 330 normal, 381 of continuous functions, 331

Index

of C ∞ (p, q)-forms, 332 of C ∞ functions, 331 of Abelian groups, 325 of finite type, 342 of holomorphic functions, 331 of meromorphic functions, 368 of modules, 330 of real analytic functions, 331 of relations, 343 of rings, 330 restriction, 326 tangent, 332, 381 simplex, 550 standard, 550 simplicial complex, 565 covering, 180 ordering, 99, 568 singular chain, 550 homology, 551 point of a holomorphic map, 11, 27 of a variety, 29 of an r-section, 128 simplex, 550 slice, 92 complex, 92 smoothing of a PL manifold, 138 spectral sequence, 528 Fr¨ olicher, 271 Grothendieck, 538 sphere, 26 stabilizer, 52 stalk, 325 standard simplex, 550 star, 103, 136, 565 open, 103, 565 star operator, 262 Stiefel manifold complex, 74 real, 76 Stokes theorem, 168 ˇ Cech-de Rham, 194, 196 stratification, 47, 190 Whitney, 47, 190, 412

589

subanalytic set, 388 triangulation, 388 submersion, 27, 45 support of a divisor, 371, 478 of a sheaf, 330 system of honeycomb cells, 189 tangent bundle holomorphic, 71 real, 69 virtual, 382 cone, 483 sheaf, 332, 381 space holomorphic, 46 real, 44 tautological bundle, 59, 81, 273 tensor product of connections, 252 Thom class as a localized Euler class, 153 ˇ in Cech-de Rham cohomology, 225 of a complex subspace, 510 of a complex vector bundle, 312 of a submanifold, 110, 225 of a subvariety, 416 of an oriented vector bundle, 111, 226 isomorphism, 110, 114, 225 morphism, 416, 510 Todd class, 498 form, 500 topological dimension, 48 manifold, 43 torsion-free module, 535 total transform, 374, 491 transversality, 90 transverse residue, 144, 149, 294

590

Complex Analytic Geometry

triangulation, 96 C ∞ , 96 subanalytic, 388 tube, 411 tubular neighborhood, 89 theorem, 89 complex, 514 unique factorization domain (UFD), 19, 539 uniqueness of analytic continuation, 13 unit, 15, 539 universal bundle, 81, 146, 273 universal coefficient theorem, 558 variety affine algebraic, 30 analytic, 28 projective algebraic, 31 quasi-projective, 32 vector bundle, 61 algebraic, 65 Hermitian, 260 Riemannian, 257

vector field, 69, 161 holomorphic, 72 virtual bundle, 250, 497 cotangent bundle, 446 index, 441 multiplicity of a function, 447 tangent bundle, 382, 440 volume form, 259, 262 Weierstrass division theorem, 17 polynomial, 16 preparation theorem, 18 Weil divisor, 370, 478 Whitney stratification, 47, 190, 412 sum, 66 formula, 155, 250, 311, 317, 458 umbrella, 30 Zariski toplogy, 32 zero section, 64