Holomorphic Foliations with Singularities: Key Concepts and Modern Results (Latin American Mathematics Series) 3030767043, 9783030767044

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Latin American Mathematics Series

Bruno Scárdua

Holomorphic Foliations with Singularities Key Concepts and Modern Results

IMCA

Instituto de Matemática y Ciencias Afines

Latin American Mathematics Series

This new series aims to gather monographs, textbooks and contributed volumes based on mathematical research conducted at/with institutions based in Latin America. This series is open to subseries, which could be topical – with contributions from a myriad of institutions – or exclusively supplied by a single Latin American research institution. Standalone books are also welcome. Books published in cooperation with research institutions may carry the name and the logo of the research institute on the cover. While it is expected that many subseries will be linked to a research institute or University located in Latin America, the creation of topical subseries, not necessarily related to institutions, is also possible. All subseries should have their own editorial board comprised of distinguished international researchers with a strong academic background. Legal permission for Springer to commercially use the institution’s brand and logo may be required.

More information about this series at http://www.springer.com/series/15993

Bruno Scárdua

Holomorphic Foliations with Singularities Key Concepts and Modern Results

Bruno Scárdua Instituto de Matemática Universidade Federal do Rio de Janeiro Rio de Janeiro, Brazil

Latin American Mathematics Series ISBN 978-3-030-76704-4 ISBN 978-3-030-76705-1 (eBook) https://doi.org/10.1007/978-3-030-76705-1 Mathematics Subject Classification: 14-XX, 37-XX, 57-XX, 14Pxx, 32Mxx, 32Sxx, 37Fxx, 57R30 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my mother Deníncia Azevedo Scardua.

Preface

The theory of foliations is one of those subjects in mathematics that gathers several distinct domains such as topology, dynamical systems, and geometry, among others. Its origins go back to the works of C. Ehresmann and Shih [28, 29] and G. Reeb [71, 72]. It provides an interesting and valuable approach to the qualitative study of dynamics and ordinary differential equations on manifolds. Although its origins are in the classical framework of real functions and manifolds, the notion of foliation is also very useful in the holomorphic world. Indeed, it has ancient origins in the study of complex differential equations. From these first problems, the introduction of singularities as an object of study is a natural step. We mention the works of P. Painlevé [64, 65] and Malmquist [54]. With P. Painlevé, the study of rational complex differential equations of the form dy P (x,y) dx = Q(x,y) has its first more specific methods and results. After Painlevé, many authors have contributed for the initial push up of the theory, among them are E. Picard, G. Darboux, H. Poincaré, H. Dulac, Briot, and Bouquet. Complex differential equations appear naturally in mathematics and in natural sciences [3, 4, 38]. For instance, we mention the theory of electrical circuits, valves, and electromagnetic waves [61]. Another motivation is the search for and study of new (classes of) transcendent functions, as the Liouvillian functions [77, 79]. With the advent of the geometric theory of foliations and the modern results of Cartan, Oka, Nishino, Suzuki, and others, on the theory of analytic functions of several complex variables and some from algebraic and analytic geometry, this field of research became quite active again. These days, it is one of the active branches of modern research in mathematics. These are the notes of a series of lectures delivered by the author in the Graduate School of Mathematical Sciences at the University of Tokyo in October 2015. They were meant to be as self-contained as possible, taking into account time and space. The basic idea was to introduce the concepts and some of the basic results in the theory of holomorphic foliations with singularities. Another goal is to guide the reader to some of the recent questions and problems in the field, providing, in this way, a motivating introduction to those who are interested in studying a new subject.

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I wish to express my gratitude to Professor Taro Asuke for his personal effort in making this project possible and for his warm hospitality. I want to thank Professor T. Tsuboi for his kind hospitality and support. I want to thank all those in the Graduate School of Mathematical Sciences at the University of Tokyo for their support and for making my stay in Tokyo such a pleasant and fruitful period. The first draft of this book was conceived during a visit to the Instituto de Matemática y Ciencias Afines – IMCA (Lima). It is my honor to present these notes in the series Monografias del IMCA. I am very much indebted to this prestigious institute for all the support during the last couple of decades. My special thanks to Professors Félix Escalante and Roger Metzger for their warm hospitality. Rio de Janeiro, Brazil April 2021

Bruno Scárdua

Contents

1

The Classical Notions of Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Definition of Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Other Definitions of Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Frobenius Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 6 9

2

Some Results from Several Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Some Extension Theorems from Several Complex Variables . . . . . . 2.2 Levi’s Global Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 14

3

Holomorphic Foliations: Non-singular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Identity Principle for Holomorphic Foliations . . . . . . . . . . . . . . . . . 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 22 23

4

Holomorphic Foliations with Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Linear Vector Fields on the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 One-Dimensional Foliations with Isolated Singularities . . . . . . . . . . . 4.3 Differential Forms and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Codimension One Foliations with Singularities . . . . . . . . . . . . . . . . . . . . 4.5 Analytic Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Two Extension Lemmas for Holomorphic Foliations . . . . . . . . . . . . . . 4.7 Kupka Singularities and Simple Singularities . . . . . . . . . . . . . . . . . . . . . . 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 26 30 31 33 34 36 37

5

Holomorphic Foliations Given by Closed 1-Forms . . . . . . . . . . . . . . . . . . . . . 5.1 Foliations Given by Closed Holomorphic 1-Forms. . . . . . . . . . . . . . . . . 5.1.1 Holonomy of Foliations Defined by Closed Holomorphic 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 40

ix

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5.2

Foliations Given by Closed Meromorphic 1-Forms . . . . . . . . . . . . . . . . 5.2.1 Holonomy of Foliations Defined by Closed meromorphic 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Reduction of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Irreducible Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Poincaré and Poincaré–Dulac Normal Forms . . . . . . . . . . . . . . . . . . . . . . . 6.3 Blow-up at the Origin (Quadratic Blow-up) . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Blow-up on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Resolution of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Blow-up of a Singular Point of a Foliation. . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Irreducible Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Separatrices: Dicriticalness and Existence . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Holonomy and Analytic Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Holonomy of Irreducible Singularities . . . . . . . . . . . . . . . . . . . . 6.8.2 Holonomy and Analytic Classification of Irreducible Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 51 53 54 56 58 61 62 65 65 67 69 71

7

Holomorphic First Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Mattei–Moussu Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Groups of Germs of Holomorphic Diffeomorphisms . . . . . . . . . . . . . . 7.3 Irreducible Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Case of a Single Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 80 81 83 83

8

Dynamics of a Local Diffeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Hyperbolic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Parabolic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Elliptic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 85 87 89

9

Foliations on Complex Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Complex Projective Plane and Foliations . . . . . . . . . . . . . . . . . . . . . . 9.2 The Theorem of Darboux–Jouanolou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Foliations Given by Closed 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Riccati Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Examples of Foliations on CP (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Example of an Action of a Low-dimensional Lie Algebra . . . . . . . . . 9.7 A Family of Foliations on CP (3) Not Coming from Plane Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 95 98 101 103 107

5.3 6

42 46

110 113

Contents

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10

Foliations with Algebraic Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Limit Sets of Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Groups of Germs of Diffeomorphisms with Finite Limit Set . . . . . . 10.3 Virtual Holonomy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Construction of Closed Meromorphic Forms . . . . . . . . . . . . . . . . . . . . . . . 10.5 The Linearization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 117 120 120 125 127 128

11

Some Modern Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Holomorphic Flows on Stein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Suzuki’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Proof of the Global Linearization Theorem . . . . . . . . . . . . . . . 11.2 Real Transverse Sections of Holomorphic Foliations . . . . . . . . . . . . . . 11.3 Non-trivial Minimal Sets of Holomorphic Foliations . . . . . . . . . . . . . . 11.4 Transversely Homogeneous Holomorphic Foliations . . . . . . . . . . . . . . 11.4.1 Transversely Lie Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Transversely Affine Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Transversely Projective Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Development of a Transversely Projective Foliation—Touzet’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Projective Structures and Differential Forms . . . . . . . . . . . . . .

131 131 132 134 137 141 142 144 144 146

12

147 148

Miscellaneous Exercises and Some Open Questions . . . . . . . . . . . . . . . . . . . . 157 12.1 Miscellaneous Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.2 Some Open Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Chapter 1

The Classical Notions of Foliations

This chapter is intended to introduce the classical notions of foliation in the real framework. The reader which is already familiar with these notions may skip to the next chapter. We refer to [11, 31, 37] or [62] for a more complete exposition of the theory of real foliations.

1.1 Definition of Foliation There are some ways of motivating the concept of foliation. Probably, the very first is given by a submersion f : M → N from a manifold M into a manifold N . If f is sufficiently differentiable (usually of class C r , r ≥ 2) then by the local form of submersions, the level sets f −1 (y), y ∈ N are embedded submanifolds of M. These fibers are locally organized as the fibers of a projection (x, y) → y. This local picture is not necessarily global, and the fibers may be disconnected. A second important example is given by a closed non-singular 1-form ω on a manifold M. Again, under sufficient differentiability conditions, by the integration lemma of Poincaré we can write locally ω = df for a submersion map f taking values on K. Here K is the field of real numbers if ω is differentiable and M is a real differentiable manifold. In case M is a complex manifold and ω is a holomorphic 1-form on M we have K = C the field of complex numbers. In this latter case f is holomorphic. Any local function f as above, defined in an open subset in M, is called a first integral for ω. Notice that two local first integrals f and f for ω in a same connected subset of M are related by f = f + constant. Therefore, they share level sets, these local sets can therefore be globalized as immersed (locally closed) submanifolds of M, again locally organized as fibers of a projection. The third and last basic example we shall mention is the one provided by a differentiable, namely C r with r ≥ 1, vector field X on a manifold M. Given a © Springer Nature Switzerland AG 2021 B. Scárdua, Holomorphic Foliations with Singularities, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-76705-1_1

1

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1 The Classical Notions of Foliations

non-singular point p ∈ M which is not a singular point of X, the flow box theorem gives a conjugation between (X, U ), where p ∈ U ⊂ M is an open neighborhood, and a constant vector field on Rm , where m = dim M. The orbits of X in U then follow the same geometrical condition of the above examples. The above examples motivate the classical definition of foliation below as follows: Definition 1.1.1 (Foliation, [11, 31, 37, 62]) Given a differentiable manifold M of dimension m and class C r , r ≥ 0; by a codimension 0 ≤ n ≤ m foliation of class C r of M, we mean an atlas F = {(Uj , ϕj )}j ∈J of M, where each coordinate chart ϕj : Uj ⊂ M → ϕj (Uj ) ⊂ Rm−n × Rn is of class C r and we have the following compatibility condition: For each non-empty intersection Ui ∩ Uj = φ, the corresponding change of coordinates  ϕj ◦ ϕi−1 ϕ (U ∩U ) : ϕi (Ui ∩ Uj ) −→ ϕj (Ui ∩ Uj ) i

i

j

preserves the natural horizontal fibration y = const. of Rm−n × Rn  (x, y). This is equivalent to say that, in coordinates (x, y) ∈ Rm−n × Rn , we have   ϕj ◦ ϕi−1 (x, y) = hij (x, y), gij (y) ∈ Rm−n × Rn . The charts ϕj : Uj → ϕj (Uj ) are called foliation charts, trivializing charts, or distinguished charts of F. The local plaques of F are the fibers of a foliation chart in F. Given a diffeomorphism ϕ : U → ϕ(U ) ⊂ Rm = Rm−n × Rn of class C r , we say that ϕ is compatible with the foliation F if for any j ∈ J such that Uj ∩ U = ∅, we also have   ϕj ◦ ϕ −1 (x, y) = h(x, y), g(y) ∈ Rm−n × Rn . In short, this is equivalent to say that F ∪ {(U, ϕ)} is still a foliation. Using this and Zorn’s lemma, we may consider the foliation atlas F as maximal, in the sense that it contains all the compatible charts of class C r of M. In M we consider the equivalence relation induced by the connected finite union of local plaques. This means that two points x, y ∈ M are equivalent x ∼ y iff x and y lie in the same plaque of F or there is a finite number of plaques P1 , . . . , Pr , r ≥ 2 of F such that x ∈ P1 , y ∈ Pr and Pi ∩ Pi+1 = ∅ for all i = 1, . . . , r − 1. Given a point x ∈ M we call the corresponding equivalence class [x] ⊂ M as the leaf of F through x. Usually we denote this leaf by Fx or by Lx . The leaf Lx ⊂ M is an immersed C r submanifold, but not necessarily embedded. These leaves then decompose M into disjoint immersed C r submanifolds. Each leaf has dimension m−n and meets a foliation chart domain along plaques of the foliation. For instance, in the case of a submersion f : M → N, the leaves of the corresponding foliation are the connected components of the level sets f −1 (y), y ∈ N. The quotient space M/ ∼ is the leaf space of F, also denoted by M/F.

1.2 Other Definitions of Foliation

3

1.2 Other Definitions of Foliation According to the literature, there are essentially three ways to define foliations in the real differentiable manifolds (cf. [11, 31, 37]). In addition to the one we just have given in Definition 1.1.1 above, we have the following. Let M be a m-dimensional manifold, m ∈ N. Let D k be the open unit ball of Rk where k ∈ N. Let 0 ≤ n ≤ m be fixed. Definition 1.2.1 A foliation of M, of codimension n and class C r , is a partition F of M consisting of pairwise disjoint immersed C r submanifolds L ⊂ M of dimension m − n, distributed as follows: for each point x ∈ M there is a neighborhood U of x, and a C r diffeomorphism ϕ : U → D m−n × D n , such that for each y ∈ D n there is L ∈ F satisfying ϕ −1 (D m−n × y) ⊂ L. The elements of the partition F are the leaves of F. The element Lx of F containing x ∈ M is the leaf of F containing x. We observe that, not every decomposition of M into immersed submanifolds with the same dimension is a foliation (see [62]). The third definition of foliation uses the notion of distinguished maps. Let F = {(Uj , ϕj ) , j ∈ J } be a foliation of a manifold M in the sense of Definition 1.1.1. Then ∀i, j the transition map ϕj ◦ (ϕi )−1 has the form ϕj ◦ (ϕi )−1 (x, y) = (fi,j (x, y), gi,j (y)). The map gi,j is a local diffeomorphism in its domain of definition. This follows from the fact that the derivative of the transition map is given by D(ϕj ◦ (ϕi )−1 )(x, y) · (v, w) = (∂x fi,j (x, y) · v, Dgi,j (y) · w), (v, w) ∈ Rm−n × Rn . We define for all i the map gi = 2 ◦ ϕi , where 2 is the projection onto the second coordinate: 2 : D m−n × D n → D n , (x, y) → y. We claim that gj = gi,j ◦ gi . j j Indeed, we have gi,j ◦ gi = gi,j ◦ i2 ◦ ϕi = 2 ◦ (ϕj ◦ ϕi−1 ) ◦ ϕi = j ◦ ϕj = gj . r m Therefore, a C foliation F of codimension n of a manifold M is equipped with an open cover {Ui }i∈I of M and C r submersions gi : Ui → D n such that for all i, j there is a local diffeomorphism gi,j : Vi ⊂ D n → Vj ⊂ D n satisfying the cocycle relations gj = gi,j ◦ gi , gi,i = Id . The gi ’s are the distinguished maps of F.  Ui such that for Conversely, suppose that M m admits an open cover M = i∈I

each i ∈ I there is a C r submersion gi : Ui → D n such that for all i, j there is a diffeomorphism gi,j : Vi ⊂ D n → Vj ⊂ D n satisfying the cocycle relations above.

4

1 The Classical Notions of Foliations

By the local form of the submersions, we can assume that for each i ∈ I there is a C r diffeomorphism ϕi : Ui → D m−n × D n such that gi = 2 ◦ ϕi . since 2 ◦ (ϕj ◦ (ϕi )−1 ) = gj ◦ (ϕi )−1 = gi,j ◦ gi ◦ (ϕi )−1 = gi,j ◦ 2 , we have that the atlas F = {(Ui , ϕi )}i∈I defines a foliation of class C r and codimension n of M. The above suggests the following equivalent definition of foliation. Definition 1.2.2 A foliation of M m of class C r and of codimension n is given by the following: 1. An open cover {Ui : i ∈ I } of M. 2. A family of C r submersions gi : Ui → D n , ∀i ∈ I ; with the following compatibility property: ∀i, j ∈ I with Ui ∩ Uj = ∅, there is a local diffeomorphism gi,j : Vi ⊂ D n → Vj ⊂ D n satisfying the cocycle relations gj = gi,j ◦ gi , gi,i = Id . The submersions gi ’s are the distinguished maps of the foliation F. This last definition leads to several interesting definitions. For instance, a foliation F of M is said to be transversely holomorphic or transversely affine depending on whether, for some convenient choice, its distinguished maps gi,j are holomorphic or affine maps. We shall resume this subject later on. In order to distinguish foliations, we shall use the following definition. Definition 1.2.3 Two foliations F and F  of manifolds M and M  respectively are C r -equivalent if there is a C r -diffeomorphism h : M → M  (h is a homeomorphism if r = 0), sending leaves of F into leaves of F  . In other words, if Fx denotes the leaf of F that contains x ∈ M and Fy denotes the leaf of F  that contains y ∈ M  then we have  , ∀ x ∈ M. h(Fx ) = Fh(x)

The above notion can be stated for the case of holomorphic objects, in the obvious way. This relation defines an equivalence in the space of foliations.

1.3 Frobenius Theorem

5

1.3 Frobenius Theorem Let X, Y be two vector fields on a manifold M and p ∈ M be fixed. Denote by Xt the local flow of X and similarly by Yt the local flow of Y assuming that X, Y ∈ C r , r ≥ 2. Given p ∈ M we define Xt∗ (Y )(p) = DX−t (Xt (p)) · Y (Xt (p)) ∈ Tp (M). Note that Xt∗ (X)(p) = X(p), ∀ t. All this holds for |t| small enough. Definition 1.3.1 The Lie bracket of X, Y is the vector field [X, Y ] on M defined at each point p ∈ M by LX (Y )(p) = [X, Y ](p) =

d  (X∗ (Y )(p)) X, Y ∈ C r , r ≥ 2. dt t=0 t

In local coordinates (x1 , . . . , xm ) ∈ M, the Lie bracket [X, Y ] has the following form: writing X=

m  i=1

 ∂ ∂ ai , Y = bi ∂xi ∂xi m

i=1

one has [X, Y ] =

m   ∂bj ∂aj ∂ ai − bi . ∂xi ∂xi ∂xj

i,j =1

When X and Y are defined in an open set of Rm , the formula above yields [X, Y ] = DY (p) · X(p) − DX(p) · Y (p). A vector field X on M is tangent to a plane field P on M (denoted by X ∈ P ) if X(p) ∈ P (p) for all p ∈ M. Definition 1.3.2 A plane field P on M is involutive if X, Y ∈ P ⇒ [X, Y ] ∈ P . Lemma 1.3.3 If F is a foliation, then its associated plane field T F is involutive. Proof Let X, Y be two vector fields tangent to T F. By using suitable local coordinates (x, y) = (x1 , . . . , xm−n , y1 , . . . , yn ) such that F is given in these coordinates by y = (y1 , . . . , yn ) = const., one can assume that X, Y are of the form X(x, y) = (f (x, y), 0), Y (x, y) = (g(x, y), 0)     f g ∂ g∂ g ∂ f ∂y f [X, Y ] = x y − x 0 0 0 0 0 0 = (f · ∂x f − f · ∂x f, 0).

6

1 The Classical Notions of Foliations

Hence [X, Y ] ∈ T F and the proof follows.

 

A plane field P on M of dimension k is completely integrable if there is a foliation F of M of codimension k such that T F = P , i.e., if for each p ∈ M we have Tp (F) = P (p). In particular, given p ∈ M, the space P (p) is the tangent space of the leaf Lp ∈ F of F that contains p. From the above lemma, a completely integrable plane field is involutive. The converse of the above fact is a well-known result in theory of foliations: Theorem 1.3.4 (Frobenius Theorem [11, 19, 31]) Involutive plane fields are completely integrable. Example 1.3.5 (Integrable Systems of Differential Forms) Let ω1 , . . . , ωr be differential 1-forms of class C r on a manifold M and assume that they are linearly independent at each point p ∈ M n . We call the set S := {ω1 , . . . , ωr } a system of 1-forms on M. We may consider the corresponding distribution P (S) of (n − r)dimensional planes defined as follows: given p ∈ M we set P (S)(p) ⊂ Tp M as P (S)(p) = {v ∈ Tp M, ωj (p) · v = 0, j = 1, . . . , r}. The system {ω1 , . . . , ωr } is called integrable if we have dωj ∧ ω1 ∧ . . . ∧ ωr = 0 for all j = 1, . . . , r. In particular, a distribution given by a 1-form ω is integrable iff ω ∧ dω = 0. This occurs for instance if we have a closed 1-form ω with ω(p) = 0, ∀p ∈ M. Claim 1.3.6 The system S is integrable if and only if the distribution P (S) is involutive. The proof of the claim is left as an exercise. Therefore, according to Frobenius theorem above, S is integrable if and only if P (S) is completely integrable. In the case of an integrable non-singular 1-form ω (i.e., a 1-form ω satisfying ω ∧ dω = 0), we have a codimension one foliation F of M which is defined by the Pfaffian equation ω = 0.

1.4 Holonomy The concept of holonomy of a foliation is motivated by the concept of return map or Poincaré map of a periodic orbit of a vector field (see Figure 1.1). Remark 1.4.1 Given a manifold  of class C r and a point p ∈  we denote by Diffr (, p) the group of germs of diffeomorphisms of class C r fixing p ∈ . Each germ is induced by a C r diffeomorphism f : (U, p) → (V , p), where U, V ⊂  are neighborhoods of p, and f (p) = p. The “return map” π : (, p) → (, p), x → π(x) above illustrated is, always under suitable differentiability conditions, a germ of diffeomorphism.

1.4 Holonomy

7

Fig. 1.1 Representation of a first return map π : (, p) → (, p), x → π(x), of a periodic orbit γ of a real vector field

Identifying the transverse section  with a disc in Rm−1 centered at the origin 0 ∈ Rm−1 (m = dimension of the ambient manifold), we may consider f as a (germ of a) diffeomorphism (Rm−1 , 0) → (Rm−1 , 0) (fixing the origin). The same kind of idea above gives the concept of holonomy group of a leaf of a foliation. Let us introduce it in a more formal way: Let F be a codimension n foliation of class C r of a manifold M m . Given a leaf L of F we fix a point p ∈ L and, by a local trivialization of F around p, choose a transverse section p ∈ p ⊂ M, diffeomorphic to a disc in Rn and transverse to all the leaves of F. Let now γ : [0, 1] → L be any closed path of class C r . Then by suitable choice of the local foliation charts, we can cover the image γ ([0, 1]) by domains of foliation charts (ϕj , Uj ), j = 0, . . . ,  with the following properties: (a) There is a partition 0 = t0 < t1 < · · · < tj < tj +1 < . . . < t = 1 of [0, 1] such that γ ([tj , tj +1 ]) ⊂ Uj . In particular, we have Uj ∩ Uj +1 = ∅. (b) For each j = 0, . . . ,  − 1 the union Uj ∪ Uj +1 is contained in some foliation chart domain. Moreover, the same holds for the union U ∪ U0 . (c) There is a neighborhood p ∈ A ⊂  such that for each y ∈ A there is a path of plaques from the plaque Py0 of (ϕ0 , U0 ) through y, to the plaque P (1)y of (U1 , ϕ1 ) that meets Py0 , then from the plaque P (1)y to the plaque Py2 of (U2 , ϕ2 ) that meets P (1)y , and so on. Thus we reach a plaque Py of (U , ϕ ) that meets the plaque Py−1 . The intersection Py ∩A is a single point fγ (y) ∈  called the γ -holonomy image of y. This defines a map germ fγ : (, p) → (, p) which has the following properties: (i) fγ has the same differentiability class as that of F. (ii) fγ does not depend on the choice of the cover {(ϕj , Uj )} of γ ([0, 1]) (as a germ). (iii) The map germ fγ only depends on the homotopy class [γ ] ∈ π1 (L; p).

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1 The Classical Notions of Foliations

(iv) The map HolL : π1 (L; p) → Diffr (; p), from the fundamental group of L based at p, into the group of germs of C r diffeomorphism of  fixing p, and given by [γ ] → fγ , is a well-defined group homomorphism. The image is denoted by Hol(F, L, ; p) ⊂ Diffr (; p). (v) The groups Hol(F, L, ; p) depend on  and p by natural conjugation by diffeomorphisms. Thus we may speak of the holonomy group of the leaf L denoted by Hol(F, L) or simply by Hol(L), and identify it with (a conjugacy class of) a subgroup of Diffr (Rn , 0), where n is the codimension of the foliation F of M. We close this chapter illustrating the concepts above introduced by means of a remarkable example. Example 1.4.2 (Foliations Generated by Closed 1-Forms) In order to illustrate the above concepts of foliation and Holonomy, we consider the class of codimension one foliations generated by closed differential 1-forms. Indeed, we prove the following: Proposition 1.4.3 A codimension one smooth foliation F tangent to a closed nonsingular C ∞ 1-form ω on a manifold M has trivial holonomy. Proof Indeed, let us fix a Riemannian metric on M. Let X be the gradient of ω, i.e., the smooth vector field on M defined by ωp (vp ) = X(p), vp , for all p ∈ M and vp ∈ Tp M. Clearly X is well-defined and non-singular since ω is non-singular. In addition X is transverse to F. Let L be a leaf of F and γ a closed curve in L. We can assume that γ : S 1 → L is an immersion. Set I = [−1, 1] and define the map φ : S 1 × I → S = φ(S 1 × I ) by φ(θ, t) = Xt (γ (θ )), where Xt (γ (θ )) stands for the integral curve of X with initial point at γ (θ ). It is clear that φ is an immersion of class C r , r ≥ 2. Then ω∗ = φ ∗ (ω) is a well-defined 1-form on S 1 × I . Because dω∗ = dφ(ω∗ ) = φ ∗ (dω) = φ ∗ (0) = 0 we have that ω∗ is closed. Hence ω∗ defines a foliation F ∗ on S 1 × I . Note that F ∗ is conjugated to F ∩ S. It follows that the curves γ ∗ = S 1 × 0 and γ have the same holonomy. Let us calculate the holonomy of γ ∗ . Fix (θ ∗ , 0) ∈ γ ∗ and  ∗ = {θ ∗ } × I . Clearly  ∗ is a transversal of F ∗ . Let f ∗ : Dom(f ∗ ) ⊂  ∗ →  ∗ be the holonomy of γ ∗ , p ∈ Dom(f ∗ ) and q = f ∗ (p). Let α be an arc in  ∗ joining p and q. Let l be a path in a leaf of F ∗ joining p, q. Let R be the closed region bounded by the curves γ ∗ , l, and α ∗ . Because

0= R

dω∗ =



ω∗ = ∂R

l

ω∗ +



ω∗ = 0 + α

α

ω∗

1.5 Exercises

9

one has

ω∗ = 0.

α

This equality implies that α is trivial and so p = q = f ∗ (p). We conclude that γ ∗ has trivial holonomy. Hence γ has trivial holonomy and the proof follows.  

1.5 Exercises 1. Show the equivalence between Definitions 1.1.1, 1.2.1, and 1.2.2 of foliation of a manifold. 2. Prove Claim 1.3.6 in Example 1.3.5. 3. Is there any dimension one C ∞ foliation F, without singularities, defined in a neighborhood of the unit disc D 2 ⊂ R2 such that F is tangent to the boundary circle S 1 = ∂D 2 ? Is it possible to have a foliation as above (i.e., smooth and non-singular) which is transverse to the boundary circle at every point? 4. Prove that a codimension one foliation F on a manifold M admits a simple closed curve γ : S 1 → M which is transverse to F at each point provided that one of the following conditions is verified: (a) F has some non-closed leaf in M. (b) M is compact and all leaves are compact. Hint: Let L0 be a non-closed leaf of F. Given p ∈ L0 \ L0 we choose a local Flow Box p ∈ Up and a small transverse section p  p p ⊂ Up . We have Lp = L0 . Given two points p1 , p2 ∈ p ∩ L0 belonging to two distinct plaques of F|Up we choose a path a : [0, 1] → L0 joining p1 to p2 . Now we choose a fibration transverse to F having basis along a(I ). Now it is easy to cut p and glue (emend) it with a piece of transverse section σ to F contained in the fibers of the fibration above. Then we may modify this construction in order to obtain a “smooth” transverse section to F. 5. Prove that a codimension one foliation of a compact manifold M always admits a simple closed curve transverse to F. 6. Prove that if F is a codimension one foliation of a manifold M such that all leaves of F are compact, then every leaf of F has finite holonomy group. Indeed, for any leaf of F the holonomy group is trivial or is cyclic of order two. 7. (Reeb foliation) The aim of this exercise is to construct the Reeb foliation on 2 the sphere S 3 . The details are left to the reader. Denote by D = {(x1 , x2 ) ∈ 2 R2 ; x12 + x22 ≤ 1} the closed unit disc on R2 . Let f : D × R → R be given C∞

by f (x1 , x2 , x3 ) = ρ(x12 + x22 ) · ex3 where ρ : R −→ R is a function such that ρ(0) = 1, ρ(1) = 0, ρ  (t) < 0, ∀ t > 0.

10

1 The Classical Notions of Foliations

Fig. 1.2 Reeb foliation of the solid torus

The foliation F of R3 given by the level surfaces of f is of codimension one and has the cylinder C  S 1 × R (given by C = {(x1 , x2 , x3 ); x12 + x22 = 1}) as leaf. Also the interior of C, given by {(x1 , x2 , x3 ); x12 + x22 < 1} is a union of leaves all diffeomorphic to R2 . The leaves except for cylinder C are (diffeomorphic to) elliptic paraboloids.   1 . In this case the For instance we may take ρ(r) := exp − exp 1−r 2 2

leaves of F outside D × R are the cylinders x12 + x22 = r 2 , r > 1 and leaves inside paraboloids, since they are graphs of  the solid cylinder are elliptic 2 1 x3 = exp 1−r 2 + cte ∈ R. Along D × [0, 1] we identify the points (x1 , x2 , 0) 2

and (x1 , x2 , 1) obtaining, in this way, a solid torus D × S 1 still equipped with 2 a foliation tangent to the boundary ∂D × S 1 ∼ = S 1 × S 1 , having other leaves 2 diffeomorphic to R . 2 2 This foliation of D × S 1 is called Reeb foliation of D × S 1 . By gluing two copies of these through the boundary, we can obtain a Reeb foliation of the three-sphere S 3 (see Figure 1.2). Complete the details. 8. Is it possible to find a decomposition of a manifold M into pairwise disjoint immersed submanifolds Sα ⊂ M, α ∈ A, for some set of indexes A, all of the same dimension, but not corresponding to a foliation of M? Can this example occur in dimension dim M = 2?

Chapter 2

Some Results from Several Complex Variables

In the course of the text, we shall refer to some results from the theory of several complex variables. For the sake of clarity, we shall now state them separately.

2.1 Some Extension Theorems from Several Complex Variables This chapter is dedicated to some useful extension theorems from several complex variables. We shall start with a statement due to Riemann. We recall that a subset X ⊂ M of a topological space is nowhere dense if the closure X has empty interior. Theorem 2.1.1 (Riemann Extension Theorem, [34]) Let M be a connected complex manifold and X ⊂ M an analytic nowhere dense subset of M. Then a holomorphic function f on M \ X which is bounded near X has a unique holomorphic extension to M. We shall now state local and simple versions of two powerful results from the theory of functions and analytic sets in several complex variables: Theorem 2.1.2 (Hartogs’ Extension Theorem, [34]) Let U ⊂ Cn be a connected open subset and W ⊂ U be a codimension ≥ 2 analytic subset. Then any holomorphic (respectively, meromorphic) function f defined on the open subset U \ W admits a unique holomorphic (respectively, meromorphic) extension to U . Remark 2.1.3 The same holds for (as an immediate consequence) vector fields and differential forms. We recall that an analytic subset of a complex space is one given locally by set of common zeros of (local) holomorphic functions. We shall refer to [35] (Chapter B pages 15–16) and [36] (Chapter II, pages 89–92) for the notion of irreducible analytic set and properties. In few words, an analytic set W ⊂ M, of a complex manifold M, is reducible if it writes © Springer Nature Switzerland AG 2021 B. Scárdua, Holomorphic Foliations with Singularities, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-76705-1_2

11

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2 Some Results from Several Complex Variables

W = W1 ∪ W2 where W1 , W2 ⊂ M are analytic sets and Wj = W, j = 1, 2. Otherwise W is called irreducible (in M). An analytic set is irreducible if and only if the set of its regular points (points where it is a complex submanifold of maximal dimension) is connected. The theorem of Thullen, Remmert, and Stein stated below gives conditions for the closure of an analytic set to be analytic. Theorem 2.1.4 (Theorem of Thullen et al. [35] page 132, [36] page 169) Let M be a complex manifold, W ⊂ M an irreducible analytic subset of M, and V an irreducible analytic subset of M \ W ; such that dim(V ) > dim(W ). Then the closure V ⊂ M is an analytic subset of dimension dim(V ). Also we need: Theorem 2.1.5 (Theorem of Chow, [35] page 150, [36] page 170 ) A (closed) analytic subvariety on a complex projective space is algebraic. And the following immediate consequence: Corollary 2.1.6 An irreducible closed analytic subset of pure codimension one of the complex projective space CP (n) is an algebraic hypersurface.

2.2 Levi’s Global Extension Theorem Recall that given an open subset U ⊂ Cn , n ≥ 2 and a map f : U → R of class C 2 we say that f is plurisubmarhomic (plush for short) or strictly plurisubmarhomic (s-plush for short) if ∀ pi ∈ U and ∀ v ∈ Cn ; the restriction u : z → f (p0 + zv) is subharmonic or strictly subharmonic, in the sense that the Laplacian satisfies u ≥ 0 or u > 0 respectively. If we denote by Hf (p0 ) the complex n × n matrix  Hf (p0 ) =

∂ 2f (p0 ) ∂zi ∂zj

i,j =1,...,n

 then Hf (p0 ) is Hermitian Hf (p0 ) = Hf (p0 )t . It well-known that: Lemma 2.2.1 Let f : U ⊂ Cn → R be a C 2 map in the open subset U . Then: 1. f is plush in U ⇔ Hf∀ p∈U is definite non-negative. 2. f is s-plush in U ⇔ Hf∀ p∈U is definite non-positive.

2.2 Levi’s Global Extension Theorem

13

As a consequence: Lemma 2.2.2 Given f ∈ C 2 (U ), U ⊂ Cn , n ≥ 2 as above, then we have: 1. f is plush in U ⇔ for each holomorphic curve γ : V ⊂ C → U the map f ◦ γ : V → R is subharmonic. 2. f is s-plush in U ⇔ f ◦ γ : V → R is strictly subharmonic for every holomorphic immersed curve γ : V ⊂ C → U . Let us see how to extend this to complex manifolds. Given f ∈ C 2 (U ) as above we define the Levi form of f as the quadratic form Lf (p) :=

n n   ∂ 2f (p) dzi dzj , p ∈ U. ∂zi ∂zj j =1 i=1

Thus we have a quadratic form at Tp Cn defined by Lf (p) · ω =

∂ 2f (p) · wi wj = w · Hf (b) · w t , ∀ ω = (w1 , . . . , wn ) ∈ Cn . ∂zi ∂zj

Using this form we can state (cf. [35, 70]): 1. f is plush in U ⇔ Lf (p) ≥ 0, ∀ p ∈ U. 2. f is s-plush in U ⇔ Lf (p) > 0, ∀ p ∈ U. Given a holomorphic map φ : V → U , V ⊂ Cm open subset, we can use the Taylor expansion of order two for f in order to prove that Lf ◦φ = φ ∗ (Lf ) for the Levi-forms of f and of f ◦ φ [35, 70]. Given now f : M → R of class C 2 , M n a complex manifold and a chart φ : U → Cn of M we can consider the Levi form of f at p ∈ M ⊂ U as the quadratic form on the (complex) tangent space Tp (M) defined by Lf (p) · v := Lf ◦φ −1 (φ(p)) · (Dφ(p) · v), ∀ v ∈ Tp (M). This is well defined according to the above remarks. Finally we reach the following definition: C2

Definition 2.2.3 ([39, 70, 80]) Given f : M −→ R, where M is a complex manifold, we say that f is plurisubharmonic (plush for short), respectively strictly plurisubharmonic (s-plush for short) if Lf (p) ≥ 0, ∀ p ∈ M (respectively, Lf (p) > 0, ∀ p ∈ M). Given 1 ≤ k ≤ n we shall say that f is k-(strictly plurisubharmonic) if ∀ p ∈ M, ∃ a subspace E ⊂ Tp (M) of (complex) dimension k such that Lf (p)E > 0, i.e., ∀ v ∈ E \ {0} Lf (p) · v > 0. Let M be a differentiable manifold. An exhaustion of M is a continuous function g : M → R such that: (a) g is bounded from below g ≥ c in M. (b) g is proper: ∀ sequence {pn } ⊂ M with no accumulation point in M, the sequence g(pn ) satisfies g(pn ) → +∞ as n → +∞.

14

2 Some Results from Several Complex Variables

A Stein manifold is a complex manifold admitting a C ∞ s-plush exhaustion (Hörmander’s theorem [39]). On the affine space Cn the function f (z) = ||z||2 is an s-plush exhaustion. The important result below is due to Levi: Theorem 2.2.4 (Levi’s Global Extension Theorem, [80]) Let M be a complex manifold admitting a k-s-plush exhaustion where k ≥ 2. If K ⊂ M is compact subset such that M \ K is connected, then any meromorphic q-form ω on M \ K admits a unique extension as a meromorphic q-form on M. We also need the following result: Proposition 2.2.5 Let X ⊂ CP (n) be an algebraic subset defined by k homogeneous polynomials in Cn+1 . Then the open manifold M = CP (n) \ X admits a -s-plush exhaustion where  = n − k + 1. In particular if X is an algebraic hypersurface (codimension one) the M = CP (n) \ X is a Stein manifold. Proof If X is defined by the homogeneous polynomials f1 , . . . , fk in Cn+1 , then we define f : M → R by setting in homogeneous coordinates (z1 , . . . , zn+1 ) ∈ Cn+1

f (z1 , . . . , zn+1 ) := ln

n+1

|zj |2 )q , k

qj 2 |fj (z)|

 (

j =1

j =1

where if dj = deg(fj ), then q1 , . . . , qk ∈ N are such that d1 q1 = · · · = dk qk = q ∈ N. Computing the Levi form of f we can conclude by using the following: Lemma 2.2.6 ∀ x ∈ Cn the quadratic form Q :=

n 

|dxj |2 +

j =1



|xi dxj − xj dxi |2

i 0, i.e., by |y|λ |x|−μ = |c| (we may assume that λ, −μ ∈ R+ still with λ/μ ∈ R \ Q). • λ/μ ∈ C\R (hyperbolic case): In this case again the leaves are not closed, except for the axes, and all leaves accumulate on both axes. Indeed, the leaves are closed outside of the two axes (meaning that a leaf accumulates only at the two axes). • λ/μ ∈ R+ : exercise (think of the radial vector field)!

4.2 One-Dimensional Foliations with Isolated Singularities From what we have seen above, a holomorphic vector field X with singular set sing(X) ⊂ M on a complex manifold M defines a dimension one holomorphic foliation F(X)∗ on the open set M ∗ = M \ sing(X) (remark: sing(X) ⊂ M is an analytic subset, assumed to be proper, so M \ sing(X) is open and connected). The leaves of F(X)∗ are the orbits of X on M ∗ , i.e., the non-singular orbits of X. A

4.2 One-Dimensional Foliations with Isolated Singularities

27

natural question is whether any one-dimensional holomorphic foliation is induced by a holomorphic vector field. The answer is no, indeed there are manifolds which may be equipped with one-dimensional foliations, but that do not admit non-trivial holomorphic vector fields. We shall see this in a while. Let us now introduce one of the main concepts in this text: In what follows, as usual, M is a connected complex manifold. Definition 4.2.1 A holomorphic foliation of dimension one of M with singularities is a pair F := (F ∗ , S) where S ⊂ M is a proper analytic subset and F ∗ is a (classical) one-dimensional holomorphic foliation of M ∗ := M \ S. We call S the singular set of F, and write S = sing(F). The leaves of F are by definition the leaves of F ∗ on M ∗ . Remark 4.2.2 Regarding the above concept: (i) We may assume that S is minimal with the property “there is a non-singular foliation of M \S that agrees with F ∗ on some open set" (cf. Propositions 4.6.2 and 4.6.1). (ii) Thanks to (i) above, in these notes we shall assume that F has isolated singularities, i.e., if dim F = 1 then sing(F) is assumed to be a discrete subset of M. The importance of the above concept also relies on the following description of isolated singularities. Lemma 4.2.3 Let F be a one-dimensional holomorphic foliation with (isolated)  singularities on a complex manifold M. Then there is an open cover M = Uj by connected subsets Uj ⊂ M such that:

j ∈J

(i) sing(F) ∩ Uj = φ or sing(F) ∩ Uj = {pj }; (ii) On each Uj , F is given (its plaques) by a holomorphic vector field Xj , with sing(Xj ) = sing(F) ∩ Uj . (iii) If Ui ∩ Uj = φ then on Ui ∩ Uj we have Xi = gij · Xj for some non-vanishing ∗ holomorphic function gij : Ui ∩ Uk → C = C \ {0}. Conversely any such data  Uj , Xj , gij defines a holomorphic, one-dimensional foliation with isolated singularities on a manifold M. Proof The proof is a consequence of the local case, i.e., we may assume that M is a connected neighborhood of the origin 0 ∈ Cm and that sing(F) = {0}. For the sake of simplicity, we assume m = 2. Given now any point p ∈ M \ {0} we consider the tangent space Tp (F) := Tp (Lp ) of the leaf Lp of F through p. Regarded as a line through the origin Tp (Lp ) identifies with an element of the  projective space C2 \ {0} C∗ = CP (1) is the Riemann sphere C = C ∪ {∞}, so that we have defined a map f : M \{0} → C. This map is holomorphic (as a map into C) and extends (by Hartogs’ extension theorem) to a meromorphic map f : M → C. Indeed, the local theory of analytic functions says that f (x, y) writes as a quotient

28

4 Holomorphic Foliations with Singularities

A(x, y) , where A, B are holomorphic functions on M (for M a small B(x, y) bidisc for instance). Now we introduce the holomorphic vector field X on M by X(x, y) =   ∂ ∂ B(x, y) + A(x, y) · Then a local integral curve x(t), y(t) , t ∈ D ⊂ C ∂x ∂y of X satisfies    x(t) ˙ = B x(t), y(t)   y(t) ˙ = A x(t), y(t) f (x, y) =

  A x(t), y(t) A dy y  (t)   , that is = = along the integral curves of X. x  (t) dx B B x(t), y(t) This shows that such integral curves are tangent to the leaves of F and by same dimension we conclude that F = F(X) in M \ {(0, 0)}.   so that

Summarizing: Proposition 4.2.4 Every isolated singularity of a holomorphic foliation of dimension one is defined by a holomorphic vector field. Thus, in a certain sense, the study of singular points of holomorphic foliations of dimension one is the study of singularities of holomorphic vector fields. The next example may require some further knowledge in theory of singularities of projective varieties. Example 4.2.5 (Implicit Complex Ordinary Differential Equations) An algebraic implicit ordinary differential equation in n ≥ 2 complex variables is given by expressions: (∗∗) fj (x1 , . . . , xn , xj ) = 0, where fj (x1 , . . . , xn , y) ∈ C[x1 , . . . , xn , y] are polynomials j = 2, . . . , n and the (x1 , . . . , xn ) ∈ Cn are affine coordinates. Clearly, any polynomial vector field X on Cn defines such an equation. In general (∗∗) defines a one-dimensional singular foliation of some algebraic variety of dimension n. For this we begin by defining Fj (x1 , . . . , xn , y2 , . . . , yn ) := fj (x1 , . . . , xn , yj ) ∈ C[x1 , . . . , xn , y2 , . . . , yn ] polynomials in n + (n − 1) = 2n − 1 variables. Put also Sj := {(x, y) ∈ Cn × Cn−1 ; Fj (x, y) = o}  {(x1 , . . . , xn , yj ) ∈ Cnx × Cyj ; fj (x1 , . . . , xn , yj ) = . 0} × Cn−2 =: j × Cn−2 (y ,...,yˆ ,...,yn ) 2

j

We consider the projectivizations Sj ⊂ CP (2n−1) and the complete intersection subvariety S := S2 ∩ . . . ∩ Sn ⊂ CP (2n − 1). Given by the differential forms ωj := yj dx1 − dxj (j = 2, . . . , n) on Cn × Cn−1 . Then {ωj = 0, j = 2, . . . , n} defines an integrable system on S. We say that the implicit differential equation (∗) is normal if S admits a normalization (desingularization) by blow-ups σ : Sˆ → S. In particular, we obtain in general a singular foliation F(∗∗) of dimension one of the

4.2 One-Dimensional Foliations with Isolated Singularities

29

algebraic n-dimensional subvariety S ⊂ CP (2n − 1). Denote by f1 : S ∩ Cn → C1 the projection in the first coordinate f1 (x1 , . . . , xn , y2 , . . . , yn ) = x1 , and extend it to a holomorphic proper mapping f1 : S → CP (1). Assume now that S admits a normalization σ : Sˆ → S. It is then possible to show that the foliation F(∗∗) lifts ˆ to a foliation by curves F(∗∗) on Sˆ and fˆ1 = f1 ◦ σ defines a holomorphic proper ˆ mapping from S over CP (1). Finally, using Stein factorization theorem [32] we can ˆ

f α find a splitting fˆ1 : Sˆ → B → CP (1) where α : B → CP (1) is a finite ramified covering and fˆ : Sˆ → B is an extended holomorphic fibration over the compact Riemann surface B such that the following diagram therefore commutes: σ Sˆ −→ S fˆ ↓ ↓ f1 α B −→ CP (1)

for a map fˆ1 : Sˆ → CP (1). Example 4.2.6 (Siegel Domain) Consider a linear vector field X=

n 

λj xj

j =1

∂ , ∂xj

on Cn , n ≥ 3. Denote by H(λ1 , . . . , λn ) the convex hull of the set of eigenvalues {λ1 , . . . , λn } ⊂ R2 . The vector field is in the Siegel domain if the origin belongs to H(λ1 , . . . , λn ). In general, given a holomorphic vector field Y in a complex manifold M with a simple singularity at a point p ∈ M, we say that the germ of Y at p is in Siegel domain if the linear part DY (p) is in the Siegel domain. Given a linear vector field X in the Siegel domain as above, the set of tangent points of X with the n n

|xj |2 = 1, X, R = λj |xj |2 = 0. unit sphere S 2n−1 (1) is given by T (X) : j =1

j =1

First we assume that λ1 = 1 and rewrite T (X) : |x1 |2 = 1 −

n

|xj |2 ,

j =2 −a 2
0 and dV = volume form of C3 . In particular, ω embeds in a local action of aff (C).

4.8 Exercises 1. Let F be a foliation of codimension one in an open subset U ⊂ Cn , such that codim(sing(F)) ≥ 2. Prove that F can be represented in U by an integrable holomorphic 1-form, if one of the conditions below is verified: (a) The second problem of Cousin has solution in U . (b) Every meromorphic function f in U can be written since the quotient of two holomorphic functions g and h, f = gh , where codim({g = 0} ∩ {h = 0}) ≥ 2. (c) U = P × (Q \ {0}), where P is a polydisc in Ck and Q is a polydisc in Cm , being m ≥ 2 and k + m = n. 2. Given an integrable holomorphic 1-form  defined in a neighborhood U of the origin 0 ∈ Cn , n ≥ 3 we shall say that the origin 0 ∈ Cn is a Kupka singularity if (0) = 0 but d(0) = 0. Using the Local form of submersions (or equivalently the Inverse map theorem) prove that if the origin is a Kupka singularity, then we can reduce the number of variables in  up to two variables: there is a holomorphic submersion ϕ : (Cn , 0) → (C2 , 0) such that ϕ(0) = 0 and a holomorphic 1-form ω(x, y) defined in a neighborhood of 0 ∈ C2 such that  = ϕ ∗ (ω) in a neighborhood of the origin 0 ∈ Cn . This is already the first part of the Kupka theorem. 3. Let M be a simply-connected manifold, dfo  dfj + λj fo fj r

ω=

j =1

38

4 Holomorphic Foliations with Singularities

a logarithmic form in M, F the logarithmic foliation induced by ω in M, and L ⊂ (fo = 0) a leaf of F. Prove that Hol(L) is conjugate to the subgroup of the group generated by the following set of linear transformations: {z → a.z ; a = e2π iλj , j = 1, . . . , r}. Hint. Consider the “multivalued" function f =fo .f1λ1 . . . frλr . Given a transverse section Σ at a point of (fo = 0) (such that fo | let a submersion and fj does not vanish in  if j ≥ 1), prove that a each element g ∈ H ol(L, ) corresponds to a “determination" of f in . 4. Let F be a logarithmic foliation of a simply-connected manifold M. Prove if L is any leaf of F then the holonomy group of L, Hol(L) is conjugate to a subgroup of the group generated by the following set of linear transformations: {z → λ.z ; λ = e

2π i λλm j

1 ≤ m ≤ r}.

5. Let M be a complex manifold of dimension 2 and F a foliation of dimension one with (isolated) singularities in M. Prove the following statement: There exist collections {Uα }α∈A , {ωα }α∈A and {hαβ }Uα ∩Uβ =φ such that: (i) (ii) (iii) (iv)

{Uα }α∈A is an open cover of M. For every α ∈ A, ωα is a holomorphic 1-form in Uα . If Uα ∩ Uβ = φ, then hαβ ∈ O∗ (Uα ∩ Uβ ) and ωα = hαβ ωβ . If p ∈ Uα is not a singularity of F then Tp F = ker(ωα (p)).

6. Prove the following claim from the text: “Let P and Q be complex manifolds, with P connected, and f : P → Q a map analytic real. Suppose that there exists a nonempty open subset U ⊂ P such  that f U is holomorphic. Then f is holomorphic.” 7. Let X∗ be a holomorphic vector field over a complex torus T = Cn / , satisfying the following invariance condition: (∗) X∗ (p + vj ) = X∗ (p), ∀j = 1, . . . , 2n, ∀p ∈ Cn . Prove that there exists a single holomorphic vector field X in T such that X∗ = π ∗ (X). 8. Let M be a complex manifold and p ∈ M a point. Let us denote by Hom(M, p) the set of germs at p ∈ M of homeomorphisms local in p that leave p fixed, and by Diff(M, p) the set of germs at p of local biholomorphisms that leave p fixed, which will be denoted by Diff(M, p). Prove that Hom(M, p) is a group with an operation of composition and that Diff(M, p) is a subgroup of Hom(M, p).

Chapter 5

Holomorphic Foliations Given by Closed 1-Forms

This chapter is dedicated to the study of an important class of codimension one foliations with singularities. It is the class of foliations given by closed 1-forms. We study their holonomy and some extension property. Starting with the holomorphic case, we are to consider the meromorphic case, making use of the extension results Propositions 4.6.1 and 4.6.2. This study also is related to the classification and normal forms of the isolated singularities of foliations, as suggested for instance by Section 6.2 and Example 4.3.2. We separate the cases of foliations given by closed holomorphic 1-forms and foliations given by closed meromorphic 1-forms.

5.1 Foliations Given by Closed Holomorphic 1-Forms Let M be a complex manifold of dimension ≥ 2 and ω be a closed holomorphic 1-form on M (that is dω = 0) assumed to be not identically zero. Then, ω is integrable ( ω ∧ dω = 0) and therefore it defines a foliation F of M. Classical integration lemma of Poincaré assures that, given any simply-connected open subset U ⊂ M, there exists a holomorphic function f : U → C, such that ω |U = df . Observe that if g : V → C is a function such that dg = ω, where U ∩ V is connected and non-empty, then g − f is constant in U ∩ V . Therefore, as we have already observed before, the corresponding foliation F can be locally defined by holomorphic functions as follows: there exist collections U = {Uα }α∈A , F = {fα }α∈A and C = {cαβ }Uα ∩Uβ =∅ , such that: (i) U is a cover of M by simply-connected open sets. (ii) If α ∈ A, then fα is a holomorphic function in Uα such that dfα = ω |Uα . (iii) If Uα ∩ Uβ = ∅, then Uα ∩ Uβ is connected, cαβ ∈ C, and fα = fβ + cαβ in Uα ∩ Uβ .

© Springer Nature Switzerland AG 2021 B. Scárdua, Holomorphic Foliations with Singularities, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-76705-1_5

39

40

5 Holomorphic Foliations Given by Closed 1-Forms

Observe that if ω is free of singularities, then the functions fα are submersions and F is a non-singular foliation. In this case, if we denote by gαβ the translation gαβ (z) = z + cαβ , then fα = gαβ ◦ fβ , and then F can be defined by local submersions as in (II) of Proposition 3.1.4, in which case the gαβ are translations. We say then that F has an additive transverse structure. In case sing(ω) = ∅, then F has an additive transverse structure in M \ sing(F). Conversely, if F is a foliation with an additive transverse structure in M \sing(F) and such that codim(sing(F)) ≥ 2, then F can be defined by a closed holomorphic 1-form ω on M. Indeed, let be given U = {Uα }α∈A , F = {fα }α∈A and also C = {cαβ }Uα ∩Uβ =φ collections satisfying (ii) and (iii), where U is an open cover of M \ sing(F). From (iii) we obtain that, if Uα ∩ Uβ = ∅, then dfα = dfβ in Uα ∩ Uβ . This implies that there exists the 1-form holomorphic ω on M \ sing(F) such that ω |Uα = dfα . It is not difficult to see that the form ω is closed and defines (induces) the foliation F in M \ sing(F). On the other hand, since codim(sing(F)) ≥ 2, the classical extension theorem of Hartogs then implies that ω extends to a holomorphic form on M, which is also closed and defines a foliation F. We can then state the following result: Proposition 5.1.1 Let M be a complex manifold and F a foliation of M, with singular set of codimension ≥ 2. Then F can be defined by a closed 1-form if, and only if, F has an additive transverse structure in M \ sing(F).

5.1.1 Holonomy of Foliations Defined by Closed Holomorphic 1-Forms Let M be a complex manifold of dimension n ≥ 2 and ω a closed holomorphic 1-form not identically zero in M. Let F be the singular foliation of codimension one defined by ω in M. Our purpose is to prove: Proposition 5.1.2 The foliation F is without holonomy. More precisely, if L ⊂ M \ sing(ω) is a leaf of F, then L has trivial holonomy. Proof We make use of the existence of an additive transverse structure for the foliation. Fix a closed regular curve γ : I → L with γ (0) = γ (1) = po . Given q ∈ γ (I ) there exists a local chart (x, y) : U → Cn−1 ×C such that U ∩L = (y = 0) and ω |U = dy (Poincaré integration lemma). We can then obtain a collection C = {((xj , yj ), Uj )}kj =1 of such charts and a partition {0 = t0 < t1 < . . . < tk = 1} of I , such that: (i) ∪kj =1 Uj = γ (I ). (ii) γ ([tj −1 , tj ]) ⊂ Uj , ∀j = 1, . . . , k. (iii) ω |Uj = dyj , ∀j = 1, . . . , k. Since γ (0) = γ (1), we can assume that:

5.2 Foliations Given by Closed Meromorphic 1-Forms

41

(iv) ((x1 , y1 ), U1 ) = ((xk , yk ), Uk ) = ((x, y), U ), where x(po ) = 0 ∈ Cn−1 . Let us consider the transverse sections j = {(xj , yj ) ∈ Uj ; xj = xjo = xj (γ (tj ))} ⊂ Uj , j = 1, . . . , k, where we shall compute the holonomy in the transverse section  = k . For the sake of simplicity, we shall denote the point (xjo , yj ) ∈ j by yj . Moreover, for the uniformization of the notation we shall put 0 =  and y0 = y = yk . Let us compute the holonomy fj : j −1 → j , j = 1, . . . , k. This holonomy is of the form yj = fj (yj −1 ). It is enough to prove that fj (yj −1 ) = yj −1 , j = 1, . . . , k. This implies that the holonomy of γ , which is the composition fk ◦. . .◦f1 , is equal to the identity of . Indeed, since ω |Uj −1 ∩Uj = dyj −1 = dyj , we obtain that d(yj − yj −1 ) = 0 in Uj −1 ∩ Uj . This implies that the difference yj − yj −1 is constant in the connected component of Uj −1 ∩ Uj that contains γ (tj −1 ), say yj = yj −1 + c. On the other hand, because Uj ∩ L = (yj = 0) and Uj −1 ∩ L = (yj −1 = 0), we obtain that c = 0 completing the proof.  

5.2 Foliations Given by Closed Meromorphic 1-Forms Let M be a complex manifold of dimension n ≥ 2 and ω a closed meromorphic 1-form (not holomorphic) on M. We shall denote the polar divisor of ω by (ω)∞ ⊂ M (see the definition in [39]). In the current case we have (ω)∞ = ∅, since ω is assumed to be not holomorphic. Since ω is closed and holomorphic in the open set N = M \ (ω)∞ , it defines a codimension one foliation of N which we shall denote F. Let us see that, indeed, F extends to a foliation of M. Proposition 5.2.1 The foliation F extends to M as a foliation F  such that (ω)∞ is invariant by F  . Proof In the proof of the extension of F we shall use the following fact (see [35, 36]): Fact 5.2.2 The set L of smooth points of (ω)∞ is open and dense in (ω)∞ . Moreover, the set S = (ω)∞ \ L is an analytic subset of M from codimension ≥ 2. The idea is to prove first that F extends to the set M \ S and then make use of Proposition 4.6.2. With this aim, let us fix a point p ∈ L. Since p is a smooth point of (ω)∞ and this set has codimension one, there exists a system of holomorphic coordinates in a neighborhood U of p, w = (x, y) : U → Cn , where w(U ) is a polydisc, x = (x1 , . . . , xn−1 ) : U → Cn−1 and such that U ∩ L = U ∩ (ω)∞ = {(x, y); y = 0}. Observe now that by definition of the polar set, there exists j > 0 such that y j ω extends to a holomorphic form on U . Let

42

5 Holomorphic Foliations Given by Closed 1-Forms

k = min{j > 0; y j ω extends to a holomorphic form in U } and let us define η=y k .ω. We can write η=an .dy +

n−1

j =1

aj dxj , where some of the

functions a1 , . . . , an do not vanish identically in U ∩ L. Note that η is integrable in U \ L and defines the same foliation that ω in this set. Henceforth, η is integrable in U , and there it defines a foliation of U that extends F |U . On the other hand, from ω = y −k .η we obtain d(y −k ) ∧ η + y −k .dη = dω = 0 ⇒ (∗) dy ∧ η = k −1 .y.dη. Then, from (*) and from Proposition 4.5.1 we get that (y = 0) = L ∩ U is invariant by the foliation defined by η. This implies then that F extends to M \ S in such a way that L is invariant by the extension, as desired.   Remark 5.2.3 Observe that the connected components of L are leaves of F  . Moreover, given a leaf L0 ⊂ L and a coordinate system w = (x, y) : U → Cn−1 × C such that U ∩ L = U ∩ L0 = (y = 0), we can consider the number k = min{j > 0; y j ω extends to a holomorphic form on U }. It is possible to prove that k only depends on L0 , i.e., does not depend on the coordinate system that we consider. We shall say then that ω has a pole of order k in L0 . Next we shall see a particular example of the situation above.

5.2.1 Holonomy of Foliations Defined by Closed meromorphic 1-Forms Let M be a complex manifold of dimension ≥ 2 and ω a closed meromorphic 1form on M. According to Proposition 5.2.1, the foliation with singularities defined by ω on M \ (ω)∞ ) can be extended to a foliation of M, which we shall denote by F, with the property that (ω)∞ is invariant by F. In other words, the smooth part of the polar set of ω is a reunion of leaves of F. Let us now see how to compute the holonomy of a leaf of this foliation. The first remark is that a leaf which is not contained in the polar set has trivial holonomy by Proposition 5.1.2. Let us then compute the holonomy of the leaves contained in the polar set (ω)∞ . For this sake we need some notation. For N  k ≥ 2 and a ∈ C, we consider the following vector field: Y k,a =

∂ yk 1 + a.y k−1 ∂y

5.2 Foliations Given by Closed Meromorphic 1-Forms

43

defined in the open set {y ∈ C; 1 + a.y k−1 = 0}. Note that Y k,a generates a local flow in a neighborhood of 0 ∈ C, which will be denoted by Yzk,a . Thus, for a fixed z ∈ C, Yzk,a is a biholomorphic map between neighborhoods of 0 ∈ C, since Yzk,a (0) = 0. Let us denote the germ of Yzk,a at the origin by [Yzk,a ]. Note that, if k ≥ 3, then, [Yzk,a ] commutes with the rotation Rλ (y) = λ.y, where k−1 λ = 1. Then, for every k ≥ 2 and every a ∈ C, the set Gk,a = {[Rλ ◦ Yzk,a ] ; z ∈ C λk−1 = 1} is an abelian group. An important particular case is when k = 2 and a = 0. In this case G2,0 is the group of homographies of the form {y →

y ; a ∈ C}, 1 + ay

2 obtained by straightforward integration of the differential equation dy dz = y . Our main result in this section is the following computation of the holonomy:

Proposition 5.2.4 Let M be a complex manifold of dimension ≥ 2 and ω a closed meromorphic 1-form on M with polar set (ω)∞ ⊂ M. Denote by F the codimension one foliation of M corresponding to ω. Let L be a leaf of F. Then: (a) If L ⊂ M \ (ω)∞ , then Hol(L) is trivial. (b) If L ⊂ (ω)∞ and ω has poles of order 1 in L, then Hol(L) is abelian and analytically linearizable, that is, Hol(L) is analytically conjugate to a subgroup of linear maps (z → λz) of C, in some neighborhood of the origin. (c) If L ⊂ (ω)∞ and ω has poles of order k ≥ 2 in L, then Hol(L) is analytically conjugate to a subgroup of Gk,a , for some a ∈ C. Proof Case (a) has already been addressed. Suppose now that L ⊂ (ω)∞ . Fix p ∈ L and a local foliation chart of F, (x, y) : U → Dn−1 × D ⊂ Cn−1 × C such that (ω)∞ ∩ U = L ∩ U = (y = 0) and the plaques of F in U are of the form y = cte . We claim that: Claim 5.2.5 We have (∗) ω |U =

g(y) .dy, yk

where g is holomorphic in D, g(0) = 0 and k is the order of the polar set of ω in L. Proof of Claim 5.2.5 Indeed, since ω defines F on M \ (ω)∞ and the plaques of F on U are of the form y = cte , we have ω |U = h(x, y)dy, where h is meromorphic in U with poles in (y = 0). On the other hand, because ω is closed we have ∂h/∂x ≡ 0, i.e., h = f (y), only depends on y. Let k be the order of the polar set of ω in L. Then f writes as in (*), as it can be easily verified.  

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5 Holomorphic Foliations Given by Closed 1-Forms

Now we need: Lemma 5.2.6 Let α be a meromorphic 1-form on a neighborhood of 0 ∈ C. Suppose that 0 is a pole of order k ≥ 1 for α. Then there exists a system of coordinates y : V → C with 0 ∈ V , y(0) = 0 and such that α writes in this system of coordinates as: (i) α = a. dy y a = 0, if k = 1. (ii) α =

1+a.y k−1 .dy, yk

with a ∈ C, if k > 1.

Proof Let us prove the case of simple pole, k = 1. The case k > 1 is similar and is a simple exercise. In the case of a simple pole we can write α = g(z) z dz, where g is holomorphic in neighborhood W of 0 and g(0) = a = 0. Note that a = Res(α, 0), that is invariant by change of coordinates (see [2]). We have then g(z) = a + z.u(z), where u is holomorphic W , so that α=a

dz + u(z)dz. z

Let ϕ(z) be a primitive (integral) of the holomorphic 1-form u(z) a dz in a neighborhood of 0. Let us consider the function y(z) = z. exp(ϕ(z)). Since y(0) = 0 and y  (0) = 0, we conclude that y is a biholomorphic map between two neighborhoods of 0. On the other hand, a

dz dz dy =a + adϕ = a + u(z)dz = α, y z z

completing the proof.

 

Let us resume the proof of the proposition. Assume first that k = 1. Fix a closed curve γ : I → L with γ (0) = γ (1) = po . From Lemma 5.2.6 and with similar arguments to the holomorphic case we have: Claim 5.2.7 There is a collection C = {((xj , yj ), Uj )}kj =1 of foliation charts of F, and a partition {0 = t0 < t1 < . . . < tk = 1} of I , such that: (i) ∪kj =1 Uj = γ (I ). (ii) γ ([tj −1 , tj ]) ⊂ Uj , ∀j = 1, . . . , k. dy (iii) ω |Uj = aj yjj , ∀j = 1, . . . , k. Since γ (0) = γ (1) we can assume that: (iv) ((x1 , y1 ), U1 ) = ((xk , yk ), Uk ) = ((x, y), U ), where x(po ) = 0 ∈ Cn−1 . Let us consider also sections j , j = 0, . . . , k as in the holomorphic case, with 0 = k = . Now we remark that if A is a connected component of Uj −1 ∩ Uj that contains dy −1 dy γ (tj −1 ), then aj −1 . yjj−1 = aj yjj in A. Comparing the residues of these two forms

5.2 Foliations Given by Closed Meromorphic 1-Forms

45 dy

dy

−1 in 0, we conclude that aj −1 = aj . We can then conclude that yjj−1 = yjj in Uj −1 ∩ Uj , for every j = 1, . . . , k. This allows us to relate the coordinates yj and yj −1 in A. Indeed, if yj = f (yj −1 ) in A, then we must have 

f (yj −1 ) dyj −1 dyj = dyj −1 = ⇒ yj f (yj −1 yj −1 

z.f (z) = f (z) ⇒ f (z) = cj z for some constant cj , as it follows from a straightforward integration of the differential equation z.f  = f . This last implies that the intermediate holonomy maps fj : j −1 → j are linear. Since the composition of linear maps is also linear, we obtain that the holonomy of γ is linear in the coordinate system that we consider. Since this system only depends on ω (not on the curve γ ), we conclude that the holonomy from L is analytically linearizable. Now we consider the case where k ≥ 2. Again we fix a closed curve γ : I → L with γ (0) = γ (1) = po . Similarly to the case k = 1, from Lemma 5.2.6 and similar arguments to the ones in the holomorphic case, we have: Claim 5.2.8 There is a collection C = {((xj , yj ), Uj )}m j =1 of foliation charts of F, and a partition {0 = t0 < t1 < . . . < tm = 1} of I , satisfying (i), (ii), (iv), and (iii) ω |Uj =

1+aj yjk−1 yjk

.dyj , ∀j = 1, . . . , m.

Observe that aj = Res(ω, yj = 0). Thus, similarly to the above argumentation, we conclude that a1 = . . . = am = a. As we have seen above, it is enough to relate yj and yj −1 . Let us do it in the case k = 2. The case k > 2 is left to the reader. For the sake of simplicity of notation, let us put yj = w and yj −1 = z, so that w = f (z) is the change of coordinates. Case a = 0 In this case, in Uj −1 ∩ Uj we have dw f  (z) z dz = 2 = .dz ⇒ z2 .f  = f 2 ⇒ f (z) = 2 1 + c.z z w (f (z))2 from where we conclude that f is in the group G2,0 . Similarly to above, since the composition of elements in G2,0 is also in G2,0 , we obtain that Hol(L) is analytically conjugate to a subgroup of G2,0 . Case a = 0 The argumentation in the case k = 2 and a = 0 is quite similar to the one above: it is enough to prove that yj and yj −1 are related by an element of G2,a . As it is easily checked, if yj = f (yj −1 ), then f satisfies the differential equation (∗) z2 .(1 + a.f (z)).f  (z) = (f (z))2 .(1 + a.z). Thus it remains to show that in this case f is in G2,a . Let us show how this is done.

46

5 Holomorphic Foliations Given by Closed 1-Forms

Step 1 Given b ∈ C, there exists a unique solution f of (*), defined in a neighborhood of 0, such that f  (0) = 1 and f  (0) = b. , then g satisfies the following Indeed, if f is a solution of (*) and g(z) = f (z)−z z2 differential equation: (∗∗) g  = g.

a + azg − g = F (z, g). 1 + az + azg

Since F is holomorphic in a neighborhood of (0, b/2), (**) has a unique solution g, defined in a neighborhood of 0 ∈ C, with g(0) = b/2. Setting f (z) = z+z2 .g(z), we obtain the required solution. Step 2 For each c ∈ C the function fc = Yck,a is a solution of (*) with initial condition fc (0) = 1 and fc (0) = 2c. The proof is a straightforward computation. Proposition 5.2.4 then follows from Steps 1 and 2 above.  

5.3 Exercises 1. Given k ≥ 2 and a ∈ C, consider the vector field Y = Y k,a =

∂ yk k−1 ∂y 1 + a.y

defined in {y ∈ C ; 1 + a.y k−1 = 0}. Denote by Yz its complex local flow defined in a neighborhood of 0 (i.e., z → Yz (p) is a solution of the differential equation y˙ = Y (y) with initial condition y(0) = p). Prove that: (a) For every z ∈ C, Yz commutes with a rotation Rβ , where β = exp(2π i/(k − 1). (b) Prove that the group Gk,a defined in Proposition 5.2.4 is abelian. (c) Complete the proof of Proposition 5.2.4 in the case k ≥ 3. Hint: The proof can be reduced to the case k = 2 by means of a ramified change of variables of the form w = y k . (d) Prove that if f ∈ Gk,a , then, f ∗ (α) = α, where α=

1 + a.y k−1 dy yk

Hint for (c): Note that LY (α) = 0 ( LY (α) denotes the Lie derivative of α with respect to Y . See [81]).

5.3 Exercises

47

2. Let ω be a closed meromorphic 1-form, in a manifold M of dimension ≥ 2, let (ω)∞ be its set of poles and L an irreducible component of (ω)∞ . Denote by L the smooth part of L. Prove that: (a) Given p ∈ L there exists a system of coordinates (x, y) : U → Cn−1 × C, in a neighborhood U of p, such that L ∩ U = (y = 0) and ω |U =

g(y) dy, yk

where k ≥ 1, k ∈ N and g is holomorphic in a neighborhood of 0 ∈ C being g(0) = 0. (b) The entire k ∈ N above does not depend on the coordinate system chosen nor on the point p ∈ L. 3. Let α be a meromorphic 1-form defined in a neighborhood of 0 ∈ C, with pole of order k ≥ 2 in 0 and such that Res(α, 0) = a ∈ C. Prove that there exists a coordinate system y, defined in a neighborhood of 0, such that the α writes: α=

1 + a.y k−1 dy. yk

4. Prove that the solution of the differential equation z2 .(1 + a.f ).f  = f 2 .(1 + a.z) such that f (0) = 0, f  (0) = 1 and f  (0) = 2c is given by f (z) = Yc2,a (z), where Yc2,a is as in Exercise 1. Hint: Use Exercise 1. 5. Given k ≥ 2 and a ∈ C, we consider the following vector field: Y k,a =

∂ yk k−1 ∂y 1 + a.y

defined in the open {y ∈ C; 1 + a.y k−1 = 0}. Let us denote by Yzk,a its local flow. Prove that if k ≥ 3, then, [Yzk,a ] commutes with a rotation Rλ (y) = λ.y, where λk−1 = 1. 6. Complete the proof of Lemma 5.2.6 in the case k ≥ 2. 7. Let λ1 , . . . , λr ∈ C \ {0} be given together with functions f1 , . . . , fr : U → C holomorphic in some connected neighborhood U ⊂ Cn of the origin. Denote r r 

df λj fjj ). Prove the following: If there is a non-constant by ω = ( fi )( i=1

j =1

holomorphic function f : U → C such that df ∧ ω = 0, then necessarily λ1 , . . . , λr are linearly dependent over Z+ , i.e., there are n1 , . . . , nr ∈ Z+ such that n1 λ1 + . . . + nr λr = 0.

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5 Holomorphic Foliations Given by Closed 1-Forms

∂ ∂ +y · ∂x ∂y is called Poincaré–Dulac normal form as we shall see later on. It is possible to see that the only invariant curve through the origin is the axis {y = 0}. Prove that the corresponding foliation is given by a closed meromorphic 1-form, with an irreducible polar set. Prove that this closed 1-form is unique up to constant multiples. Study the holonomy of the only leaf contained in the polar set. Given a foliation F by a closed meromorphic 1-form , any other closed meromorphic 1-form ω defining the same foliation is of the form ω = h for some meromorphic function h satisfying dh ∧  = 0. Given a meromorphic 1-form , an integrating factor for  is a (meromorphic) function h such that h1  is closed. In particular,  is integrable. Prove that the quotient of two integrating factors for a same 1-form  is a first integral for the foliation defined by . There is no closed holomorphic 1-form without singularities on a simplyconnected compact complex manifold. Let ω be a closed meromorphic 1-form, in a manifold M of dimension ≥ 2, let (ω)∞ be its set of poles and L an irreducible component of (ω)∞ . Denote by L the smooth part of L. Prove that:

8. Given n ∈ N \ {1}, a ∈ C∗ the vector field X(x, y) = (nx + ay n )

9.

10.

11. 12.

(a) Given p ∈ L there exists a system of coordinates (x, y) : U → Cn−1 × C, in a neighborhood U of p, such that L ∩ U = (y = 0) and ω |U =

g(y) dy, yk

where k ≥ 1, k ∈ N and g is holomorphic in a neighborhood of 0 ∈ C being g(0) = 0. (b) The entire k ∈ N above does not depend on the coordinate system chosen nor on the point p ∈ L. 13. The integration lemma states that a closed meromorphic 1-form ω defined in a neighborhood U of the origin 0 ∈ Cn can be written as ω=

r  j =1

λj

dfj g + d( r ),  nj −1 fj fj j =1

where g, f1 , . . . , fr are holomorphic functions defined in a neighborhood of the origin, λ1 , . . . , λr are complex numbers, and n1 , . . . , nr are positive natural numbers. The aim of this exercise is to give a proof of this result. First step is r  (fj = 0) in irreducible components to write the polar set of ω as (ω)∞ = j =1

(fj = 0) given by reduced holomorphic functions fj : V → C defined in some common neighborhood of 0 ∈ C. Next we introduce the complex numbers λj

5.3 Exercises

49

as the residue of ω with respect to the component  : (fj = 0) of its polar set. For this we apply some basic Milnor theory in order to assure that for V sufficiently small the fj have the origin as the only possible singularity (as a map). Using this we may consider small transverse discs j that meet j transversely at a point pj and avoid the other components. We may then choose a positively oriented simple loop γj ⊂ j \ {pj }. The residue λj is then given   by λj := 2π √1 −1 ω . Some Milnor theory proves that λj is well-defined. j r

df Finally we define the 1-form θ := ω − λj fjj . It is closed meromorphic and j =1

has no residues. This can be used to prove that θ is exact, i.e., θ = dG for some meromorphic function G defined in V sufficiently small. Analyzing the polar set of θ we conclude.

Chapter 6

Reduction of Singularities

The subject of reduction of singularities is one of the most developed and interesting in the theory of holomorphic foliations. In the last decades, various authors have given important contributions to the subject. Although the general problem of the reduction of singularities for a germ of a holomorphic foliation singularity remains open, several are the cases where it is already finished. This knowledge has been of great importance in the development of the theory of holomorphic foliations with singularities as we shall see in some of the results to be presented in this text (see Chapters 7 and 10 for instance). We would like to mention the recent work of Camacho et al. [12], Camacho and Sad [15], Cano [17], and Cano and Cerveau [18] as a partial list. In this chapter, we describe the method of reduction of singularities according to Seidenberg, based on a finite sequence of quadratic blow-ups.

6.1 Irreducible Singularities Before going into the reduction of singularities, we recall some very basic facts from the theory of singularities of holomorphic vector fields. In short, in this section we study the analytic forms of non-degenerate isolated singularities of holomorphic vector fields, according to Poincaré and Dulac.

6.2 Poincaré and Poincaré–Dulac Normal Forms In this section, we shall consider a holomorphic vector field X defined in a neighborhood U of the origin 0 ∈ Cn , n ≥ 2, with a singularity (isolated) at the origin. Since we are interested in the local analytical description of X or F(X) in

© Springer Nature Switzerland AG 2021 B. Scárdua, Holomorphic Foliations with Singularities, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-76705-1_6

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6 Reduction of Singularities

a neighborhood of 0, we shall consider U as small as necessary without further mention to this. The eigenvalues of X at 0 are those of the linear part DX(0) (here regarded as a linear map Cn → Cn ). Assume that DX(0) is non-singular. Definition 6.2.1 We shall say that the singularity 0 of X is in the Poincaré domain if the convex hull of its eigenvalues in R2  C does not contain the origin. Otherwise we say that the singularity 0 of X is in the Siegel domain. Remark 6.2.2 If m = 2 then the singularity is in the Siegel domain if, and only if, the quotient of its eigenvalues is real negative. A geometrical characterization of singularities in the Poincaré domain, as well as a rich dynamical description of them, is found in the series of works of the senior Japanese mathematician, Prof. Toshikazu Ito. An excerpt from those is the following [42]: Theorem 6.2.3 (T. Ito, 1992) A singularity of Cm of a holomorphic vector field X is in the Poincaré domain if, and only if, the foliation F(X) is transverse to every small sphere S 2n−1 (r; 0) of center 0 ∈ Cn and radius r > 0. This is the case if F(X) is transverse to some sphere contained in its definition domain. Let now m = 2. We shall say that the eigenvalues λ1 , λ2 of X are in resonance if λ1 = k λ2 or λ2 = k λ1 for some k ∈ N, k ≥ 2. For dimension two, the complete description of the analytical normal forms of singularities in the Poincaré-domain is given below: Theorem 6.2.4 (Poincaré Linearization Theorem, [4, 6, 27]) Let X be a holomorphic vector field with a singularity in the Poincaré domain at the origin 0 ∈ C2 . Suppose that the eigenvalues λ1 , λ2 of X are not in resonance. Then there is a unique holomorphic diffeomorphism ξ between neighborhoods of 0 ∈ C2 , ξ(0) = 0, such that: (i) ξ  (0) = Id. (ii) ξ∗ X = DX(0), i.e., ξ takes X into its linear part. Remark 6.2.5 (i) The above theorem is similar to a result, by Poincaré, for diffeomorphisms f : (Cm , 0) → (Cn , 0). The proof is based on the convergence of the formal (power series) solution to the linearization problem. For the case of resonances we have: Theorem 6.2.6 (Poincaré–Dulac Theorem,[27]) Let X be a holomorphic vector field with a singularity in the Poincaré domain at 0 ∈ C2 . Suppose that the eigenvalues are in resonance, λ1 = k λ2 for some k ∈ N, k ≥ 2. Then there is a unique holomorphic diffeomorphism ξ between neighborhoods of 0 ∈ C2 , ξ(0) = 0, such that:

6.3 Blow-up at the Origin (Quadratic Blow-up)

(i) ξ  (0) = Id. (ii) ξ∗ X = (λ1 x + ay n )

53

 ∂ ∂ , + λ2 y ∂x ∂y

for some a ∈ C. If a = 0 then we are in the analytically linearizable case. The proof is in the same spirit of the one for the non-resonant case. We refer to the book of Ilyashenko and Yakovenko [41] for the proofs and a detailed study in these normal forms. Example 6.2.7 (Germs of Singularities in the Siegel Domain) We consider on ∂ ∂ C2 a linear vector field X = λx ∂x + μy ∂y in the Siegel domain, i.e., λ/μ ∈ R− . We study the set T (X) of tangent points of X with the unit sphere S 3 (1). This set is given by T (X): |x|2 + |y|2 = 1 and λx x¯ + μy y¯ = 0. We obtain T (X) : μ λ |x|2 = μ−λ , |y|2 = λ−μ . Thus the set of tangent points is a real 2-torus embedded 1 × S 1 → S 3 (1). Let us compute the induced foliation by in S 3 (1), T (X) ∼ S =  

μ λ X on T (X). We parameterize the torus by x(s) = μ−λ eis , y(t) = λ−μ eit , s, t ∈ [0, 2π ]. The inducedfoliation isgiven by λx dy − μy dx = 0 #⇒  λ it   μ is  μ λ is λ μ−λ e d λμ e − μ λ−μ = 0 #⇒ λ dt − μ ds = eit d μλ e

0 #⇒ d(λt − μs) = 0. We have a linear foliation of S 1 × S 1 .   therefore     1 μ λ The foliation of the solid torus S 1 μ−λ ×  λ−μ with coordinates (s, z)  μ is  λ  1 and parametrization ψ(s, z) = μ−λ e , λ−μ z , s ∈ R, z ∈  is given by    λ   μ is  μ λ λ μ−λ eis d λ−μ z − μ λ−μ z d μ−λ e = λ dz − μz ds = 0, i.e. by, 1

x˙ = λ = 0, z˙ = μz. Again we have a linear foliation L on S 1 ×  , which   1 μ  is transverse to the fibers {x} ×  , x ∈ S 1 μ−λ and transversely holomorphic, tangent to the boundary torus. The foliation L has a compact leaf which corresponds 1 to {z = 0}, i.e., S 1 × {0} → S 1 ×  . Notice that L has a second compact leaf if and 1 only if the equation e2π in λ/μ z = z has a solution for some z ∈  , z = 0 and some n ∈ Z, n = 0. This occurs if and only if e2π i nλ/μ = 1, i.e., λ/μ ∈ Q− . In this case all leaves are compact and the foliation FX defined by X on C2 has a holomorphic n first integral: μλ = − m , n, m ∈ N, n, m = 1, FX : xdy − μλ y dx = 0 #⇒ m n m n xdy + m n y dx = 0 #⇒ d(x y ) = 0 #⇒ f = x y is a holomorphic first integral for FX .

6.3 Blow-up at the Origin (Quadratic Blow-up) The (quadratic) blow-up at the origin is defined as follows: We consider two 2 copies  of C with coordinates (a, t) and (u, y) and change of coordinates given y = tx by x = uy

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6 Reduction of Singularities

Fig. 6.1 Blow-up process of a foliation with two separatrices

In particular, ut = 1 so that we obtain a complex surface C20 that contains a projective line P →  C20 (given by (x = 0) and by (y = 0)). Define the map 2 2 π:  C0 → C by π(x, t) = (x, tx) and π(u, y) = (uy, y); then π defines a proper holomorphic projection which is a diffeomorphism between  C20 \ P and C2 \ {0}. 2 2  The manifold C0 is the blow-up of C at the origin and it is a complex (rank 1) vector bundle with basis CP (1) ≈ P, and fiber C. Roughly speaking, the “explosion” of C2 at the origin “separates” the lines though 0. It is a remarkable fact that the self-intersection P · P of the projective line P →  C20 (called exceptional divisor) is negative equal to −1. By introducing local coordinates we can introduce the notion of blow-up of complex surface M 2 at a point p ∈ M is a natural way and successively by repeating this process as many times as desired (see Figure 6.1). The final configuration of the exceptional divisor depends on the position of the blow-up centers.

6.4 Blow-up on Surfaces Let us recall the blow-up of C2 at 0. Let us consider two copies of C2 , say U and V , with coordinates (t, x) and (s, y), respectively. We define a complex manifold C˜2 , identifying the point (t, x) ∈ U \ (t = 0) with the point (s, y) = α(t, x) = (1/t, tx) ∈ V \ (s = 0). The divisor of C˜2 is defined as the submanifold D of C˜2 such that U ∩ D = (x = 0) and V ∩ D = (y = 0). Note that, since y = tx, D is well-defined and is biholomorphic to the Riemann sphere C = CP (1). Moreover, we can define a submersion P : C˜2 → D by P |U (t, x) = t and P |V (s, y) = s. Then (C˜2 , P , D)

6.4 Blow-up on Surfaces

55

is a vector fiber space with basis D, projection P , and fiber C, having D as zero section. Let us now consider the holomorphic map π : C˜2 → C2 defined by π |U (t, x) = (x, tx) and π |V (s, y) = (sy, y). Note that π is well-defined, since in U ∩ V we have y = tx and x = sy. Moreover, π has the following properties: (a) π −1 (0) = D. (b) π |C˜2 \D : C˜2 \ D → C2 \ {0} is biholomorphic. (c) π is a proper map. Then, with this structure, C˜2 is the blow-up of C2 at 0, with projection (blowdown) map π . Let us introduce this same concept in a complex surface. We consider a complex manifold of dimension two M and a point q ∈ M. The blow-up of M at q is defined as follows: take a holomorphic local chart ϕ : A → B ⊂ C2 with q ∈ A and ϕ(q) = 0. Let π : C˜2 → C2 be the blow-up map at 0, with divisor D and B˜ = π −1 (B). In the disjoint union M  = (M \ {q}) % B˜ we define an equivalence relation ∼ by setting po ∼ p1 if, and only if, po = p1 or, otherwise, po ∈ A \ {q}, p1 ∈ B˜ \ D and p1 = π −1 (ϕ(po )). The blow-up of M at q is the quotient M˜ = M  / ∼. Since B˜ is a manifold and π −1 ◦ ϕ : A \ {q} → B˜ \ D is a biholomorphic map, it follows that M˜ is a complex manifold. Roughly, M˜ is obtained from M by replacing the point q by a projective line D  C. Indeed, there is a natural embedding of the ˜ divisor D into M. ˜ we have three possibilities: (1) The equivalence class of p Given a point p ∈ M, is in D. (2) The equivalence class of p is in M \ A. (3) The equivalence class of p contains two points po ∈ A \ {q} and p1 ∈ B˜ \ D. Thus, the points of M˜ are divided into two classes: those points as in (1), that will be called points of the divisor, and the points of M˜ \ D, that will be regarded as points of M (as in (2) or (3)). A map of blow-down  : M˜ → M is defined by (p) = q in case (1), (p) = p in case (2) and (p) = po in case (3). It is not difficult to see that  has properties analogous to those of π , i.e., (a’)−1 (q) = D. (b’) |M\D : M˜ \ D → M \ {q} is ˜ a biholomorphic map. (c’) is a proper map. The above process can be iterated: start with the manifold M and a point qo ∈ M. Blowing-up M at qo , we obtain a manifold M1 and a blow-up map 1 : M1 → M with divisor D1 = −1 1 (po ). Next we consider any point q1 ∈ M1 , and perform the blow-up of M1 at q1 obtaining, in this way, a manifold M2 and a map blow-up map 2 : M2 → M1 with divisor D2 . Proceeding in this way, after k blow-ups, we obtain a manifold Mn and a blow-up map n : Mn → Mn−1 with divisor Dn . The composition n = n ◦ . . . ◦ 1 : Mn → M is a proper holomorphic map that we will call the blow-up map. The divisor D n of n is defined inductively as follows: (I) D 1 = D1 . (II) D n = n−1 ). Dn ∪ −1 n (D Note that (D n ) is a finite subset of M: indeed, this corresponds to centers of the explosions. Moreover, the map n |Mn \D n : Mn \ D n → M \ n (D n ) is a biholomorphism. The divisor D n is the union of k complex curves, each curve biholomorphic to C. For instance, for the second blow-up, if q1 ∈ D1 , then D 2 = D2 ∪ −1 2 (D1 ).

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−1 It follows that −1 2 (D1 )  C and that D2 intersects 2 (D1 ) transversally at a single point, i.e., D 2 is the union of two embedded projective lines in M2 with a single common point. For the sake of simplicity, we shall use the same notation for the Di and their successive inverse images i , . . . , n . Then, we can say that D n = ∪nj=1 Dj . If for every j = 1, . . . , n − 1 the j-th blow-up is centered at a point of D j , then n D will be a “graph free of cycles,” that is, for every i the projective Di another projective Dj transversally at a single point, called a corner of D n , in such a way that if Di1 ∩ Di2 = φ,. . . ,Dim−1 ∩ Dim = ∅, then Di1 = Dim . Such a process will be called blow-up process centered at q.

6.4.1 Resolution of Curves Let us now see in what consists the “resolution of a singularity of a curve”. Let us consider a curve C = (f (x, y) = 0) ⊂ A ⊂ C2 , where

f (0, 0) = 0, that is, 0 ∈ C. We assume that the Taylor expansion of f is f = ∞ j =k fj , where fj is a 2 ˜ homogeneous polynomial of degree j . Let π : C2 → C be the blow-down of C2 in 0. For the expression of π in the chart ((t, x), U ) of C˜2 , we obtain f ◦ π(t, x) = f (x, tx) =

∞ 

fj (x, tx)

j =k

=x . k

∞ 

x j −k .fj (1, t) = x k .fU (t, x),

j =k

so that π −1 (C) ∩ U = (x = 0) ∪ (fU (t, x) = 0). Analogously, in the chart ((s, ), we have π −1 (C) ∩ V = (y = 0) ∪ (fV (s, y) = 0), where fV (s, y) =

∞y), V j −k ˜ where C˜ = (fU = .fj (s, 1). Hence, we have π −1 (C) = D ∪ C, j =k y 0) ∪ (fV = 0). The curve C˜ is called strict transform of C. Note that C˜ ∩ D is a finite set. Indeed, C˜ ∩ D ∩ U = {(t, 0) ; fk (1, t) = 0}, while C˜ ∩ D ∩ V = {(s, 0) ; fk (s, 1) = 0}. In general, if we consider a blow-up process n : An → A with divisor D n = D1 ∪ . . . ∪ Dn , we obtain (n )−1 (C) = D n ∪ Cn , where Cn ∩ D n is a finite set. The curve Cn is called the strict transform of C by n . Definition 6.4.1 Let C be a holomorphic curve in a complex surface M. We say that the blow-up process, n : Mn → M, with divisor D n = ∪nj=1 Dj is a resolution of C, if the corresponding strict transform Cn satisfies the following properties: (i) Cn is regular. (ii) Cn meets each Dj ⊂ D n transversally. (iii) Cn ∩ D n contains no corners.

6.4 Blow-up on Surfaces

57

Let us illustrate this with an example. Example 6.4.2 We consider the singular plane curve C ⊂ C2 given by f (x, y) = y 2 − x 3 = 0. Let π1 : M1 = C˜20 → C2 be the blow-up map of C2 at the origin 0 ∈ C2 . In the local coordinates ((t, x), U ) of M1 , we obtain f ◦ π1 (t, x) = f (x, tx) = x 2 .(t 2 − x). Therefore, π1−1 (C)∩U consists of the divisor (x = 0) and of the strict transform C1 of C, with equation x −t 2 = 0. Then clearly π1−1 (C) ⊂ U , so that it is not necessary to consider the other blow-up chart. The strict transform C1 of C is regular but not transverse to the divisor D1 , since C1 ∩ D = (0, 0) ∈ U and (x − t 2 = 0) is tangent to (x = 0) at this very point. In other words, the curve is still not resolved. Then a second blow-up π2 (u, t) = (t, tu) = (t, x) is made at (0, 0) ∈ U . The divisor D 2 of this second blow-up is the union of two projective lines, D1 ∪ D2 , and in the chart (u, t), D1 is represented by (u = 0) and D2 by (t = 0). We have then f ◦ π1 ◦ π2 (u, t) = t 3 .u2 .(t − u). Thence, the strict transform C2 of C is (t − u = 0). This last curve cuts D 2 at the corner (0, 0) = D1 ∩ D2 , therefore this curve is still not resolved. With a final blow-up π3 at the point (u, t) = (0, 0), of the form t = vu (in one of the charts), we obtain a new divisor D3 , represented by (u = 0). The strict transform C3 of C with equation v − 1 = 0, cuts D3 transversally at the point (v, u) = (1, 0), which is not a corner point. Henceforth, C3 is a resolution of C. Note that, the original coordinates (x, y) are related to (v, u) by (x, y) = π1 ◦ π2 ◦ π3 (v, u) = (v.u2 , v 2 .u3 ) = π 3 (v, u). Using this and the parametrization u → (1, u) of C3 , we can obtain a parametrization u → (u2 , u3 ) of C. Theorem 6.4.3 (Resolution of Singularities for Curves [12]) Every holomorphic curve in a complex surface admits a resolution. Corollary 6.4.4 Let S be a holomorphic curve in a complex surface M. Given a point q ∈ S, there exists a neighborhood U of q and holomorphic curves S1 , . . . , Sm ⊂ U such that: (a) (b) (c) (d)

q ∈ Sj for every j = 1, . . . , m. S ∩ U ⊂ S1 ∪ . . . ∪ Sm . Si ∩ Sj = {q}, if i = j . Given j = 1, . . . , m, there exists a holomorphic injective map αj : Dr → U , where Dr = {z ∈ C ; | z |< r}, such that αj (0) = q, αj (Dr ) = Sj and the restriction αj |Dr \{0} is an embedding. In particular, each curve Sj is homeomorphic to the disc D.

Definition 6.4.5 The germs at q of curves S1 , . . . , Sm are called the local branches of S at q. For each j = 1, . . . , m, the map αj is called Puiseux parametrization of the branch Sj .

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Proof of Corollary 6.4.4 In the case where the point q is not a singular point of S the result is straightforward. Indeed, in this case, there is only one branch at q. Suppose now that q is a singularity of S. Let π : M˜ → M be a resolution of S, with divisor D = ∪nj=1 Dj , and S˜ the strict transform of S. Then S˜ cuts transversally D, at non-corner points, forming a finite set, say {q1 , . . . , qm }. Because the curve ˜ S˜ is regular, for each j = 1, . . . , m, we can obtain an embedding βj : Dr → M, which is a parametrization of a neighborhood of qj in S˜ such that βj (0) = qj . Taking the restrictions of the βj to a smaller disc if necessary, we can assume that βi (Dr ) ∩ βj (Dr ) = φ if i = j . Let us put αj = π ◦ βj and Sj = αj (Dr ). It is not difficult to check that S1 , . . . , Sm and α1 , . . . , αm satisfy (a), (c), and (d). We leave the verification of (b) as an exercise.   As a consequence of the above we obtain: Theorem 6.4.6 (cf. [33]) Let S be a holomorphic curve in a complex manifold M. Then there exists a Riemann surface S˜ and a holomorphic map φ : S˜ → M with the following properties: ˜ = S. (a) φ(S) (b) There are discrete subsets A ⊂ S˜ and B ⊂ S such that φ |S\A : S˜ \ A → S \ B ˜ is an embedding. (c) φ −1 (B) = A. Moreover, B is the singular set of S, and for every p ∈ B, φ −1 (p) is a subfinite set of A. Definition 6.4.7 The curve S˜ is called the normalization of the curve S. The theorem above implies that, given a singularity p ∈ S, we can define the branches of S at (through) p in the following way: since φ −1 (p) = {q1 , . . . , qr }, ˜ we can obtain for each j = 1, . . . , r a disc Dj ⊂ S˜ such is a finite subset of S, that pj ∈ Dj and Di ∩ Dj = φ se i = j . The germs at p of φ(D1 ), . . . , φ(Dr ) are the branches of S by p. The maps φ |Dj : Dj → S, j = 1, . . . , r, are the Puiseux parametrizations of these branches.

6.5 Blow-up of a Singular Point of a Foliation Given a foliation F of M and p ∈ M, the blow-up of F at p is the pull-back  = π ∗ (F) of F by the blow-up map π : M p → M. Both foliations are foliation F −1 −1  equivalent to M \{p} and Mp \π (p) (recall that π (p) is the exceptional divisor π −1 (p) ∼ = P) so that they have the same leaves. Eventual “new” singularities are introduced in the exceptional divisor π −1 (p). It may occur that π −1 (p) is invariant or not. If π −1 (p) is invariant by the foliation F = π ∗ (F) we say that the blow-up is non-dicritical, otherwise it is called dicritical. Let us study this procedure more closely. Let us first see what occurs with a foliation after a single blow-up. Let us consider a holomorphic foliation F in a neighborhood of 0 ∈ C2 with an isolated singularity at the origin 0. We assume that F is represented by the vector

6.5 Blow-up of a Singular Point of a Foliation

59

field X = (P (x, y), Q(x, y)) or, equivalently, by the dual 1-form ω = P (x, y)dy − Q(x, y)dx. We shall denote by F ∗ the foliation with isolated singularities F ∗ = π ∗ (ω). Thus F ∗ is the pull-back of F via the blow-up map π : C˜20 → C2 . Let us investigate its expression in local coordinates. We can write the Taylor expansion of ω at 0 as ω=

∞  (Pj dy − Qj dx), j =k

where Pj and Qj are homogeneous polynomials of degree j , with Pk ≡ 0 or Qk ≡ 0. The 1-form π ∗ (ω) writes in the chart ((t, x), U ) as π ∗ (ω) =

∞  (Pj (x, tx)d(tx) − Qj (x, tx)dx) = j =k

= xk .

∞ 

x j −k .[(tPj (1, t) − Qj (1, t))dx − xPj (1, t)dt].

j =k

Dividing the above 1-form by x k we obtain (∗) x −k .π ∗ (ω) = (tPk (1, t) − Qk (1, t))dx + xPk (1, t)dt + x.α

j −k−1 .[(tP (1, t) − Q (1, t))dx + xP (1, t)dt]. where α = ∞ j j j j =k+1 x Set R(x, y) = yPk (x, y) − xQk (x, y), in such a way that x −k .π ∗ (ω) = R(1, t)dx + xPk (1, t)dt + x.α. Analogously, computing the expression of π ∗ (ω) in the chart ((s, y), V ), we obtain (∗∗) y −k .π ∗ (ω) = R(s, 1)dy − yQk (s, 1)ds + y.β. The polynomial R(x, y) is the tangent cone of ω. We have two cases to consider: (a) R ≡ 0. In this case, we shall say that the singularity is dicritical. (b) R ≡ 0. In this case, the singularity is non-dicritical. The tangent cone has then degree k + 1. Let us take a closer look at the above cases. Case (a) In this case, the forms in (*) and (**) are still divisible by x and y, respectively. Dividing (*) by x we obtain ω1 = Pk (1, t)dt + α = Pk (1, t)dt + (tPk+1 (1, t) − Qk+1 (1, t))dx + x.α1 ,

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and this form cannot be divided by x, since Pk ≡ 0. The foliation F ∗ is then represented in this chart by ω1 and in the other chart by the form ω2 , obtained from the division of (**) by y. Note that, at the points of the divisor (x = 0), of the form (to , 0) such that Pk (1, to ) = 0, the leaves of F ∗ are transversal to the divisor. The points (to , 0) such that Pk (1, to ) = 0 will be the singular points of F ∗ , or tangency points of the leaves of F ∗ with the divisor. Note also that each leaf transversal to the divisor will originate a local separatrix of F via blow-down. Therefore, a dicritical singularity admits infinitely many separatrices. Case (b) In this case the forms in (*) and (**) cannot be divided anymore. Therefore, they already represent the foliation F ∗ in their respective charts. In particular, the divisor is invariant by F ∗ . Moreover, the singularities of F ∗ in the divisor are the points, of the (x, t) chart, of the form (0, to ) where R(1, to ) = 0, and also the point (0, 0), of the second chart, if 0 is a zero of R(s, 1) = 0. We also have that F ∗ has k + 1 singularities, counted with multiplicity, in the divisor. Note that, if some of the singularities of F ∗ have some separatrix S, then π(S) is a separatrix of F in 0. Let us now consider the blow-up process at 0 ∈ C2 , consisting of a succession of n blow-ups,  : M → C2 , with divisor D = ∪nj=1 Dj . ˜ with isolated The above argumentation proves that we can obtain a foliation F, singularities, such that M \ D  C2 \ {0} coincides with F. We shall say that the ˜ Otherwise, we shall say that the divisor Dj is non-dicritical, if it is invariant by F. divisor is dicritical. Example 6.5.1 The singularity xdy − ydx = 0 is not irreducible (λ = 1 ∈ Q+ ) and the blow-up y = tx gives xd(tx) − txdx = 0 xtdx+x 2 dt − txdx = 0 x 2 dt = 0 dt = 0. The blow-up is therefore a non-singular foliation of the surface  C20 , transverse to P. Later on, we shall see what is understood as “reduction of a singularity of a foliation.” Before that we shall introduce some notions and notations. Let X be a holomorphic vector field defined in a neighborhood of 0 ∈ C2 such that 0 is an isolated singularity of X. Let λ1 and λ2 be the eigenvalues of DX(0). Definition 6.5.2 (Simple Singularity) We say that 0 is a simple singularity of X, if: (a) λ1 = 0 and λ2 = 0 (or vice-versa). In this case, we shall say that the singularity is a saddle-node.

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61

(b) λ1 , λ2 = 0 and λ2 /λ1 ∈ Q+ . The numbers λ2 /λ1 and λ1 /λ2 are then called the characteristic numbers of the singularity. Note that the above conditions are invariant under holomorphic changes of coordinates and under multiplication of X by a holomorphic function that does not vanish at 0. Thus, the above notion may be extended to the isolated singularities of holomorphic foliations on complex surfaces. In few words, the theorem of reduction of singularities assures that given a foliation F with a finite number of (isolated) singularities on a complex surface M, there exists a (finite) blow-up process, π : M˜ → M, such that the pull-back foliation F ∗ = π ∗ (F), which is biholomorphically equivalent to F outside of the divisor of π , has only simple singularities. Indeed, it is possible to say more. Definition 6.5.3 We shall say that the blow-up process is a reduction of the singularity or resolution of the singularity, if: (i) All singularities of F˜ in D are simple. ˜ and no tangency points (ii) A dicritical divisor Dj contains no singularities of F, ˜ of F with Dj . Theorem 6.5.4 (Theorem of the Reduction of Singularities, Seidenberg [78]) Every isolated singularity of a holomorphic foliation of a complex surface admits a reduction by a blow-up process.

6.6 Irreducible Singularities The reduction of singularities theorem abovementioned can be made more accurate. Definition 6.6.1 A singularity of a holomorphic vector field X in dimension two is called irreducible if it belongs to one of the following categories: (up to a change of coordinates) ∂ ∂ + μy(1 + b(x, y)) λ/μ ∈ C \ Q+ , a(x, y), ∂x ∂y b(x, y) are holomorphic with a(0, 0) = b(0, 0) = 0. This will be  called non-degenerate irreducible case. ∂ ∂ + [y(1 + μx k ) + xb(x, y)] where λ, μ ∈ C, λ = (ii) X(x, y) = λ x k+1 ∂x ∂y 0, k ∈ N, b(x, y) is analytic of order ≥ k + 1 at 0 ∈ C2 . (i) X(x, y) = λx(1 + a(x, y))

This is the saddle-node case. A basic model for that is the formal normal form presented below:

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Theorem 6.6.2 (Martinet and Ramis [56], Hukuara et al. [40]) A germ of a saddle-node  foliation singularity is formally equivalent to a unique model Fλ,k x˙ = x k+1 given by y˙ = y(1 + λx k ) where λ = residue of the saddle-node 1 + k = multiplicity of the saddle-node. Remark 6.6.3 Not all saddle-node singularities are analytically conjugate to the formal normal form. Indeed, the formal normal form admits two separatrices (i.e., two invariant manifolds of dimension  one, through the origin), this is not the case x˙ = x 2 of the well-known Euler equation y˙ = x + y Theorem 6.6.4 (Seidenberg [78] ) Let F be a holomorphic foliation with an isolated singularity at 0 ∈ C2 . There is a finite sequence of quadratic blow-ups π(j ) : Mj → Mj −1 , (j = 1, . . . , ) such that π(1) is the blow-up of C2 at 0 ∈ C2 and π(j ) is the blow-up of Mj −1 at some point pj −1 ∈ Mj −1 , with the following properties:  := π ∗ (F), where π = π ◦ · · · ◦ π1 : M → C2 , (i.e., (a) The pull-back foliation F is a foliation with only irreducible singularities, of one of the two types): (i) xdy −λy dx +hot = 0, λ ∈ C\Q+ x k+1 dy −(y(1+λx k )+h. o. t.)dx = 0, (ii) λ ∈ C, k ∈ N (saddle-node), where h. o. t. stands for higher order terms. (b) The exceptional divisor D = π −1 (0) is a connected union of embedded   projective lines D = Pj , without triple points, transverse intersections and j =1

free of cycles.  or it is transverse to (c) A component P of D is either invariant by the foliation F  (without tangent points). F The singularity F is called non-dicritical if all components of D are invariant by  and it is called dicritical if F  exhibits some dicritical component. F, A characterization of dicritical singularities is now possible. This will be done in the next section.

6.7 Separatrices: Dicriticalness and Existence We begin with a definition that comes from the theory of (Real) ordinary differential equations. Definition 6.7.1 (Separatrix) Given a (germ of a) holomorphic singularity F at 0 ∈ C2 , a separatrix of F is (a germ of) an irreducible analytic curve   0 which is invariant by F (i.e.,  \ {0} is contained in a leaf of F).

6.7 Separatrices: Dicriticalness and Existence

63

Since a separatrix  is a germ of an irreducible analytic curve at 0 ∈ C2 , the Newton–Puiseux parametrization theorem gives a parametrization ϕ : (C, 0) → (, 0) of type (t n , t m ), n, m = 1 so that (, 0) is homeomorphic to a disc (D, 0). Example 6.7.2 (Holomorphic First Integral) Given a foliation with an isolated singularity F at 0 ∈ C2 we say that a non-constant holomorphic function f : (C2 , 0) → (C, 0) (defined in a neighborhood of the origin 0 ∈ C2 ) is a holomorphic first integral for F if f is constant along the leaves of F. If F is given by the vector field X (with sing(X) = {0}), then this is equivalent to df (X) ≡ 0. If we consider the dual 1-form ω, then this is equivalent to ω ∧ df ≡ 0. In any case, X is parallel to the Hamiltonian vector field Hf := −fy

∂ ∂ + fx ∂x ∂y

and ω is of the form ω = g ·df = g(fx dx +fy dy) for some meromorphic function g (g is holomorphic and non-vanishing if f is chosen to be reduced). We assume that f (0) = 0 . The separatrices of F are the branches of {f = 0}. It is well-known from the local theory of analytical functions [36] that f can be written (in a small r  n bidisc centered at 0 ∈ C2 ) as f = fj j , with nj ∈ N, fj holomorphic and such j =1

that: (a) fj = 0) is irreducible (b) fi and fj are relatively prime in the local ring O2 so that (fi = 0) ∩ (fj = 0) = {0}. (*) If moreover f has irreducible/connected fibers, then n1 , . . . , nr  = 1. The separatrices are then given by (fj = 0), j = 1, . . . , r. The other leaves of F are closed in a neighborhood of 0 and do not accumulate at 0. The foliation is therefore with finitely many separatrices. Example 6.7.3 (Meromorphic First Integral) A natural extension of the above definition gives as the notion of meromorphic first integral, f : M → C of a g singularity F at 0 ∈ C2 . Writing f = for g, h : (C2 , 0) → (C, 0) holomorphic h functions which (in the non-trivial case) vanish at 0 ∈ C2 , we have that the leaves of F are contained in the curves ag + bh = 0 with (a, b) ∈ C2 \ {0}. In particular, all leaves are contained in separatrices. Lemma 6.7.4 Let F be a singularity at 0 ∈ C2 . A leaf L of F is contained in a separatrix if, and only if, L \ L = {0}, i.e., L accumulates only at the singular point. Proof Let L be a leaf of F such that L \ L = {0}. By Remmert–Stein extension theorem (Theorem 2.1.4) the closure  := L ⊂ C2 is an analytic subset of dimension 1. Since  is clearly F-invariant and irreducible ( \ {0} = L is connected). We conclude that  = L = L ∪ {0} is a separatrix of F. The converse is clear.  

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As a corollary of Seidenberg’s theorem (Theorem 6.6.4) we have: Proposition 6.7.5 A foliation singularity F at 0 ∈ C2 is dicritical if, and only if, it exhibits infinitely many separatrices.   of F Proof First we observe that after the reduction of singularities, a leaf L  projects onto a (leaf contained in a) separatrix of F if, and only if, L is not contained in the exceptional divisor D. On the other hand, F is dicritical if and only if there  intersecting Pj is transverse to D is a component Pj of D for which every leaf L except may be for those at the corners Pj ∩ Pi = φ. Finally the invariant components of D only originate finitely many separatrices of F (the only possibilities come from separatrices of singularities which are not at corners, but these singularities are irreducible and therefore exhibit at most two separatrices).   As we have already mentioned, a separatrix  admits a parametrization ϕ : (C, 0) → (, 0) of type (t n , t m ), n, m = 1 so that (, 0) is homeomorphic to a disc (D, 0). In particular, the leaf L =  \ {0} has the topology of a punctured disc C∗ = C \ {0}. Its fundamental group is cyclic isomorphism to Z, generated by a loop γ  S 1 . The (local) holonomy group of the leaf L is then cyclic generated by a single diffeomorphism f : (C, 0) → (C, 0). Remark 6.7.6 It is here that we notice a drastic difference between the singular and the non-singular cases for foliations: In general it is not so common to find leaves with fundamental group in the non-singular case. On the other hand, these leaves are quite common in the singular foliations framework, thanks to the following theorem. Theorem 6.7.7 (Separatrix Theorem of Camacho and Sad, [15]) Every holomorphic foliation singularity F at 0 ∈ C2 admits some separatrix. Remark 6.7.8 The above result is typical from the holomorphic case since there are examples in the real analytic case where the foliation/vector field admits no separatrix: take X as the Hamiltonian of f = x12 +x22 in the real plane R2  (x1 , x2 ). The orbits are concentric circles, no separatrix is allowed. The Separatrix theorem is a byproduct of a suitable strategy on the reduction theorem of Seidenberg (organizing the reduction into linear chains) and a residue theorem applied to an index defined in association with a separatrix of a singularity. Just to give a few more words about this important theorem we have: Let F be a germ at 0 ∈ C2 of a singularity and  a smooth separatrix of F. In local coordinates we may assume that  : (y = 0). Then we choose ω = A(xy)dx + B(x, y)dy a holomorphic 1-form with sing ω = {0}, defining F. Since  is Finvariant we can write ω = y A1 (x, y)dx + B(x, y)dy with A1 (x, y) holomorphic. −A1 (x, 0) dx. It is a meromorphic 1-form with Then we consider the 1-form η := B(x, 0) no poles off x = 0 (notice that y  B(x, y) as a holomorphic function, otherwise ω would have non-isolated zeros). The Index of F relative to  at 0 is defined

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65

as I (F, , 0) := Res η(x = 0). The Index admits a geometrical interpretation as follows: Given x, the “inclination” of the tangent space T(x,y) F of the leaf L(x,y) of F through (x, y) is given by θx (y) =

dy −y A1 (x, y) = · dx B(x, y)

The derivative of this function θx of y at y = 0 is then θx (0)dx =

−A1 (x, 0) dx. B(x, 0)

The Index I (γ , , 0) is then the residue of this 1-form at x = 0. The index theorem of Camacho–Sad [15, 48] states that the sum of indexes of a foliation F at all the singularities in a compact analytic smooth invariant curve  on a complex surface M 2 is equal to the self-intersection (first Chern class) of  in M, does not depend therefore on the foliation F. 

I (F, p , p) =  ·  ∈ Z.

p∈sing(F )∩

6.8 Holonomy and Analytic Classification We shall now introduce the connection between the holonomy and analytic classification of a singularity. We shall start with the case of irreducible singularities and its connection with the holonomy of the separatrices.

6.8.1 Holonomy of Irreducible Singularities As we have already seen, there is always a separatrix through a holomorphic singularity in dimension two. Such a leaf has a holonomy map f : C, 0 → C, 0 and we shall study this map in some particular cases: Example 6.8.1 (Linear Case) 

x˙ = λx y˙ = μy

We fix the separatrix  : (y = 0). Choose a transverse section  : {x = 0}. Remark 6.8.2  is a (complex) disc.

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We consider the loop γ (t) = (x0 eu , 0) ⊂ 1; t ∈ [0, 2π ]. Let C be the product γ ×   S 1 × D, it is a solid torus. In this solid torus F induces a real flow given by the ordinary differential equation μy(t) dy/dx μ x  (t) dy λx(t) = =  = y(t) dt dx/dt x (t) λ x(t) dy μ = y(t).i. dt λ The solutions are y(t) = y · etiμ/λ . Therefore the first return (holonomy) map is given by f (y) = e2π iμ/λ . This is a linear map. In general for a non-degenerate irreducible singularity F : λx[1 + a(x, y)]dy− μy[1 + b(x, y)]dx = 0, the holonomy map of the separatrix  : (y = 0) is given by f (y) = e2π iμ/λ y + hot; in particular, its linear part is f  (0) = eeπ iμ/λ . Example 6.8.3 (Saddle-Node Normal Form Case) We consider a saddle-node in the normal form y k+1 dx − x(1 + λy k )dy = 0. The strong manifold  : (y = 0) has holonomy map h(y) given by a similar procedure. x(t) = x0 eit dy/dx y k+1 /x(1 + λy k ) i y k+1 (t) dy = = = dt dx/dt dx/dt 1 + λy k (t) ⎧ k+1 ⎪ ⎨y  (t) = i y (t) 1 + λy k (t) ⎪ ⎩y(0) = y f (y) = y(2π ).



For instance if k = 1 and λ = 0, i.e., for the saddle-node have d y (t) = i y (t) ⇒ dt 

2



−1 y(t)

=i

y˙ = y 2 x˙ = x

we then

6.8 Holonomy and Analytic Classification

⇒

67

−1 −1 = it + c ⇒ c = y(t) y(0)

−1 −1 y(0) = it − ⇒ y(t) = y(t) y(0) 1 − ity(0) y ⇒ f (y) = y(2π ) = this is a homography. 1 − 2π iy

⇒

For a saddle-node in the general form, the strong manifold (given by (y = 0) in the form y k+1 dx − [x(1 + λy k ) + (· · · )]dx = 0) the holonomy map of  is given by f (y) = y + ak+1 y k+1 + · · · where ak+1 = 0. It is a map tangent to the identity. Exercise 6.8.4 Calculate the holonomy map of the Poincaré–Dulac normal form F : ydx − (nx + ay n )dy = 0, n ≥ 2, for the (only) separatrix  : (y = 0).

6.8.2 Holonomy and Analytic Classification of Irreducible Singularities Let us now discuss one of the most important aspects of the concept of holonomy for singularities of foliations. It is well-known that for a pair of regular foliations a conjugation between holonomy groups of diffeomorphic leaves induces some conjugation between the foliations in neighborhoods of the given leaves. This is not immediate in case of foliations with singularities (how to extend the equivalence/conjugation to the singularities?) In the local framework we have for irreducible singularities a precise answer to this question thanks to the work of Martinet–Ramis and some other authors. Theorem 6.8.5 (Martinet and Ramis [56]) Let F1 , F2 be two germs of saddlenode singularities at 0 ∈ C2 . Assume that (y = 0) is the strong separatrix of F1 and F2 and denote by f1 , f2 : C, 0 → C, 0 the holonomy map if  with respect to F1 , F2 . Then there is a germ of a holomorphic diffeomorphism  : C2 , 0 → C2 , 0 taking the leaves of F1 onto the leaves of F2 (preserving off course the strong separatrix  : (y = 0)) if, and only if, there is a germ of a holomorphic diffeomorphism ϕ : C, 0 → C, 0 defining a conjugation ϕ ◦ f1 ◦ ϕ −1 = f2 between f1 and f2 . The same idea holds for non-degenerate irreducible singularities. Thanks to Poincaré–Dulac theorem we only need to consider singularities in the Siegel domain: Theorem 6.8.6 (Mattei and Moussu [57], Martinet and Ramis [55]) Let F1 , F2 be two germs of non-degenerate singularities Fj : xdy − λy(1 + bj (x, y))dx = 0, with bj (x, y) holomorphic, bj (0, 0) = 0, λ ∈ R− . Denote by fj : C, 0 → C, 0 the holonomy map of  : (y = 0) with respect to Fj . Then F1 and F2 are analytically

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conjugate by a holomorphic diffeomorphism  : C2 , 0 → C2 , 0 if, and only if, the holonomy maps f1 and f2 are analytically conjugate in Diff(C, 0). In the Siegel non-degenerate case, the idea of the proof is as follows: We choose vector field Xj (x, y) = x

∂ ∂ + λy(1 + bj (x, y)) ∂x ∂y

defining Fj (j = 1, 2) in a small bidisc Uj ⊂ C2 . We fix a point (x0 , 0) ∈  \ {0} close enough to origin, and consider the holonomy maps hj : (Cy , 0 → Cy , 0) defined by Fj as the first return map of the (transversely holomorphic) induced flow Lj := Fj C where C is the solid torus γ ×  ∼ = S 1 × D as above. Because Fj has a saddle like dynamics we know that for 0 < R < |x0 | the leaves of Fj intersect transversally the solid torus CR given by |z| = R, |y| < δ where δ > 0 is small enough. Remark 6.8.7 In the abstract picture of the situation above described, the leaves of Fj are visualized as real curves but actually they are complex curves, i.e., real surfaces of real dimension two. The analytic conjugation ϕ : (, (x0 , 0)) → ((x0 , 0)) between h1 and h2 satisfies ϕ ◦ h1 = h2 ◦ ϕ.  by setting First we extend ϕ to the solid torus C = C|x 0|

    ϕ y1 (t, y) = y2 t, ϕ(y) ,  where yj (t, y) is the solution of the flow Lj = Fj _C that starts from y ∈ . The above definition is consistent/valid because y1 (2π, y) = h1 (y) and y2 (2π, ϕ(y)) = h2 (ϕ(y)) and by hypothesis we have ϕ ◦ h1 (y) = h2 ◦ ϕ(y). Thus we have extended ϕ to the solid torus C. Now we show how to extend ϕ “radially” to the tori CR with 0 < R < |x0 |. We consider the induced radial flow where we consider  x (t) = e−t x and  yj (t, y) as the solution of ⎧ yj ⎨ d = Xj (e−t x,  yj ) dt ⎩  yj (0) = y .

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69

Then we extend ϕ : C → C  to the “interior” of C by setting     y2 t, ϕ(y) . ϕ  y1 (t, y) =  Some estimative shows that    y1 (t, y) ≤ eλ.(1+ε)t |y| for some constant 0 < ε & |λ|. (Recall that (important!) λ < 0). Similarly, we also have    y2 (t, y) ≤ |y|.e|λ|(1+ε)t . This shows that we have y1 (t, y)| | y2 (t, y)| ≤ A.| for some constant A > 0. Riemann extension theorem (Theorem 2.1.1) now shows that ϕ extends to the vertical axis x = 0 for |y| ≤ δ for a certain 0 < δ. We shall say that a germ F at 0 ∈ C is analytically linearizable if there is a holomorphic diffeomorphism  : C2 , 0 → C2 , 0 taking the leaves of F onto leaves of a linear foliation Fλ : xdy − λydx = 0, λ ∈ C \ {0}. In this case Fλ is unique and F is of the form F : xdy − λydx + hot = 0. Similarly a germ of a holomorphic diffeomorphism f : C, 0 → C, 0 is analytically linearizable if there is a germ of a holomorphic diffeomorphism ϕ : C, 0 → C, 0 such that ϕ ◦ f = fλ ◦ ϕ for some linear map fλ (y) = e2π iλ .y. Then, as a corollary of Theorem 6.8.6 above we have: Theorem 6.8.8 A germ of an irreducible non-degenerate singularity F : xdy − λydx + · · · = c is analytically linearizable if, and only if, its holonomy map of a given separatrix is analytically linearizable.

6.9 Examples Example 6.9.1 (The Euler Equation) Now we consider a classical example. The Euler equation is given in C2 by 

x˙ = x + y y˙ = y 2 .

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In the coordinates (u, v) = (1/x, y/x) we have: 

u˙ = −u2 (1 + v) v˙ = v 2 (1 − u) − uv.

The origin of this system is a singularity q. This singularity is reducible. Renaming the coordinates as (x, y) = (u, v) we have  x˙ = x 2 (1 + y) y˙ = xy + y 2 (x − 1). Blowing-up q as y = tx we obtain  t˙ = t 2 x˙ = −x(1 + tx). This is a saddle-node singularity with central manifold contained in the exceptional divisor and strong manifold contained in the horizontal affine axis (y = 0) ⊂ CP (2). The blow-up of q as x = uy produces  u˙ = u y˙ = y(−1 + u + uy). The origin is then a resonant singularity q2 . This singularity is not analytically linearizable and does not admit a holomorphic first integral. Now we consider the coordinate system (r, s) = (1/y, x/y). In this coordinate system we have 

r˙ = −r s˙ = −s + r(s + 1).

This is a reducible singularity q3 . Renaming coordinates as (x, y) = (r, s) we have  x˙ = x y˙ = −y + x(y + 1). Blowing-up q3 as y = tx we obtain  x˙ = −x t˙ = 1 + tx.

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71

Thus the origin is not a singular point. The blow-up x = uy gives 

u˙ = u2 1 y˙ = y[ 1+y − u].

The origin is then a saddle-node singularity, with central manifold contained in the exceptional divisor. The strong manifold is contained in the strict transform of l∞ , and is conjugated to the resonant singularity q2 . The final picture of the reduction of singularities of the Euler equation is then easily seen. Define the 1-forms ω = (x + y)dy − y 2 dx and η = 1+2y dy. Then η is y2 closed and dω = η ∧ ω. Thus, the Euler equation admits the integrating factor  1/y −1 h := exp η = y 2 e y and is given by the 1-form  = ω/ h = e y dy −d(xe1/y ) =   1/y d − xe1/y + e y dy . The Euler equation admits a Liouvillian first integral   1/y F = exp − xe1/y + e y dy which is uniform in C2 \ {y = 0}, with a line of essential singularities on (y = 0). This line is the only affine algebraic invariant curve. The other affine leaves are analytic outside of the horizontal line.

6.10 Exercises 1. Prove that a germ of a holomorphic foliation at 0 ∈ C2 , which is given by a closed meromorphic 1-form is logarithmic (i.e., given by a simple pole closed meromorphic 1-form) if, and only if, it admits no saddle-node in its reduction of singularities. 2. For a germ of foliation F at the origin 0 ∈ C2 , the following conditions are equivalent: (a) F is dicritical (i.e., not all components of the divisor in the reduction of singularities are invariant). (b) F admits infinitely many separatrices. (c) There is a nonempty germ of invariant open subset U ⊂ C2 with 0 ∈ U such that every leaf L ⊂ U is contained in a separatrix. 3. Compute the index of each separatrix in the following cases: linear case, Poincaré–Dulac normal form, and saddle-node case. 4. Use the Index theorem to prove that a foliation of the complex projective plane CP (2) having an invariant projective line E ⊂ CP (2) must have some singularity in E which is not of Siegel type. 5. Show the reduction of singularities of the foliation germ F given by x 2 dy − y 3 dx = 0. Repeat for x 2 dy − (y 3 + x 3 )dx = 0. 6. Can a germ of singularity F at 0 ∈ C2 produce only saddle-nodes in its reduction of singularities?

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7. Prove that a topological conjugacy between foliations takes separatrices onto separatrices. 8. Describe the process of resolution in 0 ∈ C2 , of the following curves: (a) y 2 − x 2 + x 3 = 0. (b) (y 2 − x 3 )(y 2 − 2x 3 ) + x 7 = 0. (c) y 2 − x 3 + f (x, y) = 0, where f and its derivatives of order 3 vanish in (0, 0). 9. Prove that the foliation of C2 defined by the form ω = (2x 2 +y 2 +xy +x 5 )dy − (2y 2 + x 7 )dx = 0 has three smooth separatrices at 0. 10. Let S = (f = 0) be a holomorphic curve in an open subset A of C2 , where 0 ∈ A is a unique singularity of df in A. Let F be the foliation of A whose leaves are the connected components of the curves (f = constant) \ {0} and  : M → C2 a blow-up process in 0 with divisor D that results in a resolution F˜ of F. (a) Prove that F has a finite number of separatrices by 0 and, therefore F˜ has the dicritical divisors. ˜ Prove that q is not a saddle-node and that (b) Let q ∈ D be a singularity of F. the characteristic numbers of F˜ in q are rational negative. (c) Prove that each singularity q ∈ D of F˜ which is not in a corner of D corresponds to a branch of S in 0 and vice-versa. (d) Prove that  gives the resolution of S at 0. 11. Consider the foliation F, defined in a neighborhood of 0 ∈ C2 by ω = d(y 2 − x 3 ) + α = 0, where α = f (x, y)dx + g(x, y)dy, and such that f and g and its derivatives of order 2 in 0 all vanish. Prove that a separatrix S has the equation w 2 − z3 = 0, at a suitable system of coordinates centered at 0 ∈ C2 . 12. Perform the reduction of singularities for the foliation germs at 0 ∈ C2 given below: (a) (b) (c) (d)

d(y 2 − x 2 + y 3 ) = 0. d[(x 2 − y 3 )(x 2 − 2y 3 ) + y 7 ] = 0. The cusp d(y 2 − x 3 ) = 0. The perturbation of the cusp given by d(y 2 − x 3 + f (x, y)) = 0, where f and its derivatives of order 3 vanish at (0, 0).

13. Discuss the reduction of singularities of the germ at 0 ∈ C2 of foliation given by F : df = 0 where f is a homogeneous polynomial of degree k ∈ {2, 3}. 14. Let 2 = −ydx + (x 2 + y)dy be the Hill form.1 Then,

1 This

1-form is connected to the classical Hill equation on the motion of the moon, [73].

6.10 Exercises

73

(i) The first blow-up originates two singularities in the exceptional divisor. One singularity is of Siegel type of the form − ydx + (1 − x + x 2 y)dy = 0.

(6.1)

The second singularity is nilpotent and given by (−y + yx + y 2 )dx + (x 2 + yx)dy = 0.

(6.2)

(ii) After performing n > 1 blow-ups starting from the second singularity and always repeating the process on the corresponding nilpotent singularity, we have: an exceptional divisor E = ∪nj=1 Ej with a nilpotent singularity at the intersection of the strict transform of the x-axis and En , which is given by (−y + nyx + nx n−1 y 2 )dx + (x 2 + yx n )dy.

(6.3)

Moreover, at each corner Ej ∩ Ej +1 , we have a Siegel type singularity given by (−y+j xy 2 +j x j −1 y j +1 )dx+(−x+(j +1)x 2 y+(j +1)x j y j )dy.

(6.4)

15. (Resonant singularities cf. Martinet and Ramis [55]) According to Martinet and Ramis [55] a resonant non-linearizable singularity is formally isomorphic to a unique equation ωp/q,k,λ := p(1 + (λ − 1)uk )ydx + q(1 + λuk )xdy where p, q, k ∈ N, λ ∈ C and u := x p y q . Moreover p/q, k, λ are the formal invariants of the equation. By introducing integral numbers n, m ∈ Z such that mp − nq = 1 we can rewrite ωp/q,k,λ = (1 + (λ − mp)uk )(pydx + qxdy)+pquk (nydx+mxdy). This last expression admits the integrating factor 1 hp/q,k,λ = pqxyuk ; this means that the 1-form hp/q,k,λ ωp/q,k,λ := p/q,k,λ is closed and meromorphic, with poles of order kp + 1 in (x = 0) and kq + 1 in (y = 0). Prove the following: Claim 6.10.1 There is a single formal meromorphic closed 1-form η with simple poles in (y = 0) such that dωp/q,k,λ = η ∧ ωp/q,k,λ . This form is η = dhp/q,k,λ / hp/q,k,λ . Hint: Since p/q,k,λ is closed we conclude that η0 := dhp/q,k,λ / hp/q,k,λ satisfies the equation dωp/q,k,λ = η0 ∧ ωp/q,k,λ . Now assume that ω is a closed meromorphic formal 1-form as in the statement. We have η − η0 = g.p/q,k,λ for some meromorphic function g such that dg ∧ p/q,k,λ = 0. If g is not constant, then p/q,k,λ admits a formal meromorphic first integral. This is not possible, because it does not admit a holomorphic first integral (see for instance [57]). Therefore g must be constant. Because both η and η0 have simple poles, this implies that gp/q,k,λ has simple poles, therefore g = 0.

Chapter 7

Holomorphic First Integrals

One of the most interesting questions about singularities of holomorphic foliations is under which conditions they admit a first integral. This goes back even to the work of Poincaré and Painlevé. In modern terms, the striking result of Mattei–Moussu can be seen as a landmark in the theory. Several are the reasons for it. For instance, its beauty and simplicity. Also because it was the first time where techniques from algebraic geometry (desingularization theory), theory of foliations (holonomy), and dynamical systems (dynamics of map germs) where used combined in a statement of holomorphic foliations. Let us present this outstanding result.

7.1 Mattei–Moussu Theorem From the structural viewpoint, the singularities with a holomorphic first integral are the most simple singularities of a holomorphic foliation. Definition 7.1.1 Given a singular foliation germ F at 0 ∈ C2 , a holomorphic function F : C2 , 0 → C, 0 (i.e., a holomorphic function F defined in some neighborhood V of 0 ∈ C2 and with F (0) = 0 ∈ C), is a holomorphic first integral of F if it is constant along the leaves of F. If F is given by the vector field X with an isolated singularity at 0, then the above condition is equivalent to df (X) ≡ 0. In terms of the dual 1-form ω, the condition becomes ω∧df ≡ 0. This latter, thanks to Saito’s division lemma [74], is equivalent to ω = gdf . As we have already seen if F admits a (non-constant) holomorphic first integral F : C2 , 0 → C, 0 then its leaves satisfy: (i) The leaves of F are closed outside of the origin. (ii) There are only finitely many leaves that accumulate at the origin.

© Springer Nature Switzerland AG 2021 B. Scárdua, Holomorphic Foliations with Singularities, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-76705-1_7

75

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7 Holomorphic First Integrals

In terms of the language of Seidenberg’s theorem (Theorem 6.6.4) we have: F admits a holomorphic first integral ⇓ (i) F is non-dicritical. (ii) The leaves are closed off the singularity. All this for a sufficiently small neighborhood of the singular points 0 ∈ C2 . Remark 7.1.2 Thanks to Remmert–Stein extension theorem (Theorem 2.1.4) a leaf which is closed off 0 ∈ C2 and is not (contained in) a separatrix, is contained in an analytic curve and is called analytic leaf. The theorem of Mattei–Moussu states a direct converse to the above and can be stated in dimension two as follows: Theorem 7.1.3 (Mattei and Moussu, [57]) Let F be a holomorphic foliation singularity at 0 ∈ C2 . Assume that for a small neighborhood V of 0 ∈ C2 we have: (i) The leaves of F in V are closed in V \ {0} ⊂ C2 . (ii) Only a finite number of leaves of F in V accumulate at 0 ∈ C2 . Then F admits a (non-constant) holomorphic first integral F : W → C in some open subset 0 ∈ W ⊂ V . The classical proof relies on the Reduction of Singularities, as well as on the dynamics of holomorphic diffeomorphisms f : C, 0 → C, 0. R. Moussu has given an alternative proof based on the classical Reeb local stability theorem [31] and on the dynamics of diffeomorphisms f : C, 0 → C, 0 [58]. The next example, due to M. Suzuki and Cerveau-Mattei, shows that there is no topological criteria for the existence of a meromorphic first integral. Example 7.1.4 (Suzuki’s Example, [84]) Consider the germ of singular foliation F at the origin 0 ∈ C2 given by:  = 0 where  = (y 3 + y 2 − xy)dx − (2xy 2 + =

xy − x 2 )dy. The germ F has the Liouvillian first integral f (x, y) = = and the following remarkable properties:

x y

exp[ y(y+1) ] x

(i) F is μ − simple, that is, it is a dicritical germ which is desingularized with only = one blow-up and the resulting foliation has no singularities on the exceptional divisor; it is transverse to this projective line everywhere except for (a unique) tangent point (see [45]). Therefore, it follows that: (i)’ Every leaf of F is a separatrix and therefore is given by some equation (f = 0) = where f ∈ O2 .

7.2 Groups of Germs of Holomorphic Diffeomorphisms

77

(ii) F does not admit a meromorphic first integral in any neighborhood of the origin =

0 ∈ C2 (see [22] for a proof, or follow our argumentation). Performing a blow-up (y = tx) at the origin 0 ∈ C2 we obtain the foliation F˜ : t 3 dx + (2xt 2 + t − 1)dt = 0 given by the vector field x˙ = 2xt 2 + t − 1,

t˙ = t 3 .

The initial foliation has the Liouvillian first integral f =

x y

exp( y(y+1) ) and x

therefore the foliation above has the Liouvillian first integral f (x, t) =

1 t

et (xt+1) . 1

Restricting this function to the projective line (x = 0) we obtain f (0, t) = 1t e t which is a Liouvillian function on C. The map σ : (C, 1) → (C, 1) defined by mapping the point p ∈ (C, 1) onto the other intersection point of the leaf Lp of F˜ though p with the projective line, is (because of the order-2 tangency) a germ of involution on (C, 1). This germ is given by the relation f (0, t) ◦ σ = f (0, t), that is, 1t et = σ 1(t) eσ (t) . This defines σ (t) as a nonalgebraic Liouvillian function on C and according to [45] this is enough to conclude that F does not admit a non-trivial = meromorphic first integral.

7.2 Groups of Germs of Holomorphic Diffeomorphisms We shall start with some basic facts. We denote by Diff(C, 0) the group of germs of holomorphic diffeomorphisms f : C, 0 → C, 0. Such a map germ has representatives given by maps fV : V → fV (V ), where 0 ∈ V ⊂ C is an open set, and fV : V → fV (V ) is a holomorphic diffeomorphism with fV (0) = 0. It can be identified (the germ f ) with a power series f (z) = f  (0)z + aj zj ∈ C{z} where f  (0) = 0.

j ≥2

Lemma 7.2.1 Let G ⊂ Diff(C, 0) be a finite subgroup. Then G is cyclic and  2π i  analytically conjugate to a group z → e ν z ν ∈ N.

g(z) · This is a well g∈G g (0) defined holomorphic map because G is finite (in particular, all elements of G have a common definition domain around 0 ∈ C). Moreover  (0) = |G| = 0 so that  ∈ Diff(C, 0). Given now any element g0 ∈ G we have

Proof We define a map  : C, 0 → C, 0 by (z) =

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7 Holomorphic First Integrals

(g0 (z)) =

 g(g0 (z))  (g ◦ g0 )(z)  = g (0) 0 g  (0) g  (0)g0 (0)

g∈G

g∈G

 (g ◦ g0 )(z) = g0 (0)(z). = g0 (0) (g ◦ g0 )(0) g∈G

Thus  ∈ Diff(C, 0) is an analytic conjugation of G with a finite subgroup of the linear group GL(1, C) = C∗ ; this ends the proof.   Given a holomorphic function (z) defined in a neighborhood of 0 ∈ C we consider the invariance group of (z) as Inv() = {g ∈ Diff(C, 0);  ◦ g = } in terms of germs. Assume that  is not constant, (0) = 0. Since up to a change of coordinates we have (z) = zν for some ν ∈ N we conclude that: Lemma 7.2.2 The invariance group Inv() is a finite cyclic group. By the two above lemmas, we have that the finite subgroups of Diff(C, 0) are the invariance groups of holomorphic functions  : C, 0 → C, 0. A final simple remark concerning the finiteness of subgroups of Diff(C, 0) is: Lemma 7.2.3 Let G ∈ Diff(C, 0) be a finitely generated subgroup such that each element g ∈ G is periodic (∃ ng ∈ N such that g ng = Id). Then G is finite. Proof First we observe that a non-trivial flat element g(z) = z + ak+1 zk+1 + · · · (ak+1 = 0) is not periodic; indeed g n (z) = z + nak+1 zk+1 + · · · . The commutator of two elements g1 , g2 ∈ G is [g1 , g2 ] = g1 g2 g1−1 g2−1 is a map tangent to the identity. Thus by hypothesis this is the identity and G is abelian. This implies that G is finite because it is finitely generated.   Now we get to the main point in our argumentation: Proposition 7.2.4 (Finiteness Condition) A germ of a holomorphic diffeomorphism f : C, 0 → C, 0 has finite order ( i.e., f n = Id for some n ∈ N) if, and only if, for some neighborhood V of 0 all the orbits of f in V are finite. The classical proof is as follows: Lemma 7.2.5 Let K ⊂ Rn be a compact connected neighborhood of 0 ∈ Rn and h : K → h(K) ∈ Rn a diffeomorphism with h(0) = 0. Then there exists a boundary point x ∈ ∂K such that the number of iterates of x by h contained in K is infinite. In general, for x ∈ K we define μK (x) := {n; hn (x) ∈ K}.

 Proof Assume by contradiction that μK is bounded in ∂K, i.e., μK ∂K < N < ∞ for some N ∈ N. Let us consider A = {x ∈ K; μK (x) < N } ⊃ ∂K ◦

B = {x ∈ Int(K); μ ◦ (x) ≥ N}, K = Int(K). K

7.2 Groups of Germs of Holomorphic Diffeomorphisms

79

Then A, B are open subset of K, A ⊃ ∂K, 0 ∈ B and A ∩ B = φ. Therefore, since K is connected, ∃ x0 ∈ K such that x0 ∈ / A ∪ B, i.e., μK (x0 ) ≥ N > μ ◦ (x0 ). Thus K there exists n0 such that y0 = hn0 (x0 ) ∈ ∂K. Then we have μK (y0 ) = μK (x0 ) ≥ N, contradiction. This proves Lemma 7.2.5.   Our strategy to prove the Finiteness condition proposition is to prove: If h ∈ Diff(C, 0) is not periodic, then there exists fundamental system of neighborhoods U of 0 ∈ C such that for each neighborhood U ∈ U the set of points x ∈ U where the U -orbit is not finite, is not countable, and contains the origin 0 ∈ C in its closure. We start by fixing a compact disc 0 ∈ D ⊂ C such that h : D → h(D) is a (holomorphic) diffeomorphism. Now we consider the 3 following sets:  • P ⊂ D is the set of periodic points of hD . • F ⊂ D is the set of non-periodic points of finite D-orbit. • I ⊂ D is the set of points with infinite D-orbit. Now we define a sequence of compact subsets An ⊂ D as follows: A0 := D A1 := D ∩ h−1 (D) .. . An := D ∩ h−1 (D) ∩ · · · ∩ h−n (D) .. . Also let Cn be the connected component of An that contains the origin 0 ∈ C and C := Cn . Then by construction, a point in C is periodic or has infinite D-orbit, n≥0

i.e., C ⊂ I ∪ P . Claim 7.2.6 If C is countable, then I is not countable. Proof of Claim 7.2.6 By hypothesis C is countable. Therefore there is a subdisc Dr ⊂ D of radius 0 < r < radius of D; such that C ∩ ∂Dr = φ (otherwise C is not countable). Therefore ∃ m ∈ N such that Cm ∩ ∂Dr = φ. Let K be a compact, connected neighborhood of Cm that does not intersect the other components of Am . In particular, we have φ = ∂K ∩ Am = ∂K ∩ D ∩ h−1 (D) ∩ · · · ∩ h−m (D) so that there must exist for every x ∈ ∂K a m ∈ N such that N  p < m and such that hp (x) ∈ / D. Consequently we get P ∩ ∂K = φ. Now we denote the sets  := h-periodic points in K P  F := non-periodic points but with finite K-orbit

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7 Holomorphic First Integrals

I := points with infinite K-orbit.   as P = n := {points periodic of period n}. n where P We stratify P P n≥0

n is finite: indeed, if x0 ∈ ∂ P n is an We observe that the boundary ∂ P n there exists a neighborhood V of x0 such that accumulation point of points ∂ P n = φ (Identity Principle). Since the orbit of hn (x) = x, ∀ x ∈ V because V ∩ P x0 does not intersect ∂K we have, for V sufficiently small, that hj (V ) ⊂ Int(K) n ), i.e., x0 ∈ Int(P n ). This actually means ∀ j = 0, 1, . . . , n − 1. Thus V ⊂ Int(P n is finite). n are isolated (⇒ ∂ P that points in ∂ P  is countable. As a consequence the boundary of P ) ∪ F  ∪ (I∪ ∂ P ). Notice The compact set K can be decomposed as K = Int(P ) is an open subset of C since by hypothesis P  ∩ ∂K = φ. By its turn, that Int(P  is open in K and may intersect ∂K. If F  = φ then ∂K ⊂ I∪ ∂ P  and since ∂ P  F is countable (as we have just seen above) and ∂K is not countable it follows that  = φ and Int(P ) = φ we conclude that for every r > 0 I is not countable. If F  intersects the boundary ∂Dr so that I is not countable. If small enough, I ∪ ∂ P ) = φ and F  = φ then, because these are disjoint open subsets, the set now Int(P ) ∪ F  is not countable. Since ∂ P  is a countable set this implies that I is K − Int(P not countable, proving Claim 7.2.6.   Lemma 7.2.7 Suppose that C is not countable. Then I is not countable. Proof By Lemma 7.2.5 for every compact disc 0 ∈ Dr ⊂ D we have (P ∪ I ) ∩ ∂Dr = φ. Therefore P ∪ I is not countable. Suppose by contradict that I is countable, then P is not countable. Since C = (C ∩ I ) ∪ (C ∩ P ) (recall that C ⊂ I ∪ P ) we have that C ∩ P is not countable (otherwise C would be countable, contradiction). Let us write C∩P =



Pn ,

n≥0

where Pn = points periodic of period n in P ∩ C. Then there exists n0 such that Pn0 is not finite with an accumulation point in Cn0 . Since hn0 is holomorphic in an open neighborhood of Cn0 we conclude from the Identity Principle that hn0 ≡ Id, contradicting the non-periodicity of h.   All together, Claim 7.2.6 and Lemma 7.2.7 prove that the set I of points x ∈ D with infinite D-orbit is not countable. This proves Proposition 7.2.4.

7.3 Irreducible Singularities We shall now address the irreducible case proving Mattei–Moussu in this situation:

7.4 The Case of a Single Blow-up

81

Lemma 7.3.1 Let F be a germ of an irreducible singularity at 0 ∈ C2 . Assume that F has closed leaves off the origin 0 ∈ C2 . Then F admits a holomorphic first integral. Indeed, F is analytically conjugate to nxdy + mydx = 0, for some n, m ∈ N, in a neighborhood of 0 ∈ C2 . Proof Since by hypothesis F is irreducible, we divide the proof into two cases: Case 1 F is a saddle-node singularity germ. In this case, we have the strong manifold say F : y k+1 dx − [x(1 + λy k ) + · · · ]dy = 0  : (y = 0). The holonomy map of the strong manifold is of the form hγ (y) = y + ak+1 y k+1 + · · ·

ak+1 = 0.

Therefore hγ is not periodic. This implies that the orbits of hγ are not all of them closed and thus F has some non-analytic leaves on small neighborhoods of the origin. Thus, this case cannot occur. Remark 7.3.2 Indeed it is possible to say much more about the dynamics of hγ : no orbit is finite except for the fixed point at the origin. Case 2 (non-degenerate case). We write F as xdy − λydx + hot = 0 λ ∈ N \ Q+ with invariant axes. Again we consider the holonomy of  : (y = 0). It is a map of the form hγ (y) = e2π iλ y + · · · . If λ ∈ / R then Poincaré Linearization theorem implies that F is analytically linearizable and therefore hγ (y) is conjugate to y → e2π iλ y. This latter map only has finite orbits when λ ∈ Q so we would have λ ∈ Q+ contradiction. Therefore λ ∈ R− and indeed because hγ is periodic we conclude that it is linearizable and the same holds for F; moreover (still because hγ is periodic) we must have λ ∈ Q− say, λ = −n/m, n, m ∈ N and F is analytically linearizable as nxdy + mydx = 0.  

7.4 The Case of a Single Blow-up In order to illustrate the main ideas in the proof we consider the following situation: F can be reduced with a single blow-up π :  C20 → C2 . Since F is non-dicritical the   = π ∗ (F). Because the leaves of F are closed exceptional divisor is F-invariant, F off the origin, the non-separatrices are analytic leaves and the same holds for the  transverse to P. leaves of F which are contained in P or the separatrices of F Remark 7.4.1 P \ sing(F) is a leaf of F.  = { Write Sing(F) p1 , . . . , p r } ⊂ P. From the irreducible case above we j of p j in  C20 where conclude that for each j ∈ {1, . . . , r} there is a neighborhood V   F admits a (non-constant) holomorphic first integral say Fj : Vj → C. We may j is a product V j = Dj × Dε of a disc p assume that V j ∈ Dj ⊂ P and a small disc

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7 Holomorphic First Integrals

0 ∈ Dε of radius ε > 0. Assume also that Di ∩ Dj = φ, ∀ i = j . Fix now a point r  p 0 ∈ P \ Dj . Since P is a 2-sphere we may choose a simply-connected domain j =1

Aj ⊂ P such that p0 , p 1 , . . . , p r } = { p0 , p j }. Aj ∩ {  in a j for F Claim 7.4.2 We may extend Fj to a holomorphic first integral F neighborhood Uj of Dj ∪ Aj . Proof Indeed, this is the classical process of extension by the holonomy, which is possible because Aj is simply-connected therefore it has no further holonomy that the one already in Dj . Let us be more precise: Fix a point  aj ∈ (Dj \ { pj }) ∩ Aj  with { and a transverse disc  qj } =  qj to F qj ∩ P. We consider a simple path δj : [0, 1] → tj with δj (0) = p 0 . δj (1) =  qj . qj of the path δj in the leaf We consider the holonomy map hδj :  →   P \ { p1 , . . . , p r } of F. j (y) := Fj (hδj (y)) for y ∈  close enough to p Then we define F 0 . This is well-defined because Aj is simply-connected and Fj is already invariant pj }. by the local holonomy map associated to a small loop γj ⊂ Dj \ { j as above. We may also Thus we can construct the (holonomy) extension F assume that Vj ⊂ Uj .    j we consider the restriction F j  and the invariance group For each extension F     j ◦ h = F k }. j  = {h ∈ Diff(, p 0 ) : F Inv F   Finally we consider the global invariance group Inv(  F, ) := subgroup of j  ) j = 1, . . . , r. Diff(, p 0 ) generated by the invariance groups Inv(F   ) is a finite group. Claim 7.4.3 Inv(F,   into leaves of F  so the j  ) takes leaves of F Proof Indeed, each map in h ∈ Inv(F   ). Since the leaves of F  are analytic curves same holds for the maps h ∈ Inv(F,  ) (except for those contained in separatrices) we conclude that each map in Inv(F,  ) is periodic (Finiteness has closed orbits. This implies that each map in Inv(F,  ) is finitely generated this implies condition Proposition7.2.4). Because Inv(F,  ) is a finite group, indeed cyclic generated by a periodic rotation z → that Inv(F, eπ i e ν z, (ν ∈ N).    ) Thus there is a holomorphic function F on  such that every map in Inv(F,  ). Because Inv(F,  ) ⊃ leaves invariant the map F ; i.e., F ◦ h = F , ∀ h ∈ Inv( F,  j  ) this implies that F is also constant along the levels of F j in Uj . Inv(F  r   in Thus we have constructed a holomorphic first integral F for F Uj . j =1

7.6 Exercises

83

Now the complementary part of

r  j =1

Aj in P is topologically a disc, i.e., it is

simply-connected. This complement then has trivial holonomy and therefore F admits a holonomy  and therefore F admits a holonomy extension as a holomorphic first integral of F 2   to a neighborhood U of P in C0 . This projects into a holomorphic first integral for F in a neighborhood of 0 ∈ C2 .

7.5 The General Case As a final word about the proof in the general case, where several blow-ups may be needed to finish the reduction of singularities, we have a similar argumentation as above by the Induction on the number of blow-ups in the reduction of singularities.

7.6 Exercises 1. Prove or give a counterexample to the following claim: A germ of foliation at the origin 0 ∈ C2 admits a holomorphic first integral if, and only if, it is logarithmic (given by a simple poles closed meromorphic 1-form), non-dicritical, and fully resonant (i.e., each singularity appearing in the reduction of singularities is non-degenerate resonant of the form xdy − λydx + . . . = 0, λ ∈ Q− ). 2. Prove or give a counterexample to the following claim: A germ of a logarithmic foliation admits a holomorphic first integral if, and only if, it is non-dicritical and each separatrix has a periodic holonomy map. 3. A topological conjugation between two germs of foliations F and F  at 0 ∈ C2 is a germ of homeomorphism ϕ : (C2 , 0) → (C2 , 0) that takes leaves of F onto leaves of F1 . Prove that if F and F  are topologically conjugate and F admits a holomorphic first integral, then F  also admits a holomorphic first integral. 4. Prove that if F is a germ of foliation at 0 ∈ C2 admitting a meromorphic first integral and F is non-dicritical, then F admits a holomorphic first integral. 5. A formal first integral for a germ of holomorphic foliation F at 0 ∈ C2 is a ∞

formal series fˆ(x, y) = fij x i y j ∈ C[[x, y]] such that d fˆ ∧ ω = 0 as a i,j =0

formal expression, where ω = P dx +Qdy is a holomorphic 1-form defining F in some neighborhood of the origin. Using the reduction of singularities prove that: (a) If F admits a formal first integral, then it is non-dicritical (b) If F admits a formal first integral, then it is a generalized curve, i.e., no saddle-nodes appear in its reduction of singularities.

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7 Holomorphic First Integrals

6. Prove that a group of germs of complex diffeomorphisms G ⊂ Diff(C, 0) is finite provided that it leaves invariant a one-variable formal function fˆ ∈ C[[z]], i.e., fˆ ◦ g = fˆ for all g ∈ G. (hint: using the formal version of the Inverse function theorem, prove that up to a change of coordinates we can write fˆ(z) = zk for some k ∈ N.) 7. Using the exercises above try to prove the following theorem due to Mattei and Moussu [57]: A germ of holomorphic foliation F at 0 ∈ C2 admitting a formal first integral also admits a holomorphic first integral. 8. Prove that if a germ of holomorphic first integral F is a non-dicritical generalized curve, then it admits a holomorphic first integral provided that: each neighborhood of the origin 0 ∈ C2 contains a nonempty invariant open subset where the leaves of F are all closed. 9. Can a germ of a Darboux foliation F at 0 ∈ C2 be dicritical without having all its leaves contained in separatrices? 10. Let F be a germ of a non-dicritical Darboux foliation at 0 ∈ C2 . If all but one of the separatrices of F exhibit periodic local holonomy maps, then the same holds for the remaining separatrix. 11. Let F be a germ of foliation at 0 ∈ C2 given by dQ(x, y) + h. o. t. = 0 where Q(x, y) is a non-degenerate quadratic form. Prove that F is analytically equivalent to the form xdy + ydx + h. o. t. = 0. Conclude that if F has some separatrix with finite holonomy map, then it admits a holomorphic first integral.

Chapter 8

Dynamics of a Local Diffeomorphism

We consider f as a germ of holomorphic diffeomorphism at 0 ∈ C fixing 0, say f (z) = λz + ak+1 zk+1 + . . . , k ≥ 1, λ ∈ C∗ . We shall describe its dynamics (i.e., the dynamics of its pseudo-orbits) according to the multiplier λ = f  (0). For a quick and effective reference in this subject we recommend [5].

8.1 Hyperbolic Case (i) Hyperbolic Case:

|λ| = 0, 1.

By Poincaré–Köenigs theorem (1884) there is a unique holomorphic diffeomorphism ϕ with ϕ  (0) = 1 that conjugates f to the linear map z → λz. Then we have if |λ| < 1 that f n (z) → 0, as n → ∞, for each z ≈ 0, following a spiraling or linear path, depending on whether λ is complex on pure real.

8.2 Parabolic Case (ii) Parabolic Case:

|λ| = 1, λk = 1 for some k ∈ N.

Proposition 8.2.1 Let h : (C, 0) → (C, 0)

be a jgerm of a holomorphic diffeomorphism tangent to the identity h(z) = z + aj z , a2 = 0. Then there exist sectors j ≥2

S + and S − with vertex at 0 ∈ C, angles π − θ0 (where 0 < θ0 < π/2), and opposite bisectrices in such a way that:

© Springer Nature Switzerland AG 2021 B. Scárdua, Holomorphic Foliations with Singularities, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-76705-1_8

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8 Dynamics of a Local Diffeomorphism

(i) h(S + ) ⊂ S + , lim hn (z) = 0, ∀ z ∈ S + h→+∞

(ii) h−1 (S −1 ) ⊂ S − , lim h−n (z) = 0, ∀ z ∈ S − n→+∞

Proof A linear change of coordinates allows us to write h(z) = z − z2 + For |z| < δ small enough we consider g(z) defined by ⎧ ⎨g(0) = 1 ⎩h(z) =

z 1 + zg(z)

j ≥3

aj zj .

.

Then g(z) is analytic (for |z| < δ small enough) and we consider the coordinate ξ = 1 , z = 0. Under these coordinates we have h as  h(ξ ) = ξ + g(1/ξ ),  h(∞) = ∞, z a diffeomorphism defined in a neighborhood of ξ = ∞. Because g(0) = 1 we have 1 1 g(1/ξ ) = 1 +  g (ξ ) with  g bounded as ξ → ∞. Therefore  h(ξ ) = ξ + 1 +  g (ξ ) ξ ξ is a map that is close to the translation T (ξ ) = ξ + 1 for |ξ | big enough. Hence we can find a sector S + of horizontal bisectrices and angle π − θ0 (0 < θ0 < π/2) and a disc D + = (ξ1 − R)2 + ξ22 ≤ (2R)2 so that if  S + :=  S+ ∩ + + + n −      (CD ) then h(S ) ⊂ S and lim h (ξ ) = ∞. Similarly we obtain S such that n→∞  S−) ⊂  S − and lim  S−. h−n (ξ ) = ∞ on  h−1 ( n→∞

The sectors S + and S − are then the images of  S + and  S − , respectively, by the change of coordinates z = 1/ξ .   Similarly, for the case, h : C, 0 → C, 0 h(z) = z +

∞

aj zj , ak+1 = 0; we j ≤k+1 Sj+ alternates with Sj− that

have sectors S1+ , . . . , Sk+ and sectors S1− , . . . , Sk− so that alternates with Sj++1 , having angles ≥ π/k, in such a way that (a) Sj+ ∩ Sj−1 = φ

− − (b) h(Sj+ ) ⊂ Sj+ , h−1 j (Sj ) ⊂ Sj k  lim hn (z) = 0, ∀ z ∈ Sj+ n→∞

j =1 k 

lim h−n (z) = 0, ∀ z ∈

n→∞

j =1

Sj−

Indeed, more can be said: Theorem 8.2.2 (Camacho, [9]) Let f (z) = λz + O(z2 ) be a holomorphic germ of a complex diffeomorphism, λn = 1 for some n ∈ N (with n minimal). Then: (i) Either f n (z) = z (ii) Or there exists k ∈ N such that f is topologically conjugate to Tk,λ,n : z → λz(1 − znk ).

8.3 Elliptic Case

87

If moreover f (z) = z + ak+1 zk+1 + O(zk+2 ) ak+1 = 0, then f is topologically conjugate to Tk : z → z + zk+1 . And also: Theorem 8.2.3 (Fatou–Leau Flower Theorem, [1]) Let f (z) = z + zk+1 + O(zk+2 ), k ∈ N. Then there exist k domains called petals, Pj , symmetric with 2π q respect to the k directions arg(z) = , q = 0, . . . , k − 1, such that: k (i) Pj ∩ Pk = φ for j = k; 0 ∈ ∂Pj and each petal Pj is holomorphic to the right-half plane H ⊂ R2 . (ii) For each z ∈ Pj we havef m (z) → 0 as m → ∞ moreover. (iii) For each j the map f P is holomorphically conjugate to the parabolic j automorphism z → z + i on H. If f (z) = z + zk+1 + O(zk+2 ) then f −1 (z) = z − zk+1 + O(zk+2 ). Therefore we get, from Leau–Fatou theorem, k attracting Qj for f −1 symmetric (zq + 1) . . . , q = 0, . . . , k − 1. These with respect to the k directions arg(z) = k directions are the bisectrices of the angles between two consecutive attracting directions for f . The Qj ’s are repelling petals for f , intersecting the Pj ’s and  Pj ∪ Qj ∪ {0} is an open neighborhood of 0 ∈ C in C. We have therefore a j

pretty clear description of the dynamics of f .

8.3 Elliptic Case Elliptic Case |λ| = 1, λ = e2π iθ for some θ ∈ R \ Q. The case is pretty rich and it is a subject of deep research. The main question first posed is whether f = λz + . . . is always analytically linearizable. It was Cremer who first gave an example of an elliptic map that is not analytically linearizable. Indeed, Cremer introduced the following: Cremer Condition, [25] For θ ∈ R \ Q if lim sup |{nθ }|−1/n = ∞, n→∞

then there exists an elliptic germ f (z) = e2π iθ z + O(z2 ) that is not linearizable. In the above statement, for a positive real number x ∈ R, we have {x} := x − [x]

[x] = the integral part of x.

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8 Dynamics of a Local Diffeomorphism

A number θ satisfying the Cremer condition above is called a Cremer number. Cremer numbers form a dense subset of R of zero Lebesgue measure [5]. On the other hand there are arithmetical conditions originally by Siegel and recently by J.C. Yoccoz and Brjuno [6] for assuring that if θ ∈ R \ Q satisfies this arithmetical condition then f is always analytically linearizable. These θ form a full Lebesgue measure subset of R. Another remarkable result is: Theorem 8.3.1 (Sigel et al. [67]) Let θ ∈ R\Q and λ = e2π iθ . If the germ pλ (z) = λz + z2 is analytically linearizable, then every germ f ∈ Diff(C, 0) with f  (0) = λ is also analytically linearizable. Regarding the dynamics we have the remarkable work of Pérez and Marco [67– 69]. He introduces the following concept: Definition 8.3.2 (Small Cycles Property) A small cycle for f is a finite orbit of f (a subset {p1 , . . . , pn } ⊂ C \ {0} such that pi = pj and f (pi ) = pj +1 (mod n)). We say that f has the small cycles property if for any open neighborhood U of 0 then exists a small cycle for f contained in U . In this case the small cycles accumulate at 0. In particular the germ f is not linearizable. Theorem 8.3.3 (Pérez-Marco) There exist elliptic germs with the small cycles property. Not all non-linearizable elliptic germs have the small cycles property. Pérez-Marco gives an arithmetic condition on θ in order to decide whether the nonlinearizable germ has the small cycles property. Actually Pérez-Marco work goes much further with the introduction of the Hedge-Hogs. He also concludes: Theorem 8.3.4 (Pérez-Marco) A non-linearizable elliptic map is always arbitrarily close to the origin some orbit that accumulates at the origin. Such an orbit cannot be closed. For an elliptic germ f : C, 0 → C, 0 we can add: Choose an open connected subset 0 ∈ U ⊂ C where f is univalued and the stable set of f in U is K(U, f ) = ∞  f −j (U ). j =0

Theorem 8.3.5 (Pérez-Marco) Let f : C, 0 → C, 0 be an elliptic map germ with stable set K(f, U ). Then: (i) K(f, U ) is compact, connected, full (i.e., U \ K(f, U ) is connected), contains the origin but is not restricted to the origin, i.e., 0 ∈ K(f, U ) = {0}. Moreover, K(f, U ) is not locally connected at any point distinct from the origin.

8.4 Exercises

89

(ii) Any point of K(f, U ) \ {0} is recurrent (i.e., it is a limit point of its orbit). (iii) There is an orbit in K(f, U ) that accumulates at the origin. (iv) No non-trivial orbit converges to the origin. The stable set K(f, U ) is called a Hedge-Hog.

8.4 Exercises 1. Let f : C, 0 → C, 0 be an elliptic map germ. Prove that f is analytically linearizable (as a germ) if and only if it is C 0 -linearizable, meaning that there is a germ of homeomorphism h : C, 0 → C, 0 conjugating f to its linear part. 2. Let f, g : C, 0 → C, 0 be germ maps with f elliptic and g = I d resonant (meaning that g  (0) is a root of 1). What can be said if we assume that f and g commute? 3. A subgroup G ⊂ Diff(C, 0) containing a hyperbolic map in its center is abelian analytically linearizable. 4. Let G ⊂ Diff(C, 0) be a not necessarily finitely generated subgroup. Assume that each map germ in G has finite order. Is it true that G is finite?

Chapter 9

Foliations on Complex Projective Spaces

Complex projective spaces are natural ambient for considering holomorphic foliations. Indeed, any polynomial differential form or vector field can be considered as a rational one in the complex projective space, giving some geometrical object in a compact manifold. Moreover, due to its very special geometry, complex projective spaces are very important in the description of the phenomena modeled by holomorphic foliations. Indeed, it is not wrong to say that in many aspects, algebraic leaves for holomorphic foliations on complex projective spaces play the role of compact leaves for non-singular foliations on compact manifolds. Other reasons are related to the natural interplay between the algebraic geometry and the theory of holomorphic foliations with singularities. Finally, we mention the abundance and richness of the examples of holomorphic foliations in complex projective spaces shedding light on what happens in other complex manifolds. In this chapter we shall study holomorphic foliations on complex projective spaces. We will be mainly concerned with the codimension one case.

9.1 The Complex Projective Plane and Foliations The complex projective plane CP (2) is the quotient space C3 \ 0 by the equivalence relation p, q ∈ C3 \ 0, p ∼ q ⇔ p = λ.q for some λ ∈ C \ {0}. Thus CP (2) is the space of lines through the origin of C3 . By introducing homogeneous coordinates [(x1 ; x2 ; x3 )] on CP (2) we conclude that CP (2) is equipped with an atlas u consisting of three affine charts (x, y), (u, v), (r, s) with the following changes of coordinates: u = 1/x, v = y/x ;

r = 1/y, s = x/y ;

s = 1/v

r = u/v.

A well-known fact is then: © Springer Nature Switzerland AG 2021 B. Scárdua, Holomorphic Foliations with Singularities, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-76705-1_9

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9 Foliations on Complex Projective Spaces

Proposition 9.1.1 The complex projective plane CP (2) is a compact, connected, and simply connected complex surface. Let us now investigate the structure of the space of holomorphic foliations with singularities (foliations of dimension one are the ones that are interesting) on the complex projective plane. Recall that a holomorphic foliation with singularities on  CP (2) is given by an open cover CP (2) = Uj such that on each open subset Uj j

the plaques of F are given by a holomorphic vector field Xj in Uj and if Ui ∩ Uj = φ then Xi U ∩U = gij Xj U ∩U for some non-vanishing holomorphic function i j i j gij : Ui ∩ Uj → C. Finally we assume that sing(Xj ) = φ or consists of a single point pj ∈ Uj . In particular we have the following example: ∂ Example 9.1.2 (Polynomial Vector Fields on C2 ) Let X(x, y) = P (x, y) + ∂x ∂ be a polynomial vector field on C2 . (P , Q ∈ C[x, y]). For our purposes Q(x, y) ∂y we may assume that P , Q have no common factor on C[x, y] and (equivalently) (P = 0) ∩ (Q = 0) is a finite subset of C2 . Thus X has isolated singularities on C2 . Let us show that the foliation with singularities induced by X on C2 extends to CP (2). For this we consider the change of coordinates u = 1/x, v = y/x . We have 

x˙ = P (x, y) y˙ = Q(x, y)

.

 1 v −x˙ 2 . Therefore u˙ = 2 = −u .P , u u x Similarly v =  yu ⇒ v˙ = yu ˙ + y u. ˙   1 v v 1 v 1 v 2 Thus v˙ = uQ , + · (−u ) · P , . Now we write P , = u u u u u u u 1  (u, v) and some n ∈ N, such that u  P (u, v). Thus · P (u, v) for a polynomial P un u˙ =

(u, v) −P −u2  · P (u, v) = · un un−2

Similarly we write 

1 v , Q u u

=

1  Q(u, v) um

 v) ∈ C[u, v]. Then with m ∈ N and u  Q(u, v˙ =

 v) uv u · Q(u, (u, v), − nP um u

9.1 The Complex Projective Plane and Foliations

⎧ (u, v) −P ⎪ ⎨u˙ = un−2  v) v P (u, v) Q(u, ⎪ ⎩v˙ = − um−1 un−1

93

.

 v) with isolated singularities in Thus we have a polynomial vector field X(u, such that in the intersection of the spaces C2(u,v) and C2(u,v) we have X =

C2(u,v)

1  X for some  ≥ 0. u  defines a foliation of C2 Remark 9.1.3 In any case we conclude that X (u,v) and this foliation coincides with the one induced by X on C2(x,y) . x 1 , s = we conclude that there is a y y polynomial vector field X(r, s), with isolated singularities on the plane C2(r,s) , such that in the intersection C2(x,y) ∩ C2(r,s) Similarly for the coordinates r =

X=

1 X rk

for some 0 ≤ k ∈ N. Therefore, just applying the definition, we conclude that X defines a foliation F(X) on the complex projective plane in a natural way. Summarizing the above example we have: A polynomial vector field X on C2 defines/induces a foliation F(X) on CP (2). Conversely we have: Proposition 9.1.4 Every holomorphic foliation with singularities on CP (2) is the one induced by a certain polynomial vector field on C2 . Proof Let F be a foliation of CP (2) with (finite) singular set sing(F) ⊂ CP (2). Denote by π : C3 \ {0} → CP (2) the  canonical projection. By definition we have CP (2) = Uj a (finite because CP (2) is compact) finite j

open cover where F has its plaques given by holomorphic vector fields Xj on Uj and having isolated singularities. We can assume that the intersections Ui ∩ Uj (i = j ) contain no singularities of F and that if Ui ∩ Uj = φ, then Xi = gij Xj with gij ∈ O ∗ (Ui ∩ Uj ). Remark 9.1.5 The idea is to “lift” F to C3 \ {0} by π : C3 \ 0 → CP (2) and then prove that F (because it comes from a foliation of CP (2)) can be given by a polynomial system. Nevertheless, the lifted foliation π ∗ F will have codimension one, i.e., dimension two. So we must pass to differential forms instead of vector fields. Let us then choose dual 1-forms  ωj on the Uj such that on each Ui ∩ Uj = φ we have ωi = gij ωj and F U is given by ωj = 0. Then we lift {ωj } and j

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9 Foliations on Complex Projective Spaces

{gij } by π : C3 \ 0 → CP (2) obtaining, in this way, 1-forms  ωj in the open sets j = π −1 (Uj ) ⊂ C3 \ 0 and such that on each intersection U i ∩ U j = φ we have U · ω  . Notice that if U ∩ U ∩ U = φ, then g · g = g on  ωi = g ij j i j k ij ik ik Ui ∩ Uj ∩ Uk . 3 \{0} =    j so that the Thus we have g g  g  = g  on U ∩ U ∩ U . Finally C U ij ij j k ik i j k j  3 1 3 j , g data U ij defines a multiplicative cocycle on C \ {0}. Because H (C \ 0) = 0 ∨

and H 2 (C3 \ 0, Z) = 0 (Cartan’s theorem, [20]) the second (multiplicative) Cousin j → problem has a solution on C3 \{0} so that there are holomorphic functions  gj : U g  i i ∩ U j = φ [34]. Therefore we have on each U C∗ with the property that g ij = gj on each gi 1 1 j = φ : ω i ∩ U i = ω j ⇒ ω i = ω j . U gj gi gj  In this way we can define a holomorphic 1-form  ω on C3 \ {0} by setting  ωU := j 1 ω j . Thanks to Hartogs’ extension theorem  ω extends as a holomorphic 1-form gj  = π ∗ (F) (induced by the pullon C3 . Now we consider the pull-back foliation F back of F) on C3 \ {0}. Claim 9.1.6 The 1-form  ω is integrable (i.e.,  ω ∧d ω = 0) and the foliation induced  by  ω coincides with F. This is quite clear since each ωj (and therefore each

1 ω j ) is integrable (in gj

dimension two any 1-form is integrable). ων+1 + · · · +  ωj + · · · where  ωj is a polynomial Now we write  ω =  ων +  homogeneous 1-form of degree j (use Taylor/power series). Then d ω = d ων + d ων+1 + · · · + d ωj + · · · and 0 =  ω ∧ d ω = ( ων +  ων+1 + ων+1 + · · · ) =  ων ∧ d ων + · · · . Therefore  ων ∧ d ων ≡ 0, i.e., · · · ) ∧ (d ων + d  ων is integrable. The details in the proof of the following claim are then lifted to the reader: Claim 9.1.7 The 1-form  ων defines the same foliation as  ω on C3 . Proof Notice that  ων is homogeneous so that it is radially saturated (if p ∈ C3 \ 0, then the leaf of  ων = 0 containing p also contains the line {λp; λ ∈ C∗ }).  which is the pull-back of a foliation of By its turn, the 1-form  ω defines F, CP (2). Therefore,  ω is also radially saturated. Thus the claim follows: given p ∈ C3 \ {0} and v* ∈ C3 we have ∀ t ∈ C ων+1 (tp) · v* + · · ·  ω(tp) · v* =  ων (tp) · v* +  ! " = tν  ων (p) · v* + t ων+1 (p) · v* + · · · .

9.2 The Theorem of Darboux–Jouanolou

95

 we have Thus for v* ∈ Tp (F) " ! ων+1 (p) · v* + · · · , ∀ t ων (p) · v* + t 0 = tν  so that  ων (p) · v* = 0 and hence v* ∈ Tp ({ ων = 0}). By comparing dimensions we conclude the proof of the claim.   Since  ων is homogeneous polynomial, it induces apolynomial 1-form ω(x, y) = P (x, y)dy−Q(x, y)dx on C2(x,y) ⊂ CP (2) so that F C2 is given by ω(x, y) = 0. (x,y)

This ends the proof of the proposition.   We shall adopt the following convention: Given an algebraic (irreducible) (not necessarily smooth) curve C ⊂ CP (2) given on C2 by an affine polynomial equation f (x, y) = 0 we consider its lift to  ⊂ C3 . Then C  is an algebraic (not necessarily C3 \ {0} and then to C3 , denoted by C smooth) hypersurface that has a homogeneous equation f(x1 , x2 , x3 ) = 0. We  denote by Z(f) the curve C on CP (2) and by (f = 0) the hypersurface C.

9.2 The Theorem of Darboux–Jouanolou Given a foliation F on CP (2) we may ask whether it has leaf that is a closed analytic subset of CP (2). In order to study this question shall use: Proposition 9.2.1 Given a foliation F on CP (2) and a leaf L ∈ F the following are equivalent: (i) (ii) (iii) (iv)

L is contained in some algebraic curve C ⊂ CP (2). L is an algebraic (invariant) curve C ⊂ CP (2). L \ L ⊂ sing(F), i.e., L only accumulate at singular points of F. L = (L ∩ sing(F)) ∪ L.

The above proposition is a consequence of the above discussion and results. We then define a leaf L ∈ F as an algebraic leaf if L = C is an algebraic curve (which is necessarily invariant by F). P be a rational function on C2 ; P (x, y), Q(x, y) ∈ Q C[x, y] have no common factor. Then R defines a foliation F on CP (2) whose leaves are algebraic contained in the algebraic curves aP + bQ = 0, (a, b) ∈ C2 = 0. In particular F has infinitely many algebraic leaves.

Example 9.2.2 Let R =

Theorem 9.2.3 (Theorem of Darboux and Jouanolou, [44]) If a foliation F on CP (2) admits infinitely many algebraic leaves, then F admits a rational first integral. In particular, all leaves are algebraic.

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9 Foliations on Complex Projective Spaces

Proof Choose a polynomial 1-form ω(x, y) = P (x, y)dy −Q(x, y)dx that defines F on C2 , with isolated singularities. Given an algebraic curve C ⊂ CP (2) with C ∩ C2 having irreducible polynomial equation f (x, y) = 0 we put C ∗ = C \ (C ∩ sing(F)). Claim 9.2.4 C ∗ is an algebraic leaf of F if, and only if,

1 ω ∧ df is a polynomial f

2-form on C2 . Proof of Claim 9.2.4 Assume that C ∗ is invariant by F. Choose a point p ∈ C ∗ and  centered at p such that f (  : ( a local chart ( x,  y) ∈ U x,  y) =  y and C ∗ ∩ U y = 0) and write ω( x,  y ) = B( x,  y )d x − A( x,  y )d y where A, B are holomorphic near ∂ ∂  x,  + B( x,  y) then defines the (0, 0). The vector field X( y ) := A( x,  y) ∂ x ∂ y . foliation of U ∗  : (  Since C ∩ U y = 0) is X-invariant, we conclude that B( x , 0) ≡ 0.  x,  y ) in local ring of holomorphic functions. Therefore we may assume that  y B( 1 1 . Thanks to Hartogs’ Therefore ω ∧ df = B( x,  y )d x ∧ d y is holomorphic in U f  y 1 extension theorem this shows that ω ∧ df is holomorphic in all points of C ∩ C2 f and therefore it is polynomial (we already have a priori that it is rational). 1 1 y Assume now that ω ∧ df is polynomial. Then similar to the above, ω ∧ d f  y   and therefore  x,  y ) so that B( x,  y) =  y · B1 ( is holomorphic in U y B( x,  y ) for  is tangent some holomorphic B1 ( x,  y ). In particular B( x , 0) ≡ 0 and therefore X . Thus C ∗ ∩ U  is F-invariant and by the Identity Principle or by to ( y = 0) in U Hartogs C is an algebraic leaf of F.   Notice that if we consider homogeneous coordinates (x1 ; x2 ; x3 ) on CP (2), then  of F to C3 can be defined by as we have seen before, the pull-back foliation F a homogeneous polynomial 1-form (x1 , x2 , x3 ) of degree ν; this form satisfying ∂ ∂ ∂  · R* ≡ 0 ( is radially saturated) where R* = x1 + x2 + x3 is the ∂x1 ∂x2 ∂x3 radial vector field. Given an algebraic curve C ⊂ CP (2) we consider an irreducible homogeneous polynomial f (x1 , x2 , x3 ) such that {f = 0} is the homogeneous equation of C = Z(f ). Let k be the degree of the coefficients of the homogeneous 1-form . Then the above claim rewrites as follows: Claim 9.2.5 C is an algebraic invariant curve by F if, and only if, there exists a 2-form θ (x1 , x2 , x3 ) such that: (i) df ∧  = f θ , (ii) the coefficients of θ are homogeneous polynomial of degree k − 1. Notice that the conclusion about the degree of (the coefficients of) θ is immediate since  ∧ df is homogeneous of degree k + deg f − 1, while f θ is homogeneous of degree deg f + deg θ .

9.2 The Theorem of Darboux–Jouanolou

97

 Let now Ek = θ ; θ is 2-form with homogeneous coefficients of degree k− on C3 . Then E is a finite dimension C-vector space of finite dimension say N(k) =  has N(k) + 1 algebraic solutions given by dim Ek . Assume that the foliation F (f0 = 0), . . . , (fN (k) = 0) where fj (x1 , x2 , x3 ) is homogeneous of degree k − 1, irreducible, and fi , fj are relatively prime if i = j . dfj ∧ = θj , j = 0, . . . , N(k) as in the claim above. Since dim Ek = We write fj N (k) the set {θ0 , . . . , θN (k) } is linearly dependent. There is (a0 , . . . , aN (k) ) ∈ N (k) CN (k)+1 \}0} such that aj θj = 0 so that j =0

⎛ ⎝

N (k)  j =0

'

N (k)

⎞ dfj ⎠∧=0 aj fj

(

N (k)

dfj · Because cod sing() ≥ 2 we fj j =0 j =0 must have f0 . . . fN · α = g for some homogeneous polynomial g(x1 , x2 , x3 ) and some N ≤ N(k) (given by the number of non-zero coefficients aj in α). N

dfj  is given by a closed Thus since α = aj is closed, we conclude that F fj j =0  admits another algebraic solution (fN (k)+1 = 0), then we write rational 1-form. If F dfN (k)+1 ∧  = fN (k)+1 θ and because {θ, θ1 , . . . , θN (k) } is linearly independent a similar argumentation shows that for some and then

fj

α ∧  = 0 for α :=

β :=

N  j =1

bj

aj

dfi(j ) fi(j )

with fi(1) . . . fi(N  ) β = h where h is a homogeneous polynomial, bj = 0, i(j ) = 0, j = 1, . . . , N  ≤ N(k). gfi(1) . . . fi(N  ) Then α = F · β where F = · Notice that F is not constant hf0 . . . fN because Res(f0 =0) α = α0 = 0 and Res(f0 =0) β = 0. Since α and β are closed, we  The have 0 = dF ∧ β and therefore dF ∧  = 0. Thus F is a first integral for F. result follows.   As a corollary of the above argumentation we have: Corollary 9.2.6 For each k ∈ N there exists N(k) ∈ N such that if a foliation of CP (2) has more than N(k) algebraic leaves, then it has a rational meromorphic first integral.

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Remark 9.2.7 Jouanolou–Darboux theorem is an algebraic parallel to Mattei– Moussu theorem.

9.3 Foliations Given by Closed 1-Forms Since CP (2) is compact and simply connected, it does not admit a non-trivial closed 1-form that is holomorphic (indeed, such an 1-form ω would be exact ω = dF for some holomorphic function F on CP (2), but then F must be constant and ω ≡ 0 because CP (2) is compact). Nevertheless there are non-trivial closed meromorphic 1-forms. Since any meromorphic function on CP (2) is already a rational function (Liouville’s theorem, [36]) we consider the class of closed rational 1-forms on CP (2) that we will denote by (CP (2)). An interesting example is the Poincaré–Dulac normal form (nx + ay n )dy − ydx = 0 (n ≥ 2, a ∈ C∗ ) that defines a foliation Fa,n on CP (2) that is also given by the closed rational 1-form a,n ∈ (CP (2)) defined by  n = (nx + ay n )dy − ydx · The poles of a,n in C are given by (y = 0) that is the polar y n+1 set of a,n . This is a general fact as we shall below. Another important example is the class of linear logarithmic foliations (also called Darboux foliations) given r r    dfj  fj · λj where fj ∈ C[x, y], λj ∈ C∗ . In this by 1-forms as ω = fj j =1 j =1 r

dfj λj ∈ (CP (2)). Next we case the foliation is given by an element  = fj j =1 describe the structure of the elements in (CP (2)). Proposition 9.3.1 (Integration Lemma—Global Version) Let ω be a closed rational 1-form on CP (2) and let  := π ∗ (ω) be its lift to C3 where π : C3 \ 0 → CP (2) is the canonical projection. Then we have  dfj g = , λj +d fj f1n1 −1 . . . frnr −1 j =1 r 

where: (a) r ≥ 2 and f1 , . . . , fr  g are homogeneous polynomials in C3 . (b) f1 , . . . , fr are irreducible and pairwise relatively prime. (c) If nj > 1, then fj  g. r

(d) deg(g) = deg(fj ) (nj − 1), i.e., deg(g) = deg(f1n1 −1 . . . frnr −1 ); j =1

(e) λ1 , . . . , λr ∈ C and

r

j =1

λj deg(fj ) = 0.

9.3 Foliations Given by Closed 1-Forms

(f) If nj = 1 then λj = 0. Moreover: (g) The polar set of ω is given by

r 

99

(fj = 0), where nj = order of (fj = 0) as a

j =1

polar curve of ω and λj = Res(fj =0) ω. Proof As we have seen, ω cannot be holomorphic, so that its polar set (ω)∞ is not r  (fj = empty. Because (ω)∞ has codimension one, it can be written as (ω)∞ = j =1

0) where f1 , . . . , fr are irreducible polynomials in C3 , pairwise relatively prime. Let λj = Res(fj =0) ω ∈ C be the residue of ω in (fj = 0) =: Z(fj ) ⊂ CP (2) and nj = order of Z(fj ) as pole of ω. Remark 9.3.2 λj can be calculated/defined as follows. Choose a point pj ∈ Z(fj ) that is not a singular point of the variety Z(fj ). Take a transverse disc Zpj centered ' ( r  Z(fi ) ∩ pj = {pj }. at pj , transverse to Z(fj ), and such that j =1

Choose a small simple loop γj ⊂ pj \ {pj } positively oriented. Then ) )  1 1 λj := ω= ω  . √ √ pj 2π −1 γj 2π −1 γj By a classical result of Deligne [26] λj is well-defined (recall that Z(fj ) is irreducible). Now we claim: Claim 9.3.3

r

j =1

λj deg(fj ) = 0.

Proof We consider a linear embedding E : CP (1) → CP (2) such that the line E(CP (1)) =: L intersects (ω)∞ only at non-singular points of the variety (ω)∞ and the intersection is always transverse.  We consider the restriction, i.e., the induced 1-form ξ := ωL = E ∗ (ω). Then ξ has polar at the points that correspond to the intersection points L∩(ω)∞ . Moreover, since L induces a transverse disc (like the discs pj ) at each intersection point p ∈ L ∩ (ω)∞ we have that if p ∈ L ∩ Z(fj ) then Resp ξ = λj . By its turn, Bezout’s theorem implies that the number of intersection points p ∈ L ∩ Z(fj ) is equal to deg(L) deg(fj ) = deg(fj ). Finally, because L is the Riemann sphere, the theorem of residues applied to ξ says that 0=

 p∈L∩(ω)∞

Res(ξ, p) =

r 

λj deg(fj ).

p=1

  Now we consider the pull-back  = π ∗ (ω) that naturally extends to C3 .

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9 Foliations on Complex Projective Spaces

The polar set of  is ()∞ =

r 

(fj = 0) ⊂ C3 and ∂ = 0 with λj =

j =1

Res(fj =0) , nj = order of (fj = 0) in ()∞ . Now we introduce the 1-form α :=

r 

λj

j =1

dfj · fj

This is a closed rational 1-form on C3 such that β :=  − α is rational, closed but with all residues equal to zero. Also (β)∞ ⊂ ()∞ . Claim 9.3.4 β is exact, i.e., β = df for some meromorphic function f on C3 . Proof Indeed, we start by proving given a closed path γ : S 1 → C3 \ ()∞ then we β = 0. Let γ : S 1 → C3 \ ()∞ be given and (by approximation theory) have γ

assume that γ is of class C ∞ . Since C3 is simply connected, there is a continuous  extension F : D → C3 (D is the closed unit disc |z| ≤ 1 in C). Such that F S 1 = γ . Again we may assume that: (i) F is of class C ∞ . (ii) F (D) avoids the singular set of ()∞ (which is a finite set). (iii) F is transverse to (the smooth part of) ()∞ . In particular F (D) ∩ ()∞ = {z1 , . . . , zm } is a finite set. Then by the theorem of residues we have



β=

γ

∗

S1

F (β) =

m 

2π i Reszj F ∗ (β) = 0.

j =1

  In order to conclude that β is exact on CP (2) it is enough to observe that we have already β = df for some meromorphic function f : C3 \ ()∞ → C. Because (β)∞ ⊂ ()∞ the function f is indeed holomorphic in C3 \ ()∞ . Once we know that β is rational we can already conclude from df = β that f admits an extension to C3 as a rational function. Thus we have proved that β is the differential of a rational function f on C3 with (f )∞ ⊂ ()∞ .   Thus we can write =

r  j =1

or else

λj

dfj + df fj

on

C3

9.4 Riccati Foliations

101

=

r 

λj

j =1

g dfj · +d fj h

9.4 Riccati Foliations The classical Riccati differential equation, put in terms of complex variables, is 

x˙ = p(x) y˙ = a(x)y 2 + b(x)y + c(x).

We will consider the case where the coefficients are polynomials p(x), a(x), b(x), c(x) ∈ C[x]. The appropriate space to study the geometry of a Riccati foliation is perhaps the surface C × C = M. Let us see why. First we recall the canonical coordinate changes in M. On M we have the natural projection π1 : M → C given by π1 (p1 , p2 ) = p1 . The fibers π1−1 (p) = {p} × C are Riemann spheres. Now we consider a Riccati foliation F of M obtained by extending from C2(x,y) to M the foliation induced by ∂ ∂ + (a(x)y 2 + b(x)y + c(x)) · We the polynomial vector field X(x, y) = p(x) ∂x ∂y claim: Claim 9.4.1 A vertical fiber π1−1 (p1 ) p1 = ∞ is invariant by F if, and only if, p1 is a zero p(x).  x˙ = p(x) 2 This is quite clear since F is given on C(x,y) by y˙ = a(x)y 2 + b(x)y + c(x) and π1 (x, y) = x. Therefore F has a finite number of invariant vertical fibers that depend on the degree of p(x). Let us change coordinates on F:



y˙ Y˙ = − 2 = −Y 2 y



a(x) b(x) + c(x) + Y Y2



Y˙ = −(c(x)Y 2 + b(x)Y + a(x)) x˙ = p(x).

Thus once again F is given by a Riccati ordinary differential equation in the C2(x,Y ) coordinate system. Therefore, we can claim: Claim 9.4.2 Given a non-invariant vertical fiber F = π1−1 (p1 ) the leaves of F are transverse to F . Proof Indeed, we may assume that p1 = ∞ and therefore the non-invariance of F equivalent to p(x) does not vanish at x = p1 .

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9 Foliations on Complex Projective Spaces

Since the expressions of F in the coordinate systems (x, y) and (x, Y ) are  x˙ = p(x) y˙ = . . .

 x˙ = p(x) Y˙ = . . .

we conclude that F is always transverse to F. Actually the above transversality is a characterization of Riccati foliations.

 

Claim 9.4.3 A foliation F on C × C = M that is transverse to some vertical fiber π1−1 (p1 ) = F is a Riccati foliation. Proof We may suppose that p1 = x1 = ∞ and choose a polynomial vector field  x˙ = A(x, y) that defines F on C2(x,y) . y˙ = B(x, y) By the transversality on C2(x,y) we conclude that A(x, y) = 0, ∀ y ∈ C. Since A is a polynomial, this implies that A(x, y) depends only on x , not on y. Now, since the x1 ) fiber F ∼ = C is compact, the foliation is also transverse to the nearby fibers π1−1 ( for  x1 ≈ x1 . By the same reasoning above, we conclude that A( x1 , y) = A( x1 ) does not depend on y, for every  x1 ≈ x1 . Because A(x, y) is a polynomial, this implies   x˙ = A(x) . that A(x, y) = A(x) depends only on x. Thus F C2 is given by y˙ = B(x, y) Now we change coordinates to (x, Y ) = (x, 1 ). Then Y˙ = −2y˙ = −Y 2 (B(x, 1 )) y

y

Y

 Y) −Y B(x,  Y ) is a polynomial, Y  B(x,  Y ), and n = where B(x, and then Y˙ = Yn ⎧ ⎨x˙ = A(x) degy B. Since F is given by and by the same arguments above,  ⎩Y˙ = − B(x, Y ) n−2 Y the transversality of F with F at the point implies that n − 2 ≤ 0, i.e., n ≤ 2.  Therefore B(x, y) = b0 (x) + b1 (x)y + b2 (x)y 2 and F is a Riccati foliation.  From the structural point of view, Riccati foliations are related to suspensions of groups of automorphisms of CP (1). For this let us recall the classical concept of Ehresmann. F

Definition 9.4.4 Given a fiber bundle space ξ(π : −→ B) with basis B, fiber F , total space E, and projection π , a foliation F on E is said to be transverse to the fibers of ξ if: (i) dim F + dim F = dim E; (ii) F is transverse to the fibers π −1 (b) ⊂ E.  (iii) Given any leaf L of F the restriction π L : L → B is a (surjective) covering map. Recall that in this case we have a natural action of π1 (B) on Diff(F ) given a base point b0 ∈ B and a path γ ∈ π1 (B, b0 ). We define a map hγ : Fb0 → Fb0 as

9.5 Examples of Foliations on CP (2)

103

follows: given y ∈ Fb0 = π −1 (b) we consider the lifted path  γy (t) ⊂ Ly obtained from the covering map π L : Ly → B (where y ∈ Ly is a leaf of F). y Then we put hγ (y) := γy (1), the final point of the lifting. The image of this group homomorphism ϕ : π1 (B, b0 ) → Diff(F ) is called the global holonomy of F in ξ . It is well-known that F is conjugate to the suspension of its global holonomy. One important fact is the following remark by Ehresmann: F

Proposition 9.4.5 ([11, 31]) Let F be a foliation of the fiber space ξ(π : E −→ B). Assume that (i) dim F + dim F = dim E and (ii) F is transverse to the fibers π −1 (b) ⊂ E. Then F is transverse to the fibers of ξ if the fiber F is compact. As a corollary we obtain: Proposition 9.4.6 Let F be a Riccati foliation of C×C and  let F1 , . . . , Fr ⊂ C×C r be the invariant (vertical) fibers of F. Then F0 = F  is a foliation  (C×C)\

transverse to the fibers of the fiber space ξ : (C × C) \

r  j =1

j =1

Fj

Fj → C \

r  j =1

Fj → C

(Fj = pj × C) and in particular F0 is holomorphically conjugate to the suspension of a finitely generated group of Möbius maps.

9.5 Examples of Foliations on CP (2) We shall now discuss some examples of foliations, holomorphic singular of dimension one, on the complex projective plane CP (2). As we have already seen, such a foliation is defined in any system (x, y) ∈ C2 ⊂ CP (2) by a polynomial vector ∂ ∂ field with isolated singularities say X(x, y) = P (x, y) ∂x + Q(x, y) ∂y . The same foliation is defined by the polynomial 1-form ω(x, y) = P (x, y)dy − Q(x, y)dx. Example 9.5.1 We consider the 1-form ω = dy−(xy+1)dx and the corresponding foliation F on CP (2). The line at infinity l∞ = CP (2) \ C2(x,y) is invariant and in the coordinates (u, v) = (1/x, y/x) we have: 

u˙ = −u3 v˙ = v(1 − u2 ) + u2 .

This is a saddle-node singularity without central manifold, and with strong manifold contained in l∞ . Indeed, rename coordinates as (u, v) = (y, x) to obtain the differential equation 

y˙ = −y 3 x˙ = x(1 − y 2 ) + y 2 .

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9 Foliations on Complex Projective Spaces

Blowing up as y = tx we obtain the resonant saddle-singularity at the origin given by (t + t 3 x)dx + x(1 − t 2 x 2 + t 2 x)dt = 0. One separatrix is contained in l∞ and the other in the exceptional divisor. Blowing up as x = uy we obtain the differential equation y 3 du + (u + y)dy = 0. This is a type of Euler equation of degree 3, i.e., a saddle-node without central manifold. The strong manifold is contained in the exceptional divisor. Now we consider the coordinate system (r, s) = (1/y, x/y). In this system the foliation is given by 

r˙ = −r(r 2 + s) s˙ = r 2 − s(r 2 + s).

We have a singularity at the origin of this system. This singularity q is not reduced. Renaming, for the sake of convenience, the coordinates (r, s) into (x, y) we get 

x˙ = x(x 2 + y) y˙ = y(x 2 + y) − x 2 .

Blowing up q to y = tx we obtain: dx + x(x + t)dt = 0, which is not singular. The blow-up x = uy produces the form 

u˙ = u3

y˙ = y(1 − u2 ) + u2 y 2 . The origin is then a saddle-node singularity with central manifold contained in the exceptional divisor, strong manifold contained in l∞ . There is no other singularity to consider. We introduce now a particular situation through the next example: Example 9.5.2 We consider the vector field X given by x˙ = −1, y˙ = y. Then the corresponding foliation is given by ydx +dy = 0. Therefore it admits the entire first integral f = yex . The only affine invariant algebraic curve is the line (y = 0). At the line at infinity l∞ there are two singularities. The one at the point (x = ∞) ∩ l∞ is given in local coordinates u = 1/x, v = y/x by u˙ = u2 , v˙ = v(1 + u). This is a saddle-node already in the irreducible form, with strong manifold contained in l∞ and central manifold contained in the projective line of the exceptional divisor. The other singularity in l∞ is at the point q = (y = ∞) ∩ l∞ , and it is given in local coordinates r = 1/y, s = x/y by r˙ = r, s˙ = r + s. This is a reducible singularity. Indeed, we rename the coordinates as (r, s) = (x, y) so that the singularity q is given by

9.5 Examples of Foliations on CP (2)

105



x˙ = x + y y˙ = y

.

Blowing up this singularity q as y = tx, we obtain: 

t˙ = t 2 x˙ = −x(1 + t)

.

Then the origin is a saddle-node singularity, with central manifold contained in the exceptional divisor and strong manifold contained in l∞ . Blowing up q as x = uy we obtain  u˙ = 1 . y˙ = y Therefore, there is no other singularity to consider. Notice that, at the origin of the blow-up (u, y) we have a normal form like the original vector field at the origin of C2 . In particular, the singularities in l∞ are non-dicritical. Example 9.5.3 We consider the 1-form ω = dy −(y +x)dx and the corresponding foliation F on CP (2). The line l∞ is invariant and in coordinates (u, v) = (1/x, y/x) we have 

u˙ = −u2 v˙ = 1 + v(1 − u).

Thus the origin is not a singular point. There are two singularities in l∞ . Let us call q1 the singularity given by u = 0, v = −1. Also, we call q2 the one at the origin of the coordinate system (r, s) defined by r = 1/y, s = x/y. This singularity q2 is given by 

r˙ = −r(1 + s) s˙ = r − s(1 + s)

.

We need to blow up this singularity q2 . Putting it into (x, y) = (s, r) coordinates (this should then cause no confusion) we have 

y˙ = y(1 + x) x˙ = −y + x(1 + x)

.

106

9 Foliations on Complex Projective Spaces

The blow-up y = tx produces 

t˙ = t 2

.

x˙ = x(1 + x − t)

This is a saddle-node singularity, with strong manifold contained in l∞ and central manifold contained in the exceptional divisor. Blowing up q to x = uy (again this should cause no confusion in the notation), we get the 1-form dy + y(1 + uy)du = 0. So, there are no other singularities to consider in the reduction of q2 . Let us study q1 . It is given by u = 0, v = −1 in the coordinate system u = 1/x, v = y/x and the corresponding differential equation is 

u˙ = −u2 v˙ = 1 + v(1 − u).

Changing coordinates into (u, V ) where V = v + 1 we obtain 

u˙ = −u2 V˙ = u + V (1 − u).

The origin, which corresponds to q1 , is then a saddle-node with strong manifold contained in l∞ . Let us study the existence of a central manifold. For this sake we rename coordinates as (x, y) = (v, u). Therefore we obtain the differential equation: 

x˙ = x(1 − y) + y

.

y˙ = −y 2 Blowing up this singularity as y = tx we obtain 

t˙ = t + t 2

.

x˙ = −x(1 + t − tx)

The origin of this (x, t) system is then a resonant saddle-type singularity. Now we blow up q1 as x = uy to obtain 

u˙ = −(1 + u) y˙ = y 2

.

9.6 Example of an Action of a Low-dimensional Lie Algebra

107

Thus, in coordinates U = 1 + u and y we have U˙ = −U, y˙ = y 2 , which is a normal form saddle-node with strong manifold contained in the exceptional projective line and with invariant central manifold transverse to l∞ . The final picture of the reduction of singularities is as follows. There is a saddlenode central manifold transverse to l∞ . Also, the foliation F has an affine algebraic leaf, given by the line x + y + 1 = 0. Observe that, indeed this foliation F : dy − (x + y)dx = 0 is affine equivalent to the one given by dy − ydx = 0, via the automorphism T : C2 → C2 , T (x, y) = (x, x + y + 1).  Finally, the foliation dy −(x +y)dx = 0 is given by the rational 1-form  := d(x+y+1) x+y+1

dy −(x +y)dx

1 x+y+1

=

− dx, which is closed.

Remark 9.5.4 Notice that Examples 9.5.1 and 9.5.3 only produce saddle-nodes tangent to l∞ in the reduction of singularities of l∞ . In the case of Example 9.5.1 we also have: there is no closed rational 1-form that defines the foliation, no affine invariant algebraic curve, and no central manifolds transverse to l∞ .

9.6 Example of an Action of a Low-dimensional Lie Algebra Next we give an example of foliation of CP (3) that is given by a kind of local action of the affine group Aff(C). We consider affine coordinates (x1 , x2 , x3 , x4 ) ∈ C4 and the vector fields, R, S, X given by ⎛ x1 ⎜ x2 O R=⎜ ⎝O x3

⎞ ⎟ ⎟, ⎠

⎛

⎞

x1

⎜ 2x2 O S=⎜ ⎝O 3x3

⎟ ⎟, ⎠ 5x4

x4

⎞

⎛ a1 x1 x3 + a2 x22 ⎜ bx2 x3 O X=⎜ ⎝ O c1 x32 + c2 x1 x4

⎟ ⎟. ⎠ x3 x4

These are degree 2 homogeneous vector fields satisfying [R, S] = 0,

[R, X] = X,

[S, X] = 3X.

Thus we have an action on C4 that defines an integrable 1-form ω = iR iS iX (dV ) where dV is the volume 4-form in C4 . Here by iX (α) we denote the contraction of the differential form α by the vector field X as in the usual sense.

108

9 Foliations on Complex Projective Spaces

Claim 9.6.1 ω is given by ω = [(3c1 − 2b − 1)x2 x32 x4 + 2c2 x1 x2 x42 ]dx1 − [(−2a1 + 4c1 − 2)x1 x32 x4 − 2a2 x22 x3 x4 + 4c2 x12 x42 ]dx2 + [(−3a1 + 4b − 1)x1 x2 x3 x4 − 3a2 x23 x4 ]dx3 − [(−a1 + 2b − c1 )x1 x2 x)32 − a2 x23 x3 − c2 x12 x2 x4 ]dx4 . Proof We have    dx1 dx2 dx3 dx4    a x x + a2 x 2 bx2 x3 c1 x 2 + c2 x1 x4 x3 x4  2 3 . ω =  1 1 3 x1 2x2 3x3 5x4    x1 x2 x3 x4    Let F denote the foliation induced by ω = 0 in CP (3). Claim 9.6.2 F has a degree 3. Moreover: (i) {x2 = x4 = 0} induces invariant planes on CP (3). (ii) {x1 = 0} and {x2 = 0} are non-invariant. (iii) If [3a1 + 6c1 = 1 + 8b], then ω admits a logarithmic derivative; indeed, the 1-form ω1 = −x4 [(−2a1 + 4c1 − 2)x32 − 2a2 x22 x3 + 4c2 x4 ]dx2 + x2 x4 [(−3a1 + 4b − 1)x3 − 3a2 x22 ]dx3 − x2 [(−a1 + 2b − c1 )x32 − a2 x22 x3 − c2 x4 ]dx4 has a singular set of codimensions 2 and admits the generalized integrating factor or logarithmic derivative η1 =

11 dx2 4 dx4 + , 3 x2 3 x4

which is a closed rational 1-form that satisfies dω1 = η1 ∧ ω1 . Proof We rewrite ω as ω = x2 x4 [(3c1 − 2b − 1)x32 x32 + 3c2 x1 x4 ]dx1 − x4 [(−2a1 + 4c1 − 2)x1 x32 − 2a2 x22 x3 + 4c2 x12 x4 ]dx2 + x2 x4 [(−3a1 + 4b − 1)x1 x3 − 3a2 x22 ]dx3 − x2 [(−a1 + 2b − c1 )x1 x32 − a2 x22 x3 − c2 x12 x4 ]dx4 .

9.6 Example of an Action of a Low-dimensional Lie Algebra

109

Therefore ω1 is given by ω1 = 0x4 [(−2a1 + 4c1 − 2)x32 − 2a2 x22 x3 + 4c2 x4 ]dx2 + x2 x4 [(−3a1 + 4b − 1)x3 − 3a2 x22 ]dx3 − x2 [(−a1 + 2b − c1 )x32 − a2 x22 x3 − c2 x4 ]dx4 . We compute d1 where 1 = We have d1 = [ +[

ω1 x2 x4

instead of computing dω1 .

2x3 (−2a1 + 4c1 − 2) − 8a2 x2 ]dx2 ∧ dx3 x2 x2 x3 4c2 + 2a2 ]dx2 ∧ dx4 x2 x4

+ [(a1 − 3b + c1 )

a2 x22 2x3 + ]dx3 ∧ dx4 . x4 x4

dx4 2 We look for a 1-form η1 = λdx x2 + μ x4 , with λ, μ ∈ C, such that d1 = η1 ∧ 1 . We have , x3 η1 ∧ 1 = λ (−3a1 + 4b − 1) − 3a2 x2 dx2 ∧ dx3 x2 . ' ( x32 2a2 x2 x3 4c2 + μ (−2a1 + 4c1 − 2) − + x2 x4 x4 x2 ' (/ x32 x2 x3 c2 −λ (−a1 + 2b − c1 ) − a2 − dx2 ∧ dx4 x2 x4 x4 x2 . / x22 x3 + μ 3a2 + (3a1 − 4b + 1) dx3 ∧ dx4 . x4 x4

Therefore we have d1 = η1 ∧ 1 if, and only if, the following equations are verified: (1st )

2x3 x3 (−2a1 + 4c1 − 2) − 8a2 x2 = λ((−3a1 + 4b − 1) − 3a2 x2 ) x2 x2

(2nd )

4c2 2a2 x2 x3 x2 x3 + = (−2a2 μ + λa2 ) x2 x4 x4 + (4μc2 + λc2 )

1 + (μ(−2a1 + 4c1 − 2) x2

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9 Foliations on Complex Projective Spaces

+ λ(a1 − 2b + 4))

(3rd )

x32 . x2 x4

a2 x22 x3 + 2(c1 + a1 − 2b) x4 x4 ' ( 3a2 x22 x3 =μ − (−3a1 + 4b − 1) . x4 x4

Thus we obtain the following equivalent equations: (a) (b) (c) (d) (e) (f) (g)

λ(−3a1 + 4b − 1) = 2(−2a1 + 4c1 − 2). 3λa2 = 8a2 . (4μ + λ)c2 = 4c2 . (λ − 2μ)a2 = 2a2 . λ(−a1 + 2b − c1 ) = μ(−2a1 + 4c1 − 2) 3a2 μ = a2 . μ(3a1 − 4b + 1) = 2(c1 + a1 − 2b).

(S).

Let us search some solutions of (S). Assume a2 = 0. Then λ=

8 , 3

μ=

1 3

and we obtain

3a1 + 6c1 = 8b + 1. Thus we obtain items (i), (ii), (iii).

 

Notions such as Liouvillian first integral and generalized integrating factor are extensively discussed in [79] and [16] as well as in [77].

9.7 A Family of Foliations on CP (3) Not Coming from Plane Foliations Next we present a family of foliations of degree 4 on CP (3) whose elements are not pull-back from plane foliations. Let ei ∧ ej , 1 ≤ i < j ≤ 4 be the canonical base for ∧2 C4 , which defines homogeneous coordinates [(pij )i