Multidimensional Residue Theory and Applications 9781470471125, 9781470474898

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Mathematical Surveys and Monographs Volume 275

Multidimensional Residue Theory and Applications Alekos Vidras Alain Yger

Multidimensional Residue Theory and Applications

Mathematical Surveys and Monographs Volume 275

Multidimensional Residue Theory and Applications Alekos Vidras Alain Yger

EDITORIAL COMMITTEE Alexander H. Barnett Michael A. Hill Bryna Kra (chair)

David Savitt Natasa Sesum Jared Wunsch

2020 Mathematics Subject Classification. Primary 13Pxx, 14Q25, 32-02, 32A05, 32A10, 32A26, 32A27, 32Cxx, 32C05, 42Bxx.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-275

Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: https://doi.org/10.1090/surv/275

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2023 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

28 27 26 25 24 23

To the memory of our teacher and friend, Carlos A. Berenstein, who shared with us enthusiastically his ideas about the project of the present monograph from the day we started it nearly fourteen years ago. Such ideas inspired and motivated us since then by influencing the content and organization of the text.

Contents Preface

xiii

Chapter 1. Residue calculus in one variable 1.1. An analytic approach 1.1.1. Laurent expansions and meromorphic functions 1.1.2. Residue of a (1, 0) meromorphic form 1.1.3. Jordan’s lemma and Mellin transform 1.2. The geometric point of a view 1.2.1. Factorization of integration currents through residue currents 1.2.2. Residue currents and Lagrange’s interpolation formula 1.2.3. Residue currents and Bergman–Weil developments 1.2.4. Residue currents and multiplicative calculus 1.2.5. Residues of sections of hermitian bundles 1.3. The algebraic point of view 1.3.1. Cauchy’s formula in A1ℂ = ℂ or ℙ1ℂ and division in 𝕂[X] 1.3.2. Currential Cauchy’s formula on compact Riemann surfaces 1.3.3. Green currents and heights of arithmetic cycles 1.3.4. Residues, residue currents, and Gorenstein 𝕂-algebras

1 1 1 4 5 11 12 21 25 27 32 38 38 44 49 58

Chapter 2. Residue currents: a multiplicative approach 2.1. De Rham complex and iterated residues 2.1.1. Division of forms and de Rham’s lemma 2.1.2. Leray coboundary morphism, cohomological residue formulae 2.1.3. Meromorphic multilogarithmic forms 2.2. Coleff–Herrera sheaves of currents 2.2.1. Integration currents, pseudo-meromorphicity, and holonomy 2.2.2. The sheaf CHX ,V 2.2.3. Coleff–Herrera currents with prescribed polar set 2.2.4. Extension-restriction of pseudo-meromorphic currents 2.3. Coleff–Herrera’s original construction revisited 2.3.1. Essential intersection and Coleff–Herrera’s original construction 2.3.2. Coleff–Herrera currents attached to sections of line bundles 2.3.3. Coleff-Herrera theory in the complete intersection setting

63 63 63 69 72 75 76 82 90 93 95 95 101 110

Chapter 3. Residue currents: a bundle approach 3.1. A toric complex geometry digest 3.1.1. Rational fans 3.1.2. Complete simplicial fans and homogeneous coordinate rings 3.1.3. K¨ahler cone and moment maps 3.1.4. Projective toric manifolds

117 117 117 119 122 123

vii

viii

CONTENTS

3.2. Bochner–Martinelli currents attached to bundle sections 3.2.1. The currents R|s | and P|s | for s ∈ O(X , E), E = (E, | |) 3.2.2. The complete intersection setting revisited 3.2.3. Transformation laws from the geometric point of view 3.3. Bochner–Martinelli currents and generically exact complexes 3.3.1. Generically exact Koszul and Eagon–Northcott complexes 3.3.2. The currents RF• and P F• for a metrized complex F• 3.3.3. Residual obstruction for exactness and duality theorems 3.4. Bochner–Martinelli residue currents: structural results 3.4.1. Decomposition of R|s | ([V ]) along distinguished varieties

124 125 128 131 136 136 139 145 150 150

3.4.2. RF• and the Buchsbaum–Eisenbud–Fitting sequence 3.4.3. Residue currents and structure forms 3.4.4. The Cohen–Macaulay case

153 158 161

Chapter 4. Bochner–Martinelli kernels and weights 4.1. Diagonal submanifold and Koszul complex K• D 4.1.1. Koszul complex over the duplication of a good manifold 4.1.2. Nabla operators and Lν sheaves of currents for good manifolds 4.2. Affine and projective Bochner–Martinelli kernels 4.2.1. The affine case X = ℂN 4.2.2. The projective case X = ℙN ℂ 4.3. Chern connection and Koppelman’s formulae on good manifolds 4.4. Bochner–Martinelli weights, definition, and constructions 4.4.1. Global weights 4.4.2. Local weights 4.4.3. Global weights attached to metrized complexes 4.5. Bochner–Martinelli weighted integral representation formulae 4.5.1. The affine case X = Ω ⊂ ℂN 4.5.2. The projective case X = ℙN ℂ 4.5.3. Weighted Koppelman’s formulae in the general setting 4.6. Solving the F• -Hefer problem in specific situations 4.6.1. Complexes of trivial bundles over Stein manifolds 4.6.2. Projective Hefer forms and Koszul complex over ℙN ℂ 4.6.3. Hefer problem, syzygies for homogeneous polynomial ideals

167 167 167 169 171 171 173 175 179 179 182 185 188 188 193 194 196 196 201 205

Chapter 5. Integral closure, Brian¸con–Skoda type theorems 5.1. Integral closure of ideals and valuative criterion 5.1.1. Integral closure of ideals, the analytic reduced case 5.1.2. The nonreduced case, valuative criterion, nœtherian operators 5.2. Brian¸con–Skoda theorem in 𝔸 = O0 5.3. Brian¸con–Skoda theorem in 𝔸 = (O/IV )0 5.4. Brian¸con–Skoda theorem for purely dimensional 𝔸 = (O/I)0 5.5. Primary ideals in O0

211 211 211 212 215 221 230 236

Chapter 6. Residue calculus and trace formulae 6.1. Trace in analytic polyhedra 6.1.1. Analytic Weil polyhedron, nondegeneracy, skeleton 6.1.2. The case when m = n 6.1.3. The case when m > n

243 243 243 245 250

CONTENTS

6.1.4. Hardy spaces of strongly nondegenerate analytic Weil polyhedra 6.2. Algebraic approach to residue symbols 6.2.1. Quasi-regular sequences in a commutative 𝕂-algebra 𝔸 6.2.2. Algebraic residue symbols as traces 6.2.3. Wiebe’s theorem and the algebraic transformation law 6.3. Residue symbols in 𝕂-polynomial algebras 6.3.1. Residue symbols Res𝕂[X]/𝕂 6.3.2. Residue symbols Res𝕂𝕁 [X,Y ]/𝕂 6.3.3. Residues symbols in 𝕂[X1±1 , . . . , Xn±1 ] 6.4. Trace in 𝕂[X] or 𝕂𝕁 [X, Y ] 6.4.1. The algebraic trace function in the 𝕂[X]-setting 6.4.2. The algebraic trace function in the 𝕂𝕁 [X, Y ] setting 6.5. The characteristic zero case 6.5.1. Algebraic residue symbols realized as currents 6.5.2. Properness and algebraic residue symbols in characteristic 0 6.5.3. Properness and toric residue symbols (characteristic 0) 6.6. Residue symbols and arithmetics 6.6.1. The univariate case 6.6.2. Geometric and arithmetic markers for complexity 6.6.3. Size estimates of rational algebraic residue symbols 6.6.4. Rational algebraic residue symbols: a dynamic approach Chapter 7. Miscellaneous applications: intersection, division 7.1. Hilbert’s nullstellensatz in 𝕂[X1 , . . . , Xn ] and residue calculus 7.1.1. Rabinowitsch’s trick and Grete Hermann’s algorithmic procedure 7.1.2. Lipman–Teissier theorem and N -solvability of B´ezout identity 7.1.3. N -solvability of B´ezout identity in ℂ[X] and Bochner–Martinelli currents 7.1.4. N -solvability in ℂ[X] of B´ezout identity via a diagram chase 7.1.5. Sharp geometric nullstellensatz 7.1.6. An arithmetic Perron theorem in its parametric form 7.1.7. Hilbert’s nullstellensatz and P = N P 7.2. Jacobi–Lagrange–Kronecker (JLK) parametric identities 7.2.1. Parametric B´ezout identity of the JLK form 7.2.2. Parametric membership identities of the JLK form 7.3. Effective geometric Brian¸con–Skoda–Huneke type theorems 7.3.1. Global algebraic Brian¸con–Skoda–Huneke exponents μ(𝕏 : ℙN ℂ) 7.3.2. An effective Brian¸con–Skoda–Huneke type theorem on 𝕏 ⊂ ℂN 7.3.3. Global Brian¸con–Skoda or Lipman–Teissier theorems in 𝕂n 7.4. Algebraic residues and tropical considerations 7.4.1. Rational polyhedral complexes and tropical cycles 7.4.2. The tropical current trop∗ Z 7.4.3. Tropicalization and (p, q)-supercurrents on ℝn 7.5. Jacobian and socle, radical and top radical 7.6. Multivariate residue calculus and the exponential function 7.6.1. Weighted algebras of entire functions, tempered currents 7.6.2. Trace formulae in weighted algebras of entire functions 7.6.3. Ehrenpreis–Palamodov’s fundamental principle revisited 7.7. Residue calculus outside the commutative setting

ix

253 254 254 257 260 263 264 268 269 270 270 277 283 284 289 297 300 301 302 310 316 323 323 323 325 328 331 333 337 340 346 347 353 361 361 365 369 372 372 376 378 385 389 390 392 401 410

x

CONTENTS

7.7.1. Univariate quaternionic setting and relevant regularity concepts 7.7.2. Principal value and residue currents in the ℍ-setting

410 417

Appendix A. Complex manifolds and analytic spaces A.1. Complex manifold, structural sheaf OX , sheaves of OX -modules A.1.1. Definitions A.1.2. Examples (ℂN , ℙN ℂ , X (Σ)) A.2. Coherence, Stein manifolds, free resolutions A.2.1. Coherent sheaves of OX -modules A.2.2. Stein manifolds and Cartan Theorems A and B A.2.3. Free resolutions of coherent sheaves A.3. Closed analytic subsets of a complex manifold A.3.1. Local dimension A.3.2. Splitting V = Vsing ∪ Vreg , global dimension A.3.3. Free resolutions of sheaves and codimension of supports A.3.4. Local presentation in the purely dimensional case A.3.5. Weakly holomorphic functions and Oka universal denominator A.3.6. Example: closed analytic subsets of ℙN ℂ , Chow’s theorem A.4. Complex analytic spaces, normalization, and log resolutions A.4.1. Complex analytic spaces A.4.2. Normalization and blowup of complex analytic spaces A.4.3. Log resolution and the Hironaka theorem

421 421 421 423 425 425 425 427 429 429 430 430 431 432 434 435 435 436 441

Appendix B. Holomorphic bundles over complex analytic spaces B.1. Analytic cocycles, holomorphic bundles, isomorphism classes B.1.1. Cocycles on a complex analytic space, examples B.1.2. Linear algebra and construction of cocycles B.1.3. Holomorphic bundles over a complex analytic space B.1.4. Isomorphism classes of holomorphic bundles B.1.5. Local holomorphic frames and sheaves of holomorphic sections B.1.6. The concepts of local versus global complete intersection ∗(p,q) over a complex manifold X B.1.7. The bundles TX B.2. Sheaves of bundle-valued differential forms or currents B.2.1. Differential forms and currents on a complex manifold B.2.2. Differential forms and currents on a complex analytic space B.2.3. Dolbeault complex, Serre duality on a complex manifold B.2.4. Strongly holomorphic (p, 0) forms on V → X B.2.5. The Barlet sheaves on V → X B.2.6. Dolbeault complex, Serre duality on a complex analytic space B.3. Hermitian bundles B.3.1. Hermitian complex bundles over a complex manifold B.3.2. Hermitian holomorphic bundles over a complex analytic space B.3.3. Chern and Segre forms and classes

443 443 443 444 444 446 447 448 449 451 451 456 458 459 460 462 463 463 464 465

Appendix C. Positivity on complex analytic spaces C.1. Positive forms, positive currents C.1.1. Positivity of differential forms on a ℂ-vector space C.1.2. Positive currents on a complex manifold C.1.3. Trivial extension of positive closed (p, p)-currents

469 469 469 470 472

CONTENTS

C.2. Lelong numbers and integration currents C.2.1. Lelong numbers of positive closed currents on Stein manifolds C.2.2. Lelong numbers of positive closed currents on analytic spaces C.2.3. Analytic subvarieties and corresponding integration currents C.2.4. Integration currents attached to sections of holomorphic bundles C.2.5. Holonomy of integration currents and consequences C.3. Positivity of holomorphic hermitian bundles

xi

473 473 474 475 477 478 479

Appendix D. Various concepts in algebraic or analytic geometry D.1. Cycles on a complex analytic space D.1.1. Cycles, effectivity, splitting along a dimension and cutting out D.1.2. Lelong–Poincar´e equation and classes of generalized cycles D.1.3. Local multiplicities of classes of generalized cycles ˇ D.2. Cartier divisors and Cech cohomology D.2.1. Cartier divisors and Picard group on a complex analytic space D.2.2. Toric Cartier divisors on a toric variety ∗ ˇ D.2.3. Picard group in terms of Cech cohomology with values in OX D.2.4. The first Chern characteristic class of a complex line bundle D.3. Weil divisors and Chow groups D.3.1. Weil divisors on complex analytic spaces or algebraic varieties D.3.2. Chow groups on complex analytic spaces or algebraic varieties D.4. Ampleness versus positivity for holomorphic line bundles D.4.1. Globally generated holomorphic bundles D.4.2. Semi-ampleness of holomorphic bundles D.4.3. Very ampleness/ampleness of line bundles (compact setting) D.4.4. Very ampleness and ampleness of higher rank bundles D.4.5. Examples D.5. Local versus global intersection theory D.5.1. Local extended multi-index of intersection, Crofton formula D.5.2. Crofton’s averaging and metrized toric divisors D.5.3. The join product in the projective setting D.5.4. Global intersection in ℙN ℂ and classes of generalized cycles

483 483 483 484 488 489 489 490 491 492 492 492 493 494 494 495 495 496 497 498 498 500 502 503

Bibliography

507

Index

527

Preface The first glimpses of multi-dimensional residue theory within the framework of complex analytic geometry and its applications were already apparent in the work of Henri Poincar´e [P1885, P1887, P1906]. The concluding sentence of his 1887 memoir [P1887], We may summarize the preceding by saying that the integral taken along a closed surface S only depends on the singular curves contained in the interior of this surface. . . , announces the birth not so much of the concept of residue but of a new way of thinking that nearly half a century later would lead Georges de Rham [DR32, DR36] and Jean Leray [Le59] to consider in terms of homology and cohomology groups the problem of integrating a singular differential form. Let us describe Leray’s approach in the simplest possible case of a product manifold X = ℂ × Y, where Y is a complex (N − 1)-dimensional manifold (N ≥ 2), in which one considers the smooth complex hypersurface S = {0} × Y ⊂ X . Let  𝔻y U = 𝔻ℂ (0, 1) × Y = y∈Y

be the simplest open neighborhood of S considered here with its trivial fibration over S. For every y ∈ Y the fiber 𝔻y = {(ζ, y) : |ζ| < 1} ⊂ ℂ is a complex disc, whose boundary is the circle Γy = {(ζ, y) : |ζ| = 1}. Furthermore, the closures 𝔻y of all fibers are pairwise disjoint. Consider the projection map defined by μ : (ζ, y) ∈ U −→ (0, y) ∈ S and the map  (ζ, y) if (ζ, y) ∈ /U  ν : (ζ, y) ∈ X \ S −→  ζ/|ζ|, y if (ζ, y) ∈ 𝔻y \ {(0, y)} with values in ∈ X \ U . Both maps are deformation retracts. For every k ∈ ℕ the deformation retract ν induces isomorphisms between homology groups ν∗ : Hk (X \ S) −→ Hk (X \ U ). The same is true for the deformation retract μ inducing isomorphisms between homology groups μ∗ : Hk (S) −→ Hk+2 (X , X \ U ) for every k ∈ ℕ.  Here the construction is more subtle and is described as follows. Let c = ι νι {σι } ∈ Ck (S) be a k-chain with coefficients νι in ℤ representing c¯ ∈ Hk (S). Then μ−1 (Supp σι ) = 𝔻ℂ (0, 1) × Supp σι for each ι. The set 𝔻ℂ (0, 1) × σι inherits the orientations of 𝔻ℂ (0, 1) and σι and thus defines a (k + 2)-chain in X with support in U . Therefore, the class of the (k + 2)-chain  ˇ∗ (c) in ι νι {𝔻ℂ (0, 1) × σι } in Ck+2 (X ) modulo Ck+2 (X \ U ) defines an element μ xiii

xiv

PREFACE

Ck+2 (X , X \ U ) := Ck+2 (X )/Ck+2 (X \ U ). It follows that ∂(ˇ μ∗ (c)) = μ ˇ∗ (∂c), since ˇ∗ induces a homomorphism μ∗ ∂𝔻ℂ (0, 1) × σι ∈ Ck+1 (X \ U ) for any ι. Thus μ from Hk (S) to Hk+2 (X , X \ U ). The support of each (k + 2)-simplex in X intersects S transversally, ensuring that the intersection of a chain C ∈ Ck+2 (X ) with S leads to a k-chain S ∩ C ∈ Ck (S) satisfying ∂(S ∩ C) = S ∩ ∂C. The transversal intersection with S induces, at the level of homology classes, the construction of the inverse homomorphism μ−1 ∗ . Thus, μ∗ is an isomorphism. Now, consider the Mayer–Vietoris long exact sequence ι

π

∂

ι

∗ ∗ ∗ ∗ · · · −→ Hk+2 (X ) −→ Hk+2 (X , X \ U ) −→ Hk+1 (X \ U ) −→ Hk+1 (X ) −→ · · · .

Most interesting is the connecting homomorphism δ = ν∗−1 ◦ δ∗ ◦ μ∗ : Hk (S) → Hk+1 (X \ S) and its dual homomorphism Res = δ∗ : H k+1 (X \ S) −→ H k (S). The dual homomorphism δ∗ on the de Rham cohomology groups corresponds precisely to taking the residue on S of closed differential forms φ on X \ S representing classes in H k+1 (X \ S). If φ ∈ Z k+1 (X \ S) is a meromorphic form with poles of order 1 along S and γ ∈ Z k (S), then the residue formula   1 φ= Res φ 2iπ δγ γ holds. This procedure iterates when an ordered sequence (S1 , . . . , Sm ) of smooth hypersurfaces intersecting transversally on X replaces S. The important monograph by L. A. Aizenberg and A. P. Yuzhakov [AY] elaborates on applications of Leray iterated residues to the cohomological reduction of semimeromorphic and rational forms. It is of crucial importance when solving, for example, systems of nonlinear algebraic equations, and it applies to elimination theory. Thinking in terms of geometry, one realizes that the theory of distributions, hence of currents, brought a different, promising approach towards the concept of residue in complex analytic geometry since the early 1970s. Its origin lies in the efforts directed at division by holomorphic or real analytic functions as described in the work of Laurent Schwartz [Sc55], Stanislas L  ojasiewicz [Lo59], and, more recently of Michael Atiyah [At70]. This point of view also profits from the fundamental result by Heisuke Hironaka [Hir64] about the resolution of singularities over a field with characteristic 0. We should mention here the pioneering works in this direction by Miguel Herrera and David Liebermann [HL71], Pierre Dolbeault and Jean-Baptiste Poly [Pol72, Pol74], and Fran¸cois Norguet [Nor74, Nor96]. The ground-breaking monograph by Nicolas Coleff and Miguel Herrera [CoH78] describes what is meant by taking a residue of a semimeromorphic form ψ/(f1 · · · fm ), where the smooth form ψ is compactly supported, along an analytic set Z(f ) = {f1 = · · · = fm = 0} defined by a holomorphic map f = (f1 , . . . , fm ) : ℂN −→ ℂm . That is, in this monograph, instead of residue, iterated residue, or Leray morphism, the central object is the residue current, together with its principal value current companion. The work of J. Solomin [Sol77], completing the relevant circle of ideas, becomes a

PREFACE

xv

junction point for their continuation in a highly stimulating thesis by Mikael Passare [Pas88A] and his subsequent work [Pas88B] more than ten years later. Let s be a section of a holomorphic line bundle L over a complex analytic space X . Assuming that the line bundle L is smoothly metrized as L = (L, | |), the two fundamental objects in the Coleff–Herrera multiplicative calculus are currents P s and Rs (principal value and residue currents, respectively) defined by (0.1) P s : ϕ ∈ D(N,N ) (X , L)  

   s∗  s∗

 −→ lim+ , ϕ = |s|2λ ,ϕ = P s,λ , ϕ 2 2 2iπ|s| ε→0 λ=0 λ=0 |s|≥ε 2iπ|s| X and (0.2) Rs = ∂P s : ϕ ∈ D(N,N −1) (X , L)  

   s∗ 

 s∗ 2λ s,λ −→ lim+ , ϕ = ∂|s| , ϕ = R , ϕ , 2 2iπ|s|2 ε→0 λ=0 λ=0 |s|=ε 2iπ|s| X where s∗ denotes the conjugate section of s with respect to the smooth hermitian metric on L. Mellin transform relates here both of the proposed representations. A multiplicative formalism involving such objects is relatively easy to describe thanks to the second representation (0.2) involving Atiyah’s approach to division by holomorphic functions through the analytic continuation principle. Keeping in mind the problem of division by holomorphic or real analytic functions, we point out the key role that Michael Atiyah’s view plays in our presentation from the beginning of our monograph, unifying his novel approach with the point of view of Poincar´e, de Rham, and Leray. This is a deliberate choice. Let us explain briefly why. Firstly, this approach makes it relatively easier to use multiplicative formalism, as explained briefly above. Secondly, our choice is essentially motivated by the following important result. Let OℂN ∂/∂ζ be the sheaf of differential operators with analytic coefficients on ℂN . Given f1 , . . . , fm ∈ OℂN ,0 , a crucial fact is that there exists a system of formal identitities [Bj74, Sab87, Gy93] ∂ (fj f λ ), b(λ1 , . . . , λm ) f λ = Qj λ, ζ, ∂ζ

(0.3)

j = 1, . . . , m,

where λ1 , . . . , λm are independent parameters, Qj ∈ OℂN ,0 ∂/∂ζ ⊗ℂ ℂ[λ] for any m λ j = 1, . . . , m, f λ = j=1 fj j , and b ∈ ℚ[λ], realizing a transcendental basis of ℂ[λ]. The polynomial b ∈ ℚ[λ] is of the form (0.4)

b(λ) =

   nι,0 + m (nι,0 ∈ ℕ∗ , (nι,1 , . . . , nι,m ) ∈ ℕm \ {0}). j=1 nι,j λj ι

The archetypal example for such a system of formal identities is realized by N 

(λκ + 1)

κ=1

N

 κ=1

ζκλκ =



 κ=1,...,N

κ=j

(λκ + 1)

N

∂  ζj ζκλκ , j = 1, . . . , N, ∂ζj κ=1

xvi

PREFACE

after substituting fj = ζj for j = 1, . . . , N in (0.3), thus taking m = N . Formal identities (0.3) lead to true identities (0.5)

|f |2λ ∂ |f |2λ rj = Qj λ, ζ, r −1 , ∂ζ fj fj j |f |2λ ∂ |f |2λ b(λ1 , . . . , λj − rj , . . . , λm ) rj = Qj λ, ζ, rj −1 , ∂ζ fj fj

b(λ1 , . . . , λj − rj , . . . , λm )

j = 1, . . . , m, rj ∈ ℕ,

involving germs of distributions at the origin of ℂN , complex parameters λ1 , . . . , λm , m 2λ and using the notation |f | = j=1 |fj |2λj . Then, division by the fj ’s, by the fj ’s,  m or even by f 2 = j=1 |fj |2 (or their powers) is solved using integration by parts, combined with the analytic continuation principle, thanks to true identities (0.5) in the sense of germs of distributions. In the global algebraic setting, where OℂN ,0 is replaced by 𝕂[X1 , . . . , XN ], when 𝕂 is a field with characteristic 0, analogues of (0.3) persist with polynomial b ∈ 𝕂[λ] and the operator Q belonging the Weyl algebra 𝕂[λ, X] d/dX; see [Be72, Lic88]. Using algebraic substitutes to formal equations (0.3), we shall be able to express the action of principal value or residue currents arising from elements in ℚ[X1 , . . . , XN ] or ℚ[X0 , . . . , XN ], respectively, N on K¨ahler differential forms (similarly on 𝔸N ℚ or ℙℚ ), as values at the origin of arithmetic ζ-functions, thus making them quite special complex numbers; see for example [BY98,CaM00]. Finally, we mention another reason, of more prospective flavor, for why we keep using Atiyah’s approach. Although the pairing of holomorphic coordinates with ghost antiholomorphic ones (which is essential for us in the present monograph) may be lost, some of the ideas behind such an approach may persist in settings totally different from that of complex analytic geometry. For example, this is the case with Igusa calculus [Igu] on 𝕂-prehomogeneous vector spaces, where 𝕂 is a p-adic field. The above considerations lead us to pursue consistently Atiyah’s approach to representing principal values and residues from the current point of view. We will focus on principal value and residue currents attached to smoothly metrized, generically exact complexes of holomorphic bundles. Among those of particular interest to us are the algebraically tractable complexes of Koszul and of Eagon and Northcott. Alternatively, we apply our approach to (unfortunately) untractable syzygies for 𝕂[X1 , . . . , XN ]/𝕁, where 𝕁 is an ideal in 𝕂[X1 , . . . , XN ], or to the untractable syzygies 𝕂[X0 , . . . , XN ]/𝕁, where 𝕁 is a homogeneous ideal in 𝕂[X0 , . . . , XN ]. In both cases, 𝕂 is a subfield of ℂ. Note that the field 𝕂 could be ℂ, ℚ, or a number field, which justifies the need to keep track of Atiyah’s approach to profit from the closed-form expressions of formal identities that are global analogues of (0.3). Furthermore, bundle valued currents introduced using this point of view proved to be quite flexible, since one profits from 2N parameters ζ1 , . . . , ζN , ζ1 , . . . , ζN instead of just the holomorphic coordinates ζ1 , . . . , ζN to metrize holomorphic bundles or complexes. We intend in this monograph to illustrate this flexibility by adapting it to study questions ranging from multi-dimensional complex analysis and commutative algebra to intersection theory. In some sense, we update here the content of our text Residue Currents and B´ezout Identities ([BGVY]) with Carlos Alberto Berenstein and Roger Gay, published almost 30 years ago (see also [BGY89]), by

PREFACE

xvii

incorporating research developments that took place in multi-dimensional residue theory over these years. Chapter 1 of the monograph contains the presentation in the univariate setting of most of the tools and methods extended in this monograph to the multi-variate frame. Section 1.1 presents in one complex variable the machinery of analytic continuation used in a more complex form later in the book. We elaborate there on Mellin transformation leading us consistently to Atiyah’s approach. Formal identities (0.3), followed by true identities (0.5) in the sense of distributions, are of course trivial in the univariate context. In §1.2.1, §1.2.4, and §1.2.5, we present in the univariate setting the machinery that will be extensively carried to the multi-variate situation throughout Chapters 2 and 3. The contents of §1.2.2, §1.2.3, §1.3.1, §1.3.3, and §1.3.4 in the univariate case foreshadow the development of research themes of a more algebraic nature which will be carried out later on starting from Chapter 5. Even though the triviality of some results in the univariate context could make them a bit disappointing, we highly advise the reader to start with this preliminary chapter before entering the inherently highly technical world of those that follow. Chapter 2 establishes the bridge between the Poincar´e–de Rham–Leray geometric approach towards the concept of residue and its more analytic, by nature, realization through division by holomorphic or antiholomorphic functions leading to the construction of principal value and residue currents as proposed by N. Coleff, M. Herrera and D. Liebermann [HL71, CoH78]; see §2.1 and §2.3.1. As always, this point of view is realized through Atiyah’s approach. Although we keep to division by holomorphic functions, we will nevertheless comment on the alternative construction of such employing integration on semianalyic chains. We will point out then that the Mellin transformation, as presented in Chapter 1 in the univariate setting (see §1.1.3 and §1.2.1), connects the two presentations (see the end of §2.3.2). The class of so-called Coleff–Herrera currents is identified and studied in §2.2. The contribution of Jan-Erik Bj¨ ork is towering here. Since all the currents involved in the present monograph are pseudo-meromorphic and they obey the crucial dimension principle, we describe them, together with their properties, in §2.2.1. The case of complete intersections, important in the sequel, will be detailed in 2.3.3. The aim of Chapter 3 is to present a fundamental pairing of smoothly metrized generically exact complexes F• of holomorphic bundles over a complex manifold or over a complex analytic space X (think for example about Koszul or Eagon– Northcott complexes, or complexes related to syzygies) with a couple of objects {residue current, principal value} =(RF• , P F• ). When X is a Stein manifold and the complex of sheaves of sections of holomorphic bundles corresponding to F∗• happens to be exact, each current RF• will play the role of a duality marker. This construction and its consequences culminate in §3.3. The complementary role of P F• as a marker for the quotient (instead of the remainder) in the realization through closed-form division formulae of such duality will be exemplified later in Chapter 4. It is in Chapter 3, namely in §3.2.3 and §3.4.4, that a key tool in multi-variate residue calculus appears. This tool is the geometric transformation law and its variants, extensively used in the part of this monograph devoted to applications, namely in Chapters 6 and 7. The extrinsic and intrinsic structures

xviii

PREFACE

of RF• are described in §3.4.2 and §3.4.3 when the complex F• is induced by a minimal resolution of (OX /IV )0 for a purely dimensional closed analytic subset V singular at 0. Such an intrinsic description proves to be essential in §5.3 and §5.4, as well as in §7.3.2 in the global algebraic affine setting. Chapter 4 has a pivotal role. Suppose X is a complex manifold (which embeds as the diagonal submanifold in X = X × X ) and that the diagonal submanifold is defined (at the order exactly one) as the zero set of a section Δ of a holomorphic bundle, whose rank equals dim X . It implies that the Koszul complex K•Δ is exact (except at order 0). This class of manifolds includes two main examples, namely ℂN and ℙN ℂ , which are of major importance for algebraic questions. Bochner–Martinelli kernels on such manifolds, whose construction was initiated by Rudolf Fueter [Fu39, Mar43], enter the stage in §4.2. Weights are then introduced, as was already done in the univariate case for the Cauchy–Pompeiu formula (see §1.2.2 and §1.2.3), and they lead to the weighted integral representation formula. The contributions of Mats Andersson, Bo Berndtsson, and Mikael Passare [AndB82, AndP88, And03, And06A] are essential here. Solving the so-called Hefer problem attached to a generically exact complex of holomorphic bundles over such complex manifold X will be the almost constant objective from §4.4.3 and on in this chapter. Once this is done (if possible), explicit weighted division formulae, where the principal value current P F• is a marker for the quotient and the residue current RF• is a marker for the remainder, will follow either on a Stein manifold, in particular on ℂN , or on the compact manifold ℙN ℂ ; see Theorems 4.44, 4.51, and 4.54. These closed-form division formulae realize explicitly division, whenever RF• is a marker for duality. They will play a dominant role in §7.6.1 and §7.6.3, which nearly concludes the core of this monograph. The point of view inherited from the current approach towards the concept of residue opened new inroads for its use. It allows us to answer questions related to ideals in polynomial rings and to solve problems related to under-determined systems of equations. It applies to many other problems formulated in ℂN or in complex manifolds as described in the highly engaging monograph by August Tsikh published in 1988 (Russian edition) [Ts], having almost the same title as the monograph we present here. Starting with Chapter 5, we focus on different, but still relevant, aspects residue. This time, we view it through an algebraic instead of a geometric (or analytic) point of view. The fundamental result established by Jo¨el Brian¸con and Henri Skoda in [BriS74]—which states that given any ideal I in the local ring OℂN ,0 , one has that I N +k−1 ⊂ I k for any k ∈ ℕ∗ where J denotes the integral closure of an ideal in a commutative ring—came as a surprise. It took nearly seven years before an alternative proof of a somewhat more algebraic nature, that did not rely on a ∂-resolution, appeared in [LipS81, LipT81]. The residue, this time with its algebraic interpretation, plays a decisive role in the results of the Brian¸con–Skoda type. The algebraic point of view on residue theory was developed and elaborated in the work by E. Kunz, R. Waldi, M. Kreutzer, and R. H¨ ubl [Ku92, KuW88, KK87, HK90] and of J. Lipman [Lip87], and we will consider it as well in the sequel of this monograph. Our goal in Chapter 5 is to present effective realizations of the Brian¸con–Skoda theorem in a regular local ring and of its companion theorem by Brian¸con, Skoda, and Huneke in a nœtherian nonregular

PREFACE

xix

local ring (O/IV )0 , where V is a germ of the purely dimensional singular analytic subset at the origin. We also present some effective realizations in the nonreduced setting, once the formulation of the statement is correctly settled. The effectiveness of the proofs is provided here by the tools introduced in Chapter 4. The results presented in this chapter are due to Mats Andersson and Jakob Sznajdman. The notion of trace appears as a common denominator between the algebraic, analytic, and geometric interpretations of the concept of residue. The aim of Chapter 6 is to make this evident. Throughout this chapter, we will manipulate the three visions, either as algebraists or analysts and geometers. Cauchy’s formula, established by Andr´e Weil in 1935 [We35], is our starting point. We will recall the interpretation of residues as traces by Joseph Lipman [Lip87] and then illustrate this interpretation in polynomial algebras 𝕂[X1 , . . . , XN ] or in their quotient by a purely dimensional radical ideal 𝕁. We chose to as a guideline in this chapter the notion of properness, especially through §6.3–§6.5. It seems to be a good setting for showing how to profit from the smoothness of tools inspired by the current approach. The Euler–Jacobi formula, relative to the vanishing of total sums of residues in the affine (nonprojective) setting, illustrates how powerful such a flexible approach can be when compared to a more rigid algebraic one. Chapter 6 ends with arithmetic considerations about the arithmetic complexity of generalized Taylor-type developments involved in expanded versions of the algebraic Cauchy–Weil formula. We consider here the residue calculus in ℚ[X1 , . . . , XN ] or in ℚ[X1 , . . . , XN ]/𝕁 and introduce markers for both the geometric and arithmetic complexities. We continue analyzing how multi-variate residue calculus behaves with respect to both geometric and arithmetic intersection theory in §6.6. In Chapter 7, we pursue questions in interpolation and elimination theory, as in the previous chapter, and focus again on the role of multi-variate residue calculus there. It means that we return to a circle of ideas, whose origin lies in the contributions of Carl Jacobi, Leopold Kronecker, Arthur Cayley, Francis Macaulay, and David Hilbert. Hilbert’s nullstellensatz, as well as the Brian¸con–Skoda global theorem in ℂN , are presented under various facets: Lagrange–Kronecker’s parametric closed form formulae, diagram chase, explicit division formulae that can be interpreted as averaged ones (through Crofton’s averaging formula) respecting the field over which polynomial entries are defined. We provide in §7.6 trace or division formulae in the affine setting, where the transcendental function exp is involved. For example, we present trace or division formulae in the weighted Paley–Wiener algebra of entire functions. In the same section, we also reformulate the Euler–Ehrenpreis–Palamodov fundamental principle asserting that any solution of a system of partial differential equations (PDEs) in a convex open subset of ℝN is a “sum” of elementary solutions. The same strategy applies to representations of mean-periodic functions according to the same principle under some specified hypothesis. Summarizing, one could say that Chapter 7 contains applications of multi-dimensional residue theory to a broad spectrum of mathematical problems. Moreover, it also contains a short insight (§7.7) into the noncommutative setting of the univariate quaternionic algebra ℍ. This could be interpreted both as a modest attempt to transpose some pieces of residue machinery to the noncommutative world and as a tribute to Rudolf Fueter, who introduced Bochner–Martinelli kernels

xx

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[Fu39] that are present throughout this monograph. In some sense, the material contained in this chapter is intended to illustrate our point of view that residue currents are a powerful, flexible tool to be applied in many, seemingly unrelated, areas of mathematical research. The monograph concludes with four appendices containing the material needed to make the presentation of the main subjects as self-contained as possible. It required that we rework already known notions and adapt them into our setting. We hope that the material presented in the monograph illustrates the strength and the applicability of the methods developed in this particular area of multi-variate complex analysis. The authors are indebted to the referees for essential comments on and corrections to earlier versions of the manuscript. Their critical reading of the original text contributed much to its clarity and presentation. Many thanks are also due to Ina Mette for her patience and support.

CHAPTER 1

Residue calculus in one variable In this introductory chapter, we recall briefly the classical residue in complex analysis, analytic geometry, and commutative algebra. We focus in particular on some analytic aspects of this notion that will be crucial when we introduce and study its multi-dimensional analogue in several complex variables. Even though some of the points about residue theory that are discussed below look irrelevant in the univariate setting, they are extremely important in the multi-variable framework. We conclude the chapter with some nontypical applications preparing the reader for the problems to be studied in the following chapters. 1.1. An analytic approach 1.1.1. Laurent expansions and meromorphic functions. Let U be an open connected subset of ℂ, α a point in U , and h a holomorphic function in U \ {α}. Let also r > 0 be small enough so that the disk D = D(α, r) is relatively compact in U . For all ζ ∈ D(α, r), ζ = α, one can then write the Laurent series  ak (ζ − α)k , (1.1) h(ζ) = k∈ℤ

where the Laurent coefficients ak , k ∈ ℤ, are given by  2π 1 h(α + reiθ )e−ikθ dθ. (1.2) ak = 2πr k 0 Note that the coefficients ak do not depend on r. Let A = {k ∈ ℤ : ak = 0}. Then the order of h at α is defined to be the element θα (h) ∈ ℤ ∪ {−∞, +∞}: ⎧ ⎪ ⎨+∞ if A = ∅, (1.3) θα (h) = inf A if A = ∅ and is bounded from below, ⎪ ⎩ −∞ if A = ∅ and is not bounded from below. The singularity of h is removable if θα (h) ≥ 0. In this case, there is a unique holomorphic extension g of h on U . In particular, if θα (h) = +∞, the analytic continuation principle ensures that g ≡ 0 on U . Note that the singularity α is removable if and only if |h| is bounded in D(0, r) \ {α} for r > 0 small enough. If θα (h) is a strictly positive (resp., strictly negative) integer, then we say that α is a zero (resp., a pole) of order θα (h) (resp., −θα (h)) for h. The singularity α is essential if θα (h) = −∞. Otherwise, if θα (h) = ν ∈ ℤ, then one can factorize h in the open subset U \ {α} as h(ζ) = (ζ − α)ν u(ζ), u being holomorphic on U with u(α) = 0. Assume that U is an open connected subset of ℂ, W is a closed discrete subset of U , and h is a holomorphic function in U \ W . We say that h is meromorphic 1

2

1. RESIDUE CALCULUS IN ONE VARIABLE

in U if, for any α in W , θα (h) > −∞. This amounts to saying that h is the restriction to U \ W of a (necessarily) unique holomorphic function from U to the complex projective line ℙ1ℂ . We adopt here an alternative point of view: a holomorphic function h : U \ W → ℂ is meromorphic in U if and only if it extends as a distribution in U . If this is the case, then the extension of h is not unique. One way to construct a particular one (denoted as h|U\W and called a standard extension) in a neighborhood of α ∈ W is to set

 dζ ∧ dζ  (1.4) h|U\W , ϕ := |(ζ − α)|2λ h(ζ)ϕ(ζ, ζ) , 2i λ=0 U where the support of the smooth test function ϕ(ζ, ζ) is contained in a neighborhood Dα of α satisfying Dα ∩ W = {α}. If Re λ  1, the interior part of the brackets in (1.4) is perfectly well defined. Then one applies to it the analytic continuation as a holomorphic function of λ to some half-plane Re λ > −η0 , η0 > 0. Its value at λ = 0 is described by (1.4). Here the integration is understood with respect to the complex variable ζ. The ability to obtain such a suitable analytic continuation is based on Stokes’s theorem, as we shall explain next. If W = {z ∈ U : f (z) = 0}, where f is holomorphic in U , then the standard extension h|U\W is defined as the principal value distribution, denoted by PV[h],

 dζ ∧ dζ  (1.5) PV[h] : ϕ ∈ D(U ) → h|U\W , ϕ := |f (ζ)|2λ h(ζ)ϕ(ζ, ζ) . 2i λ=0 U This definition does not depend on the choice of f . Any other holomorphic function g in U that satisfies W = {g = 0}, without taking into account possible multiplicities, will generate the same principal value distribution. The right-hand side in (1.5) amounts in fact to taking the limit  dζ ∧ dζ h(ζ) ϕ(ζ, ζ) lim →0 |f |≥ 2i when  > 0. The approach in (1.5) through the analytic continuation process has an advantage that will be more evident later on in the multi-variate context. It also appears to be more coherent with the classical definition of meromorphic functions as holomorphic functions from U to ℙ1ℂ given the role played by analytic continuation. To compute the value at λ = 0 in (1.4), or more generally in (1.5), one can profit (locally, about each point z ∈ U , in particular about each z = α ∈ W ) from the formal local Bernstein–Sato equation for f , identified here with its germ in Oℂ,z , (1.6)

Qz (λ, ζ, ∂)[f λ+1 ] = Qz (λ, ζ, ∂)[f ⊗ f λ ] = bz (λ) f λ ,

where bz ∈ ℂ[λ] is a polynomial with strictly negative rational roots, and Qz is a differential operator in ∂ whose coefficients are polynomial in λ and analytic in ζ in a neighborhood of z. Here the differential operator ∂ = ∂/∂ζ acts on f ⊗ f λ according to Leibniz’s rule in Oℂ,z . Furthermore, ∂(f λ ) = λ(f  /f ) ⊗ f λ belongs to MU,z [λ] ⊗ f λ , where MU denotes the sheaf of meromorphic functions on U . Any element in OU,z ∂ acts then from OU,z ⊗ f λ to MU,z [λ] ⊗ f λ . In particular ∂(f λ+1 ) = ∂(f ⊗ f λ ) = (λ + 1) f  ⊗ f λ . The monic polynomial bz , which is minimal among all b ∈ ℂ[λ] involved in such formal identity, is called the local Bernstein– Sato polynomial of f at z (or of the germ fz ∈ Oℂ,z ). Observe that the existence of formal equation (1.6) about z = α ∈ W follows trivially from the fact that f

1.1. AN ANALYTIC APPROACH

3

can be considered there as f (ζ) = (ζ − α)μα (f ) (up to some local biholomorphism about α) with μα (f ) ∈ ℕ∗ and that we have for any α ∈ ℂ, q1 , q2 ∈ ℤ, the formal relations (1.7) 

1 |ζ − α|2λ (ζ − α)q1 (ζ − α)q2 dζ = ∂ |ζ − α|2λ (ζ − α)q1 +1 (ζ − α)q2 , λ + q1 + 1 

1 |ζ − α|2λ (ζ − α)q1 (ζ − α)q2 dζ = ∂ |ζ − α|2λ (ζ − α)q1 (ζ − α)q2 +1 . λ + q2 + 1 ¯ ζ, −∂/∂ζ), then for any test function ϕ, whose is support If Q∗z (λ, ζ, ∂/∂ζ) = Qz (λ, sufficiently close to z, one has that

1  dζ ∧ dζ  λ+1 (1.8) h|U\W , ϕ = f λf h Q∗z (λ, ζ, ∂/∂ζ)(ϕ) = ··· b(λ) U 2i λ=0    −1 M

1 ∧ dζ dζ 2λ M = M −1 |f | f h Q∗z (λ + j, ζ, ∂/∂ζ) (ϕ) 2i j=0 b(λ + j) U j=0 1 = M −1 j=0

λ=0

 b(j)

f U

M

h

−1 M j=0

dζ ∧ dζ , Q∗z (j, ζ, ∂/∂ζ) (ϕ) 2i

where M ∈ ℕ∗ and satisfies M ≥ maxα (max(0, −θα (h)/μα (f ))). Here maxα denotes the maximum with respect to α ∈ W ∩ supp ϕ. Let us point out what will be the main advantage of the representation (1.8) for the standard extension h|U\W if one compares it to (1.5). If 𝕂 is a field of characteristic 0 and p ∈ 𝕂[X], then there is an algebraic formal global Bernstein–Sato equation (1.9)

Q(λ, X, ∂/∂X)[pλ+1 ] = Q(λ, X, ∂/∂X)[p ⊗ pλ ] = b(λ) pλ .

In (1.9) the Weyl algebra 𝕂 X, ∂/∂X acts from 𝕂[X] ⊗ pλ to 𝕂(X) ⊗ pλ according to Leibniz’s rule and (∂/∂X)(pλ ) = λ (p (X)/p(X)) ⊗ pλ , while b ∈ 𝕂[λ]. The polynomials b ∈ 𝕂[λ] involved in the formal identity (1.9) define a principal ideal, whose monic generator is called the global Bernstein–Sato polynomial of p ∈ 𝕂[X] [Be72], see also [Bor87, V.1.16.4]. Whenever 𝕂 embeds to ℂ, such minimal polynomial b belongs to ℚ[λ] and factorizes into a product of first order factors X + γ, with γ ∈ ℚ+ since b is the least common multiple of all local Bernstein–Sato polynomials bα , α ∈ p−1 ({0}) [BiM98]. Using the global Bernstein–Sato equation (1.9) to represent h|U\W then allows us to keep track of some algebraic, or even arithmetic, information. Consider, for example, the case when U = ℂ, h = H = g/p ∈ ℚ(X), and f = p ∈ ℤ[X] with coprime coefficients. It means that Q and b in (1.9) can be taken with integer coefficients. In this case, using (1.9), one expresses the action of the duality bracket as a polynomial in z with explicit convergent integrals over ℂ as coefficients   PV[H] (1 + ζz)N  1 g(ζ) (1 + ζz)N dζ ∧ dζ = lim , 2 N +1 2π (1 + |ζ| ) 2i →0+ 2π |p|≥ p(ζ) (1 + |ζ|2 )N +1 

N dζ ∧ dζ  g(ζ) (1 + ζz) 1 |p(ζ)|2λ = 2π ℂ p(ζ) (1 + |ζ|2 )N +1 2i λ=0 ¯ X, −∂/∂X), then for N ∈ ℕ large enough and z ∈ ℂ. If Q∗ (λ, X, ∂/∂X) = Q(λ, such an expression is the coefficient a0 in the Laurent expansion with respect to λ

4

1. RESIDUE CALCULUS IN ONE VARIABLE

about the origin of  (1 + ζz)N dζ ∧ dζ 1 (1.10) λ −→ pλ pλ−1 g(ζ) 2π ℂ (1 + |ζ|2 )N +1 2i  1 1 (1 + ζz)N dζ ∧ dζ = pλ pλ Q∗ (λ − 1, ζ, ∂/∂ζ) g(ζ) 2π b(λ − 1) ℂ (1 + |ζ|2 )N +1 2i  λ log |p(ζ)|2 N dζ ∧ dζ 1 e (1 + ζz) = . Q∗ (λ − 1, ζ, −∂/∂ζ) g(ζ) b(λ − 1) ℂ 2π (1 + |ζ|2 )N +1 2i Convergent integrals of the form   dζ¯ ∧ dζ 1 ζ¯ν ν (1.11) (log |p(ζ)|)  2π ℂ (1 + |ζ|2 )ν 2i  ∞  2π

ρν  +1  1 (log |p(ρeiθ |)ν e−iν θ dθ dρ = 2π 0 (1 + ρ2 )ν  0 (with ν, ν  , ν  ∈ ℕ, ν  − 2ν  < −2), which appear (once multiplied by rational numbers) as coefficients of the polynomial expression in z defined by (1.10), are indeed of analytic nature. Nevertheless, one could recognize in the particular case where ν = 1, ν  = 0, and ν  > 2 that the integral expression (1.11) involves the Ronkin function  2π 1 + ρ ∈ ℝ → log |p(ρeiθ )| dθ, 2π 0 whose value at ρ = 1 is the so-called Mahler measure of the polynomial p; see also §D.5.2. Mahler measure is a marker for the arithmetic complexity of the algebraic hypersurface p−1 ({0}) considered here as an algebraic subvariety of the affine algebraic variety 𝔸ℚ . 1.1.2. Residue of a (1, 0) meromorphic form. Let h be a meromorphic function on a simply connected domain U , let α be a pole of h, and let α > 0 be small enough so that D(α, α ) ⊂ U . Then the local residue of h(ζ)dζ at the pole α is, by definition, the quantity  1 h(ζ)dζ. (1.12) Res [h(ζ)dζ, α] := 2πi |ζ−α|=α Denoting by γ a continuous loop in U which does not contain any pole of h, and  dζ writing iα (γ) = (1/2πi) for the index of γ about α, we have the classical γ ζ −α residue formula   1 (1.13) h(ζ)dζ = iα (γ) Res [h(ζ)dζ, α]. 2πi γ α∈U α pole of h

Indeed, from the homological equivalence  γ∼ iα (γ)∂D(α, α ) = α∈U

α pole of h

(where ∂D(α, α ) has the usual orientation), it follows that    h(ζ)dζ = iα (γ) h(ζ)dζ. γ

α∈U

α pole of h

|ζ−α|=α

1.1. AN ANALYTIC APPROACH

Since

5

 h(ζ)dζ = 2πiRes [h(ζ)dζ, α] , |ζ−α|=α

we recover (1.13). It is also interesting to point out that the expression Res [h(ζ)dζ, α] does not seem to have any true connection with the integration. Besides the fact that α has to be taken small enough, the choice of α is irrelevant in (1.12). We just use the integration symbol to materialize the duality of homology-cohomology. More precisely, if c is a cycle (i.e., a smooth chain without boundary) and ω is a cocycle (i.e., a C 1 closed differential form), then the duality bracket c, ω is expressed by  c, ω := ω . c

Before going on in this brief expository survey of the theory, let us recall how one can classically perform the computation of residues. Let D be a disk centered at α, let h be a holomorphic function in D \ {α} which has a simple pole at α, i.e., h(ζ) =

1 f (ζ) ζ −α

with f holomorphic  in a neighborhood of α, and let f (α) = 0. The Taylor expansion of f (ζ), f (ζ) = k≥0 ak (ζ − α)k gives us a0 = f (α) for the coefficient of 1/(ζ − α) in the Laurent expansion of h. If α is a pole of h of order ν, then we have the formula,

∂ ν−1

 1 × . (ζ − α)ν h(ζ) (1.14) Res [h(ζ)dζ, α] = (ν − 1)! ∂ζ ζ=α This formula can be somehow misleading since it creates the impression that division by the factor (ν − 1)! happens to be integrated within the computation. This could be a stumbling block when one works in fields of positive characteristic (as we will do at some point). It is much better to forget about (1.14) and just interpret the residue of hdζ at α as the Laurent coefficient a−1 in the Laurent development of h about α. Note anyway that computing directly the Laurent development of h about α is usually more efficient from the computational point of view than using (1.14). 1.1.3. Jordan’s lemma and Mellin transform. We describe in this section some very classical applications of residue calculus in a global context that we will need later on, and we refer the reader to [BG] for complex analysis prerequisites. Let h be a nonconstant meromorphic function in an open set U , and let ∂K be the oriented boundary of a compact set K ⊂ U , where |∂K| denotes its support. If h has no poles on ∂K and a is a complex number outside h(|∂K|), then for any holomorphic function f in U , the integral  h (ζ) 1 dζ f (ζ) 2πi ∂K h(ζ) − a equals the difference between the sum of the values of f at all zeroes of h − a in K (counted with their multiplicities) and the sum of the values of f at all poles of h − a, that is, all poles of h in K (counted also with their orders). In particular, if

6

1. RESIDUE CALCULUS IN ONE VARIABLE

f (ζ) = 1, then we have 1 2πi

 ∂K

h (ζ) dζ = Nh+ (a) − Nh− (a), h(ζ) − a

where Nh+ (a) is the sum of the multiplicities of the distinct roots of the equation h(z) − a = 0 contained in K, and Nh− (a) = Nh− (0) is the sum of the orders of the poles of h − a contained in K. Let f be a continuous function on τ + iℝ, with τ ∈ ℝ. We set   τ +iρ −ζ f (ζ)t dζ := lim f (ζ)t−ζ dζ PV ρ→∞

τ +iℝ

τ −iρ

for any t > 0 for which the limit exists. Let us recall the classical Jordan lemma. Lemma 1.1 (Jordan’s lemma in a half-plane). Let f be a continuous function on τ +iℝ, where τ ∈ ℝ. Suppose that f can be extended to a meromorphic function, denoted also by f , on the half-plane Re ζ < τ . that is also continuous up to the boundary Re ζ = τ . Furthermore, we assume that this extension f satisfies the following condition: max

lim

n→∞ {ζ : |ζ−τ |=ρn ,Re ζ τ0 . We focus here on some properties of the function θ (essentially related to its behavior near the origin) that are deduced from the study of its Mellin transform Mθ . The Fourier inversion formula provides one way to recover the function θ from the knowledge of its Mellin transform Mθ . Let τ be a real number, τ > τ0 , such that the function ω → f (ω) = Mθ (τ + iω) belongs to the space L2 (ℝ). Changing the variable t = es , we can write  +∞ f (ω) = (τ + iω) eiωs eτ s θ(es )ds . −∞

Then the function ω → f (ω)/(τ + iω) appears as the Fourier transform of the function s → eτ s θ(es ), belonging to the space L1 (ℝ). The Fourier transform inversion

8

1. RESIDUE CALCULUS IN ONE VARIABLE

formula applied to the function ω → f (ω)/(τ + iω), belonging to the space L1 (ℝ), implies that the function s → eτ s θ(es ) is continuous in ℝ and that one has  ∞ Mθ (τ + iω) −isω 1 τs s e dω, e θ(e ) = 2π −∞ τ + iω for any s ∈ ℝ and for any τ > τ0 . For any real number τ > τ0 , the last identity is equivalent to  Mθ (ζ) −ζ 1 t dζ . (1.15) θ(t) = 2iπ τ +iℝ λ The formula (1.15) is known as the Mellin transform inversion formula. Proposition 1.2. Assume that there exist two constants C > 0 and η > 0 such that the Mellin transform Mθ extends to the holomorphic function f in the half-plane Re λ > −η and −η/2 ≤ Re λ ≤ τ0 + η =⇒ |f (λ)| ≤ C,  |f (τ + iω)| η dω < +∞. − ≤ τ ≤ τ0 + η =⇒ 2 1 + |ω| ℝ Then the function θ extends to [0, ∞[ continuously as θ(0) = f (0). Proof. Let t be a real strictly positive number. The function λ → f (λ)t−λ /λ is meromorphic in the half-plane Re λ > −η and has a unique simple pole at the origin. The residue of λ → f (λ)t−λ /λ at the origin is equal to f (0). Mellin’s inversion formula implies for τ = τ0 + η that  f (ζ) −ζ 1 t dζ = θ(t). 2iπ τ +iℝ ζ Note that the second hypothesis implies the convergence of the integral and the validity of the Mellin inversion formula. One can use the residue theorem as follows. Let Cρ be the loop consisting of the concatenation of the paths Cρ,1 = [τ −iρ, τ +iρ], Cρ,2 = [τ + iρ, −η/2 + iρ], Cρ,3 = [−η/2 + iρ, −η/2 − iρ], Cρ,4 = [−η/2 − iρ, τ − iρ] (for ρ > 0). For t > 0 we have that  1 f (ζ)t−ζ dζ = f (0) . 2iπ Cρ ζ One can see easily that the integrals on Cρ,2 and Cρ,4 of the differential form f (λ)t−λ dλ/λ tend to zero when ρ tends to infinity. The integral along Cρ,1 tends to θ(t) when ρ tends to +∞. Therefore, one has  f (ζ) −ζ 1 t dζ + f (0) . θ(t) = 2iπ − η2 +iℝ ζ Since the integral in the formula above is bounded by (constant) tη/2 , we have the result we wanted to prove.  In the same vein, we can obtain an asymptotic expansion for θ when t tends to 0+ . The precision of the expansion increases as τ moves to the left.

1.1. AN ANALYTIC APPROACH

9

Proposition 1.3. Assume that there exist three constants C > 0, R > 0, and η > 0 such that the Mellin transform Mθ extends to a holomorphic function f in the half-plane Re λ > −η and −η/2 ≤ Re λ , |Im λ| ≥ T =⇒ |f (λ)| ≤ C  η |f (τ + iω)| dω < +∞. − ≤ τ ≤ τ0 + η =⇒ 2 1 + |ω| |ω|≥R Then, for all τ ∈ ] − η/2, τ0 ] such that f has no poles on τ + iℝ,   dζ 1 (1.16) θ(t) = Res[t−ζ f (ζ)dζ/ζ, α] + f (ζ)t−ζ 2iπ ζ τ +iℝ α pole of f Re α>τ

with



(1.17)



lim

t→0

f (ζ)t−ζ dζ/ζ

τ +iℝ

Res[t−ζ f (ζ)dζ/ζ, α]

= 0.

α pole of f

Re α>τ

The following two propositions involving the Gamma function naturally complete this subsection devoted to the Mellin transform and its inversion. Proposition 1.4. Let t > 0, let β be a complex number such that Re β > 0, and let 0 < τ < Re β. Then, we have the formula  1 Γ(ζ)Γ(β − ζ)t−ζ dζ. (1.18) (1 + t)−β = 2iπΓ(β) τ +iℝ Proof. To prove (1.18), we notice that the function ζ → Γβ (ζ) = Γ(ζ)Γ(β − ζ) is a meromorphic function in ℂ, with poles are 0, −1, −2, . . . and β, β + 1, . . . . On the other hand, Stirling’s formula √ Γ(ζ) ∼ 2πe−ζ ζ ζ−1/2 ensures the rapid decrease of Γβ at infinity along any vertical line in the complex plane. Let n be a strictly positive integer and let η ∈ ]0, 1/2[ such that Γβ has no poles on the line −(n + η) + iℝ. Using Cauchy’s formula as in the proof of Proposition 1.2 implies that   1 1 Γ(ζ)Γ(β − ζ)t−ζ dζ = Γ(ζ)Γ(β − ζ)t−ζ dζ 2iπΓ(β) τ +iℝ 2iπΓ(β) −n−η+iℝ n  Res[Γ(ζ)Γ(β − ζ)t−ζ dζ, −k] + 1 = 2iπΓ(β)

k=0

 −n−η+iℝ

+1+

Γ(ζ)Γ(β − ζ)t−ζ dt

n  (−β)(−β − 1) · · · (−β − k + 1)tk k=1

k!

.

10

1. RESIDUE CALCULUS IN ONE VARIABLE

As we notice immediately Rn (t) = 1 +

n  (−β)(−β − 1) · · · (−β − k + 1)tk

k!

k=1

represents the principal part of the Taylor development at order n for the function t → (1 + t)−β about t = 0. If for t > 0 we denote by  1 Ψ(t) = Γ(ζ)Γ(β − ζ)t−ζ dζ, 2iπΓ(β) τ +iℝ then it follows from    1   Γ(ζ)Γ(β − ζ)t−ζ dζ  ≤ Cn tn+η  2iπΓ(β) −n−η+iℝ that +

when t tends to 0 for any n ∈ ℕ∗ , that

Ψ(t) − Rn (t) = o(tn ), for any n ∈ ℕ∗ . Using the functional equation of Γ, one has, Γ(1 − η + it) Γ(−n − η + it) =  n (−k − η + it)

(1.19)

k=0

and Γ(β + n + η − it) = Γ(β + η − it)

(1.20)

n−1 

(β + k + η − it).

k=0

For any  > 0, there exists T () in ℕ∗ such that, for any (t, k) ∈ ℝ × ℕ satisfying |t| + k ≥ T (), one has  η − it + β + k     ≤ (1 + ).  η − it + k Using (1.19) and (1.20), one obtains, as soon as n > T (), the estimate  |Γβ (−n − η + it)|dt |t|≥T ()

≤ (1 + )n−1

 |t|≥T ()

|Γ(β + n − it)Γ(1 − η + it)| dt ≤ C()(1 + )n−1 . |η − it + n|

Therefore, one has, for n > T (), a more precise estimate    1   Γ(ζ)Γ(β − ζ)t−ζ dζ  ≤ C()tn+η (1 + )n ,  2iπΓ(β) −n−η+iℝ which allows us to conclude that    1   Γ(ζ)Γ(β − ζ)t−ζ dζ  = 0 lim  n →∞ 2iπΓ(β) −n−η+iℝ whenever t(1 + ) < 1. Therefore, for any such t, one has lim Rn (t) = Ψ(t),

n→∞

and the proposition is proved when 0 < t < 1. To prove the result when t > 1, we just need to move the contour of integration to the right instead of moving it to the left. In order to get the result for t = 1, it is enough to apply Lebesgue’s theorem and verify that the function Ψ is continuous at t = 1. 

1.2. THE GEOMETRIC POINT OF A VIEW

11

Proposition 1.5. Let m ∈ ℕ so that m ≥ 2, let t1 , . . . , tm be m strictly positive numbers, let β be a complex number with positive real part Re β > 0, and let τ1 , . . . , τm−1 be m − 1 strictly positive numbers satisfying τ1 + · · · + τm−1 < Re β. Then one has (1.21) (t1 + · · · + tm )−β   1 −ζm−1 ζ ∗ 1 = · · · Γ∗m (ζ)t−ζ · · · tm−1 tm dζ1 · · · dζm−1 , 1 m−1 (2iπ) Γ(β) τ1 +iℝ τm−1 +iℝ where Γ∗m (ζ) = Γ(ζ1 ) · · · Γ(ζm−1 )Γ(β − ζ1 − · · · − ζm−1 ) , ζ ∗ =

m−1 

ζk − β .

k=1

Proof. If t1 , t2 are two strictly positive numbers, then for any complex number β such that Re β > 0 and any τ satisfying 0 < τ < Re β, Proposition 1.4 implies that  1 t1 −β −β s−β −β (t1 + t2 ) = t2 1 + = Γ(s)Γ(β − s)t−s ds . 1 t2 t2 2iπΓ(β) τ +iℝ Proposition 1.5 is then proved when m = 2. This procedure can be iterated, which leads to (1.21).  Remark 1.6. Formula (1.21) is a continuous version of the binomial formula (t1 + · · · + tm )k =

k  κ1 +···+κm−1

k! tκ1 1 · · · tκmm , κ ! · · · κ ! 1 m =m

k ∈ ℕ∗ , t1 , . . . , tm > 0,

when the positive integer exponent k is replaced by the complex exponent with strictly negative real part −β. It transforms the additive operation between the −ζm−1 ζ ∗ 1 · · · tm−1 tm in the intetj in (t1 + · · · + tm )−β into a multiplicative one in t−ζ 1 grand. Multi-dimensional residue theory is intrinsically based on this multiplicative λm ). On the other hand, scheme (think about the formal expression f1λ1 ⊗ · · · ⊗ fm the geometric point of view naturally appeals to an additive scheme. For example, if f = (f1 , . . . , fm ) is a section of the trivial bundle ℂm equipped with its canonical hermitian over some complex manifold X , then the additive ex metric 2 |f | plays the central role, instead of the multiplicative one pression f 2 = m j 1 λm . This observation explains why Proposition 1.5 plays an interesting f1λ1 ⊗ · · · ⊗ fm algebraic role and therefore will have consequences for us later, despite its analytic nature. 1.2. The geometric point of a view In this section, we start from an analytic point of view. We analyze how the integration current on the analytic cycle [V (F )], where F = (f1 , . . . , fm ) is a vector of holomorphic functions in some domain U in ℂ, can be factorized through residue currents and the differential forms dfj . Such factorization will be a crucial operational tool in multi-variate residue calculus. Then we slightly change our point of view and adopt a more geometric one. We think of U as a connected open subset in a Riemann surface X . In this case, we consider the collection F = (f1 , . . . , fm ) to be a holomorphic section of the trivial hermitian m-bundle (ℂm , | |) → X over X .

12

1. RESIDUE CALCULUS IN ONE VARIABLE

This vector bundle is assumed to be equipped with its canonical metric, inherited from the canonical scalar product on ℂm (Z, W ) ∈ ℂm × ℂm −→ Z · W =

m 

Zj Wj .

j=1

We need also to introduce the ℂ-bilinear form (1.22)

(Z, W ) ∈ ℂm × ℂm −→ ⟪Z, W ⟫ =

m 

Zj Wj

j=1 m corresponding to the duality bracket when one of the two copies m of ℂ 2 is identified 2 with its dual. The necessity of the introduction of F  = j=1 |fj | within this more invariant frame becomes more evident in the present section.

1.2.1. Factorization of integration currents through residue currents. It is well known that the field of meromorphic functions defined in a domain U ⊂ ℂ, equipped with the natural addition and multiplication of functions is the quotient field of the ring H(U ) of holomorphic functions in U . Consider a meromorphic function h = g/f , where f ≡ 0 and g are holomorphic functions in some domain U ⊂ ℂ, and let α be a zero of f . We denote by χ−1 any bi-holomorphic transform between a neighborhood Dα of α in the ζ-plane and a disk Δ centered at χ−1 (α) = 0 in the w-plane such that its pullback χ∗ (f ) equals χ∗ (f )(w) = f (χ(w)) = wν , where ν = μα (f ) is the multiplicity of α as a zero of f . Its inverse is denoted by χ. It is clear that for any  > 0 sufficiently small, {ζ ∈ ℂ : |f |2 = } ∩ Dα = χ({|w|2 = }) is the support of a cycle homologous to some circular loop ∂D(α, α ) about α. Therefore we have   1 1 g(ζ) 1 dζ = Res[h(ζ)dζ, α] = g(ζ)f (ζ)dζ 2iπ {|f |2 =} f (ζ) 2iπ  {|f |2 =} 1 1

= lim+ (1.23) g(ζ)f (ζ)dζ . 2iπ  {|f |2 =} →0 Now let F ∈ (H(U )\{0})m , where U is a domain in ℂ. Let α be a zero of F , that is a common zero of the entries of F , and let ν = μα (F ) = να be the multiplicity of α as a zero of F . That is, να is the minimum of the multiplicities να,1 , . . . , να,m of α respectively as a zero of the entries of F . Since V (F ) = F −1 ({(0, . . . , 0)}) is a discrete subset of U , Weierstrass’s theorem implies that there exists f ∈ H(U ) such that f −1 ({0}) = V (F ) and μα (f ) = να for any α ∈ V (F ) = V (f ). The analytic cycle associated with the proper ideal generated by the entries of F in H(U ) is  div(f ) = να {α}. α∈V (F )

It is also known as the intersection cycle div(f1 ) • · · · • div(fm ) if one refers to the entries fj of F in H(U ) \ {0}, but one needs to observe that our situation is a situation of improper intersection as soon as m > 1. Here {α} denotes the

1.2. THE GEOMETRIC POINT OF A VIEW

13

complex number α considered as a 0-cycle (i.e., a Weil divisor1 in the one variable context) in U . The integration current on this cycle is the (1, 1)-current in U which associates to any test form ϕ ∈ D(U ) = D(0,0) (U ) the value  να ϕ(α), [F = 0], ϕ = [div(f )], ϕ = α

/ V (F ). More information regarding currents (that is, differwhere να = 0 when α ∈ ential forms with distributions as coefficients) is to be found in §B.2.1. In particular, §C.2.3 is devoted to the notion of integration current. For any α ∈ V (F ) and any j in {1, . . . , m}, we may write the jth entry of F as (1.24)

fj (ζ) = (ζ − α)να qF,α,j (ζ).

where qF,α = (qF,α,1 , . . . , qF,α,m ) is a vector of holomorphic functions in U such that at least one of its coordinates does not vanish at α. Alternatively, we may also write (1.25)

fj (ζ) = f (ζ) uj (ζ),

where u1 , . . . , um ∈ H(U ) and the set of zeroes of u = (u1 , . . . , um ) is empty, i.e., V (u) = ∅. We fix α ∈ V (F ) for the moment and set ν = να , q = qF,α for the sake of simplicity. We write F 2 =

m 

fj fj , q2 =

j=1

m 

qj qj ,

j=1

so that F 2 = |(ζ − α)|2ν q2 . The function F 2 has an isolated zero at the point α, while the function q2 is strictly positive in a neighborhood of the point α. We define formally two differential forms of bidegree (1,0) in U by setting ∂ log(F 2 ) = F −2 ∂F 2 , ∂ log(q2 ) = q−2 ∂q2 . Observe that the first one is smooth in U \ V (F ), while the second one is smooth in U \ V (q). In particular, it means that it is smooth in a neighborhood of the point α in U . Note also that νdζ ∂ log(F 2 ) = + ∂ log q2 . ζ −a Define now a real change of variables by setting w = (ζ − α)q1/ν . This induces a real diffeomorphism ζ → w between some open neighborhood Dα of α and some open disk Δ about the origin in the w-plane. Recall that we denote by χ its inverse diffeomorphism. That is, (1.26)

ζ = χ(w) = α + wA(w) , w ∈ Δ,

where A(w) = q(χ(w))−1/ν . We have dζ = A(w) dw + w dA(w) , dζ = A(w) dw + w dA(w),  is, a formal sum να {α} of points α in U , where each of them is equipped with a multiplicity να ∈ ℤ. This formal sum is locally finite in the following sense: for any compact K in U , card{α ∈ K : να = 0} < ∞. The divisor is called effective when all να are positive. The support of a Weil divisor is the set of points α where να = 0. For more details, see §D.3.1. 1 That

14

1. RESIDUE CALCULUS IN ONE VARIABLE

and therefore the pullback of the form ∂ log F 2 equals χ∗ (∂ log F 2 ) =

ν(A(w) dw + w dA(w)) νdw + χ∗ (∂ log q2 ) = + ω, w A(w) w

where ω is a smooth form in Δ. For any test function in the ℂ vector space D(Dα ) = D (0,0) (Dα ) of smooth complex-valued functions with compact support in Dα and for any  > 0 small enough, let  1 θ(ϕ; ) = ϕ(ζ, ζ)∂ log F (ζ)2 2iπ F 2 =  1 = (ϕ ◦ χ)(w, w)χ∗ (∂ log F 2 ). 2iπ |w|2ν = Following previous remarks, we have lim θ(ϕ; ) = ν ϕ(α) .

→0+

It means that for any test function ϕ ∈ D(Dα ),  1 ϕ(ζ, ζ)∂F 2 ν ϕ(α) = lim+ F 2 →0 2iπ F 2 =  1 (1.27) = lim+ ϕ(ζ, ζ)∂F 2 . →0 2iπ F 2 = We state the following lemma. Lemma 1.7. Let F ∈ (H(U ) \ {0})m , where U is a domain in ℂ. For any test function ϕ ∈ D(0,0) (U ), we have  1 [F = 0] , ϕ = lim+ ϕ(ζ, ζ) ∂ log F 2 →0 2iπ F 2 =  τ ∂F 2 ∧ ∂F 2 (1.28) = lim+ ϕ(ζ, ζ) . F 2 (F 2 + τ )2 τ →0 2iπ U Proof. The proof is rather simple. If ϕ is a test function, we can consider the function  1 (1.29)  → θ(ϕ; ) = ϕ(ζ, ζ)∂ log F 2 . 2iπ F 2 = It is clear that lim θ(ϕ; ) = 0

→0+

if the support of ϕ does not intersect V (F ). On the other hand, if the support of ϕ lies in Dα for some α ∈ V (F ), then one has from (1.27) that lim θ(ϕ; ) = να ϕ(α) = [F = 0] , ϕ.

→0+

To recover the second formula in (1.28), we just notice that, for any integer κ ≥ 1, one has, for τ > 0  ∞ tκ−1 1 (1.30) τ dt = . p+1 (t + τ ) κ 0

1.2. THE GEOMETRIC POINT OF A VIEW

15

Moreover, if σ is a locally integrable function on [0, ∞[, which is continuous at the origin and compactly supported, then, it follows from (1.30), that  ∞ κ−1 t σ(t) dt . σ(0) = lim+ τ (t + τ )κ+1 τ →0 0 Apply this for κ = 1 and σ(·) = θ(ϕ; ·) defined in (1.29), where ϕ is a test function in U whose support lies in an arbitrary small neighborhood of α ∈ V (F ). We get  ∞ θ(ϕ; t) dt. [F = 0] , ϕ = θ(ϕ; 0) = lim+ τ (t + τ )2 τ →0 0 Fubini’s theorem implies that   ∞ θ(ϕ; t) ∂F 2 ∧ ∂F 2 τ dt = ϕ(ζ, ζ) . (1.31) τ 2 (t + τ ) 2iπ Dα F 2 (τ + F 2 )2 0 Let us give a detailed proof of the last claim. Using the change of variables (1.26), that is ζ = α + wq(ζ)−1/ν , w ∈ Δ, ζ ∈ Dα , where ν = να , we have ∂F 2 = ν |w|2ν−2 w dw , ∂F 2 = ν |w|2ν−2 w dw. Thus, the last integral in (1.28) becomes   ∂F 2 ∧ ∂F 2 (ϕ ◦ χ)(w, w)|w|2(2ν−2)+1 dw ∧ dw τ τ ν2 ϕ(ζ, ζ) = 2iπ U F 2 (F 2 + τ )2 2iπ Δ |w|2ν (|w|2ν + τ )2 2  dw ∧ dw τν = (ϕ ◦ χ)(w, w)|w|2ν−3 . 2iπ Δ (|w|2ν + τ )2 Passing to the polar coordinates w = r 1/ν eiξ , r > 0, ξ ∈ [0, 2π], so that |w|2ν = r, and remarking that i  1 2(1−ν) iξ r 2ν dw ∧ dw = e dr ∧ dξ, ν we imply that  dw ∧ dw τ ν2 (ϕ ◦ χ)(w, w)|w|2ν−3 2iπ D (|w|2ν + τ )2 1 2ν−3+2−2ν  +∞  2ν  2π  1 iξ  iξ r τν = (ϕ ◦ χ) r 2ν e ie dξ dr 2iπ 0 (r + τ )2 0  +∞ θ(ϕ; t) dt. =τ (t + τ )2 0 This proves (1.31), and so we have proved the second formula in (1.28) when the support of ϕ lies in an arbitrary small neighborhood of some α ∈ V (F ). If the support of ϕ does not intersect V (F ), then it is clear that  τ ∂F 2 ∧ ∂F 2 ϕ = 0 = [F = 0] , ϕ. lim+ F 2 (F 2 + τ )2 τ →0 2iπ U The proof of the lemma is complete.



16

1. RESIDUE CALCULUS IN ONE VARIABLE

Inspired by the result stated in Lemma 1.7, one can associate to a vector F ) of F ∈ (H(U ) \ {0})m , where U is a domain in ℂ, a vector RF = (R1F , . . . , Rm (0, 1)-currents in U called residue currents, whose action can be expressed in terms of F . This is a vector of continuous linear functionals, namely (0, 1)-currents, on the space of smooth (1, 0) differential forms with compact support in U , which is denoted by D(1,0) (U ), while its dual is denoted by  D(0,1) (U ). The next proposition provides such construction. Proposition 1.8. Let U be a domain in ℂ, let F ∈ (H(U ) \ {0})m with entries f1 , . . . , fm , and let ϕ ∈ D (1,0) (U ). For each j ∈ {1, . . . , m}, the limit  1 1 (1.32) lim+ fj (ζ)ϕ(ζ, ζ) →0 2iπ  F 2 = exists, and the map (1.33)

RjF

:ϕ∈D

(1,0)

1 1 (U ) −→ lim+ →0 2iπ 

 F 2 =

fj (ζ)ϕ(ζ, ζ)

defines a (0, 1)-current on D(1,0) (U ). Moreover, for any  ∈ D(0,0) (U ), the function  f j dF 2 λ F 2(λ−1) ∧ (ζ, ζ) dζ λ −→ 2iπ U F 2 extends from {λ ∈ ℂ : Re λ  1} as a meromorphic function in ℂ with (possible) poles in −ℚ+ , and one has (1.34) RjF , dζ

λ   f j dF 2 = F 2(λ−1) ∧ (ζ, ζ) dζ = [ RjF,λ , dζ]λ=0 2iπ U F 2 λ=0

∂ νF,α −1 q  1 F,α,j  (α). = (νF,α − 1)! ∂ζ qF,α 2 α∈V (F )

Proof. Let ϕ ∈ D(1,0) (U ) and take j = 1. We compute first the Mellin transform Mθ1 of the function  1 1 θ1 (ϕ; ) :  → f1 (ζ) ϕ(ζ, ζ). 2iπ  F 2 = When the support of ϕ lies in some arbitrary small neighborhood of a zero α of F 2 , the computation of this Mellin transform can be performed using Fubini’s theorem and a local change of coordinates as before. It is always possible to reduce our problem to this situation since  1 1 lim+ f1 (ζ)ϕ(ζ, ζ) = 0 →0 2iπ  F 2 = when the support of ϕ does not intersect V (F ). For Re λ  1 large enough, one has  f dF 2 λ Mθ1 (λ) = F 2(λ−1) 1 ∧ ϕ(ζ, ζ). 2iπ U F 2 It is clear from the definition that the function Mθ1 is holomorphic in {Re λ > 1/2}. We assume from now on that the support of ϕ lies in Dα for some α ∈ V (F ). Suppose also that ν = να , Dα and Δ have been defined correspondingly in (1.24) or according to (1.26). We want to show that the function Mθ1 can be analytically

1.2. THE GEOMETRIC POINT OF A VIEW

17

continued as a meromorphic function of λ to the whole complex plane ℂ with (possible) poles in −ℚ+ and hence is holomorphic in the half-plane Re λ > −τ for some τ ∈ ℚ+ . This is easily done after one notices that F 2 = |ζ − α|2ν × q2 , where q2 = qα 2 is a smooth nonvanishing function in U . One then applies the Bernstein–Sato equation (1.6) for f = (ζ − α)ν . This will be the procedure one will systematically employ when dealing with several variables, because it opens a way to bypass the difficulties inherent in division thanks to repeated use of Stokes’s theorem (integration by parts really). Let us give here an elementary and explicit approach leading in particular to (1.34). We use again the local real change of variables w = (ζ − α)q1/ν = χ−1 (ζ) and write χ∗ ((q1 /q2 )ϕ) = ξ dw + η dw. Since d|w|2ν = ν|w|2ν−2 (w dw + w dw), we have  dF 2 λ Mθ1 (λ) = F 2(λ−1) f 1 ∧ ϕ(ζ, ζ) 2iπ Dα F 2  q

λ d|w|2ν 1 ∗ = |w|2ν(λ−1) w ν ∧ χ ϕ (w, w) 2iπ Δ |w|2ν q2  νλ = |w|2(νλ−ν−1) w ν (w dw + w dw) ∧ (ξ dw + η dw) 2iπ Δ  νλ = |w|2(νλ−ν−1) w ν (w ξ − w η) dw ∧ dw 2iπ Δ  

νλ ν(λ−1) νλ−1 = w w ξ dw ∧ dw − wν(λ−1)−1 w νλ η dw ∧ dw 2iπ Δ Δ = λ(I(λ) − J(λ)). The possible poles for the function  ν I : λ → wν(λ−1) wνλ−1 ξ dw ∧ dw 2iπ Δ are the points λ satisfying 2νλ − ν − 1 = −|1 − ν| − 2,  ∈ ℕ∗ , or equivalently, the points λ = (1 − )/ν,  ∈ ℕ∗ . In particular, there exists a simple pole at λ = 0. Similarly, the possible poles for the function  ν J : λ → wν(λ−1)−1 wνλ η dw ∧ dw 2iπ Δ are the points λ such that 2νλ − ν − 1 = −ν − 1 − 2,  ∈ ℕ∗ or, which is the same, the points λ = −/ν,  ∈ ℕ∗ . Therefore, the function Mθ1 can be continued as a meromorphic function having poles in the half-plane Re ζ ≤ −1/ν, and we have Mθ1 (0) = Res [I(λ)dλ, α]. We can even be more precise and get ∂ ν−1 q 1  1 2 (α) Mθ1 (0) = Res [I(λ)dλ, α] = (ν − 1)! ∂ζ q if ϕ = dζ. The proof of the proposition is complete.



Remark 1.9. When m = 1, the term with index α in the right-hand side of formula (1.34) becomes that of formula (1.14), where the meromorphic function h is replaced by the semimeromorphic2 one /f . When m > 1, one can rewrite 2 The terminology used here comes from the fact that the denominator f is holomorphic while the numerator  is smooth (more precisely here in D(U )).

18

1. RESIDUE CALCULUS IN ONE VARIABLE

(1.34) for j = 1, . . . , m as (1.35)

RjF =

uj f R . u2

 (f ) Here f ∈ H(U ) is such that div(f ) = α∈V (F ) νF,α {α}, Rf = R1 , while the holomorphic functions uj inU , satisfying V (u) = ∅, have been introduced in (1.25). m Also, note that u2 = j=1 |uj |2 . The equality (1.35) is understood in the sense of currents, that is, as an equality between distributions which appear as coefficients of differential forms. The key point here is that the action of Rf , and hence the action of all residue currents RjF , j = 1, . . . , m, involves only computations of holomorphic derivatives. Expressions depending only on the antiholomorphic coordinate ζ, though they are essential to keep track of positivity for F 2 ≥ 0 or q2 > 0, are treated as constants in such residue computations. Since positivity is a basic concept in multi-variate complex analysis, this remark will play a major role all through this monograph. Let us conclude this section with the fundamental observation summarizing the results obtained in Lemma 1.7 and Proposition 1.8. Proposition 1.10. Let F = (f1 , . . . , fm ) as in Proposition 1.8. The integration current on the intersection cycle [F = 0] = div(f1 ) • · · · • div(fm ) factorizes through the formula (1.36)

[F = 0] =

m 

RjF ∧ dfj .

j=1

Remark 1.11. For the sake of simplicity, we use the same notation for an analytic 0-cycle and the corresponding integration (1, 1)-current. The factorization formula (1.36) is known as Lelong–Poincar´e formula. It still holds in the more general geometric context where F is a section of a holomorphic bundle as in §1.2.5; see §C.2.4 (in particular Proposition C.32). F ) that was introduced in Proposition The vector of currents RF = (R1F , . . . , Rm 1.8 inherits a very important property: for any h ∈ H(U ) vanishing on V (F ), one has

(1.37)

∀ϕ ∈ D (1,0) (U ), RF , hϕ = 0.

This can be written in the abridged way hRF = 0 as ℂm -valued (0, 1)-current on U . Moreover, let h be a holomorphic function in U such that |h| ≤ CF . This inequality is equivalent to saying that h vanishes at any point α ∈ V (F ) with multiplicity at least equal to νF,α . It then follows from (1.34) that (1.38)

∀ϕ ∈ D (1,0) (U ), RF , hϕ = 0,

or, in an abridged way, hRF = 0 as ℂm -valued (0, 1)-current on U . In the particular case when m = 1 and F = (f ), the action of the current RF (also denoted then Rf ) on a test form ϕ ∈ D(1,0) (U ) is given as    2λ  1

1 1 1 ϕ |f | = ∧ϕ f ϕ = lim+ ∂ . lim+ 2iπ U f →0 2iπ  |f |2 = →0 2iπ |f |2 = f λ=0

1.2. THE GEOMETRIC POINT OF A VIEW

19

Stokes’s formula allows rewriting this action as   2λ  |f | 1 1 1

f ∂ϕ = − . Rf , ϕ = − lim+ ∧ ∂ϕ 2iπ U f →0 2iπ  |f |2 ≥ λ=0 If ϕ =  dζ, where  ∈ D(0,0) (U ), then one recognizes from (1.5) that ∂  1 . (1.39) Rf ,  dζ = − PV[1/f ], π ∂ζ Denote as P0f ∈  D(0,0) (U ) the (0, 0)-current on U acting on smooth (1, 1)-forms θ ∈ D1,1 (U ) as 1 P0f , θ dζ ∧ dζ = PV[1/f ], θ. π It follows from (1.39) that one has ∂(P0f ) = Rf

(1.40)

in the sense of currents in U . If we denote alternatively P0f as [1/f ], it makes sense to denote Rf as ∂(1/f ). Proposition 1.12 below extends from the case where m = 1 to the case where m ∈ ℕ∗ the definition of P0f = [1/f ]. Proposition 1.12. Let F = (f1 , . . . , fm ) ∈ (H(U ) \ {0})m , where U is a domain in ℂ and ϕ ∈ D1,1 (U ). For each j ∈ {1, . . . , m}, the limit  1 fj (ζ) (1.41) lim+ ϕ(ζ, ζ) →0 2iπ F 2 ≥ F (ζ)2 exists and the map (1.42)

F P0,j : ϕ ∈ D(1,1) (U ) −→ lim

→0+

1 2iπ

 F 2 ≥

fj (ζ) ϕ(ζ, ζ) F (ζ)2

defines a (0, 0)-current on U . Moreover, for any ϕ ∈ D(1,1) (U ), the function  1 λ −→ F 2(λ−1) fj ϕ 2iπ U extends from {λ ∈ ℂ; Re λ  1} as a meromorphic function with (possible) poles in −ℚ+ , and one has   1

F,λ F (1.43) P0,j , ϕ = F 2(λ−1) fj ϕ = [ P0,j , ϕ]λ=0 . 2iπ U λ=0 Proof. It is enough to take j = 1. If Supp(ϕ) ∩ V (F ) = ∅, then the function  1 λ ∈ ℂ −→ F 2(λ−1) f1 ϕ 2iπ U is entire and Lebesgue’s theorem implies that   1 f1 (ζ) f1 (ζ) 1 lim+ ϕ(ζ, ζ) = ϕ(ζ, ζ) 2 2iπ U F (ζ)2 →0 2iπ F 2 ≥ F (ζ)   1

= F 2(λ−1) f1 ϕ . 2iπ U λ=0

20

1. RESIDUE CALCULUS IN ONE VARIABLE

One can therefore assume that Supp(ϕ) ⊂ Dα , where α ∈ V (F ). From now on ν = να , Dα , χ, Δ, q = qα are taken as in the proof of Proposition 1.8, according to (1.24) and (1.26). For any λ ∈ ℂ with Re λ  1, one has then that   1 |χ−1 (ζ)|2νλ q1 (ζ) 1 (1.44) F 2(λ−1) f1 ϕ = ϕ(ζ) 2iπ U 2iπ Dα (χ−1 (ζ))ν q(ζ)2  |w|2νλ ∗ q 1 1 χ ϕ (w). = 2iπ Δ wν q2 On the other hand, for any ε > 0 sufficiently small one has that   1 f1 (ζ) 1 q1 (ζ) 1 ϕ(ζ) = ϕ(ζ) 2 −1 ν 2iπ F 2 ≥ F (ζ) 2iπ {ζ∈𝔻α : |(χ−1 (ζ))ν |2 ≥ε} (χ (ζ)) q(ζ)2  1 1 ∗ q1 = χ ϕ (w). 2iπ {w∈Δ : |wν |2 ≥ε} wν q2 It follows then from (1.44) and the Bernstein–Sato equation (1.6) with f = wν that the function  1 λ ∈ {λ ∈ ℂ : Re λ >> 1} −→ F 2(λ−1) f1 ϕ 2iπ U extends as a meromorphic function with rational poles in ] − ∞, −1/ν]. On the other hand, one can see using polar coordinates w = ρeiθ that 

1  |w|2νλ q

 1 1 ∗ q1 1 ∗ χ ϕ (w) = χ ϕ (w) . lim ε→0+ 2iπ {w∈Δ;|wν |2 ≥ε} w ν q2 2iπ Δ wν q2 λ=0 This concludes the proof of the proposition.



f Remark 1.13. It follows  from (1.35) and from the definition of P0 , where f ∈ H(U ) satisfies div(f ) = α∈V (F ) νF,α {α}, that

(1.45)

F = P0,j

uj Pf u2 0

in the sense of currents for any j = 1, . . . , m. Furthermore, the holomorphic functions uj ∈ H(U ) satisfying V (u) = ∅ have been introduced in (1.25). Corollary 1.14. For any j = 1, . . . , m and any ϕ ∈ D(1,0) (U ), the function  f 1 j (1.46) λ −→ F 2λ ∂ ∧ϕ 2iπ U F 2 extends from {λ; Re λ  1} as a meromorphic function in ℂ with (possible) poles in −ℚ+ . Its value ϕ ∈ D (1,0) (U ) −→ SjF , ϕ = [ SjF,λ , ϕ]λ=0 at λ = 0 defines the action of a current SjF such that (1.47)

¯ F = RF + S F , ∂P 0,j j j

which extends (1.40) to the case m > 1.

1.2. THE GEOMETRIC POINT OF A VIEW

21

Proof. It follows from Leibniz’s formula and Stokes’s theorem that for λ ∈ ℂ such that Re λ  1, 1 2iπ

 F 

2λ

U



f 1 fj j 2(λ−1) 2λ ∧ ϕ = ∧ϕ ∂ ∂(F  f ) − ∂F  j F 2 2iπ U F 2  

1 fj =− F 2(λ−1) f j ∂ϕ + ∂F 2λ ∧ ϕ . 2iπ U F 2 U

It follows from Proposition 1.12 that the function 1 λ ∈ {λ ∈ ℂ : Re λ  1} −→ − 2iπ

 F 2(λ−1) f j ∂ϕ

U

extends as a meromorphic function with poles in −ℚ+ , whose value at λ = 0 equals ∂PjF , ϕ. On the other hand, one has 1 2iπ

 ∂F 2λ ∧ U

fj ϕ = RjF,λ , ϕ; F 2

see (1.34). Such a function of λ extends also by Proposition 1.8 as a meromorphic function with poles in −ℚ+ , whose value at λ = 0 equals RjF , ϕ. Corollary 1.14 is proved.   Remark 1.15. Let f ∈ H(U ) satisfy div(f ) = α∈V (F ) νF,α {α} and let u1 , . . . , um in H(U ) be such that V (u) = ∅, that is, they have been chosen as in (1.25). It follows from (1.35), (1.45), and Corollary 1.14 that for any j = 1, . . . , m, SjF = ∂

(1.48)

u j P0f u2

in the sense of currents. 1.2.2. Residue currents and Lagrange’s interpolation formula. Suppose that U is a bounded domain in ℂ with piecewise C 1 -boundary and that f is a function holomorphic in U that extends continuously to K = U . Assume also that f = 0 on the boundary of U . That is, we assume that f = 0 on the support of the piecewise C 1 cycle ∂K equipped with the counterclockwise (positive) orientation. The function (z, ζ) ∈ U × K −→ bf (z, ζ) =

f (ζ) − f (z) ζ −z

belongs to C 1 (U × K) and is holomorphic in U × U as a function of two complex variables. According to the terminology used in the algebraic context (see Corollary 1.37 in §1.3.1), such a function bf is called a b´ezoutian attached to f . It is well known that any function h, holomorphic in U and continuous up to K, admits in U the Cauchy integral representation formula. What is less known is that this integral representation formula can be transformed in order to become Lagrange’s

22

1. RESIDUE CALCULUS IN ONE VARIABLE

interpolation formula (1.49)

 h(ζ) dζ 1 2πi ∂K ζ − z  

h(ζ) bf (z, ζ) dζ 1 + f (z) h(ζ) = 2πi ∂K f (ζ) ∂K f (ζ)(ζ − z)  1 

h(ζ)b (z, ζ)  h(ζ) f dζ, α + f (z) dζ Res = f (ζ) 2πi ∂K f (ζ)(ζ − z) {α∈U : f (α)=0}   1

 1 h(ζ) = ∂ , h(ζ) bf (z, ζ) dζ + f (z) dζ . f (ζ) 2iπ ∂K f (ζ)(ζ − z)

h(z) =

Formula (1.49) will play an essential role throughout this monograph. As a consequence, one gets the trace formula  1  (1.50) h(z) = ∂ , h(ζ) bf (z, ζ) dζ modulo f H(U ) f (ζ) for any h ∈ H(U ). Let us consider in this section, instead of f ∈ H(U ) ∩ C 1 (K), which does not vanish on the boundary ∂U , a vector F = (f1 , . . . , fm ) of nonidentically zero functions belonging to H(U ) ∩ C 1 (K) with set of common zeroes V (F ) ⊂ U . As a consequence V (F ) is finite. For any invertible element u of H(U ) ∩ C 1 (K) the divisor div(f ) of the function  (ζ − α)νF,α ∈ H(U ) ∈ C 1 (K) (1.51) f : ζ ∈ U −→ u(ζ) α∈V (F )

equals the intersection cycle div(f1 ) • · · · • div(fm ) in U . From now on we fix such a function f and rewrite F = (f1 , . . . , fm ) as F = (f u1 , . . . , f um ), where fj = f uj , for j = 1, . . . , m and u = (u1 , . . . , um ) is a vector of functions in H(U ) ∩ C 1 (K), whose components do not share any common zero in K = U . The corresponding b´ezoutian is then equal to bF = (bf1 , . . . , bfm ). Let also σj = uj /u2 ∈ C ∞ (U ) ∩ C 1 (K) for any j = 1, . . . , m with the respective vector map σ = (σ1 , . . . , σm ). If λ denotes a complex parameter with sufficiently large real part Re λ  1 and N ∈ ℕ, then one 1 defines a function ΦF,λ N ∈ C (U × K) as F (ζ)

N F,λ (z, ζ))N = 1 + (z − ζ) F (ζ)2λ bF (z, ζ) · ΦF,λ N (z, ζ) = (Φ F (ζ)2

N F (ζ) = 1 − F (ζ)2λ + F (ζ)2λ F (z) · F (ζ)2 |f (ζ)|2λ N (1.52) = 1 − u(ζ)2λ |f (ζ)|2λ + u(ζ)2λ ⟪σ(ζ), F (z)⟫ . f (ζ) The function (1.52) is holomorphic in its first variable z. Observe that for any N ∈ ℕ∗   ∂ ζ ΦF,λ F,λ N +1 (z, ζ) dζ = −(N + 1) ΦF,λ N (z, ζ) ∂ ⟪P0 (ζ), bF (z, ζ)⟫ 2iπ F,λ (1.53) (ζ) + S F,λ (ζ), bF (z, ζ)⟫, = −(N + 1) ΦF,λ N (z, ζ)⟪ R

1.2. THE GEOMETRIC POINT OF A VIEW

23

F,λ where P0F,λ , RF,λ , S F,λ denote respectively the vectors of C 1 differential forms P0,j , F,λ F,λ Rj , and Sj defined in K by Proposition 1.8, Proposition 1.12, and Corollary 1.14. In the above statements, these forms are described as corresponding currents in U . The starting point in order to transform Cauchy’s formula and involve the vector of holomorphic functions F = (f1 , . . . , fm ) instead of a single holomorphic function is the Cauchy–Pompeiu integral formula in U [Pom1909]. This was previously already done within establishing (1.49) in the case when m = 1. Namely, if φ is a C 1 function in K, then for any z ∈ U ,   1 1 φ(ζ) dζ ∂(φ(ζ) dζ) (1.54) φ(z) = − . 2iπ ∂K ζ − z 2iπ K ζ − z

Weighted Koppelman’s formulae (in either the analytic or geometric context), which we will study later on (in §4.3 and §4.5.3), can be derived from the following modification of (1.54). Let Φ : (z, ζ) −→ Φ(z, ζ) in C 1 (U ×K) be such that ∂¯z Φ(z, ζ) = 0, which means that Φ is holomorphic in z. Then, for any h ∈ H(U ) ∩ C 1 (K) and for any z ∈ U , one has that   ∂ z,ζ (Φ(z, ζ) dζ) Φ(z, ζ) dζ 1 − h(ζ) h(ζ) h(z) Φ(z, z) = 2iπ ∂K ζ −z ζ −z K   1 Φ(z, ζ) dζ ∂Φ dζ ∧ dζ (1.55) − . = h(ζ) h(ζ) (z, ζ) 2iπ ∂K ζ −z ζ −z ∂ζ K Thus one can formulate the following Lagrange-type division formula. Proposition 1.16. Let U , K = U , F = (f1 , . . . , fm ) be as before. We assume that f is of the form (1.51), where fj = f uj and u = (u1 , . . . , um ) > 0 in K. If σ = (σ1 , . . . , σm ) = u/u2 , then for any h ∈ H(U ) ∩ C(K), the function  h(ζ) ΦF,λ (z, ζ) ∂⟪P F,λ (ζ), bF (z, ζ)⟫ λ −→ K

extends from the half-plane {λ ∈ ℂ : Re λ  1} to the complex plane ℂ as a meromorphic function with (possible) poles in −ℚ+ . Moreover, for any z ∈ U , 

 h(ζ) ΦF,λ (z, ζ) ∂ ⟪P0F,λ (ζ), bF (z, ζ)⟫ (1.56) h(z) = 2 λ=0 K  F (z) · F (ζ) 2 dζ 1 . + h(ζ) 2iπ ∂K F (ζ)2 ζ−z The first term in the sum on the right-hand side of (1.56) splits as the sum of the residue term  1  1 ∂¯ (1.57) + ⟪σ, F (z)⟫ ∂¯ 2 , h ⟪σ, bF (z, ·)⟫ dζ f f with the spread integral    2 (1.58) 2 χK ⟪σ, F (z)⟫ P0f , h ∂ ⟪σ, bF (z, ·)⟫ dζ . Proof. Observe that one has for any ζ ∈ K and any λ ∈ ℂ that the identity u(ζ)2λ = eλ log u(ζ) = 1 + λ 2

∞ 

λκ (log u(ζ)2 )κ+1 = 1 + λ ρλ (ζ) (κ + 1)! κ=0

24

1. RESIDUE CALCULUS IN ONE VARIABLE

holds. Moreover, the function (λ, ζ) → ρλ (ζ) is continuous on ℂ × K and is holomorphic in λ. We rewrite the last equality in (1.52) as (1.59) ΦF,λ (z, ζ) = 1 − |f (ζ)|2λ + ⟪σ(ζ), F (z)⟫

|f (ζ)|2λ f (ζ)

+ λ ρλ (ζ) |f (ζ)|2λ and split each RjF,λ and SjF,λ as |f |2λ σj |f |2λ + λ ∂ρλ , (1.60) RjF,λ = ∂ 2iπ f f

⟪σ(ζ), F (z)⟫ f (ζ)

−1

2λ ¯ j + λ σj ∂ρλ |f | . SjF,λ = ∂σ 2iπ f

Let θ ∈ D(U ) be such that θ ≡ 1 about V (F ). For each continuous (1, 0)-form ϕ on K, and each j = 1, . . . , m, the functions  ¯ F,λ (ζ) ∧ ϕ(ζ) (1 − θ(ζ)) h(ζ) ΦF,λ(z, ζ) ∂P λ −→ 0,j K

and

 (1 − θ)

λ −→ K

σj |f |2λ |f |2λ 1 − |f |2λ + ⟪σ, F (z)⟫ ∧ϕ ∂ 2iπ f f  |f |4λ + (1 − θ)⟪σ, F (z)⟫ ∂σj ∧ ϕ. 2iπ f 2 K

are entire, since F , and hence f , does not vanish on the support of 1 − θ. Moreover, relations (1.59), (1.60), and the fact that λ −→ ρλ (ζ) is entire for any ζ ∈ K show that they share the same value at λ = 0. Repeating the proofs of Proposition 1.8 and Proposition 1.12, one shows that whenever the (1, 0)-form ϕ is in addition smooth in U , the two functions  F,λ θ(ζ) ΦF,λ (z, ζ) ∂P0,j (ζ) ∧ ϕ(ζ) λ −→ K

and



λ −→

θ K

σj |f |2λ |f |2λ 1 − |f |2λ + ⟪σ, F (z)⟫ ∧ϕ ∂ 2iπ f f  |f |4λ + θ⟪σ, F (z)⟫ ∂σj ∧ ϕ. 2iπ f 2 K

extend from the half-plane {λ ∈ ℂ : Re λ  1} to meromorphic functions in ℂ with (possible) poles in −ℚ+ and still share the same value at λ = 0. This proves the first assertion in the proposition. To prove the second assertion, one observes that for any z ∈ U and any complex number λ satisfying Re λ  1 it holds that ΦF,λ (z, z) = 1. Then it follows from the Cauchy–Pompeiu formula (1.55) that for any such λ and any z ∈ U ,  

∂ z,ζ (ΦF,λ 1 dζ F,λ 2 (z, ζ) dζ) − . h(z) = h(ζ) Φ2 (z, ζ) h(ζ) 2iπ ∂K ζ −z ζ −z K To obtain the integral representation formula (1.56), one needs to use (1.53) for N = 2 first and then follow the analytic continuation of both sides of this identity as meromorphic functions of λ. Finally, one identifies the evaluations of both sides

1.2. THE GEOMETRIC POINT OF A VIEW

25

at λ = 0. The final assertion follows from the following equalities between (0, 1)currents:

|f |2λ  1 1 (1.61) (1 − |f |2λ ) ∂ = ∂ 2iπ f λ=0 2 f

|f |2λ |f |2λ  1 1 |f 2 |2λ = .  ∂ = ∂ ∂ f 2iπ f λ=0 2 2iπ f 2 f2 Remark 1.17. The right-hand side of the integral representation formula (1.56) is congruent, as a function of z, to  1  z −→ ∂ , h(ζ) ⟪σ, bF (z, ·)⟫ dζ = ⟪RF (ζ) , h(ζ) bF (z, ζ) dζ⟫ f modulo the ideal generated by f1 , . . . , fm in H(U ). The major difference between the Lagrange division-interpolation formula (1.49) (when m = 1) and the formula (1.56) is contained in the description of the defect term h(z) − ⟪RF (ζ) , h(ζ) bF (z, ζ) dζ⟫ = ⟪F (z), q(z)⟫. When m = 1, q is expressed in terms of the restriction of h to the boundary of K, while in the case when m > 1, the expression of q involves all values of h in K given the contribution (1.58). 1.2.3. Residue currents and Bergman–Weil developments. Let U , K, F , f , σ be as in the previous subsection. Let ρF > 0 be the minimum of F  on the boundary of U and let U  = {ζ ∈ U : F (ζ) < ρF }. Cauchy’s integral representation formula for h ∈ H(U ) ∩ C(K) (equation (1.49), line 1) can be transformed, when z ∈ U  , to   m 1 fj (ζ) − fj (z)

fj (ζ) h(z) = dζ h(ζ) 2iπ ∂K j=1 (F (ζ) − F (z)) · F (ζ) ζ −z ∞  m F (z) · F (ζ) k  fj (ζ) − fj (z)

fj (ζ) 1  h(ζ) dζ = 2iπ F (ζ)2 F (ζ)2 ζ −z ∂K j=1 k=0

(1.62)

=

k ∞ 

1   ⟪σ, F (z)⟫ |f k+1 |2λ h ⟪σ, bF (z, ·)⟫ dζ . k+1 2iπ f λ=0 ∂K k=0

It follows from the definition of the principal value and residue currents for k ∈ ℕ (see (1.39) and (1.40)),

|f k+1 |2λ  1 |f k+1 |2λ  k+1 f k+1 = ∂ = , R = ∂ , P0f 2iπf k+1 λ=0 f k+1 2iπf k+1 λ=0 and the use of Stokes’s formula, that the right-hand side of the integral representation formula (1.61) truncates as follows: for any N ∈ ℕ∗ and z ∈ U , (1.63) h(z) =

N −1  

1  ⟪σ, , h b (z, ·)⟫ dζ F f k+1 k=0    k+1 k , h ∂ ⟪σ, F (z)⟫ ⟪σ, bF (z, ·)⟫ dζ + χK P0f  F (z) · F (ζ) N dζ 1 . h(ζ) + 2iπ ∂K F (ζ)2 ζ−z k

⟪σ, F (z)⟫ ∂

26

1. RESIDUE CALCULUS IN ONE VARIABLE

When m = 1 and F = f , formula (1.63) simplifies as 1  1  1 dζ N k (z, ·) dζ f (z) + , h b f (z) f k+1 N f 2iπ ∂K f (ζ) ζ − z k=0 1  dζ N 1 f (z). = LN [h; f ](z) + 2iπ ∂K f N (ζ) ζ − z

h(z) = (1.64)

N −1  

∂

When m > 1, the following proposition provides a generalization of (1.64) that respects mthe fact that the remainder term LN [h; F ] modulo the N th power of the ideal 1 fj H(U ) involves only the action of (0, 1)-currents supported by V (F ). This was not the case in the N -truncated version (1.63) of (1.62). Proposition 1.18. Let U , K, F , f , σ be as in Proposition 1.16 and let N in ℕ∗ . For any h ∈ H(U ) ∩ C 1 (K), the function  F,λ h(ζ) ΦF,λ λ −→ N +1 (z, ζ) ∂ ⟪P0 (ζ), bF (z, ζ)⟫ K

extends from the right half-plane {λ ∈ ℂ : Re λ  1} to ℂ as a meromorphic function with (possible) poles in −ℚ+ . Moreover, for any z ∈ U ,

 (1.65) h(z) = (N + 1)

 F,λ h(ζ) ΦF,λ N (z, ζ) ∂ ⟪P0 (ζ), bF (z, ζ)⟫ λ=0 K  F (z) · F (ζ) N +1 dζ 1 . + h(ζ) 2iπ ∂K F (ζ)2 ζ−z

The first term in the sum on the right-hand side of (1.65) splits as the sum of the residue term (1.66)

N   k=0

k

⟪σ, F (z)⟫ ∂

1  ⟪σ, , h b (z, ·)⟫ dζ F f k+1

and of the spread integral    N +1 N , h ∂ ⟪σ, bF (z, ·)⟫ dζ . (1.67) (N + 1) χK ⟪σ, F (z)⟫ P0f Proof. The first assertion is proved in the same way as the first assertion in Proposition 1.16. To prove the integral representation formula (1.65), one applies again the Cauchy–Pompeiu formula (1.55) with Φ(z, ζ) = ΦF,λ N (z, ζ), where the complex parameter λ has sufficiently large real part Re λ  1. Formula (1.65) follows by applying the analytic continuation principle up to λ = 0. For the last assertion, one first observes that the equalities between (0, 1)-currents (1.61) can be easily generalized as follows: for any N ∈ ℕ∗ and 0 ≤ k ≤ N , one has    1  

1 N N |f |2kλ |f |2λ  (1 − |f |2λ )N −k k ∂ = tk (1 − t)N −k dt ∂ k+1 k k f 2iπ f λ=0 f 0 1 1 = (1.68) . ∂ N + 1 f k+1

1.2. THE GEOMETRIC POINT OF A VIEW

This implies that 

 F,λ ΦF,λ , bF (z, ·)⟫ dζ (N + 1) N (z, ·) ⟪R K

=

N  

27

λ=0 k

⟪σ, F (z)⟫ ∂

k=0

1  ⟪σ, , h b (z, ·)⟫ dζ . F f k+1

Since one has also

  F,λ ⟪S (N + 1) ΦF,λ (z, ·) , b (z, ·)⟫ dζ F N λ=0 K    N +1 N = (N + 1) χK ⟪σ, F (z)⟫ P0f , h ∂ ⟪σ, bF (z, ·)⟫ dζ because

|f |2(k+1)λ  (1 − |f |2λ )N −k = 0 for k = 0, . . . , N − 1 f k+1 λ=0

|f |2(N +1)λ  N +1 = P0f (with singular support V (F ) ⊂ U ), 2πif N +1 λ=0 the last assertion of the proposition follows.



Remark 1.19. The right-hand side of the integral representation formula (1.65) is congruent to the reminder contribution, as a function of z, N −1  1   k ⟪σ, F (z)⟫ ∂ k+1 , h ⟪σ, bF (z, ·)⟫ dζ , (1.69) z −→ LN [h; F ](z) := f k=0

modulo the N th power of the ideal generated by the F in H(U ). When m > 1 one observes in the expression for the quotient contribution of h − LN [h; F ] as an N homogeneous form m   [N ] κ (1.70) h − LN [h; F ] = ⟪F, qN [h]⟫ = ( fj j ) qN,κ [h], |κ|=N

1

deduced from (1.65), that the quotient vector qN [h] = (qN,κ [h])|κ|=N fails to be described in terms of boundary values of h. This occurs because for m > 1 the [N ] spread integral (1.67) contributes to ⟪F, qN [h]⟫ . Definition 1.20. Let U , K, F , f , σ be as in Proposition 1.16. The sequence (LN [h; F ], qN [h])N ≥1 , defined by (1.69) and (1.70), is a marker for iterated euclidean division in H(U ) with respect to the ideal generated by F . Such a sequence is called a Bergman–Weil expansion for h in H(U ). 1.2.4. Residue currents and multiplicative calculus. Let U be a domain in ℂ. In this section, we formalize a multiplicative commutative tensorial calculus involving as fundamental bricks (or objects) the vector-valued residue currents RF and principal value currents 2iπP0F , where F ∈ (H(U ) \ {0})m . These currents are defined in Propositions 1.8 and 1.12, respectively. This calculus has been already used in the proofs of Proposition 1.16 and Proposition 1.18 in the previous subsections. Although multiplication of distributions, hence of currents, remains a stumbling block, one profits here from the fact that our fundamental objects RF or 2iπP0F are realized by employing analytic continuation as RF = [RF,λ ]λ=0 or

28

1. RESIDUE CALCULUS IN ONE VARIABLE

2iπP0F = [2iπP0F,λ ]λ=0 . Let us start with a result that ensures the robustness of the multiplication of principal value currents 2iπP0F . Proposition 1.21. Let N ∈ ℕ∗ , (m1 , . . . , mN ) ∈ (ℕ∗ )N , F ∈ (H(U ) \ {0})m

for 1 ≤  ≤ N , and M = m1 · · · mN . The function λ ∈ {λ ∈ ℂN : Re λ  0 for  = 1, . . . , N } −→ 2iπP0F1 ,λ1 ⊗ · · · ⊗ 2iπP0FN ,λN ∈ (C(U ))M induces a  D(0,0) (U, ℂM )-valued map which extends to ℂN as a meromorphic function in N complex variables. Its polar set is the union of affine hyperplanes defined over ℚ that are disjoint from {λ ∈ ℂN : Re λ ≥ 0 for  = 1, . . . , N }. Moreover  (1.71) 2iπP0F1 ,λ1 ⊗ · · · ⊗ 2iπP0FN ,λN λ=0 = 2iπ P0F1 ⊗···⊗FN . Proof. Let f ∈ H(U ) be such that [F = 0] = [div(f )] every  = 1, . . . , N . That is, F = f U , where U ∈ H(U ) and V (U ) = ∅. Let also Σ = U /U 2 . For any λ in ℂN satisfying Re λ >> 0 for  = 1, . . . , N , one has N N N  ! ! |f |2λ  1 + ⟪rλ (ζ), λ⟫ 2iπP0F ,λ = Σ , f =1

=1

=1

where rλ (ζ) ∈ ℂ depends smoothly on ζ ∈ U and holomorphically on λ ∈ ℂN . N Here, following the convention for notation (1.22), ⟪rλ (ζ), λ⟫ := =1 rλ, (ζ) λ . "N "N Let α ∈ =1 V (F ) = =1 V (f ) and let ν,α be the order of f at α for any  = 1, . . . , N . Denote also by ν = να := (ν1,α , . . . , νN,α ). Then one has in a small disk Dα centered about α that N

N  |f |2λ

|ζ − α|2⟪ν,λ⟫ 1 + ⟪λ, ρλ (ζ)⟫ = , N f (ζ − α)|ν| =1 v =1  N where ⟪ν, λ⟫ = N =1 ν λ , |ν| = =1 ν , the v are invertible holomorphic functions in Dα with f (ζ) = (ζ − α)ν v (ζ) for  = 1, . . . , N , and ρλ (ζ) ∈ ℂN depends smoothly on ζ ∈ Dα and holomorphically on λ ∈ ℂN . Using repeatedly the second identity in (1.7) to perform integration by parts in Dα , one gets the first assertion of the proposition. It is immediate to check that (1.71) holds. 

Let us now incorporate into our finite collection of elementary principal value currents 2iπP0F ,  = 1, . . . , N (as in Proposition 1.21), the residue current RF0 , where F0 ∈ (H(U ) \ {0})m0 . Given G ∈ (H(U ) \ {0})M and α ∈ V (F0 ), we need to define first the localization of RF0 ⊗G at α by (1.72)

(RF0 ⊗G )α = θ RF0 ⊗G ,

where θ ∈ D(U ) equals identically 1 about α and Supp(θ) ∩ V (F0 ⊗ G) = {α}. Observe that the vector-valued current defined by (1.72) is supported by {α} and does not depend on the choice of θ. Although the robustness of the multiplicative approach within the context of Proposition 1.21 does not subsist anymore within this new context, the following useful result needs to be mentioned. Proposition 1.22. Let N ∈ ℕ∗ , (m0 , . . . , mN ) ∈ (ℕ∗ )N +1 , let for 0 ≤  ≤ N , F ∈ (H(U ) \ {0})m and let M = m1 · · · mN . The function λ ∈ {λ ∈ ℂ; Re λ  0} −→ RF0 ,λ ⊗ 2iπP0F1 ,λ ⊗ · · · ⊗ 2iπP0FN ,λ ∈ (C(U ))m0 M

1.2. THE GEOMETRIC POINT OF A VIEW

29

induces a  D(0,1) (U, ℂm0 M )-valued map which extends to ℂ as a meromorphic function of λ with poles in −ℚ+ . Moreover, if G = F1 ⊗ · · · ⊗ FN , then   F0 ,λ νF0 ,α (1.73) R ⊗ 2iπP0F1 ,λ ⊗ · · · ⊗ 2iπP0FN ,λ λ=0 = (RF0 ⊗G )α . νF0 ⊗G,α α∈V (F0 )

Proof. Let f ∈ H(U ) be such that [F = 0] = [div(f )], for each  = 0, . . . , N . Then F = f U , where U ∈ H(U ) and V (U ) = ∅. Define Σ = U /U 2 . Let us now introduce N + 1 complex parameters λ ,  = 0, . . . , N , and then we set λ = (λ0 , λ1 , . . . , λN ) = (λ0 , λ ). If Re λ  0 for  = 0, . . . , N , one has (1.74) RF0 ,λ0 ⊗

N ! =1

N N

! λ0  |f |2λ df0  1+⟪rλ (ζ), λ⟫ 2iπP0F ,λ = Σ , 2iπ f f0 =0

=0

where rλ (ζ) ∈ ℂN +1 depends smoothly on ζ ∈ U and holomorphically on the parameter λ ∈ ℂN +1 . It follows from Proposition 1.21 that the current-valued function N

! 2iπP0F ,λ

λ −→ RF0 ,λ0 ⊗ =1

|U\V (F0 )

∈  D(0,1) (U \ V (F0 ))

extends to ℂN +1 as a meromorphic map, whose polar set is the union of ℚhyperplanes (in the λ coordinates) that are disjoint from {λ ∈ ℂN +1 : Re λj ≥ 0 for j = 0, . . . , N }. Moreover, because of the factor λ0 in front of the right-hand side of (1.74), one has N

! RF0 ,λ0 ⊗ (2iπP0F ,λ

=1



|U\V (F0 ) λ=0

= 0.

It remains to study what happens about a point α ∈ V (F0 ). Let α be such a point and let ν0 = ν0,α be the order of f0 at α. Let also ν  = να = (ν1,α , . . . , νN,α ) where ν,α is the order of f at α, ν = (ν0 , ν  ). Then one has in a small disk Dα about α that (1.75)

N λ0  |f |2λ df0 ν0 λ0 1 + ⟪λ, ρλ (ζ)⟫ ¯ 1 |ζ − α|2⟪ν,λ⟫ = ∂ N 2iπ f ⟪ν, λ⟫ 2iπ (ζ − α)|ν| f0 =0 v =0 λ0 |ζ − α|2⟪ν,λ⟫ 1 + ⟪λ, ρλ (ζ)⟫ dv0 , + N 2iπ (ζ − α)|ν| v0 =0 v

where the v are invertible holomorphic functions in Dα with f (ζ) = (ζ − α)ν v (ζ), and ρλ (ζ) depends smoothly on ζ ∈ Dα and holomorphically on λ ∈ ℂN +1 . If one takes all λ to be equal to a complex parameter λ satisfying Re λ  0, then the right-hand side of (1.75) becomes 2λ  1 + λρ (ζ) ν 1 (ζ − α)|ν| 2λ λ (ζ − α)|ν|  dv0 λ 0 ¯ + . ∂ N |ν| 2iπ (ζ − α)|ν| 2iπ (ζ − α)|ν| v0 =0 v This expression, considered a current-valued function in λ, extends the whole complex plane to a meromorphic function with a polar set in −ℚ+ . The first assertion

30

1. RESIDUE CALCULUS IN ONE VARIABLE

of the proposition follows. Moreover one has

1 + λρ (ζ) λ (ζ − α)|ν| 2λ dv  λ 0 =0 N |ν| 2iπ v (ζ − α) λ=0 0 =0 v in view of the factor λ. Formula (1.73) follows from (1.74), (1.75), together with the fact that N 1 (ζ − α)|ν| 2λ ! 

1 ¯ Σ = (RF0 ⊗G )α , ∂ ∧ N  |ν| 2iπ (ζ − α) λ=0 v  =0 =0 where G = F1 ⊗ · · · ⊗ FN .



We present below another multiplicative procedure (which happens to be this time dis-symmetric) to multiply the residue current RF0 with the principle value current 2iπP0F for  = 1, . . . , N . One needs to introduce for that purpose N + 1 strictly positive integer weights γ0 , . . . , γN such that (1.76)

γ0 < min(γ1 , . . . , γN ) or γ0 > max(γ1 , . . . , γN ).

Proposition 1.23. Let N ∈ ℕ∗ , (m0 , . . . , mN ) ∈ (ℕ∗ )N +1 , let for 0 ≤  ≤ N , F ∈ (H(U ) \ {0})m and let M = m1 · · · mN be as in the previous proposition. Let also γ = (γ0 , . . . , γN ) ∈ (ℕ∗ )N +1 satisfying (1.76) and η < (γ∞ )−1 . The function π (1.77) λ ∈ {reiθ ∈ ℂ : |θ| < η , r  1} 2 γ1 γN F0 ,λγ0 −→ R ⊗ 2iπP0F1 ,λ ⊗ · · · ⊗ 2iπP0FN ,λ ∈ (C(U ))m0 M induces a  D(U, ℂm0 M )-valued map extending to an open neighborhood of the closed sector S with aperture π γ−1 ∞ , bisected by the positive real axis, as a holomorphic map. In particular, it means that the map (1.23) extends holomorphically up to a neighborhood of the origin in the complex plane. Moreover, one has 

γ1 γN γ0 (1.78) RF0 ,λ ⊗ 2iπP0F1 ,λ ⊗ · · · ⊗ 2iπP0FN ,λ λ=0  F0 ⊗G (R )α when γ0 < min(γ1 , . . . , γn ) α∈V (F0 )

 = G F0 when γ0 > max(γ1 , . . . , γn ), (R ) 2iπP0 ⊗ α α∈V (F0 )\V (G) where G = F1 ⊗ · · · ⊗ FN . Proof. One repeats the proof of Proposition 1.22 up to (1.75) about the local extension of the mapping for some α ∈ V (F0 )). From now on we keep the notation used from this stage in the proof of Proposition 1.22. If one takes λ = λγ for  = 0, . . . , N , then the right-hand side of (1.75) splits as the sum of two expressions, both multiplied by a factor 1 + λ ρ˜λ (ζ) . N =0 v Within the above factor v are the invertible holomorphic functions in Dα present in (1.75) and the function ρ˜λ (ζ) still depends smoothly on ζ ∈ Dα and holomorphically on λ ∈ ℂ, but now depends also on the choice of γ. One of these expressions in λ is γ λγ0 |ζ − α|2⟪ν,λ ⟫ dv0 . 2iπ (ζ − α)|ν| v0

1.2. THE GEOMETRIC POINT OF A VIEW

31

Once considered as a (0, 1)-current valued function of the complex parameter λ, it extends holomorphically to an open neighborhood of S as a holomorphic function vanishing at the origin. The second expression in λ is γ ν0 λγ0 1 |ζ − α|2⟪ν,λ ⟫ ∂ . ⟪ν, λγ ⟫ 2iπ (ζ − α)|ν| If γ0 < min1≤≤N γ , then one has ν0 λ γ 0 1 = . N ⟪ν, λγ ⟫ 1 + λ =1 (ν /ν0 )λγ −γ0 −1 The above quotient extends holomorphically as a function of λ to an open neighborhood of the closed sector S that takes value 1 at the origin. Since 1 |ζ − α|2μ μ −→ ∂ 2iπ (ζ − α)|ν| extends from {μ ∈ ℂ; Re μ  0} as a meromorphic current valued map with poles in −ℚ+ , the first assertion of the proposition follows in this case. Moreover, one can check easily that the value of the holomorphic continuation of the map (1.77) at λ = 0 is equal to the current

N 1 |ζ − α|2μ !  Σ = (RF0 ⊗G )α ∧  |ν| 2iπ μ=0 (ζ − α) v  =0 =1

1 N

in Dα . When γ0 < min1≤≤N γ , the formula (1.78) follows. Suppose now that γ0 > max1≤≤N γ . If α ∈ V (F0 ) ∩ V (F ) for some  ∈ {1, . . . , N } with γ minimal, then one has ν0 λγ0 − ν0 λ γ 0  = ⟪ν, λγ ⟫ χ + λ { ∈{0,...,N } : γ  >γ } ν λγ  −γ −1 for some χ ≥ ν in ℕ∗ . Such a function of λ is holomorphic in a neighborhood of S and vanishes at the origin. As a consequence, the assertions in the proposition follow provided U is replaced by Dα . It remains to consider the case where U is replaced by Dα with α ∈ V (F0 ) \ V (G), in which case the conclusion also holds.  The proposition is therefore also proved when γ0 > max1≤≤N γ . Given Propositions 1.22 and 1.23, let us propose the following definition and thus summarize the main results obtained in this subsection. Definition 1.24. Let F, G be two vectors of nonidentically zero holomorphic functions in a domain U ⊂ ℂ. The currents P0G ⍟ RF = RF ⍟ P0G , RF ⊗ P0G , and P0G ⊗ RF are vector-valued currents defined and realized respectively through analytic continuation by (1.79)



P0G ⍟ RF = RF ⍟ P0G = P0G0 ,λ ⊗ RG,λ

=

λ=0



RF ⊗ P0G = RF,λ ⊗ P0G,λ P0G

⊗R = F



P0G,λ

⊗R

2

F,λ2

 = λ=0

α∈V (F )

νF,α (RF ⊗G )α , νF ⊗G,α

(RF ⊗G )α ,

α∈V (F )

 = λ=0







α∈V (F )\V (G)

(RF ⊗G )α = P0G ⊗ (RF )|U\V (G) .

32

1. RESIDUE CALCULUS IN ONE VARIABLE

The commutator [RF , P0G ] for the ⊗ operation equals  [RF , P0G ] = RF ⊗ P0G − P0G ⊗ RF =

(RF ⊗G )α .

α∈V (F ) ∩ V (G)

1.2.5. Residues of sections of hermitian bundles. Let us consider now a hermitian holomorphic bundle E = (E, | |) with rank m over some Riemann surface X . Recall that | | denotes a smooth hermitian metric on E. The reader, who is not familiar with the geometric point of view of hermitian bundles with finite rank on a complex manifold, is urged to refer to §B.3. To any global holomorphic section s ∈ OX (X , E) one attaches its (smooth) conjugate section s∗ with respect to the # ∗ hermitian metric | |; see (B.59) in §B.3.2.1. We adopt the notation E = 0 # ∗ # ∗ $ $m # ∗ − if  ∈ ℤ and E = E otherwise. Contraction ∈ℤ # E = ℂ # ⊕ =1 or interior multiplication s : E ∗ → E ∗ is the morphism between complex bundles, whose action is described locally as follows: # • s  E ∗ = 0 for any  ≤ 0; • if  ≥ 1 and (e1 , . . . , em ) is a local holomorphic frame for E, then, for any ordered multi-index J : 1 ≤ j1 < · · · < j ≤ , (1.80) s e∗J = s (e∗j1 ∧ · · · ∧ e∗j ) =

 

∗ ∧ · · · ∧ e∗ , (−1)κ−1 e∗jκ , s ej1∗ ∧ · · · ∧ e% j

jκ

κ=1

where the symbol & means omission and the brackets ,  stand for the description of the duality (E ∗ , E). # # The contraction morphism s : E ∗ → E ∗ , whose action is defined above, presides over the definition of the Koszul complex K•s . Definition 1.25. Let X , E, s, and s be as above and let V (s) = s−1 ({0}). The Koszul complex Ks• over X is the complex of holomorphic bundles over X given by (1.81)

0 −→

m '



s E ∗ −→

m−1 '





s s ··· → E∗ →

 '



s E ∗ −→

−1 '







s s s ··· → E ∗ −→ ℂ, E∗ →

which is exact over the open subset X \ V (s). Let



' ' ' D(0,•) (U, E ∗ ) open =  D(0,0) (U, E ∗ ) ⊕  D(0,1) (U, E ∗ ) open U ⊂ X U ⊂ X # ∗ be the sheaf of E -valued (0, •)-currents on X . The action of s as a homomorphism from the sheaf LE into LE is described locally, in terms of representations with respect to a local holomorphic frame, by m n 

 Tj e∗j =  s j Tj , s LE =





j=1

(1.82) s

m  j=1

j=1

n m



 Tj e∗j ∧ e∗J =  sj Tj e∗J − Tj e∗j ∧ s e∗J j=1

j=1

for any ordered multi-index J with length 0 ≤  ≤ m. If T has pure bidegree (0, 0), then  = 1. Otherwise,  = −1, if T has pure bidegree (0, 1). The sheaf LE is

1.2. THE GEOMETRIC POINT OF A VIEW

33

decomposed into a direct sum as LE =

(

LνE =

1 (

LνE ,

ν=−m

ν∈ℤ

where LνE =

(



γ  '  D(0,γ+ν) U, E ∗

(1.83)

=



open

U ⊂ X

γ∈ℤ −ν  '  D(0,0) U, E ∗

open

U ⊂ X

⊕





'  1−ν  D(0,1) U, E∗

open

U ⊂ X

,

for each ν ∈ ℤ. The decomposition (1.83) leads to the following definition of the homorphism (operator) ∇s along the section s ∈ O(X , E). Definition 1.26. The sheaf homomorphism ∇s : LE → LE is defined to be the difference between the anticommuting homomorphism 2iπ s and Dolbeault ¯ 3 That is, homomorphism ∂. ¯ ∇s = 2iπ s − ∂.

(1.84)

The homomorphism ∇s acts from Lν−1 into LνE for each ν ∈ ℤ. E The goal of the following proposition is to extend and complete the definitions of the vector-valued currents RF = [RF,λ ]λ=0 and P0F = [P0F,λ ]λ=0 in the framework of hermitian holomorphic bundles using the homomorphism (1.84). These currents were previously described in Propositions 1.8 and 1.12, respectively, for the particular case where s = F was a section of the trivial bundle U × ℂm over a domain U ⊂ ℂ. Proposition 1.27. Let E = (E, | |) be a hermitian holomorphic bundle with rank m over a Riemann surface X and let s ∈ OX (X , E). The holomorphic L0E (X ) (respectively L−1 E )-valued maps λ −→ R|s,λ| = ∂ |s|2λ (1.85)

s∗ , 2iπ|s|2

s,λ s,λ 2λ λ −→ P|s,λ | = P| |,0 + P| |,1 = |s|

s∗ s∗ s∗ 2λ + |s| ∧ ∂ 2iπ|s|2 2iπ|s|2 2iπ|s|2

defined in {λ ∈ ℂ : Re λ  1} extend to ℂ as meromorphic functions with poles in −ℚ+ . Moreover, their values R|s | and P|s | at λ = 0 satisfy ∇s R|s | = 0,

(1.86)

∇s P|s | = [1] − R|s | .

The current R|s | is called the residue current attached to the section s of E, while the current P|s | is known as the principal value current attached to the section s of the bundle E. 

E ∗ is a holomorphic bundle with finite rank, the Dolbeault sheaf homomorphism ∂¯ : LE → LE is well defined. 3 Since

34

1. RESIDUE CALCULUS IN ONE VARIABLE

Proof. Because of the correspondence between Cartier divisors on X and holomorphic line bundles over X (see Example B.2), the section s factorizes as s = f ⊗ u. In this factorization, f is a holomorphic section of the line bundle L corresponding to the Cartier divisor attached to s and u is a nonvanishing section of the holomorphic bundle L−1 ⊗E with rank m. The nonvanishing section u induces an isomorphism between L and a subbundle of E having rank 1. This isomorphism allows us to carry the metric | | on E as a metric | | on L such that |f| = |s|. Intuitively it means that one finds locally a holomorphic function, whose zero set realizes V (s) = s−1 (0). Let s = f u1 e1 +· · ·+um em = f u along a local holomorphic frame e = (e1 , . . . , em ) for E over a trivialization local chart U . Suppose that the metric | | on E is defined over U by ∀ x ∈ U, ∀ ξ ∈ Ex , ξ2x =

m  m 

hj,k (x)ξj ξk ,

j=1 k=1

so that s∗ = f

m  m j=1

k=1

 hj,k uk e∗j in U . For Re λ  1 one has in U that

|f |2λ

s∗ s∗ 2λ = |f| = ) χ∗u , (1 + λρ λ 2iπ|s|2 2iπ|f|2 2iπf s∗ s∗ 2λ = (∂|f| ) (∂|s|2λ ) 2iπ|s|2 2iπ|f|2

|f |2λ |f |2λ +λ = (1 + λρλ ) ∂ ∂ρλ χ∗u , 2iπf 2iπf

∗ ∗ ∗ ∗ s s s s = |f|2λ ∧∂ ∧∂ |s|2λ 2iπ|s|2 2iπ|s|2 2iπ|f|2 2iπ|f|2 |f |2λ

χ∗ u = (1 + λρλ ) χ∗u ∧ ∂ , 2 2iπf 2iπ |s|2λ

where m  m ∀ x ∈ U,

(1.87)

χ∗u (x)

=

j=1 m j=1



∗ k=1 hj,k (x)uk (x) ej (x) , m k=1 hj,k (x) uj (x)uk (x)

while ρλ (x) depends smoothly on x ∈ U and holomorphically on λ ∈ ℂ. Similar arguments to those used in the proofs of Propositions 1.8 and 1.12 show that these three functions of λ, considered respectively as  D(0,0) (X , E ∗ ),  D(0,1) (X , E ∗ ),  (0,1) D (X , E ∗ ∧E ∗ )-valued, extend to the whole complex plane as meromorphic functions with poles in −ℚ+ . The first assertion of the proposition follows. Moreover, the identities (1.88) 1 χ∗ s,λ u ∗ s f ∗ f2 ∗ χu ∧ ∂ R|s | = [R|s,λ | ]λ=0 = ∂ f χu and P| | = [P| | ]λ=0 = P χu + P 2iπ are valid in the local chart U , where χ∗u is defined in (1.87). Note that analytic continuation commutes with the action of ∇s . Thus, it follows from (1.82) that

∇s R|s,λ |

 λ=0

s∗ 

 = − ∂ |s|2λ s = ∂ |s|2λ 2 |s| λ=0

λ=0

= ∂[1] = 0.

1.2. THE GEOMETRIC POINT OF A VIEW

Since

35

s∗ , 2iπ|s|2 s∗

2λ 2iπs (P|s,λ 1+∂ , | ) = |s| 2iπ|s|2

s,λ s,λ 2λ ∂P|s,λ ∂ | = ∂P| |,0 = R| | + |s|



one has that 2λ ∇s (P|s,λ − R|s,λ | ) = |s| | .

Hence ∇s (P|s | ) = [1] − R|s | , 

when evaluating the analytic continuation at λ = 0. The proposition is proved.

Remark 1.28. When s is a section of a line bundle, the L−1 -valued current is independent of the choice of the metric. In this case, it is also denoted as Rs = ∂(1/s). Moreover, if s ∈ O(X , L ) and s ∈ O(X , L ) are such that s = as , where a is a global holomorphic section of Homℂ (L , L ), then one has   Rs = aRs as (L )−1 -valued (0, 1)-currents on X . This is a formulation of the so-called transformation law in the univariate setting. Such transformation law will be later a crucial tool for multi-variate residue calculus. R|s |

The Dolbeault complex of sheaves bundles   

'  '   '  ι ∂ ∂ −→  D(0,0) U, E ∗ −→  D(0,1) U, E ∗ −→ 0 0 −→ OX (U, E ∗ ) U

U

U

is locally exact as a consequence of Dolbeault’s lemma, for any  = 0, . . . , m; see §B.2.3. The next proposition relies on the following chasing diagram of sheaves. m   #   #   s s  (0,0)  (0,0)  (0,0) U, E ∗ U, E ∗ U) D → ··· D → ··· D U

∂ ↓ m  #   (0,1) X , E∗ D ∂ ↓ 0

U



U

s → ···

∂ ↓   #   (0,1) X , E∗ D ∂ ↓ 0

U

U



s → ···

∂ ↓   (0,1) U) D ∂ ↓ 0

U

Proposition 1.29. Suppose that X is a noncompact Riemann surface and that s is a holomorphic section of a hermitian holomorphic bundle E = (E, | |) with rank m. Given a holomorphic function h ∈ OX (ℂ), one has (1.89)

h R|s | = 0 ⇐⇒ h ∈ s (OX (X , E ∗ )).

Proof. The implication (⇐=) follows from the local expression of R|F | with respect to a local orthonormal frame (∗1 , . . . , ∗m ) for E ∗ ; see (1.88) and (1.38). Since any noncompact Riemann surface is a Stein manifold (see §A.2.2) it is enough to prove the direct implication (=⇒), when X is replaced by U . Here, (U, ζ : U → ℂ) is a local chart over which E is a trivial bundle. Since the Dolbeault complex of sheaves of currents ( D(0,•) (U, E ∗ ))U is exact in this case, there exists q ∈  D(0,0) (X , E ∗ ) such that ∂q = P|s |,1 . Let a = h · P|s |,0 + h · (2iπs q).

36

1. RESIDUE CALCULUS IN ONE VARIABLE

One has (1.90)

2iπs a = h · (2iπs (P|s |,0 )) = [h].

On the other hand, one has, according to the computation rules (1.82) and to the first identity in (1.86), that the assumption h · R|s | = 0 implies     ∂a = h · (2iπ s − ∇s )(P|s |,0 ) − h · 2iπs (∂q)   = [h] − h · (∇s P|s | − ∇s P|s |,1 ) − h · 2iπs P|s |,1     = [h] − h · [1] − R|s | − 2iπs P|s |,1 − h · 2iπs (P|s |,1 ) = 0. This shows that 2iπa is represented as a (0, 0)-current by a holomorphic section of E ∗ . Hence h ∈ s (OX (X , E ∗ )) in view of (1.90).  Proposition 1.10 can be reformulated within the geometric context of this subsection. One needs to introduce for such purpose a (smooth) connection D on the holomorphic bundle E; see §B.2.1.2. The connection D is acting on C ∞ (X , E) and ∞ ∞ has values in the space C1∞ (X , E) = C1,0 (X , E) ⊕ C0,1 (X , E) of E-valued onedifferential forms on the underlying two-dimensional differentiable manifold Xℝ . Given such connection D on E, one can consider as well the left action D of D as a contraction on E ∗ -valued currents with degree 1 on X . An important example is the following: the metrized bundle E inherits a Chern connection DE = D| | , which is compatible with both the hermitian metric | | and the complex structure on E; see Definition B.44. Proposition 1.30. Let E = (E, | |) be a hermitian holomorphic bundle with finite rank m over a Riemann surface X and let s ∈ OX (X , E). Let also s = f ⊗ u be any factorization of s as the product of a holomorphic section of the line bundle L corresponding to the Cartier divisor attached to s with a nonvanishing section of the holomorphic bundle L−1 ⊗ E having rank m. The (1, 1) integration current [s = 0] corresponding to the divisor div(f) factorizes as (1.91)

[s = 0] = [div(f)] = D(s) R|s | .

Proof. Let (U, ζ) be a local chart about α ∈ V (s) over which E is trivialized and let (e1 , . . . , em ) be a local holomorphic frame for E over U . Let us start with the local representation s = f u1 e1 + · · · + um em of the section s in U , where f ∈ H(U ), u ∈ (H(U ))m with V (u) = ∅. One has D(s) = ω1 e1 + · · · + ωm em , where m

 ωj = (df ) uj + f duj + ωj,k uk k=1

for j = 1, . . . , m. Furthermore, the smooth 1-forms ωj,k are the entries of the matrix M(D; e; e) of the connection D expressed in the local frame e; see (B.18) and (B.19) in §B.2.1.2. Since 1 R|s | = ∂ χ∗ f u (see (1.88)) and f ∂(1/f ) = 0 in the sense of currents in U , one has that 1 1 D(s) (R|s | ) = D(s) ∂ χ∗u = (df ) m uj ej ∂ χ∗u 1 f f 1 =∂ ∧ df = [div(f )] = [div(f)] f in U , taking into account Proposition 1.10.



1.2. THE GEOMETRIC POINT OF A VIEW

37

One may transpose within the above geometric context the multiplicative calculus of currents as introduced in §1.2.4. Thus, Definition 1.24 can be reformulated as follows. Definition 1.31. Let u, v be respective holomorphic sections of two hermitian holomorphic bundles (E  , | |), (E  ,  ), satisfying E = E  ⊕ E  . One defines three global sections of L−1 E through analytic continuation as follows:

(1.92)

 u,λ , Pv  ⍟ R|u | = R|u | ⍟ Pv  = Pv,λ  ⊗ R| | λ=0 

v,λ2 R|u | ⊗ Pv  = R|u,λ , | ⊗ P  λ=0 

u,λ2 Pv  ⊗ R|u | = Pv,λ .  ⊗R λ=0

Considerations leading to constructions in Proposition 1.27, when s is a holomorphic section of a hermitian holomorphic bundle E = (E, | |), motivate the following definition. Definition 1.32. Let X be a connected Riemann surface and let E be a holomorphic bundle with a finite rank over X . Let also W be a polar set, that is, a discrete subset of X so that there exists a holomorphic line bundle L and a meromorphic section f of L defined on X satisfying V (f) = |div+ (f)| = W . A semimeromorphic section h of E over X is a smooth section of E over the maximal open subset X \ W , denoted by def(h), so that f ⊗ h extends from X \ W to X as a smooth section of L ⊗ E. The set W is denoted by Pol(h) in this case. Any semimeromorphic section h of E admits a standard extension h|X \Pol(h) as an element of  D(0,0) (X , E). Such extension is defined in a similar way to the extension of h|U\W in (1.4). Definition 1.33. Let h be a semimeromorphic section of E with W and f as in Definition 1.31. Let also div− (f) = div(f) − div+ (f), with support |div− (f)| being equal to the polar set of f. The standard extension h|X \W ∈  D(0,0) (X , E) is the (0, 0)-current defined as  ∀ ϕ ∈ D(1,1) (X \ |div+ (f)|, E ∗ ), h|X \W , ϕ = ϕ h, X (1.93)

   ∀ ϕ ∈ D(1,1) (X \ |div− (f)|, E ∗ ), h|X \W , ϕ = ϕ  f 2λ h , X

λ=0

where   is an arbitrary smooth metric on L|X \|div− (f)| and ϕ denotes the interior contraction by the E ∗ -valued (1, 1)-form ϕ to an E-valued (0, 0)-current. The definition depends neither on the choice f nor on the choice of the metric   on L|X \|div− (f)| , and the two definitions (1.93) coincide on X \ |div(f)|). When h is a meromorphic section over X of a holomorphic line bundle L, one has that (1.94)

h

|X \|div− (h)| Supp ∂ = |div− (h)|. 2iπ

38

1. RESIDUE CALCULUS IN ONE VARIABLE

When h is a global meromorphic section of the canonical bundle, here it is the ∗(1,0) (1,0) cotangent holomorphic bundle TX = (TX )∗ , whose global sections are (1, 0)differential forms (see Example B.13), one has moreover that h

|X \|div− (h)| ∂ ∈  D(1,1) (X ). 2iπ If X is compact, then the identity  h|X \|div− (h)|

= 0, ∂ (1.95) 2iπ X derived using Stokes’s theorem, is known as the residue formula. Assume that α is a pole of h. Let (U, ζ) be a local chart about α, with centered local coordinate ζ : ζ(α) = 0. If

 ak ζ k dζ (1.96) h(ζ) dζ = k∈ℤ

is the expression of h in the local centered coordinate ζ, then none of the Laurent coefficients ak (for k = −1) is geometrically intrinsic. These coefficients depend on the choice of the local coordinate ζ. This is not the case for the coefficient  h|X \|div− (h)|

a−1 = ∂ , 2iπ α X where ( )α means, as it was already the case in (1.72), the localization at α. That is,

h h |X \|div− (h)| |X \|div− (h)| , (1.97) ∂ = θ∂ 2iπ 2iπ α where θ ∈ D(X ), θ ≡ 1 about α and Supp θ ∩ |div− (h)| = {α}. It is important to observe that all Laurent coefficients ak for k = −1 fail to be geometrically intrinsic. However, the geometrically intrinsic, localized current (1.97) encodes the Laurent coefficients ak , k < 0, involved in the representation (1.96) of h along the local centered coordinate ζ. The essence of this approach will be exemplified in the algebraic setting; see §1.3.2, in particular, Lemma 1.43 in the case X = ℂ. 1.3. The algebraic point of view 1.3.1. Cauchy’s formula in A1ℂ = ℂ or ℙ1ℂ and division in 𝕂[X]. Let 𝕂 be a subfield of ℂ. Observe that any field of characteristic 0 and cardinality at most 2ℵ0 is isomorphic to such a subfield. Let us also assume throughout this section that the polynomial p ∈ 𝕂[X] is fixed, has degree d = deg p ≥ 1 and that its dominant coefficient equals γ0 . For any a dX, with a ∈ 𝕂[X], the K¨ahler differential ω(X) = a(X) dX/p(X) ∈ Ω𝕂(X)/𝕂 induces a meromorphic section h=

a(ζ) dζ p(ζ)

of the canonical bundle of ℙ1ℂ . One can express h in homogeneous coordinates as (1.98)

h=

A(X0 , X1 ) X1 dX0 − X0 dX1 , P (X0 , X1 ) X deg a−(d−2) 0

where P ∈ Oℙ1ℂ (d) and A ∈ Oℙ1ℂ (deg a) denote the homogenizations of p and a, respectively. Let h|ℂ\p−1 ({0}) be the standard extension of h, when considered as a

1.3. THE ALGEBRAIC POINT OF VIEW

39

meromorphic form on the open chart U = ℂ Let also h|ℙ1ℂ \|div− (h)| be its standard extension, when considered as a meromorphic form on ℙ1ℂ . Proposition 1.34. One has in  D(1,1) (ℂ) that

h 1 |ℂ\p−1 ({0}) =∂ ∧ a(ζ) dζ. ∂ (1.99) 2iπ p If deg a ≤ d − 2, then one has in  D(1,1) (ℙ1ℂ ) that h|ℙ1 \|div− (h)| h|ℙ1 \|div− (h)|

ℂ ℂ = ∂ ∂ = 2iπ 2iπ |ℂ



{α∈ℂ ; p(α)=0}

1

∂ ∧ a(ζ) dζ, p α

where ( )α denotes, as usual, the localization at α. Therefore, using Stokes’s formula on the compact projective space ℙ1ℂ implies the Euler–Jacobi formula (1.100)   1

  1 h|ℙ1 \|div− (h)|  ℂ ∧a(ζ)dζ = = 0. ∂ , a(ζ) dζ = ∂ ∂ p α p 2iπ ℂ ℙ1ℂ

{α∈ℂ : p(α)=0}

Proof. The first assertion is an immediate consequence of the definition of Rf = ∂(1/f ). The second assertion follows from the fact that, if deg a − (d − 2) ≤ 0,  then the support of div− (h) does not contain ∞ = [0 : 1]; see (1.98). Corollary 1.35. Let 𝕂 be a subfield of ℂ. Let d−1

 a = pq + r = pq + ρκ (a; p)X d−1−κ κ=0

be the euclidean division formula of the polynomial a ∈ 𝕂[X] by the polynomial d p = κ=0 γκ X d−κ ∈ 𝕂[X], when d ≥ 1 and γ0 ∈ 𝕂∗ . If h is the meromorphic (1, 0)-form on ℙ1 (ℂ), whose restriction to ℂ equals a(ζ) dζ/p(ζ), then one has    1

h 1

 |ℙ (ℂ)\|div− (h)| ∂ , a(ζ) dζ = − ∂ p α 2iπ ∞ ℙ1 (ℂ) {α∈ℂ : p(α)=0}

(1.101)

=

ρd−1 (a; p) ∈ 𝕂. γ0

Proof. The first equality in (1.101) follows from Stokes’s theorem. The fact that p · ∂(1/p) = 0 in  D(1,0) (ℂ) (see (1.38) with F = (f )), and Proposition 1.34 imply that to prove the second equality, it is enough to prove that  1

  1 ∂ , ζ d−1 dζ = . p α γ0 {α∈ℂ : p(α)=0}

dζ = dp/(γ0 d) − b(ζ) dζ, when deg b ≤ d − 2. Thus, Propositions Observe that ζ 1.34 and 1.10 imply that    1

 1 1 1 ∂ , a(ζ) dζ = × ∂ ∧ dp p α γ0 d ℂ p {α∈ℂ : p(α)=0}  1 1 {α∈ℂ : p(α)=0} νp,α = , × = γ0 d γ0  when divℂ (p) = {α∈ℂ : p(α)=0} νp,α {α}, which concludes the proof.  d−1

40

1. RESIDUE CALCULUS IN ONE VARIABLE

It follows from (1.14) that for any fraction a ∈ 𝕂(X) and any α ∈ p−1 ({0}), which is not a complex pole of a,   1

∂ , a(ζ) dζ ∈ 𝕂, p α where 𝕂 denotes the integral closure of 𝕂 in ℂ. In the particular case where a is regular on p−1 ({0}), it is convenient to introduce the following definition and notation. Definition 1.36. Let p ∈ 𝕂[X] with d = deg p ≥ 1 and let a ∈ 𝕂(X) be regular on p−1 ({0}). The global residue of the K¨ ahler differential a(X) dX with respect to p is the element of 𝕂 defined and denoted4 by ) *  1

  a dX := ∂ , a(ζ) dζ . (1.102) Res𝕂[X]/𝕂 p p α {α∈ℂ : p(α)=0}

One can reformulate and at the same time extend Corollary 1.35 as follows. Corollary 1.37. Assume that p ∈ 𝕂[X] has degree d = deg p ≥ 1. Let the fraction a = a1 /a0 ∈ 𝕂(X), where a0 , a1 ∈ 𝕂[X], be such that a0 is coprime with p. If u ∈ 𝕂[X] satisfies a0 u ≡ 1 modulo p 𝕂[X] in 𝕂[X], then * ) * ) ρd−1 (a1 u; p) a(X) dX a1 u dX (1.103) Res𝕂[X]/𝕂 ∈ 𝕂. = = ResK[X]/𝕂 p p γ0 Proof. The assertion follows from Corollary 1.35, once one invokes the fact  that p · ∂(1/p) = 0 in  D(0,1) (ℂ). Taking into account the material from §1.2.5, Cauchy’s formula in the affine space 𝔸1ℂ = ℂ admits the following current formulation. For any z ∈ ℂ,

dζ  = [1] − [{z}], (1.104) 2iπ (ζ−z)∂/∂ζ − ∂ 2iπ (ζ − z) ℂ\{z} where (ζ, z) −→ (ζ − z) ∂/∂ζ plays the role of a section of a holomorphic line bundle over ℂζ × ℂz . The zero set of this section equals precisely the diagonal subset ζ = z and is of multiplicity 1. As usual, [{z}] denotes the integration (1, 1)current attached to the divisor {z}, and [1] describes the (0, 0)-current induced by  the constant function 1. Given p = dκ=0 γκ X d−κ in 𝕂d [X] with γ0 ∈ 𝕂∗ as before, let d−1 κ

 p(Y ) − p(X)   = (1.105) b(X, Y ) = γκ Y κ−κ X d−1−κ ∈ 𝕂[X, Y ]. Y −X  κ=0 κ =0

For a fixed z ∈ ℂ one has that (1.106)

    2iπ(ζ−z)∂/∂ζ − ∂ ζ b(z, ζ) dζ = 2iπ(ζ−z)∂/∂ζ b(z, ζ) dζ = 2iπ [p(ζ) − p(z)]. According to the terminology already used in §1.2.2 within the analytic context, the polynomial b(X, Y ) is called a b´ezoutian for p. The tautological identity

dζ b(z, ζ) dζ   = , 2iπ (ζ − z) ℂ\{z} 2iπ(p(ζ) − p(z)) ℂ\p−1 ({p(z)}) 4 The

notations used here were introduced by R. Hartshorne and J. Lipman [Ha66, Lip87].

1.3. THE ALGEBRAIC POINT OF VIEW

41

allows Cauchy’s formula to bridge euclidean division in 𝕂[X] with algebraic residue calculus with 𝕂 as field of constants. Proposition 1.38. Let p ∈ 𝕂d [X], the 𝕂 vector space of polynomials with degree d ∈ ℕ∗ , and let b ∈ 𝕂[X, Y ] be as in (1.105). For any a ∈ 𝕂[X] and N ∈ ℕ∗ , the quotient and the remainder in the euclidean division formula a = pN qN + rN in 𝕂[X] can be expressed by )

a [ deg d ]−N



qN (X) =

Res𝕂[Y ]/𝕂

k=0

(1.107)

=

a [ deg ]−N d−1 d 

k=0

Res𝕂[Y ]/𝕂

rN (X) =

) Res𝕂[Y ]/𝕂

k=0

(1.108)

=

N −1 d−1  

κ =0 k+N +1

X d−1−κ pk (X)

p

κ=0

and N −1 

* a(Y ) b(X, Y ) dY pk (X) pk+N +1 ⎡ ⎤ 

κ κ−κ  γκ Y dY ⎦ ⎣a(Y )

* a(Y ) b(X, Y ) dY pk (X) pk+1 ⎡ ⎤

 κ κ−κ  dY a(Y ) γ Y κ ⎣ ⎦

Res𝕂[Y ]/𝕂

κ =0

p

k=0 κ=0

X d−1−κ pk (X),

k+1

respectively. Proof. Let R > 0 be such that p−1 ({0}) ⊂ D(0, R). As seen in §1.2.3 (see (1.62) applied for F = (p) and h = a), one has that for any z ∈ D(0, R) the identity ) * ∞  a(Y ) b(z, Y ) dY k Res𝕂[Y ]/𝕂 (1.109) a(z) = p (z) pk+1 k=0

holds, provided that |p(z)| < min|ζ|=R |p(ζ)|. On the other hand, it follows from Proposition 1.34 that ) * a(Y ) b(X, Y ) dY kd > deg a =⇒ Res𝕂[Y ]/𝕂 ≡ 0. pk+1 Then the analytic identity (1.109), valid for z ∈ D(0, R) which satisfies the inequality |p(z)| < min|ζ|=R |p|, truncates as the algebraic identity a [ deg d ]

a(X) =



k=0

Res𝕂[Y ]/𝕂

) * a(Y ) b(X, Y ) dY pk (X). pk+1

The assertion of the proposition follows from the uniqueness of the quotient and remainder in the euclidean division.  Corollary 1.39. Let p0 and p1 be two coprime polynomials in 𝕂[X], with respective degrees d0 ≥ 1, d1 ≥ 1. B´ezout’s identity5 1 = u0 p0 + u1 p1 , where 5 Such identity will play a central role in the multi-variate setting later on in the present monograph; see Chapters 6 and 7. Although it goes back to Euclid and is not explicitly formulated ´ in such a way by Etienne B´ezout (1730–1783) himself in his work (see, for example, the excellent

E. B´ ezout biography by Liliane Alfonsi [Alf]), its pivotal role in B´ ezout’s contributions justifies the terminology we will use from now on.

42

1. RESIDUE CALCULUS IN ONE VARIABLE

deg u0 ≤ d1 − 1 and deg u1 ≤ d0 − 1 is equivalently expressed in 𝕂[X] as   ⎡ ⎤ 1 b0 (X, Y ) p0 (X) dY ⎦ 1 = Res𝕂[Y ]/𝕂 ⎣ p1 (Y ) b1 (X, Y ) p1 (X) p0   ⎡ ⎤ 1 b0 (X, Y ) p0 (X) dY ⎦. = −Res𝕂[Y ]/𝕂 ⎣ p0 (Y ) b1 (X, Y ) p1 (X) (1.110) p1 The b´ezoutians b0 , b1 ∈ 𝕂[X, Y ] for the polynomials p0 and p1 , respectively, are determined by (1.105). Proof. One has in 𝕂(X, Y ) that    1 b0 (X, Y ) p0 (X) b0 (X, Y ) = p1 (Y ) b1 (X, Y ) p1 (X) b1 (X, Y )

 p0 (Y )/p1 (Y ) , 1

  p0 (X) b0 (X, Y ) = p1 (X) b1 (X, Y )

  1 . p1 (Y )/p0 (Y )

 1 b0 (X, Y ) p0 (Y ) b1 (X, Y )

Since pj · ∂(1/pj ) = 0 in  D(0,1) (ℂ), we have that  ⎡ ) * 1 b0 (X, Y ) b (X, Y ) dY Res𝕂[Y ]/𝕂 0 = Res𝕂[Y ]/𝕂 ⎣ p1 (Y ) b1 (X, Y ) p0 p0

 ⎤ p0 (X) dY ⎦ p1 (X) ,

  ⎡ ⎤ ) * 1 b0 (X, Y ) p0 (X) b (X, Y ) dY dY ⎦ . Res𝕂[Y ]/𝕂 1 = −Res𝕂[Y ]/𝕂 ⎣ p0 (Y ) b1 (X, Y ) p1 (X) p1 p1 The assertion of the corollary then follows from Proposition 1.38 with a = 1 and N = 1.  Let us realize the above ideas in the setting of the compact Riemann surface ℙ1ℂ . This compact surface is the Riemann sphere, whose genus is equal to 0 and whose first homology group is trivial H 1 (ℙ1ℂ ) = 0. Its canonical bundle6 is O(−2). The diagonal subspace of the product space (ℙ1ℂ )z × (ℙ1ℂ )ζ , equipped with the projections πz and πζ on the coordinate spaces, is defined once again as the zero set of a section of a holomorphic line bundle L. Namely, let (1,0)

L = Homℂ ((O(1))ζ , (O(1))z ) ⊗ (Tℙ1

ℂ

)ζ

 (O(1))z ⊗ (O(−1))ζ ⊗ (O(2))ζ = (O(1))z ⊗ (O(1))ζ , where Lz = πz∗ L and Lζ = πζ∗ L for any holomorphic line bundle over ℙ1ℂ . The section of L expressed in homogeneous coordinates as s([z0 : z1 ], [ζ0 : ζ1 ]) = −z0

(1.111)

∂ ∂ − z1 ∂ζ0 ∂ζ1

is such that V (s) = {(z, ζ) ∈ ℙ1ℂ ×ℙ1ℂ : z0 ζ1 −z1 ζ0 = 0}. Thus V (s) is the diagonal subspace of ℙ1ℂ × ℙ1ℂ and the vanishing order of the section s is equal to 1. The 6 We

denote Oℙ1 (k) by O(k) for any k ∈ ℤ. ℂ

1.3. THE ALGEBRAIC POINT OF VIEW

43

L∗  L−1 -valued meromorphic form over the open subset U = {(z, ζ) ∈ ℙ1ℂ × ℙ1ℂ ; z0 ζ1 − z1 ζ0 = 0}, defined by (1.112)

(z, ζ) ∈ U −→

ζ 1 ζ1 dζ0 − ζ0 dζ1 ζ02 1 = d , 2iπ ζ0 z1 − ζ1 z0 ζ0 z1 − ζ1 z0 ζ0

is such that for all z = [z0 : z1 ] ∈ ℙ1ℂ the equality

1 ζ dζ − ζ dζ

  1 0 0 1  = [1] − [{z}] ⊗ Id (O(−1))z (1.113) 2iπs − ∂ ζ  1 2iπ ζ0 z1 − ζ1 z0 ℙℂ \{z} holds. The relation (1.113) is the current formulation of Cauchy’s formula on ℙ1ℂ to be compared with its current analogue (1.104) on the affine space 𝔸1ℂ . For d ≥ 1, let P be a global section of O(d) defined over 𝕂. It means that P is a homogeneous polynomial with degree d in X0 , X1 . Let B = (B0 , B1 ) be a pair of (nonunique) global sections Bj (X0 , X1 , Y0 , Y1 ) of Oℙ3ℂ (d − 1) defined over 𝕂 such that (1.114)

P (Y0 , Y1 ) − P (X0 , X1 ) = B0 (X, Y )(Y0 − X0 ) + B1 (X, Y )(Y1 − X1 ).

This amounts to saying that B0 and B1 are homogeneous polynomials with degree d − 1 in four variables with coefficients in 𝕂 satisfying (1.114). At this point, one needs to equip holomorphic bundles, among them the O(1), with hermitian metrics. One considers the line bundle O(1) equipped with the Fubini–Study metric7 having its first Chern form equal to 1 (ζ1 dζ0 − ζ0 dζ1 ) ∧ (ζ1 dζ0 − ζ0 dζ1 ) 2iπ ζ4 ζ ζ

1 0 1 + dζ dζ0 ∧ ∂ =− ∧ ∂ 1 2iπ ζ2 ζ2 and Chern connection D satisfying c = ddc log ζ2 =

Dζj = dζj − ζj ∂ log ζ2 for j = 0, 1; see §B.3.2.2. For (z, ζ) ∈ ℙ1ℂ × ℙ1ℂ , let w(z, ζ) =

z0 ζ 0 + z1 ζ 1 + c(ζ). ζ2

  One defines in this way a smooth form with values in ℂ ⊕ Hom (O(1))ζ , (O(1))z . This form is holomorphic in z, but unfortunately not in ζ. Nevertheless, it satisfies ζ ζ 0 1 (1.115) (2iπs − ∂ ζ )(w) = −z0 ∂ − z1 ∂ 2 ζ ζ2 ζ ζ

0 1 + dζ = 0. − s dζ0 ∧ ∂ ∧ ∂ 1 ζ2 ζ2 For a fixed z in ℙ1ℂ , the algebraic identity in z, ζ, ζ, dζ, dζ



  (1.116) 2iπs − ∂ ζ B0 (z, w ζ) Dζ0 + B1 (z, w ζ) Dζ1 = 2iπ wd P (ζ) − P (z) 7 In some sense, as will be specified in this monograph, such a metric is the most “algebraic” that is available on O(1) given Crofton’s formula; see §D.5.1 and §D.5.2. The canonical metric (ζ0 , ζ1 )∞ instead of (ζ0 , ζ1 )2 would certainly be a better choice given diophantine geometry. Unfortunately, it is just continuous and fails to be smooth as the Fubini–Study metric is.

44

1. RESIDUE CALCULUS IN ONE VARIABLE

holds. The field 𝕂 is still considered the field of coefficients since Bj ∈ 𝕂[X, Y ] for j = 0, 1. The identity (1.116) resulting from (1.115) and from (1.114) plays in (ℙ1ℂ )ζ the same role as the identity (1.106) plays in the affine space (𝔸1ℂ )ζ , when z ∈ 𝔸1ℂ is fixed. It allows Cauchy’s formula in ℙ1ℂ in its current formulation (1.113) to bridge geometric residue calculus on ℙ1ℂ , as introduced in §1.2.5, and division problems. The major difference is the necessity to introduce the Fubini–Study metric on O(1), its Chern connection, and the related weight w, to keep track of homogenization. Unlike what happens in 𝔸1ℂ , the introduction of the above machinery in our calculus prevents us to respect in this context the basis field 𝕂. One needs to invoke averaging to interpret algebraically closed-form division formulae that are obtained. This approach will be carried through in ℙN ℂ , see §4.4, but we wanted to introduce it briefly in the univariate setting as soon as possible. 1.3.2. Currential Cauchy’s formula on compact Riemann surfaces. We consider in this section a connected compact Riemann surface X with genus g ≥ 1 and Euler characteristic 2(1−g) ≤ 0. To formulate Cauchy’s formula in terms of bundle-valued currents on X , as we did in §1.3.1 for X = ℂ or X = ℙ1ℂ (the case of genus g = 0), we need to recall some basic facts about compact Riemann surfaces. For more details, we suggest to the reader the following references [Fa73, Mu84, FK, Bos92]. A connected compact Riemann surface X inherits the structure of an algebraic projective nonsingular curve embedded in ℙ3ℂ . If g = 0 the Riemann surface X equals the Riemann sphere, which may be also identified with the projective line ℙ1ℂ . When the genus g = 1, X is the complex torus, while X is a hyperbolic compact Riemann manifold, when g > 1. Since we assume here g ≥ 1, Riemann’s uniformization theorem implies the existence of a universal covering  ℂ for g = 1 (1.117) π : X& → X , where X& = 𝔻 for g > 1. ∗(1,0)

The canonical bundle TX = ωX has a degree equal to 2(g − 1). The first object one needs to introduce is a canonical ℤ-basis {a, b} for the 2g-lattice H 1 (X , ℤ). The notation means that a = {a1 , . . . , ag } and b = {b1 , . . . , bg } are two families of g elements in the first homology group H 1 (X , ℤ) such that the intersection  numbers8 of (aκ , bκ ), (aκ , aκ ), (bκ , bκ ) for any 1 ≤ κ, κ ≤ g are equal to δκκ , 0, 0 respectively. We let also {1 , . . . , g } be a basis of the g-dimensional ℂ-vector space Ω(X) = O(X , ωX ) of abelian differentials on X normalized with respect to a. That is,   1 (1.118) κ = δκκ , 2iπ ακ for each 1 ≤ κ, κ ≤ g. Let also



1  κ H= 2iπ bκ 1≤κ,κ ≤g

be the symmetric Riemann period matrix with positive definite imaginary part. Let J(X ) = ℂg /(ℤg + Hℤg ) (with projection quotient morphism πJ ) be the jacobian 8 For

intersection theory between elements in H 1 (Xℝ , ℤ), we refer the reader to [FK, III.1].

1.3. THE ALGEBRAIC POINT OF VIEW

45

of X , o be the marked origin on X and μ : X → Γ be the Abel–Jacobi map defined by  x

 x modulo ℤg + H ℤg . (1.119) μ(x) = 1 , . . . , g o o   The map (1.119), acting on Weil divisors by μ( ι νι {αι }) = ι νι μ(αι ), realizes a homomorphism from the Picard group Pic(X ) into jacobian J(X ) of X . This homomorphism induces an isomorphism between the subgroup Pic0 (X ) of elements with degree9 0 and J(X ), thanks to Abel’s inversion theorem. The Abel–Jacobi map μ is lifted to the universal covering X& as the morphism μ & : X& → ℂg satisfying μ ◦ π = πJ ◦ μ &.

(1.120)

The fundamental entire function of g complex variables attached to the positive definite matrix H, hence attached to the canonical ℤ-basis {a, b} of H 1 (X , ℤ), is its thˆeta function θ = θH defined by g g  

 exp iπ h, κ κ + 2iπ ⟪κ , w⟫ , ∀ w ∈ ℂg . (1.121) θ(w) = =1  =1

κ∈ℤg

Let w0 = v0 +Hu0 ∈ ℂg , where u0 , v0 ∈ ℝg be such that θ(w0 ) = 0. The normalized thˆeta function with characteristics θ[w0 ]/(θ[w0 ](0)) is the entire function on ℂg defined by   θ(w + w0 ) θ[w0 ] (w) = exp 2iπ ⟪u0 , w⟫ , ∀ w ∈ ℂg . (1.122) θ[w0 ](0) θ(w0 ) Normalization, which will be important for us, corresponds to the fact that θ[w0 ] (0) = 1. θ[w0 ](0) Let us make the first observation. Proposition 1.40. Let j ∈ J(X ) be such that θ(w0 ) = 0 for some w0 in πJ−1 ({j}). Then θ(w) = 0 for any w ∈ πJ−1 ({j}). Moreover, θ defines a Cartier divisor divJ(X ) (θ) on the g-dimensional complex manifold J. All entire functions in g variables defined by w ∈ ℂg −→

θ[w0 ] (w), where w0 ∈ πJ−1 ({j}) θ[w0 ](0)

coincide with the entire function denoted by w ∈ ℂg −→

θ[j] (w). θ[j](0)

For any such j, the multi-valent function (1.123)

ζ ∈ X −→

θ[j] θ[j] (ζ) := (& μ(ξ)), θ[j](0) θ[j](0)

where ξ ∈ π −1 ({ζ})

is interpreted as a section of the unitary flat10 line bundle with class μ−1 ({j}) belonging to Pic0 (X ). 9 The degree of L ∈ Pic(X ) is defined as the degree of the Weil divisor attached to any Cartier divisor representing L. 10 An holomorphic line bundle L over X is said to be unitary flat if and only if it can be equipped with a hermitian metric | | with first Chern class c1 (L) = 0; see (B.3.3).

46

1. RESIDUE CALCULUS IN ONE VARIABLE

Proof. We give here a sketch of the proof. Its details can be found, for example, in [Fa73, I]. One has the quasi-periodicity relations (1.124) g g 

 θ(w + k ) = θ(w), θ(w + H · k ) = exp − iπ h, k k  − 2iπ⟪k , w⟫ θ(w) =1  =1 



for any w ∈ ℂ and k , k ∈ ℤ . The first assertion follows. Take now two elements ξ0 and ξ1 in π −1 ({ζ}). Then it follows from (1.120) that g

g

μ &(y0 ) − μ &(y1 ) = k + H · k ∈ ℤg + H · ℤg . Quasi-periodicity relations (1.124) imply that, if w0 = v0 +H ·u0 , where u0 , v0 ∈ ℝg , then  θ[w0 ]   θ[w0 ] (& μ(ξ1 )) = exp 2iπ ⟪k , v0 ⟫ − ⟪k , u0 ⟫ (& μ(ξ0 )). θ[w0 ](0) θ[w0 ](0) As a consequence, the multi-valent function (1.123) can be understood as a meromorphic section over X of a unitary flat bundle representing an element of Pic0 (X ). Observe finally that the image of a such element of Pic0 (X ) by the Abel–Jacobi  map μ equals the class of v0 + H · u0 modulo ℤg + H · ℤg , that is, j. Riemann’s theorem for half-periods, see [Fa73, Theorem 1.1], asserts that there √ exists an element ωX ∈ Pic(X ), whose degree equals √ √ deg ωX = g − 1 = μ( ωX ), √ √ √ so that ωX ⊗ ωX = ωX in Pic(X ). Meromorphic sections of ωX are naturally called (meromorphic ) half-order differentials on X . Furthermore, there are 22g elements in Pic(X ), whose meromorphic sections have the properties that allow them to be also called (meromorphic) half-order differentials on X . We are ready now to formulate the following definition. Definition 1.41. An element of Pic(X ), whose meromorphic section is called √ (meromorphic) half-order differentials on X , is of the form ωX ⊗(μ−1 ({j})), where 0 μ−1 ({j}) ∈ Pic (X ) is attached by Abel’s inversion theorem to any j ∈ J(X ) so that ˙ There are 22g elements of this form in Pic(X ), all of them having their 2 j = 0. degree equal to g − 1 and all of them satisfying  √  √ ωX ⊗ (μ−1 ({j})) ⊗ ωX ⊗ (μ−1 ({j})) = ωX ⊗ (μ−1 ({2 j})) = ωX . (1.125) A fundamental fact is that the line bundle √ √ (1.126) L = (πz∗ ( ωX ))−1 ⊗ (πζ∗ ( ωX ))−1 ⊗ ρ∗ (0J(X ) ), over Xz × Xζ , where

  ρ : (z, ζ) ∈ X × X −→ μ {z} − {ζ} ∈ J(X )

and 0J(X ) is the null bundle over J(X ), admits an holomorphic section E, whose zero set is equal exactly to the diagonal {(z, ζ) ∈ X × X : z = ζ} and it vanishes to order 1 on this diagonal. To be more specific, given a local holomorphic coordinate t about z, one has, for ζ close to z in X , that   t(ζ) − t(z) / (1.127) E(z, ζ)  / 1 + O (t(ζ) − t(z))2 . dt(z) dt(ζ)

1.3. THE ALGEBRAIC POINT OF VIEW

47

Moreover, E is antisymmetric. Namely, (1.128)

E(z, ζ) = −E(ζ, z),

∀ (ζ, z) ∈ X × X .

We refer to [Mu84, IIIb-§1] for construction of E, together with its list of properties, among them (1.127); see also [Fa73, II]. Note in particular, that this construction can also be carried out in the case where g = 0. That is, it can be realized in the case when X = ℙ1ℂ , as explained on page 3.209 in [Mu84]. Moreover, it can be considered as an alternative way for the construction proposed in §1.3.1. Such section E of L is called the prime form of X with respect to the ℤ-canonical basis {a, b}, the corresponding matrix H, and thˆeta function θ = θH . Then a Cauchy-type formula in terms of distributions holds for smooth sections √ of any line bundle ωX ⊗μ−1 ({j}), when j ∈ J(X ) is such that the θ does not vanish −1 on πJ ({−j}).11 This holds in particular for smooth sections of the line bundles introduced in Definition 1.41, provided that j ∈ J(X ), satisfies 2 j = 0˙ and is such that θ does not vanish on πJ−1 ({j}). Proposition 1.42. Let j ∈ J(X ) be such that θ does not vanish on πJ−1 ({−j}) √ and let ϕ be a smooth section of ωX ⊗ μ−1 ({j}) on X . For any z ∈ X , the section θ[−j]

1 ζ ∈ X −→ (ζ − z) ⊗ θ[−j](0) E(z, ζ) √ is a meromorphic section of ωX ⊗ μ−1 ({−j}). Furthermore, the section

θ[−j] 1 (ζ − z) ⊗ ζ ∈ X −→ ϕ(ζ) ⊗ θ[−j](0) E(z, ζ) is also a semimeromorphic section of √    √  ωX ⊗ μ−1 ({j}) ⊗ μ−1 ({−j} ⊗ ωX = ωX , whose standard extension to X satisfies  θ[−j]

1 1  . (ζ − z) ⊗ ∂ ζ ϕ(ζ) ⊗ (1.129) ϕ(z) = − 2iπ X θ[−j](0) E(z, ζ) X \{z} Proof. Proposition 1.40 implies that θ[−j] ζ −→ (ζ) θ[−j](0) is a section of μ−1 ({−j}). The first assertion follows from the definition of the bundle L, see (1.126), whose section E(z, ζ) is. Formula (1.129) follows from the fact the meromorphic section

θ[−j] 1 (ζ − z) ⊗ ζ −→ θ[−j](0) E(z, ζ) √ of ωX ⊗ μ−1 ({−j}) admits z as a single simple pole, that the prime form is expressed in terms of a local coordinate t about z as (1.127), and that the entire function θ[−j] (w) w ∈ ℂg −→ θ[−j](0)  √  condition on j is equivalent to the fact that OX X , ωX ⊗μ−1 (−{j}) = 0. Meromor√ phic sections of some ωX ⊗ μ−1 ({j}), with j ∈ X , are sometimes called multiplicative half-order differentials on X ; see [AlPV20, §2.1] for such terminology. 11 Such

48

1. RESIDUE CALCULUS IN ONE VARIABLE

has been specifically normalized so that its value at 0 equals 1. Observe that for ζ fixed in X , 1 θ[−j] 1 θ[−j] (ζ − z) ⊗ =− (ζ − z) ⊗ z −→ θ[−j](0) E(z, ζ) θ[−j](0) E(ζ, z) √ is a section of ωX ⊗ μ−1 ({j}), which is coherent with the current Cauchy repre√  sentation (1.129) of ϕ ∈ C ∞ (X , ωX ⊗ μ−1 ({j}). Let h be a meromorphic function on X , and let U be an open subset of X . If W is the finite polar set of h in U , then h|U\W ∈  D(0,0) (U ) is the standard √ extension of h from U \ W to U . Let also f ∈ OX U, ωX ⊗ μ−1 ({j}) , where j is such that θ does not vanish on πJ−1 ({−j}). For each z ∈ U \ W we consider the (1, 0)-meromorphic form in U having single, simple pole at z:

θ[−j] 1 (ζ − z) ⊗ . (1.130) ζ −→ f (ζ) ⊗ θ[−j](0) E(ζ, z) The residue current ∂(h|U\W /(2iπ)) ∈  D(0,1) (U ) with support W acts by duality on the (1, 0) meromorphic form (1.130), since this form is holomorphic in the neighborhood U \ {z} of W in U . Moreover

θ[−j]

 h 1  |U\W (ζ) , f (ζ) ⊗ (ζ − z) ⊗ (1.131) z ∈ U \ W −→ ∂ 2iπ θ[−j](0) E(ζ, z) √ −1 defines a meromorphic section of ωX ⊗ μ ({j}), with polar set contained in W . Observe that the order of such meromorphic section at a point α ∈ W equals, at most, to the order of h at α. In the particular case where any α ∈ W is a pole with order 1 of h, it follows from the normalization θ[{−j}]/θ[{−j}](0) = 1 at the origin in ℂg and from (1.127) that the meromorphic extension of (1.131) to U coincides with the polar part of hf in U . In other words, the section

θ[−j]

 h 1  |U\W (ζ − z) ⊗ (ζ) , f (ζ) ⊗ z ∈ U \ W −→ h(z)f (z) − ∂ 2iπ θ[−j](0) E(ζ, z) √ −1 is the restriction to U \ W of a holomorphic section of ωX ⊗ μ ({j}) in U . This idea has been extensively used in operator theory concerning the concept of resolvent; see for example [Vin98, §3], [AlV02] or [AlPV20]. The situation is by far more involved in the general case when the poles of h in U are not supposed to be simple anymore. If such is the case, then coefficients ak (h; α) for  the Laurent k dt in a local coordinate k ≤ −1 involved in the expansion of h = k∈ℤ ak (h; α)t t centered at the pole α ∈ W with order > 1 depend precisely on the choice of the local coordinate t about α. This prevents them to be defined intrinsically, while the definition of the meromorphic function (1.131) is intrinsic. Let us mention also here the following elementary local lemma about germs of meromorphic functions at the origin in ℂ. Lemma 1.43. Let f ∈ Oℂ,0 be a germ of holomorphic function at the origin 0 ∈ ℂ. Let also h be a germ of meromorphic function at the origin such that o(h; α) = ν ∈ ℕ∗ , with standard extension [h] ∈ ( D(0,0) (ℂ))0 as a germ of (0, 0)current. Then   dζ  z ∈ ℂ∗ −→ ∂[h] 0 (ζ) , f(ζ) z−ζ extends as a meromorphic function with single pole ζ = 0, whose germ at this pole is exactly the polar part of hf about the origin.

1.3. THE ALGEBRAIC POINT OF VIEW

49

Proof. Let f = u/ζ ν , where u is an invertible germ in Oℂ,0 . Then, for any z ∈ ℂ∗ ,   dζ   1

dζ  = ∂ ν ∂[h] 0 (ζ) , f(ζ) , (uf)(ζ) z−ζ ζ z−ζ 0 ν−1 1 1   1

∂ ν−1−κ , (uf )(ζ) dζ κ = z κ=0 ζ z 0 =

ν−1 ν−1 1  aν−κ−1 (u f; 0) 1  = aκ (u f; 0) z κ . z κ=0 zκ z ν κ=0

 1.3.3. Green currents and heights of arithmetic cycles. This section is inspired by S. Lang’s presentation of Arakelov’s theory of arithmetic surfaces [Ar74, L88]. In what follows, X is a connected compact nonsingular Riemann surface, which inherits a structure of the complex algebraic projective nonsingular curve. 1 We will focus mainly on the   case X = ℙℂ or on the case of the arithmetic scheme 𝕏 = Spec Proj(ℤ[X0 , X1 ]) , when arithmetic considerations are  of importance. A 0-cycle (or Weil divisor) on X is a formal finite sum Z = α∈X να {α}, where α ∈ X and να ∈ ℤ∗ . Its support |Z| equals the finite subset  (1.132) |Z| = {α ∈ X : Z = να {α} with να ∈ ℤ∗ }. α

This 0-cycle is decomposed to the sum of a positive and negative parts as 



 Z = Z+ + Z− = να {α} + να {α} . {α: να ∈ℕ∗ }

It is said to be effective if Z defined to be

−

{α: να ∈−ℕ∗ }

= ∅. The geometric complexity of the cycle Z is deg Z =



να .

α

If L ∈ Pic(X ), then deg L is the degree of the Weil divisor (or 0-cycle) Z = ZD , corresponding to any representant D of L as a Cartier divisor; see §D.2. The degree deg L does not depend on the representative D. Let D = (D, | |) be the line bundle corresponding to D, equipped with an hermitian continuous metric | |, whose Chern current is the (1, 1)-current c1 (D); see Definition B.46. Proposition 1.44. Let L, D, and D be as above. For any meromorphic section s of D, one defines the current G(D; s) ∈  D(0,0) (X ) as follows:12

(1.133)

G(D; s) = [− log |s|2 ] about ζ ∈ / |ZD |

−|s|2λ  + G(D; s) = about ζ ∈ |ZD | λ λ=0

|s|−2λ  − G(D; s) = about ζ ∈ |ZD |. λ λ=0

12 The almost everywhere (a.e.) defined section − log |s|2 is locally integrable on X and the current G(D; s) defined in (1.133) equals the current [− log |s|2 ] everywhere. Nevertheless, it will be convenient to exploit in the multivariate situation the analytic continuation approach presented here.

50

1. RESIDUE CALCULUS IN ONE VARIABLE

 The notation means that one considers the meromorphic continuation in λ λ=0 from {λ ∈ ℂ : Re λ  1} to a complex plane ℂ of a  D(0,0) (X )-valued map (1.133) and then takes the coefficient a0 in the Laurent development of such continuation about λ = 0. The current G(D, s) satisfies in  D(1,1) (X ) the Lelong–Poincar´e–Green equation     1 ∂∂ G(D; s) + [div(s)] = c1 (D). (1.134) ddc G(D; s) + [div(s)] = 2iπ Proof. Let us examine the second case in (1.133). In particular, we are interested to see in detail what happens in a local chart about α ∈ ℤ+ . One has, for Re λ  1, that ∞  [(log |s|2 ])κ exp(λ log |s|2 ) 1 −|s|2λ =− = − − [log |s|2 ] − λ λ λ κ! κ=1

in  D(0,0) (X ), because all functions (log |s|2 )κ are locally integrable on X and hence integrable, since X is compact, for κ ∈ ℕ∗ . Let t be a local coordinate centered at α (t(α) = 0) and e be a local frame for X about α. Then s is factorized as s(t) = h(t)e(t), where h is a meromorphic function. One has −|s|2λ |h|2λ

∂ 2λ ∂h |e|2λ = − + ∂ log |e|2 |e|2λ ddc = ddc − |h| λ λ 2iπ h |h|2λ ∂h ∂h + ddc log |e|2 = −|e|2λ λ ∧ 2iπ h h

∂ log |e|2 ∂h + ∂ log |e|2 ∧ . + λ |h|2λ h 2iπ The meromorphic continuation of such current-valued section s of λ is holomorphic up to the origin and its value at λ = 0 equals −[div(h)] − ddc log |e|2 , given Proposition 1.10 (with m = 1). Since c1 (D) = −ddc log |e|2 (see Definition B.46), the Lelong–Poincar´e formula (1.134) is proved.  Proposition 1.44 suggests the following definition. Definition 1.45. Let L be a holomorphic line bundle over X and let s be a meromorphic section of L. A Green current with respect to (L, s) is an element G ∈  D(0,0) (X ) satisfying (1.135)

ddc G + [div(s)] = [ϕ] where

∞ ϕ ∈ C1,1 (X ).

Given a continuous metric | | on L, a Green current G with respect to (L, s) is said to be normalized with respect to (L, s), where L = (L, | |), if and only if [ϕ] = c1 (D), where c1 (D) is the first Chern current of L. Example 1.46. Let s be a meromorphic section of a metrized holomorphic bundle E = (E, | |) with rank(E) > 1. Assume also that s is factorized in the form s = f ⊗ u, where f is a meromorphic section of the line bundle L corresponding to the Cartier divisor attached to s and u is a nonvanishing section of L−1 ⊗ E; see Proposition 1.30. Let | |s be the unique hermitian metric on L such that |f|s = |s|. One has

|s|2λ    = −[div(f)] − c1 L, | |s . (1.136) ddc ([− log |s|2 ]) = ddc − λ λ=0

1.3. THE ALGEBRAIC POINT OF VIEW

51

Given two holomorphic line bundles (L, s) and (L , s ) with respective meromorphic sections s and s , the 0-cycles div(s) and div(s ) are said to intersect properly on X (or define a proper intersection on X ) if and only if the intersection of their supports |div(s)| ∩ |div(s )| = ∅. We describe the proper intersection in terms of 0-cycles or (1, 1)-currents respectively as (1.137)

div(s) • div(s ) = div(1) (with |div(1)| = ∅) [div(s)] ∧ [div(s )] = 0 (for dimension reasons).

Let now G and G be normalized Green currents with corresponding line bundles (L, s) and (L , s ). Recall that the hermitian metrics on L and L are smooth. It appears illuminating13 to express the fact that the (1, 1)-current (1.138)

G  G := G [div(s )] + c1 (L) G ,

satisfies the trivial identity ddc (G  G ) = 0 (for dimension reasons), as ddc (G  G ) = ddc G ∧ [div(s )] + c1 (L) ∧ ddc G     = c1 (L) − [div(s)] ∧ [div(s )] + c1 (L) ∧ c1 (L ) − [div(s )] = −[div(s)] ∧ [div(s )] + c1 (L) ∧ c1 (L ) = −[div(1)] + c1 (L) ∧ c1 (L ). Thus G  G is interpreted as a normalized Green current for the section s ≡ 1 of the trivial bundle X × ℂ, equipped with the flat hermitian structure induced by the canonical hermitian metric on ℂ. This leads to the following definition. 

Definition 1.47. Let L and L be two holomorphic line bundles equipped with smooth metrics. Let also G and G be normalized Green currents corresponding to the line bundles (L, s) and (L , s ). The Chow arithmetic14 product of the pairs (div(s), G) and (div(s ), G ) is defined by       div(s), G  div(s ), G = div(1), G  G

(1.139) = div(1), G [div(s )] + c1 (L) G . From now on let us consider the case of X = ℙ1ℂ . We equip the line bundle L = O(1) = Oℙ1ℂ (1) with the Fubini–Study metric  1 defined as follows: for any (ξ0 , ξ1 ) ∈ ℂ2 and ζ ∈ ℙ1ℂ , [0]

(1.140)

|ξ0 ζ0 + ξ1 ζ1 | ξ0 ζ0 + ξ1 ζ1 1 = / . |ζ0 |2 + |ζ1 |2

13 This formal reformulation of a known fact will be justified in the higher dimensional setting. There, multiplication between currents is realized using analytic continuation method, inspired by those described in §1.2.4. It also makes sense in the univariate setting, but within the arithmetic frame that will be presented later in this subsection. 14 The use of the word arithmetic here is not out of place, although there is no arithmetic involved up to this point, since algebraic projective curves are considered over the basis field ℂ. The justification of the term arithmetic used is based on the fact that such an operation will play its full role within the arithmetic setting, for example, when algebraic projective curves considered could be as well assumed as defined over ℚ or ℚ.

52

1. RESIDUE CALCULUS IN ONE VARIABLE

0 [0] = (L[0] ,   ) equals Then the first Chern form of L 1 (1.141)

& c = ddc (log(|ζ0 |2 + |ζ1 |2 )) =

1 d(ζ1 /ζ0 ) ∧ d(ζ1 /ζ0 )  2 . 2iπ 1 + |ζ1 /ζ0 |2

It coincides with the K¨ahler form on ℙ1ℂ generating the ℂ-vector space of harmonic (1, 1)-forms on ℙ1ℂ . Proposition 1.44, together with Stokes’s theorem, implies that    c 2 & c= dd (− log ζ0 1 ) + [div(ζ0 )] (1.142)

ℙ1ℂ

ℙ1ℂ

ℙ1 (ℂ)

 =

[div(ζ0 )] = deg(div(ζ0 )) = 1. ℙ1 (ℂ)

More generally, for any δ ∈ ℕ∗ , the (Fubini–Study) metric  δ on the line bundle O(δ) (with degree δ) is defined as follows: for (ξ0 , . . . , ξδ ) ∈ ℂδ+1 and ζ ∈ ℙ1ℂ ,   d δ κ δ−κ   1 1 κ=0 ξκ ζ0 ζ1 κ δ−κ 1 1 (1.143) ξκ ζ0 ζ1 = / . δ ( (|ζ0 |2 + |ζ1 |2 )δ κ=0 %δ the metrized line bundle (O(δ),  δ ), so that For any δ ∈ ℕ∗ , we denote by L 0 %1 . L[0] = L Given any T ∈  D(0,0) (ℙ1ℂ ), the current c T ⊥c = T − T, &

(1.144) satisfies (1.145)

T



⊥ c

,& c  = T, & c  − T, & c

ℙ1ℂ

& c = 0,

taking into account (1.142). The current (1.144) is said to be the projection of T orthogonally to harmonic forms. The following lemma provides a complement to Definition 1.45, when one works on a k¨ ahlerian compact one-dimensional manifold c). such as (ℙ1ℂ , & Lemma 1.48. Let L = (L, | |) be a holomorphic line bundle on ℙ1ℂ equipped with a continuous metric and let s be a meromorphic section of L. The current

   − log |s(ζ)|2 & c(ζ) & c [− log |s|2 ]⊥c = [− log |s|2 ] − ℙ1ℂ

is the unique normalized Green current with respect to (L, s) orthogonal to harmonic (1, 1)-forms. Proof. The difference T between two such Green currents is necessarily a c = 0, one has ddc -closed current, hence a harmonic current. Since it satisfies T, & T = 0.  Now, fix an integer d ∈ ℕ∗ and let s = P be a holomorphic section of the line bundle L = Oℙ1ℂ (d) = O(d). That is, P is a homogeneous polynomial (for the time being with complex coefficients), with degree d in homogeneous variables z = (z0 , z1 ). From now on we shall use the notation z = [z0 : z1 ] for points in ℙ1ℂ

1.3. THE ALGEBRAIC POINT OF VIEW

53

expressed in terms of homogeneous coordinates. We will assume that P (X0 , X1 ) is the homogenization of a polynomial p ∈ ℂ[x] with deg p = d ∈ ℕ∗ , that is, (1.146) p(x) =

d 

γκ x

d−κ

∗

with γ0 ∈ ℂ ,

P (X) = P (X0 , X1 ) =

κ=0

d 

γκ X0κ X1d−κ .

κ=0

One has then



div(P ) = (div(P ))+ =

νp,ξ {[1 : ξ]},

{ξ∈ℂ : p(ξ)=0}

where νp,ξ denotes the multiplicity of ξ as a zero of p. Thus   (1.147) p(x) = γ0 (x − ξ)νp,ξ , P (X) = γ0 {ξ∈ℂ : p(ξ)=0}

(X1 − ξX0 )νp,ξ .

{ξ∈ℂ : p(ξ)=0}

Therefore, the geometric complexity of the 0-cycle div(P ) is  (1.148) deg(div(P )) = νp,ξ = d. ξ∈V (p)

Another important geometric object attached to a 0-cycle on ℙ1ℂ is its Chow cycle on a dual copy of ℙ1ℂ . See, for example, Remark D.13 for the related concept of sheaf of Chow ideals of a cycle Z on a complex manifold of arbitrary dimension. Naturally, the above observation applies to div(P ). Definition 1.49. Let (ℙ1ℂ )u be a copy of (ℙ1ℂ )z , whose homogeneous variables are denoted by (u0 , u1 ). That is, the current point is denoted by u = [u0 : u1 ] instead of z = [z0 : z1 ]. If  να {[α0 : α1 ]} Z= α=[α0 :α1 ]∈ℙ1ℂ

is a 0-cycle on (ℙ1ℂ )z , then its Chow cycle Z  is the cycle div(Φ ) on (ℙ1ℂ )u , where Φ is the holomorphic section of O(ℙ1ℂ )∗ (deg Z) defined, up to multiplication by some χ ∈ ℂ∗ , as  (u0 α0 + u1 α1 )να = Φ (u0 , u1 ). (1.149) Φ ([u0 : u1 ]) = α=[α0 :α1 ]∈|Z|

Φ is taken to be a Chow form for div(P ). Example 1.50. If Z = div(P ), where P (X0 , X1 ) is the homogenization of p ∈ ℂ[x] with degree exactly d ∈ ℕ∗ , then one can take Φ (u0 , u1 ) = P (u1 , −u0 ). 1 1  Remark 1.51. Consider the product space  X = (ℙℂ )ζ × (ℙℂ )u . Its subset consisting of points (z, u) = [z0 : z1 ], [u0 , u1 ] such that the supports of divisors |Z| and |div(ℙ1ℂ )ζ (u0 ζ0 + u1 ζ1 )| intersect at [z0 : z1 ] on (ℙ1ℂ )ζ is called an incidence set and the points [z0 : z1 ] ∈ |Z| and [u0 : u1 ] ∈ (ℙ1ℂ )u are then said to be incident. Taking into account the multiplicity νZ,[z0 :z1 ] for such a pair of incident points, one gets a 1-cycle IncZ on the two-dimensional product manifold X. Then the Chow cycle Z  is the direct image of IncZ through the proper projection (ζ, u) → u; see §D.1.1 for the definition of the direct image of a cycle.

54

1. RESIDUE CALCULUS IN ONE VARIABLE

The concept of a Chow cycle is of major interest in the arithmetic setting. To illustrate this, we assume for the rest of the current section that the polynomial p in (1.147), whose homogenization equals P , belongs to ℤ[X], with gcd(γ0 , . . . , γd ) = 1. Moreover, one has that Φ (U0 , U1 ) = P (−U1 , U0 ) ∈ ℤ[U0 , U1 ] with jointly coprime coefficients. Then P ∈ ℤ[X0 , X1 ] defines an arithmetic cycle divarithm (P ) with pure codimension 1 on the arithmetic scheme   𝕏 = Spec Proj(ℤ[X0 , X1 ]) . Note that arithmetic cycles with pure codimension 2 (or dimension −1) on 𝕏 are formal (finite) linear combinations  νp {p} with νp ∈ ℤ∗ . p prime

Let us now explain how to measure the arithmetic complexity of divarithm (P ) for 0 [0] . For that purpose, we will need first the following the metrized line bundle L 15 lemma. Lemma 1.52. Let δ ∈ ℕ∗ . For any holomorphic section Q(z) =

δ 

(ακ0 z1 − ακ1 z0 ), with ακ = (ακ,0 , ακ,1 ) ∈ ℂ2 \ {(0, 0)},

κ=1

of the line bundle O(δ), one has  (1.150) ℙ1ℂ



δ   − log Q(ζ)2δ & c(ζ) = δ − log ακ 2 . κ=1

Proof. It is enough to prove the assertion for δ = 1. The assertion for δ ∈ ℕ∗ follows from the fact that log is a homomorphism from (ℝ+ , ×) to (ℝ, +). Let then (u0 , u1 ) ∈ ℂ2 \ {(0, 0)} and Q(z) = ⟪u, z⟫ = u0 z0 + u1 z1 . It follows from the invariance of the K¨ ahler form & c under the action of U (2) on ℙ1ℂ that one can assume that (u0 , u1 ) = u(0, 1). Then, (1.142) implies that   12  1  1 |t|2 dt ∧ dt − log 1⟪u, ζ⟫11 & c(ζ) = − log u2 − log 2 2 )2 1 2iπ 1 + |t| (1 + |t| ℙℂ ℂ  ∞ r dr =− − log u2 log 1 + r (1 + r)2 0  1 =− log ρ dρ − log u2 = 1 − log u2 .  0

Let us introduce the unit sphere 𝕊 = 𝕊3 in ℂ2  ℝ4 with its canonical normalized measure σ𝕊 . We now state the following proposition. 15 This lemma corresponds to one, among many, of the formulations of Crofton’s averaging formula in the univariate setting; see §D.5.2.

1.3. THE ALGEBRAIC POINT OF VIEW

55

Proposition 1.53. Let P be the homogenization of p ∈ ℤd [x] as in (1.147). %d , P ). FurLet G be the unique (0, 0) Green current normalized with respect to (L thermore, assume that G is orthogonal to harmonic forms, that is, G, & c  = 0. Let Φ (u) = P (u1 , −u0 ) as Chow form for div(P ). Then  d + log |P (0 , 1 )|2 dσ𝕊 () (1.151) 2 𝕊  12 1   1 − log 1⟪u, ζ⟫11  G (ζ) ≥ 0. = log |Φ (u)| + 2 ℙ1ℂ Proof. Since G, & c  = 0, one has (1.152)   1 12 1 12     − log 1⟪u, ζ⟫11  G (ζ) = − log 1⟪u, ζ⟫11 [div(P )] ≥ −d log u2 . ℙ1ℂ

ℙ1ℂ

The final inequality follows from the positivity of 1 1 1⟪u, ζ⟫12 1 (u, ζ) −→ − log u2 on (ℙ1ℂ ) × ℙ1ℂ because of the Cauchy–Schwarz inequality. Moreover,16 one has (1.153)   12   1 − log 1⟪u, ζ⟫11 [div(P )] = d + log |P (0 , 1 )|2 dσ𝕊 (). log |Φ (u)|2 + ℙ1ℂ

𝕊

The equality in (1.151) follows. Observe now that the left-hand side does not depend on the choice of u. The positivity of the expression in (1.151) follows then from the inequality in (1.152) with a convenient choice of u. We need to use here only the following immediate observation: since p ∈ ℤ[x], any nonzero coefficient γ  of P is such that |γ| ∈ ℕ∗ , hence |γ| ≥ 1. Proposition 1.53 suggests the following definition. Definition 1.54. Let p ∈ ℤ[X] be as in (1.146) with jointly coprime coefficients γκ , κ = 0, . . . , d. The height of the codimension 1 0-cycle  projective logarithmic  divarithm (P ) in Spec Proj(ℤ[X0 , X1 ]) is the positive real number defined by (1.154)      hproj divarithm (P ) = hproj divarithm (p) = d + log |P (0 , 1 )|2 dσ𝕊 (). 𝕊

Remark 1.55. Let u = [u0 : u1 ] ∈ where (u0 , u1 ) ∈ ℤ2 \ {(0, 0)},   be such that div z → ⟪u, z⟫ and div(P ) intersect properly on ℙ1ℂ . This implies that the corresponding arithmetic  cycles intersect properly along a codimension 2  cycle on Spec Proj(ℤ[X0 , X1 ]) . For such u = [u0 : u1 ] ∈ (ℙ1ℚ ) the first contribution log |Φ (u0 , u1 )| on the right-hand side of (1.151) is a marker for the degree of the intersection for codimension 2 (or dimension −1) cycle of divarithm (P ) and   of divarithm z → ⟪u, z⟫ . Both entries involved in the cup product (1.139), where L = Ld , L = L[0] , s = P , s = ⟪u, z⟫ and the metrics are Fubini–Study metrics, contribute to the splitting (1.151) of the projective logarithmic height. This interpretation of (1.151) is valid, provided that the intersection is understood on (ℙ1ℚ ) ,

16 Formula (1.151) is the incarnation in the univariate situation of a more general important statement; see [Ph91, Theorem 2].

56

1. RESIDUE CALCULUS IN ONE VARIABLE

  Spec Proj(ℤ[X0 , X1 ]) instead of on ℙ1ℂ . The above construction, as described here, is the univariate version of the approach developed in [GS90, BGS94] in the multivariate projective setting. 0 [0] as a standard measurement tool (a The use of the metrized line bundle L marker, see Definition 1.54), for the arithmetic complexity of the codimensional cycle divarithm (P ), where P ∈ ℤ[X0 , X1 ] is the homogenization of p ∈ ℤd [X] as in (1.146) having jointly coprime coefficients, has an alternative. This alternative is based on the natural choice to equip the holomorphic line bundle L[0] = O(1) with its canonical metric defined by |ξ0 ζ0 + ξ1 ζ1 |can =

(1.155)

|ξ0 ζ0 + ξ1 ζ1 | . max(|ζ0 |, |ζ1 |)

The metric | |can is not smooth as the Fubini–Study metric | |1 was. It is just continuous and the first Chern current of the metrized line bundle L[0] = (L[0] , | |can ) has its support included in ℂ. Furthermore, it equals, when expressed in the affine complex coordinate t = ζ1 /ζ0 , ccan = c1 (L[0] ) = σ𝕊1

(1.156)

dt ∧ dt , 2i

where σ𝕊1 is the normalized Haar measure on the unit circle 𝕊1 in ℂ  ℝ2 . Observe that the positive (1, 1)-current ccan is also realized as the limit of positive (1, 1)currents on ℙ1ℂ (1.157)

ccan

 ∗  c   TN (& dd log |ζ0 |2N + |ζ1 |2N c) = lim , = lim N →+∞ N →+∞ N N

where TN ([ζ0 : ζ1 ]) = [ζ0N : ζ1N ]. For anyN ∈ ℕ∗ , the direct  image of the arithmetic, codimension 1 cycle divarithm (P ) of Spec Proj(ℤ[X0 , X1 ]) by TN is the codimension 1 cycle divarithm (P [N ] ). Recall that P [N ] is the holomorphic section of O(N d) defined by (1.158)

P

[N ]

(ζ) =

γ0N



N −1 

  ζ1 − exp(2iπκ/N ) ξ ζ0 ∈ ℤ[X0 , X1 ].

{ξ∈ℂ : p(ξ)=0} κ=0

Formula (1.157) suggests the following notion for normalized logarithmic projective height, known also as canonical logarithmic height. Definition 1.56. Let p ∈ ℤd [x], P ∈ ℤd [X0 , X1 ] be as in (1.146), and let their coefficients be jointly coprime. The following limit     1 hproj (TN )∗ (divarithm (P )) 1 hproj divarithm (P [N ] ) = lim lim N →+∞ N N →+∞ N Nd Nd

1.3. THE ALGEBRAIC POINT OF VIEW

57

exists and the positive real number hcan (divarithm (P )) = hcan (divarithm (p)) is defined asymptotically by

(1.159)

hcan (divarithm (P )) hcan (divarithm (p)) = d d   1 hproj (TN )∗ (divarithm (P )) = lim N →+∞ N Nd   1 hproj divarithm (P [N ] ) . = lim N →+∞ N Nd

The canonical logarithmic height of divarithm (P ) (or divarithm (p)) is a marker for the arithmetic complexity of the codimension 1 arithmetic cycle divarithm (P ), if one takes as a standard measurement tool the metrized line bundle L[0] . What we described above is an Arakelov-type approach to complexity through a normalization process operating on the projective logarithmic height (Definitions 1.54 and 1.56) in the univariate setting. Another approach, involving Berkovich analytifications of ℙ1ℚ at p-adic (ultrametric) places, as companions of the algebraic Riemann surface ℙ1ℂ (at the archimedean place), will be briefly presented in §6.6.2. The construction of the projective logarithmic height in the multivariate setting (see [GS90, BGS94]), together with the realization of normalized Green currents through the analytic continuation approach, provides the opportunity to express in some particular cases the projective logarithmic height of a hypersurface of ℙnℂ defined over ℚ as the value at the origin of an Igusa zeta function; see for example [CaM00, BY98]. On the other hand, the normalized projective logarithmic height   of an arithmetic codimension 1 cycle in Spec Proj(ℤ[X0 , . . . , Xn ]) is related to a very important convex (or concave, depending of the point of view one adopts) function on the real affine space ℝn ; see [Ro00,PasR04,Mai00,Gua18,GuaS23]. Let us state here this result in the univariate case. Proposition 1.57. Let p ∈ ℤd [X] be as in (1.146), whose coefficients are jointly coprime. The functions  2π 1 log |p(ex+iθ )| dθ, Rp : x ∈ ℝ −→ 2π 0 (1.160)  2π 1 log |p(e−x−iθ )| dθ ρp : x ∈ ℝ −→ − 2π 0 are respectively convex and concave, piecewise affine functions. Furthermore, one has hcan (divarithm (p)) = Rp (0) = −ρp (0)  (1.161) νp,ξ max(0, log |ξ|). = log |γ0 | + 



{ξ∈ℂ : p(ξ)=0}

The real number exp hcan (divarithm (p)) = exp(Rp (0)) = exp(−ρp (0)) is called the Mahler measure of p ∈ ℤd [x], whose coefficients are coprime. Proof. The function Rp is piecewise affine, increasing, and hence convex. This claim follows from the fact that for any α ∈ ℂ∗ and any r > 0,  2π 1 log |reiθ − α| dθ = (log |α|) χ]0,|α|] (r) + (log r) χ]|α|,+∞[ (r). 2iπ 0

58

1. RESIDUE CALCULUS IN ONE VARIABLE

The last equality in (1.161) is a direct consequence of the above identity. The first equality in (1.161) follows from the definition of the projective logarithmic height (see Definition 1.54), then of its normalization (see Definition 1.56), combined with (1.157).  1.3.4. Residues, residue currents, and Gorenstein 𝕂-algebras. According to the algebraic point of view we adopt in this section, it appears that the concept of residue appeals to that of Gorenstein 𝕂-algebra. Definition 1.58. Let 𝕂 be a field and let P be a commutative 𝕂-algebra with finite dimension as a 𝕂-vector space. The 𝕂-algebra P is said to be a Gorenstein 𝕂-algebra if and only if there exists a 𝕂-linear map ρ : P → 𝕂 such that the symmetric bilinear form (1.162)

(a, b) ∈ P −→ ρ(ab) ∈ 𝕂

(where P is considered with its structure of 𝕂-vector space) is nondegenerate. We present in this section some examples of Gorenstein 𝕂-algebras, where the symmetric bilinear nondegenerate form ρ is specified in terms of residue theory in the univariate case. Such examples illustrate either local, global, or semilocal situations. Proposition 1.59. Let f be a nonzero element in the maximal ideal M of O = Oℂ,0 . The ℂ-algebra O/(f ), with finite dimension νf,0 ∈ ℕ∗ , is Gorenstein, with the symmetric bilinear form ρ taken as  1

 (1.163) ρ : (a, b) ∈ O/(f ) ⊗ O/(f ) −→ ρ(ab) = ∂ , (AB)(ζ) dζ , f 0 where A, B are arbitrary representatives of the classes a and b modulo f in O. Proof. The result is an immediate consequence of Proposition 1.16 (see in particular Remark 1.17), in the particular case m = 1. In fact, it is enough to invoke here formula (1.50).  Remark 1.60. Since

  1

∂ , (AB)(ζ) dζ f 0

depends only on the germ of 1-cohomology class of ζ → (AB)(ζ) dζ at the origin, the duality bracket realized by the symmetric nondegenerate bilinear form ρ in (1.162) is also called the O/(f ) dualizing local cohomological residue. When 𝕂 is a field, elements in 𝕂[X, X −1 ] are called Laurent polynomials with coefficients in 𝕂. Moreover, if 𝕂 is a subfield of ℂ, then let p ∈ 𝕂[X] be a polynomial with degree d ∈ ℕ∗ and let F ∈ 𝕂[X, X −1 ] be a Laurent polynomial with card(Supp(F )) ∈ ℕ∗ . Then the finite sets V (p) = {α ∈ ℂ : p(α) = 0}, respectively V𝕋 (F ) = {reiθ ∈ ℂ∗ : F (reiθ ) = 0}, are nonempty subsets of the algebraic closure of 𝕂 in ℂ, and one has the following proposition. Proposition 1.61. The 𝕂-algebras 𝕂[X]/(p) and 𝕂[X, X −1 ]/(F ) are finitely dimensional as 𝕂-vector spaces and Gorenstein, with symmetric bilinear form ρ

1.3. THE ALGEBRAIC POINT OF VIEW

59

defined by

   1

∂ , (AB)(ζ) dζ p α α∈V (p) (1.164) ) * (AB)(X) dX = Res𝕂[X]/𝕂 p ¯ B ¯ in 𝕂[X] in the first case and by for any representatives A, B of A,   1

dζ  ρ(a, b) = ∂ , (AB)(ζ) F reiθ ζ reiθ ∈V𝕋 (F ) (1.165) ) * (AB)(X) dX := Res𝕂[X,X −1 ]/𝕂 F 𝕋 ρ(a, b) =

¯ B ¯ in 𝕂[X, X −1 ] in the second case. for any representatives A, B of A, Proof. In the first case, when p ∈ 𝕂[X], the vector space P = 𝕂[X]/(p) is finite dimensional, with dimension dim P = deg p = d. Moreover, it follows from Corollary 1.35 that ) *    1

(AB)(X) dX := ∂ , (AB)(ζ) dζ Res𝕂[X]/𝕂 p p α α∈V (p)

(see Definition 1.36) belongs to 𝕂 and depends only on the classes a and b of A and B in 𝕂[X]/(p). The fact that the symmetric bilinear map defined by (1.164) is nondegenerate follows from Proposition 1.38. Any F ∈ 𝕂[X, X −1 ] is factorized as the product F = p(X)/X μ , with p ∈ 𝕂[X], p(0) ∈ 𝕂∗ , and μ ∈ ℤ. The algebra 𝕂[X, X −1 ]/(F ) is a finitely dimensional 𝕂-algebra with dimension deg p, that is, the length of the closed convex hull of Supp(F ) in ℝ. For any Laurent polynomials A, B ∈ 𝕂[X, X −1 ], let (AB)(X) = q(X)X ν with q ∈ 𝕂[X], q(0) ∈ 𝕂∗ and ν ∈ ℤ. Then   1

dζ  ∂ , (AB)(ζ) F reiθ ζ reiθ ∈V𝕋 (F )    1

μ+ν−1 = ∂ , q(ζ)ζ dζ . p reiθ iθ re ∈V𝕋 (F )

Let us discuss the following alternatives: • if μ + ν − 1 ∈ ℕ, then   1

dζ  ∂ , (AB)(ζ) F reiθ ζ reiθ ∈V𝕋 (F ) * 

)  1 q(X) X μ+ν−1 dX = Res𝕂[X]/𝕂 , q(ζ)ζ μ+ν−1 dζ ; − ∂ p p 0 • if μ + ν − 1 = −χ ∈ ℤ− , then   1

dζ  ∂ , (AB)(ζ) F reiθ ζ iθ

re ∈V𝕋 (F )

= Res𝕂[X]/𝕂

)

*   1

q(X) dX , q(ζ) dζ . − ∂ χ χ X p ζ p 0

60

1. RESIDUE CALCULUS IN ONE VARIABLE

In both cases, the complex number ρ(a, b) defined by (1.165) is independent of the choice of the representatives A, B of a, b modulo (F ) respectively and belongs to 𝕂 according to Corollary 1.35 and formula (1.14). The nondegeneracy of the bilinear symmetric form ρ follows again from Proposition 1.38, since F divides G  in 𝕂[X, X −1 ] if and only if G(ζ)/F (ζ) ∈ H(ℂ∗ ).   The fact that for F ∈ 𝕂[X, X −1 ] satisfying card Supp(F ) ≥ 1 the vector space K[X, X −1 ]/(F ) is a Gorenstein algebra with symmetric bilinear form given by (1.165) suggests the following definition. Definition 1.62. Let 𝕂 be a subfield of ℂ. For any pair of Laurent polynomials (F, G) assumed to be coprime in 𝕂[X, X −1 ], the element of 𝕂 defined by ) *   1

dζ  G(X) dX := (1.166) Res𝕂[X,X −1 ]/𝕂 ∂ , G(ζ) F F reiθ ζ 𝕋 iθ re ∈V𝕋 (F )

is called the (global) toric residue with respect to F ∈ 𝕂[X, X −1 ] of the K¨ ahler differential G(X) dX ∈ Ω𝕂[X,X −1 ]/𝕂 . Now, let E = (E, | |) be a hermitian holomorphic bundle with rank m over a noncompact Riemann surface X and s ∈ OX (X , E). For h in OX (X , ℂ) and u ∈ D(X , E), let  ρ(u, h) = R|s | , h u , where the (0, 1)-current

R|s | = ∂|s|2λ

(1.167)

s∗  2iπ |s|2 λ=0

has been introduced in Proposition 1.27. On observes that ρ(u, h) depends only on the class u of u modulo s D(1,0) (X , E ∗ ∧ E ∗ ) and on the class h of h modulo s OX (X , E ∗ ). Let ρ :

D (1,0) (X , E) OX (X ) −→ ℂ × s D(1,0) (X , E ∗ ∧ E ∗ ) s OX (X , E ∗ )

be the bilinear form defined by (1.168)

 ρ(u, h) = R|s | , h u ∈ ℂ,

where u ∈ D (1,0) (X , E ∗ ) is any representative of u modulo s D(1,0) (X , E ∗ ∧ E ∗ ) and h is any representative of h modulo s OX (X , E ∗ ). It follows from Proposition 1.29 that

D (1,0) (X , E ∗ ) (1.169) ρ(u, h) = 0 for any u ∈ =⇒ h = 0. s D(1,0) (X , E ∗ ∧ E ∗ ) The ring OX (X )/s OX (X , E ∗ ) is always Cohen–Macaulay as a zero-dimensional ring; see §A.3.3 for the notion of Cohen–Macaulay ring or sheaf of ideals. Nevertheless, in the case where it is finitely dimensional as a ℂ-vector space, namely when card (s−1 ({0})) < +∞, the ℂ-algebra OX (X )/s OX (X , E ∗ ) fails, in general,

1.3. THE ALGEBRAIC POINT OF VIEW

61

to be Gorenstein when m > 1.17 Duality is materialized instead by the implication (1.169), where the bilinear form ρ acts this time asymmetrically according to (1.168). The current R|s | is not ∂-closed when m > 1, but a positive point is that its expression (1.167) in terms of the holomorphic section s does not require any information about s−1 ({0}). The situation will be drastically different in the higher dimensional setting. Let us consider the case of quotients of ℂ[X1 , . . . , Xn ] as an example. It remains true that if p1 , . . . , pm ∈ ℂ[X1 , . . . , Xn ] are such that ℂ[X1 , . . . , Xn ]/(p1 , . . . , pm ) is finitely dimensional as a ℂ-vector space, then ℂ[X1 , . . . , Xn ]/(p1 , . . . , pm ) is Cohen– Macaulay. But such a ℂ-algebra fails in general to be ℂ-Gorenstein, since being ℂ-Gorenstein relies on the quasi-regularity of the sequence (p1 , . . . , pm ), which, geometrically speaking, appeals to the concept of complete intersection; see §6.2.1. The situation is much worse if dim(ℂ[X1 , . . . , Xn ]/(p1 , . . . , pm )) > 0, since in this case, the existence of embedded associated primes could prevent in general ℂ[X1 , . . . , Xn ]/(p1 , . . . , pm ) from being Cohen–Macaulay; see §A.3.3. For all these reasons, the current materialization of duality, as in (1.169) or at least some intermediate result but with a tractable current R|s | , will be of major interest. It is therefore now time to jump from the univariate situation to the multivariate one.

nevertheless that if X = U is a domain in ℂ and s is a section of the trivial m bundle ℂm , then one has s O(U, (ℂm )∗ ) = p O(U ) for some polynomial map p. This implies that the ℂ-algebra O(U )/ s O(U ) = O(U )/p O(U ) is Gorenstein. Similarly, 𝕂[X]/(p1 , . . . , pm ) and 𝕂[X, X −1 ]/(F1 , . . . , Fm ) remain Gorenstein 𝕂-algebras when the pj (respectively the Fj ) are polynomials (respectively Laurent polynomials) with coefficients in 𝕂. 17 Observe

CHAPTER 2

Residue currents: a multiplicative approach 2.1. De Rham complex and iterated residues We recall briefly in this section the cohomological theory of residue (through the formulation of residue formula (2.23)), which goes back to the pioneering work of H. Poincar´e [P1887]. This theory was developed later by G. de Rham [DR32, DR36], and more recently by J. Leray [Le59]. For an updated overview of H. Poincar´e’s contribution to multivariate residue calculus, accompanied by updated bibliography, the reader may refer to [Y10]. Let us quote here once more one of the fundamental remarks of H. Poincar´e [P1887]: the integral over a closed surface depends only on singular curves which lie on this surface. This remark is our guideline throughout the whole section. We focus here on bridging the cohomological and current approaches to the residue. The current approach to the residue is developed and presented in §2.2 and §2.3.3. 2.1.1. Division of forms and de Rham’s lemma. Let X be a complex N -dimensional manifold.1 Let also U ⊂ X be an open subset in X and let S be a codimension 1 submanifold in X . That is, for any z ∈ S, one can find a neighborhood U ⊂ X of z in X , a holomorphic function s on U such that ds = 0 in U , and S ∩ U = {z ∈ U : s(z) = 0}. The function s ∈ OX (U ) is called a defining function for the complex submanifold S (with codimension 1) in U . Definition 2.1. Let ω ∈ Ck∞ (X \ S) be a ℂ-valued, smooth differential kform on the real, 2N -dimensional, underlying differentiable manifold (X \ S)ℝ , 0 ≤ k ≤ 2N . The k-form ω is called semimeromorphic on X with polar set contained in S if and only if, for any z ∈ S there exist a neighborhood U of z and a defining holomorphic function s for S in U , together with a positive integer ν = ν(z) such that the k-form sν ω extends as an element in Ck∞ (U ), that is, sν ω is a smooth differential k-form on Uℝ . When ν(z) = 1 for any z ∈ S, ω is called semimeromorphic on X , with at most simple poles along the hypersurface S. Let us recall here the crucial division lemma, due to G. de Rham (see [DR32, DR36]) but used as a crucial operational tool in [Le59]. 1 From now on in this chapter, X will denote either an N -dimensional complex manifold or an irreducible reduced complex analytic space with pure dimension N ; see respectively §A.1.1 or §A.4.1. We will treat similar semilocal problems (think of X , for example, as a neighborhood of a specified point, or more generally of a closed analytic subset V in some ambient manifold or irreducible reduced analytic space) or global ones. Of course, obstructions for the definition of nontrivial global objects (such as holomorphic functions on X , or more generally global holomorphic sections for some holomorphic vector bundle E → X ) have always to be taken into account.

63

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Lemma 2.2. Let X , S be as above. Let also ω be a semimeromorphic k-form (k ≥ 1) on X with at most simple poles along the hypersurface S, such that dω = 0 in X \ S. Then, for any z ∈ S one can find an open neighborhood U = Uz ⊂ X of z, a holomorphic function s = sz on U , such that one has simultaneously ds = 0 in U and S ∩ U = {z ∈ U : s(z) = 0}. Furthermore, one can also find a smooth (k − 1)-form r[ω] = rz [ω] and a smooth k-form σ[ω] = σz [ω] on U , such that (2.1) Moreover, (r[ω])|S∩U z ∈ S, let (2.2)

ds ∧ r[ω] + σ[ω] . s ∞ ∈ Ck−1 (S ∩ U ) and satisfies d((r[ω])|S∩U ) = 0. For each ω=

(resS∩Uz [ω])z := (r[ω])|S∩Uz

be a locally defined, smooth, (k − 1)-form. All locally defined forms (resS∩Uz [ω])z on S fit together as a d-closed, smooth (k − 1)-form on S, called the Poincar´e–Leray residue of ω on S, and is denoted as Res[X ;S] (ω). Proof. Let s be the local defining function of S in a neighborhood U of a point z ∈ S, with ds = 0 in U . Since ω is d-closed, one has d(s ω) = ds ∧ ω in U \ S, which implies that d(s ω) ∧ ds = 0 in U \ S. This holds also in U , since s ω extends as a smooth k-form in U , as a consequence of the assumption that the differential form ω ∈ Ck∞ (U \ S) is semimeromorphic with at most simple poles along the hypersurface S. This implies, since ds = 0 on U , that there exists σ[ω] ∈ Ck∞ (U ) such that d(s ω) = ds ∧ σ[ω]

(2.3) in U . Thus, one has in U that

ds ∧ (s ω − s σ[ω]) = s (d(s ω) − ds ∧ σ[ω]) = 0. This in turn implies, following the same argument as above and using again the ∞ (U ) satisfying the identity fact that ds = 0 on U , that there exists r[ω] ∈ Ck−1 (2.4)

s ω − s σ[ω] = ds ∧ r[ω]

in U . This leads to the division formula (2.1). Next, applying the differential operator d to (2.4), we get ds ∧ ω − ds ∧ σ[ω] − s d(σ[ω]) = −ds ∧ d(r[ω]), that is d(s ω) = ds ∧ σ[ω] + s d(σ[ω]) − ds ∧ d(r[ω]). If one uses (2.3), then it follows that s d(σ[ω]) = ds ∧ d(r[ω]) in U holds, implying that the restriction (resS∩U [ω])|S∩U of r[ω] to the submanifold S∩U is d-closed. The same reasoning shows that, if z and w are two distinct points in S with Uz ∩Uw = ∅, then (2.5)

(dsz )|S ∧ (rz [ω] − rw [ω])|S = 0

in Uz ∩ Uw . Therefore, one has in Uz ∩ Uw that (rz [ω])|S∩Uz = resS∩Uz [ω] = (rw [ω])|S∩Uw = resS∩Uw [ω], which shows that the collection of {rz [ω], z ∈ S}, represents the collection of restrictions to different Uz , z ∈ S, of a smooth d-closed, (k −1)-form on the complex submanifold S of X . This completes the proof of the lemma. 

2.1. DE RHAM COMPLEX AND ITERATED RESIDUES

65

When ω is a semimeromorphic k-form on X with poles of order at most ν0 , ν0 > 1 along S, a division formula similar to the first assertion in Lemma 2.2 exists. Lemma 2.3. Let X , S be as above. Let also ω be a semimeromorphic k-form (k ≥ 1) on X , with poles with order at most ν0 , ν0 > 1, along the hypersurface S, such that dω = 0 in X \ S. For any point z ∈ S, one can find an open neighborhood U = Uz ⊂ X of z, a holomorphic defining function s = sz for S in U which ∞ (U ), σν0 [ω] = satisfies ds = 0 there, and differential forms rν0 [ω] = rz,ν0 [ω] ∈ Ck−1 ∞ σ[z,ν0 ] [ω] ∈ Ck (U ) such that one has in U (2.6)

ω=

ds σν [ω] ∧ rν0 [ω] + ν00 −1 . sν 0 s

Proof. Take U and s in a neighborhood of z to be as in the proof of Lemma 2.2. One has in U \ S that d(sν0 ω) = ν0 d(sν0 −1 ) ∧ ω. Hence, ds ∧ d(sν0 ω) = 0 in U , since sν0 ω extends to U as a smooth k-form. Since ds = 0 on U , it follows that d(sν0 ω) = ν0 ds ∧ σν0 [ω] for some σν0 [ω] ∈ Ck∞ (U ). Then d(sν0 ω)

  − ds ∧ σν0 [ω] = 0 ds ∧ sν0 ω − s σν0 [ω] = s ν0 ∞ in U . This implies, since again ds = 0 there, the existence of rν [ω] ∈ Ck−1 (U ) such ν0 that s − s σν0 [ω] = ds ∧ rν0 [ω]. 

Lemma 2.2 suggests strengthening the concept of semimeromorphic form with polar set along a smooth complex hypersurface S and with at most simple poles along S (see Definition 2.1) as follows. Definition 2.4. Let S be a smooth complex hypersurface in X . A semimeromorphic k-form ω ∈ Ck∞ (X \ S) with at most simple poles along S is said to be ∞ S-logarithmic if and only if dω ∈ Ck+1 (X \ S) is a semimeromorphic (k + 1)-form with at most simple poles on S. Remark 2.5. If a semimeromorphic form ω ∈ Ck∞ (X \ S) happens to be Slogarithmic, then one can repeat the argument used to prove Lemma 2.2 and express ω in a neighborhood U of z ∈ S as in (2.1), using the defining function s for S in the open neighborhood U of z. Moreover, the local forms (rz [ω])|S∩Uz , for z ∈ S, ∞ still correspond to the local expressions of Res[X ;S] (ω) ∈ Ck−1 (S), which is also called the Poincar´e–Leray (in short Leray) residue of ω on S; see [Pol72]. In this case, the Poincar´e–Leray residue is not d-closed anymore as in Lemma 2.2, unless dω = 0 in U \ S. This remark justifies the terminology used in Definition 2.4. Example 2.6 (Poincar´e–Leray residue of (N, 0)-meromorphic forms). Let Ω be a domain in ℂN and let f ∈ OℂN (Ω) be such that df ≡ 0 on any irreducible component of the reduced hypersurface f −1 ({0}). Consider X := Ω \ (f −1 ({0}))sing = Ω \ {z ∈ Ω ; f (z) = df (z) = 0}. Then the set S ⊂ X defined by S = {z ∈ X : f (z) = 0} is a complex submanifold of X . For any holomorphic (N, 0)-form ω = h(z) dz1 ∧ · · · ∧ dzN

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in Ω one can realize Res[X :S]

ω(z) f (z)

in the open set uj = {z ∈ S : (∂f /∂zj )(z) = 0}, for any j = 1, . . . , N , as follows: ω(z)

h dz ∧ · · · ∧ dz %j ∧ · · · ∧ dzN 1 (2.7) ResS = (−1)j−1 . f (z) |uj ∂f /∂zj |uj Example 2.7 (The Bochner–Martinelli kernel). Let & c (z) = ddc log z2 be 2 N the K¨ ahler form on the projective space ℙℂ , where zj ’s denote the homogeneous coordinates, namely z = [z0 : · · · : zN ]. The K¨ahler form & c (z) is also the first  = (O(1),  1 ), where the metric Chern form of the metrized line bundle O(1) is the Fubini–Study metric, as defined by (1.140) in the univariate setting. The Fubini–Study normalized volume form & c N factorizes in ℙN ℂ as (& c (z))N = g(z, z) E(z) ∧ E(z) ,

(2.8) where E(z) =

N 

%j ∧ · · · ∧ dzN (−1)j−1 zj dz0 ∧ · · · ∧ dz

j=0

denotes the Euler form in ℙN ℂ . The coefficient (z, z) → g(z, z) is a smooth (−2N )homogeneous function in ℂN +1 \ {(0, . . . , 0)}, which contributes to make the form ω, defined below, invariant under the torus action (z0 , . . . , zN ) → (tz0 , . . . , tzN ), +1 t ∈ ℂ∗ , on ℂN +1 \ {(0, . . . , 0)}. In the punctured projective space X = ℙN \ ℂ N +1 N \ {(0, . . . , 0)}) ∪ ℙℂ consider, for each j = {(0, . . . , 0)} which splits as (ℂ 0, . . . , N , the open set Uj defined by  Uj = {t(z0 , . . . , zN ) : t ∈ ℂ∗ } ∪ {[z0 : · · · : zN ]}. (z0 ,...,zN )∈ℂN +1 zj =0

One interprets3 here the point [z0 : · · · : zN ] as the intersection of the projective +1 , materialized line {(tz0 , . . . , tzN ) : t ∈ ℂ∗ } with the hyperplane at infinity in ℙN ℂ "N N N here as ℙℂ . One has X = j=0 Uj . In each Uj \ ℙℂ , consider the (N + 1, N ) semimeromorphic form with at most simple poles along ℙN ℂ defined by ωj = Since

dzj ∧ (& c (z))N . zj

dzj ∧ E(z) = dz0 ∧ · · · ∧ dzN zj

in Uj \ ℙN ℂ , it follows from (2.8) that ωj (z) = dz0 ∧ · · · ∧ dzN ∧ g(z, z) E(z) 2 We refer to Example A.1.2(2), for a first introduction to the (complex) projective space ℙN , ℂ based on the point of view that will be developed in this monograph. 3 This reflects the point of view on which the concept of perspective during the Renaissance was based; see [ShcTY06].

2.1. DE RHAM COMPLEX AND ITERATED RESIDUES

67

there. Thus the forms ωj , j = 0, . . . , N , fit together in X \ ℙN ℂ as the (N + 1, N ) closed, smooth differential form N zj dzj j=0 zj dzj N ω(z) = ∧ (& c (z)) = ∧ (& c (z))N = ∂[log z2 ] ∧ (& c (z))N . 2 |zj | z2 N This smooth closed form ω ∈ ℂ∞ 2N +1 (X \ ℙℂ ) is semimeromorphic in X , with at N most simple poles along ℙℂ . Moreover, one has that

(2.9)

c N. Res[ℙN +1 ;ℙN ] (ω) = & ℂ

ℂ

∞ N +1 The (N +1, N )-differential form ω/(2iπ) ∈ C2N \{(0, . . . , 0)}) extends as a +1 (ℂ N +1 locally integrable form to ℂ . This extension is known as the Bochner–Martinelli kernel in ℂN +1 and it will play an essential role throughout this monograph; see in particular §4.2.1. The nonstandard approach to the derivation of the Bochner– Martinelli kernel presented above was presented in [ShcTY06, §3].

It will be important for us to deal later on with the case of the closed hypersurface S ⊂ X failing to be smooth. Namely, it will be the case when the closed analytic subset Ssing ⊂ S, with codimension at least 2 (see §A.3.2), fails to be empty. Since we deal with meromorphic (p, 0)-forms for the moment, as in our two previous Examples 2.6 and 2.7, we need to specify the notion of meromorphic logarithmic form on X with poles along a closed analytic hypersurface S; see [AlT01, AlT08]. We use for that purpose the terminology already used in Definition 2.4. Definition 2.8. Let X be an N -dimensional complex manifold, and let S be a closed analytic hypersurface in X . A meromorphic (p, 0)-form on X , 1 ≤ p ≤ N , that is, a meromorphic section of the sheaf Ωp , is said to be S-logarithmic if and only if, for any z ∈ S, one can find an open neighborhood U = Uz of z in X and a reduced defining holomorphic function4 s in U , such that the meromorphic forms (s ω)|U and (s dω)|U (or, which is equivalent, the meromorphic forms (s ω)|U and (ds ∧ ω)|U ) are pole-free in U . Let X ◦ = X \ Ssing and S ◦ = S \ Ssing = Sreg . Then S ◦ is a codimension 1 submanifold of the N -dimensional complex manifold X ◦ . Given a meromorphic S-logarithmic form, one can define (see Remark 2.5) the Poincar´e–Leray residue Res[X ◦ ;S ◦ ] (ω|X ◦ ) as a holomorphic (p − 1, 0)-form on S ◦ ⊂ X ◦ . Such holomorphic (p − 1, 0)-form on S ◦ induces the definition of a (p, 1)-current on X ◦ , supported by S ◦ , and such an action is defined as  (2.10) Res[X ◦ ;S ◦ ] (ω|X ◦ ) :   ϕ ∈ D(N −p,N −1) (X ◦ ) −→ Res[X ◦ ;S ◦ ] (ω|X ◦ ) ∧ ϕ = Res[X ◦ ;S ◦ ] (ω|X ◦ ) ∧ ϕ. S◦

S

Proposition 2.9. Let X , S and ω beas in Definition 2.8. Let X ◦ = X \ Ssing and S ◦ = Sreg . The (p, 1)-current T ◦ = Res[X ◦ ;S ◦ ] (ω|X ◦ ) ∈  D(p,1) (X ◦ ) defined in (2.10) extends as a current T ∈  D(p,1) (X ) supported by S and ∂-closed. The extension T is defined locally around S, in terms of any reduced defining function 4 By this we mean a defining function s for S in U , such that ds ≡ 0 on any irreducible component of S ∩ U .

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2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

s for S in a neighborhood U of z ∈ S, by (2.11)

T = RS ∧ [ω] : ϕ ∈ D

(N −p,N −1)





(U ) −→ −

|s|2λ U

 ω ∧ ∂ϕ , 2iπ λ=0

 where the meaning of was introduced in Chapter 1, through the analytic λ=0 continuation method from {λ ∈ ℂ ; Re λ  1}. In other words 

|s|2λ

λ  ∧ω |s|2λ ∂ log |s|2 ∧ ω = . (2.12) T|U = ∂ 2iπ 2iπ λ=0 λ=0 Proof. For z ∈ S let U be an open neighborhood of z in X , and let s be a reduced holomorphic defining function for S in U . One may also assume that ζ : U → 𝔻N is a centered system of coordinates on X with ζ(z) = 0 so that there is a formal Bernstein–Sato equation5 Q(λ, ζ, ∂/∂ζ)(s(ζ))λ+1 = b(λ)(s(ζ))λ ,

(2.13)

where b(λ) is a polynomial, called minimal local Bernstein–Sato polynomial for the germ of s(ζ) at the origin. In this case the roots of b(z) belong −ℚ+ . The differential operator Q is a differential operator in (∂/∂ζ1 , . . . , ∂/∂ζN ) (in short ∂/∂ζ), whose coefficients are polynomial in λ and analytic in local coordinates ζ in a neighborhood of the origin. Namely, for any meromorphic coefficient w(ζ)/s(ζ) of ω(ζ) along the dζJ with cardinality |J| = p, one deduces from (2.13) the following equality in the sense of distributions:

s(ζ) w(ζ) Q(λ, ζ, ∂/∂ζ) |s(ζ)|2λ w(ζ) (2.14) = b(λ) × |s(ζ)|2λ . s(ζ) s(ζ) Since the roots of b lie in −ℚ+ , the  D(p,0) (U )-valued map 

λ ∈ {λ ∈ ℂ : Re λ  1} −→ |s|2λ ω|U extends meromorphically to ℂ, with (possible) poles in ℚ− . Its value at the origin λ = 0 is well defined. Moreover, such value does not depend on the choice of the reduced holomorphic function s, since any two such defining functions differ by a multiplicative factor consisting of an invertible holomorphic function χ in U , such that λ → [|χ(ζ)|2λ ] is an entire, current-valued function of λ that is equal to [1] at λ = 0. It remains to prove that



∂ |s|2λ ω|U = (T ◦ )|U\Ssing . λ=0

◦

|U\Ssing

◦

Fix a point z ∈ S ∩ U and take s|U ◦ as a holomorphic defining equation for S ◦ in a neighborhood of z ◦ . Let us assume that U ◦ ⊂ U with ds = 0 in U ◦ . Since ω is a meromorphic logarithmic form with poles on S, one can express (see Remark 2.5) ω in U ◦ as ds ω= ∧ r ◦ [ω] + σ ◦ [ω], s where r ◦ [ω] and σ ◦ [ω] are holomorphic forms with respective bidegrees (p − 1, 0) and (p, 0) in U ◦ . Therefore, for λ satisfying Re λ  1, one has in U ◦ that ∂(|s|2λ ω) = ∂|s|2λ ∧

ds ∧ r ◦ [ω] + ∂|s|2λ ∧ σ ◦ [ω]. s

5 For the existence of such a local formal equation, analogous to (1.6), valid in ζ(U ) provided U is small enough around 0, we refer to [Be72, Bj74, Ka76].

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69

For any ϕ◦ ∈ D (N −p,N −1) (U ◦ ), one can check immediately that 

  ds ∧ r ◦ [ω] , ϕ◦ ∂(|s|2λ ω), ϕ◦ λ=0 = ∂|s|2λ ∧ s λ=0  ◦ ◦ ◦ = 2iπ r [ω] ∧ ϕ = T , ϕ◦ , U ◦ ∩S ◦

which concludes the proof of the proposition.



Proposition 2.9 suggests the following definition. Definition 2.10. Let ω be a logarithmic (p, 0)-form in X , with polar set along the closed hypersurface S. The (p, 0)-current, expressed locally as

  (2.15) |s|2λ ω , λ=0

U

where s is any reduced holomorphic defining function for S in U , is called the standard extension of [ω] from X \ S and is denoted by 1X \S · [ω]. Note that one can take s = 1 in case S ∩ U = ∅. One has then 1

X \S · [ω] (2.16) ∂ = RS ∧ [ω], 2iπ according to Proposition 2.9. We point out here that the standard extension current 1X \S · [ω] can also be realized locally as 

 (2.17) 1X \S · [ω] , ϕ = lim+ ω ∧ ϕ , ∀ ϕ ∈ D (N −p,N ) (U ), ε→0

U ∩{|s|≥ε}

where s is any reduced defining holomorphic function for S in U . The Mellin transform realizes the bridge between the locally explicit formulas (2.15) and (2.17) describing the action of the current 1X \S · [ω]. 2.1.2. Leray coboundary morphism, cohomological residue formulae. As a preamble let us first introduce some pieces of notation. Given a complex manifold Y, we denote by Ck∞ (Y, ℂ), for any 0 ≤ k ≤ 2 dim Y, the group of k-dimensional smooth chains c = ι νι {γι }, νι ∈ ℂ on the corresponding differentiable manifold Yℝ . Similarly, we denote by   ∞ Zk∞ (Y, ℂ) = Ker ∂ : Ck+1 (Y, ℂ) → Ck∞ (Y, ℂ) ,  ∞  Bk∞ (Y, ℂ) = ∂ Ck+1 (Y, ℂ)) ⊂ Zk∞ (Y, ℂ) the subgroups of (smooth) k-cycles and k-boundaries respectively, where ∂ denotes  ∞ the boundary morphism. The support of a k-chain c = ν {γ ι } ∈ Ck (Y) is ι ι defined as the union of closed subsets γι (Δk ), which are smooth images of the simplexes Δk = {t ∈ ℝk : t1 + · · · + tk ≤ 1}. The support of the k-chain c is denoted by |c|. The notions of boundary and coboundary of a k-chain lead to the description of the kth homology group Hk (Y, ℂ) 

Zk∞ (Y, ℂ) , Bk∞ (Y, ℂ)

for any 0 ≤ k ≤ 2 dim Y. Given a complex N -dimensional manifold X , together with a codimension 1 complex submanifold S ⊂ X , the Leray coboundary morphism from H• (S, ℂ) to H• (X \ S, ℂ), is defined as follows.

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2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

Definition 2.11. Let γ¯ ∈ Hk−1 (S, ℂ) be a (k − 1)-cycle for 1 ≤ k ≤ 2N − 1. A k-cycle Γ ∈ Zk (X \ S, ℂ) represents δ [X ;S] (¯ γ ), that is, δ [X ;S] (¯ γ ) = Γ if and only ∞ if there exists c ∈ Ck+1 (X , ℂ) satisfying |∂c| ⊂ X \ S and a representative γ of γ¯ in ∞ Zk−1 (S, ℂ), such that • |c| intersects transversally S along |γ|, • ∂σ − Γ ∈ Bk (X \ S, ℂ). Example 2.12. (1) When X = ℂ and S = {0}, δ [ℂ;{0}] ({0}) is the 1-homology class of the cycle {t ∈ [0, 1] → e2iπt }. +1 and S = ℙN (2) When X = ℙN ℂ is its hyperplane at infinity (see Example ℂ N +1 N N +1 \ℙN 2.7) a representative for the class δ [ℙℂ ;ℙℂ ] ([ℙN ℂ ) is ℂ ]) in H2N +1 (ℙℂ 2N +1 (1) is a parametrization the (2N + 1)-cycle {Γ}. Here Γ : Δ2N +1 → 𝕊 +1 of the unit sphere centered at the origin in ℂN +1  ℝ2(N +1) = ℙN \ℙN ℂ. ℂ Fix now m ∈ {1, . . . , N }. Let S1 , . . . , Sm be a collection of m codimension 1 complex submanifolds (called also hypersurfaces) in X intersecting there transversally . It means that for any z ∈ S1 ∪ · · · ∪ Sm there exists an open neighborhood U of z in X , together with holomorphic defining functions s1 , . . . , sm for S1 , . . . , Sm respectively such that, for any subset J ⊂ {1, . . . , m}, 3  4 2 ' Sj ∩ U : dsj (z) = 0 = ∅. (2.18) z∈ j∈J

j∈J

An ordering of {S1 , . . . , Sm } given by (S1 , . . . , Sm ) and setting S0 = X lead to the embedding of 3 3





Sκ \ Sκ → Yk = Sκ \ Sκ (2.19) Σk = κ≤k

κ>k

κ≤k−1

κ>k

as a codimension 1 complex submanifold into a ((N − (k − 1))-dimensional complex manifold for any k = 1, . . . , m. One has also that Yk \ Σk = Σk−1 by construction. Then any γ¯ ∈ Hk (Σm , ℂ), δ [Ym ;Σm ] (¯ γ ) ∈ Hk+1 (Ym \ Σm ) = Hk (Σk−1 , ℂ), for any 1 ≤ k ≤ 2N −m. If S = (S1 , . . . , Sm ), then one can iterate this construction and define   γ) γ ) = δ [Y1 ;Σ1 ] ◦ · · · ◦ δ [Ym ;Σm ] (¯ (2.20) δ [X ;S] (¯   ∈ Hk+m (Σ0 , ℂ) = Hk+m X \ (S1 ∪ · · · ∪ Sm ), ℂ . As we introduced for semimeromorphic form in X the concept of being Slogarithmic when S was a smooth complex hypersurface of X (see Definition 2.4), one needs to introduce the concept of being S-multilogarithmic, when one has m ≤ N and S1 , . . . , Sm are m codimension 1 complex submanifolds which intersect transversally on X . Let us for that denote, for j = 1, . . . , m, % = S1 ∪ · · · ∪ S %j ∪ · · · ∪ Sm . |S| j

Definition 2.13. Let m ≤ n, S1 , . . . , Sm be m codimension 1 complex submanifolds intersecting transversally on X , according to condition (2.18) around each point z ∈ |S|. Given 0 ≤ k ≤ 2N , a semimeromorphic k-form with polar set contained in |S| is an element in Ck∞ (X \ |S|) such that each z ∈ |S| admits

2.1. DE RHAM COMPLEX AND ITERATED RESIDUES

71

a neighborhood U , with reduced holomorphic defining functions s1 , . . . , sm respectively for S1 , . . . , Sm in U , such that (s1 · · · sm )ν ω extends from U \ |S| as a smooth form in U for some positive integer ν = ν(z). Such a semimeromorphic form with polar set contained in |S| is said to be S-multilogarithmic, if and only if, for each z ∈ |S|, there exists an open neighborhood of z in X , together with holomorphic defining functions s1 , . . . , sm respectively for S1 , . . . , Sm in U , such that, for any j = 1, . . . , m, the restrictions of sj ω and sj dω to U are of the form (2.21)

m 

ωj,κ ,

% . ωκ semimeromorphic, with Pol(ωj,κ ) ⊂ |S| κ

κ=1

Given S1 , . . . , Sm m codimension 1 complex submanifolds which intersect transversally in X , and a k semimeromorphic form ω, with m ≤ k ≤ 2N , which is S-multilogarithmic and such that dω = 0 in X \ |S|, one can define iteratively, starting from ω ∈ Ck∞ (X \ |S|), (2.22)



Res[X ;S] (ω) = Res[Ym ;Σm ] ◦ Res[Ym−1 ;Σm−1 ] ◦ · · · ◦ Res[Y2 :Σ2 ] ◦ Res[Y1 ;Σ1 ] (ω) ∞ (S1 ∩ · · · ∩ Sm ), ∈ Ck−m

where the Σk and the Yk were defined in (2.19). Such construction depends on the ordering of the collection {S1 , . . . , Sm }. Within this geometric context, we formulate the cohomological residue formula. Proposition 2.14 (Cohomological residue formula). Let X be an N -dimensional complex manifold, and let S = (S1 , . . . , Sm ) with 1 ≤ m ≤ N be an ordered sequence of codimension 1 complex submanifolds which intersect transversally on X . Let ω be a semimeromorphic k-form which is S-multilogarithmic and moreover is supposed to be d-closed on X \ |S|, with m ≤ k ≤ 2N . Let also γ ∈ Zk−m (S1 ∩ · · · ∩ Sm , ℂ) and suppose that Γ ∈ Zk (X \ |S|, ℂ) represents δ [X ;S] (¯ γ) ∈ Hk (X \ |S|, ℂ). Then,   1 (2.23) Res[X ;S] (ω) = ω, (2iπ)m Γ γ where the iterated morphisms δ [X ;S] and Res[X ;S] have been defined in (2.20) and (2.22), respectively. Proof. The fact that the hypersurfaces Sj intersect transversally makes possible the construction of the iterated operator Res[X ;S] through an iteration of Lemma 2.2. The proof of the proposition reduces then to the proof in the case m = 1, which is a direct consequence of Lemma 2.2, together with the fact that ∂(1/ζ) = δ0 (dζ ∧ dζ)/(2iπ) in the univariate setting. For more details, we refer the reader to [AY].  Let us conclude this subsection with a few comments relative to the case m = 1. It follows from Proposition 2.14 and de Rham’s duality theorem that Res[X ;S] (ω) depends only on the cohomology class of ω in X \ S. When 1 ≤ k ≤ 2N and ω is any semimeromorphic k-form with at most poles with order ν0 > 1 along S and such that dω = 0 in X \ S, one can write, using a partition of unity (θι )ι in X

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subordinate to a covering of X by open sets Uι , where a division formula such as (2.6) holds,  dsι σν ,ι [ω] θι ν0 ∧ rν0 ,ι [ω] + ν00 −1 ω= sι sι ι

 1 1  d(θι rν0 ,ι [ω])  σν0 ,ι [ω] 0 = + d s1−ν θι rν0 ,ι [ω] − θι ν0 −1 . ι 1 − ν0 1 − ν0 sιν0 −1 sι ι ι ι Iterating this procedure, one gets a representant ωlog of the cohomology class of ω in H k (X \ S, ℂ), which is semimeromorphic with simple poles along S. The residue formula (2.23) now becomes   1 (2.24) Res[X ;S] (ωlog ) = Res[X ;S] (ωlog ). 2iπ Γ γ In fact any smooth d-closed k-form ω in X \ S with 1 ≤ k ≤ 2N , where S is a smooth complex hypersurface in X , admits in its cohomology class in H k (X \ S, ℂ) a representative ωlog with simple poles along S. This can be proved indirectly, using a duality argument based on de Rham’s theorem [DR55]; see for example [AY, Theorem 16.5]. Since such a proof fails to be constructive, we will admit the ω ) to each result here. As a consequence, one can attach a Leray residue Res[X ;S] (¯ cohomology class ω ¯ ∈ H k (X \ S, ℂ), namely Res[X ;S] (ωlog ). Residue formula (2.24) remains valid whenever Γ represents δ [X ;S] (¯ γ ) in Hk (X \ S, ℂ). 2.1.3. Meromorphic multilogarithmic forms. For m ≤ N , we consider now an ordered sequence S = (S1 , . . . , Sm ) of complex hypersurfaces S1 , . . . , Sm in an N -dimensional complex manifold X . Let, as before, |S| = S1 ∪ · · · ∪ Sm and % = S1 ∪ · · · ∪ S %j ∪ · · · ∪ Sm , for, j = 1, . . . , m. |S| j It is important to realize here that the closed analytic subsets (S1 )sing , . . . , (Sm )sing , whose codimension is greater than or equal to 2, are not assumed anymore to be empty, as was the case for m = 1 in Definition 2.8 and Proposition 2.9. If 0 ≤ p ≤ N , then the (p, 0) meromorphic forms ω, whose polar set Pol(ω) ∞ (X \ |S|) by the is contained in |S|, are characterized among elements ω in Cp,0 following two facts: • one has that ∂ω = 0 in X \ |S|; • any z ∈ |S| admits an open neighborhood U = Uz , with reduced holomorphic defining functions s1 , . . . , sm in U for hypersurfaces S1 , . . . , Sm respectively, so that for some positive integer ν = νz the (p, 0) holomorphic differential form (s1 · · · sm )ν ωU\|S| extends as a (p, 0) holomorphic form in U . This characterization follows from Hilbert’s nullstellensatz in the local nœtherian ring OℂN ,0 . The notion of S-multilogarithmic meromorphic form was introduced in [Pol72, AlT01] as an extension of Definition 2.8 to the case where m > 1. This notion does not contradict Definition 2.13 in the case where the hypersurfaces Sj were smooth.

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Definition 2.15. A meromorphic (p, 0)-form with polar set contained in |S| is said to be S-multilogarithmic if and only if any z ∈ |S| admits an open neighborhood U = Uz , with reduced holomorphic defining functions s1 , . . . , sm in U for the hypersurfaces S1 , . . . , Sm respectively, such that: • the meromorphic forms (sj ω)|U and (sj dω)U or, equivalently, the forms (sj ω)|U and (dsj ∧ ω)|U are of the form m 

(2.25)

ωj,κ

% ∩ U, with Pol(ωj,κ ) ⊂ |S| κ

κ=1

for any j = 1, . . . , m; # • the differential form dsJ = j∈J dsj does not vanish identically on any 5 irreducible component of S J ∩ U , where S J = j∈J Sj for every ordered subset J of {1, . . . , m}. " Let X ◦ = X \ J⊂{1,...,m} (S J )sing and let the meromorphic (p, 0)-form ω be S-multilogarithmic as in Definition 2.15. Then ω|X ◦ is S ◦ -multilogarithmic in X ◦ , ◦ ) of hypersurfaces, where S ◦ is defined as the ordered sequence S ◦ = (S1◦ , . . . , Sm ◦ ◦ satisfying Sj = Sj ∩ X for every j = 1, . . . , m. When m ≤ p ≤ N and the meromorphic (p, 0)-form ω is S-multilogarithmic form in X , its restriction to X ◦ , with polar set contained in |S ◦ |, is S ◦ -multilogarithmic as a meromorphic (p, 0)form in X ◦ . Then one can attach to it its Leray residue Res[X ◦ ;S ◦ ] (ω|X ◦ ), which 5m is a holomorphic (p − m, 0) form on j=1 Sj◦ ; see (2.22). Similarly one can attach 5 ◦ to it the (p, m)-current on X ◦ supported by m j=1 Sj ⊂ S {1,...,m} , whose action is defined as follows:  (2.26) Res[X ◦ ;S ◦ ] (ω|X ◦ ) : ϕ ∈ D(N −p,N −m) (X ◦ )   −→  Res[X ◦ ;S ◦ ] (ω|X ◦ ) ∧ ϕ = Res[X ◦ ;S ◦ ] (ω|X ◦ ) ∧ ϕ. m j=1

Sj◦

S {1,...,m}

◦

Proposition 2.16. Let X , S, X ◦ , S , and ω be as above. The (p, m)-current  ◦ T = Res[X ◦ ;S ] (ω|X ◦ ) ∈  D(p,m) (X ◦ ) defined in (2.26) extends as the ∂-closed (p, m)-current T = RS ∧ ω supported in |S|. This extension is defined6 in a neighborhood of each point z ∈ |S|, independently of the choice of the reduced defining functions s1 , . . . , sm for the hypersurfaces Sj that fulfill the requirements in Definition 2.15 in an open neighborhood U of the point z, as follows: ◦

(2.27) T|U



|s |2λ |s |2λm m 1 ∧ ···∧ ∂ ∧ ω|U = ∂ 2iπ 2iπ λ=0 m



m(m+1)  2 m+1−κ ∂ log |sm | ∂ log |s1 |2 ∧ ···∧ ∧ ω|U = λ 2 |sκ |2λ . 2iπ 2iπ λ=0 κ=1

Moreover, when m < N , given any reduced holomorphic function sm+1 in U , which is not identically zero on any irreducible component of S J , for some specific ordering   meaning of the symbol in (2.27) remains the same as in many places in our λ=0 introductory first chapter: one considers first the current-valued function of λ as defined from the beginning for Re λ 1 in a conic sector with aperture < π/(m + 1) bisected by the real positive axis ℝ+ , then takes the meromorphic continuation as a function of λ, which remains pole-free around ℝ+ , and finally evaluates the value at λ = 0. 6 The

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J = {1, . . . , m}, one has also that (2.28) T|U =

m(m+1) m−1

∂ log |s |2   m+1−κ ∂ log |s1 |2 m λ 2 |sm+1 |2λ ∧···∧ ∧ ω|U |sκ |2λ . 2iπ 2iπ λ=0 κ=0 Proof. Let us prove first that the definition of T|U is licit in a neighborhood U of z ∈ |S| and that it does not depend on the reduced defining functions s1 , . . . , sm in U for S1 , . . . , Sm respectively, and that moreover (2.28) holds. We will assume first that m = 2 < N . We need at this point the following lemma. Lemma 2.17. Let f, g, h, u, v be holomorphic functions in a neighborhood Ω of the origin in ℂN , N > 2, and let q, r ∈ ℕ∗ . Suppose that u, v are invertible in Ω and that the hypersurfaces Sf = V (f ), Sg = V (g), Sh = V (h) intersect properly. Then the  D(0,1) (Ω)-valued map |ug|2μ |vf |2λ  μ −→ Tμ = ∂ gr fq λ=0 extends from {μ ∈ ℂ : Re μ  1} to the complex plane ℂ as a meromorphic map with poles in −ℚ+ . Its value at μ = 0 is independent of u and v. Furthermore,the  (0,2) D (Ω)-valued map ν −→ |h|2ν ∂T0 extends from {ν ∈ ℂ : Re ν  1} as a  meromorphic map with poles in −ℚ+ , with |h|2ν ∂T0 ν=0 = ∂T0 . In addition, both  (0,2) D (Ω)-valued functions |vg|2λ |uf |2λ2 |vg|2λ2 |uf |2λ3 2λ ∧ , λ −  → |h| ∧ λ −→ ∂ ∂ ∂ ∂ gr fq gr fq extend from {ν ∈ ℂ : Re ν  1} to the complex plane ℂ as meromorphic maps with poles in −ℚ+ . Their value at λ = 0 is independent of the invertible functions u and v and coincides there with the value of ∂T0 . Proof. Hironaka’s resolution of singularities theorem (see Theorem A.41), reduces the proof to the case where  f , g, h are monomial maps. Then, all assertions are easily proved. The assertion |h|2ν ∂T0 ν=0 = ∂T0 is the only one that requires the fact that codim(V (f ) ∩ V (g) ∩ V (h)) = 3, while the current involved is a (0, 2)-current. We will state in §2.2 a dimension principle that will cover this situation.  Lemma 2.17 can be iterated and thus provides the justification needed for the definition of T|U regarding its independence on the choice of the reduced defining holomorphic functions sj . Moreover, iteration of Lemma 2.17 shows that for ω|U =

ΨU , (s1 · · · sm )q

where q ∈ ℕ∗ and ΨU ∈ Ωp (U ), one has (2.29) T|U =  

|s |2λm |s |2λ2 |s |2λ1   m 2 1 ∧ · · · ∧ ∂ ∂ ∂ · · · ∧ΨU 2iπ sqm 2iπ sq2 2iπ sq1 λ1 =0 λ2 =0 λm−1 =0 λm =0  and |sm+1 |2λ T|U λ=0 = T|U . Let us now show that T extends T ◦ in U . Let z ◦ ∈ |S ◦ | ∩ U and take (s1 )|U ◦ , . . . , (sm )|U ◦ as defining functions for the smooth

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75

hypersurfaces Sj◦ around z ◦ . The hypothesis that ω is S-multilogarithmic implies that ω|U ◦ is expressed as ω|U ◦ =

m ' dsj ∧ r ◦ [ω] + σ ◦ [ω], s j j=1

 ◦ where r ◦ [ω] is a (p−m, 0) holomorphic form in U ◦ and σ ◦ [ω] is of the form m κ=1 κ , % respectively. As a where the forms κ◦ have their polar sets contained in |S| κ consequence

|s |2λ |s |2λm  m 1 ∂ = 0. ∧ ···∧ ∂ ∧ σ ◦ [ω] 2iπ 2iπ λ=0 Since

|s |2λ |s |2λm  m 1 ∂ ∧ r ◦ [ω] ∧ ··· ∧∂ ∧ (ds1 ∧ · · · ∧ dsm ) 2iπ sm 2iπ s1 λ=0    = r ◦ [ω] ∧ S {1,...,m} |U ◦ = Res[X ◦ ;S ◦ ] (ω|X ◦ ) |U ◦ ,  T coincides with T ◦ = Res[X ◦ ;S◦ ] (ω|X ◦ ) on U ◦ , and the proposition is proved.  Remark 2.18. When U is an open neighborhood of an arbitrary point in |S|, the fact that |sm+1 |2λ T|U = T|U for any reduced holomorphic function in U , which is not identically zero on any irreducible component of the closed analytic subset S {1,...,m} ∩ U = S1 ∩ S2 ∩ · · · ∩ Sm ∩ U , can be interpreted as follows. The current T supported on S {1,...,m} is the standard extension of its restriction to the (N − m)dimensional complex manifold X ◦ ∩ S {1,...,m} . The current T coincides with the current induced by the iterated Leray residue Res[X ◦ ;S ◦ ] (ω|X ◦ ) on the manifold X ◦ ∩ S {1,...,m} . Actually, the action of T , as a current in X supported by S {1,...,m} , involves nontangential derivations, while the Leray residue Res[X ◦ ;S ◦ ] (ω|X ◦ ), which it extends from the point of view, is a holomorphic (p − m, 0) form on the complex manifold X ◦ ∩ S {1,...,m} , and is considered an open subset of S {1,...,m} . 2.2. Coleff–Herrera sheaves of currents The notion of the Coleff–Herrera sheaf of currents on a complex manifold, with support in a prescribed closed analytic subset V , and its importance were pointed out by J.E. Bj¨ ork in [Bj96, Bj04]. The original construction of sections of such a sheaf is due to N. Coleff and M. Herrera in [CoH78]. In this section, following the approach of J.E. Bj¨ ork, M. Andersson, H. Samuelsson Kalm [Bj96, Bj04, And09, BjS10], we will introduce the various concepts and then revisit (with this point of view) the original theory developed in 1978 by N. Coleff and M. Herrera in [CoH78]. Let us fix from now on some notations that will be used throughout this section. Let X be an N -dimensional complex manifold and let z ∈ X . Its structural sheaf will be denoted by O. For any sheaf F of O-modules on X , we will denote by Fz its stalk at a point z ∈ X . For a neighborhood U of z and f ∈ F(U ), ∞ the germ of f at z will be denoted by fz ∈ Fz . For any 0 ≤ p, q ≤ N , we let Cp,q  (p,q) be the sheaf of smooth (complex-valued) (p, q)-forms on X and D be that of complex-valued currents with bidegree (p, q) on X . We let also OX ∂/∂ζ be the sheaf of holomorphic differential operators. It is a sheaf, whose stalk at z consists

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of germs of operators

∂ ∂ Q ζ1 , . . . , ζn , ,..., ∂ζ1 ∂ζN

at z, where Q is holomorphic in ζ1 , . . . , ζN and polynomial in the ∂/∂ζj , j = 1, . . . , N . As usual, a system of local coordinates in a neighborhood of z ∈ X will be denoted by (ζ1 , . . . , ζN ). It is said to be centered if ζ(z) = 0. 2.2.1. Integration currents, pseudo-meromorphicity, and holonomy.  Let V0 = {z : v1 (z) = · · · = vK (z) = 0} 0 be the germ at the origin of a purely dimensional closed analytic set V in a neighborhood of the origin in ℂN with codim0 V = M . The local presentation of closed analytic subsets (see Proposition A.24 in §A.3.4) allows us to assume that V admits around 0 the following representation. There exist f1 , . . . , fM so that df1 ∧ · · · ∧ dfM does not vanish identically on any irreducible component of the germ of a closed analytic set 5 −1 W = M j=1 V (fj ), where V (fj ) is by definition fj ({0}). Furthermore, we assume that V is the union of a finite number of irreducible components W1 , . . . , WL among the irreducible components W1 , . . . , WL of W . Namely, one has for some 1 ≤ L ≤ L, (2.30)

V =

M 3

M

3

V (fj ) \ (WL +1 ∪ · · · ∪ WL ) = V (fj ) \ V (fM +1 ),

j=1

j=1

K

where fM +1 = κ=1 λκ vκ does not vanish identically WL +1 \ V, . . . , WL \ V , so that V = W V (fM +1 ) according to the notation in §D.1.1; see formula (D.4). It follows from Proposition 2.16 that the integration (M, M )-current [W ] factorizes in a neighborhood of the origin as [W ] = R(V (f1 ),...,V (fM )) ∧ (2.31)

M ' dfj f j=1 j

|f |2λ |f |2λM  M 1 ∧··· ∧ ∂ = ∂ ∧ df 2iπ fM 2iπ f1 λ=0 M

M (M +1) M |f |2λM +1−κ ' dfj  κ=1 κ = λ 2 ∧ df, f1 · · · fM f λ=0 j=1 j

where df = df1 ∧ · · · ∧ dfM . Moreover, it follows from (2.28) that (2.32)

M −1 M +1−κ M

M (M +1) ' |fκ |2λ dfj  2λ [V ] = λ 2 (1 − |fM +1 | ) κ=0 ∧ df f1 · · · fM f λ=0 j=1 j

|f |2λM |f |2λ1 

   M 1 = (1 − |fM +1 |2λM +1 ) ∂ ∧ ··· ∂ ··· ∧ df. fM f1 λ1 =0 λM +1 =0

Let us consider such a factorization in a neighborhood Ω of the origin in ℂN and 6 → Ω be a log resolution for the hypersurface "M +1 V (fj ); see Definition let π : Ω j=1 A.40 and Theorem A.41. Consider the local centered coordinates t = (t1 , . . . , tN ) 6 Then, in a neighborhood of a point z˜ (with t(˜ z ) = 0) on the complex manifold Ω.

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77

one has, for any j = 1, . . . , M + 1, in the corresponding local chart that π ∗ fj (t) = uj (t) tαj , j = 1, . . . , M + 1,

N α where tαj = κ=1 tj j,κ , αj,κ ∈ ℕ, and uj is invertible. The last representation implies that in a neighborhood of z˜ and outside t1 · · · tN = 0 one has that M M '  ' dfj dtjκ (t) = γJ , π∗ t f κ=1 jκ j=1 j 1≤j m. For any z ∈ V ∩ S and any local defining function s for S around z, the map λ −→ (1 − |s|2λ )T extends from {λ ∈ ℂ : Re λ  1} as a

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current-valued map whose value at λ = 0 belongs to PM(X ); see Proposition 2.21 and Remark 2.22. Moreover, one has in this case and for any z ∈ X that   (2.39) (1 − |s|2λ ) T λ=0 z = 0 independently of the choice of the defining section s. We rephrase (2.39) as   (2.40) 1S · T z = 0.  Proof. The germ at z of current T − |s|2λ T λ=0 is supported by V ∩ S since |s|0 =1 on V \ S. Since codim(V ∩ S) > m, it follows from Proposition 2.26 that Tz − |s|2λ T λ=0 z = 0.  Remark 2.28. One can restate Corollary 2.27 as follows: any (•, m) pseudomeromorphic current, whose support is contained in a purely dimensional closed analytic subset of codimension m in X , is realized as the standard extension TV \S→V from its restriction to V \ S. The codimension of the hypersurface S satisfies codim(V ∩ S) > m. For example, such was the case for the current RS ∧ ω in Proposition 2.16. Keeping its notation, consider V = S {1,...,m} and choose the hypersurface S to be such that V \ S = S {1,...,m} ∩ X ◦ . Remark 2.29. The concept of a pseudo-meromorphic current on a complex manifold X introduced in Definition 2.19 can be carried through to a purely dimensional complex analytic space; see for example [AndW18, AndW21]. We stick here to the most general definition, as formulated in [AndW18, §2.1] and rephrased in [AndW18, Theorem 2.15(2)]. Suppose that π & : X& → X is a smooth modification around z ∈ X ; see Theorem A.41 in §A.4.3. Then a germ of a current at z is a germ of a pseudo-meromorphic current at z on X if it is the germ at z of a current π &∗ (& μ), where μ & is a pseudo-meromorphic current on X&. It will be enough for our purposes in this monograph to follow a more restrictive definition. Such a restricted version corresponds to the original definition introduced in [AndW10, §2] and with the following formulation: a current is pseudo-meromorphic in a neighborhood of z ∈ X if it is the sum of currents (Πι )∗ τι , where τι is of the form &Nι ◦ · · · ◦ π %1 is a tower of successive either smooth modifications (2.33) and Πι = π or open inclusions. Propositions 2.21, 2.24, 2.26 (dimension principle) as well as Corollary 2.27 remain valid in these more general contexts; see [AndW18, §2.3] and [AndW21, §2.1]. 2.2.2. The sheaf CHX ,V . We introduce in this section the important notion of the Coleff–Herrera current for a purely dimensional, closed analytic subset in an N -dimensional complex manifold. These currents induce the Coleff–Herrera sheaf of currents. We start for that purpose with a preliminary definition. Definition 2.30. Let 0 ≤ M ≤ N and z ∈ X . A germ Qz of a holomorphic ∞ ∞ differential operator from the stalk (CN,N −M )z to the stalk (CN −M,N −M )z is a ℂlinear operator between these two ℂ-vector spaces. Its action is described in local coordinates for the basis {dζ J ∧ (dζ1 ∧ · · · ∧ dζN ) : |J| = N − M }, {dζ J ∧ dζI : |I| = |J| = N − M }  N 2  N  , M matrix with entries in Oz ∂/∂ζ. Its adjoint operator Q∗z acts, by a M at the level of corresponding stalks, from ( D(M,M ) )z to ( D(0,M ) )z as follows. If

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83

T is a representative of Tz in an infinitesimal neighborhood Uz of z, where Qz is represented by Q, then Q∗z (Tz ) = (Q∗ (T ))z . Recall that  ∗ Q (T ) , ϕ = T, Q(ϕ) . (2.41) ∀ϕ ∈ D(N −M,N −M ) (U ), Proposition 2.31. Let z ∈ X and let Tz ∈ (PM(M,M ) )z be such that one has Supp Tz ⊂ Vz , where Vz is the germ at z of a closed analytic subset V . We also assume for the sake of simplicity10 that Tz equals the germ at z of a finite sum of germs of currents realized in a neighborhood of z as direct images Tι = (Πι )∗ τι , where Πι is a tower of either smooth modifications or open inclusions and each elementary current τι is of the form (2.33) (hence Nι = N ). If Qz is any germ of ∞ ∞ a holomorphic differential operator from (CN −M,N )z to (CN −M,N −M )z , then one has that Supp (Q∗z (Tz )) ⊂ Vz and Q∗z (Tz ) ∈ (PM(0,M ) )z . Proof. The first assertion is immediate: consider a small neighborhood U of z with a representative V of Vz in U , and a representative Q of Qz . For any ϕ in D(N,N −M ) (U ), one has that Supp(ϕ) ⊂ U \ V =⇒ Supp(Q(ϕ)) ⊂ U \ V . Let us prove the second assertion. One may assume that a representative T of Tz in some neighborhood U of z is the direct image of a current τ of bidimension equal to (N − M, N − M ). The current τ is supposed to be defined in a neighborhood U of the origin in ℂN . Then one has T = Π∗ τ , where Π is a holomorphic map Π : U → U which equals a tower of smooth modifications or open inclusions. Moreover, τ is of the form

1  ' 1

! ⊗ ∧ (t), PV γκ ∂ γκ τ= tκ tκ   1≤κ≤k

k M , such that Tz admits the representation |δ |2λ

 z . (2.42) Tz = Q∗z [Vz ] δz λ=0 Then T ∈ PM(0,M ) (X ). Furthermore, the current T satisfies around each point z ∈ V the two following properties: (1) (2.43)

∀ h ∈ I(Vz ) = {h ∈ Oz : Vz ⊂ V (h)}, h · Tz = 0;

(2) given any germ of a hypersurface Sz at z such that codimz (Sz ∩ Vz ) > M , and a reduced defining germ s for it in a neighborhood of z,   2λ (2.44) |s| T λ=0 z = Tz . Proof. The first assertion follows from the fact that [V ] ∈ PM(M,M ) (X ); see Example 2.20. The second assertion follows from Propositions 2.31 and 2.24, together with Corollary 2.27.  For a purely dimensional closed analytic subset V with codim V = M > 0 in the complex manifold X , the sheaf of ∂-closed currents of bidegree (•, M ) supported by V will play a major role. Its definition is suggested by Corollary 2.32. This sheaf is called the Coleff–Herrera sheaf, referring to the pioneering contribution of Nicolas R. Coleff and Miguel E. Herrera [CoH78]; see also [DS85]. Its definition was formalized by J.E. Bj¨ ork in [Bj96, Bj04]. The variation of the definition we present here is based on the analytic continuation approach used by M. Andersson in [And09]. Definition 2.33 (Coleff–Herrera sheaf). Let X be an N dimensional manifold and let V be a closed analytic subset of pure codimension M > 0. A Coleff–Herrera current T in an open subset U ⊂ X , with respect to V , is a ∂-closed (•, M )-current T in U , with Supp T ⊂ V satisfying the following properties: (1) for each z ∈ U ∩ V , its germ Tz at z satisfies (2.43); (2) for any s ∈ Oz such that codimz (V (s) ∩ Vz ) > M ,11 λ −→ |s|2λ Tz 11 This additional property is also usually reformulated as follows : T satisfies the standard extension property (S. E. P) locally around V ∩ S from its restriction to the open subset V \ S, when S is a closed hypersurface so that codim(V ∩ S) > codim V = M .

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85

extends from {λ ∈ ℂ : Re λ  1} to the complex plane ℂ as a ( D(•,M ) )z valued meromorphic function for which (possible) poles have strictly negative real part and which satisfies (2.44). The sheaf, whose sections are (p, M ) Coleff–Herrera currents, is denoted by CHpX ,V , with N ( CHpX ,V . CHX ,V = p=0

Remark 2.34. Let V be a closed analytic subset with pure codimension M of an N -dimensional complex manifold X . Given z 0 ∈ V , consider at the level of germs at z 0 two local embeddings ι : V ∩Uz0 → U ⊂ ℂNι and ι : V ∩Uz0 → U ⊂ ℂNι . As in Remark A.16, one may assume that ι = ιmin is minimal, thus U = U × ℂνw with ν = Nι − Nιmin and ι = ι : z → (ιmin (z), 0) in a neighborhood of z 0 on V . One has −M (2.45) Tιmin (z0 ) ∈ (CHN U ,ιmin (V ∩U

)ιmin (z0 )

z0 )

−M +ν =⇒ (T ∧ [w = 0])ι(z0 ) ∈ (CHN U ×ℂν ,ι(V ∩U

z0 )

As a consequence, the morphism N −M +ν (CHU  ,ι(V ∩U

(2.46)

)ι(z0 ) z0 )

 −M → [w = 0] ∧ CHN U ,ιmin (V ∩U

)ι(z0 ) .



0 z 0 ) ι(z )

is injective. Such a morphism fails to induce an isomorphism of sheaves since, for example, m '   1 −M N −M +ν dw ∧ ∂ γj +1 ∧ CHN U ,ιmin (V ∩Uz0 ) ι(z 0 ) ⊂ (CHU ,ι(V ∩Uz0 ) )ι(z 0 ) w j j=1 while (2.47) ν '  dw ∧ ∂ j=1

1 γ +1 wj j

−M ∧ CHN U ,ιmin (V ∩U

z0 )

 ι(z 0 )

 −M ⊂ [w = 0] ∧ CHN U ,ιmin (V ∩U

z0 )

 ι(z 0 )

⇐⇒ (γ1 , . . . , γν ) = 0. The possibility of taking derivatives in additional directions, such as those corresponding here to the wj ’s, prevents as a consequence the (germs of) Coleff– Herrera sheaves (CHX ,V )z , where z ∈ V , being independent of the local embedding −M V = CHdim Vz → Xz . Despite this fact, the Coleff–Herrera sheaf CHN X ,V , once X ,V combined with the structural sheaf on V , plays a key role in the representation of the dualizing sheaf 12 ωVdimV as −M N −M (2.48) ωVN −M = ωVdim V = HomOX (OX /IX ,V , CHN X ,V ) = HomOX (OV , CHX ,V ).

The reason why the sheaf on the right-hand side of (2.48) is intrinsically defined is that  −M HomOU OU /Iιmin (V ∩Uz0 ),U , CHN U ,ιmin (V ∩Uz0 ) )   N −M +ν ,  HomOU OU /Iι(V ∩Uz0 ),U , CHU  ,ι(V ∩U 0) z

12 See

Definition B.34 and Remark B.35.

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2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

where ι = ιmin is a local minimal embedding around z 0 , thus U = U × ℂνw with ν = Nι −Nιmin and ι = ι : z → (ιmin (z), 0) in a neighborhood of z 0 . Instead of the −M −p structural sheaf OV , one may as well use the sheaves ΩN , p = 1, . . . , N − M , V of strongly holomorphic (N − M − p, 0)-forms on V which are also intrinsically defined.13 It follows then from the isomorphisms  −M −p −M HomOU ΩN /(Jιmin (V ∩Uz0 )→U )K¨ah , CHN U U ,ιmin (V ∩Uz0 ) )  −M −p  N −M +ν /(Jι(V ∩Uz0 )→U )K¨ah , CHU  HomOU ΩN  ,ι(V ∩U U 0) z

involving the sheaves of K¨ ahler-Grothendieck differentials14 under the same conditions on the local embeddings ι = ιmin and ι = ι, together with the fact that Coleff–Herrera currents satisfy the standard extension property (which allows to quotient out the torsion) that for any p = 0, . . . , N − M − 1, (2.49)

−M −p −M , CHN ωVp  HomOX (ΩN X ,V ), V

where the Barlet sheaf ωVp ⊂X = ωVp is the sheaf which sections are (p, 0)-forms holomorphic in the sense of Barlet; see Definition B.34. We refer the reader for more details about such isomorphisms to [Sam21, proof of Proposition 4.3]. Example 2.35. Proposition 2.24 shows that any pseudo-meromorphic current T in U ⊂ X , whose support Supp T is contained in U ∩V , satisfies (2.43). Corollary 2.27 shows that any such pseudo-meromorphic current T , with bidegree (•, M ), satisfies also the standard extension property required in Definition 2.33. As a consequence,15 any ∂-closed pseudo-meromorphic (•, M )-current in U supported by the purely dimensional closed subset U ∩ V belongs to CHX ,V (U ). Remark 2.36. Given a holomorphic bundle E over X , the sheaf of E-valued Coleff–Herrera currents for a closed analytic subset V with pure codimension M is the sheaf whose sections on any open subset U are E-valued (•, M )-currents in U supported by U ∩ V . The coordinates of these sections expressed in a local frame in a neighborhood of any z ∈ U ∩ V have their germs belonging to the corresponding stalk (CHX ,V )z . The sheaf is denoted by CHX ,V (·, E) and is decomposed as CHX ,V (·, E) =

N (

CHpX ,V (·, E),

p=0

according to the bidegrees (p, M ) of currents, for p = 0, . . . , N . We state next an important local structure theorem for sections of CH0X ,V . Besides the fact that such a result (see [Bj96, Bj04, And09, BjS10]) shows their pseudo-meromorphicity, it also emphasizes the characteristic property that only holomorphic differential operators are involved in the action of such currents. This explains why they do play a role of an algebraic nature, despite of their analytic structure. These differential operators with analytic coefficients are, as we will see later on in this monograph, reminiscent of the so-called Nœtherian operators involved in the algebraic formulation of the Ehrenpreis–Palamodov fundamental 13 See

Definition B.32 and Remark B.33. Remark B.33. 15 One could view this assertion as the residue analogue to the second assertion in the dimension principle for normal currents (see Proposition C.26) or the second assertion in the dimension principle for generalized cycles (see Proposition D.5). 14 See

2.2. COLEFF–HERRERA SHEAVES OF CURRENTS

87

principle, [Ehr,Pal,Bj74,AndW07] or §5.1.2 and §7.6.3 in this monograph. Local structure results of this kind originally go back to the work of P. Dolbeault [Dol93]. The result below is due to Jan-Erik Bj¨ ork [Bj04], who formalized for the first time the concept of the Coleff–Herrera current and showed the major role it plays. Proposition 2.37 (Structure of CH0X ,V -germs). Let X and V be as above, where codim V = M , z ∈ V , and Tz ∈ (CH0X ,V )z . There exists a germ qz of a ∞ ∞ holomorphic differential operator from (CN,N −M )z to (C0,N −M )z , together with a germ ωz of an (N − M, 0) holomorphic form, and δz ∈ Oz with codim(V (δ) ∩ Vz ) > M , such that

 |δ|2λ [V ] (2.50) Tz = Q∗ , δ λ=0 z ∞ ∞ where Qz : (CN,N −M )z → (CN −M,N −M )z is the germ of a holomorphic differential operator defined by ∞ ∀ ϕ ∈ (CN,N −M )z ,

Qz (ϕ) = qz (ϕ) ∧ ωz .

Proof. We follow here the argument from [And09]. First, consider the case when V is smooth around z. Thus V is defined in local, centered coordinates (ζ1 , . . . , ζN ) in a neighborhood of this point as V = {ζ1 = · · · = ζM } = {ζ = 0}. We set ω to be the differential form ω = dζM +1 ∧ · · · ∧ dζN . Let also NT be the order of the representative T around z. For j = 1, . . . , M , one has ζ j Tz = 0, because of (2.43). This implies dζj ∧ Tz = 0,

(2.51)

∀ j = 1, . . . , M,

since T is assumed to be ∂-closed. From now on, we fix a small neighborhood U = Uz of z, namely a polydisk Δζ ×Δζ  , where we separate the local coordinates in two blocks ζ = (ζ1 , . . . , ζM ) and ζ  = (ζM +1 , . . . , ζN ), and keep T as representative for Tz of order NT in U . Let 

ϕJ dζJ ∧ (dζ1 ∧ · · · ∧ dζN ) ∈ D (N,N −M ) (U ). ϕ= |J|=N −M

It follows from (2.51) and the property ζj T = 0 for j = 1, . . . , M , that for any θ ∈ D(U ), which equals identically to one around z,    θT, ϕ = T, θ ϕ{M +1,...,N } dζ  ∧ dζ ∧ ω(ζ  ) = T, θ ψ dζ  ∧ dζ ∧ ω(ζ  ) ∂ κ    ζκ , dζ  ∧ ω(ζ  ) ∧ dζ , [ψ](0, ζ  ) =± T,θ ∂ζ κ! M {κ∈ℕ

: |κ|≤NT }

where dζ = dζ1 ∧ · · · ∧ dζM and dζ  = dζM +1 ∧ · · · ∧ dζN . We use here notations that will be used throughout this monograph, namely, for any κ ∈ ℕM , κ! =

M  j=1

κj !,

M ∂ κ  ∂ κj = , ∂ζ ∂ζj j=1

ζκ =

M 

κ

ζj j ,

|κ| = κ1 + · · · + κM .

j=1

If π denotes the projection (ζ, ζ  ) −→ ζ  , then the direct images of (0, 0)-currents (that is, of distributions) in Δζ  , defined by   aκ (ζ  ) = π∗ ζ κ T (ζ, ζ  ) ∧ dζ , κ ∈ ℕM with |κ| ≤ NT ,

88

2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

are represented by holomorphic functions aκ in Δζ  . This follows from the fact that T is ∂-closed. Therefore, if ∂  aκ (ζ  ) ∂ κ (ψ) dζ  , (ϕ) = ± q ζ, ∂ζ κ! ∂ζ M {κ∈ℕ

then one has

: |κ|≤NT }

  Tz , ϕz  = [ζ = 0] , q(ϕ) ∧ ω = [V ]z , Q(ϕz ) ,

which proves the result in this case. Let us now consider the case where z ∈ Vsing . We recall here the following fact; see Proposition A.24 in §A.3.4. In a convenient neighborhood U = Uz of z, V ∩ U is the union of finitely many components W1 , . . . , WL of a reduced complete intersection W . More precisely W = {ζ ∈ U : f1 (ζ) = · · · = fM (ζ) = 0}, where fj ∈ O(U ) for j = 1, . . . , M , and df1 ∧· · ·∧dfM ≡ 0 on any irreducible component of W , in particular on W1 , . . . , WL . Furthermore, because of the Nœther presentation of V , which follows from Proposition A.24 (see §A.3.4), one can choose centered local coordinates (ζ, ζ  ) ∈ Δζ × Δζ  in a neighborhood of z, where ζ = (ζ1 , . . . , ζM ) and ζ  = (ζM +1 , . . . , ζN ) such that the projection π : (ζ, ζ  ) → ζ  is proper. Moreover,16  ∂f (ζ, ζ  )    (2.52) J(ζ, ζ  ) =   ≡ 0 on (Δζ × Δζ  ) ∩ W for  = 1, . . . , L. ∂ζ As before, let NT be the order of a representative T of Tz in an arbitrarily small neighborhood U of z. In a neighborhood of any point z0 = (ζ 0 , ζ 0 ) in the open  subset (Δζ × Δζ  ) ∩ V \ V (J), one can use (f , ζ  ) as local coordinates. The chain rule implies that ⎞ ⎞ ⎛ ⎛ ∂/∂f1 ∂f −1 ∂/∂ζ1 ⎜ .. ⎟ ⎜ .. ⎟   t (ζ, ζ  ) (2.53) ⎝ . ⎠ (ζ, ζ ). ⎝ . ⎠(ζ, ζ ) = ∂ζ ∂/∂fM ∂/∂ζM Since T is ∂-closed, the direct images of (0, 0)-currents aκ , with κ ∈ ℕM satisfying |κ| ≤ NT , defined in Δζ  by   aκ (ζ  ) = π∗ f κ (ζ, ζ  ) df (ζ, ζ  ) ∧ T (ζ, ζ  ) (where df = df1 ∧· · ·∧dfM ), are ∂-closed. Thus, these (0, 0)-currents are represented by holomorphic functions aκ in Δζ  . Moreover, for ν ◦ ∈ ℕ∗ large enough,  ◦ aκ (ζ  ) ∂ κ q ◦ : (ζ, ζ  ) −→ J ν (ζ, ζ  ) κ! ∂f M {κ∈ℕ

: |κ|≤N }

is a holomorphic differential operator in Δζ ×Δζ  . Observe now that any differential (N, N − M ) form ϕ(ζ, ζ  ) = ψ dζ  ∧ dζ1 ∧ · · · ∧ dζN in Δζ × Δζ  can be rewritten as ϕ(ζ, ζ  ) =

ψ(ζ, ζ  )  dζ ∧ df (ζ, ζ  ) ∧ dζ  . J(ζ, ζ  )

16 Note that the jacobian determinant J in (2.52) can be used as a universal Oka denominator for holomorphic functions on U ∩ Vreg that extend as locally bounded functions on U ∩ V ; see §A.3.5.

2.2. COLEFF–HERRERA SHEAVES OF CURRENTS

Since, for ν ∈ ℕ∗ large enough, q : ψ −→ J ν q ◦

89

ψ

J remains a holomorphic differential operator in Δζ × Δζ  , one can represent T in (Δζ × Δζ  ) \ V (J) as

|J|2λ  (2.54) T = Q∗ [V ] , ◦ J ν +ν λ=0 where Q : ψ dζ  ∧ (dζ1 ∧ · · · ∧ dζN ) −→ ± q(ψ) dζ  . Finally we claim that both the current T and the pseudo-meromorphic (0, M )current

|J|2λ  Q∗ [V ] ◦ J ν +ν λ=0 in Δζ × Δζ  inherit the standard extension property (2.44). Actually, for μ ∈ ℂ with Re μ  1, |J|2λ 

[V ] (|J|2μ T )z = |J|2μ Q∗ ◦ J ν +ν λ=0 z as a consequence of the equality of (0, M )-currents in (Δζ × Δζ  ) \ V (J). The assertion of the proposition follows by applying the analytic continuation principle. The proof of Proposition 2.37 is completed.  Remark 2.38. One should point out that the denominator δ that appears in the local representation (2.50) in the above proof can be chosen as a power of the Oka universal denominator attached to V around the point z; see §A.3.5. Proposition 2.37 for sections of CH0X ,V can be slightly modified to show that (CHpX ,V )z ⊂ PMz for any z ∈ V and for any 0 ≤ p ≤ N . In fact, one can summarize all results established up to now and conclude this subsection with the following statement. Theorem 2.39. Let X be an N -dimensional complex manifold and let V be any purely dimensional, closed analytic subset of X of codim V = M > 0. For each p = 0, . . . , N , the sheaf CHpX ,V coincides with the sheaf of (p, M ) pseudomeromorphic currents supported by V and ∂-closed. Proof. As we have already seen (see Example 2.35), any pseudo-meromorphic (p, M )-current in U ⊂ X supported by V and ∂-closed in U belongs to CHpX ,V (U ). It follows from Proposition 2.37, together with Example 2.20 and Proposition 2.31, that sections of CH0X ,V are also sections of PM(0,M ) . Thus Theorem 2.39 holds for p = 0. Let us consider Tz ∈ CHpX ,V and show that it is a germ at z of pseudomeromorphic (p, M )-current. Let us express the germ Tz as

 Tz = (dζi1 ∧ · · · ∧ dζip ) ∧ TI . 1≤i1 M . Recall that one defines meromorphic functions with a polar set included in a closed hypersurface S ⊂ X starting from the definition of meromorphic functions on X as holomorphic functions h in X \ S that are such that [h] extends as a ∂-closed distribution to X . Then it is natural to use the Cauchy–Riemann operator ∂ to introduce the sheaf CHS X ,V of meromorphic Coleff–Herrera currents with prescribed polar set along S. This is done as follows. Definition 2.40 (The sheaf CHS X ,V ). Let U be an open subset of X . A (•, M )current supported by V is said to be a meromorphic Coleff–Herrera current with prescribed polar set along S if it satisfies both conditions (1) and (2) as in Definition 2.33, together with the condition (2.55) The corresponding sheaf

Supp (∂ T ) ⊂ V ∩ S. CHS X ,V

decomposes as

CHS X ,V =

N (

CHS,p X ,V

p=0

according to the bidegrees (p, M )-currents for p = 0, . . . , N . Remark 2.41. Let E be a holomorphic bundle and let U an open subset of X . An E-valued current T ∈  D(•,M ) (U, E) is said to be a meromorphic E-valued current with prescribed polar set along S if its coordinates in a local frame in a neighborhood of any z ∈ U ∩ V have their germs at z lie in (CHS X ,V )z . This sheaf is denoted by CHX ,V (·, E) and splits as CHS X ,V (·, E) =

N (

CHS,p X ,V (·, E),

p=0

according to the bidegrees of (p, M )-currents for p = 0, . . . , N . Theorem 2.39 implies the following useful result. Theorem 2.42. Let X be an N -dimensional complex manifold and let V ⊂ X be a purely dimensional closed analytic subset with codimension M . Assume also that S is a closed analytic hypersurface in a neighborhood of V such that one has codim(V ∩ S) > M . The sheaf CHS X ,V coincides with the sheaf of (•, M ) pseudomeromorphic currents T supported by V and satisfies Supp (∂T ) ⊂ V ∩ S. Proof. Any pseudo-meromorphic (•, M )-current in U ⊂ X , with Supp T included in V , satisfies conditions (1) and (2) in Definition 2.33; see Proposition 2.24 and Corollary 2.27. Then, any pseudo-meromorphic (•, M )-current T supported by V and which satisfies (2.55) belongs to CHS X ,V (U ). Let us now prove the converse inclusion. In order to do that, consider Tz ∈ CHS X ,V . That is, consider the germ

2.2. COLEFF–HERRERA SHEAVES OF CURRENTS

91

Tz of Coleff–Herrera current with prescribed polar set along S at a point in z ∈ V . Since (CHS X ,V )z = (CHX ,V )z for any z ∈ V \ S and Theorem 2.39 holds in that case, we are led to consider the case where z ∈ V ∩ S. Let s be a reduced defining holomorphic function for S around z. Then for ν ∈ ℕ∗ strictly larger than the order of (∂T )z around z, one has that (sν T )z ∈ (CHX ,V )z ⊂ PMz , where the last inclusion follows from Theorem 2.39. Furthermore, Proposition 2.21 asserts that the ( D)z valued map

 |s|2νλ (sν T ) λ −→ ν s λ=0 z extends from {λ ∈ ℂ : Re λ  1} to the complex plane ℂ as a meromorphic map with (possible) poles in −ℚ+ . Its value at λ = 0 belongs to PMz . Since   2νλ |s| T λ=0 z = Tz , because T satisfies condition (1) as in Definition 2.33, one gets that Tz ∈ PMz . Since Tz satisfies also that Supp (∂ T )z ⊂ Vz ∩ Sz , the converse inclusion is proved. This concludes the proof of the theorem.  Sections of CHS X ,V induce via the action of the ∂ operator sections of CHX ,V as follows.

∩S

Theorem 2.43. Let X be an N -dimensional complex manifold and V ⊂ X be a purely dimensional, closed analytic subset with codimension M . Let also S be a closed analytic hypersurface in a neighborhood of V such that codim(V ∩ S) > M . Then the following inclusion of sheaves is valid: (2.56)

∂ (CHS X ,V )z ⊂ (CHX ,V ∩S )z , for any z ∈ V.

Proof. Theorem 2.42 and the obvious fact that ∂ acts from PM(•,M ) to imply the inclusions PM (•,M +1)

(•,M ) ∂ (CHS )z ⊂ (PM(•,M +1) )z . X ,V )z ⊂ ∂ (PM

Given z ∈ V and Tz ∈ (CHS X ,V )z , one has by definition Supp (∂ Tz ) ⊂ Vz ∩ Sz . Since codimz (V ∩ S) > M , ∂ Tz ∈ (PM(•,M +1) )z , and ∂(∂ Tz ) = 0, it follows from Theorem 2.39 that ∂ Tz ∈ (CHX ,V ∩S )z , which concludes the proof of the theorem.  If Tz ∈ (CHX ,V )z is a germ of a Coleff–Herrera current at z ∈ V and Sz is a germ of hypersurface at z with reduced defining function s around z, then it follows from Theorem 2.43 that

|s|2λ  T ∂ ∈ (CHX ,V X \S ∩ S )z . (2.57) s λ=0 z Here V X \S denotes, in terms of germs of closed analytic subsets at the point z, the union of the irreducible components of V which do not lie entirely in S. When 1 Tz belongs instead to (CHS X ,V )z , such that (S1 )z is a germ of a hypersurface with codimz (V ∩ S1 ) = M + 1, we have a slightly more involved formulation of a result, which could be iterated as well.

92

2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

Proposition 2.44. Let X be an N -dimensional complex manifold and let V ⊂ X be a purely dimensional, closed analytic subset with codim V = M > 0. Let also S1 be a fixed closed analytic hypersurface such that codim(V ∩S1 ) > M , s2 ∈ O(X ), and S2 = V (s2 ). Furthermore, assume that (1) V X \S2 is the union of irreducible components V of V with codim(V ∩ S2 ) > M; V \S (2) S1 2 is the closed analytic subset of S1 defined as the union of the irreducible components of S1 , whose intersection with V does not lie entirely in V X \S2 ∩ S2 .   (•,M +1) 1 -valued map Then, for each z ∈ V and each Tz ∈ (CHS X ,V )z , the ( D z

|s |2λ 2 ∧T λ −→ ∂ s2 z extends from {λ ∈ ℂ : Re λ  1} to the complex plane ℂ as a meromorphic map with poles in −ℚ+ . Moreover, one has for any z ∈ V that

|s |2λ      S1V \S2 2 1 ∧ CHS ∂ ⊂ CHX ,V X \S2 ∩S z . (2.58) X ,V z 2 s2 λ=0 1 Proof. The first assertion follows from the inclusion (CHS X ,V )z ⊂ PMz for any z ∈ V ; see Theorem 2.42, combined with Proposition 2.21. Proposition 2.21 1 implies that for any z ∈ V and Tz ∈ (CHS X ,V )z ,

|s |2λ 

2 ∧T ∂ ∈ PMz . s2 λ=0 z Since V is purely (N − M )-dimensional and codim(V ∩ S1 ) > M , it follows from the fact that for any z ∈ V , (V X \S2 )z is a union of some irreducible components of Vz having codimz (V X \S2 ∩ S1 ) = M + 1 at any z ∈ V ∩ S1 . The principle of analytic continuation implies that one has equality between sheaves of currents

|s |2λ  

|s |2λ 2 2 S1 1 ∧ CHS ∧ CH ∂ = ∂ . X ,V X ,V X \S2 λ=0 s2 s2 λ=0 Given a germ

  1 Tz ∈ CHS , X ,V X \S2 z

one has for the same reasons as above (with V replaced by V X \S2 ) that for any z∈V 

|s |2λ 2 ∧T Tz = ∂ ∈ PMz . s2 λ=0 z Moreover, the germ Tz is the germ of a (•, M + 1) pseudo-meromorphic current T at z, whose support lies in V X \S2 ∩ S2 . This support lies in a closed analytic subset of V having codimension in X at least equal to M + 1. Let us now take z ∈ V X \S2 ∩ S2 , so that we have codimz V X \S2 ∩ S2 = M + 1. To prove that   S V \S2 Tz ∈ CHX ,V1 X \S2 ∩S z , 2

as the assertion (2.58) requires, it is sufficient to show that (2.59)

V \S2

Supp (∂ T)z ⊂ S1

,

2.2. COLEFF–HERRERA SHEAVES OF CURRENTS

93

after invoking once again Theorem 2.42. It follows from Theorem 2.43 that one has (∂ T )z ∈ (CHX ,V X \S2 ∩ S1 )z . On the other hand, one has |s |2λ 

  2 ∂T z = ∂ . ∧∂T s2 λ=0 z Clearly the support of ∂Tz lies in (S1 ∩ S2 )z since Supp (∂ T ) ⊂ V ∩ S1 . It is therefore enough, in order to prove (2.59), to suppose that z ∈ (V X \S2 ∩ S2 ) ∩ S1 . V \S Assume, in addition, that z ∈ S1 2 ∩S2 , in other words z does not belong to any of the irreducible components Σ of S1 which is such that codim (Σ ∩ V ∩ S2 ) ≥ M + 2. This means that any irreducible component of (S1 )z lies entirely in (S2 )z . As a consequence, one has

|s |2λ  2 ∧∂T ∂ =0 s2 λ=0 z for Re λ  1. Then, the analytic continuation principle allows us to conclude that (∂ T)z = 0. This completes the proof of the proposition.  2.2.4. Extension-restriction of pseudo-meromorphic currents. Let U be an open subset of the N -dimensional complex manifold X and let T be a complexvalued pseudo-meromorphic current in U . Let S be a complex hypersurface in X , whose reduced local defining (holomorphic) function s determines the reduced defining germ sz ∈ Oz in a neighborhood of each z ∈ S. It follows from Proposition 2.21 that for any z ∈ U ∩ S, the ( D)z -valued map λ −→ (1 − |sz |2λ ) Tz

(2.60)

extends from {λ ∈ ℂ : Re λ  1} to ℂ as a meromorphic map with poles in −ℚ+ , whose value (1−|sz |2λ )Tz λ=0 at λ = 0 belongs to PMz . This value is independent of the reduced  holomorphic defining germ sz for S around z; see Remark 2.22. All such germs (1 − |sz |2λ )Tz λ=0 globalize as a pseudo-meromorphic current in U with support contained in the closed analytic subset S ∩ U of U . This current is naturally denoted by 1U∩S · T . On the other hand, the current T − 1U∩S · T has several alternative notations, such as (2.61)

T − 1U∩S · T = 1U\S · T = T(U\S)→U .

In the particular case when T is a (•, m)-current supported by a closed analytic subset W with pure codimension equal to m, the dimension principle (see Proposition 2.26 and Corollary 2.27) imply that as soon as the codimension of the hypersurface S satisfies codim (W ∩ S) > m, then (2.62)

1U∩S · T = 0,

or alternatively, T(U\S)→U = T.

A current T ∈ PM(U ), as in (2.62), is said to be a standard extension of its restriction to the open subset U \ S of U . This explains the notation T(U\S)→U for 1U\S T from above. The procedure described previously can be iterated. Suppose that Wz is a germ at z of closed analytic subset of X . Let (hz,1 , . . . , hz,mz ) be any finite sequence of elements in OX ,z such that Wz = V (hz,1 ) ∩ · · · ∩ V (hz,mz ). Given T ∈ PM(U ), where z ∈ U , let  (Πι )∗ τι . Tz = ι

94

2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

Each Πι : Uι ⊂ ℂNι → Uz above is a holomorphic mapping17 and τι is an elementary current in Uι which is of the form (2.33). It follows by iterated use of Proposition 2.25 that, according to the notation previously introduced in (2.40),      (2.63) 1V (hz,mz ) · 1V (hz,mz −1 ) · · · · 1V (hz,1 ) · T · · · z   (Πι )∗ τι z . = {ι : Suppess τι ⊂Π−1 ι (Wz )}

The fact that the right-hand side of (2.63) depends only on Wz and not on the sequence (hz,1 , . . . , hz,mz ), such that Wz = V (hz,1 ) ∩ · · · ∩ V (hz,mz ), suggests the following definition. Definition 2.45. Let T ∈ PM(U ), where U is an open subset of X , and let W be a closed analytic subset of X . The restriction of T to the closed analytic subset W is the pseudo-meromorphic current 1U∩W · T ∈ PM(U ) (with support included in U ∩ W ) whose germ at a point z ∈ U ∩ W equals      (2.64) (1U∩W · T )z := 1V (hz,mz ) · 1V (hz,mz −1 ) · · · · 1V (hz,1 ) · T · · · , z

where (hz,1 , . . . , hz,m ) is an arbitrary sequence of elements in Oz such that one has Wz = V (hz,1 ) ∩ · · · ∩ V (hz,mz ) and the right-hand side of (2.64) is defined according to (2.63). The extension of T ∈ PM(U ) from the open subset U \ W ⊂ U is defined as the pseudo-meromorphic current T − 1U∩W · T = TU\W →U = 1U\W · T ∈ PM(U ). Given W  and W  two closed analytic subsets of X and T ∈ PM(U ), where U is an open subset of X , the transitivity equality 1W  · (1W  · T ) = 1W  ∩W  · T in PM(U ) follows from Definition 2.45. Example 2.46. Suppose that W has pure codimension equal to m. It follows from Proposition A.24 (see §A.3.4) that for any z ∈ U ∩ W there exists m + 1 reduced germs sz,1 , . . . , sz,m , sz,m+1 ∈ Oz such that dsz,1 ∧ · · · ∧ dsz,m ≡ 0 on any irreducible component of V (sz,1 ) ∩ · · · ∩ V (sz,m ) and (1) the germ Wz is the finite union of Vz,1 , . . . , Vz,L among the irreducible components Vz,1 , . . . , Vz,L of V (sz,1 ) ∩ · · · ∩ V (sz,m ), where L ≥ L ; (2) one has sz,m+1 ≡ 0 on Wz but codimz (V (sz,m+1 ) ∩ (Vz, \ Wz )) > m for any  > L . Given T ∈ PM(U ), the sequence (sz,1 , . . . , sz,m+1 ) can be used in order to express (1U∩W · T )z when z ∈ U ∩ V according to (2.64). Proposition 2.47. Let X be an N -dimensional complex manifold and let W be a closed analytic subset of pure codimension m. Let also U be an open subset of X and T ∈ PM(U ) with bidegree (•, m). Assume that there exists a closed analytic subset W  such that codim W  > m and Supp T ∩ W ⊂ W  . Then, one has (2.65)

1U∩W · T = 0,

or alternatively T(U\W )→U = T.

Proof. It is an immediate consequence of the dimension principle for pseudomeromorphic currents; see Proposition 2.26 and Corollary 2.27.  17 It is enough to consider the case where each Π is a tower of one of smooth modifications, ι open inclusions, or simple projections; see [AndW18, Theorem 2.15].

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In particular, restriction or extension operations defined by T ∈ PM(U ) −→ 1U∩W · T ∈ PM(U ), T ∈ PM(U ) −→ T(U\W )→U ∈ PM(U ) can be applied to sections of the Coleff–Herrera sheaves of currents CHX ,V or CHS X ,V described by Definitions 2.33 and 2.40, respectively. They extend the same operations performed on integration currents (as specific examples of (m, m) positive closed currents, see Example C.13) or generalized cycles (see Definition D.4), which are current re-interpretations of gap-sheaves construction; see §D.1.1. We conclude this brief subsection by formalizing a principle that we already met in several examples. This principle relates to the concepts of Coleff–Herrera current or meromorphic Coleff–Herrera current with prescribed poles along a hypersurface. Definition 2.48 (Standard extension property (SEP)). Let X be an N -dimensional complex manifold and let U be its open subset. A pseudo-meromorphic current T ∈ PM(U ), whose support is included in a closed analytic subset V ⊂ U , is said to be a standard extension for V or it is said to satisfy the standard extension property (SEP ) across closed hypersurfaces of V ⊃ Supp T if and only if for any complex hypersurface S in U the following implication holds:   (2.66) codimU (V ∩ S) > codimU (V ) =⇒ T(U\S)→U = T. Example 2.49. Given a purely dimensional closed analytic subset V of X and given a hypersurface S satisfying codim(V ∩ S) > codim V , any section of the X ,V Coleff–Herrera sheaves CHX ,V , CHS (·, E), CHS X ,V or CH X ,V (·, E), when E is a holomorphic bundle over X (see Remark 2.36 and 2.41, respectively) satisfies the SEP across any closed hypersurface of U ∩ V . This is condition (2), formulated in the definition of the sheaves CHX ,V or CHS X ,V . 2.3. Coleff–Herrera’s original construction revisited In this section, X is a complex N -dimensional manifold, and V ⊂ X denotes a purely dimensional closed analytic subset with codimension M in X . We start in §2.3.1 with a presentation of Nicolas Coleff and Miguel Herrera’s construction in [CoH78]. We will then reformulate it in §2.3.2 in more geometric terms by introducing holomorphic sections of holomorphic line bundles equipped with hermitian metrics. Finally, in §2.3.3, we conclude Chapter 2 with an up-to-date presentation of Coleff–Herrera theory within the complete intersection setting. 2.3.1. Essential intersection and Coleff–Herrera’s original construction. Let V ⊂ X be as before and let S1 , . . . , Sm be complex hypersurfaces of X , when 1 ≤ m ≤ N − M , such that codimX (V ∩ Sj ) > M for j = 1, . . . , m. We recall that, given a closed analytic subset W , together with a complex hypersurface S in X , the closed analytic subset W X \S ⊂ W is defined locally as follows. For any z ∈ W such that Wz,1 , . . . , Wz,L are the irreducible components of Wz ,  (2.67) WzX \S = Wz, . {1≤≤L : Wz, ⊂Sz }

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When 1 ≤ m ≤ N − M , let us first recall the notion of essential intersection18 of a purely dimensional closed analytic subset V ⊂ X with an ordered sequence of hypersurfaces S = (S1 , . . . , Sm ) in X such that codimX (V ∩ Sj ) > M for any j. Definition 2.50. Let 1 ≤ m ≤ N − M = N − codimX V . Given an ordered sequence S = (S1 , . . . , Sm ) of complex hypersurfaces  5  {S1 , . . . , Sm } of X with codimX (V ∩ Sj ) > M , the essential intersection V S ess is the closed analytic subset of X , either with pure codimension M + m or empty, defined through the following iterated process: V [0] = V, X \S

V [j+1] = Vj j+1 ∩ Sj+1 for j = 0, . . . , m − 1,  3  S ess . V [m] := V  5  Remark 2.51. The definition of V S ess highly depends on the ordering of S1 , . . . , Sm . However, if the hypersurfaces S1 , . . . , Sm are in such position19 that they satisfy codim(V ∩ S1 ∩ · · · ∩ Sm ) = M + m or +∞, then one has  3   3  (2.69) V S ess = V s(S) ess

(2.68)

for any permutation s of {S1 , . . . , Sm }. The reason why we mention here the concept of essential intersection, besides its intimate relation with the Coleff–Herrera construction of residue currents that we describe in this subsection, is that it appears crucially in the methods developed towards algebraic or analytic improper intersection theory ([StV82, Tw95, AndSWY17B, AndESWY21A, AndESWY21B]). It is therefore deeply connected with the concept of gap sheaf ; see §D.1.120 or, e.g., [Mas03] for more details. We will come back to these important concepts later on in this monograph. Adapting a slightly different approach to the original one, we present the construction of residue currents proposed by Coleff and Herrera in [CoH78]. These currents, also known as Coleff–Herrera residue currents, are attached to a closed purely (N − M )-dimensional analytic subset V ⊂ X , to an ordered sequence of hypersurfaces S = (S1 , . . . , Sm ), for 1 ≤ m ≤ N − M , in X with codim(V ∩ Sj ) > M for j = 1, . . . , m, and to a meromorphic (p, 0)-form ω in an open neighborhood U of V in"X . The polar set of (p, 0)-form ω, 0 ≤ p ≤ N − M , lies in U ∩ |S|, where m |S| = j=1 Sj . For each z ∈ V ∩ |S|, Hilbert’s nullstellensatz in Oz implies that for reduced defining germs of holomorphic sections sj = sz,j , j = 1, . . . , m, of the hypersurfaces Sj around z there exist an integer ν = νz ∈ ℕ∗ such that (2.70)

((s1 · · · sm )ν ω)z = z ∈ (Ωp )z .

Observe that the sj ’s are nonidentically zero on any irreducible component of Vz since codimX (V ∩ Sj ) > M . 18 See also §D.1.1, in particular (D.4) and (D.5), for the related concept of a cutting-out operation performed on cycles. For what concerns this notion of essential intersection (deeply connected with our approach toward analytic residue theory), we refer the reader to [CoH78] and [Sol77]. 19 One says then they are in position to define a complete intersection on V ; see §2.3.3. 20 We refer in particular here to (D.4) and(D.5), where the cutting-out operation on cycles is described.

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Definition 2.52. Let V , S, ω be as above. The current T = RS ∧ ω ∧ [V ] is the pseudo-meromorphic ∂-closed current in X , with bidegree (p + M, M + m), / Vz ∩ |S|. In case z ∈ V ∩ |S|, defined as follows. It is defined locally as Tz = 0, if z ∈ it is defined by the iterative procedure

(2.71)

(T [0] )z = ( ∧ [V ])z ,

|s |2νλ  j+1 (T [j+1] )z = ∂ ∧ (T [j] )z ν 2iπ sj+1 λ=0   [m] RS ∧ ω ∧ [V ] z = (T )z

for j = 0, . . . , m − 1,

following Proposition 2.21. Here sj = sj,z is the germ at z of a reduced defining function for Sj for every j = 1, . . . , m and z , ν = νz are taken as in (2.70). This current has its support included in V ∩ |S| and it depends only on [V ], S, and ω. Remark 2.53. The definition of the sheaf of (p, m)-currents on the (reduced) complex analytic space (V, O/IV ), in particular their interpretation through the identification T ←→ (ιV )∗ T , where ιV : V → X (see (B.33) in §B.2.2.2), allows one to interpret RS ∧ ω ∧ [V ] as the direct image through the inclusion ι of a (p, m)-current on V , denoted by Rι−1 (S) ∧ ι∗V ω. V

Proposition 2.54 (Lelong–Poincar´e type factorization formulae). Let X , V , S, ω be as above. One has

p+M (2.72) RS ∧ ω ∧ [V ] ∈ CHX ,(V  S)ess (X ). 5 S)ess is defined as the union of Moreover, assume that the germ Vz at z ∈ (V a finite number of some irreducible components W1 , . . . , WL of the intersection germ Wz = V (σz,1 ) ∩ · · · ∩ V (σz,M ), where σz,1 , . . . , σz,M ∈ Oz fulfill the following condition: dσz,1 ∧ · · · ∧ dσz,M ≡ 0 on any of the L ≥ L irreducible components W of Wz . Then, there is a germ of a hypersurface ΣV,S,z with 3   S  )ess ∩ ΣV,S  ,z > M + m codimz (V such that 21

0   ΣV,S,z  (2.73) RS ∧ ω ∧ [V ] z ⊂ CHX ,(V ∧ dσ1,z ∧ · · · ∧ dσM,z ∧ (Ωp )z . S)ess z

If L = L , then Vz = Wz , and one has

  (2.74) RS ∧ ω ∧ [V ] z ⊂ CHX ,(V



0 S)ess

z

∧ dσ1,z ∧ · · · ∧ dσM,z ∧ (Ωp )z .

Proof. Let us first prove assertion (2.72). One can see from the construction of the current T [j] appearing at step j in the iterative procedure (2.71) that it is supported by the closed analytic subset V [j] . The subset V [j] appears at the same step j of the iterative procedure (2.68). It follows that RS ∧ω ∧[V ] is a (p+M, 5 m+ M ) pseudo-meromorphic current, whose support is included in V [m] = V S ess . [m] [m] = ∅, RS ∧ ω ∧ [V ] = 0. Otherwise, V has pure codimension M + m and If V  (X ). In order to prove the Theorem 2.39 implies that RS ∧ ω ∧ [V ] ∈ CHp+M  5  X ,(V S)ess 5 S)ess = ∅ and other two assertions, let us fix z ∈ V S ess . It means that (V 21 Multiplicative factorization formulas, such as (2.73) and (2.74), are known as factorization formulae of the Lelong–Poincar´ e type. The most classical one is the formula (2.76) for the germ of an integration current [V ]z .

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that the defining reduced holomorphic germs sz,1 , . . . , sz,m for S1 , . . . , Sm around z, are not all being identically equal to zero on any irreducible component of Vz . Specifically, the defining reduced holomorphic germs sz,1 , . . . , sz,m for S1 , . . . , Sm around z are not all being identically equal to zero on any W for  = 1, . . . , L . Let z ∈ (Ωp )z and ν ∈ ℕ∗ such that (sz,1 · · · sz,m )ν ωz = z . Let Tz be the germ of a pseudo-meromorphic (0, M )-current at z defined through the iterative procedure

(2.75)

T[0] = [1], 

|σ 2λ z,κ+1 | ∧ (T[κ] )z (T[κ+1] )z = ∂ for κ = 0, . . . , M − 1, 2iπ σz,κ+1 λ=0

 Tz = (1 − |σz,M +1 |2λ ) (T[M ] )z , λ=0

where σz,M +1 ∈ I(Vz ) is chosen such that codimz ((W \ V ) ∩ V (σz,M +1 )) = M + 1 for  = L + 1, . . . , L, when L < L, or is taken to be equal to 0, if L = L. It follows from Theorem 2.39 that Tz = (T[M ] )z ∈ (CHX ,V )z . Similarly, Theorem 2.42 implies that Tz ∈ CHX ,V . Moreover, it follows from Proposition 2.16 and formula (2.29) that (2.76)

[V ]z = Tz ∧ dσz,1 ∧ · · · ∧ dσz,M .

Repeated use of Proposition 2.44 shows that the iterated procedure

(2.77)

(U[0] )z = Tz , 

|s 2νλ z,j+1 | ∧ (U[j] )z (U[j+1] )z = ∂ ν sz,j+1 λ=0

for j = 0, . . . , m − 1,

Uz = (U[m] )z

  ΣV,S,z  leads to the construction of an element Uz ∈ CHX ,(V S)ess z for some ΣV,S,z , depending on V (σz,M +1 ) and on the germs at z of the Sj , in case L < L. In the case where L = L, one has that σz,M +1 = 0 and thus Uz ∈ CHX ,(V  S)ess z . The  factorization formulae (2.73) (when L < L) or (2.74) m(whenν L = L) follow from the definition of (RS ∧ ω ∧ [V ])z = RS ∧ [V ] ∧ /( 1 σz,j ) z , together with the  factorization (2.76) for [V ]z . Following Coleff–Herrera in [CoH78], one may define also mixed principal value-residue currents. Definition 2.55. Let V , S = (S1 , . . . , Sm , Sm +1 , . . . , Sm ) = (S  , S  ), and let ω be as above, where 1 ≤ m ≤ m. The current T = RS  PS  ∧ ω ∧ [V ] = RPS  S  ∧ ω ∧ [V ] is the pseudo-meromorphic current in X , with bidegree (p + M, M + m ). In case z ∈ / Vz ∩ |S  |, then it is defined locally as Tz = 0. On the other hand, if z ∈ V ∩ |S  |, then it is defined by the iterative procedure (2.78)



RPS  S 

(T [0] )z = ( ∧ [V ])z , 

|s |2νλ j+1 ∧ (T [j] )z (T [j+1] )z = ∂ for j = 0, . . . , m − 1, ν 2iπ sj+1 λ=0

|s |2νλ  j+1 [j] (T [j+1] )z = (T ) for j = m , . . . , m − 1, z 2iπ sνj+1 λ=0  ∧ ω ∧ [V ] z = (T [m] )z

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following Proposition 2.21, where sj = sj,z is the germ of a reduced defining function for the hypersurface Sj in a neighborhood of z for every j = 1, . . . , m, and z , ν = νz are taken to be as in (2.70). This current has its support included in V ∩ |S  | and it depends only on [V ], S and ω. Remark 2.56. The definition of the sheaf of (p, m)-currents on the (reduced) complex analytic space (V, O/IV ) allows one to interpret, as in Remark 2.53, the mixed current RPS  S  ∧ ω ∧ [V ] as the direct image through the inclusion ιV of a (p, m )-current on V , denoted by RPι−1 (S  S  ) ∧ ι∗V ω. V

Proposition 2.57. Let X , V , S = (S  , S  ), ω be as above. Let, as usual,  5  X \|S  | 5  (V S )ess be the union of the irreducible components of (V S )ess which do not lie entirely in |S  |. One has (2.79)

RPS  S  ∧ ω ∧ [V ] ∈



|S  | CH 

X , (V



S  )ess

X \|S |

p+M (X ).

Moreover, assume that the germ Vz is defined as the union of a finite number of  some irreducible components 5  W1 , . . . , WL of Wz = V (σz,1 ) ∩ · · · ∩ V (σz,M ) in a neighborhood of z ∈ (V S )ess , where σz,1 , . . . , σz,M ∈ Oz are such that one has dσz,1 ∧ · · · ∧ dσz,M ≡ 0 on any of the L ≥ L irreducible components W of Wz . Then, there is a germ of a hypersurface ΣV,S,z ⊃ |S  |z , whose codimension satisfies   3  X \|S  | codimz ( V S )ess ∩ ΣV,S,z > M + m , such that (2.80)

  Σ RPS  S  ∧ ω ∧ [V ] z ⊂ CH V,S,z X, V

S  )ess

X \|S |

0 z

∧ dσ1,z ∧ · · · ∧ dσM,z ∧ (Ωp )z .

If L = L, then one can take ΣV,S,z = |S  |z . Proof. The first assertion follows from Theorem 2.42. The proof of the second assertion is based on the same arguments present in the proof of the second assertion in Proposition 2.54.  The currents RS ∧ ω ∧ [V ] or RPS  S  ∧ ω ∧ [V ] were originally introduced in [CoH78] through the integration on semi-analytic chains and the related notion of the admissible path of integration with respect to (V, S) in a neighborhood of z ∈ V ∩ S1 ∩ · · · ∩ Sm . We recall the definitions of these objects below. Definition 2.58 (Semi-analytic chains ΓV,s (ε) around z ∈ V ). Let X be an N -dimensional complex manifold and let V ⊂ X be a purely dimensional closed analytic subset with codim V = M . For z ∈ V let s = (s1 , . . . , sm , sm +1 , . . . , sm ) be an ordered sequence of m reduced elements in Oz , which are not identically equal to 0 on any irreducible component of Vz . Assume that the elements of this ordered sequence are represented by reduced elements (also denoted by sj ) in some neighborhood Uz of z in X . For any ε = (ε1 , . . . , εm , εm +1 , . . . , εm ) such that

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(ε1 , . . . , εm ) is not a critical value for (|s1 |2 , . . . , |sm |2 )|(V ∩Uz )ℝ , the closed semianalytic22 subset 2 4 ζ ∈ V ∩ Uz : |sj (ζ)|2 = εj for 1 ≤ j ≤ m , |sj (ζ)|2 > εj for m + 1 ≤ j ≤ m induces a (2(N − M ) − |m |)-dimensional semi-analytic integration chain on Uz by     2(N −M )−m (Uz )ℝ −→ [ΓV,s (ε)] , Φ = Φ (2.81) ΓV,s (ε) : Φ ∈ D = lim

→+∞

  V

j>m

ΓV,s (ε)

|s |2

|s  |2

|s |2

j m 1 ∧ · · · ∧ ∂ χ ∧ Φ, χ ∂ χ  ε ε ε j m 1 

where (χ )∈ℕ∗ , with Supp χ ⊂ [1 − 1/, +∞[, is a smooth regularization by convolution with an approximation of unity of the characteristic function χ[1,+∞[ . Definition 2.59. Let X and V be as above and let S = (S  , S  ) be an ordered sequence of m hypersurfaces as in Definition 2.55. Let s = (s1 , . . . , sm , sm +1 , . . . , sm ) be a sequence of reduced defining germs for the hypersurfaces S1 , . . . , Sm at z ∈ V respectively. A map t : ]0, 1[−→ ε(t) ∈ (ℝ+ )m is said to be m-admissible in a neighborhood of z ∈ V ∩ (S1 ∩ · · · ∩ Sm ) with respect to (V, S) if and only if (1) Im δ avoids the set of critical values of (|s1 |2 , . . . , |sm |2 )|(V ∩Uz )ℝ for some sufficiently small neighborhood Uz of z; (2) when t tends to 0+ , then one has that for all k ∈ ℕ∗ ,   (2.82) εj (t) = O (εj+1 (t))k for j = 1, . . . , m − 1. The presentation of germs of Coleff–Herrera currents (RS ∧ ω ∧ [V ])z , when m = m or of the germs (RPS S  ∧ ω ∧ [V ])z , when m < m, at z ∈ (V ∩ S  )ess for a meromorphic (p, 0)-form ω, whose polar set is along |S| around z, is as follows (see [CoH78]). Theorem 2.60. Suppose that t → ε(t) is m-admissible with respect to z and that Uz fulfills condition (1) in Definition 2.59. Then, when m = m,  Γ (ε(t))   V,s (2.83) RS ∧ ω ∧ [V ] , ϕ = lim , ω∧ϕ m t→0+ (2iπ) for any ϕ ∈ DN −M −p,N −M −m (Uz ). Furthermore, if m < m, then  Γ (ε(t))   V,s (2.84) RPS  S  ∧ ω ∧ [V ] , ϕ = lim , ω ∧ ϕ t→0+ (2iπ)m 

for any ϕ ∈ DN −M −p,N −M −m (Uz ). To avoid the technical condition (1) in Definition 2.59, one can fix a smooth regularization χ of χ[1,+∞[ on the real line and for each ε ∈ (ℝ+ )m , substitute the semi-analytic integration chain ΓV,S (ε) with bidimension (N − M, N − M − m ) defined in (2.81) by the (M, M + m )-current  |s |2

|s  |2

|s |2

j m 1 (2.85) [ΓχV,s (ε)] = ∧··· ∧∂ χ ∧ [V ]. χ ∂ χ  ε ε ε j m 1  j>m

22 The terminology semi-analytic comes from the fact that such subset is defined in terms of either equalities sι = 0 or inequalities sι > 0, where the sι are germs at z of real analytic functions on Xℝ .

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This is done in [SamL13, Theorems 2 and 11]. The following result also holds [SamL13, Theorems 2 and 11]. Theorem 2.61. Suppose that t : ]0, 1[−→ ε(t) ∈ (ℝ+ )m fulfills condition (2) in Definition 2.59. Let V , S, ω and z ∈ (V ∩ S  )ess be as in Theorem 2.60. Let also s = (s1 , . . . , sm , sm +1 , . . . , sm ) = (s , s ) be a sequence of reduced defining germs for an ordered sequence (S1 , . . . , Sm ) = (S  , S  ) at z. Then one has 

Γχ (ε(t))   V,s RS ∧ ω ∧ [V ] z = lim ∧ ω when m = m, t→0+ (2iπ)m z  (2.86) Γχ (ε(t))

  V,s RPS  S  ∧ ω ∧ [V ] z = lim ∧ ω when m < m t→0+ (2iπ)m z with respect to the weak convergence in ( D)z . In this monograph we propose systematically an alternative approach for the definition of residue or mixed principal value–residue currents which illustrates in this subsection that of the currents RS ∧ ω ∧ [V ] or RPS  S  ∧ ω ∧ [V ]. Our point of view leads to different representations of the above objects to those on the righthand sides of equalities in Theorem 2.60 or Theorem 2.61. The approach presented in the present text is based on the analytic continuation principle as a substitute for integration on semi-analytic chains along admissible paths (as in Theorem 2.60) or weak limits along admissible paths after regularization (as in Theorem 2.61). The reason for our choice is that besides the fact that it avoids appealing to the intricate condition (2) in Definition 2.59, from our point of view it makes the construction more algebraic. Therefore, it gives some hope of transposing it within an algebraic or even arithmetic frame. To invite the reader towards potential further developments, let us mention here the references [Igu] and [Den92] related to Igusa calculus as well as [Jea74, Bar82, BarM85] related to Mellin’s inversion. 2.3.2. Coleff–Herrera currents attached to sections of line bundles. In this section, we extend the approach developed in the univariate setting in §1.2.5. Until now, the currents RS ∧ ω ∧ [V ] or RPS  S  ∧ ω ∧ [V ] we introduced in §2.3.1, involve, besides geometric objects such as the purely dimensional closed analytic subset V or the ordered sequence of hypersurfaces S, (S  , S  ) which all intersect V properly, algebraic objects such as the meromorphic form ω in the ambient manifold X , with poles along |S|. We present in this subsection an alternative point of view, where sections of Cartier divisors on X play a central role. We plan to consider, besides problems of geometric nature related, for example, to proper or improper intersection theory, questions of algebraic nature. These are problems that appear, for example, in division or interpolation theory. In order to obtain a coherent picture of the working frame we are dealing with, all currents T , meromorphic forms ω, or test forms ϕ should be treated as sections of holomorphic bundles over X . The bundle approach reveals to be useful yet for another reason. It sustains, as will be apparent later from Chapter 3, averaging procedures where Hilbert–Samuel multiplicity of an ideal in Oz is expressed as a Lelong number (Proposition C.24), or where Crofton’s formula, as stated in §D.5.1, is used. We will deal in this section with holomorphic sections of (holomorphic) line bundles on N -dimensional complex manifold X , in correspondence with Cartier divisors on X ; see §D.2.1. Similar constructions involving holomorphic sections of complex vector bundles of higher rank will be discussed in the forthcoming Chapter 3. The geometric point of view

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of the Coleff–Herrera theory, as we present it here, was introduced by M. Andersson [And04, And05, And07]. For the background on bundle theory we refer the reader to Appendix B. Let E be a holomorphic bundle over X . We start by introducing homomorphisms between sheaves of bundle-valued pseudo-meromorphic currents on X in compliance with the definitions of R|s | , P|s | given in Proposition 1.27. Definition 2.62. Let s ∈ OX (X , L) = O(X , L), where L = O(D) is the holomorphic line bundle corresponding to the Cartier divisor D on X . Let | | be a hermitian metric on L and let s∗ ∈ O(X , O(−D)) be its complex conjugate.23 The operators [R|s | ] = [R|s |,0 + R|s |,1 ], [P|s | ] = [P|s |,0 ] and [R|s | + P|s | ] are respectively defined as follows. For any open subset U of the N -dimensional complex manifold X and for any section T of PM(U, E), we define (2.87)

 s∗ ⊗T 2 2iπ|s| λ=0

 ∗   s = 1V (s) T + ∂|s|2λ ⊗T ∈ PM U, (ℂ ⊕ O(−D)) ⊗ E , 2 2iπ|s| λ=0



∗   s ⊗T (T ) = |s|2λ ∈ PM U, O(−D) ⊗ E , 2 2iπ|s| λ=0     (T ) = R|s | (T ) + P|s | (T ) ∈ PM U, (ℂ ⊕ O(−D)) ⊗ E .

 s R| | (T ) = 1 − |s|2λ + ∂|s|2λ

 s P| |

 s R| | + P|s |

Remark 2.63. If | | and | | are two metrics of O(D), one has (|s| )2 = ρ(|s| )2 , where ρ ∈ C ∞ (X , ℝ+ ). As a consequence (λ, z) → ρλ (z) depends on the parameter λ as an entire function of λ, whose value at λ = 0 equals identically 1 for z ∈ X . As a consequence, given T ∈ PM(U, E), the pseudo-meromorphic currents R|s | (T )  or P|s | (T ) do not depend on the choice of the metric | | on O(D). Therefore, the    currents defined by (2.87) will be denoted by Rs (T ), P s (T ), and (R + P )s (T ) from now on. Proposition 2.64. Let V be a purely dimensional closed subset of X and let s ∈ O(X , O(D)) be such that codim (V ∩ V (s)) > codim V . If T ∈ CHX ,V (U, E), one has 

   s s∗ ⊗ T ∈ CHX ,V ∩V (s) U, O(−D) ⊗ E , R (T ) = ∂|s|2λ 2 2iπ|s| λ=0 (2.88)   s V (s)  P (T ) ∈ CHX ,V U, O(−D) ⊗ E . If T ∈ CHΣ X ,V (U, E) for some closed hypersurface Σ in a neighborhhod of V such that codim(V ∩ Σ) > codim V , then one has

  s   V \V (s) s∗ R (T ) = ∂|s|2λ ∈ CHΣ ⊗ T X ,V X \S ∩V (s) U, O(−D) ⊗ E , 2 2iπ|s| λ=0 (2.89)  s  (Σ∪V (s))  P (T ) ∈ CHX ,V U, O(−D) ⊗ E , 23 See §B.3.2. Note that since L = O(D) is a line bundle, s∗ /|s|2 is a meromorphic section of O(−D).

2.3. COLEFF–HERRERA’S ORIGINAL CONSTRUCTION REVISITED

103

where V X \V (s) is the union of the irreducible components of S on which s ≡ 0 and ΣV \V (s) is the union of the irreducible components of Σ, whose intersection with V does not lie entirely in V X \V (s) ∩ V (s). Proof. The first assertions follow from Theorems 2.39 and 2.42, together with the fact that Coleff–Herrera currents for a closed analytic set V with pure dimension inherit the standard extension property across any proper hypersurface of V . The  second assertion follows from Proposition 2.44 with S1 = Σ and S2 = V (s). Proposition 2.65. Let ∇s be the operator24 ∇s = 2iπs − ∂. For any current T ∈ PM(U, E), one has       ∇s Ps (T ) = Id − Rs (T ) + Ps (∂ T ), (2.90)    ∇s Rs (T ) = − Rs (∂ T ). Proof. For λ ∈ ℂ with Re λ  1, ∇s |s|2λ



s∗ s∗ s∗ 2λ 2λ 2λ ⊗ T + |s| ⊗ (∂ T ) ⊗ T = |s| T − ∂|s| 2iπ|s|2 2iπ|s|2 2iπ|s|2 s∗ ⊗T = T − 1 − |s|2λ + ∂|s|2λ 2iπ|s|2 s∗ ⊗ (∂ T ). + |s|2λ 2iπ|s|2

The first equality in (2.90) follows from the analytic continuation principle. One has also

s∗ ⊗ T = −(∂|s|2λ ) T, 2iπs 1 − |s|2λ + ∂|s|2λ 2 2iπ|s|

s∗ ⊗ T = −(∂|s|2λ ) T + (1 − |s|2λ ) ∂ T ∂ 1 − |s|2λ + ∂|s|2λ 2iπ|s|2 s∗ + (∂|s|2λ ) 2 ⊗ (∂ T ). |s| Furthermore, subtracting the second equality from the first and then applying the analytic continuation principle leads to the second equality in (2.90).  Observe that the action of any operator [Rs ] or [P s ] preserves pseudo-meromorphicity. Thus, given T ∈ PM(•,M ) (U, E) and given m Cartier divisors D1 , . . . , Dm on X with global holomorphic sections sj ∈ O(X , O(Dj )), one defines the following (ℂ ⊕ O(−D1 )) ⊗ · · · ⊗ (ℂ ⊕ O(−Dm )) ⊗ E-valued pseudo-meromorphic currents in 24 Such

an operator was already introduced in the univariate setting; see Definition 1.26. We extend its action in the new context, respecting the following Leibniz rule: if T is a pseudomeromorphic current and Φ is an O(−D)-valued k differential form, then ∂(Φ ⊗ T ) = ∂Φ ⊗ T + (−1)k+1 Φ ⊗ ∂T.

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2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

the open subset U : R(s1 ,...,sm ) (T ) =



  Rsm ◦ · · · ◦ Rs1 (T )

min(m,N −M )

= (2.91) (R + P )(s1 ,...,sm ) (T ) =







r=0

#J=r

(s ,...,sm )

RJ 1

(T ).

  (R + P )sm ◦ · · · ◦ (R + P )s1 (T )

min(m,N −M )

=





r=0

#J=r

(s ,...,sm )

(R + P )J 1

(T ),

where, for each subset J of {1, . . . , m} with cardinal |J| = r   (s ,...,sm ) (2.92) RJ 1 (T ) ∈ PM(•,M +r) U, O(−|D|J ) ⊗ E and (s ,...,sm )

(2.93) (R + P )J 1

(T ) =



(s ,...,sm )

(R + P )J,J1 

J  ⊂{1,...,m}\J

∈ PM(•,M +r) U,

(

(T )

O(−|D|J∪J  ) ⊗ E .

J  ⊂{1,...,m}\J

The notation |D|K denotes the sum of the Cartier divisors Dj for j ∈ K. Moreover, given a multi-index 1 ≤ j1 < · · · < jr ≤ m, one has  (s ,...,sm )  (2.94) Supp RJ 1 (T ) ⊂ WJ (T ; s), [m]

where WJ (T ; s) = WJ (T ; s) is the closed analytic subset of Supp T obtained iteratively by [0]

(2.95)

WJ (T ; s) = Supp T,  [j] /J (WJ (T ; s))V (sj+1 ) = if j + 1 ∈ [j+1] WJ (T ; s) = [j] X \V (sj+1 ) (WJ (T ; s)) ∩ V (sj+1 ) if j + 1 ∈ J for j = 0, . . . , m − 1.

Given a subset J  ⊂ {1, . . . , m} \ J, one has   (s ,...,s ) (2.96) Supp (R + P )J,J1  m (T ) ⊂ WJ,J  (T ; s), [m]

where WJ,J  (T ; s) = WJ,J  (T ; s) is the closed analytic subset of Supp T obtained iteratively by [0]

(2.97)

WJ,J  (T ; s) = Supp T, ⎧ [j] ⎪ (W (T ; s))V (sj+1 ) if j + 1 ∈ / J ∪ J ⎪ ⎨ J,J  [j+1] [j] X \V (sj+1 ) WJ,J  (T ; s) = (WJ,J if j + 1 ∈ J   (T ; s)) ⎪ ⎪ ⎩ [j] (WJ,J  (T ; s))X \V (sj+1 ) ∩ V (sj+1 ) if j + 1 ∈ J for j = 0, . . . , m − 1.

2.3. COLEFF–HERRERA’S ORIGINAL CONSTRUCTION REVISITED

105

Proposition 2.66. Assume that V is a purely dimensional closed subset of X with codimension M . Assume also that T ∈ CHS X ,V (U ) for some hypersurface S satisfying codim (V ∩ S) > M . If J and J  are any disjoint subsets of {1, . . . , m} so that codim WJ,J  ([V ]; s) ≤ N , then WJ,J  ([V ]; s) has pure codimension M + |J| and there is a closed hypersurface ΣJ,J  = ΣJ,J  (V, S, s), in X with codim(WJ,J  ([V ]; s) ∩ ΣJ,J  ) > codim WJ,J  ([V ]; s) satisfying (s ,...,sm )

(R + P )J,J1 

Σ

J,J (T ) ∈ CHX ,W



J,J  ([V

];s)

  U, O(−|D|J∪J  ) . 

Proof. It follows from Proposition 2.44.

Definition 2.67. If T = [V ] and codim (V ∩ V (sj )) > codim V for any j = 1, . . . , m, then the bundle-valued current R(s1 ,...,sm ) ([V ]) is called the Coleff– Herrera residue current on V attached to the the sections (sj )|V of the line bundles (O(Dj ))|V , defined as restrictions to the complex analytic space (V, O/IV ) of the Cartier divisors O(Dj ). The current R(s1 ,...,sm ) ([V ]) in the ambient manifold X is the direct image through the embedding ιV : V → X of a bundle-valued current ∗ ∗ R(ιV s1 ,...,ιV sm ) on the complex analytic space (V, O/IV ). Remark 2.68. If V is locally defined in a neighborhood of z ∈ WJ ([V ]; s) as the union of a finite number of irreducible components of V (σz,1 ) ∩ · · · ∩ V (σz,M ) such that the differential form dσz,1 ∧ · · · ∧ dσz,M does not vanish identically on any irreducible component of V (σz,1 )∩· · ·∩V (σz,N ), then there exists a germ of a hyper (s ,...,sm )  surface Σz,J at z with codimz (WJ (V ; s)∩Σz ) > M +r such that RJ 1 ([V ]) z factorizes as   (s ,...,sm )   Σz,J · , O(−|D| (2.98) RJ 1 ([V ]) z ∈ (dσz,1 ∧ · · · ∧ dσz,m ) CHX ,W ) . J J (T ;s) z

This follows from Proposition 2.44 by using arguments similar to those used in the proof of the second assertion in Proposition 2.54. Let us point out here that the construction of the current R(s1 ,...,sm ) ([V ]) through an iterated analytic continuation process, which is not so easy to describe and thus exploit, can be also realized through a direct (noniterative anymore) approach involving just one complex parameter. Proposition 2.69 ([AndSWY17A]). Let X , V ⊂ X with pure codimension M as before, and let sj ∈ O(Dj ), j = 1, . . . , m, where D1 , . . . , Dm are Cartier divisors on X . Suppose codim (V ∩ V (sj )) > M for j = 1, . . . , m. For any j = 1, . . . , m, let | | be an arbitrary hermitian metric on O(Dj ). Let also γ1 > γ2 > · · · > γm ≥ 1

(2.99)

be m strictly positive integers and let η > 0. The current-valued map (2.100) γm γm s∗m s∗1 2λγ1 2λγ1 λ −→ 1−|sm |2λ +∂|sm |2λ | +∂|s | · · · 1−|s [V ] 1 1 2iπ|sm |2 2iπ|s1 |2 extends from



λ ∈ ℂ : |arg]π,π[ (λ)|
0. Assume also that Ω is a neighborhood of the origin in −M , χ1 , . . . , χm are m smooth strictly positive functions in Ω, and ζ β1 , . . . , ζ βm ℂN ζ are m monomial functions in the variables ζ = (ζ1 , . . . , ζN −M ). For any integer r such that 0 ≤ r ≤ min(m, N − M ), the holomorphic current-valued map  |χ ζ βj |2μs(j) ' |χ ζ βj |2μs(j) j j ∂ (2.103) Θ : (μ1 , . . . , μm ) −→ βj ζ ζ βj j>r j≤r

in {μ ∈ ℂm : Re μj  1 for j = 1, . . . , m} is such that the map λ −→ Θ(λγ1 , . . . , λγm ) extends from



 π , |λ|  1 2(γ1 + η) to ℂ as a meromorphic map, whose polar set does not intersect the closed sector (2.101). Its value at λ = 0 equals the pseudo-meromorphic current (2.104)   



 

= · · · Θ(μ1 , . . . , μm ) ··· . Θ(λγ1 , . . . , λγm ) λ ∈ ℂ : |arg]π,π[ (λ)|
r κ=1 extends from {μ ∈ ℂm : Re μj  1 for j = 1, . . . , m} to ℂm as a meromorphic function, whose polar set does not intersect {μ ∈ ℂm : Re μj ≥ 0 for j = 1, . . . , m}. On the other hand, if det[βj,κ ]1≤j,κ≤r = 0, then the polar set of the univariate rational function r  λγs(κ) m λ −→ γs(j) j=1 βκ,j λ κ=1 does not intersect the closed sector (2.101), for any permutation s of {1, . . . , m}. Moreover, its value at μ = 0 equals r



     μs(κ) m ··· ··· . λm−1 =0 λm =0 j=1 βj,κ μs(j) λ1 =0 λ2 =0 κ=1 Since this is valid for any permutation s of {1, . . . , m}, with this procedure one can treat all nonzero terms in the development of Θ(μ) in (2.105). Thus, the proof of the lemma follows in that particular case where all χj equals identically the constant 1. When the χj are arbitrary strictly positive functions, one needs to introduce m additional complex parameters τ1 , . . . , τm and consider instead of Θ the  D(Ω)-valued map  |χ |2τj |ζ βj |2μs(j) ' |χ |2τj |ζ βj |2μs(j) j j Θ : (μ1 , . . . , μm , τ1 , . . . , τm ) −→ ∂ . βj ζ ζ βj j>r j≤r

108

2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

From the reasoning above, the  D(Ω)-valued map (λ, τ1 , . . . , τm ) −→ Θ(λγ1 , . . . , λγm , τ1 , . . . , τm ) extends from  (λ, τ1 , . . . , τm ) ∈ ℂm+1 : |arg]π,π[ (λ)|
r j≤r

extends from {μ ∈ ℂ : Re μj  1 for j = 1, . . . , m} to ℂm as a meromorphic function, whose polar set intersects ([−η, +∞[)m along a finite union of hyperplanes  r m satisfy the condition j=1 αι,j μj = 0. The coefficients αι ∈ ℕ m

card({κ ∈ {1, . . . , r} : αι,κ ∈ ℕ∗ }) ≥ 2. Moreover η > 0, as well as the polar hyperplanes ⟪αι , μ⟫ = 0, are independent from the exponent ν. Proof of Lemma 2.77. If μ ∈ ℂm is such that Re μj  1 for any j, then it is enough to apply the current Ξ(μ) on test forms of the form ψ ∧ ϕ, where

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2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

ϕ ∈ D (N −M,0) (Ω) and ψ is a (N − M − r, 0)-form with constant coefficients, for linearity reasons. Let, as usual, ιV : V → Ω be the inclusion embedding and π : V = VY −→ V be a log resolution,27 where Y is the hypersurface of V defined as the union of V ∩V (s1 · · · sm ) with any hypersurface containing Vsing . As in the proof of Proposition 2.69, one chooses centered local coordinates ζ = (ζ1 , . . . , ζN −M ) in a neighborhood of any point z ∈ π −1 (V ). With respect to these centered coordinates and in a convenient neighborhood U of z in V, one has (ιV ◦ π)∗ sj = σj ζ βj

N −M β for any j = 1, . . . , m, with σj ∈ (O(U ))∗ and ζ βj = κ=1 ζκ j,κ ≡ 1 a true of unity on the complex manifold V subordinate monomial. Let (θ ι )ι be a partition " to a covering (ιV ◦ π)−1 (Supp ϕ) ⊂ ι U ι of the compact subset (ιV ◦ π)−1 (Supp ϕ). Then, one has28 (2.113) ι ι    |σjι ζ βj |2μj ' |σjι ζ βj |2μj

∧ θ ι (ιV ◦ π)∗ (ψ ∧ ϕ). Ξ, μ = ∂ ι )ν ζ νβjι ι )ν ζ νβjι ι (σ (σ U j j ι j>r j≤r It is enough to consider the open charts U ι around points z = z ι in π −1 (Y). Fixing such a chart, we drop the index ι for a moment to simplify the notation. Therefore, the term from the right-hand side of (2.113) will be   |σj ζ βj |2μj ' |σj ζ βj |2μj

∧ θ (ιV ◦ π)∗ (ψ ∧ ϕ). ∂ (2.114) ν ζ νβj ν ζ νβj (σ ) (σ ) j j U j>r j≤r

After possible re-indexation of s1 , . . . , sr , we us assume that for some ρ ≤ r the set {β1 , . . . , βρ } is a maximal subset of {β1 , . . . , βr }, whose elements are ℚ-linearly independent in ℚN −M . Up to a holomorphic change of centered local coordinates ζ in the chart U , one may assume then in addition that σ1 = · · · = σρ ≡ 1 in U . Thus (2.114) becomes βj ρ r m

  |σ |2λj   |ζ βj |2μj ' dζ j ∧ ω ∧ θ (ιV ◦ π)∗ ϕ, (2.115) μj ν βj νβj σ ζ U j ζ j=1 j>ρ j=1 j=1 where ω=

r ' dσj ∧ (ιV ◦ π)∗ ψ ∈ ΩN −M −ρ (U ). σ j j=ρ+1

Let K ⊂ {1, . . . , N − M } be the set of indices of coordinates ζ1 , . . . , ζN −M that divide the monomial j>r ζ βj . Observe that K = ∅ if r = m. For each subset K ⊂ K, let ιK

ΠK = {ζ ∈ U : ζκ = 0 for κ ∈ K} → U. Then the pullback ωK = ι∗K ω of ω extends constantly to U . If   ω 6= ω{k1 } − ω{k1 ,k2 } + · · · + (−1)|K|−1 ωK , k1 ∈K

27 See 28 For

chart Uι .

{k1 0 being independent of ν. Lemma 2.77 shows this indeed holds for r = 1. Let r ≥ 2 and assume that the result holds when r is replaced by r − 1. Using Stokes’s theorem, we rewrite (2.112) as   |sj |2μj ' |sj |2μj ∧∂ϕ ∂ (2.123) Ξ(μ), ϕ = (−1)r sνj sνj V j>r 2≤j≤r   |sj |2μj |sj  |2λj  '  |sj |2μj  − ∧ ∧ ϕ. ∂ ∂ sνj sνj sνj V   j >r

{j>r : j=j }

2≤j≤r

By Lemma 2.77, the function of μ in the left-hand side of (2.123) extends to a meromorphic function to {μ ∈ ℂm : Re μj > −η for j = 1, . . . , m} for some η > 0, independent  of ν. Moreover, its polar set is included in a union of hyperplanes with equation rj=1 αι,j μj = 0 and is also independent of ν. Due to the inductive hypothesis (for the first term) and because of Lemma 2.77 (for the terms indexed with j  ), the function of μ in the right-hand side of (2.123) also extends to a meromorphic funcη for j = 1, . . . , m} for some η6 > 0, which is indepention to {μ ∈ ℂm : Re μj > −6 dent on ν. Its polar set is included in a union of hyperplanes, with equations of the r 6ι,j μj + α 6ι μj  = 0 for some j  > r. Then, the meromorphic continuation form j=2 α of the function (2.112) in {μ ∈ ℂm ; Re μj > − min(η, η6) for j =  1, . . . , m} has its r polar set truly included in a union of hyperplanes with equation j=2 αι,j μj = 0. Using the same argument and restricting eventually η (still independent of ν), one can see that for any ρ∈ {1, . . . , r}, this polar set lies in fact in a union of hyperplanes with equation 1≤j≤r,j=ρ αι,j μj = 0. Since this is true for any ρ, it implies that the function (2.112) can be analytically continued as a meromorphic function to {μ ∈ ℂm : Re μj > −ˇ η for j = 1, . . . , m} for some ηˇ > 0. This concludes the proof of the inductive argument and thus the proof of Theorem 2.75.  Theorem 2.75 suggests introducing the following important definition. Definition 2.78. Let V → X be a closed analytic subset with pure codimension M of an N -dimensional complex manifold X , let D1 , . . . , Dm , 1 ≤ m ≤ N − M , be m Cartier divisors on the manifold X , and let |D| be the Cartier divisor D1 + · · · + Dm . Given a vector of holomorphic sections s = (s1 , . . . , sm ) in O(D1 ) × · · · × O(Dm ) such that   (2.124) codim V ∩ V (s) = M + m, where V (s) = |div(s1 )| ∩ · · · ∩ |div(sm )|, the Coleff–Herrera (M, M + m)-current   Rs ∧ [V ] ∈ CHX ,V X , O(−|D|) is called the Coleff–Herrera residue current attached to the ordered sequence s defining a complete intersection on V . We will denote it from now on in this monograph

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2. RESIDUE CURRENTS: A MULTIPLICATIVE APPROACH

by29

1

:= Rs ∧ [V ]. s V We conclude this subsection and Chapter 2 with the following major result.

(2.125)

∂

Theorem 2.79 ([CoH78,Pas88A]). Let X be a complex, N -dimensional manifold with structural sheaf O, let V ⊂ X be a closed analytic subset with pure codimension M , and let s ∈ O(D1 ) × · · · × O(Dm ) satisfying (2.124) as in Definition 2.78. For any permutation s of {s1 , . . . , sm } with signature ε(s), one has (2.126)

∂(1/s(s))V = ε(s) ∂(1/s)V .

Moreover, in addition to the properties satisfied by any section of the Coleff–Herrera  sheaf CHV ∩V (s) · , O(−|D|) , the residue current ∂(1/s)V also satisfies, in terms of sheaves, the annihilation property m

1 ( (2.127) IV ⊕ Osj ∂ = 0, s V j=1 where IV is the radical ideal sheaf of V and Osj is the subsheaf of the sheaf of sections of O(Dj ) generated by sj . Proof. The first assertion follows immediately from Theorem 2.75. This theorem, once combined with the analytic continuation principle, implies also that 1 IV ∂ = 0. s V The equality between continuous differential forms m ' s∗j

∧ [V ] s j0 ∂ |sj |2μj |sj |2 j=1 ' s∗j

= (−1)j0 −1 ∂|sj0 |2μj0 ∧ ∂ |sj |2μj ∧ [V ] |sj |2 j=j0

'

s∗j

∧ [V ] , ∂ |sj |2μj = (−1)j0 −1 ∂ |sj0 |2μj0 2 |sj | j=j0

when Re μj  1 for j = 1, . . . , m, Theorem 2.75, Stokes’s theorem, and the analytic continuation principle imply that Osj0 ∂(1/s)V = 0.  Remark 2.80. Assertion (2.127) constitutes the direct (=⇒) part of a result to be completely settled in §3.3.3. This result is known as the duality theorem for complete intersections (see Theorem 3.39) and it is of great importance.

has to be careful with such a notation if one interprets ∂(1/s) as ∂(1/s1 )∧· · ·∧∂(1/sm ). The reason is the following: according to Theorem 2.75, one has m 

s∗j ∧ [V ] ∂(1/s)V = (−1)m(m−1)/2 ∂ |sj |2μj 2 μ1 =···=μm =0 2iπ|sj | j=1 29 One

independently of the hermitian metrics | | which equip the line bundles O(Dj ). One needs to take care of the correction of the sign. In other words, one should better interpret ∂(1/s) as the formal wedge product ∂(1/sm ) ∧ · · · ∧ ∂(1/s1 ).

CHAPTER 3

Residue currents: a bundle approach 3.1. A toric complex geometry digest Toric varieties play an important role in this monograph. They are fundamental for the construction of local log resolutions used, as we have already seen, to prove the existence of the residue currents. They play also a crucial role in the construction of related integral representations of residue symbols. Therefore, before introducing Bochner–Martinelli currents, let us review first some necessary preliminary material. Naturally, our presentation is aimed at conforming with the scope of this monograph. 3.1.1. Rational fans. We begin with some basic definitions from convex geometry. The brackets ⟪ , ⟫ denote here the duality between a ℤ-lattice N (respectively a ℝ-vector space E such as ℝN ) and its dual N  = Homℤ (N , ℤ) (respectively its ℝ-dual E ∗ ). Our reference lattice N will be ℤN , and we use the identification (ℤN )  ℤN . Definition 3.1 (ℤN -rational cones, duality, faces). A ℤN -rational cone σ in ℝ is a finite intersection of closed half-subspaces {x ∈ ℝN : ⟪a , x⟫ ≥ 0}, where a ∈ (ℤN )  ℤN . Its dimension dim σ is that of the ℝ-affine subspace it spans in ˜ , defined by ℝN = ℤN ⊗ℤ ℝ. The dual cone σ N

σ ˜ = {ξ ∈ ℝN : ⟪ξ , x⟫ ≥ 0 for any x ∈ σ}, ˇ = N, is a ℤN -rational cone as well. The cone σ is said to be strict if and only if dim σ which amounts to saying that σ does not contain any ℝ-linear subspace of ℝN . A face τ of a rational cone σ is any subset of σ of the form τ = {x ∈ σ : ⟪ξ , x⟫ = 0} for some (nonunique) ξ ∈ σ ˜ . Any face τ of σ (one writes τ ≺ σ) is also a rational cone with dimension between 0 and dim σ. The set of faces τ of a rational cone σ is denoted by τ (σ). Remark 3.2. ℤN -rational cones are particular examples of ℤN -rational generalized polyhedra; see Definition 7.51. The notion of a face of a ℤN -rational cone is a particular case of the notion of a face of a ℤN -rational generalized polyhedron; see Definition 7.52. Definition 3.3. A ℤN -rational fan Σ in ℝN is a finite collection of strict rational1 cones in ℝN , so that any face τ ∈ τ (σ) of any cone σ ∈ Σ remains in Σ, and any intersection of two cones in Σ is a face of both of them. The fan is 1 We mean here ℤN -rational. More generally, one should take instead of ℤN a free ℤ-module N with rank N (thus isomorphic to ℤN ) and consider N -rational cones as cones in N ⊗ℤ ℝ  ℝN which are of the form {a1 v1 + · · · + as vs : aj ≥ 0}, where v1 , . . . , vs are generators lying in N . The dual lattice Homℤ (N , ℤ) = N  remains isomorphic, but not equal, to ℤN . For the sake

117

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3. RESIDUE CURRENTS: A BUNDLE APPROACH

" called projectively complete (respectively affinely complete) when σ∈Σ σ = ℝN " (respectively when σ∈Σ σ = (ℝ+ )N ).2 A rational fan in ℝN is called simplicial if any cone in the fan is generated by a set of vectors in ℤN that can be completed to a basis of ℝN . Any such cone is called simplicial. Furthermore, the fan is called primitive if any cone in the fan is generated by a set of primitive vectors in ℤN (that is, vectors with coprime coordinates) that can be completed to a basis of ℤN . Such a cone is said to be primitive. Example 3.4 (Normal fan). Let β1 , . . . , βM be a collection of points in ℕN and Δ+ β = convex hull

M    βj + (ℝ+ )N j=1

with set of vertices (ℝ+ )N , defined by

Δ+ β (0).

Equivalence classes for the equivalence relation R+ β in

2 + (3.1) x R+ β y ⇐⇒ u ∈ Δβ (0) : ⟪u, x⟫ = 2

4 min ⟪v, x⟫ +

v∈Δβ (0)

= u ∈ Δ+ β (0) : ⟪u, y⟫ =

4 min ⟪v, y⟫ , +

v∈Δβ (0)

If are ℤN -rational cones forming an affinely complete ℤN -rational fan Σ+ β. β1 , . . . , βM are points in ℤN such that the convex hull Δβ of {β1 , . . . , βM } has a nonempty interior, then Δβ (0) denotes the corresponding set of vertices. The equivalence classes for the equivalence relation Rβ in ℝN , defined as in (3.1) but N with Δβ (0) in place of Δ+ β (0), are ℤ -rational cones forming a projectively complete ℤN -rational fan Σβ . The fan Σ+ β (respectively Σβ ) is called the normal fan to Δ+ (respectively to Δ ). β β Among subfamilies of cones in a ℤN -rational fan Σ, the most important are the following. (1) The family of the one-dimensional cones or rays (denoted by Σ(1)). This family is generated by the vectors ξ1 , . . . , ξd , where ξj = (ξj,1 , . . . , ξj,N ) ∈ ℤN denotes a primitive generator of the ray ξj ℝ+ ∈ Σ(1) for every j = 1, . . . , d. (2) The family of the N -dimensional cones denoted by Σ(N ). From the family Σ(N ) it is possible to reconstruct the whole fan based on the operation of taking faces. The notion of convexity for a rational fan in ℝN plays an important role also. A rational fan Σ in ℝN is said to be convex if and only if the endpoints of the primitive vectors ξ1 , . . . , ξd generating its rays lie on the boundary of their closed convex hull conv(ξ1 , . . . , ξd ). Any complete convex rational fan Σ in ℝN is dual to a reflexive of simplicity, one takes here N = ℤN . In this subsection rational will always mean ℤN -rational (unless specified). 2 Projectively complete rational fans are crucial for studying global questions once nice geometric presentations are constructed for them (see for example Theorem 3.13), while affinely complete rational fans are, on the other hand, quite useful when constructing (toric) log resolutions of the form we will precisely need in this chapter (see Theorem 3.11 and Example 3.12).

3.1. A TORIC COMPLEX GEOMETRY DIGEST

119

integral N -dimensional polyhedron.3 In particular, if the endpoints of the primitive vectors ξ1 , . . . , ξd generating the rays of Σ are vertices of an N -dimensional reflexive polyhedron P = PΣ containing the origin as an interior point, then Σ is the normal fan, as defined in Example 3.4, to the dual integral polyhedron (3.2) ΔΣ = Pˇ := {x∗ ∈ (ℝN )∗  ℝN : ⟪x∗ , y⟫ ≥ −1 ∀ y ∈ P }, containing the origin as an interior point also. 3.1.2. Complete simplicial fans and homogeneous coordinate rings. Let Σ be a ℤN -rational complete simplicial fan and let Σ(1) = {ξ1 ℝ+ , . . . , ξd ℝ+ } be its set of rays generated by primitive vectors ξj ∈ ℤN , where d = N + r. Let us associate to each ray ξj ℝ+ a homogeneous coordinate zj , j = 1, . . . , N + r; see [Cx95] or also [CLO]. Definition 3.5. The polynomial ring ℂ[z1 , . . . , zd ] is called the homogeneous coordinate ring attached to the rational fan Σ. Inside this ring, the monomial ideal I irr (Σ) ⊂ ℂ[z1 , . . . , zd ] defined by

  (3.3) I irr (Σ) = zj ℂ[z1 , . . . , zd ] σ∈Σ(N )

{1≤j≤d : ξj ℝ+ ≺σ}

is called the irrelevant ideal attached to Σ. If 𝕋d = ((ℂ∗ )d , ×) denotes the ddimensional complex torus with pointwise multiplication, then its multiplicative subgroup d    ξ d tj j,k = 1 for k = 1, . . . , N (3.4) G(Σ) = t = (t1 , . . . , td ) ∈ 𝕋 : j=1

is called the toric action group attached to Σ. Remark 3.6. If # denotes the pointwise multiplication in ℂd , then one has G(Σ) # I irr (Σ) ⊂ I irr (Σ) . In the particular case where Σ(N ) contains at least one primitive cone, then G(Σ) is isomorphic to the complex torus 𝕋r , where r = d − N . Such is the case when the fan Σ itself is primitive; see Definition 3.3. In general, the toric action group G(Σ) factorizes as the product of a factor isomorphic to 𝕋r , with a group of finite order. Speaking additively instead of multiplicatively, G(Σ) is in general the direct sum of a free group Gfree (Σ) with rank r and of a finite abelian torsion group Gtors (Σ), which disappears as soon as Σ(N ) contains at least one primitive cone. A description of the zero set of the irrelevant ideal I irr (Σ) is given in [Bat93, §2]. Proposition 3.7. A subset of Σ(1) is said to be primitive if the cone it generates in ℝN does not belong to Σ, while it is the case for the cone generated by any of its proper subsets. Then V (I irr (Σ)) is the union of all coordinate planes {zi1 = · · · = zik = 0} such that {ξi1 , . . . , ξik } is a primitive subset of Σ(1). Having these tools within hands, one can associate to the fan Σ an N -dimensional complex analytic space; see §A.4.1. polyhedron P is called (ℤN ) integral if its vertices are in ℤN . An integral polyhedron P is said to be reflexive if its dual P˜ := {y ∈ ℝN : ⟪ξ, y⟫ ≥ −1 for any ξ ∈ Δ} has its vertices in ℤN also. 3A

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Definition 3.8 ([Ful93, Cx95, CLO]). The compact complex N -dimensional analytic space XΣ is defined as the geometric quotient (3.5)

XΣ =

ℂd \ I irr (Σ) . G(Σ)

It is a complex manifold4 if and only if Σ is primitive. The complex analytic space XΣ is always normal; see Definition A.32. Moreover, the torus 𝕋N acts on it and at the same time embeds in it as a dense Zariski open orbit under its action. Remark 3.9 (Algebraic toric variety 𝕏Σ ). The complex analytic space XΣ is classically realized as the set of closed points of the algebraic toric variety 𝕏Σ constructed from the rational fan Σ (complete or not, simplicial or not) as follows.  σ ∩ (ℤN ) ] . For any τ ≺ σ, there is For each cone σ ∈ Σ, let 𝕏σ = Spec ℂ[ˇ a scheme embedding ισ,τ : 𝕏τ → 𝕏σ . Moreover,  this embedding is compatible ±1 with the action of 𝕋N = Spec >ℂ[X1±1 , . . . , XN ] = 𝕏{0} on each 𝕏σ , σ  ∈ Σ. Starting with the disjoint union σ∈Σ 𝕏Σ , then identifying for each pair (σ  , σ  ) the algebraic varieties 𝕏σ and 𝕏σ along 𝕏σ ∩σ , using the embeddings ισ ,σ ∩σ and ισ ,σ ∩σ , one constructs an algebraic variety 𝕏Σ (defined over ℚ), together with an action on it of 𝕋N = 𝕏{0} ; see [Ful93] for more details. Furthermore, each 𝕏σ can be decomposed into a set of orbits under the action of the complex torus 𝕏{0} as " " 𝕏σ = τ ≺σ Orbτ , where Orbτ  𝕋N −dim τ . Thus, one has that Orbτ = τ ≺σ Orbσ for any τ ∈ Σ in terms of Zariski topology. If τ is a ray, then Orbτ is in particular an algebraic hypersurface ℍ of 𝕏Σ . The orbit Orb{0} = 𝕏{0} = 𝕋N is therefore an open dense orbit in 𝕏Σ acting on it. Observe also that 𝕏Σ is normal. The algebraic variety 𝕏Σ is called the toric variety attached to the fan Σ. We are interested in this monograph in the N -dimensional complex analytic space XΣ , whose points are its closed points. The complex analytic space XΣ defined by (3.5) is a useful alternative (geometric) presentation of the set of closed points XΣ of 𝕏Σ , when the fan Σ is complete and simplicial. See also [CxLS11] for an alternative presentation. Each irreducible hypersurface Hj = {zj = 0} ⊂ XΣ (modulo the action of G(Σ)) corresponds from the set-theoretical point of view to the (N −1)-dimensional complete simplicial toric variety attached to the ℤN /ξj ℤ-rational complete simplicial fan πj (Σ) in ℝN /ξj ℝ, where πj : ℝN → ℝN /ℝ ξj denotes the quotient map. It is a closed orbit under the action of the dense torus 𝕋N on XΣ ; see Remark 3.9. Then {Hj } is a Weil divisor on XΣ , which is also invariant under the action of the torus 𝕋N on XΣ . Hence {Hj } is called a toric Weil divisor. Let   (3.6) ν : (m1 , . . . , mN ) ∈ (ℤN )  ℤN −→ ⟪m, ξ1 ⟫, . . . , ⟪m, ξd ⟫ ∈ ℤd . The following important result relates the toric action group G(Σ) attached to the simplicial fan with the (N − 1)-Chow group AN −1 (XΣ ) of the complex analytic space XΣ ; see §D.3.2. Theorem 3.10 ([Cx95]). One has AN −1 (XΣ )  ℤd /ν((ℤN ) ) and, as a consequence   (3.7) G(Σ)  Homℤ AN −1 (XΣ ), 𝕋 . 4 Its

complex structure is, in general, an orbifold complex structure.

3.1. A TORIC COMPLEX GEOMETRY DIGEST

121

In order to describe the action of the factor of G(Σ), which is isomorphic to 𝕋r , it is useful to introduce a collection of d − N = r linear, ℚ-independent relations with integer coefficients (3.8)

aj,1 ξ1 + · · · + aj,d ξd = 0 , j = 1, . . . , r,

between the rays ξ1 , . . . , ξd of the fan Σ. Note that since the fan is complete, one can choose all the aj,k ≥ 0, j = 1, . . . , r, k = 1, . . . , d (see [Ky05, Proposition 1]). Relations (3.8) describe the kernel of the map ν defined by (3.6). Thus, the factor of G isomorphic to 𝕋r induces the following equivalence relation on ℂd \ V (I irr (Σ)): z  z  ⇐⇒ ∃ t ∈ 𝕋r , z  =

r  

r

 a  a  tj j,1 z1 , . . . , tj j,d zd .

j=1

j=1

If Σ(N ) contains at least one primitive cone, then AN −1 (XΣ )  ℤr and G  𝕋r . In this case, an isomorphism between AN −1 (XΣ ) and ℤr is realized by (3.9)

  ℤd ˙ ∈ AN −1 (XΣ ) = ∈ ℤr .  − → a m , . . . , a m m 1,k k r,k k ν((ℤN ) ) d

d

k=1

k=1

Before concluding this subsection, let us mention two results that will be of importance to us. The first one concerns fans in general, independently if they are simplicial or not, complete or not; see [Ful93, §2.6]. Recall that for a ℤN -rational 6 as well and such that each cone fan Σ, a refinement of Σ is a ℤN -rational fan Σ 6 of Σ is a union of cones belonging to Σ. If the fan Σ is primitive, then so is any 6 In case Σ fails to be primitive, the following result holds. refinement Σ. Theorem 3.11 (Resolution of singularities within the toric varieties world). 6 of Σ, Let Σ be a ℤN -rational fan. One can always find a primitive refinement Σ → 𝕏 , which is birational and proper. together with a morphism π : 𝕏Σ Σ  As a consequence, such a proper birational morphism from 𝕏Σ  to 𝕏Σ induces a proper bimeromorphic5 morphism of complex analytic spaces from the N -dimensional complex manifold XΣ  to the N -dimensional complex analytic space XΣ , which is in general singular. This provides a resolution of singularities for XΣ ; see §A.4.3. Example 3.12. Theorem 3.11 holds in particular for the fan Σ+ β associated with a finite collection of points β1 , . . . , βM in ℕN , as explained in Example 3.4. This fan is affinely complete, but not projectively. It is not in general simplicial. Theorem 3.11, once combined with Hironaka’s theorem (see Theorem A.41) allows us for example to realize the following: given f1 , . . . , fm ∈ OX (U ), where X is an N -dimensional complex analytic space and U ⊂ X , one may construct a log-resolution π : X6 → U subordinate to a hypersurface Y in U containing both U ∩ Xsing and the V (fj ) ∩ U for j = 1, . . . , m, in such a way that (f1 , . . . , fm )(OX )|U := π ∗ ((f1 , . . . , fm )OX )|U is a principal sheaf of OX -ideals, whose zero set is a hypersurface with normal crossings on the complex analytic manifold X6. 5 In fact birational, since complex analytic spaces considered here come from algebraic varieties 𝕏Σ  and XΣ defined over ℚ.

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3. RESIDUE CURRENTS: A BUNDLE APPROACH

The second result concerns ℤN -rational complete simplicial fans, where the set Σ(N ) contains at least one primitive cone; see [ShcTY06, Theorem 1]. It completes the presentation of the complex analytic space XΣ given by (3.5) and extends the concept of perspective as introduced in the Italian Renaissance. The N +1 in concept of perspective is re-interpreted in terms of the embedding ℙN ℂ → ℙℂ N +1 N +1 such a way that ℙℂ is realized as the disjoint union of the affine space ℂ with its hyperplane at infinity, namely H = ℙN ℂ , as already mentioned in Example 2.7. Theorem 3.13 (Multiperspective). Let Σ be a complete ℤN -rational simplicial fan with d = N + r rays. If Σ(N ) contains at least one primitive cone, then there exists a d-lattice N and a complete simplicial fan Σ in N ℝ = N ⊗ℤ ℝ such that (1) X Σ = ℂd % (H 1 ∪ · · · ∪ H r ) for some complex hypersurfaces H j , j = 1, . . . , r; (2) there exists an isomorphism of complex spaces φ : H 1 ∩ · · · ∩ H r ←→ XΣ ; (3) for any z ∈ ℂd \ V (I irr (Σ)), one has     (3.10) G(Σ)(z) ∩ H 1 ∩ · · · ∩ H r = φ−1 z mod G(Σ) . 3.1.3. K¨ ahler cone and moment maps. Let Σ be a ℤN -rational complete primitive fan. The compact analytic space XΣ defined by (3.5) (or defined utilizing the algebraic construction described in Remark 3.9) is a compact N -dimensional complex manifold. Consider the set Σ(1) = {ξ1 ℝ+ , . . . , ξd ℝ+ }, where d = N + r. Consider also r linear relations between the d rays of Σ with coefficients aj,k in ℕ as in (3.8), that are ℚ-linearly independent. Given a set of such relations with matrix [a] ∈ Md,r (ℕ), let Θ[a] : γ = (γ1 , . . . , γd ) ∈ ℝd −→

d 

a1,k γk , . . . ,

k=1

d 

ar,k γk ∈ ℝr .

k=1

For any γ = (γ1 , . . . , γd ) ∈ ℝ , the map from Σ(1) to ℝ sending ξj to −γj extends in a unique way as a Σ-piecewise linear map Φγ from ℝN to ℝ. That is, Φγ is a continuous map from ℝN to ℝ, whose restriction to any σ ∈ Σ is ℝ-linear. d

Definition 3.14. The K¨ ahler cone of the primitive fan Σ is defined by 2 4 (3.11) K(Σ) = Θ[a] γ ∈ ℝd : Φγ is strictly convex (independently of the matrix [a]). Let us provide an alternative presentation of K(Σ) in terms of the primitive + subsets of Σ(1); see Proposition  3.7. For any primitive subset {ξ}J = {ξj ℝ , j ∈ J} of Σ(1), the vector |ξ|J = j∈J ξj belongs to some σJ ∈ Σ(N ), which is generated by the primitive vectors ξJ,1 , . . . , ξJ,N , that is |ξ|J =

N 

αJ,ν ξJ,ν

ν=1

for some rational coefficients αJ,ν ≥ 0. The ℚ-independent r relations (3.8) generate the ℚ-vector space of ℚ-linear relations between the primitive vectors ξ1 , . . . , ξd . Thus, there exist unique rational numbers t1 (J), . . . , tr (J) satisfying N r d

   (3.12) |ξ|J − αJ,ν ξJ,ν = tρ (J) aρ,k ξk . ν=1

ρ=1

k=1

3.1. A TORIC COMPLEX GEOMETRY DIGEST

123

The alternative presentation of K(Σ), arising from the description of V (I irr (Σ)) (see Proposition 3.7) is then (3.13)

r  2 K(Σ) = γ ∈ (ℝ+ )r : tρ (J) γρ > 0

∀ {ξ}J

primitive

⊂

4 Σ(1) .

ρ=1

Since the symbols |z1 |2 , . . . , |zd |2 satisfy the system of ℚ-linearly independent relations d  aj,k |zk |2 = 0, j = 1, . . . , r, k=1

as well as the ξk (k = 1, . . . , d), do, one has 

|zj | − 2

N 

αJ,ν |zJ,ν | = 2

ν=1

j∈J

r 

tρ (J)

ρ=1

d 

aρ,k |zk |2

k=1

for any j ∈ J. As a consequence, K(Σ) is also defined by (3.14) K(Σ) = μ

2

z∈ℂ : d

|z|2J

−

n 

αJ,ν |zJ,ν |2 > 0

∀ {ξ}J

primitive

⊂

Σ(1)

4 ,

ν=1

where the map (3.15)

μ : z ∈ ℂd −→

d  k=1

a1,k |zk |2 , . . . ,

d 

ar,k |zk |2 ∈ (ℝ+ )r

k=1

is known as a moment map with respect to the fan Σ; see [Shc07, §1.4] or [Ky05, §1]. The moment map can be seen from the point of view of symplectic geometry; see §3.1.4. 3.1.4. Projective toric manifolds. Assume now that the ℤN -rational fan Σ, where d = N + r = |Σ(1)|, is complete, primitive, and such that XΣ is projective as an algebraic compact manifold. Being projective amounts to saying that there d exists an ample6 toric Weil divisor j=1 aj {Hj }, with aj ∈ ℤ, on XΣ . Let D = d j=1 aj Dj be the corresponding Cartier divisor. It is a well known fact that when XΣ is a complex toric manifold, then a line bundle O(D) over XΣ is ample if and only if it is very ample [Dem70]. If, in addition, XΣ is projective, then Σ can be realized as the normal fan of an absolutely simple, N -dimensional, convex polyhedron ΔΣ = ΔD ⊂ (ℝN )∗ ; see [Oda, §2.4, Theorem 1.10] and Example 3.4. A convex polyhedron ΔΣ = ΔD ⊂ (ℝN )∗ is said to be absolutely simple if it satisfies the following properties: (1) Any vertex of ΔΣ belongs to exactly N edges of ΔΣ ; (2) Primitive vectors defining the directions of N edges meeting at a vertex of ΔΣ form a basis of the lattice ℤN . Note that when Σ is a convex fan, one can take as ΔΣ the reflexive polytope defined in formula (3.2) as the dual polyhedron of the convex hull of ξj , j = 1, . . . , N + r. 6 See §D.4, in particular §D.40, for the notion of ampleness of Cartier divisors on a complex algebraic variety, in particular toric divisors on a complete toric algebraic variety.

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3. RESIDUE CURRENTS: A BUNDLE APPROACH

The divisor D = D1 + · · · + DN +r is very ample in this case. One can also prove that when XΣ is smooth and projective, the subset {z ∈ ℂd : |z|2J −

N 

αJ,ν |zJ,ν |2 > 0

∀ {ξ}J

primitive

⊂

4 Σ(1) ⊂ ℂd ,

ν=1

whose image by the moment map z −→

d 

a1,k |zk |2 , . . . ,

k=1

d 

ar,k |zk |2

k=1

equals the K¨ahler cone K(Σ) (see (3.14)) does not intersect V (I irr (Σ)). Detailed description of the K¨ahler cone K(Σ) in terms of primitive subsets of Σ(1) is proposed in [Bat93, §2]; see also [Cx97, Theorem 4.1]. Moreover, for any γ = (γ1 , . . . , γr ) in K(Σ), the maximal compact subgroup r r

    a a Gℝ (Σ) = tj j,1 , . . . , tj j,d , tj ∈ 𝕊1 , j = 1, . . . , r j=1

j=1

of G(Σ) induces a group action on the real manifold μ−1 (γ) such that μ−1 (γ)/Gℝ (Σ) is another representation of the toric projective manifold XΣ . To be more specific, the quotient μ−1 (γ)/Gℝ (Σ) can be identified with XΣ through a ℝ-diffeomorphism that preserves the symplectic form ω = (1/2i)

d 

dzj ∧ dzj .

1

When XΣ is a projective toric manifold, its K¨ ahler cone K(Σ) coincides, modulo the identification of ℝr with AN −1 (XΣ ) ⊗ℤ ℝ induced by the isomorphism (3.9) tensored with ℝ, with the set of cohomology classes of closed positive (1, 1)-forms in the cohomology group H 1,1 (XΣ , ℝ) = H 2 (XΣ , ℝ)  AN −1 (XΣ ) ⊗ℤ ℝ. That is, the K¨ ahler cone K(Σ) coincides with the cone of K¨ ahler forms inducing K¨ ahler metrics on the complex K¨ ahler manifold XΣ ; see, e.g., [Cx97, §3]. This justifies the terminology used to denote by extension K(Σ) as the K¨ ahler cone of the projective toric manifold XΣ . The moment map (3.15) from ℂd to (ℝ+ )r is surjective, due to the fact that rank [aj,k ]j,k = r. It follows from [Cx97, Theorem 4.1] that when XΣ is a projective toric manifold, then the isomorphism (3.9), with all entries aj,k chosen to be nonnegative, once tensored with ℝ, provides the representation  (3.16) K(Σ)  μ ℂd \ I irr (Σ)) ⊂ (ℝ+ )r , where μ is the moment map (3.15). 3.2. Bochner–Martinelli currents attached to bundle sections In this section, we consider an N -dimensional complex manifold X with structural sheaf O with a closed analytic subset V ⊂ X with pure codimension M . We also let E be a holomorphic bundle E → X over X with rank m ≥ 1, and we let s ∈ O(X , E) be a global holomorphic section of E over X . The Koszul complex attached to s is defined as in §1.2.5, namely (3.17)

0 −→

m '



s E ∗ −→

m−1 '





s s ··· → E∗ →

 '



s E ∗ −→

−1 '







s s s ··· → E ∗ −→ ℂ, E∗ →

3.2. BOCHNER–MARTINELLI CURRENTS ATTACHED TO BUNDLE SECTIONS

125

where s denotes the interior multiplication. It is denoted by Ks• . We introduce also the holomorphic bundle '

(3.18)

E∗ =

 ('

E∗ = ℂ ⊕

∈ℤ

m '  (

E ∗.

=1

Similarly to (1.83), we consider the sheaves of currents LM,ν E , for ν ∈ ℤ, defined by7 γ (  '  M,ν  (M,M +γ+ν) U, E ∗ (3.19) LE = D open . U ⊂ X

γ∈ℤ

As in §1.2.5, the operator ∇s = 2iπs − ∂ acting from LM,ν−1 to LM,ν for each E E ν ∈ ℤ will play an important role. Note that, since s is holomorphic, s and ∂ 2 2 anticommute, which implies ∇2s = s = ∂ = 0. We equip the holomorphic bundle E with a smooth hermitian metric | | and then denote by E = (E, | |) the metrized bundle. The metric | | induces the definition of a hermitian metric on the dual bundle E ∗ and therefore induces the definition of a hermitian metric | | on each # ∗ # ∗ entry E of the Koszul complex (3.17). The metric induced on each entry E will still be denoted for the time being by | |, since we try to avoid overweighing notations. Recall that the conjugate section s∗ ∈ C ∞ (X , E ∗ ) is defined by s∗ (z)(ξ) = ξ, s(z)z

(3.20)

∀ z ∈ X , ∀ ξ ∈ Ez ;

see (B.59), §B.3.2.1. It is also the section of E ∗ → X with pointwise minimal norm | |z among those satisfying s s∗ = |s|2 in X . 3.2.1. The currents R|s | and P|s | for s ∈ O(X , E), E = (E, | |). We M,−1 introduce the LM,0 (X )-valued maps defined respectively as8 E (X ) and LE

(3.21)

 λ −→ R|s,λ ([V ]) := 1 − |s|2λ + (∂ |s|2λ ) |

λ −→ P|s,λ ([V ]) := |s|2λ |

s∗ s∗

r−1 ∂ [V ], 2iπ|s|2 2iπ|s|2 r≥1  s∗ s∗

r−1 ∂ [V ] 2iπ|s|2 2iπ|s|2 r≥1

for {λ ∈ ℂ : Re λ  1}. Observe that  2λ ∂ 2iπs R|s,λ | ([V ]) = − (∂ |s| )

s∗

r [V ], 2iπ|s|2 r≥0  s∗

r ∂ R|s,λ − ∂ |s|2λ − (∂ |s|2λ ) ∂ [V ]. | ([V ]) = 2iπ|s|2 r≥1

This implies (3.22)

∇s R|s,λ | ([V ]) = 0.

M = 0, we just set LνE instead of L0,ν E . is here understood as wedge multiplication. Moreover, one needs to under       stand a product T  T  of two respectively  E ∗ and  E ∗ -valued currents as the  + E ∗ valued current T  ∧ T  . 7 When

8 Multiplication

126

3. RESIDUE CURRENTS: A BUNDLE APPROACH

One has also

 2λ ([V ]) = |s| ∂ 2iπs P|s,λ | ∂ P|s,λ ([V ]) = |s|2λ |

s∗

r [V ], 2iπ|s|2 r≥0  s∗

r   [V ] + R|s,λ ∂ ([V ]) − (1 − |s|2λ ) [V ] , | 2 2iπ|s| r≥1

which implies (3.23)

 s,λ  ∇s P|s,λ | ([V ]) = Id − R| | ([V ]).

Observe that (3.22) follows from (3.23) if one takes the action of ∇s on both sides and exploits the fact that ∇2s = 0. One can state now a multivariate version of Proposition 1.27. Proposition 3.15. Let X , V , E = (E, | |), s be as above. The LM,0 E (X ) and maps λ −→ R|s,λ| ([V ]) and λ −→ P|s,λ ([V ]), respectively, extend | from {λ ∈ ℂ Re λ  1} to ℂ as meromorphic maps, whose polar sets lie in −ℚ+ . Moreover, one has   ' R|s | ([V ]) = R|s,λ| ([V ]) λ=0 ∈ PM X , E ∗ ),   ' ∗ E ), P|s | ([V ]) = P|s,λ | ([V ]) λ=0 ∈ PM X , M,−1 (X )-valued LE

(3.24)

Supp R|s | ⊂ V ∩ V (s),

∇s R|s | = 0,

∇s P|s | = [V ] − R|s | ([V ]),

0 ≤ r < codimV (V ∩ V (s))  (M,M +r) =⇒ R|s | ([V ]) = Proj∧r E ∗ R|s | ([V ]) = 0. Furthermore, for any r ≥ codimV (V ∩S), for any z ∈ V ∩V (s) and for any f ∈ Oz ,     f = 0 on V ∩ V (s) around z =⇒ f · R|s | ([V ]) z = df ∧ R|s | ([V ]) z = 0, (3.25)   (M,M +r)  = 0. |f | = O(|s|r ) on V around z =⇒ f · R|s | ([V ]) z Proof. Since the integration current [V ] is pseudo-meromorphic, it follows from Proposition 2.21 that λ −→ (1 − |s|2λ )[V ] extends to ℂ as a  D(M,M ) (X )valued meromorphic map, whose poles are in −ℚ+ . Its value at λ = 0 equals the integration current 1X \V (s) · [V ]. Such integration current acts over the set V X \V (s) , where V X \V (s) is the union of the irreducible components of V that are not contained entirely in V (s). This is a particular case of restriction of pseudomeromorphic currents; see §2.2.4 or, for this specific situation, see §D.1.1. Then it is clear that one can replace V by V X \S and hence one can suppose that codimV (V ∩V (s)) > 1. Furthermore, it is also clear that it is enough to prove the result locally around V ∩V (s). Thus, we fix a point a ∈ V ∩V (s). Let π : V → V ∩Ua be a log resolution as in Theorem A.41, applied here for X = V ∩ Ua . Let also Y be the union of the proper closed analytic subset (V ∩ Ua ) ∩ V (s) with a closed hypersurface of V ∩ Ua containing Vsing ∩ Ua . Then V is an (N − M )-dimensional complex manifold. Let us suppose that z ∈ V admits a neighborhood U such that (ιV ◦ π)∗ s = s0 ⊗ s , where s0 is a section of a Cartier divisor D = Dz on U and s is a nonvanishing section of O(−D) ⊗ (ιV ◦ π)∗ (E|V ). One has s∗ 1  = σ, (ιV ◦ π)∗ 2 |s| s0

3.2. BOCHNER–MARTINELLI CURRENTS ATTACHED TO BUNDLE SECTIONS

127

∗ where σ  is a smooth section of O(D) ⊗ (ιV ◦ π)∗ (E|V ) in U . For any r ≥ 1 and λ ∈ ℂ such that Re λ  1, one has s∗ s∗ r−1

1 = (∂ s0 s 2λ ) r σ  (∂σ  )r−1 (ιV ◦ π)∗ (∂ |s|2λ ) 2 ∂ 2 |s| |s| s0 s s 2λ 0 σ  (∂σ  )r−1 , =∂ sr0 s∗ s∗ r−1

s0 s 2λ  = σ (∂σ  )r−1 . (ιV ◦ π)∗ |s|2λ 2 ∂ |s| |s|2 sr0

The  D(U, O(−rD))-valued maps s s 2λ 0 , λ −→ ∂ sr0

λ −→

s0 s 2λ sr0

extend to ℂ as meromorphic maps with poles in −ℚ+ as a consequence of results established in §2.3.2; see also Proposition 2.21. Moreover, their values at λ =  r 0 equal the pseudo-meromorphic currents 2iπ ∂ 1/sr0 and 2iπ P s0 , respectively; see §2.3.2. The direct images of these currents by ι ◦ π define germs of pseudomeromorphic currents at a. The first assertion # of the proposition is proved, and the fact that both R|s | ([V ]) and P|s | ([V ]) are E ∗ -valued pseudo-meromorphic currents is established. It is immediate that R|s | ([V ]) is supported by V ∩ V (s). The final two assertions in the third line of (3.24) follow from (3.22) and (3.23) and the application of the analytic continuation principle. Observe that the current r  '  (M,M +r)  s ∈ PM(M,M +r) X , E ∗ R| | ([V ]) is supported by a closed analytic subset with codimension M + codimV (V ∩ V (s)). Then, it follows from the dimension principle for pseudo-meromorphic currents (see Proposition 2.26) that this current is the null current as soon as M + r < M + codimV (V ∩ V (s)), that is, when r < codimV (V ∩ V (s)). The assertion in the fourth line of (3.24) is then proved. The first assertion in (3.25) follows from the pseudo-meromorphicity of R|s | ([V ]), together with the fact that this current is supported by V ∩ V (s); see Proposition 2.24. The second assertion in (3.25) follows from the fact that |f | = O(|s|r ) on V around a =⇒ |(ιV ◦ π)∗ f | = O(|s0 |r ) around z ∈ V, if we refer here to our previous proof, which implies that 1 (ιV ◦ π)∗ f · ∂ r = 0. s0 This concludes the proof of Proposition 3.15.



Example 3.16 (The case of trivial bundles). Suppose that s1 , . . . , sm are m holomorphic functions on X , so that (s1 , . . . , sm ) is considered as a holomorphic section of the trivial m-bundle ℂm over X , equipped with its standard hermitian metric. Let the frame (1 , . . . , m ) be the canonical basis of ℂm . Then, for any r ≥ 1, the current (R|s | ([V ]))M,M +r = Proj∧r (ℂm )∗ R|s | ([V ]) equals  R|s |;j1 ,...,jr ([V ]) ∗j1 ∧ · · · ∧ ∗jr , 1≤j1 m ∀ κ = 1, . . . , k.

Furthermore, we assume that there exists a meromorphic section H of Homℂ (E, E) such that S = Hs, together with positive integers q1 , . . . , qk , with q = (q1 , . . . , qk ) such that (3.43)

  Δ = tq11 tq22 · · · tqkk det H ∈ O X , O(⟪q, T ⟫) ⊗ Homℂ (det E, det E) .

Proposition 3.23. Let χ1 , . . . , χk ∈ ℕ. Under the previous hypothesis and the additional hypothesis V ∩ V (s) ⊂ V ∩ V (S), one has

(3.44)

1 ∂  1+χ1 1+χk  V s 1 , . . . , s m , t1 , . . . , tk

1 = Δ ∂  . 1+χ1 +q1 1+χk +qk  V S1 , . . . , Sm , t1 , . . . , tk

Proof. Choose metrics | | on each line bundle O(D) involved in the setting. Each such metric induces a metric on all O(D) for  ∈ ℕ∗ . Given these metrics,

3.2. BOCHNER–MARTINELLI CURRENTS ATTACHED TO BUNDLE SECTIONS

135

μ ∈ ℂm , ν ∈ ℂk , consider the following expressions: |s |2μm |s |2μ1 m 1 ∧ · · · ∧ , Rs,μ = ∂ ∂ | | sm s1 |S |2μm |S |2μ1 m 1 ∧ ···∧ ∂ , RS,μ | | = ∂ Sm S1 (3.45) |t |2νk |t |2ν1 k 1 Rt,ν ∧ · · · ∧ ∂ 1+χ1 , 1+χk | | = ∂ tk t1 |t |2νk |t |2ν1 k 1 Rν| | = ∂ 1+χk +qk ∧ · · · ∧ ∂ 1+χ1 +q1 Δ . tk t1 It follows from Theorem 2.75 that in a neighborhood of V ∩ V (s + t), the map s,μ (μ, ν) −→ Rt,ν | | R| | [V ]

extends as a meromorphic map on ℂm × ℂk , whose polar set does not intersect {(μ, ν) : Re μj ≥ 0, Re νκ ≥ 0, 1 ≤ j ≤ m, 1 ≤ κ ≤ k}. Moreover, its value at (0, 0) equals the current

1 ∂  .  1 k V s1 , . . . , sm , t1+χ , . . . , t1+χ 1 k Fix now ν ∈ ℂk with Re νκ  1 for any κ = 1, . . . , k. Consider the map μ −→ Rν| | RS,μ | | [V ]. It follows from Theorem 3.21 that its value at μ = 0 equals the current

 s,μ [V ], Rt,ν | | R| | μ=0

which is supported by V ∩ V (s). On the other hand, Theorem 2.75 implies that in a neighborhood of V ∩ V (S) ∩ V (t), containing V ∩ V (s) ∩ V (t) by hypothesis, the map (μ, ν) −→ Rν| | RS,μ | | [V ] extends as a meromorphic map to ℂm × ℂk . The polar set of the meromorphic extension does not intersect {(μ, ν) : Re μj ≥ 0, Re νκ ≥ 0 for 1 ≤ j ≤ m, 1 ≤ κ ≤ k}. Moreover, its value at (0, 0) equals the current

1 Δ ∂  .  1 +q1 k +qk V S1 , . . . , Sm , t1+χ , . . . , t1+χ 1 k Then, one has, as a consequence of the principle of analytic continuation, that the equality



 1 t,ν s,μ R ∂  = R [V ]  | | | | 1 k V μ=0 ν=0 s1 , . . . , sm , t1+χ , . . . , t1+χ 1 k

 1  = Rν| | RS,μ [V ] = Δ ∂   | | 1 +q1 k +qk μ=ν=0 V S1 , . . . , Sm , t1+χ , . . . , t1+χ 1 k holds in a neighborhood of any point in V ∩ V (s) ∩ V (t). On the other hand, both currents equal to 0 in a neighborhood of a point in V ∩ (V (S) \ V (s)) ∩ V (t). This concludes the proof of the proposition. 

136

3. RESIDUE CURRENTS: A BUNDLE APPROACH

3.3. Bochner–Martinelli currents and generically exact complexes Let (X , O) be an N -dimensional complex manifold equipped with its structure sheaf. We consider in this section, in place of a hermitian vector bundle (E, | |), a complex F• of holomorphic vector bundles E , together with morphisms F between such bundles, (3.46)

F

FL−1

F

F

F

L 2 1 0 0 −→ EL −→ EL−1 −→ · · · −→ E1 −→ E0 −→ E−1 =

E0 −→ 0. Im F1

As usual, F0 denotes the quotient map, Im(F ) ⊂ Ker (F−1 ) for any  = 0, . . . , L, and 0 stands for the null vector bundle X × {0}. We assume that the complex F• of holomorphic bundles is metrized. More precisely, each entry E ,  ≥ 0, is equipped with a smooth hermitian metric12 | | = | | . Such a metrized complex F• is paired with a closed analytic subset V with pure codimension M in X . Under some conditions to be specified and illustrated in the next subsection, one will attach in §3.3.2 to such a complex F• of holomorphic bundles and to V a residue and a principal value current R|F•| ([V ]),

P|F|• ([V ]),

of the Bochner–Martinelli type. Our reference model is the Koszul complex Ks• attached to a section s of a holomorphic bundle E, once it is equipped with a smooth hermitian metric. In §3.3.3, we will consider the particular case V = X to profit from Dolbeault’s lemma there and be able to explain why the residue current R|F•| = R|F•| ([X ]) materializes the obstruction for exactness of the complex F• over X . Both construction and reasoning are inspired by the strategy outlined previously in the univariate case; see §1.2.5. 3.3.1. Generically exact Koszul and Eagon–Northcott complexes. The concept of a generically exact complex and its connection to the construction of attached residue or principal value currents was introduced and described in [And04]. Definition 3.24. A complex F• of holomorphic bundles (E , F ), where r denotes the rank of E for  ≥ −1, is said to be generically exact if and only if there exists a closed analytic subset V (F• ), which we will always assume minimal with respect to inclusion, satisfying  (−1)ν r+ν . (3.47) ∀ z ∈ X \ V (F• ), ∀  = 1, . . . , L, rank F (z) = ν≥0

This amounts to saying that the restriction of F• to the open subset X \V (F• ) is an exact complex of holomorphic bundles, namely Im (F ) = Ker(F−1 ) over X \V (F• ) for  = 1, . . . , L.

12 The

metric on E−1 is induced by that on E0 .

3.3. BOCHNER–MARTINELLI CURRENTS AND GENERICALLY EXACT COMPLEXES 137

Given a generically exact complex F• , one denotes by E(F• ) = E = (3.48) F =

(

E ,

≥−1

(

∇F• = 2iπ F − ∂.

F ,

≥−1

Furthermore, we assume that the bundle E is equipped with 𝔽2 -grading

(

( (3.49) E = E+ ⊕ E− = E2 ⊕ E2−1 . ≥0

≥0

  This grading induces a natural 𝔽2 -grading on Homℂ E(F• ), E(F• ) . Then, the morphism F sending E + to E − and E − to E + is an odd map as the operator ∂ is, when both morphisms are considered as acting in either of the four ways 

F,∂

D(U, E) −→  D(U, E) or F,∂

∞ ∞ C•,• (U, E) −→ C•,• (U, E) or

  F ,∂   D U, Homℂ (E, E) −→  D U, Homℂ (E, E) ,   F ,∂ ∞   ∞ U, Homℂ (E, E) −→ C•,• U, Homℂ (E, E) C•,• 

when U is an open subset of X . Note that both morphisms are denoted either as F, ∂ or F , ∂, depending whether they are considered as acting on E-valued sections or acting on Homℂ (E, E)-valued ones. Since F is holomorphic, one has (3.50)

F ∂ = −∂ F, F ∂ = −∂ F ,

2

∇2F• = (2iπ F − ∂)2 = −2iπ (F ∂ + ∂ F ) = F 2 = ∂ = 0, 2

∇2F• = (2iπ F − ∂)2 = −2iπ (F ∂ + ∂ F ) = F 2 = ∂ = 0

for such actions. Note also that ∇F• (IdE(F• ) ) = 0. Example 3.25 (Koszul complex). Given a holomorphic bundle E with rank m and s ∈ O(X , E), the Koszul complex (3.51) m m−1 +1  ' ' ' (s )1 (s ) +1 ' (s )2 (s ) −1 ∗ (s )m ∗ ∗ (s )

E −→ E · · · −→ E −→ E ∗ −→ · · · −→ E ∗ −→ ℂ 0→ is by far the most tractable among the generically exact complexes which will be considered in this monograph. It is denoted by Ks• , and s is the interior multiplication at each level  such as introduced in (1.80). One has V (Ks• ) = V (s). The operators F, F , ∇F• , ∇F• are denoted in this case as s , s , ∇Ks• = 2iπs − ∂, ∇Ks• = 2iπs − ∂. Example 3.26 (Eagon–Northcott complex [EaN62]). Let E be a holomorphic vector bundle with rank m and s ∈ O(X , E). Fix k ∈ ℕ∗ . Let (Ks• )1 , . . . , (Ks• )k be k copies of the Koszul complex Ks• . Then one has for each κ = 1, . . . , k and for each  ∈ ℕ that ((Ks• )κ )

=

 '

E∗

. κ

138

3. RESIDUE CURRENTS: A BUNDLE APPROACH

For each  ∈ ℕ∗ , let k ν' κ +1 '

(

(ENs,k • ) =

E∗

κ

{ν∈ℕk : |ν|=−1} κ=1

(3.52)

(

=

k '

((Ks• )κ )νκ +1 .

{ν∈ℕk : |ν|=−1} κ=1

In particular, for  = 1, ∗ ∗ (ENs,k • )1 = (E )1 ∧ · · · ∧ (E )k . s,k The morphism Fs,k : (ENs,k • ) −→ (EN• )−1 is defined by

F1s,k = (s )1 ∧ · · · ∧ (s )k : (ENs,k • )1 −→ ℂ, and for any  ≥ 2, Fs,k =

k     (s )κ ν κ=1

|(ENs,k • )

ν≥2

.

Thus, we construct a generically exact complex ENs,k • (3.53)

s,k F +1

s,k F −1

F s,k

s,k · · · −→ (ENs,k • ) −→ (EN• )−1 −→ · · ·

F s,k

F s,k

3 2 s,k (ENs,k · · · −→ • )2 −→ (EN• )1 =

k '

F s,k

1 (E ∗ )κ −→ ℂ.

κ=1

Observe that again V sections

(ENk,s • )

= V (s) and that one has in terms of sheaves of

 k O( · , Im (F1s,k )) = s O( · , E ∗ ) ,

relating the Eagon–Northcott complex ENs,k to the kth power of the ideal sheaf • O( · , s E ∗ ). Example 3.27 (Locally free resolutions of coherent sheaves of O-modules). Let F be a coherent sheaf of O-modules; see §A.2.1. For every z ∈ X , one can associate to F (see Definition A.8) a free resolution for F over some neighborhood U of z. That is, one can associate to F a generically exact sequence of trivial vector bundles over U (3.54) FL−1

F

F

F

F

L 3 2 1 U × ℂrL−1 −→ · · · −→ U × ℂr2 −→ U × ℂr1 −→ U × ℂr 0 , 0−→ U × ℂrL −→

where L and the ranks r ,  = 0, . . . , L, the morphisms F ,  = 1, . . . , L depend on z and are such that the corresponding sequence of locally free sheaves of sections ⊕ (3.55) 0−→ OU

rL

L ⊕ −→ OU

F

rL−1

FL−1

3 ⊕ −→ · · · −→ OU

F

r2

2 ⊕ −→ OU

F

r1

1 ⊕ −→ OU ,

F

r0

where coker(F1 )  F|U , is exact. If the coherent sheaf F admits at the global level a finite, locally free resolution13 (3.56)

0 −→ LXrL −→ LXrL −1 −→ · · · −→ LXr2 −→ LXr1 −→ LXr0 fL

fL−1

f3

f2

f1

13 Such is the case, for example, when U = U is a convenient neighborhood of a holomorν phically convex compact subset taken from a sequence (Kν )ν≥1 of compact subsets exhausting a Stein manifold; see Proposition A.12.

3.3. BOCHNER–MARTINELLI CURRENTS AND GENERICALLY EXACT COMPLEXES 139

with coker(f1 )  F, then one can consider each locally free sheaf LXr with constant rank r as the sheaf of sections O(· , E ) of a holomorphic r -bundle E . Then, one can attach to the above finite free resolution for F the corresponding generically exact complex (3.46) with V (F• ) = Supp F. A particularly important case appears r when X is a Stein manifold and F = O⊕ /J , where J is a coherent subsheaf ⊕r ideals. In this setting, if one replaces X by an open subset Uν from an of O exhausting sequence, then one chooses the finite, locally free resolution (3.56) which r obeys the conditions LXr0 = O⊕ and Im f1 = J ; see Proposition A.12. Then the corresponding generically exact complex of trivial bundles is such that one r has V (F• ) = V (J ) = Supp (O⊕ /Im f1 ). When s ∈ O(X , E), with rank E = codim V (s) in addition, the Koszul complex Ks• (see Example 3.25) provides an example a locally free resolution of O/Is . If the sequence of locally free sheaves (see Example 3.26) of sections attached to the Eagon–Northcott complex ENs,k • happens to be exact, it provides a locally free resolution for O/(Is )k , as is the case when codim V (s) = rank (E). More generally, in polynomial algebra, Eagon– Northcott complexes are used to provide locally free resolutions for determinantal sheaves of ideals, when the associated sequence of locally free sheaves of sections attached to them is exact. Recall that a determinantal sheaf of ideals in polynomial algebra is defined by the minors of the maximal rank of a matrix with polynomial entries. 3.3.2. The currents RF• and P F• for a metrized complex F• . Let X be an N -dimensional complex manifold, let V be a closed analytic subset with pure codimension equal to M < N , let E be a holomorphic bundle, and let s ∈ O(X , E). Let F : ℂ → E be the morphism of bundles over X defined by ∀z ∈ X , ∀ t ∈ ℂz .

F (z)(t) = t s(z) ∈ Ez

Let also F0 be the quotient morphism from E to E/Im F . Then the short complex F

F

0 E/Im F −→ 0 Fs• : 0 −→ ℂ −→ E −→

is the most elementary model of the generically exact complex, with V (Fs• ) = V (s). Let | | be a smooth hermitian metric on E, which allows us to metrize the complex Fs• as Fs• . In this subsection we adapt the construction of the currents R|s | ([V ]) and P|s | ([V ]) described in Proposition 3.15 to the case of generically exact complex (F• , | |) of metrized holomorphic bundles. From the prototypical examples where F• = Fs• or F• = Ks• , M. Andersson and E. Wulcan in [And04] proposed the construction of Bochner–Martinelli type residue or principal value currents attached to a metrized generically exact complex F• . This construction will prove to be quite helpful when translating division problems in terms of multivariate residue calculus. We present it here, following essentially their reasoning, as it was presented in [AndW07, §2]. Let F

FL−1

F

F

F

L 2 1 0 (3.57) F• : 0 −→ EL −→ EL−1 −→ · · · −→ E1 −→ E0 −→ E−1 =

E0 −→ 0 Im F1

be a metrized generalized complex. It means that each entry E is equipped with a smooth hermitian metric | |, where the metric on E−1 is the quotient metric induced by the metric on E0 . Thus, E = (E , | |). For each  ∈ ℕ, we denote as ρ

140

3. RESIDUE CURRENTS: A BUNDLE APPROACH

the generic rank of F (see (3.47)) and let ρ

 '  ρ

• := det ΔF F ∈ O X , Homℂ (E , E−1 ) .  

(3.58)

Observe that such a definition ensures that14  • (3.59) V (F• ) = V (ΔF  ), ≥1

where V (F• ) is the smallest closed analytic subset of X such that the restriction of the complex F• to the open subset X \ V (F• ) is exact; see Definition 3.24. The with a metric motivates us to introduce fact that each bundle E is now equipped  the morphism σ ∈ Homℂ (E−1 , E ) |X \V (F• ) for each  ≥ 0. This morphism is defined as the minimal inverse of F for the choice of the hermitian structures E−1 and E . Namely, the minimality of F is expressed in terms the orthogonal splitting given by (E−1 )z = (Im F )z ⊕ ((Im F )⊥ )z , so that, for all z ∈ X \ V (F• ), (3.60) σ (z)(ξ)  0 when ξ ∈ ((Im F )⊥ )z   = η such that |η| = min{ˇη∈(E )z : F (z)(ˇη)=ξ} |ˇ η | when ξ ∈ Im F z .   Note that σ ∈ C ∞ X \ V (F• ), Homℂ (E−1 , E ) . Let

   (3.61) σ := σ ProjE −1 ∈ C ∞ X \ V (F• ), Homℂ (E(F• ), E(F• )) . ≥0

|X \V (F• )

One has, by construction, that σ is an odd smooth section of Homℂ (E(F• ), E(F• )) over X \ V (F• ) and that it satisfies F (σ) = σ ◦ F + F ◦ σ = IdE(F• ) , (3.62)

∂σ := ∂(σ) = ∂ ◦ σ + σ ◦ ∂, ∇F• (σ) = 2iπ IdE(F• ) − ∂σ, σ ∂σ = ∂σ σ

over X \ V (F• ). From the third equality in (3.62), one has that 1  σ

r−1 (∇F• (σ))−1 = ∂ 2iπ 2iπ r≥1  σ  −1 σ

r−1 =⇒ ∂ = σ ∇F• (σ) 2iπ 2iπ r≥1  σ σ

r−1 =⇒ ∂ ∇F• (σ) = σ 2iπ 2iπ r≥1

holds on X \ V (F• ). Next, one computes the action of ∇F• on both sides of the above equality using the fact that (∇F• )2 = 0. After right-multiplication by 14 Note

that V (Δ0 ) = ∅, since F0 is surjective.

3.3. BOCHNER–MARTINELLI CURRENTS AND GENERICALLY EXACT COMPLEXES 141

 −1 ∇F• (σ) of the results obtained, one gets the following important identity on X \ V (F• ):  σ σ

r−1 = IdE(F• ) . ∂ (3.63) ∇F• 2iπ 2iπ r≥1

For λ ∈ ℂ such that Re λ  1, let (3.64)

|ΔF• |2λ =



• 2λ IdE(F• ) . |ΔF | 

≥1

If E = Homℂ (E(F• ), E(F• )), then one introduces, motivated by the construction s,λ of the maps λ −→ R|s,λ | ([V ]) and λ −→ P| | ([V ]) in (3.21), the E-current valued maps (3.65)   σ σ

r−1  ∂ RF• ,λ ([V ]) : λ −→ (IdE(F• ) − |ΔF• |2λ )+ ∂ |ΔF• |2λ [V ], 2iπ 2iπ r≥1  σ σ

r−1 [V ], ∂ P F• ,λ ([V ]) : λ −→ |ΔF• |2λ 2iπ 2iπ r≥1

that are well defined for λ ∈ ℂ satisfying Re λ  1. The splitting15 over X \ V (F• )   σk ∂σk−1 · · · ∂σ+2 ∂σ+1  σ σ

r−1 ∂ = 2iπ 2iπ (2iπ)k− r≥1 ≥−1 k> (3.66)   ∂σk ∂σk−1 · · · ∂σ+2 σ+1 = , (2iπ)k− ≥−1

k>

implies that the E-current valued maps λ −→ RF• ,λ ([V ]) and λ −→ P F• ,λ ([V ]) split respectively as

  F ,λ   R,k• ([V ]) , RF• ,λ ([V ]) = IdE(F• ) − |ΔF• |2λ [V ] + (3.67)

P F• ,λ ([V ]) =

  ≥−1

F• ,λ P,k ([V

]) ,

≥−1

k>

k>

where

(3.68)

 σk ∂σk−1 · · · ∂σ+2 ∂σ+1  F• ,λ ([V ]) = ∂ |ΔF• |2λ [V ] R,k (2iπ)k−  ∂σk ∂σk−1 · · · ∂σ+2 σ+1  [V ] = ∂ |ΔF• |2λ (2iπ)k−   ∈  D(0,k−) X , Homℂ (E , Ek ) F• ,λ ([V ]) = |ΔF• |2λ P,k

σk ∂σk−1 · · · ∂σ+2 ∂σ+1 [V ] (2iπ)k−

∂σk ∂σk−1 · · · ∂σ+2 σ+1 [V ] (2iπ)k−   ∈  D(0,k−−1) X , Homℂ (E , Ek ) , = |ΔF• |2λ

15 The

equality comes from the fact that σ (∂σ) = (∂σ) σ.

142

3. RESIDUE CURRENTS: A BUNDLE APPROACH

for any k >  ≥ −1. For Re λ  1, it follows from (3.63) that (3.69)

 σ σ

r−1  [V ] + |ΔF• |2λ [V ] ∂ ∇F• P|F|• ,λ ([V ]) = ∇F• (|ΔF• |2λ ) 2iπ 2iπ r≥1  σ σ

r−1   [V ] + |ΔF• |2λ [V ] ∂ = − ∂ |ΔF• |2λ 2iπ 2iπ r≥1

= IdE(F• ) [V ] −

R|F•| ,λ ([V

]).

Taking the action of ∇F• on both sides and using the fact that ∇2F• = 0 leads to   (3.70) ∇F• R|F•| ,λ ([V ]) = 0 for Re λ  1. In the spirit of Proposition 3.15, we obtain the following result [AndW07]. Proposition 3.28. Let X , V , F• , E = Homℂ (E(F• ), E(F• )), λ −→ |ΔF• |2λ , σ be as above. For each k >  ≥ −1, the two Homℂ (E , Ek )-current valued maps F• ,λ λ −→ R,k ([V ]),

F• ,λ λ −→ P,k ([V ])

extend to ℂ as meromorphic maps with polar set in −ℚ+ . Their values at λ = 0 are Homℂ (E , Ek )-valued pseudo-meromorphic currents with bidegrees (M, M + (k − )) and (M, M + (k −  − 1)), respectively. Correspondingly, we denote these currents F• F• ([V ]) and P,k ([V ]). Moreover, the E-valued currents as R,k (3.71) RF• ([V ]) = IdE(F• ) 1V (F• ) [V ] +

  ≥−1

F• R,k ([V ])

k>

   σ σ

r−1  = IdE(F• ) 1V (F• ) [V ] + ∂ |ΔF• |2λ ∂ [V ], 2iπ 2iπ λ=0 r≥1

  F• P F• ([V ]) = P,k ([V ]) ≥−1

k>

 σ σ

r−1 

∂ [V ] = |ΔF• |2λ 2iπ 2iπ λ=0 r≥1

satisfy (3.72)

  ∇F• RF• ([V ]) = 0,

  ∇F• P F• ([V ]) = IdE(F• ) [V ] − RF• ([V ]).

One has in addition (3.73)

F• 1 ≤ k −  < codimV (V ∩ V (F• )) =⇒ R,k ([V ]) = 0.

Furthermore, for any , k with  ≥ −1 and k −  ≥ codimV (V ∩ V (F• )), for any z ∈ V ∩ V (F• ) and any f ∈ Oz ,     F• F• = df ∧ R,k = 0. (3.74) f = 0 on V ∩ V (F• ) around z =⇒ f R,k z z " • Proof. For each  ≥ −1, let VF• = k> V (ΔF k ) and π : V → V be a log resolution of V for the complex analytic subset Y of V defined as the union of V ∩ VF• with any hypersurface of V containing Vsing ; see Theorem A.41. Let

3.3. BOCHNER–MARTINELLI CURRENTS AND GENERICALLY EXACT COMPLEXES 143

a ∈ V ∩ VF• and z ∈ V be in the pre-image through ιV ◦ π of a neighborhood of a in X . One has in a local chart U = Uz around z that for any κ > ,  • (ιV ◦ π )∗ ΔF κ = sκ,0 Sκ ,  where sκ,0 is a holomorphic section of a line bundle  #ρκLκ and Sκ is a nonvanishing  ∗ section of the holomorphic bundle L−1 ⊗(ι ◦π ) Hom (E V  ℂ κ , Eκ−1 ) . A linear κ algebra argument (see [And06D, shows that sκ,0 (ιV ◦ π )∗ σκ is then  Lemma 4.1]) ∗ a smooth section ψκ of Homℂ Lκ ⊗ (ιV ◦ π ) Eκ−1 , Lκ ⊗ (ιV ◦ π )∗ Eκ . For each k >  consider now the maps 

 λ −→ (ιV ◦ π)∗ ∂|ΔF• |2λ σk ∂σk−1 · · · ∂σ+2 ∂σ+1 ,

λ −→ (ιV ◦ π)∗ |ΔF• |2λ σk ∂σk−1 · · · ∂σ+2 ∂σ+1 .

Since ψκ+1 ψκ = 0 for any κ > , as this holds outside the union of the hypersurfaces V (s0,κ ) for κ > , these maps equal  2λ ∂ (ιV ◦ π)∗ ΔF•  λ −→ ψk ∂ψk−1 · · · ∂ψ+2 ∂ψ+1 , s0,k · · · s0,+1   (ιV ◦ π)∗ ΔF• 2λ ψk ∂ψk−1 · · · ∂ψ+2 ∂ψ+1 , λ −→ s0,k · · · s0,+1 respectively. Then, it follows, as in the proof of Proposition 3.15, that these maps extend to ℂ as meromorphic maps with polar set contained in −ℚ+ and whose values at λ = 0 equal pseudo-meromorphic currents. The first assertion of the proposition then follows. Formulae (3.72) follow from (3.63) and (3.64) if one invokes the analytic continuation principle. Assertion (3.65) follows from the dimension principle (Proposition 2.26) taking into account the fact that the pseudoF• ([V ]) has bidegree (M, M + (k − )) and that its support meromorphic current R,k " F• equals V ∩ κ> Vκ ⊂ V ∩ V (F• ). One has the slightly more precise result

 F• (3.75) 1 ≤ k −  < codimV V ∩ VκF• =⇒ R,k ([V ]) = 0. κ>

Assertion (3.66) follows from Proposition 2.24, since RF• ([V ]) is a pseudo-meromorphic current supported by V ∩ V (F• ). Proposition 3.28 is thus completely proved.  Remark 3.29. If codimV (V ∩ V (F• )) > 0, then the complex of holomorphic bundles E0|V (F2 )|V (FL )|V (F1 )|V (F0 )|V 0 −→ (EL )|V −→ · · · −→ (E1 )|V −→ (E0 )|V −→ −→ 0 Im (F1 )|V can be interpreted as a generically exact complex of holomorphic bundles over the complex analytic space (V, OV ). The proof of Proposition 3.28 shows that the currents RF• ([V ]) and P F• ([V ]) can be interpreted as the direct images through ιV of pseudo-meromorphic currents on the complex analytic space (V, OV ), in accordance to Definition B.44 in §B.2.2.2. Therefore, the currents RF• |V and P F•|V satisfy (3.76)

∇VF RF• |V = 0,

∇VF P F•|V = IdE((F• )|V ) − RF• |V ,

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3. RESIDUE CURRENTS: A BUNDLE APPROACH

V where the action of the operator ∇ F• = 2iπF |V − ∂ V is now understood from    D V, Homℂ (E((F• )|V ), E((F• )|V )) into itself.

Example 3.30 (Koszul complex). Let E be a holomorphic bundle with rank m over X and s ∈ O(X , E). One can attach to s the generically exact Koszul complex Ks• ; see (3.51) in Example 3.25. Given a closed analytic subset V of X with pure dimension M , one can also attach to s and to the hermitian bundle E = (E, | |) the Bochner–Martinelli residue and principal value currents R|s | ([V ]), P|s | ([V ]), respectively, as in Proposition 3.15. The currents

s∗ r−1   s∗ R|s | ([V ]) = (1 − |s|2λ ) + (∂|s|2λ ) ∂ [V ], 2iπ|s|2 2iπ|s|2 λ=0 r≥1 (3.77)

s∗ r−1   s∗ P|s | ([V ]) = |s|2λ ∂ [V ] 2iπ|s|2 2iπ|s|2 λ=0 r≥1

can be re-interpretedas follows. For each  ≥1, the  #−1  smooth section of the metrized # E ∗ ,  E ∗ , | | , whose metric is induced by the holomorphic bundle Homℂ metric | | on E, defined by (3.78)

−1 '  s∗ (z)  s σ (z, ξ) = z, E∗ , ξ ∀ z ∈ X \ V (K ), ∀ ξ ∈ • 2 |s(z)| z

is the minimal inverse. This follows from (3.60) and the fact that  −1 ' '   ∗ E , E∗ (s ) ∈ O X , Homℂ

on X \ V (Ks• ). The currents R|s | ([V ]) and P|s | ([V ]), considered as acting by multiplication from E(Ks• ) into itself, define E(Ks• ) = Homℂ (E(Ks• ), E(Ks• ))-valued s s currents. These currents coincide with RK• ([V ]) and P K• ([V ]), where the Koszul s complex K• is equipped with the metrics induced by the original metric | | on E. In the particular case where codimV (V ∩ V (s)) = m, one has for any  ≥ 0,

(3.79)

 m ' ' RF• = Homℂ ( E ∗ , E ∗ )-component of

s∗ m−1  s∗ (∂|s|2λ ) ∂ [V ] = 0 unless  = 0. 2iπ|s|2 2iπ|s|2 λ=0

Example 3.31 (Eagon–Northcott complex). Let E = (E, | |) and s ∈ O(X , E) as in Example 3.30. Let ENs,k • be the Eagon–Northcott complex constructed from k copies (Ks• )κ, where κ = 1, . . . , k,of Ks• ; see Example 3.26. For each copy (Ks• )κ , let (σ)κ ∈ C ∞ X \ V (s), E((Ks• )κ ) be the minimal inverse of (s )κ , as defined in (3.78). For each κ = 1, . . . , k, let  (σ)κ (σ)κ

r−1 s   (3.80) Q(K• )κ = ∂ ∈ C ∞ X \ V (s), E (Ks• )κ . 2iπ 2iπ r≥1

Expression (3.80) is considered in terms of its action by multiplication from O(X \ V (s), E((Ks• )κ ) into itself as in Example 3.30. It follows from simple multilinear algebra (see [And06C, §3]) that, if one assumes codimV (V ∩ V (s)) > 0, then 

s s s,k [V ]. (3.81) REN• ([V ]) = (∂ |s|2λ ) Q(K• )1 · · · Q(K• )k λ=0

3.3. BOCHNER–MARTINELLI CURRENTS AND GENERICALLY EXACT COMPLEXES 145

In particular, one has that s,k

EN• ([V ]) = Homℂ (ℂ, ENs,k (3.82) R0,r r )-valued component of



k  1 2λ (∂ |s| ) (σ)κ (2iπ)r κ=1

 {ρ∈ℕk : |ρ|=r−1}

(σ)

ρk  (σ)

ρ1 1 k ∂ ··· ∂ [V ] 2iπ 2iπ λ=0

for any r ≥ 1. It follows from assertion (3.73) in Proposition 3.28 that such a current equals 0 as soon as r < codimV (V ∩ V (s)). Moreover, since the m coordinates of ∂σ (where σ denotes the minimal inverse of s) in a local frame for E|V in a neighborhood of z ∈ V are linearly dependent since s σ = 1 outside V (f ), one has s,k

EN• ([V ]) as soon as r > min(N − M, m). Therefore, in the particular case where R0,r codimV (V ∩ V (s)) = m, the only nontrivial currents from the list (3.82) are the ones where r = m.

3.3.3. Residual obstruction for exactness and duality theorems. We consider in this section a generically exact metrized complex of holomorphic bundles E0 −→ 0 Im F1 over an N -dimensional complex manifold X . For E(F• ) = Homℂ (E(F• ), E(F• )), let RF• = RF• ([X ]) and P F• = P F• ([X ]) be the E(F• )-valued currents introduced in Proposition 3.28. Their decompositions are described by

     F• F• R,k ([X ]) = R,k RF• , = R F• = FL−1

F

F

F

F

L 2 1 0 F• : 0 −→ EL −→ EL−1 −→ · · · −→ E1 −→ E0 −→ E−1 =

≥−1

(3.83) P

F•

=

k>

 

≥−1

k>

F• P,k ([X ])

≥−1

=

k>

 

≥−1

k>

F• P,k

≥−1

=



PF• .

≥−1

We formulate in this subsection Propositions 3.32 and 3.36 to illustrate the following important fact: the current RF• , through its components RF• ,  ≥ 0, materializes the obstruction for the associated complex of locally free sheaves of sections (3.84)

F

FL−1

F

F

L 2 1 0 −→ O(·, EL ) −→ OX (·, EL−1 ) −→ · · · −→ OX (·, E1 ) −→ OX (·, E0 )

to be exact. Namely, one has the following proposition. Proposition 3.32. Let  ≥ 0 and h ∈ O(X , E ) such that either F h = 0 when  ≥ 1   (3.85) or hz ∈ O(·, Im F1 ) z ∀ z ∈ X \ V (F• ) when  = 0. Then (3.86)

    RF• h = 0 =⇒ ∀ z ∈ X , hz ∈ O(·, Im F+1 ) z .

F• Conversely, in the case R+1 = 0,     (3.87) ∀ z ∈ X , hz ∈ O(·, Im F+1 ) z =⇒ RF• h = 0.

Proof. The proof of assertion (3.86) is inspired by the approach developed to establish the same claim in the univariate setting; see Proposition 1.29. Let us first prove (3.86), when  ≥ 1. Since h ∈ O(X , E ), RF• h = 0 ⇐⇒ RF• h = 0.

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3. RESIDUE CURRENTS: A BUNDLE APPROACH

It follows from (3.72) that (3.88)

∇F• (P F• h) = (∇F• P F• )(h) − P F• (∇F• h) 

= (IdE(F• ) − RF• )(h) − P F• (2iπF h − ∂h) = h.

uκ , with uκ ∈  D(0,κ−−1) (X , Eκ ). One has  2iπF+1 u+1 = h (2iπF − ∂) u = h =⇒ 2iπ Fκ uκ = ∂uκ−1 ∀ κ >  + 1.

Let u = P F h =

κ≥+1

Applying Dolbeault’s lemma in a neighborhood of any point z ∈ X , one solves locally the sequence of ∂ equations



∂qκ,z = uκ + 2iπ Fκ+1 qκ+1 , qκ,z ∈  D(0,κ−−2) (·, Eκ ) , z

z

taking into account that Fκ+1 = 0 for κ  1 and that any (0, N )-form is locally ∂-exact. Let az = (u+1 )z + 2iπ F+2 (q+2 )z . One has (F+1 a)z = hz , since F 2 = 0 and    (∂a)z = (∂u+1 )z − F+2 ∂q+2,z = 2iπF+2 u+2 − ∂q+2 z   = − 2iπ) (2iπF+2 F+3 q+3 z = 0, using again the property that F 2 = 0. This proves the assertion (3.86) in the case where  ≥1. Consider now the case where  = 0. Let h ∈ O(X , E0 ). We assume that hz ∈ O(·, Im F1 ) z in a neighborhood of any point z ∈ V (F• ). Relation (3.88) and the fact that F0 F1 = 0 (since F 2 = 0) imply that ∇F• (P F• (h)) = h. Therefore, the proof in the case  = 0 can be carried along the same lines as in the case  ≥ 1. In order to prove assertion (3.87), it is enough to prove that under the hypothesis F• = 0 the implication R+1   hz ∈ O(·, Im F+1 ) z =⇒ (h RF• )z = 0   holds for any point z ∈ X . If hz ∈ O(·, Im F+1 ) z , then there exists an element   az ∈ O(·, E+1 ) z satisfying hz = (∇F∗• (a))z . Then, one has   F•   F•         R h z = RF• h z = RF• (∇F• a) z = ∇F• RF• a z = ∇F• R+1 a z = 0, since ∇F• (RF• ) = 0 in view of (3.72).



Remark 3.33. If X is a Stein manifold, assertion (3.86) stands as follows: (3.89)

RF• h = 0 =⇒ h = F+1 a for some a ∈ O(X , E+1 ).

Example 3.34 (Koszul complex, Example 3.30 continued). An important particular case, where Proposition 3.32 applies, is that of the Koszul complex introduced in Example 3.30. If E = (E, | |) is an m-dimensional, hermitian holomorphic bundle over a complex manifold X , whose section s ∈ O(X , E) has its conjugate section equal to s∗ ∈ C ∞ (X , E ∗ ), then one has that the implication 

s∗ m−1 s∗ (3.90) h R0F• = (∂|s|2λ ) ∂ (h) = 0 2iπ|s|2 2iπ|s|2 |E0 (Ks• )=ℂ λ=0   =⇒ hz ∈ O(·, Im s ) z ∀ z ∈ X ,

3.3. BOCHNER–MARTINELLI CURRENTS AND GENERICALLY EXACT COMPLEXES 147

where s : E ∗ → ℂ is the interior multiplication by the section s, holds for any h ∈ O(X , E0 (Ks• )) = O(X ). Moreover, if codim V (s) = m = rank E, then it follows from (3.79) in Example 3.30, together with assertion (3.87) in Proposition 3.32, that



    (3.91) hz ∈ O(·, Im s ) z ∀ z ∈ X ⇐⇒ hz ∈ O(·, Im s ) z ∀ z ∈ V (s) 

s∗ m−1 s∗ ∂ (h) = 0. =⇒ h R0F• = (∂|s|2λ ) 2iπ|s|2 2iπ|s|2 |E0 (Ks• )=ℂ λ=0 Example 3.35 (Eagon–Northcott complex, Example 3.31 continued). Let X , E = (E, | |), and s ∈ O(X , E) be as in the previous example. Let k ∈ ℕ∗ . One has, for any h ∈ O(X , E0 (ENs,k • )) = O(X ), that



s,k s s (3.92) h R0EN• = (∂ |s|2λ ) Q(K• )1 · · · Q(K• )k (h) = 0 k,s |E0 (EN• )=ℂ λ=0

=⇒ hz ∈ (Is )kz , 16 wheres Is is the coherent sheaf of ideals attached to s.   The smooth sections (K• )κ s s Q of the vector bundle Homℂ E((K• )κ ), E((K• )κ ) over X \ V (s), defined for k copies of the Koszul complex Ks• , have been described in (3.81). s

The fact that the equality R1K• = 0 holds for any section s ∈ O(X , E) satisfying codim V (s) = rank E (see Example 3.34) suggests the following proposition. Proposition 3.36. Let E0 −→ 0 Im F1 be a metrized generically exact complex of holomorphic bundles over a complex N dimensional manifold (X , O). If the associated complex of locally free sheaves of sections FL−1

F

F

F

F

L 2 1 0 F• : 0 −→ EL −→ EL−1 −→ · · · −→ E1 −→ E0 −→ E−1 =

(3.93)

F

FL−1

F

F

L 2 1 O(·, EL−1 ) −→ · · · −→ O(·, E1 ) −→ O(·, E0 ) 0 −→ O(·, EL ) −→

is exact, then R1F• = 0. Proof. The exactness of the complex of locally free sheaves of sections (3.93) is equivalent to the so-called Buchsbaum–Eisenbud–Fitting conditions • ∀ κ > 0, codim V (ΔF κ ) ≥ κ,   #rank Fκ • Homℂ (Eκ , Eκ−1 ) have been defined for where the sections ΔF κ ∈ O X, any κ ∈ ℕ∗ in (3.58) (see Theorem A.18).17 It is pointed out in Remark A.19 that, under the hypothesis that the complex (3.93) of locally free sheaves is exact, the inclusion

(3.94)

F• F• • κ ≥ κ ≥ 1 =⇒ V (ΔF κ ) ⊂ V (Δκ ) ⊂ V (Δ1 ) = V (F• )   • holds. The decreasing sequence of closed analytic subsets V (ΔF κ ) κ≥1 depends only on the sheaf F = coker(F1 ), whose free resolution over X is given by (3.93).

(3.95)

hz ∈ (Is )z ⇐⇒ hz = s az for some az ∈ (O(·, E ∗ ))z . reader will also find the related references there. In any case, we invite him/her to consult §A.2.3 and §A.3.3 before reading the proof that follows. 16 Namely, 17 The

148

3. RESIDUE CURRENTS: A BUNDLE APPROACH

This follows from Theorem A.9 quoted in Appendix A. Relations (3.67) and (3.68) imply that  

∂σκ ∂σκ−1 · · · ∂σ3 σ2  F • 2λ (∂|ΔF R1F• = R1,κ = . 1 | ) (2iπ)κ−1 λ=0 κ≥2

κ≥2

The current

σ2  F• • 2λ R1,2 = (∂|ΔF 1 | ) 2iπ λ=0 • is pseudo-meromorphic, with bidegree (0, 1). It is supported by V (ΔF 2 ). The dimension principle for pseudo-meromorphic currents (see Proposition 2.26) and F• • the fact codim V (ΔF 2 ) ≥ 2 imply that R1,2 = 0. Let us now consider the pseudomeromorphic current

∂σ3 σ2  F• • 2λ = (∂|ΔF | ) . R1,3 1 2iπ λ=0 • In X \ V (ΔF 3 ), one has

F• (R1,3 )X \V (ΔF• ) = (∂σ3 )|X \V (ΔF• ) 3

3



• 2λ (∂|ΔF 1 | )

σ2  . (2iπ)2 λ=0 |X \V (ΔF3 • )

• The first factor (∂σ3 )|X \V (ΔF• ) is a smooth form in X \ V (ΔF 3 ). The second one, 3 namely



σ2  • 2λ , (∂|ΔF 1 | ) (2iπ)2 λ=0 |X \V (ΔF3 • ) • is also a pseudo-meromorphic current X \ V (ΔF 3 ) with bidegree (0, 1). Its support F• • is included in the closed analytic subset (X \V (ΔF 3 ))∩V (Δ2 ), whose codimension is at least 2. Then, the dimension principle implies that



σ2  • 2λ (∂|ΔF =0 1 | ) (2iπ)2 λ=0 |X \V (ΔF3 • )

F• F• • in X \ V (ΔF 3 ). But this means that Supp R1,3 ⊂ V (Δ3 ). Buchsbaum–Eisenbud– • Fitting conditions (3.94) imply for κ = 3 that codim V (ΔF 3 ) ≥ 3. Therefore, the F• dimension principle applies once more for the pseudo-meromorphic current R1,3 F• with bidegree (0, 2), and one concludes that R1,3 = 0. The procedure iterates from step κ ≥ 3 to step κ + 1, following the same reasoning, which relies on (3.94), (3.95), and the fact that all currents involved are pseudo-meromorphic, and F• =0 therefore conform to the dimension principle. Thus, we have shown that R1,κ

for any κ ≥ 2. But, this is equivalent to R1F• = 0. Proposition 3.36 is thus proved.  Remark 3.37. If the complex of locally free sheaves (3.93) is exact, it follows from (3.95) that (3.96)

• V (F• ) = V (ΔF 1 ) = {z ∈ X : rank F1 (z) < ρ1 } 2 4 O(·, E0 ) = z∈X : F = is not locally free at z . F1 O(·, E1 )

Repeating the proof of Proposition 3.36, which relies on (3.94), (3.95), and the fact that all currents appearing in the picture are pseudo-meromorphic, one can show,

3.3. BOCHNER–MARTINELLI CURRENTS AND GENERICALLY EXACT COMPLEXES 149

using the dimension principle, that (3.97)

R F• =





min(N,L)

RF• = R0F• =

F• R0,κ ,

• κ=codim V (ΔF 1 )

≥0

under the hypothesis in Proposition 3.36. Observe that in the particular case where F1 is generically surjective, one has O(·, E0 ) • . (3.98) V (F• ) = V (ΔF 1 ) = zero locus of the annihilator of F = F1 O(·, E1 ) We conclude the section by formulating its main result. Theorem 3.38 (Duality theorem). Let E0 −→ 0 Im F1 be a generically exact complex of metrized holomorphic bundles over a complex N dimensional manifold (X , O). Assume that the associated complex of locally free sheaves of sections (3.93) is exact. Then, for any h ∈ O(X , E0 ) satisfying   (3.99) hz ∈ O(·, Im F1 ) z ∀ z generic in X , F

FL−1

F

F

F

L 2 1 0 F• : 0 −→ EL −→ EL−1 −→ · · · −→ E1 −→ E0 −→ E−1 =

the following are equivalent:

   σ σ

r−1  (3.100) R0F• h = ∂ |ΔF• |2λ ∂ (h) = 0 2iπ 2iπ |E0 λ=0 r≥1   ⇐⇒ hz ∈ O(·, Im F1 ) z ∀ z ∈ X . Proof. The direct assertion (=⇒) follows from assertion (3.86) for  = 0 in Proposition 3.32. Since R1F• = 0, by Proposition 3.36, assertion (3.87) for  = 0 applies, which proves the converse assertion (⇐=). This concludes the proof of Theorem 3.38.  We point out that most important are the cases where generically exact complexes of bundles lead to explicitly tractable exact complexes of locally free sheaves of sections. Theorem 3.39 (Duality theorem for complete intersections). Let E = (E, | |) be a metrized holomorphic bundle over a complex manifold X . Let also the section s ∈ O(X , E) be such that codim V (s) = rank E. If Is is the sheaf of ideals attached to the section s, then for any k ∈ ℕ∗ and any h ∈ O(X ) one has that s,k   k  (3.101) hz ∈ Is z ∀ z ∈ V (s) ⇐⇒ R0EN• h = 0. Furthermore, if rank E = m, codim V (s) = m and there are Cartier divisors D1 , . . . , Dm on X such that s = s1 ⊕ · · · ⊕ sm , sj ∈ O(X , O(Dj )), then one has that (3.102)   k hz ∈ Is z

sκ    ∀ z ∈ V (s) ⇐⇒ R0K• h = 0 ∀ κ ∈ (ℕ∗ )m with |κ| = k + m − 1   1 ⇐⇒ h ∂ κ = 0 ∀ κ ∈ (ℕ∗ )m with |κ| = k + m − 1 s for any k ∈ ℕ∗ , where for any multi-index κ ∈ (ℕ∗ )m we used the notation

sκ = sκ1 1 ⊕ · · · ⊕ sκmm ,

sκ = (sκ1 1 , . . . , sκmm ).

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3. RESIDUE CURRENTS: A BUNDLE APPROACH

Proof. Observe that Is is locally an example of a determinantal ideal sheaf in a neighborhood of any z ∈ V (s) [EaN62]. Thus, the generically exact complex of sheaves associated with the Eagon–Northcott complex ENs,k • , as in (3.93), is exact. Then Theorem 3.38 applies. The second equivalence in (3.102) follows from Proposition 3.18. The first equivalence in (3.102) follows from the fact that if codim V (s1 ⊕ · · · ⊕ sm ) = m, then, for any k ∈ ℕ∗ and for any z ∈ V (s1 ⊕ · · · ⊕ sm ), one has 3  k Is z = (Isκ )z . {κ∈(ℕ∗ )m : |κ|=k+m−1}

This follows, for example, by the now classical algebraic argument due to Melvin Hochster found in [LipT81, §3]. It remains to apply assertion (3.87) in Proposition κ 3.36 when F• = Ks• and  = 0, which is licit in our case according to (3.79) in Example 3.30.  Our last proposition in this section concerns the Cohen–Macaulay situation18 and Coleff–Herrera sheaves of bundle-valued currents which have been introduced previously in Definition 2.33 and Remark 2.36. Proposition 3.40. Let X and F• be as in Theorem 3.38, where F1 is generically surjective and L = codim V (F• ) = M . Then   F• ∈ CH0X ,V [M ] (F• ) X , Homℂ (E0 (F• ), EM (F• ) , (3.103) RF• = R0F• = R0,M where V [M ] (F• ) denotes the union of irreducible components of V with codimension M = L. Proof. The two first equalities follow from assertion (3.97) stated in Remark 3.37. The fact that RF• is ∂-closed follows from the fact that ∇F• RF• = 0 (see (3.72)) combined with the fact that F• with M = L =⇒ F RF• = 0. RF• = R0,M

The current RF• is an E(F• )-valued current (in fact a Homℂ (E0 (F ), EM (F• ))valued current) with bidegree (0, M ). Thus RF• is supported by the union V [M ] (F• ) of irreducible components of V (F• ) with codimension M , according to the dimension principle for pseudo-meromorphic currents; see Proposition 2.26. It is then a Coleff–Herrera current with respect to V [M ] (F• ) according to Proposition 2.24 and Corollary 2.27.  3.4. Bochner–Martinelli residue currents: structural results 3.4.1. Decomposition of R|s | ([V ]) along distinguished varieties. In this section, we consider a purely n-dimensional complex analytic space (V, OV ), together with its embedding ιV : V → X in some ambient N = n + M -dimensional complex manifold X equipped with its structural sheaf O. The structural sheaf OV equals the quotient sheaf OV /IV , where (IV )z = {h ∈ Oz : h = 0 on Vz }.19 More 18 We refer the reader to Definition A.21 for the notion of the Cohen–Macaulay (coherent) sheaf, which we will revisit in the forthcoming §3.4.2, §3.4.3, and 3.4.4. 19 Important examples for us will be the cases where X = ℂN or ℙN and V is a purely ℂ n-dimensional algebraic affine or projective subvariety. Naturally, V is defined as the set of common zeroes of a finite collection of polynomials p ∈ ℂ[X1 , . . . , XN ] or homogeneous polynomials P in ℂ[X0 , . . . , XN ], expressed in affine coordinates z1 , . . . , zN or homogeneous coordinates

3.4. BOCHNER–MARTINELLI RESIDUE CURRENTS: STRUCTURAL RESULTS

151

specifically, we focus in this section on a metrized holomorphic bundle E over X and its holomorphic section s # ∈ O(X , E) satisfying codimV (V ∩ V (s)) > 0. As pointed in Definition 3.17, the E ∗ -valued current R|s | ([V ]), whose definition and # ∗ properties have been listed in Proposition 3.15, can be also considered a ( E|V )valued current R|s,V | on the (embedded) complex analytic space (V, OV ). We denote by ∗ ) J = s|V OV ( · , E|V

(3.104)

the coherent sheaf of OV ideals on the complex analytic space (V, OV ) attached to s|V ∈ OV (V, E|V ). Let us first recall the following definition; see also Definition A.35 together with Remark A.38 in §A.4.2. Definition 3.41 (Distinguished varieties for J ). Let πJ : V& → Vbe the normalized blowup of V along the coherent ideal sheaf J and let E (J ) = γ mγ {Hγ } be the Weil (exceptional) divisor on the normal complex analytic space V& , which is associated20 to the principal ideal sheaf J OV . For any r = 0, . . . , n, let Vr (J ) (respectively Hr (J )) be the set of irreducible closed analytic subsets of V (respectively irreducible hypersurfaces H of V& ) defined by (3.105) Vr (J ) = {& πJ (Hγ ) : codimV (& πJ (Hγ )) = r} = {& πJ (Hγ ) : Hγ ∈ Hr (J )}. The family Vr (J ), whose indexation by γ is highly redundant in general, is called the collection of distinguished varieties with codimension r for the coherent subsheaf of OV -ideals J . We will denote by Vr (J ) the closed analytic subset of V with codimension r defined as  (3.106) Vr (J ) = V. V ∈Vr (J )

The current R|s,V | admits then the following decomposition. Proposition 3.42. Let (X , O), (V, OV ) be as above. Assume also that the embedding ιV : V → X , the vector bundle E = (E, | |), with | | smooth and the section s ∈ O(X , E) are as above. Then, the Bochner–Martinelli residue current R|s,V | is decomposed, as a bundle-valued current on V , in the following manner R|s,V| =

(3.107) S|s,V |,r

N|s,V |,r

#r

n  s,V (S|s,V |,r + N| |,r ), r=1

∗ E|V

where and are -valued pseudo-meromorphic currents with bidegree (0, r), whose supports satisfy  (3.108) Supp S|s,V Supp N|s,V Vρ (J ). |,r ⊂ Vr (J ), |,r ⊂ 1≤ρ codimV W ≥ r. 1≤ρ 0 on V (𝕁) ∩ ∂(R 𝔹N ), there exists η > 0 such that for any t ∈ η 𝔹n , V (𝕁) ∩ V (p − t) ∩ B(0, R) ⊂ interior(Supp χ). Moreover, it follows from Stokes’s theorem and Proposition 3.18 that for any holomorphic function h in R 𝔹N , for any t ∈ η 𝔹N with 0 < η ) 1, n    σ σ n−1  1 ' t t , χ h dζ = ∂ ([V (𝕁)]) , ∂χ ∧ hdζ j , (6.116) ∂ p − t V (𝕁) 2iπ 2iπ j=1 where σ t : (z, w) ∈ ℂN \ V (p − t) −→

n   pj (z, w) − tj ∗  j = σt,1 (z, w), . . . , σt,n (z, w) . 2 p(z, w) − t j=1

Let also, for (z, w) in a neighborhood of Supp(∂χ) ∩ V (𝕁), ˇ t (z, w) = σ (6.117) =

n 

pj (z, w)

j=1 ⟪p(z, w), p(z, w) ∞ 

⟪

k=0

− t⟫

  ˇt,1 (z, w), . . . , σ ˇt,n (z, w) ∗j = σ

n p(z, w) k  pj (z, w) , t⟫ ∗ . 2 j p(z, w)2 p(z, w) j=1

Since σ t (z, w)

n 

n 

ˇ t (z, w) (pj (z, w) − t) j = σ (pj (z, w) − t) j ≡ 1

j=1

j=1

for (z, w) in a neighborhood of Supp(∂χ) ∩ V (𝕁) and t ∈ η 𝔹n , a classical cohomological deformation argument based on Stokes’s formula on the complex analytic space V (𝕁) → ℂN ensures that one also has, as well as (6.116), that n   σ  1 ' ˇ t n−1 ˇt σ , χ h dζ = ∂ ([V (𝕁)]) , ∂χ ∧ hdζ j . (6.118) ∂ p − t V (𝕁) 2iπ 2iπ j=1 After some easy combinatorial computations, one has for t ∈ η 𝔹n that  1  ∂ , χ h dζ p − t V (𝕁)  n − 1 + |κ| = n − 1, κ κ∈ℕn (6.119) n   ' σ σ n−1 × σκ ∂ ([V (𝕁)]) , ∂χ ∧ h dζ j tκ 2iπ 2iπ j=1  κ = aχ,κ [h dζ] t κ∈ℕn n as a convergent power series in t in Oℂn (η 𝔹n ). Let V = V (𝕁) × ℂn ⊂ ℂℕ z,w × ℂt n and θ ∈ D(ℂt , ℂ) identically equal to one in a neighborhood of t = 0. It follows

286

6. RESIDUE CALCULUS AND TRACE FORMULAE

from the geometric transformation law (see Theorem 3.21) together with Theorem 2.75 that for each κ ∈ ℕn ,

  1 aχ,κ [h dζ] = ∂ , h χ θ dζ ∧ dt p − t, tκ1 1 +1 , . . . , tnκn +1 V  

1 κ , h t χ θ dζ ∧ dt = ∂ κ +1 (6.120) κ +1 κ +1 κ +1 V p 1 , . . . , pnn , t1 1 , . . . , tnn 

 1 = ∂ κ +1 , h dζ ∧ dt . p 1 , . . . , pnκn +1 V |κ|

The second equality in (6.115) then holds for any κ, with h = (D [p0 ]q)|R 𝔹N . To prove the first equality, let us introduce univariate polynomials φ1 , . . . , φn with coefficients in 𝕂 such that n  φj (Xj ) − aj,k pk ∈ 𝕁, aj,k ∈ 𝕂[X, Y ], deg(aj,k pk ) ≤ (deg 𝕏) D; k=1

see [Jel05, Theorem 1.6]. Let us start with ) * ) * ¯ p+1 ,κ q¯ dX/¯ 0 Res𝕂𝕁 [X,Y ]/𝕂 κ1 +1 = Res𝕂𝕁 [X,Y ]/𝕂 κ1 +1 , p¯1 , . . . , p¯nκn +1 p¯1 , . . . , p¯nκn +1 |κ|

¯ see Definition 6.32. It follows from Theorem 6.24 and where ,κ = q¯ D [¯ p0 ] dX; Remark 6.25 (see (6.46)) that ) *  (6.121) κ! Res𝕂𝕁 [X,Y ]/𝕂 κ1 +1 ,κ κn +1 p¯1 , . . . , p¯n ) *  ¯ Aκν ,κ (det A) = ν! Res𝕂𝕁 [X,Y ]/𝕂 ν1 +1 ¯ ¯n) , φ1 (X1 ), . . . , φνnn +1 (X n {ν∈ℕ : |ν|=|κ|}

where the Aκν are the classes modulo 𝕁 of the polynomials Aκν defined as in (6.47). Let us prove first that one also has in parallel that 

 1 (6.122) κ! ∂ κ1 +1 , ω ,κ p1 , . . . , pnκn +1 V (𝕁)

   1 = ν! ∂ ν1 +1 , (det A) Aκν ω,κ , νn +1 φ1 (ζ1 ), . . . , φn (ζn ) V (𝕁) {ν∈ℕn : |ν|=|κ|} |κ|

where ω,κ (z, w) = qD [p0 ](z, w) dz. In order to prove (6.121), we introduce (6.123) Σ : (z, w) ∈ ℂN \ V (φ) n n  k=1 aj,k (z, w) φk (zk ) ∗ j = (Σ1 (z, w), . . . , Σn (z, w)), −→ φ(z)2 j=1 so that, for any (z, w) ∈ V (𝕁) \ φ−1 ({0}), n n 



pj (z, w) j = σ(z, w) pj (z, w) j ≡ 1. (6.124) Σ(z, w) j=1

j=1

Let χ0 and χ1 in D(ℂ , ℝ) be both identically equal to one in a neighborhood of V (𝕁) ∩ φ−1 ({0}), such that Supp (∂χ0 ) ∩ Supp (∂χ1 ) = ∅. For any ψ ∈ D(ℂN , ℂ) N

6.5. THE CHARACTERISTIC ZERO CASE

287

which is identically equal to 1 in a neighborhood of Supp (∂χ0 ) and identically equal to 0 in a neighborhood of Supp (∂χ1 ), the (n, n − 1) form Θ=

n   κ ψ σ + (1 − ψ) Σ ψσ + (1 − ψ)Σ

n−1 ψ σj + (1 − ψ)Σj j ∂ 2iπ 2iπ j=1

induces a closed form on the open subset V (𝕁) \ {0} of the n-dimensional complex analytic space V (𝕁) → ℂN . Stokes’s theorem on V (𝕁) therefore implies   (6.125) ∂χ0 ∧ Θ = ∂χ1 ∧ Θ. V (𝕁)

V (𝕁

Since the left-hand side of (6.122) is independent on χ, if follows from the second equality in (6.115) and from (6.125) that  (6.126) κ! ∂

 1 , ω ,κ p1κ1 +1 , . . . , pnκn +1 V (𝕁) n  ' (n − 1 + k)!  κ Σ Σ n−1 = Σ ∂ ([V (𝕁)]) , ∂χ ∧ ω,κ j . (n − 1)! 2iπ 2iπ j=1

Since Σκ (z, w) =

n  n  j=1

aj,k (z, w)

j=1

φj (zj ) κj = φ(z)2



Aκν

{ν∈ℕn : |ν|=|κ|}

n  φj (zj ) νj φ(z)2 j=1

according to the definition of the Aκν (see (6.47)), equality (6.122) follows. Given the similarity of computation rules (6.121) and (6.122), proving the first equality in (6.115) amounts to proving that for any polynomials u(X1 ), . . . , u(Xn ), v(X, Y ) in 𝕂[X, Y ],35 one has (6.127) *  )

 ¯ 1 v¯ dX Res𝕂𝕁 [X,Y ]/𝕂 ∂ , v(ζ) dζ . = ¯ 1 ), . . . , un (X ¯n) u1 ( X u1 (ζ1 ), . . . , un (ζn ) V (𝕁) Let R = (R1 , . . . , Rn ) ∈ (ℝ+ )n , such that the polydisc Δ = R1 𝔻 × · · · × Rn 𝔻 of ℂn contains {α ∈ ℂn : u1 (α1 ) = · · · = un (αn ) = 0} and is such that the support of its Shilov boundary Γ(Δ) does not intersect the discriminant locus of the proper projection π : (z, w) ∈ V (𝕁) −→ z ∈ ℂn . It follows from Stokes’s formula on V (𝕁), as in the proof of Proposition 6.8, that  n

  ' dζj 1 1 . (6.128) ∂ , v(ζ) dζ = v u1 (ζ1 ), . . . , un (ζn ) V (𝕁) (2iπ)n π∗ (Γ(Δ)) j=1 uj (ζj ) One has in 𝕂[t] (after specification of t on an element of ℂn ) the equality * )

   ¯ 1 v¯ dX = ∂ , v dζ = v(ξ). ¯ −t X ζ1 − t1 , . . . , ζn − tn V (𝕁) 𝕏 −1 ξ∈π

35 Let

({t})

us say, for example, given ν ∈ ℕn such that |ν| = |κ|, ν +1

uj (Xj ) = φj j

(Xj ) for j = 1, . . . , n,

|κ|

v = qAκ ν D [p0 ](X, Y ).

288

6. RESIDUE CALCULUS AND TRACE FORMULAE

Then, one has for any κ ∈ ℕn that ) *  

¯ 1 v¯ dX , v dζ = Res (6.129) ∂ κ1 +1 𝕂𝕁 [X,Y ]/𝕂 ¯ κ1 +1 , . . . , X ¯ nκn +1 . X ζ1 , . . . , ζnκn +1 V (𝕁) 1 Let

 1 1 = n cκ ζ −κ deg u j u (ζ ) j j ζ j=1 κ∈ℕn j=1 j n be the convergent Laurent development of 1/ j=1 uj (ζj ) on the support of the cycle π ∗ (Γ(Δ)). One has then the (finite, since v is a polynomial function) expansion  n 

  ' 1 dζj 1 v c ∂ , v dζ = κ (2iπ)n π∗ (Γ(Δ)) j=1 uj (ζj ) ζ1deg u1 +κ1 , . . . , ζndeg un +κn V (𝕁) κ∈ℕn ) *  ¯ v¯ dX = cκ Res𝕂𝕁 [X,Y ]/𝕂 ¯ deg u1 +κ1 ¯ ndeg un +κn , X1 ,...,X n n

κ∈ℕ

where the second equality follows from (6.129). Since ) *  ¯ v¯ dX cκ Res𝕂𝕁 [X,Y ]/𝕂 ¯ deg u1 +κ1 ¯ ndeg un +κn X1 ,...,X n κ∈ℕ

* ¯ v¯ dX ¯ 1 ), . . . , un (X ¯n) u1 ( X

)

= Res𝕂𝕁 [X,Y ]/𝕂

as a consequence (for example) of Proposition 6.44, equality (6.127) is proved, which concludes the proof of Theorem 6.49.  Remark 6.50. Since n n

' ' σ σ n−1 (n − 1)!  j−1  ∂ = (−1) σ ∂σ ∗j , j j 2iπ 2iπ (2iπ)n j=1  j=1 j =j

one can also rewrite the second equality in (6.115) as (6.130) ) *   ¯ p+1 n − 1 + |κ| q¯ dX/¯ |κ| 0 ∂χ ∧ σ κ Ω ∧ (D [p0 ]q) dζ, =− Res𝕂𝕁 [X,Y ]/𝕂 κ1 +1 p¯1 , . . . , p¯nκn +1 n − 1, κ V (𝕁) where (6.131)

n

' (−1)n(n−1)/2 (n − 1)!  j−1  . (−1) σ ∂σ Ω= j j (2iπ)n  j=1 j =j

Let us mention here an important corollary of this result, also known as the Euler–Jacobi residue formula, already mentioned in the context of multivariate algebraic residue calculus over 𝕂[X] instead of 𝕂𝕁 [X, Y ]; see Remark 6.42 (see also [KK87]). Corollary 6.51. Let 𝕂 ⊂ ℂ, 𝕁 ⊂ 𝕂[X, Y ], 𝕏 = Spec 𝕂𝕁 [X, Y ]. Let also p1 , . . . , p¯n ) is a quasi-regular sequence in p1 , . . . , pn ∈ 𝕂[X, Y ] be such that p¯ = (¯ 𝔸 = 𝕂𝕁 [X, Y ], with 𝕀p¯ = 𝔸. Let P1 , . . . , Pn be the homogenizations of the pj’s in 𝕂[X0 , X, Y ] and let 𝕏ℂ be the Zariski closure of 𝕏ℂ in ℙN ℂ . Suppose that (6.132)

𝕏ℂ ∩ V (P1 ) ∩ · · · ∩ V (Pn ) = ∅.

6.5. THE CHARACTERISTIC ZERO CASE

289

Then, for any q ∈ 𝕂[X, Y ] and any subsets I ⊂ {1, . . . , n}, J ⊂ {1, . . . , N − n} whose sum of cardinals equals n, ) * n  ¯ Y¯ ) dX ¯ I ∧ dY¯J q(X, deg pj − n =⇒ Res𝕂𝕁 [X,Y ]/𝕂 (6.133) deg q < = 0. p¯1 , . . . , p¯n j=1

Proof. Let dj = deg pj . The homogenizations P1 , . . . , Pn of the polynomials p1 , . . . , pn induce holomorphic sections respectively of the holomorphic bundles (dj ) over ℙN OℙN ℂ . Consider the homogeneous coordinates ℂ [z0 : Z1 : · · · : ZN ], Zj = z0 zj for j = 1, . . . , n, Zn+j = z0 wj for j = 1, . . . , N − n, on ℙN ℂ . Since P1 , . . . , Pn do not share any common zero on V (𝕁) ∩ {z0 = 0}, one has

1 Supp ∂ ⊂ V (𝕁) ∩ V (p), P1 , . . . , Pn V (𝕁) where the Coleff–Herrera current on the left-hand side was introduced in Definition 2.78. The condition (6.133) on the degrees ensures that the rational n-form q(Z1 /z0 , . . . , Z0 /z0 )

n '

d(Zj /z0 ) p (Z1 /z0 , . . . , ZN /z0 ) j=1 j

does not have the hyperplane {z0 = 0} as polar hyperplane in ℙN ℂ . Then the assertion results from Stokes’s theorem on the complex analytic space V (𝕁), together with the first equality in (6.115), since one can find a system of coordinates (z  , w ) subset K ⊂ {1, . . . , N } with cardinal n, the cardinal of the in ℂN such that for any 5  intersection of 𝕏ℂ with j∈K {zk = 0} equals deg 𝕏. Remark 6.52. Suppose that the geometric condition (6.112) is fulfilled. Take q = X κ Y χ , I = {1, . . . , n}, J = ∅, where κ ∈ ℕn and χ ∈ ℕN −n . If the conditions in Corollary 6.51 are satisfied, then one has ) κ χ * ¯ Y¯ dX ¯ X (6.134) deg ≤ |κ| + |χ| ¯ −t X 𝕏 instead of the upper estimate by |κ| + |χ| deg 𝕏 which was provided by Proposition 6.43. 6.5.2. Properness and algebraic residue symbols in characteristic 0. Let (p1 , . . . , pn ) be a sequence of polynomials in 𝕂[X, Y ] whose classes p¯j modulo the purely n-dimensional radical ideal 𝕁 define a quasi-regular sequence p¯ in 𝕂𝕁 [X, Y ]  ojasiewicz such that 𝕀p¯ = 𝕂𝕁 [X, Y ]. Following the ideas introduced by Stanislas L [Lo59, Lo, JKS92] (see also [Boc75]), we introduce the following definition. Definition 6.53. Let δj , j = 1, . . . , n be nonnegative rational numbers. The sequence p is said to admit δ as L  ojasiewicz multi-exponent for the algebraic variety 𝕏 = Spec 𝕂𝕁 [X, Y ] if and only if for any ε = (ε1 , . . . , εn ), where 0 < εj ) 1 for j = 1, . . . , n, there exist strictly positive constants Cε and cε such that (6.135) (z, w) ∈ 𝕏ℂ and max(|z|, |w|) ≥ C |pj (z, w)| (max(|z|, |w|))δj +εj ≥ cε , 1≤j≤n (max(|z|, |w|))deg pj

=⇒ max

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6. RESIDUE CALCULUS AND TRACE FORMULAE

where |z| = max(|z1 |, . . . , |zn |), |w| = max(|w1 |, . . . , |wN −n |). Such a L  ojasiewicz multi-exponent is called a multi-index of properness for 𝕏 in the particular case when δj < deg pj for j = 1, . . . , n. Remark 6.54. Definition 6.53 holds when 𝕂 is an arbitrary field of any characteristic, ℂ being replaced by an algebraic closure 𝕂, 𝕏ℂ by 𝕏𝕂 , and | | by a nontrivial archimidean absolute value on 𝕂, as in §6.4.1. The notion of a p-admissible L  ojasiewicz multi-exponent with respect to 𝕏 extends then the notion of a p-admissible (δ1 , . . . , δn ) properness multi-exponent such as introduced in Definition 6.38. Proposition 6.55. Let 𝕂 be a field with arbitrary characteristic, let 𝕁 ⊂ 𝕂[X, Y ] be purely n-dimensional and radical, and let 𝕏 = Spec 𝕂𝕁 [X, Y ], p1 , . . . , pn ∈ 𝕂[X, Y ]  as above. Let deg 𝕏 = deg 𝕏𝕂 and D = nj=1 deg pj . Then   (6.136) (δ1 , . . . , δn )◦ = D deg 𝕏, . . . , D deg 𝕏 is a L  ojasiewicz multi-exponent for p with respect to the algebraic variety 𝕏. Moren over p induces a proper polynomial morphism from 𝕏𝕂 to 𝕂 if and only if it admits any (δ1 , . . . , δn ) with δj < deg pj for j = 1, . . . , n, such that ⟪κ, d⟫ ≤ D deg 𝕏 =⇒ ⟪κ, d − δ⟫ < 1

(6.137)

as a multi-index exponent of properness for 𝕏. Proof. Up to a 𝕂-linear change of coordinates, one may assume that the geometric hypothesis (6.68) holds. The first assertion follows from the fact that there exist univariate polynomials φ1 (X1 ), . . . , φn (Xn ) such that, for j = 1, . . . , n, φj (Xj ) −

n 

aj,k pk ∈ 𝕁 with aj,k ∈ 𝕂[X, Y ],

k=1

max deg(aj,k pk ) ≤ D deg 𝕏; k

see [Jel05, Theorem 3.8]. Therefore, one has (x, y) ∈ 𝕏𝕂

and

|x| ≥ C

 =⇒ c|φ(x)| ≤ max 1 + max(|x|, |y|))D deg 𝕏−deg pj |pj (x, y)| 1≤j≤n

for some C > 0 and c > 0, which implies that (δ1 , . . . , δn )◦ is a L  ojasiewicz exponent. The second assertion follows from the fact that p¯1 , . . . , p¯n are 𝕂-algebraically independent in 𝕂𝕁 [X, Y ] (since p¯ is quasi-regular such that 𝕀p¯ = 𝕂𝕁 [X, Y ]). Then, n if p realizes a proper polynomial morphism from 𝕏𝕂 to 𝕂 , then there exists for  νj uj,ρ (t) τ νj −ρ with any j = 1, . . . , n, a polynomial Ej (t, τ ) = τ νj + ρ=1 degt Ej (tdeg p1 , . . . , tdeg pn , t) ≤ D deg 𝕏,

Ej (p1 , . . . , pn , Xj ) ∈ 𝕁;

see [Jel05, Theorem 3.3]. Then the second claim of the proposition is proved similarly to Proposition 6.39.  If 𝕂 ⊂ ℂ, the following important result generalizes Corollary 6.51; see [BVY05, Proposition 4.1]. Theorem 6.56. Let 𝕂 ⊂ ℂ, 𝕁 ⊂ 𝕂[X, Y ], p1 , . . . , pn ∈ 𝕂[X, Y ] such that p¯ is a quasi-regular sequence in 𝕂𝕁 [X, Y ] with 𝕀p¯ = 𝕂𝕁 [X, Y ]. Suppose that p admits a multi-exponent of properness (δ1 , . . . , δn ) with respect to 𝕏. Then, for any q in

6.5. THE CHARACTERISTIC ZERO CASE

291

𝕂[X, Y ] and any subsets I ⊂ {1, . . . , n}, J ⊂ {1, . . . , N − n} whose sum of cardinals equals n, ) * n  ¯ Y¯ ) dX ¯ I ∧ dY¯J q(X, (6.138) deg q < (dj − δj ) − n =⇒ Res𝕂𝕁 [X,Y ]/𝕂 = 0. p¯1 , . . . , p¯n j=1

Moreover the element in 𝕂[t] defined as in Proposition 6.48 and (6.111) equals ) * ) *  ¯ ¯ q¯ dX q¯ dX κ (6.139) = Res𝕂𝕁 [X,Y ]/𝕂 κ1 +1 κn +1 t , p¯ − t 𝕏 , . . . , p ¯ p ¯ n 1 d ¯ κ∈Eδ [¯ q dX]

where, if d = (deg p1 , . . . , deg pn ) and |d − δ| = |d| − |δ|, 2 4 ¯ = κ ∈ ℕn : ⟪κ , d − δ⟫ + |d − δ| ≤ deg q + n . (6.140) Eδd [¯ q dX] Proof. Let us explain why the first assertion (6.138) implies the second one (6.139). Let χ ∈ D(ℂn , ℂ) be identically equal to one in a neighborhood of 𝕏ℂ ∩ V (p). Since p defines a proper polynomial morphism from 𝕏ℂ to ℂn , it follows from Theorem 6.49 and specification of parameters t as complex numbers in η 𝔹n with 0 < η ) 1 that one has in ℂ[[t1 , . . . , tn ]] ) * ) *

  ¯ ¯ 1 q¯ dX q¯ dX = = ∂ , q χ dζ p¯ − t 𝕏 p¯ − t 𝕏 p1 − t1 , . . . , pn − tn V (𝕁) ℂ

   1 ∂ κ1 +1 , q χ dζ tκ = κn +1 V (𝕁) p , . . . , p n n 1 κ∈ℕ ) * (6.141)  ¯ q¯ dX = Res ℂ𝕁 [X,Y ]/ℂ κ1 +1 tκ , . . . , p¯nκn +1 p¯1 n κ∈ℕ ) *  ¯ q¯ dX = Res 𝕂𝕁 [X,Y ]/𝕂 κ1 +1 tκ . , . . . , p¯nκn +1 p¯1 n κ∈ℕ

It follows from (6.138) that the development on the right-hand side of (6.141) ¯ defined as (6.140). Proving (6.138) amounts to q dX] truncates as (6.139), with Eδd [¯ proving it when I = {1, . . . , n}, J = ∅ and condition (6.112) is satisfied, since, up to a 𝕂-linear change or coordinates, one can always assume that 2 4  Z = (Z1 , . . . , Zn ) ∈ 𝕏ℂ ⊂ 𝕂N : Zi = · · · = Zi = 0  = deg 𝕏 1

n

for any 1 ≤ i1 < · · · < in ≤ N . It is also clear that one may replace δj by δj + 1/ν with ν ∈ ℕ∗ arbitrarily large and then suppose that for some C > 0 and c > 0, one has (6.142) |pj (z, w)| (z, w) ∈ V (𝕁) and max(|z|, |w|) ≥ C =⇒ max ≥ c. 1≤j≤n (max(|z|, |w|))deg pj −δj The proof we give here is inspired by that presented in [Z05, Theorem 1] when n = N . Let, for j = 1, . . . , n,  mj , mj , m ∈ ℕ∗ with mj ≥ 3. (deg pj − δj ) = m  j =j

Denote m = (m1 , . . . , mn ), M = m

n 

(deg pj − δj )

j=1

292

6. RESIDUE CALCULUS AND TRACE FORMULAE

and let ρ1 , . . . , ρn the C 1 functions in ℂN defined by N m (z, w) −→ ρj (z, w) = |pj j (z, w)| for j = 1, . . . , n. Let Σm be the C 1 ℂn -valued map defined as follows Σm : (z, w) ∈ ℂN \ V (p) −→

n    ρ2j (z, w) ∗j = Σm,1 (z, w), . . . , Σm,n (z, w) . 2 ρ(x, w) pj (z, w) j=1

Since Σm (z, w)

n 

n



pj (z, w) j = σ(z, w) pj (z, w) j ≡ 1,

j=1

j=1

where σ is defined as in (6.114), the same cohomological argument as that used in the proof of Theorem 6.49, with (σ, Σm ) instead of (σ, Σ) as in (6.123) and κ = 0 there, together with the second equality in (6.115), shows that, if χ ∈ D(ℂN , ℂ) is identically equal to one in a neighborhood of V (𝕁) ∩ V (p), (6.143)

)

Res𝕂𝕁 [X,Y ]/𝕂

*  n  ' ¯ σ σ n−1 q¯ dX ∂ ([V (𝕁)]) , q (∂χ ∧ dζ) j = p¯1 , . . . , p¯n 2iπ 2iπ j=1 =

n  Σ n−1 ' m ∂ ([V (𝕁)]) , q (∂χ ∧ dζ) j . 2iπ 2iπ j=1

Σ

m

It follows from (6.142) that for some cm > 0 (6.144) (z, w) ∈ V (𝕁) and max(|z|, |w|) ≥ C =⇒ ρ(z, w)2 ≥ cm (max(|z|, |w|))M . Let θ be an even function such that

c c   2c 2c

m m m m , , [0, 1] , θ ≡ 1 on , . (6.145) θ∈D − 3 3 3 3 Let R > C and χR (z, w) = θ(ρ2 (z, w)/RM ). Since pj Σm,j ≡ (ρj /ρ)2 for j = 1, . . . , n on ℂN \ V (p), one can rewrite (6.143) when χ = χR as (see also (6.130)) ) *  n  ¯ |pj | 1−2/mj q¯ dX Ωm ∧ q dζ, ∂χR ∧ ρ (6.146) Res𝕂𝕁 [X,Y ]/𝕂 =− p¯1 , . . . , p¯n pj j V (𝕁) j=1

where

(−1)n(n−1)/2 (n − 1)! 2 = (2iπ)n

n−1

Ωm

 n

j=1 (−1)

j−1

ρ2n

ρj

# j  =j

 ∂ρj  .

Since ∂χR = R−M θ  (ρ2 /RM ) ∂ρ2 , one gets from (6.146) that ) * ¯ q¯ dX (6.147) Res𝕂𝕁 [X,Y ]/𝕂 p¯1 , . . . , p¯n #n n n n−1  2  |pj |

 1−2/mj j=1 dρj 2  ρ qθ ∧ dζ. ρj =− M R RM pj ρ2(n−1) V (𝕁) j=1 j=1

6.5. THE CHARACTERISTIC ZERO CASE

293

For any (z, w) ∈ V (𝕁), max(|z|, |w|) ≥ R > C =⇒ ρ2 (z, w) ≥ cm (max(|z|, |w|))M ≥ cm RM according to (6.144). It follows then from the choice of θ (see (6.145)) that ρ2

Supp θ  ∩ V (𝕁) ⊂ {(z, w) ∈ ℂN : max(|z|, |w|) ≤ R}. RM There is a positive constant Cθ (q) such that (6.148) n  ρ2  |pj |   ∀ (z, w) almost everywhere ∈ V (𝕁), q θ   ≤ Cθ (q) Rdeg q . RM p j j=1 Given I, I  ⊂ {1, . . . , n} whose sum of cardinals equals n and ζj = xj + i yj for j = 1, . . . , n, let us study the asymptotic behavior when R → +∞ of #n 1−2/mj  dρj j=1 ρj R −→ ΘI,I  (R) := ΘR ∧ dxI ∧ dyI  , 2(n−1) ρ V (𝕁) where ΘR

a.e. on V (𝕁)

=

q θ

n ρ2  |pj | . RM pj j=1

If Wπ denotes the algebraic hypersurface of ℂn defined as the discriminant locus of the proper projection π : (z, w) ∈ V (𝕁) → z ∈ ℂn , one has #n 1−2/mj  dρj j=1 ρj ΘR ∧ dxI ∧ dyI  2(n−1) ρ V (𝕁) #n 1−2/mj  dρj j=1 ρj = ΘR ∧ dxI ∧ dyI  . 2(n−1) ρ V (𝕁)\π −1 (Wπ )   The 2n-dimensional real manifold V (𝕁) \ π −1 (Wπ ) ℝ defined as the underlying real manifold of the n-dimensional complex manifold V (𝕁) \ π −1 (Wπ ) is a union of deg 𝕏 sheets Vι . Each such sheet is a 2n-differentiable manifold on which each map πI,I  ,ι : (Re z, Im z, Re w, Im w) ∈ Vι −→ (ρ1 , . . . , ρn , xI , yI  ) ∈ ℝ2n  is proper with degree deg πI,I  ,ι = dι (I, I  ). Let |d(I, I  )| = ι dι (I, I  ). Since #n 1−2/mj dρj j=1 ρj [m] dvn = 2(n−1) ρ is a volume form on (ℝ+ )n , one has then, taking also into account estimates (6.148), that #n 1−2/mj   dρj   j=1 ρj  ΘR ∧ dx ∧ dy  I I 2(n−1) ρ −1 V (𝕁)\π (Wπ )  Rdeg q+n [m]  n dv = C (q) , ≤ Cθ (q) Rdeg q+n−M |d(I, I  )| n θ ρ2 cm 2cm R j=1 M/mj 3 ≤ RM ≤ 3 where Cθ (q) =

Cθ (q) (cm /3)n−1

 cm 3

≤ρ2 ≤ 2c3m

dvn[m] .

294

6. RESIDUE CALCULUS AND TRACE FORMULAE

When deg q + n
1 and m < n, ⎪ n−2 ⎪ ⎩ if n > 1 and m ≥ n > 1. j=0 dm−j d0 There exists a significant difference between the geometric marker d (p) and the marker 

j dj n (7.26) N ({p0 , . . . , pm }) = min(m + 1, n + 1) max min dm , (m+1−n) , dm , + d0 which dominates the N -solvability of B´ezout identity with respect to {p0 , . . . , pm } on the algebraic variety 𝕏 = 𝕂n according to Theorem 7.8. To be more specific, the difference lies in that the marker (7.26) involves the factor min(m + 1, n + 1) reflecting the role played by the Lipman–Teissier theorem, while the definition of d (p) reflects only B´ezout’s geometric intersection theorem; see §6.6.2. One that this setting was already considered in §6.3.2, §6.4.2 or §6.5.2 and in §6.6.2, §6.6.3, and §6.6.4 as well. In the two last cases, the context was the arithmetic one, where 𝕂 = ℚ was the field of algebraic numbers. 6 Note

334

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

may state the following sharp version of Hilbert’s geometric nullstellensatz ([Ko88, Theorem 1.5] and [Som99, Theorem 3.19] in the case where N = n and 𝕏 = 𝕂n , [Jel05, Theorem 1.1] in the general case). Theorem 7.16. Let 𝕂[X, Y ], 𝕏, 𝕁, {p0 , . . . , pm } be as above. There are polynomials q0 , . . . , qm ∈ 𝕂[X, Y ] such that  m  d (p) when m < n, qj pj ∈ 𝕁 with max deg qj pj ≤ (7.27) 1 − 0≤j≤m 2 d (p) − 1 when m ≥ n. j=0 Moreover, if m, n and the dj ’s are prescribed with in addition m < n, then (7.28)

1−

m 

qj pj ∈ 𝕁 =⇒ max deg qj pj ≥ d (p). 0≤j≤n

j=0

When 𝕏 = 𝕂n and m ≥ n, the upper bound 2 d (p) − 1 in (7.27) can be (possibly) refined as (3/2)ιd d (p), where ιd = card({j : dj = 2}). In the particular case where 𝕏 and (m, d0 , . . . , dn ) are prescribed, then assertion (7.28) holds in the three following situations: n = 1, m < n, m ≥ n with ιd = 0. Proof. The proof of this result, as presented in [Jel05], is based on considerations that we introduced in §6.4 when exploring the concept of an algebraic trace in the polynomial setting of 𝕂[X] or, more generally, of the quotient setting of 𝕂𝕁 [X, Y ] = 𝕂[X, Y ]𝕁. Consider first the case when m < n. We sketch here Jelonek’s proof of [Jel05, Theorem 3.6]. It follows from Hilbert’s nullstellensatz (see Theorem 7.1) that there ◦ ∈ 𝕂[X, Y ] such that exist q0◦ , . . . , qm (7.29)

1−

m 

qj◦ pj ∈ 𝕁.

j=0

Observe that from the very beginning this initial step prevents the construction of q0 , . . . , qm ∈ ℤ[X, Y ] in (7.27) from being explicit in terms of the input data besides the variables X, Y . The 𝕏, p0 , . . . , pm . Let t be an additional parameter  polynomial map Φ : (x, y, t) ∈ 𝕏 × 𝕂 −→ x, y, t p0 (x, y), . . . , t pm (x, y) ∈ 𝕂N +m+1 m has a closed image in 𝕏 × 𝕂m+1 ⊂ 𝕂N +m+1 since t = 0 qj◦ (X, Y )(tpj (X, Y )) in 𝕂[X, Y ][t] (with respect to the Zariski topology) according to (7.29). Moreover, Φ embeds 𝕏×𝕂 as an affine, purely (n+1)-dimensional algebraic subvariety Φ(𝕏×𝕂) of the affine algebraic variety 𝕏 × 𝕂m+1 ⊂ 𝕂N +m+1 . Let us complete the sequence of m + 1 polynomials (pm , . . . , p0 ) as the sequence with length n defined as p = (p0 , . . . , pn−1 ) := (pm , . . . , p0 , 0, . . . , 0), which is also ordered in such a way. Let π : Φ(𝕏 × 𝕂) → 𝕂n+1 be a generic projection. It amounts to saying that the extension (π ◦ Φ)∗ : 𝕂[w0 , . . . , wn ] → 𝕂[X, Y, t]/𝕁(𝕏 × 𝔸1𝕂 ) is integral. Since 𝕂 is algebraically closed, hence infinite, it follows from Carl Siegel’s box principle (see for example [MW83, §4,Lemma 1] or [Jel05, Lemma 3.5]) that one can choose π  to satisfy π(Φ(x, y, t)) = u0 (x, y, t), . . . , un (x, y, t) , with

(7.30)

uj (x, y, t) = j (x, y) + t

n−1  κ=j

un (x, y, t) = n (x, y),

aj,κ pκ (x, y)

for

j = 0, . . . , n − 1,

7.1. HILBERT’S NULLSTELLENSATZ IN 𝕂[X1 , . . . , Xn ] AND RESIDUE CALCULUS

335

where 0 , . . . , n are n + 1 generic linear forms in 𝕂[X, Y ] and the coefficients aj,κ belong to 𝕂. We now consider 𝕏 as defined over 𝕂(t) instead of 𝕂. Finiteness of the map ut = (u0 , . . . , un ) : 𝕏 → (𝕂(t))n+1 and the generalized version of the Perron theorem [Per, Satz 57, S. 129]7 imply that there exists a polynomial E ∈ 𝕂(t)[W0 , . . . , Wn ] such that E(ut (X, Y )) ∈ 𝕂(t)[X, Y ] 𝕁,  deg ut,0 deg , . . . , wn X,Y degw0 ,...,wn E w0 X,Y

ut,n 

≤ d (p).

That is, (7.31)

(x, y) ∈ 𝕏 =⇒ E(u(x, y, t)) = 0,   d0 degw0 ,...,wn E w0dm , . . . , wm , wm+1 , . . . , wn ≤ d (p).

Since Φ is an embedding and π is a generic projection, (x, y, t) → u(x, y, t) realizes a finite morphism from 𝕏 × 𝕂 to 𝕂n+1 . It follows from the fact that (π ◦ Φ)∗ is an integral extension, that t admits for this extension a monic minimal polynomial satisfying ν  γρ (u(X, Y, t)) tν−ρ ∈ 𝕂(t)[X, Y ] 𝕁. tν + ρ=1

It follows  from the minimality of such a monic minimal polynomial and (7.31) that tν + νρ=1 γ(u(X, Y, t)) tν−ρ divides E(u(X, Y, t)) in 𝕂(t)[X, Y ] 𝕁. Then (7.32)

d0 , wm+1 , . . . , wn ) ≤ d (p) for ρ = 1, . . . , ν. degw0 ,...,wn γρ (w0dm , . . . , wm

If one adds all terms of the  form tν q(X, Y ) for some q ∈ 𝕂[X, Y ], which occur in the ν ν polynomial expression t + ρ=1 γρ (u(X, Y, t)) tν−ρ, then one gets in view of (7.32) m the relation 1 − j=0 qj (X, Y )pj (X, Y ) ∈ 𝕁 with max0≤j≤m deg(qj pj ) ≤ d (p). It is the result we were looking for. As a subordinate step, let us consider the extreme case when m + 1 = n, but where dim 𝕏 ∩ V (p) = 0 instead of −∞. That is, 𝕏 ∩ V (p) = {α1 , . . . , αμ }, where the α are distinct points8 in 𝕏. One repeats the construction of the polynomial map Φ : 𝕏 × 𝕂 −→ 𝕏 × 𝕂n ⊂ 𝕂N × 𝕂n , which now becomes a nonclosed embedding outside {α1 , . . . , αμ }. Let (x, y) ←→ ξ = (ξ1 , . . . , ξN ) be a generic linear change of coordinates in 𝕂N . For each coordinate ξ, let π ξ be a generic projection from the Zariski closure of Φ(𝕏 × 𝕂) in 𝕂N × 𝕂n onto 𝕂n+1 such that, if uξ = π ξ ◦ Φ, then uξj (ξ, t) = ξj (ξ) + t

n−1 

aξj,κ pn−1−j (ξ) for

0 ≤ j ≤ n − 1,

κ=j

uξn (ξ, t) = ξ. 7 See also [Pl05, Theorem 1.1 and §2] and [Jel05, Theorem 3.3] for the generalized formulation already quoted and used in §6.4.1 as well as in §6.4.2; see, e.g., Proposition 6.43 there. It is [Jel05, Theorem 3.3] that we invoke again here with 𝕂(t) being a reference field instead of 𝕂. It would be enough for ut to be in such context a generically finite morphism from 𝕏 (considered as being defined over 𝕂(t)) to (𝕂(t))n+1 . The important point to observe is that the Perron theorem is an immediate consequence of of two facts: B´ ezout’s geometric intersection theorem on one side (which is the most important side), and on the other side the fact that any substitution of variables (w0 , . . . , wn ) −→ (tδ00 + t0 , . . . , tδnn + tn ) with δj ∈ ℕ∗ preserves the reducibility of elements E in 𝕂[w] provided E −1 ({0}) does not contain hyperplanes of the form wj = aj with aj ∈ 𝕂; see [Jel05, Corollary 3.2]. 8 We do not care here about multiplicities.

336

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

Here ξ0 , . . . , ξn−1 are generic linear forms and the coefficients aτj,κ belong to 𝕂. Perron’s theorem [Jel05, Theorem 3.3] still applies since (uξ0 (·, t), . . . , uξn−1 (·, t)) can be considered as a generically finite morphism from 𝕏 (viewed now as being defined over 𝕂(t)) to (𝕂(t))n+1 . Let E ξ (uξ0 (·, t), . . . , uξn−1 (·, t)) ∈ 𝕂(t)[ξ] 𝕁 be the Perron algebraic dependency relation with a weighted degree in (w0 , . . . , wn ) less than d (p) (ξ, t) ∈ 𝕏 × 𝕂 −→ uξ (ξ, t) ∈ 𝕂n+1 admits the "μas in (7.31). Since  hypersurface =1 V (wn −ξ(α )) as the set of nonproperness, the minimal equation of t, considered as an element of 𝕂[𝕏][t], with respect to uξ , is not monic any more, but of the form ν  ξ ξ γ0ξ (uξ (X, Y, t)) tν + γρξ (uξ (X, Y, t)) tν −ρ ∈ 𝕂(t)[X, Y ]𝕁, ρ=1

with

γρξ

∈ 𝕂[w0 , . . . , wn ] for ρ = 0, . . . , ν ξ . Moreover, V (γ0ξ ) ⊂ {w ∈ 𝕂n+1 :

μ 

(wn − ξ(α )) = 0}.

=1

We now repeat the msame argument as in the previous paragraph. In place of relation of the form 1 − j=0 qj pj ∈ 𝕁, with the degree estimates max deg(qj pj ) ≤ d (p)

0≤j≤m

deduced in the case studied previously, one derives now the relation μ 

(ξ − ξ(α )) −

n−1 

qjξ pj ∈ 𝕁,

j=0

=1

max0≤j≤n−1 deg(qjξ pj )

where ≤ d (p). This result is stated in [Jel05, Theorem 3.8] and is presented there as an elimination theorem, which is natural since it provides a cartesian product that contains {α1 , . . . , αμ } = 𝕏 ∩ V (p). Observe that one can interpret such relation as a sharp version of Max Nœther’s AF + BG division theorem [No1873] concerning the degree estimates. We can finally consider the case m ≥ n, with the hypothesis 𝕏 ∩ V (p) = ∅. To begin with observe that it is enough to consider the case where m = n. Actually, p0 , . . . , p˜m ), where if one re-organizes the sequence (p0 , . . . , pm ) as (˜ d˜1 ≥ d˜2 ≥ · · · ≥ d˜m ≥ d˜0 when d˜j = deg p˜j , then one can again invoke Carl Siegel’s box principle (see [MW83, §4, Lemma 1] or [Jel05, Lemma 3.5]) to consider n + 1 generic combinations  pˇ0 = p˜0 , pˇj = γj,κ p˜κ for j = 1, . . . , n κ≥j

which do not share any common on 𝕏 (as the original pj ’s). Then we replace the sequence (p0 , . . . , pm ) by the sequence obtained from the re-organization of the sequence of polynomials (ˇ p0 , . . . , pˇn ) along their increasing degrees. The proof of assertion (7.27) for the remaining case m = n goes as follows; see [Jel05, Theorem 3.10]. Extract two distinct n-valued polynomial maps p[0] and p[1] from the (n + 1)valued polynomial map p. Then 𝕏 ∩ V (p[0] ) and 𝕏 ∩ V (p[1] ) are both of dimension

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less than or equal to 0. If one of these algebraic subsets is empty, then there is nothing to prove because of the first case considered in this proof. Accordingly, one can assume both of them are truly zero dimensional. A generic linear change of coordinates (x, y) ↔ (x , y  ) in 𝕂N , if necessary, allows us to chose a coordinate ξ among the new coordinates x , y  such that the images by ξ of 𝕏 ∩ V (p[0] ) and of 𝕏 ∩ V (p[1] ) are disjoint. We now conclude with the elimination result, whose proof is in our second step, together with the fact that if φ[0] and φ[1] are two univariate polynomials with respective degrees δ0 and δ1 and without common zeroes, then Euclid’s algorithm provides an identity 1 = ψ [0] φ[0] + ψ [1] φ[1] with max deg(ψ [j] φ[j] ) ≤ δ0 + δ1 − 1. As for the proof of the last assertion of the theorem in case 𝕏 = 𝕂n , which is due to Janos Koll´ ar, we refer the reader to [Ko88, Theorem 1.5] or to the instructional presentation by Bernard Teissier [T90] in the Bourbaki seminar. We refer to [Jel05] and [Ko88] for the proofs of the assertions relative to the sharpness (optimality). It is also worth comparing the optimality of the present results to the  optimality assertion in Theorem 7.8 (stated for the case 𝕏 = 𝕂n ). 7.1.6. An arithmetic Perron theorem in its parametric form. Let 𝕂 be an algebraically closed field and let 𝕏 be a purely n-dimensional affine algebraic variety defined over 𝕂 and embedded in 𝕂N = Spec (𝕂[X, Y ]). As we observed in §7.1.5, the Perron theorem, in its generalized form [Jel05, Theorem 3.3], appears to be the cornerstone leading to the formulation of an almost sharp version of Hilbert’s nullstellensatz in 𝕂𝕁(𝕏) [X, Y ], in terms of d = deg 𝕏 and the degrees of the polynomial entries p0 , . . . , pm ; see Theorem 7.16. This approach is realized through d (p) or (2 d (p) − 1)-solvability of B´ezout identity for {p0 , . . . , pm }, combined with the Rabinowitsch trick, as stated in Proposition 7.4. We return in this section to the arithmetic frame of §6.6.3 and §6.6.4. We assume that 𝕂 = ℚ and that the algebraic variety 𝕏 ⊂ 𝕂N = Spec (𝕂[X, Y ]) is defined over ℚ. Namely, we assume that elements of ℤ[X, Y ] generate the radical ideal 𝕁 = 𝕁(𝕏). Then the arithmetic marker for the arithmetic complexity of 𝕏 is its canonical global logarithmic height h, as defined in Proposition 6.65. To analyze how the ideas introduced in the proof of Theorem 7.16 can be transposed to the arithmetic setting and to see why the Perron theorem plays a pivotal role, one needs to review the proof of Theorem 7.16 from its beginning, which we do now. Let (p0 , . . . , pn ) be a sequence of elements in ℤ[X, Y ] such that {(x, y) ∈ 𝕏 :, p0 (x, y) = · · · = pn (x, y) = 0} = ∅ N

in ℚ . It follows from Hilbert’s nullstellensatz in ℚ[X, Y ] and from the fact that 𝕏 is defined over ℚ (see also 7.9) that there exist γ ◦ ∈ ℕ∗ and q1◦ , . . . , qn◦ Remark n ◦ ◦ in ℤ[X, Y ] such that γ − j=0 qj pj ∈ 𝕁. As a consequence, one can repeat the initial argument in the proof of Theorem 7.16 and claim that the mapping   N +n+1 Φ : (x, y, t) ∈ 𝕏 × ℚ −→ x, y, t p0 (x, y), . . . , t pn (x, y) ∈ ℚ realizes a closed embedding of the (n + 1)-dimensional algebraic variety 𝕏 × ℚ as an (n + 1)-dimensional algebraic subvariety Φ(𝕏 × ℚ) ⊂ 𝕏 × ℚ

n+1

.

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The left inverse Φ−1 left : Φ(𝕏 × ℚ) → 𝕏 × ℚ is defined by q0◦ (x, y)w0 + · · · + qn◦ (x, y)wn Φ−1 ((x, y), w , . . . , w ) = x, y, . 0 n left γ◦ Let τj = (τj,1 , . . . , τj,N ), for j = 0, . . . , n, be n+1 sets of parameters, whose cardinal is N . Specifying the parameters as generic integers, which means that (τ0 , . . . , τn ) is a generic element in ℤ(n+1)N , the projection   n+1 π [τ ] : ((x, y), z0 , . . . , zn ) ∈ Φ(𝕏×ℚ) → z0 + τ0 , (x, y), . . . , zn + τn , (x, y) ∈ ℚ is a finite projection.9 Then (7.33) (x, y, t) −→ u[τ ] (x, y, t) = u[τ ] (x, y, t) := (π [τ ] ◦ Φ)(x, y, t)   = t p0 (x, y) + τ0 , (x, y), · · · , t pn (x, y) + τn , (x, y) = (ℚ(τ ))n+1 . realizes a finite morphism from 𝕏ℚ(τ ) ×𝔸1ℚ(τ ) = 𝕏ℚ(τ ) ×ℚ(τ ) to 𝔸n+1 ℚ It means that the inclusion of algebras (7.34)

(π [τ ] ◦ Φ)∗ : ℚ(τ )[w0 , . . . , wn ] → ℚ(τ )[X, Y, t]/𝕁(𝕏ℚ(τ ) × 𝔸1ℚ )

is an integral extension. Let P (τ, w, T ) ∈ ℤ[τ, W ][T ] be the minimal polynomial of t with respect to this extension. It is primitive with respect to ℤ[τ ], is necessarily square-free, and is also primitive for ℤ[τ, T ]. It is of the form ν 

 (7.35) P (τ, w, T ) = γ0 (τ )T ν + γρ,κ (τ )wκ T ν−ρ , ρ=1

κ∈ℕn+1

where γ0 and γρ,κ belong to ℤ[τ ] = ℤ[τ0 , . . . , τn ], and one also has that (7.36) γ0 (τ )tν +

ν   ρ=1

γρ,κ (τ )

n   κ t pj (X, Y ) + τj , (x, y) j tν−ρ j=0

κ∈ℕn+1

∈ ℚ(τ )[X, Y ] 𝕁. Moreover,

  [τ ] (x, y) −→ ut (x, y) = t p0 (x, y) + τ0 , (x, y), . . . , t pn (x, y) + τn , (x, y)

is a generically finite morphism from 𝕏ℚ(τ,t) ; that is, the variety 𝕏 is considered n+1 . Then, one has, over ℚ(τ, t) instead of ℚ, onto its image in 𝔸n+1 ℚ(τ,t) = (ℚ(t, τ )) taking the Zariski closure of such image in ((ℚ(τ, t))n+1 , that (7.37)

[τ ]

Im ut = { ∈ (ℚ(τ, t))n+1 : P (τ, , t) = 0};

see [AKS13, Lemma 4.8]. The coefficient of tν on the left-hand side of (7.36) is of the form n  γ0 (τ ) − qj (τ, X, Y ) pj (X, Y ), j=0 9 The notion of a “Chow ideal” of a cycle (see Remark D.13) as well as that of a “Chow form” already introduced in the univariate setting (see Definition 1.49) remain constantly present in the approach presented here. Deep machinery relying on the concept of multi-Chow form (attached to multi-projective cycles) is developed extensively within the frame of arithmetic intersection theory by Yuri V. Nesterenko, Dale W. Brownawell, Patrice Philippon; see, e.g., [Bro88], [PhS08] and [AKS13, §2.3] for up-to-date bibliography about it. This machinery is involved in the approach developed in [AKS13], sustaining the presentation we sketch here.

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where qj (τ, X, Y ) ∈ ℤ[τ ][X, Y ]. Take any α ∈ ℕ(n+1)N in the support of the polynomial in (n + 1)N variables γ0 (τ0 , . . . , τn ) and let γ0,α ∈ ℤ∗ be its coefficient. α Then, n the coefficient of τ in the development as a polynomial in (τ0 , . . . , τn ) of ezout identity j=0 qj (τ, X, Y ) pj (X, Y ) provides, when subtracted from γ0,α , B´ γ−

(7.38)

m 

qj (X, Y )pj (X, Y ) ∈ 𝕁.

j=0

One needs now to concentrate on upper bounds for the degrees of the qj ’s and the logarithmic sizes of absolute values of their coefficients, including the estimate of log |γ0,α |. To deal with such estimates, one needs the following parametric version of the Perron generalized theorem [AKS13, Theorem 3.15].10 Theorem 7.17. Let 𝕏 be as above, and let τ = {τι }ι be a finite collection of parameters realizing a transcendental basis of ℚ(τ ). Let also U0 , . . . , Un be n + 1 elements in Uj ∈ ℤ[τ, X, Y ] such that the mapping   (x, y) −→ U τ (x, y) = U0 (τ, x, y), . . . , Un (τ, x, y) realizes a generically finite morphism from 𝕏 ℚ(τ ) onto its image in the affine space  κ 𝔸n+1 κ∈ℕn+1 Eκ (τ )w , reduced in ℚ(τ )[X, Y ] and primitive ℚ(τ ) . Let E ∈ ℤ[τ, w] = in ℤ[τ ], such that in terms of Zariski topology in (𝔸ℚ(τ ) )n+1 , one has 2 4 Im U =  ∈ (ℚ(τ ))n+1 : E(τ, ) = 0 . Then, the following degree and logarithmic size estimates hold:  deg U0 deg Un  ≤ d (degX,Y U ), degw0 ,...,wn E τ, w0 X,Y , . . . , wn X,Y m    degτι Uj degτι E τ, U (τ, X, Y ) ≤ d (degX,Y U ) , deg X,Y Uj j=0

(7.39)

and, for any κ ∈ ℕn+1 , (7.40) h(Eκ ) + κ, h(U ) ≤ d (degX,Y U ) +

n 

1

j=0

degX,Y Uj

h d

+ log(n + 2)



   h(Uj ) + log card (Supp(Uj )) + 2 + log(|τι | + 1) degτι Uj , ι

where degX,Y U := (degX,Y U0 , . . . , degX,Y Un ) and h(U ) := (h(U0 ), . . . , h(Un )). Remark 7.18. Observe that estimates for the degrees in X, Y (respectively for the degrees in the parameters τι or logarithmic sizes estimates) in the first line of (7.39) (respectively in the second line of (7.39) or in (7.40)) fit with B´ezout markers (6.180) for geometric intersection theory (respectively with B´ezout markers (6.197) or (6.198) for arithmetic intersection theory) as introduced in §6.6.2. Theorem 7.17 is pivotal for the formulation of the sharpest version for the arithmetic nullstellensatz known up to now, namely Theorem 6.70, following the reasoning that we described previously in this subsection. Technical estimates needed to justify degree and logarithmic size estimates for log |γ| and the qj in 10 We admit this result here since proving it would require the whole machinery of Chow forms of multi-projective cycles, together with control on their logarithmic heights in intersection theory; see [AKS13, §1,2]. This would take us too far from the purpose of the present monograph.

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(7.38) become routine, once one invokes Theorem 7.17 with {τ, t} as parameters, since then U (τ, t, X, Y ) = u[τ ] (X, Y, t) as in (7.33); see [AKS13, §4.3]. As for the proof of the arithmetic elimination theorem (Theorem 6.70) quoted previously in this monograph from [SomY21, Theorem 5.1 and Corollary 5.6], it follows essentially along the same lines as those carried through in the argument developed for the second step of the proof of Theorem 7.16 to get an effective sharp geometric elimination (or Max Nœther’s AF + BG theorem). This theorem provides then a nearly optimal effective arithmetic elimination theorem. We refer to [SomY21, §5] for more details. Remark 7.19. As we have already seen in §6.6.3 and §6.6.4, the arithmetic Perron theorem, either in its parametric version stated as Theorem 7.17 or in its version without parameters τ , remains crucial for estimating the arithmetic complexity of multivariate residue calculus over ℚ. 7.1.7. Hilbert’s nullstellensatz and P = N P . Let A be a commutative ring. Within this subsection, A is one of the rings ℤ, ℝ, ℚ, ℚ, ℂ or one of their transcendental extensions ℤ[τ ] = ℤ[τ0 , . . . , τm ], ℝ[τ ], ℚ[τ ], ℚ[τ ], ℂ[τ ],11 all with characteristic 0, as well as their p-adic companions ℤp , ℝp , ℂp or ℤp [τ ], ℝp [τ ], ℂp [τ ] with p a prime number. Within the characteristic 0 setting, one could consider as well 𝔽p or 𝔽p [τ ]. The concept of logarithmic height on such ring A is the pendant of the notion of entropy. It is assigned to play the same role as that of a chaos (or disorder) quantifier in physics or dynamics. For example, when A = ℤ, the logarithmic height defined by h(a) = log(|a| + 1) quantifies the number of necessary bits required to encode a ∈ ℤ∗ , besides its sign. Such is the case also for various notions of naive logarithmic height, Mahler measure, etc., that we introduced in §6.6.1 or §6.6.2. When A = ℝ or A = ℂ, it is natural to consider h(a) = 1 for any a ∈ A to take into account that the cost of the multiplication by a does not depend on the size of the real or complex number a. Let A be a ring, which is also a domain, equipped with a logarithmic size h. Let also p0 , . . . , pm ∈ A[X1 , . . . , Xn ] be m + 1 polynomials with coefficients in A. The ability to decide whether or not the polynomial functions x → pj (x), j = 0, . . . , m, share a common zero in an integral closure (in fact any) of the fraction field Frac(A) of A is a typical example of a decision problem. This is the case for the following more general problem: given the polynomials p0 , . . . , pm as mbefore and q ∈ A[X1 , . . . , Xn ], decide whether or not q belongs to the ideal j=0 pj Frac(A)[X1 , . . . , Xn ] or at least to the integral closure of such ideal in Frac(A)[X]. The first problem mentioned above is the nulltellensatz decision problem over A, while the second one is known as the membership decision problem over A. Let us describe the problems a little more conceptually, following, for example, the pioneering paper [BlSS89] devoted specifically to the case where A = ℝ. A decision problem over A is a pair of subsets (D, D yes ), with D yes ⊂ D, of the space A (here the integer  may be finite or infinite). The space A plays the role of the space of inputs for some machine12 𝕄 over A. An algorithm solving the decision 11 The field ℂ is here visualized as its subjacent ℝ-vector space ℝ2 in order to profit from the fact that ℝ is ordered, as ℤ and ℚ are, which ℂ is not. 12 A machine over A consists of inputs space Ainit , with init ∈ ℕ∗ , possibly infinite, a space of outputs Afin , and a space of transient states Atrans , with fin, trans ∈ ℕ∗ , together with a graph 𝔾𝕄 having a finite number of nodes N are classified into computation nodes, branching nodes, and exit nodes. We refer the reader to [BlSS89, §2] for a didactic presentation of the

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problem is a machine 𝕄 initiated from any input e ∈ D, which eventually stops, then providing an output s𝕄 (e) such that s𝕄 (e) = 1 ⇐⇒ e ∈ D yes . More precisely, given an element e in Ainit , the machine is said to stop when initiated at e when, as soon as it starts having e as input, then some exit node is reached after some minimal time t𝕄 (e) (the time is being indexed here with the number of steps the machine proceeds). Whenever e is such an entry, the cost of the machine (when initiated in e) is by definition the quantity c 𝕄 (e) = t𝕄 (e) × h𝕄 (e) .

(7.41)

The quantity h𝕄 (e) denotes the maximal logarithmic height of all elements in the ring A that have been involved in all transient states, which appear as intermediate states before the machine reaches the exit node that will lead to the output s𝕄 (e) (in the output space) within the time t𝕄 (e). The machine 𝕄 is said to operate in polynomial time if its cost function satisfies for any e ∈ Ainit  N (7.42) c 𝕄 (e) ≤ C𝕄 |e| + h(e) 𝕄 for some N𝕄 ∈ ℕ∗ , C𝕄 > 0. Here |e| denotes the length of the support of e. That is, it denotes the number of nonzero entries of e as an element in Ainit𝕄 . The nullstellensatz decision problem over A = ℂ is known to be NP-complete; see [BlSS89, §6, Main theorem]. The whole content of this paper is an introduction to the NP complexity class of decision problems over A = ℝ. The terminology NP stands for decidable in Nondeterminist Polynomial time. Let us recall what it means, at least heuristically. A decision problem (D, Dyes ) with D yes ⊂ D ⊂ A ( ∈ ℕ∗ , possibly infinite) over the ring A is said to be in the NP class if there exist two integers  ∈ ℕ∗ , possibly infinite, a machine 𝕄 over A with input space Ainit 𝕄 such that   D × A ⊂ Ainit ⊂ A × A , 𝕄 

where A plays the role of a probability space, such that  • the machine eventually stops when it is initiated at any (e, ω) in D × A ;  • one has s𝕄 (e, ω) ∈ {0, 1} for any (e, ω) ∈ D × A ; • s𝕄 (e, ω) = 1 ⇐⇒ e ∈ D yes ; • most importantly, there exists N𝕄 ∈ ℕ and C𝕄 > 0 such that (7.43) ∀ e ∈ D yes , ∃ ω ∈ A



such that s 𝕄 (e, ω) = 1

 N with c 𝕄 (e, ω) ≤ C 𝕄 |e| + h(e) 𝕄 .

Given such a decision problem (D, D yes ) belonging to the NP class over A, it is said to be NP-complete if it satisfies some universal property within the category of decision problems in the NP class over A. This means the following: given any ˜ D ˜ yes ) over A, there exists a map Ψ : D → D ˜ such that decision problem (D, ˜ ˜ Ψ(˜ e) ∈ D yes ⇐⇒ e˜ ∈ D yes and Ψ is the restriction to D of the output e˜ → s𝕄 (˜ e) ˜ ⊂ Ainit𝕄 ) that operates over A in polynomial time. of a machine 𝕄 (with D role devoted to such nodes within the class to which it belongs. Each node comes either with one edge to the next mode and no incoming edge (input node), with incoming edges and exactly one outgoing edge (computation node), incoming edges and two outgoing edges together with a polynomial map from Atrans to itself governing the switching of the machine towards one of the two alternative nodes (branching node), or no outgoing edge and thus a unique linear map from Atrans to Afin (exit node).

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A decision problem (D, Dyes ) over the ring A is said to be in the class P if there is a machine 𝕄 over A that solves it in polynomial time. Any decision problem in the P class over A is in the NP class over A. It is still unknown and it remains a highly challenging question whether the P class over any of the classical commutative rings A is strictly included or equal to the NP class over A [Sm98]. Given what we discuss within the present monograph,13 let us quote here the following results [BlCSS96, §7.1, Theorems 1 and 2] or [ShuS96, Main Theorem] about the conjecture P = NP (or P = NP) over the rings ℂ (where the logarithmic size is taken as constant equal to 1) or ℚ (where the logarithmic size derives from the logarithmic size a → h(a) = log(|a| + 1) on ℤ). Theorem 7.20. If P = NP is true over ℂ, then P = NP is true over ℚ. The converse is true as well. Moreover, if one makes the hypothesis that the sequence (k!)k≥0 is ultimately hard to compute,14 then the NP nullstellensatz decision problem over ℂ fails to be in the P class over ℂ. This implies that P = NP over ℂ or, which is equivalent, P = NP over ℚ. Sharp versions of the arithmetic Hilbert’s nullstellensatz formulated as B´ezout identity described in Theorem 6.70, with (nearly) sharp estimates both in degree and logarithmic size, are obtained (as we have seen already in §7.1.6 and will make more precise in §7.1.7) through a nonalgorithmic procedure relying on the use of a parametric version of the Perron theorem; see Theorem 7.17. The statement requires starting with an a priori solution of B´ezout identity. Consequently, such proofs do not provide proof of Hilbert’s nullstellensatz since they rely on it from the beginning; see, for example, the proof of Theorem 7.16. On the other hand, surprisingly, multivariate residue calculus, inspired originally by a more analytic point of view than algebraic, will provide (as we will see in §7.2.1, Theorem 7.27), a closed formula of the Lagrange–Kronecker type solving explicitly with reasonable bounds in terms of degrees and logarithmic sizes (fitting with geometric and arithmetic B´ezout type control in intersection theory) the B´ezout arithmetic identity, that is, Hilbert’s nullstellensatz over ℚ. The sequence (k!)k≥0 , whose complexity is involved in Theorem 7.20, is also constantly present there as the sequence of denominators that appear when Leibniz’s differential calculus (governed by Leibniz’s formula) is reformulated in terms of multivariate residue calculus (dominated by the transformation law or its generalizations presented in §3.2.3); see Remark 6.25. For these reasons, a dynamical approach to the arithmetic complexity of multivariate residue symbols defined over ℚ or ℚ, as sketched in §6.6.4, would need to be pursued. A deformation argument of Arnold’s type (see [AVGZ]) could be invoked instead of appealing to the somehow unexplicit Perron reasoning. It is hard to see at this point whether some algorithmic procedure could lead to the closed formula 13 It

should be added here that a large part of this heuristical section arose from correspondence with Lenor Blum. The starting point for the correspondence was the content of the informal notes of an introductory lecture A.Yger gave in Bordeaux in 2001. Here we take the opportunity to refer the reader to the seminal Emmy Nœther lecture [Bl04]. 14 Computation with length  of k ∈ ℤ is a sequence of integers [ν ν , . . . , ν ], initiating at 0 1  ν0 = 1, ending at ν = k, so that any intermediate νι is expressed as νι • νι with ι < ι, ι < ι and • denotes the addition, the substraction, or the multiplication, in which case, the minimum length is denoted as min (k). Given a sequence (uk )k≥0 of integers, such a sequence is said to be ultimately easy to compute if there exists a sequence of nonzero integers (mk )k≥0 and N ∈ ℕ∗ such that (mk uk ) ≤ (log k)N for any k ≥ 2. Otherwise, the sequence (uk )k≥0 is said to be ultimately hard to compute.

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that we shall propose in the following subsection, but this remains a challenge about the key test role of the NP-complete nullstellensatz decision problem over ℂ or ℚ concerning the P=NP (or P= NP) conjecture over one of these two rings. Even though a closed formula is far from sustaining an algorithmic procedure, it seems significant to us to make such heuristic observations here. Let us go on with some additional heuristical considerations. Another trick one can use to encode the complexity of a system of polynomial inputs in ℤ[X1 , . . . , Xn ] is to encode the construction process itself. Then continue by encoding the process of evaluation of the different entries of the system. For example, the univariate polynomial 2n

x2

− 1 ∈ ℤ[x]

is quite easy to encode this way (iterating 2n times the process x → x2 ), although it has a doubly exponential degree. This idea arose in the 1970s through the works of J. Heintz, J. Morgenstern, C. P. Schnorr, and others. Its development has been swift since that time. For example, one refers to the works [KP94,GHMMPM97, GHMMP98] to find both a presentation and a prospective outlook about the role this approach could have for diophantine approximation questions in ℚ[X1 , . . . , Xn ]. The key notion here is the concept of straight line program on ℤ[X1 , . . . , Xn ] = ℤ[X]; see e.g [GHMMPM97, Definition 18]. This concept is inspired by the notion of simplicity (where “simplicity” means “easy to compute” algorithmically) of a numerical sequence already mentioned when asking whether the factorial sequence (k!)k≥0 is (or is not) ultimately hard to compute; see Theorem 7.20. Such a program 𝕊𝕃 consists of the following data: a directed acyclic graph15 𝔾𝕊𝕃 paired with a list of instructions 𝕀𝕊𝕃 , all of them defining some kind of protocol, one for each different entry gate of the graph. Together with (𝔾𝕊𝕃 , 𝕀𝕊𝕃 ), a finite set τ𝕊𝕃 of integer parameters is also given. Moreover, the graph 𝔾𝕊𝕃 presents exactly n + 1 entry gates labeled as X1 , . . . , Xn and 1. The depth of a gate g of 𝔾𝕊𝕃 denotes the length of the longest path from g until one of the entry gates. One can label the gates of 𝔾𝕊𝕃 by pairs (k, ), where k ∈ ℕ means the depth of the gate and  ∈ ℕ denotes another index. This index is used to label the gates 𝔾𝕊𝕃 with fixed depth arbitrarily. Gates are then classified along the lexicographic order ≺ on ℕ × ℕ; that is, all gates gk, of 𝔾𝕊𝕃 sharing k as depth are ordered along lexicographic order with respect to l. To each such gate g = gk, of 𝔾𝕊𝕃 , one may associate (7.44) qk, =









κ ,λ τk, qκ ,λ

×

0≤κ 0. If not, let 4 2 r(z) = max ρ ∈ {1, . . . , n} : |pjρ (z)| z d (p)+ε < c . It follows from (7.60) that  5  r(z) dist z, r(z) ρ=1 V (pjρ ) (7.62) ≤ z−ε/ ρ=1 djρ (z) . z 5r(z) Let ζz ∈ ρ=1 V (pjρ ) such that r(z) z − ζz  ≤ z−ε/ ρ=1 djρ (z) . z

It follows from (7.59) that   1 + t[jρ ] , ζz  ζz  z − ζz  3 ≥ c(τ ) ≥ c(τ ) − c(τ ) ≥ c(τ ), max z z z 4 r(z)+1≤ρ≤n provided z is assumed to be large enough, though depending on c(τ ), C(τ ), c, the dj ’s, and ε. Without loss of generality, such an assumption could have been 16 See Definition 6.53 for what it means, for a ℂn -valued polynomial map on an n-dimensional algebraic subvariety 𝕏ℂ of ℂN , to admit (δ1 , . . . , δn ) as a multi-exponent of properness with respect to 𝕏ℂ . 17 Although the proof of this result is closely related to that of [BY91, Lemma 5.3] and relies, as in [BY91, §5], on the use of Nœther’s preparation theorem, we present below a different and more direct approach to such types of results.

350

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

made from the beginning of our discussion. For any ρ between r(z) + 1 and n, one also has that      1 + t[jρ ] , z − 1 + t[jρ ] , ζz   r(z)

c(τ ) z, 4 again assuming z to be large enough and depending on c(τ ), τ , the dj ’s as well as ε. Thus, one has for such z that   1 + t[jρ ] , ζz  ≥ c(τ ) z. max 2 r(z)+1≤ρ≤n ≤ τ  z − ζz  ≤ τ  z−ε/

ρ=1

djρ (z)

z ≤

Therefore max

r(z)+1≤ρ≤n

|φjρ ,ν (t[jρ ] , z)| ≥ c (τ, ε) zν− d (p)−ε .

Assuming that ε > 0 is arbitrarily close to 0, this concludes the proof of the proposition, taking into account the definition of δ-properness for polynomial maps  from ℂn to ℂn as formulated in Definition 6.53. Now, let Δνp ∈ A[τ, X, Y ] be the (n + 1) × (n + 1) determinant defined by (7.63) Δνp (τ, X, Y ) :=   φ0,ν (t[0] , X) p1 (X) φ2,νp (t[2] , X) ··· p  H [0] [2]  φ0,νp ,1 (t , X, Y ) Hp1 ,1 (Y ) Hφ2,νp ,1 (t , X, Y ) · · ·  .. .. .. ..  . . . .   .. .. .. ..   . . . .  .. .. .. ..   . . . .  Hφ0,νp ,n (t[0] , X, Y ) Hp1 ,n (Y ) Hφ2,νp ,n (t[2] , X, Y ) · · ·

 φn,νp (t[n] , X)  Hφn,νp ,1 (t[n] , X, Y )   ..  .  , ..   .  ..   .  Hφn,νp ,n (t[n] , X, Y )

where νp := (n + 1) d (p) + |d|

(7.64)

and the Hefer forms H0,νp , Hp1 , H2,νp , . . . , Hn,νp have been defined in (7.51), (7.52). The following result is a realization of B´ezout identity using Jacobi–Lagrange– Kronecker interpolation. It extends to the multivariate setting the result formulated in Corollary 1.39 in the univariate setting. When A = ℤ, such a result was stated in a nonparametric form as [BY91, Theorem 5.1]. Theorem 7.27. Let A, 𝕂, p = (p0 , . . . , pn ), τ = {t[j] }1≤j,k≤n , νp , and Δνp in A[τ, X, Y ] be as above. Then, one has in 𝕂(τ )[X] the parametric B´ezout identity ⎤ ⎡ 1 Δνp (τ, X, Y ) dY1 ∧ · · · ∧ dYn ⎦ 1 = −Res𝕂(τ )[Y ]/𝕂(τ ) ⎣ p1 (Y ) [0] [2] [n] φ (t , Y ), φ (t , Y ), . . . , φ (t , Y ) 0,ν 2,ν n,ν p p p (7.65)  [j] = q1 (τ, X)p1(X) + qj (τ, X) φj,νp (t , X), j=1

where (7.66)

  max degX (q1 (τ, ·)p1 ), degX (qj (τ, ·)φj,νp (t[j] , ·)) ≤ |d| + n(νp − 1). j=1

7.2. JACOBI–LAGRANGE–KRONECKER (JLK) PARAMETRIC IDENTITIES

351

Proof. Let ν > d (p) and bν (τ, X, Y ) ∈ A[τ, X, Y ] be such that Hφ0,ν (t[0] , z, ζ) ∧ Hφ2,ν (t[2] , z, ζ) ∧ · · · ∧ Hφn,ν (t[n] , z, ζ) = bν (τ, z, ζ) Since degζ bν ≤

 j=1

n '

dζj .

j=1

dj + n(ν − 1), it follows from Proposition 7.26 and Theorem 2

6.56 (once τ has been specified on an element in ℂn \ V (Φ)) that (7.67) |κ|(ν − d (p)) − n d (p) * )  bν (τ, X, Y ) dY1 ∧ · · · ∧ dYn > dj =⇒ Res𝕂(τ )[Y ]/𝕂(τ ) κ1 +1 [0] κ2 +1 [2] κn +1 [n] φ0,ν (t , Y ), φ2,ν (t , Y ), . . . , φn,ν (t , Y ) j=1

= 0 in 𝕂(τ )[X]. If ν = νp , then one has in particular that (7.68) κ = (0, . . . , 0) =⇒ |κ|(νp − d (p)) − n d (p) ≥ νp − (n + 1) d (p) = |d| >

dj

j=1

* bνp (τ, X, Y ) dY1 ∧ · · · ∧ dYn =0 κ2 +1 [2] κn +1 [n] 1 +1 φκ0,ν (t[0] , Y ), φ2,ν (t , Y ), . . . , φn,ν (t , Y ) p p

) =⇒ Res𝕂(τ )[Y ]/𝕂(τ )



in 𝕂(τ )[X], according to (7.67). It then follows from Theorem 6.28 that one has in 𝕂(t)[X] the equality of polynomials * ) bνp (τ, X, Y ) dY1 ∧ · · · ∧ dYn (7.69) 1 = Res𝕂(τ )[Y ]/𝕂(τ ) . φ0,νp (t[0] , Y ), φ2,νp (t[2] , Y ), . . . , φn,νp (t[n] , Y ) It follows from (7.58) that one can rewrite (7.69) as ⎤ ⎡ 1 b (τ, X, Y ) p (Y ) dY ∧ · · · ∧ dY ν 1 1 n ⎦ . (7.70) 1 = Res𝕂(τ )[Y ]/𝕂(τ ) ⎣ p1 (Y ) p [0] [2] [n] φ0,νp (t , Y ), φ2,νp (t , Y ), . . . , φn,νp (t , Y ) Since the (n+1)×(n+1) determinant Δνp (τ, X, Y ) remains invariant after adding to the first line the linear combination of the n last rows with coefficients respectively follows from the formal construction of local residue symbols Y1 −X1 , . . . , Yn −Xn , it5 in Oℂn ,z (where z ∈ j=1 V (φν,j (t[j] , ·)) as traces,18 combined with the rule of Sarrus, that one can rewrite the algebraic identity (7.70) as (7.65) with degree estimates (7.66).  Example 7.28. Suppose that A = ℤ. If one specifies the set of parameters τ 2 on an element of ℤn \ V (Φ), where logarithmic sizes of the entries are controlled according to Remark 7.9, then the algebraic identity (7.65) provides, after lifting n ∗ q denominators, B´ezout identity γ = j=0 j pj with γ ∈ ℕ , q j ∈ ℤ[X], and maxj deg(q j pj ) ≤ |d| + n(νp − 1). Since some (probably not sharp) control on the 18 One

needs to invoke here the fact that (ωa,(0,...,0) (h))T = 0, when h belongs to

in the formal algebraic construction (6.29).

n j=1

fj 𝔸

352

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

arithmetic complexity of residue symbols ⎡

⎤

q dY p1

⎢ (7.71) Resℚ[X]/ℚ ⎣ ν

ν

⎥ ⎦ ν

L0p p0 , L2p p2 , . . . , Lnp pn

⎡

⎢ ⎢ = Resℚ[Y ]/ℚ ⎢ ⎣

q



νp j=1 Lj

p1

⎤ dY

⎥ ⎥ ⎥ ⎦

(L0 p0 )νp , (L2 p2 )νp , . . . , (Ln pn )νp is provided according to the transformation law and the arithmetic nullstellensatz Theorem 6.70 or to the Perron generalized theorem, Theorem 7.17 (see §6.6.3 and §6.6.4) the approach described in the present subsection leads to an effective arithmetic nullstellensatz. One should mention that a primitive result [BY91, Theorem 5.1] conjectured the fact that B´ezout’s arithmetic intersection theorem controls the N -solvability of B´ezout’s identity in ℚ[X1 , . . . , Xn ]. We remark though that the logarithmic size estimates obtained there were far from being sharp19 when compared to, for example, those specified in Theorem 6.70. The positive point here 2 remains that the identity (7.65), when τ is specified in ℤn with Φ(τ ) = 0, provides a formulation of Hilbert’s nullstellensatz as a closed formula in terms of algebraic residue symbols Resℚ[X]/ℚ [ ]. One should compare this approach with the result stated as Theorem 7.21, although this last result does not seem to be able to be supported by an algorithmic process. It seems reasonable to guess that A. Cayley, C. Jacobi, L. Kronecker, and F. Macaulay could have been already aware of a kind of division-interpolation path towards an explicit solution of B´ezout identity; see for example, Eugen Netto’s monographs [Ne, Ne1910]. Let us conclude this subsection with a particular case, confirming the intimate connection between Macaulay determinants and resultant theory [Mac] using multivariate residue calculus. Proposition 7.29 is the most immediate extension of Corollary 1.39 from the univariate to the multivariate setting. Proposition 7.29. Suppose that the homogenization P0 , P2 , . . . , Pn of the polynomials pj ’s define a regular sequence in the local ring ℂ[[X0 , . . . , Xn ]].20 Moreover, assume that 3 |divℙnℂ (Pj )| = ∅. (7.72) {ζ0 = 0} ∩ j=1

19 They were nevertheless sharp enough to conjecture that B´ ezout’s geometric and arithmetic intersection theorems rule jointly the effective solvability of B´ezout identity. A rough upper estimate for max(log γ, h(q j )) is expressed in terms of κ(n)(d (p))2 h c(n) (p), where the arithmetic markers h c(n) (p) were introduced in (6.197). This follows from the size estimates in [SomY21, Theorem 6.7] in a straightforward way. Note that the logarithmic size estimates originally given in [BY91, Theorem 5.1] are improved substantially in this parametric presentation. 20 This amounts to saying that R ∗ n+1 : d0 ,...,dn [P0 , . . . , Pn ] ∈ A , where the Pj ’s are the homogenization of the pj ’s in A[X0 , . . . , Xn ] and Rn+1 : d0 ,...,dn denotes the Macaulay resultant of n+1 homogeneous forms in n+1 variables with respective degrees d0 , . . . , dn ; see [Mac1902,Mac], [GKZ, Chapter 13, §1], or [L].

7.2. JACOBI–LAGRANGE–KRONECKER (JLK) PARAMETRIC IDENTITIES

353

Then one has B´ezout identity (7.73)

  ⎤  p0 (X) p1 (X) ··· pn (X)   ⎢ 1  Hp0 ,1 (X, Y ) Hp1 ,1 (X, Y ) · · · Hpn ,1 (X, Y )  ⎥   ⎢ ⎥ dY   ⎢ ⎥ .. .. .. ..  ⎢ p1 (Y )  ⎥ . . . . 1 = −Res𝕂[X]/𝕂 ⎢   ⎥ Hp ,n (X, Y ) Hp ,n (X, Y ) · · · Hp ,n (X, Y ) ⎢ ⎥ 0 1 n ⎢ ⎥ ⎣ ⎦ p0 (Y ), p2 (Y ), . . . , pn (Y ) n  = qj (X)pj (X), ⎡

j=0

and the following degree estimates, expressed in terms of the so-called Macaulay exponent |d| − (n + 1) + 1 = |d| − n, max deg qj pj ≤ |d| − n

(7.74)

0≤j≤n

hold. Proof. Proposition 6.40 implies that the map (p0 , p2 , . . . , pn ) admits (0, . . . , 0) as a multiproperness exponent. It follows from Theorem 6.28 and Theorem 6.56 that the identity * ) b0 (X, Y ) dX (7.75) 1 = Res𝕂[Y ]/𝕂 p0 (Y ), p2 (Y ), . . . , pn (Y ) ⎡ ⎤ 1 b0 (X, Y )p1 (Y ) dX ⎦ , = Res𝕂[Y ]/𝕂 ⎣ p1 (Y ) p0 (Y ), p2 (Y ), . . . , pn (Y ) where (Hp0 ∧ Hp2 ∧ · · · ∧ Hpn )(z, ζ) = b0 (z, ζ) dζ holds in 𝕂[X]. The reformulation of the identity (7.75) as (7.73) relies on the same argument as that used to deduce the identity (7.70) from (7.65), with straightforward degree estimates (7.74).  7.2.2. Parametric membership identities of the JLK form. We consider in this subsection the case of the sequence p = (p1 , . . . , pm ) when m ≤ n. We still  deg pj . As before, one introduces the multiparameter let |d| = m j=1   [j] (7.76) τ = tk : j = 1, . . . , n, k = 1, . . . , n , whose length equals n2 , and we let t[j] = (t1 , . . . , tn ). Given ν ∈ ℕ∗ , one extends the definition of the φj,ν (t, X) ∈ A[t, X], where t = {t1 , . . . , tn }, as given in (7.50) when 1 ≤ j ≤ m, by letting  ν 1 + t, X pj (X) for j = 1, . . . , m, ν (7.77) φj,ν (t, X) =  1 + t, X for m < j ≤ n. [j]

[j]

Then we extend the definition of the Hefer forms for m < j ≤ n by (7.78)

Hφj,ν (t, z, ζ) =

ν−1  ν  ν−1−ρ t, dζ. 1 + t, z 1 + t, ζ ρ=0

354

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

For j = 1, . . . , m we consider the Hefer forms Hφν,j (t, z, ζ) introduced in (7.52). Let also m    ν 1 + t[j] , X (7.79) Dν (τ, X) := j=1

and let (7.80)



bν (τ, X, Y ) = {∈ℕn

Y  bν, (ν, X)

: |κ|≤|d|+(n−1)ν}

be the polynomial with degree in X, Y degX,Y bν (τ, X, Y ) = |d| + n ν − n defined using the product of Hefer forms (7.81)

Hφ1,ν (t[1] , z, ζ) ∧ · · · ∧ Hφn,ν (t[n] , z, ζ) = bν (τ, z, ζ)

n '

dζj .

j=1

Since the supports of the divisors divℂn (pj ) intersect properly in the affine space ℂn (see (7.49)) E. Nœther’s preparation theorem (see for example [L, VdW]) admits now the following reformulation. Proposition 7.30. Let τ be as in (7.76). There exists a polynomial Ψ ∈ ℂ[τ ], whose degree satisfies (7.82)

max

j=0,2,...,n

degt[j] Ψ ≤ 2m+1 (d (p))2 , 2

such that, if τ is specified as an element in ℂn , then Ψ(τ ) = 0

 3  =⇒  V (pj ) ∩

(7.83)

j∈J

= deg

3

3

2

4

  ζ ∈ ℂn : 1 + t[j ] , ζ = 0 

j  ∈{1,...,n}\J

V (pj )

j∈J

for any J ⊂ {1, . . . , m}, with multiplicities taken into account. Proof. The argument of the proof relies on [KPS01, Proposition 4.5], as the construction of the polynomial J⊂{0,2,...,n} ΨJ in the proof of Proposition 7.23 does.  Remark 7.31. It follows from the estimates (7.82) for the degree of Ψ in each 2 block t[j] for j = 1, . . . , n, that one can specify τ as an element in ℤn with a control 2 of logarithmic size. The logarithmic sizes of the entries of at least one τ ∈ ℤn such that Ψ(τ ) = 0 are bounded by (m + 1) log 2 + 2 log d (p). Both assertions on the right-hand side of the implication (7.55) are satisfied for such specialization of τ such that Ψ(τ ) = 0. 2

Corollary 7.32. Let the parameter τ be specified on an element of ℂn such that Ψ(τ ) = 0. There exist strictly positive constants C(τ ) and c(τ ) such that for any subset J ⊂ {1, . . . , m}, 3   1 + t[j  ] , ζ ≥ c(τ ) ζ. V (pj ) and ζ ≥ C(τ ) =⇒  max (7.84) ζ ∈ j∈J

j ∈{1,...,n}\J

7.2. JACOBI–LAGRANGE–KRONECKER (JLK) PARAMETRIC IDENTITIES

355

Proof. It relies on the same argument as the proof of assertion (7.59) in Corollary 7.25 does.  One can transpose in such a new context Proposition 7.26 as follows. Proposition 7.33. Let the multiparameter τ , as in (7.76), be specified as an 2 element of ℂn satisfying Ψ(τ ) = 0, where Ψ is given by Proposition 7.30. For any integer ν such that ν > d (p), the polynomial map   ζ ∈ ℂn −→ φ1,ν (t[1] , ζ), . . . , φn,ν (t[n] , ζ) ∈ ℂn   admits d1 + d (p), . . . , dm +d (p), d (p), . . . , d (p) as a multi-exponent of properness with respect to the affine space 𝔸nℂ . Proof. Fix from now on21 0 < ε < ν − d (p). Sharp nullstellen inequalities, as formulated in [Cy05, Theorem 5.1], show that there exists c ∈ ]0, 1] such that one has    dist z, 5

j∈J dj j∈J V (pj ) (7.85) max |pj (z)| ≥ c j∈J max(1, z2 ) for any subset J ⊂ {1, . . . , n}. Moreover, for any z ∈ ℂn the relations (7.86)     z ≥ C =⇒ max max |pj (z)| z d (p)+ε , max 1 + t[j] , z z d (p)+ε ≥ c 1≤j≤m

j>m

also hold. Let z ∈ ℂ such that z ≥ C. Note that we will need to re-inforce this constraint later. We now distinguish two cases. Suppose first that   (7.87) max 1 + t[j] , z z d (p)+ε ≥ c. n

j>m

Either max1≤j≤m |pj (z)| z d (p)+ε < c or max1≤j≤m |pj (z)| z d (p)+ε ≥ c. • If max1≤j≤m |pj (z)| z d (p)+ε < c, then it follows from the estimate (7.85) with J = {1, . . . , m} that  5m  dist z, j=1 V (pj ) ≤ z−ε/d (p) . (7.88) z  5m  5 Let ζz ∈ m j=1 V (pj ) be such that z − ζz  = dist z, j=1 V (pj ) . Since   max 1 + t[j] , ζz  ≥ c(τ ) ζz  j>m

(7.89)

according to (7.84), the inequality  c(τ )  max 1 + t[j] , z ≥ ζz  j>m 2 also holds, in view of (7.88), provided the hypothesis z ≥ C is re-inforced as |z| ≥ Cτ,ε ≥ C sufficiently large. Therefore, one has   max φν,j (t[j] , z) ≥ (c(τ ) c)ν zν− d (p)−ε = c (τ ) zν− d (p)−ε j>m

according to (7.87). 21 Although the proof of the proposition is closely related to that of [El93, Lemma 2.2] (see also [BGVY, Proposition 5.8]) and relies (as in [BY91, §5]) on the use of Nœther’s preparation theorem, we present a different and more direct approach.

356

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

• If max1≤j≤m |pj (z)| z d (p)+ε ≥ c, then let us order in increasing size the m positive numbers |pj (z)| z d (p)+ε , namely (7.90) |pj1 (z) (z)| z d (p)+ε ≤ |pj2 (z) (z)| z d (p)+ε ≤ · · · ≤ |pjm (z) (z)| z d (p)+ε . One repeats now the argument, already used in the proof of Proposition 7.26, starting from the reordering (7.90) instead of (7.61). Let 2 4 r(z) = max ρ ∈ {1, . . . , m} : |pjρ (z)| z d (p)+ε < c .

(7.91)

(7.92)

It follows from (7.85) that  5  dist z, r(z) ρ=1 V (pjρ )

≤ z−ε/

r(z)

ρ=1 djρ ≤ z−ε/d (p) . z  5r  5r Let again ζz ∈ ρ=1 V (pjρ ) be such that z − ζz  = dist z, ρ=1 V (pjρ ) . Since   1 + t[j  ] , ζz  ≥ c(τ ) ζz  max 

j ∈{1,...,n}\{j1 ,...,jr(z) }

according once again to (7.84), it follows from (7.91) that the inequality   1 + t[j  ] , z ≥ c(τ ) z 2 j  ∈{1,...,n}\{j1 ,...,jr } max

(7.93)

also holds, provided, once again, that the initial constraint z ≥ C is re-inforced as |z| ≥ Cτ,ε ≥ C sufficiently large. Then, taking into account the definition of r(z), one has that   φν,j  (t[j  ] , z) ≥ c (τ ) zν− d (p)−ε . max  j ∈{1,...,n}\{j1 ,...,jr(z) }

Suppose now that we are in the second case, namely, we assume that   max 1 + tj , z z d (p)+ε < c j>m

holds in place of (7.87). This implies according to (7.86) that max |pj (z)| z d (p)+ε ≥ c.

1≤j≤m

From now on, we repeat once more the argument used to treat the situation in the second subcase above or that we invoked when proving Proposition 7.26. It leads this time, provided the constraint on |z|  1 is re-inforced, to the lower estimate (7.94)

max |φj,ν (t[j] , z)| ≥ c (τ ) zν− d(p)−ε .

1≤j≤m

Since the inequalities (7.89), (7.93), or (7.94) are satisfied by z for z ≥ Cτ,ε (depending on the situation among the three we analyzed) and since ε is arbitrarily small, Proposition 7.33 is proved.  We may now state, as a consequence of Proposition 7.33, a parametric realization with denominators of the membership of q ∈ A[X] to the polynomial ideal (p1 , . . . , pm ) 𝕂[X].22 22 A nonparametric version of such result (with M = 3) was first proved by Mohamed Elkadi [El93, Theorem 3.1] within the arithmetic setting where A = ℤ.

7.2. JACOBI–LAGRANGE–KRONECKER (JLK) PARAMETRIC IDENTITIES

357

Theorem 7.34. Let m ≤ n and p1 , . . . , pm ∈ A[X] = A[X1 , . . . , Xn ] be such that the supports of the divisors divℂn (pj ) intersect properly in ℂn , according to (7.49). Assume also that q ∈ A[X] ∩ (p1 , . . . , pm ) 𝕂[X]. Fix M ∈ ℕ∗ with M ≥ 2 and let (7.95) M deg q + |d| + n d (p) +m d (p), ν := νp (M ) = M d (p), K = Kp,q (M ) := M −1 M −1 and let τ be the set of n2 parameters as in (7.76). Then the following algebraic identity holds in 𝕂(τ, X): (7.96) q(X) =

1 Dν (τ, X)

 {κ∈ℕn : |κ| d (p) ≤Kp,q , min1≤j≤m κj ≥1}

)

* n q(Y ) Dν (τ, Y ) bν (τ, X, Y ) dY κ φj,νj (t[j] , X) κ1 +1 [1] κn +1 [n] φ1,ν (t , Y ), . . . , φn,ν (t , Y )

Res𝕂(τ )[Y ]/𝕂(τ )

j=1

=

1 Dν (τ, X)

m 

qj (τ, X) pj (X).

j=1

The polynomials qj in (7.96) belong to 𝕂(τ )[X] and satisfy the degree estimates

M + 1 (7.97) max degX (qj (τ, X)pj (X)) ≤ deg q + |d| + (mM + n) d (p) . 1≤j≤m M −1 Proof. It follows Proposition 7.33 and Theorem 6.56, applied once τ is spec2 ified as an element in ℂn \ V (Ψ), that |κ| d (p) > Kp,q ⇐⇒ (M − 1)(|κ| + n) d (p) > deg q + |d| + M (m + n) d (p)

(7.98) =⇒ (M − 1)(|κ| + n) d (p) > degY q(Y )Dν (τ, Y ) bν (τ, X, Y ) + n ) * q(Y ) Dν (τ, Y ) bν (τ, X, Y ) dY =⇒ Res𝕂(τ )[Y ]/𝕂(τ ) κ1 +1 [1] = 0, κn +1 [n] (t , Y ) φ1,ν (t , Y ), . . . , φn,ν # where dY = nk=1 dYk . Furthermore, Theorem 6.28 implies that one has the polynomial identity  (7.99) q(X) Dν (τ, X) = {κ∈ℕn : |κ| d (p)≤Kp,q }

)

Res𝕂(τ )[Y ]/𝕂(τ )

* n q(Y ) Dν (τ, Y ) bν (τ, X, Y ) dY κ φj,νj (t[j] , X) κn +1 [n] 1 +1 φκ1,ν (t[1] , Y ), . . . , φn,ν (t , Y ) j=1

in 𝕂(τ )[X]. The formal construction of algebraic symbols as traces in (6.29), namely n  the fact that (ωa,(0,...,0) (h))T = 0, when h belongs to j=1 fj 𝔸, implies that for any κ ∈ ℕn such that κ1 = · · · = κm = 0, ) * q(Y ) Dν (τ, Y ) bν (τ, X, Y ) dY Res𝕂(t)[Y ]/𝕂(t) κ1 +1 [1] = 0, κn +1 [n] φ1,ν (t , Y ), . . . , φn,ν (t , Y ) since the polynomial q(X)Dν (τ, X) lies in the ideal generated by the polynomials φ1,ν (τ, X), . . . , φm,ν (τ, X) in 𝕂(τ )[X]. The algebraic identity (7.96) then follows.

358

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

Let us split bν (τ, X, Y ) as in (7.80) and consider for κ ∈ ℕn with min1≤j≤m κj ≥ 1 the contribution (7.100) ) * n q(Y ) Dν (τ, Y ) Y  dY κ φj,νj (t[j] , X) bν, (τ, X)Res𝕂(t)[Y ]/𝕂(t) κ1 +1 [1] κn +1 [n] φ1,ν (t , Y ), . . . , φn,ν (t , Y ) j=1

in the numerator of the right-hand side of the algebraic identity (7.96). Since min1≤j≤m κj ≥ 1, such a contribution splits as m j=1 qj, (τ, X)pj (X). It remains then to estimate max1≤j≤m degX (qj, (τ, X)pj (X)). Observe first that such a contribution (7.100) equals 0 unless |κ| d (p) ≤

(7.101)

deg q + n d (p) + || + M (m − n) d (p) + n , M −1

according to Theorem 6.56, which we assume from now on. Then relation (7.101) implies (7.102) degX

n 

1 κ φj,νj (t[j] , X) ≤ |κ| (M d (p) + dm ) ≤ |κ| d (p) M + m−1 1

j=1

≤

dj

 M +1 deg q + n d (p) + || + M (m − n) d (p) + n . M −1

On the other hand   degX bν, (τ, X) ≤ |d| + n M d (p) − 1 − ||.

(7.103)

It follows from the estimates (7.102) and (7.103) that   max degX qj, (τ, X)pj (X) 1≤j≤n



2 M +1 M +1 deg q + (|| + n) + m M − n d (p) M −1 M −1 M −1

M + 1 ≤ deg q + |d| + (mM + n) d (p) , M −1 ≤ |d| +

since || + n ≤ |d| + nM d (p). This provides the required estimates (7.97), and thus concludes the proof of Theorem 7.34.  Consider now n + 1 sets of parameters τ0 , . . . , τn of the form   [j] (7.104) τρ = tρ,k : j = 1, . . . , n, k = 1, . . . , n , ρ = 0, . . . , n, such that T = {τ0 , . . . , τn } is a transcendental basis of 𝕂(T ). Let Rn+1 : m,...,m be the Macaulay resultant of n+1 homogeneous forms in n+1 variables with respective degrees m, . . . , m. Let also Rn : m,...,m be the Macaulay resultant of n homogeneous forms in n variables with corresponding degrees m, . . . , m. We recall that Rn+1 :m,...,m (respectively Rn : m,...,m ) is a multi-homogeneous polynomial with integer coefficients in n) blocks of variables. All  1) (respectively n−1+m   (n + (respectively ). The polynomials blocks of variables have length n+m m m

7.2. JACOBI–LAGRANGE–KRONECKER (JLK) PARAMETRIC IDENTITIES

359

Rn+1 : m,...,m (respectively Rn : m,...,m ) are homogeneous with degree mn (respectively mn−1 ) in each block. For each ρ = 0, . . . , n, let Hρ,ν (T, X, Y ) =

n 

Hρ,ν,k (T, X, Y ) dYk =

k=1

n 

Hρ,k (τρ , X, Y ) dYk

k=1

be the K¨ ahler 1-form with coefficients in ℤ[T, X, Y ] corresponding to a Hefer algebraic form for ζ −→ Dν (τρ , ζ). It follows from Proposition 7.29 that B´ezout identity (7.105) 1 = −Res ℚ(T )[X]/ℚ(T )   ⎤ ⎡  Dν (τ0 , X) Dν (τ1 , X) ··· Dν (τn , X)    H0,ν,1 (T, X, Y ) H1,ν,1 (T, X, Y ) · · · Hn,ν,1 (T, X, Y )  ⎥ ⎢ 1   ⎥ ⎢ dY   ⎥ ⎢ .. .. .. ..  ⎥ ⎢ Dν (τ1 , Y )  . . . .   ⎥ ⎢ H0,ν,n (T, X, Y ) H1,ν,n (T, X, Y ) · · · Hn,ν,n (T, X, Y ) ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ Dν (τ0 , Y ), Dν (τ2 , Y ), . . . , Dν (τn , Y ) n  = Qν,ρ (T, X) Dν (τρ , X), ρ=0

holds in ℚ(T )[X] for any ν ∈ ℕ∗ . Furthermore, Macaulay degree estimates (7.106)

max

0≤ρ≤n



  degX Qν,ρ (T, X) Dν (τρ , X) ≤ (n + 1)m ν − n,

hold in ℚ(T )[X] for any ν ∈ ℕ∗ also. Moreover, (7.105) provides a B´ezout identity 2 in ℚ(X), provided T is specified on an element of ℤn (n+1) such that (7.107)

Rn : m,...,m [(D1 (τ0 , ·))m , (D1 (τ2 , ·))m , . . . , (D1 (τn , ·))m ] = 0, (τ0 , ·), Dhom (τ1 , ·), . . . , Dhom (τn , ·)] = 0, Rn+1 : m,...,m [Dhom 1 1 1

where the (D1 (τρ , ·))m ’s denote the homogeneous parts of higher degree m for the polynomials D1 (τρ , X) considered as polynomials in X, and the Dhom (τρ , X)’s de1 note their homogenization. We may now conclude this subsection with the following main result. Theorem 7.35. Let m ≤ n and let the polynomials p1 , . . . , pm ∈ A[X] be such that the supports of the divisors divℂn (pj ) intersect properly in ℂn as in (7.49). Let also q ∈ A[X] ∩ (p1 , . . . , pm ) 𝕂[X]. Fix M ∈ {2, 3, . . . , } and let (7.108) M deg q + |d| + n d (p) +m d (p). ν := νp (M ) = M d (p), K = Kp,q (M ) := M −1 M −1

360

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

If T = {τ0 , . . . , τn } is a set of n2 (n + 1) parameters as in (7.104), then the following membership parametric identity holds in 𝕂(T )[X]: (7.109) q(X) =

n 



Qν,ρ (T, X)

{κ∈ℕn : |κ| d (p) ≤Kp,q , min1≤j≤m κj ≥1}

ρ=0

)

Res𝕂(τ )[Y ]/𝕂(τ )

* n q(Y ) Dν (τρ , Y ) bν (τρ , X, Y ) dY  κj [j] φj,ν (tρ,· , X) κ1 +1 [1] κn +1 [n] φ1,ν (tρ,· , Y ), . . . , φn,ν (tρ,· , Y ) j=1

=

n 

Qν,ρ (T, X)

ρ=0

m 

qj (τρ , X) pj (X) =

j=1

m 

Qj (T, X) pj (X).

j=1

The polynomials qj (τρ , X) ∈ A[τρ , X] and Qν,ρ (T, X) ∈ ℤ[T, X] are defined for ρ = 0, . . . , n and j = 1, . . . , n by (7.96) and (7.105), respectively. Moreover, the polynomials Qj (T, X) ∈ 𝕂(T )[X] satisfy the degree estimates (7.110)

max degX (Qj (T, X)pj (X))

1≤j≤m

≤

M + 1 deg q + |d| + (mM + n) d (p) + (n + 1) M d (p) − n. M −1

Proof. The results follow from B´ezout identity (7.105) in ℚ(T )[X], once combined with the parametric realization with denominators of the membership of q with respect to (p1 , . . . , pm ) A[X] given in Theorem 7.34 by (7.96).  Example 7.36. Let A = ℤ. The parameters T introduced in Theorem 7.35 2 can be specified on an element T ◦ = (τ0 , . . . , τn ) ∈ ℤn (n+1) such that n 

Ψ(τρ ) = 0 (7.111)

ρ=0

Rn : m,...,m [(D1 (τ0 , ·))m , (D1 (τ2 , ·))m , . . . , (D1 (τn , ·))m ] = 0 (τ0 , ·), Dhom (τ1 , ·), . . . , Dhom (τn , ·)] = 0. Rn+1 : m,...,m [Dhom 1 1 1

Moreover, one deduces from the estimates for deg Ψ (see (7.82)) and from the fact that the degrees of Rn+1 : m,...,m and Rn : m,...,m in each block of variables equal respectively mn and mn−1 , an upper estimate for the logarithmic size of such specialization of T ; see for example Remark 7.31. With T specialized as such 2 an element in ℤn (n+1) , (7.96) provides a formula m  q j (X) pj (X), γ q(X) = j=1

after lifting denominators, where γ ∈ ℕ∗ and q j ∈ ℤ[X] for j = 1, . . . , m. Thus, one could compare Theorem 7.35 to [KP94, Theorem 2] (in the case where m = n), which carries an algorithmic process materialized by the straight line program 𝕊𝕃 ((p1 , . . . , pm )) on ℤ[X]. Although Theorem 7.35, which produces after specialization of the parameters a closed formula for the membership problem in ℚ(X) in the case of proper intersections, is not accompanied by such an algorithmic process, it is worthwhile to observe that multivariate residue calculus (that is, the Jacobi– Lagrange–Kronecker approach) is deeply involved in the realization of the closed formula it produces. Estimates for log γ and h (q j ) in terms of the geometric and

7.3. EFFECTIVE GEOMETRIC BRIANC ¸ ON–SKODA–HUNEKE TYPE THEOREMS

361

arithmetic markers d (p) and h c (p) follow from Theorem 6.72 or Theorem 6.76 (in the case where N = n and 𝕏 = 𝔸nℚ ). One needs to use for that the observation (7.71). The quantities log γ and the h (q j ) are roughly estimated from above by κ(n) (d (p))3 h c(m,n) . Such identities, together with logarithmic size estimates, are due to Mohamed Elkadi [El93, Theorem 3.1]. Bounds obtained in [El93, Theorem 3.1] can be improved in view of the results established since then. Namely, the estimates can be made sharper using the Perron parametric theorem (Theorem 7.17) and its arithmetic implications, once combined with the transformation law or its generalizations (Theorem 3.21 or Proposition 3.23). Nevertheless, they still remain probably far from optimal. The main point to insist on is the fact that Theorem 7.35, as Theorem 7.27 does for B´ezout identity, leads to closed formulae while (almost) respecting jointly geometric and arithmetic intersection theory. 7.3. Effective geometric Brian¸ con–Skoda–Huneke type theorems 7.3.1. Global algebraic Brian¸ con–Skoda–Huneke exponents μ(𝕏 : ℙN ℂ ). It is assumed throughout this section that 𝕏 is a purely n-dimensional algebraic subvariety of ℂN . Let 𝕁 = 𝕁(𝕏) be its radical ideal in ℂ[X1 , . . . , XN ] and let 𝕏 be its n-dimensional Zariski closure in ℙN ℂ . Let also 𝕀 = 𝕀(𝕏) be the homogeneous radical ideal in the graded algebra 𝕊 = ℂ[X0 , . . . , XN ] = ℂ[X] satisfying (7.112)

Φ ∈ 𝕀 ⇐⇒ Φ(z) = 0 ∀ z = [z0 : · · · : zN ] ∈ 𝕏.

Following the presentation in §4.6.3, we consider a finite collection of homogeneous polynomials {Φι }ι in the graded algebra 𝕊 = ℂ[X0 , . . . , XN ], with corresponding degrees Dι and such that  (7.113) 𝕀= Φι ℂ[X]. ι

Consider the mapping 𝔽Φ :

(

(

Qι ∈

ι

𝕊−Dι −→

ι



Qι Φι ∈ 𝕊,

ι

where the graded modules 𝕊−δ for δ ∈ ℕ are defined in (4.131). The 𝕊-module 𝕄−1 = 𝕊/Im 𝔽Φ is finitely generated and the complex of graded free 𝕊-modules (7.114) 𝔽

𝔽 +1

𝔽 −1

𝔽

𝔽

𝔽

Π

L

2 Φ 𝕄Φ • : 0 −→ 𝕄L −→ · · · −→ 𝕄 −→ 𝕄−1 −→ · · · −→ 𝕄1 −→ 𝕊 −→ 𝕄−1 −→ 0,

where (7.115)

𝕄 =

(

𝕊−d[ ] , ι

 = 1, 2, . . . ,

ι

and Π denotes the quotient map, is exact. Hilbert’s syzygy theorem [Hil1890] (see also [Eis05, Chapter 1A]) implies that this complex has length L ≤ N + 1. Moreover, the complex (7.114) becomes unique to an isomorphism (see [Eis05, §1A and §1B]) if one takes it minimal, that is Im 𝔽 ⊂

N  

Xk 𝕊−d[ −1] ι

k=0

ι

362

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

for any  ≥ 1.23 From now on the resolution (7.114) will be taken as minimal. Since 𝕏 is purely dimensional, the depth of the quotient ring satisfies depth(ℂ[X]/𝕀) ≥ 1 and the length L of such a minimal resolution in (7.114) is bounded from above by N = (N + 1) − 1, improving the upper bound N + 1 provided by Hilbert’s syzygy theorem. Besides the marker deg 𝕏 = deg 𝕏 = d for the geometric complexity of 𝕏 (see §6.6.2) defined intrinsically and independently of the embedding 𝕏 → ℙN ℂ, one needs to introduce the Castelnuovo–Mumford regularity r(𝕏), more precisely the Castelnuovo–Mumford regularity of 𝕏 → ℙN ℂ . It is a geometric invariant of 𝕏 → ℙN , which depends on the embedding. On the other hand, r(𝕏) is a marker for ℂ the algebraic complexity of 𝕏 as embedded in ℙN . Let us recall here the definition ℂ N of this geometric invariant of 𝕏 → ℙℂ . Definition 7.37.24 The Castelnuovo–Mumford regularity r(𝕏) of 𝕏 ⊂ 𝕏 → ℙN ℂ is defined analogously to the regularity of the quotient of 𝕊 by the homogeneous ideal 𝕀. That is, given any set Φ = {Φι }ι of generators for 𝕀 in ℂ[X] and any minimal exact complex of graded free 𝕊-modules 𝕄 = 𝕄Φ • for ℂ[X0 , . . . , XN ]/𝕀 as in (7.114),   (7.116) reg(𝕏) = reg(𝕏) := reg ℂ[X0 , . . . , XN ]/𝕀 + 1 = max(d[] ι − ) + 1. ,ι

[]

The degrees dι in (7.115) for  = 1, 2, . . . are called the graded Betti numbers of 𝕀 (or of 𝕊/𝕀) and are independent of the minimal exact complex 𝕄. Both markers d and reg(𝕏), defined for the geometric complexity of 𝕏 (intrinsically defined) and for its algebraic complexity (when considered as embedded in ℙN ℂ ) respectively, will play a key role throughout §7.3. It is now time to come back to multidimensional residue theory. We proceed as follows. Consider the correspondences ( ( []∗ 𝕊−d[ ] ←→ E = O(−d[]  = 1, 2, . . . , 𝕄 = ι ) ι , ι

ι

ι

[] {ι }ι

where the notation denotes, as usual, a copy of the canonical basis of ℤrank 𝕄 , indexed by . We construct, as in §4.6.3, from 𝕄Φ • a generically exact complex of holomorphic bundles (7.117) ℂ FL−1 F2 F0 FL F1 =FΦ EL−1 −→ · · · −→ E1 −→ ℂ −→ E−1 = −→ 0. F• : 0 −→ EL −→ Im FΦ []

Let us metrize F• as F• by equipping with the Fubini–Study metric each O(−dι ) for  = 1, . . . , L and any ι. Let us also fix, for the time being, a smooth nonvanishing #n 1,0 = T on ℙN section of the inverse of the canonical bundle KℙN ℂ . Namely, we ℙN ℂ ℂ consider N # k 0≤k ≤n : k =k dzk k=0 (−1) zk ∞ ∈ CN,0 Υ= (ℙN (ℂ)). χ(z0 , . . . , zn ) Then the equality of currents (7.118)

RF• ∧ Υ = R0F• ∧ Υ = (ι𝕏→ℙN )∗ ωΥ ℂ

23 We refer the reader for more details and relevant bibliography to §4.6.3, where these concepts are described. 24 For more details about the assertions contained in the following definition, we refer the reader to [Eis05, §4A and §4E, Exercice 4.3].

7.3. EFFECTIVE GEOMETRIC BRIANC ¸ ON–SKODA–HUNEKE TYPE THEOREMS

363

defines in the local setting the structure current, as described in §3.4.3. The structure current is also known as structure form since it is represented by an almost semimeromorphic form on the complex analytic space 𝕏. The structure form is attached to the complex F• over the ambient manifold ℙN ℂ , to its Fubini–Study metrization, and to the choice of the smooth section Υ. Moreover, one has locally on 𝕏 (see Proposition 3.49) that (7.119) ω = ωF• ,Υ = & )∗ = (Π 𝕏

n−1 

ωr

r=0 n−1 

ρ=0

|e |2μ1 |e |2μ0     |eρ |μρ

1 &∗ f ··· sρ · · · s1 0  s0 Π 𝕏 eρ e1 e0 μ0 =0 μρ−1 =0 μρ =0

is realized as the pushforward of semimeromorphic forms through iterated normalized blowups, as specified in (3.133). Actually, the eρ ’s in (7.119) are holomorphic sections of holomorphic line bundles, while the sρ ’s are smooth bundle-valued (0, 1) forms, s0 is a smooth (bundle-valued) function, and f is a regular (bundle valued) differential n-form on 𝕏 in the sense of Barlet (this concept is defined in §B.2.5). Moreover, the realization (7.119) of ω is uniquely determined, since Υ does not vanish. Observe that ω is smooth on 𝕏reg and that the holomorphic sections & in (7.119) depend only on the Fitting ideal e0 , e1 , . . . , en−1 of line bundles over Π 𝕏 sheaves of I𝕏 , as introduced in Definition 3.44; see §3.4.3. Hence the holomorphic sections e0 , e1 , . . . , en−1 of line bundles in (7.119) depend only on the radical ideal sheaf I𝕏 , that is, only on the homogeneous ideal 𝕀 of 𝕊. One concludes from the preceding discussion that the singularities of the structure form ω, appearing in its realization as the pushforward in (7.119), depend only on the embedding 𝕏 → ℙN ℂ. Let P1 , . . . , Pm ∈ ℂ[X] with respective degrees d1 , . . . , dm . The corresponding $m Koszul complex KP ℙN is attached to the section P = • over j=1 Pj εj of the $m ℂ O N (d ), where each O N (d ) is equipped with a holomorphic m-bundle j j ℙℂ j=1 ℙℂ Fubini–Study metric. The principal value and residue currents attached to such a metrized Koszul complex are of the form

P P K• = |P |2λ

min(m,N +1)

= |P |2λ

min(m,N +1)

 r=1

(7.120)



P ∗ r−1  P∗ ∂ 2iπ|P |2 2iπ|P |2 λ=0 QP r



r=1

P RK• = ∂|P |2λ



min(m,N )

QP r

r=1

, λ=0

 , λ=0

described in Proposition 3.28 and Example 3.30. Then, given the structure form denoted by ω = ωΥ attached by (7.118) to the residue current RF• = R0F• , the current (7.121)



KP R𝕏 • ω := ι∗𝕏→ℙN ∂|P |2λ ℂ



min(m,N )

r=1

 ∧ ω QP r

λ=0

364

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

is well defined as a pseudo-meromorphic bundle-valued current on the complex analytic space 𝕏 by (7.119) and Proposition 2.21; see also Remark 2.29. We already used such a construction locally in the more general situation where the metrized Eagon–Northcott complex replaced the metrized Koszul complex; see §5.3. In fact, while proving Theorem 5.10 we established a local result that can be made global on the compact complex analytic space 𝕏. It is the content of the following theorem. Theorem 7.38. Let 𝕏 ⊂ 𝕏 ⊂ ℙN ℂ and 𝕀 be as above. There is a uniform ) ∈ ℕ such that for any F• constructed from a minimal minimal exponent μ(𝕏 : ℙN ℂ (D) resolution of ℂ[X]/𝕀, any Υ, any P = (P1 , . . . , Pm ) as above, and any Θ ∈ OℙN ℂ ∗ with D ∈ ℕ , (7.122)

sup z∈𝕏\V (P )

|Θ(z)| |P (z)|

μ(𝕏 : ℙN ℂ )+n

 KP  < +∞ =⇒ R𝕏 • ω ι𝕏→ℙN Θ ≡ 0 ℂ

as a bundle-valued current on the complex analytic space 𝕏. Moreover, one has N μ(𝕏 : ℙN ℂ ) = 0 if 𝕏 is a complex n-dimensional submanifold of ℙℂ . Proof. Such a uniform integer was defined locally as   /I𝕏 z μ𝔸(z) with 𝔸(z) = OℙN ℂ for any z ∈ 𝕏, while proving Theorem 5.10 with k = 1. The globalization of all μ𝔸(a) ’s as μ(𝕏 : ℙN ℂ ) follows from a compactness argument. Such a global exponent N μ(𝕏 : ℙℂ ) ∈ ℕ is finally chosen to be minimal. The fact that μ(𝕏 : ℙN ℂ ) = 0 when 𝕏 is smooth follows from Brian¸con–Skoda’s theorem (Theorem 5.7 with k = 1) in the local regular N -dimensional ring OℙN for any z ∈ 𝕏.  ℂ ,z Theorem 5.7 is due to Jo¨el Brian¸con and Henri Skoda [BriS74], while Theorem 5.10 is due to Craig Huneke [Hu92]. From the point of multivariate residue theory, the following definition makes sense. Definition 7.39. The exponent μ(𝕏 : ℙN con–Skoda– ℂ ) is called the Brian¸ Huneke exponent of 𝕏 with respect to the embedding 𝕏 ⊂ ℙN . ℂ We introduce now the direct sum KP • ⊕ F• of the complex F• with the Koszul P,1 complex KP = EN as in (5.42). Thus, we extend the action of the morphisms • • P ( ) and F as in (5.43) and define the induced actions25    N   P  P D ℙN ℂ , E(K• ⊕ F• ) −→ D ℙℂ , E(K• ⊕ F• ) . In particular, which is important here for us, it induces the action of the operator P P ∇KP• ⊕F• from the space of E(KP • ⊕F• ) = Homℂ (E(K• ⊕F• ), E(K• ⊕F• ))-valued currents on ℙN ℂ to itself. Let us recall the current identity:

P

P 

 (7.123) ∇KP• ⊕F• P F• + P K• ,λ R0F• = IdE(KP• ⊕F• ) − RK• ,λ R0F• λ=0

λ=0

(see (5.45) with a = P and k = 1, combined with RF• = R0F• ). 25 We refer the reader to §3.3.1 and §3.3.2 for the notation used here and the definition of the ∇ operator.

7.3. EFFECTIVE GEOMETRIC BRIANC ¸ ON–SKODA–HUNEKE TYPE THEOREMS

365

Corollary 7.40. For any 𝕏, F• , P = (P1 , . . . , Pm ), D, and Θ ∈ O(D) as in Theorem 7.38, one has that (7.124)

sup z∈𝕏\V (P )

|Θ(z)| |P (z)|μ(𝕏 : ℙℂ )+n N

< +∞

=⇒ Θ = ∇KP• ⊕F•

P  P F• + P K• ,λ R0F•

Θ

λ=0

in the sense of currents on ℙN ℂ. Proof. Since 

P RK• ,λ R0F•

λ=0

 KP  Θ = 0 ⇐⇒ R𝕏 • ω ι𝕏→ℙN Θ = 0, ℂ

by (7.118) and (7.121), the result follows from Theorem 7.38.



Remark 7.41. In fact, the integer μ(𝕏 : ℙN ℂ ) could be chosen as an integer μ(𝕏) depending intrinsically on the projective algebraic variety 𝕏, since the singularities of any structure form ωF• ,Υ (a priori attached to some embedding such as 𝕏 → ℙN ℂ , as is the case here) do not depend on such an embedding; see [AndW15, Proposition 2.5 and §3]. We refer the reader to these results for more details. We will not enter into such considerations here and will always remain in the more restrictive embedded situation, which had been our point of view throughout the present monograph. 7.3.2. An effective Brian¸ con–Skoda–Huneke type theorem on 𝕏 ⊂ ℂN . We present in this subsection an effective global version of Brian¸con–Skoda– Huneke’s theorem formulated in Theorem 5.10 for the local ring (OℂN /IV )0 . In such a global setting, 𝕏 is, as in the previous subsection, a purely n-dimensional algebraic subvariety of ℂN . We attach to its closure 𝕏 in ℙN ℂ the Fubini–Study metrized complex F• with length L ≤ N as in (7.117). Let p = (pι )ι : ℂN −→ ℂm be a polynomial map and let |d(p)|∞ := max deg pι .

(7.125)

ι

Recall Definition 7.5 as formulated within the context of a general algebraically N closed field 𝕂 instead of ℂ but with n = N and the affine space 𝔸N in place 𝕂 =𝕂 of 𝕏. Instead, we consider here the subvariety 𝕏 as above and introduce the notion N of L  ojasiewicz exponent on 𝕏 along 𝕏 ∩ (ℙN ℂ \ ℂ ). We consider for that purpose the normalized blowup & −→ 𝕏 π &P : 𝕏

(7.126)

of the purely n-dimensional complex analytic space 𝕏 with respect to the coherent N ideal sheaf generated by the pullbacks via the embedding of 𝕏 into ℙ ℂ , of the sections z → Pι (z), where the Pι ’s is the homogenization of the pι ’s. Let γ mγ Hγ be its exceptional divisor. Normality in codimension 1 implies that the localizations O𝕏,H  γ,

where

mγ ∈ ℕ ∗

and

N π &P (Hγ ) ⊂ 𝕏 ∩ (ℙN ℂ \ ℂ ),

are discrete valuation rings. Then we can reformulate Definition 7.5 in this new context as follows.

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Definition 7.42. The strictly positive rational number (7.127) μ∞ (𝕏 , p) :=

max

N {γ : mγ ∈ℕ∗ , π P (Hγ )⊂𝕏 ∩ (ℙN ℂ \ℂ )}

mγ   length OℙN ,H /(z0 ) OℙN ,H ℂ

γ

ℂ

γ

N is called the L  ojasiewicz exponent of p on 𝕏 along 𝕏 ∩ (ℙN ℂ \ ℂ ).

It follows from the fact that  mγ degO N (|d(p)|∞ ) π &(Hγ ) ≤ degO ℙ

γ

ℙN ℂ

ℂ

(|d(p)|∞ )

𝕏,

see [EinL99], [Laz, (5.20)], [AndW15, p. 641], and from Wirtinger’s formula (6.175), that one has the important inequality  ∞ r ∞ (p) (7.128) μ (𝕏 , p)] + 1 ≤ d ∞ (p) = d |d(p)|∞𝕏 ≤ d |d(p)|n∞ , where (7.129) ∞ (p) r𝕏 N N := max{codim𝕏 𝕏 ∩ (ℙN &P (Hγ ) ⊂ 𝕏 ∩ (ℙN ℂ \ ℂ ) : γ such that π ℂ \ℂ ) ≤ n.

4

We can now state the main result in this section; see [AndW15, Theorem A]. Theorem 7.43. Let 𝕏 be a purely n-dimensional algebraic subvariety of ℂN with degree d and Castelnuovo–Mumford regularity reg(𝕏), when embedded into N con– ℙN ℂ . Let 𝕁 be its defining ideal in ℂ[X] and let μ(𝕏 : ℙ (ℂ)) be the Brian¸ Skoda–Huneke exponent as described in Definition 7.39. Let also p be a polynomial ∞ (p) be defined by (7.125) and (7.129), map from ℂN to ℂm , with |d(p)|∞ and let r𝕏 respectively. Suppose that q ∈ ℂ[X] is such that the almost everywhere defined function (7.130)

φ : z ∈ 𝕏 −→

is locally bounded on 𝕏. Then q¯ ∈ membership as (7.131)

q(X) =

 ι



|q(z)| p(z)μ(𝕏 : ℙℂ )+n N

p¯ι in ℂ[X]/𝕁 ℂ[X], and one can realize this qι pι

modulo

𝕁

ι

with the degree estimates (7.132)

max deg qι pι ι

∞   r𝕏 (p) ≤ max deg q + μ(𝕏 : ℙN , (n + 1)(|d(p)|∞ − 1) + reg(𝕏) , ℂ ) + n d |d|∞

Proof. Let Q ∈ ℂ[X0 , . . . , XN ] be the homogenization of q,

∞   r𝕏 (p) M = max deg q + μ(𝕏 : ℙN , (n + 1)(|d(p)|∞ − 1) + reg(𝕏) ℂ ) + n d |d|∞

7.3. EFFECTIVE GEOMETRIC BRIANC ¸ ON–SKODA–HUNEKE TYPE THEOREMS

367

& −→ 𝕏 be the norand let Θ(X0 , . . . , XN ) = Q(X0 , . . . , XN )X0M −deg q . Let π &P : 𝕏 malized blowup of the n-dimensional complex analytic space 𝕏 along the coherent sheaf of ideals generated in O𝕏 by the sections ι∗𝕏→ℙN (z → Pι (z)) ℂ

as in (7.126). We repeat at this point the argument already used in the proof of our previous Proposition 7.10. Let γ mγ Hγ be the corresponding exceptional divisor & It follows from the definition of μ∞ (𝕏 , p) on the normal complex analytic space 𝕏. with its upper estimate (7.128) that ∞     r𝕏 (p) mγ,∞ ≥ μ(𝕏 : ℙN μ(𝕏 : ℙN ℂ ) + n d |d|∞ ℂ ) + n mγ for all the irreducible hypersurfaces Hγ such that π &P (Hγ ) lies entirely in the algeN braic hypersurface 𝕏 ∩ (ℙN ℂ \ ℂ ) of 𝕏. On the other hand, it follows from the fact that φ is locally boundedon 𝕏 that Q vanishes with some order greater than &(Hγ ) hits or equal to μ(𝕏 : ℙN ℂ ) + n mγ on any hypersurface Hγ such that π the open subset 𝕏 of 𝕏. As a consequence, the condition on the left-hand side of (7.124) is satisfied with Θ, and we conclude from Corollary 7.40 that 



P Θ (7.133) Θ = ∇KP• ⊕F• P F• + P K• ,λ R0F• λ=0

ℙN ℂ.

From this point, the diagram chase argument used in the sense of currents on to prove the first assertion in Proposition 3.32 carries through. It concludes that that given any z ∈ 𝕏 ∩ V (P ), one has in OℙN ℂ ,z   Θz = θι,z (Pι )z + ι (Φι )z , ι

ι

    26 N (deg Θ − deg Φι ) where θι,z ∈ OℙN (deg Θ − deg p ) and  ∈ O . It ι ι,z ℙ z z ℂ ℂ remains to add the constraint (7.134)

deg Θ ≥ (n + 1)(|d(p)|∞ − 1) + reg (𝕏)

to ensure that the argument carries through at all steps, including, in particular the final one, corresponding to  = N , with respect to a global application of Dolbeault’s lemma in the ambient manifold ℙN ℂ instead of a local application of such a lemma (δ)) = 0 for in a Stein neighborhood of each point in 𝕏∩ V (P ). Since H 0,r (ℙN ℂ , OℙN ℂ any δ ∈ ℤ as soon as r < N (see §B.2.3, Example B.31(2)) one needs to concentrate only on the step  = N . The only ∂-global obstruction that remains to be solved globally on ℙN ℂ is then 



P (7.135) ∂ Q = P F• + P K• ,λ R0F• Θ, λ=0 N +1

where



P P F• + P K• ,λ R0F•

λ=0 N +1

26 Observe that this argument has already been used in the proof of Theorem 5.11 and could have been used in order to prove Theorem 5.10 as well, while our proof of Theorem 5.10 relied on the use of integral representation formulae. The original proof of Theorem 5.10 by Mats Andersson [AndSS, Theorem 1.1] was based on the chase diagram argument we keep following this time in the present approach.

368

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

is the component with bidegree (0, N ) of the current 

P . P F• + P K• ,λ R0F• λ=0

F• Since R0,r = 0 for r < N − n according to the dimension principle for PM currents (see Proposition 2.26) and L ≤ N , where L denotes the length of F• , one has



P  P F• + P K• ,λ R0F•

=

λ=0 N +1

n+1 

F• |P |2λ Qr R0,N +1−r

λ=0

r=1

=

n+1 

F• |P |2λ Qr R0,N +1−r

r=1



+ [P F• ]



N +1

 . λ=0

$ [N +1−r] F• For each r ≤ n + 1, R0,N (−dι )-valued while Qr +1−r is EN +1−r = ι OℙN ℂ  #  r is a section of Homℂ E(KP ), ( O N (− deg p )) . Therefore, if ι • ι ℙℂ +1−r] ≥ −N deg Θ − r|d(p)|∞ − d[N ι

(7.136)

for any 1 ≤ r ≤ n + 1 and any ι, the ∂-problem (7.135) is globally solvable in ℙN ℂ since H 0,N (ℙN (δ)) = 0 for any δ ≥ −N (see §B.2.3, Example B.31(2)). Since ℂ , OℙN ℂ +1−r] − (N + 1 − r)) ≤ reg(𝕏) − 1 max(d[N ι

for any r = 1, . . . , n + 1 according to Definition 7.37 for r(𝕏), condition (7.136) is satisfied as soon as deg Θ ≥ (n + 1)(|d(p)|∞ − 1) + reg(𝕏).  One concludes from the chase diagram argument that Θ ∈ ι Pι ℂ[X] modulo 𝕁, which implies the existence of theΘι ’s with Θι an homogeneous polynomial with degree M − deg pι such that Θ = ι Θι Pι modulo 𝕁. The existence of the qι ’s such that (7.131) holds with the degree estimates (7.132) follows by dehomogenization.  r ∞ (p)

as the smallest Remark 7.44. One can always refine μ(𝕏 : ℙN ℂ ) + n) d |d|∞  ∞ integer μ(𝕏 : ℙN ) + n) μ (𝕏, p) + 1 satisfying the inequality (7.128) and then ℂ get max deg qι pι ι   ∞  ≤ max deg q + μ(𝕏 : ℙN ℂ ) + n μ (𝕏 , p) + 1 , (n + 1)(|d(p)|∞ − 1) + reg(𝕏)

(7.137)



instead of (7.132). Since |d(p)|∞ was our marker for the geometric complexity of p, we did not account for this refinement but only mention it here. We will be more careful in the next subsection, where n = N and 𝕏 = ℂN . In the particular case where 𝕏 is smooth (in which case μ(𝕏 : ℙN ℂ ) = 0 according to the last assertion in Theorem 7.38) and m ≤ n, it follows from Theorem 5.8 when k = 1 that the function φ in (7.130) can be replaced by z ∈ 𝕏 −→

|q(z)| p(z)m

7.3. EFFECTIVE GEOMETRIC BRIANC ¸ ON–SKODA–HUNEKE TYPE THEOREMS

369

and the estimate (7.132) be replaced by (7.138)

max deg qι pι ι

   ≤ max deg q + m μ∞ (𝕏, p) + 1 , m(|d(p)|∞ − 1) + reg(𝕏) .

7.3.3. Global Brian¸ con–Skoda or Lipman–Teissier theorems in 𝕂n . We begin this subsection with the particular case of Theorem 7.43, where one has N = n. It means that 𝕏 = 𝔸nℂ = ℂn and that the degree deg 𝕏 = d is equal to one. Furthermore, the Castelnuovo–Mumford regularity reg(𝕏) equals also 1 for 𝕏 = ℂn embedded into ℙnℂ , with Zariski closure equal to ℙnℂ . It follows from Theorem 7.43 together with Remark 7.44 that if p = (pι )ι is a polynomial map from ℂn to ℂm and q ∈ ℂ[X] is such that the function z ∈ 𝕏 −→

|q(z)| p(z)min(m,n)

n is locally bounded realize in an effective way the membership on ℂ , then one can of q to the ideal ι pι (X) ℂ[X] as p = ι qι pι with

(7.139)

max deg qι pι ι   ≤ max deg q + [min(m, n) μ∞ (p)] + 1, min(m, n + 1)(|d(p)|∞ − 1) + 1 ,

where μ∞ (p) = μ∞ (ℂn , p) (see Definition 7.5 generalized as Definition 7.42). In fact, one may refine the degree estimates above thanks to the two following observations. • When m ≤ n, the additional constraint M ≥ min(m, n + 1)(|d(p)|∞ − 1) + reg(𝕏) = m(|d(p)|∞ − 1) + 1 required for the choice on M = deg Θ in the proof of Theorem 7.43 in order to solve the ∂-problem at any step until the final one ( = N = n) is not necessary since H 0,r (ℙnℂ , OℙN (δ)) = 0 for any δ ∈ ℤ. Hence we drop ℂ it. • When m > n, the fact that 𝕏 = ℙnℂ implies that the initiating current identity for the diagram chase (a cornerstone in the proof of Theorem 7.43) is P 

P K• ,λ Θ Θ = ∇KP• λ=0

instead of (7.133). The precise constraint on deg Θ = M required in order for the ∂-problem at the final step ( = n) in the diagram chase to be globally solvable is M ≥ |d(p)|1,n+1 − (n + 1) + 1 = |d(p)|1,n+1 − n, where (7.140)

|d(p)|1,n+1 =

max

ι0 n the results produced by Theorems 7.45 or 7.47 with q n instead of q. Namely, one can compare the estimates     qι pι , with max deg qι pι ≤ max deg q + n μ∞ (p) + 1, |d(p)|1,n+1 − n qn = ι

ι

produced by Theorems 7.45 or 7.47 to the estimates (formulated slightly differently)  q n+1 = qι pι with max deg qι pι ≤ (n + 1) max(deg q + μ∞ (p), |d(p)|∞ ) ι

derived by Theorem 7.49. 7.4. Algebraic residues and tropical considerations 7.4.1. Rational polyhedral complexes and tropical cycles. Let N be an n-dimensional lattice and let N ℝ = N ⊗ℤ ℝ be the corresponding n-dimensional vector space. The archetypal example for this setting is the lattice N = ℤn with N ℝ = ℝn . We refer to [MiR18, §2.1.2] for the following definition, see also [BabH17, §2.4], when N ⊂ ℚn and hence N ℝ ⊂ ℝn . 28 Condition (7.145) is equivalent to the fact that q belongs to the integral closure of the ideal p ι ι 𝕂[X] in 𝕂[X], following Definition A.36.  29 As a matter of fact q min(m,n) is already known to lie in ι pι 𝕂[X] in view of the Lipman– Teissier theorem in each n-dimensional regular local ring (𝕂[X])X−x , x ∈ 𝕂n .



7.4. ALGEBRAIC RESIDUES AND TROPICAL CONSIDERATIONS

373

Definition 7.51. A subset Δ ⊂ N ℝ , defined as a finite intersection of halfspaces {x ∈ N ℝ : a , x ≥ c}, where a ∈ N  = Homℤ (N , ℤ) and c ∈ ℝ, is called a generalized30 N -rational polyhedron. Its dimension is defined to be the dimension of the ℝ-affine subspace AΔ (whose direction31 is denoted by VΔ ) that Δ generates in N ℝ and it satisfies 0 ≤ dim Δ ≤ n = rank N . Its relative interior Δ◦ is its interior in the ℝ-affine subspace AΔ . Let us state a few basic definitions introduced already for N -rational cones (see §3.1.1), which are N -rational generalized polyhedra stable under the homotheties x → λx of N ℝ . Definition 7.52. Let Δ be a generalized N -rational polyhedron. A subset δ ⊂ Δ is said to be a face of Δ, which we denote by δ ≺ Δ, if and only if there exist a ∈ N  and cδ ∈ ℝ (not necessarily unique) so that δ = {x ∈ Δ : a , x = cδ }. Any generalized N -rational cone Δ of dimension m generates a collection of generalized N -rational polyhedra, namely its faces, whose dimensions vary between 0 and m. Faces of dimension 0 are called vertices of Δ, faces of dimension 1 are called rays of Δ, and faces having dimension m − 1 are called facets of Δ. Thus, we are ready to introduce the following notion. Definition 7.53. A generalized N -rational polyhedron Δ is said to be N entire if and only if its vertices belong to the lattice N . Example 7.54. Given an N -rational polyhedron with exactly two distinct vertices a, b ∈ N , its pairing Δ with an ordering  of {a, b} is called an oriented ray of the lattice N . The N -dimension of such an oriented ray is defined as the number of points of the lattice N on the oriented ray [a, b] minus one. That is, (7.148)

dN (Δ ) := |([a, b] ∩ N ) − 1| ≥ 1.

Definition 7.55. Given a generalized N -rational polyhedron Δ with a face δ ≺ Δ, the N -rational cone of Δ from its face δ is defined by (7.149)

σδ≺Δ := {λ (x − y) : x ∈ Δ, y ∈ δ, λ ≥ 0}.

Example 7.56. Let δ ≺ Δ be a facet of the generalized N -polyhedron Δ with dimension dim Δ = m ≥ 1. Let also Vδ be the (m − 1)-dimensional ℝsubspace directing the affine subspace Aδ generated by δ in N ℝ . Let us consider the (n−(m−1))-lattice N /N Vδ defined as the quotient lattice of N by the (m−1)trace lattice N Vδ := N ∩ Vδ . Then         (7.150) σδ≺Δ mod Vδ ⊂ VΔ mod Vδ ⊂ N ℝ mod Vδ = N /N Vδ ℝ defines a N /N Vδ -rational (oriented) one-dimensional cone (that is a ray) in the ℝ   quotient vector space N /N Vδ ℝ . Let ¯ = u(σδ≺Δ mod Vδ ) ∈ N /N Vδ u 30 By “generalized”, we mean that Δ is not assumed to be bounded. In other words, the polyhedron under consideration is not defined necessarily as the convex envelope of a finite subset in N ℝ . For example, in the case where N = ℤn , Δ is not assumed to be the Newton polyhedron ±1 ] (when 𝕂 is a commutative field) as it was the case for the Δj ’s of an element in 𝕂[X1±1 , . . . , Xn in §6.5.3. 31 This means the underlying dim A -dimensional ℝ-vector space. Δ

374

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

be a N /N Vδ -primitive vector of such ray,32 which one can lift as a unique N primitive element in N denoted as uτ ≺Δ . Generalized N -rational polyhedra will play the role of elementary building blocks in the construction of so-called tropical cycles or tropical subspaces in ℝn (see [MiR18, §2.1.3 and §4.2]) as closed irreducible analytic subsets of irreducible algebraic subvarieties analogously do in analytic  or algebraic geometry; see §D.1. Let us introduce a notion similar to a cycle ι {Vι } with no multiplicities. Definition 7.57. An N -rational polyhedral complex Z in N ℝ is a finite collection P = {Δι }ι of generalized N -rational polyhedra in N ℝ satisfying δ ≺ Δι for some ι =⇒ δ ∈ P (7.151) Δι and Δι ∈ P =⇒ Δι ∩ Δι ≺ Δι and Δι ∩ Δι ≺ Δι . The dimension of P is defined to be (7.152)

dim P = max dim Δι . ι

P is said to be purely dimensional if all the maximal polyhedra Δι (also known as maximal cells) ordered by inclusion share the same dimension in N ℝ . Moreover, ∀ r ∈ {0, . . . , dim P}, P(r) := {Δι ∈ P : dim Δι = r},  (7.153) Δ. |P| = Supp P := ι

ι

Example 7.58. N -rational fans, as introduced in Definition 3.3, provide typical examples of N -rational polyhedral complexes, whose cells Δ are all N -rational cones σ. Let us now introduce the multiplicities. Definition 7.59. A purely dimensional, weighted N -rational polyhedral complex is a formal combination    ν(Δι ) {Δι } = P, ν , Z = ι

where {Δι }ι = P is a N -rational polyhedral complex and ν : Δ ∈ P → ℤ is a weight function. A weighted, purely dimensional N -rational polyhedral complex is said to be balanced if and only if it satisfies the balancing conditions  ν(Δ) uδ≺Δ = 0, (7.154) δ ∈ P(dim P − 1) =⇒ {Δ∈P(dim P) : δ≺Δ}

where the primitive elements of uδ≺Δ of N are defined as in Example 7.56. Next, we introduce an equivalence relation between weighted, purely dimensional N -rational polyhedral complexes. Two weighted purely (n − m)-dimensional weighted N - rational polyhedral complexes (P, ν) and (P  , ν  ) are equivalent, that is, they represent the same object as an equivalence class, if and only if the implications (7.155)

Δ ∈ P(n − m), Δ ∈ P  (n − m), Δ◦ ∩ (Δ )◦ = ∅ =⇒ ν(Δ) = ν  (Δ )

hold. This amounts to saying that (P, ν) and (P  , ν  ) inherit a common weighted refinement (P  , ν  ); see [MiR18, Definition 4.2.1 and Exercice 4.2.10] or also 32 See

Definition 3.3 with ℤn replaced by the lattice N /N Vδ .

7.4. ALGEBRAIC RESIDUES AND TROPICAL CONSIDERATIONS

375

[BabH17, Definition 2.6]. Then, let us propose the following definition when N = ℤn . Definition 7.60. Let 0 ≤ m ≤ n. A tropical (n − m)-cycle on ℝn (or also a tropical (n − m)-subvariety of ℝn , according to the terminology in [BabH17]), is the equivalence class Z (modulo the equivalence relation above) of a weighted, balanced, purely (n − m)-dimensional ℤn -rational polyhedral complex  Z = ν(Δι ) {Δι }. ι

Such a tropical (n − m)-cycle Z is said to be effective if it admits an effective representative. That is if it admits as a representative a weighted, balanced ℤn  rational polyhedral complex such as Z = ι ν(Δι ) {Δι } with ν(Δι ) > 0 for any ι. Before discussing the particular example of (n − 1)-tropical cycles in ℝn , one needs to recall the definition of the semiring (Trop,  ,  ). On [−∞, ∞[, let us define two operations, namely the tropical addition and the tropical multiplication: (7.156)

∀ a, b ∈ [−∞, ∞[,

a  b := max(a, b),

∀ a, b ∈ [−∞, ∞[,

a  b := a + b.

The role of zero is now played by −∞ since a  (−∞) = (−∞)  a = a for any a ∈ [−∞, ∞[, while the role of the unit element is played by 0 since a0 = 0a = a + 0 = 0 + a = a

∀ a ∈ [−∞, ∞[.

Tropical addition is idempotent, that is (7.157)

a  a = max(a, a) = a

∀ a ∈ [−∞, ∞[.

When a ∈ ℝ, then there cannot be any b ∈ [−∞, ∞[ such that a  b = −∞, which means that no element (besides −∞) admits an inverse for tropical addition. Hence it does not make sense to speak about tropical subtraction either: the idempotency relation (7.157) is a substitute for the lack of subtraction. Nevertheless, we note that one can define tropical division, namely a/trop b := a − b, when a ∈ [−∞, ∞[ and b ∈ ℝ. The tropical semiring33 thus constructed is denoted as (Trop,  ,  ). We will from now on interpret ℝn as (Trop \ {−∞})n . Example 7.61. The following result was established in [Mi04, Proposition 2.4]; see also [MiR18, §2, Theorem 2.4.10]. Given an effective (n − 1)-tropical cycle Z , there is a tropical Laurent polynomial34 (7.158)

p(X) =

α∈A (−v(α))  X1

α1

 · · ·  Xn

αn

,

where A is a finite subset of ℤn , whose closed convex envelope Δ is an n-dimensional ℤn -entire polyhedron, and v : A → ℝ = Trop \ {−∞} is such that Z admits as 33 One may call it sometimes “semifield” since any nonzero element a admits an inverse (−a) for tropical multiplication. Note anyway that 0 is different from the neutral element (−∞) with respect to tropical addition  . 34 Which is unique, up to the tropical multiplication by a tropical monomial.

376

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

particular representative (P, ν). Namely, the support |P| of P is the corner locus35 of the evaluation function   (7.159) x ∈ ℝn −→ p(x) = max α, x − v(α) α∈A

of p in ℝ = (Trop \ {−∞}) . The (n − 1)-dimensional ℤn -rational polyhedral complex P itself is realized from p; see [PasR04, Proposition 1]. Let n

n

Φ : (x, ξ) ∈ ℝn × Δ −→ p(x) + pˇ(ξ) − ξ, x, where pˇ is the Legendre–Fenchel transform36 of p, so that Φ is convex in ℝ × Δ. One has 2 4 (7.160) P = Δξ := {x ∈ ℝn : Φ(x, ξ) = 0} , ξ ∈ Δ such that dim Δξ ≤ n − 1 . As for the weight function ν : P → ℕ, it is realized from the tropical polynomial (7.158): for any Δ ∈ P(n − 1), 3 (7.161) Δ = Δx , x∈Δ

= {ξ ∈ Δ : Φ(x, ξ) = 0}, is a one-dimensional ℤn -integer polyhedron where lying in Δ, whose ℤn -dimension dℤn ((Δx ) ), defined in (7.148), equals ν(Δ). The two families of generalized ℤn -rational polyhedra Δx

{Δξ : ξ ∈ Δ} , {Δx : x ∈ ℝn } define two n-dimensional ℤn -rational polyhedral complexes P and P  of ℝnx and ℝnξ , respectively. Their corresponding supports ℝn and Δ are dual to each other; see [PasR04, Proposition 1] for details. It is natural to denote the effective (n − 1)dimensional tropical cycle which is represented by such (P, ν) by (7.162)

(P, ν) = div(Trop\{−∞})n (p).

7.4.2. The tropical current trop∗ Z . For n ∈ ℕ∗ , let trop be the tropicalization morphism, namely (7.163) trop : z = (z1 , . . . , zn ) ∈ (ℂ∗ )n = Homℤ ((ℤn ) , ℂ∗ ) −→ trop(z) = −(log |z1 |, . . . , log |zn |) = −Log z ∈ ℝn = Homℤ ((ℤn ) , ℝ) (see §D.5.2) and let arg be its companion morphism (7.164) arg : z = (z1 , . . . , zn ) ∈ (ℂ∗ )n = Homℤ ((ℤn ) , ℂ∗ ) −→ arg(z) = (arg(z1 ), . . . , arg(zn )) ∈ (𝕊1 )n = Homℤ ((ℤn ) , 𝕊1 ). We recall the exact sequence of ℤ-modules ι

∗

→ℂ (7.165) 𝕍ℤn : 0 −→ Homℤ ((ℤn ) , 𝕊1 ) = (𝕊1 )n 𝕊−→ Homℤ ((ℤn ) , ℂ∗ ) 1

trop

−→ Homℤ ((ℤn ) , ℝ) = ℝn −→ 0. corner locus of a real-valued continuous function is the subset of points in ℝn in a neighborhood of which it fails to be smooth. 36 That is, p n ˇ is the convex function defined on a dual copy ℝn ξ of ℝx by 35 The

pˇ(ξ) = sup (ξ, y − p(y)). y∈ℝn

The polyhedron Δ = conv(A ) is the stability domain of pˇ equal to pˇ−1 (ℝ).

7.4. ALGEBRAIC RESIDUES AND TROPICAL CONSIDERATIONS

377

One has similar exact sequences 𝕍N , when one replaces ℤn by any n-lattice, namely (7.166) ι 1 →ℂ∗ trop Homℤ (N  , ℂ∗ ) −→ Homℤ (N  , ℝ) −→ 0. 𝕍N : 0 −→ Homℤ (N  , 𝕊1 ) 𝕊−→ For any (n − m)-dimensional subspace V of ℝn , let ℍℤn (V ) be the exact sequence of ℤ-modules (7.167) ℍℤn (V ) : 0 −→ (ℤn )V ⊗ℤ ℝ = V

ι V →ℝn

−→ ℤn ⊗ℤ ℝ = ℝn

π

V −→ (ℤn /(ℤn )V ) ⊗ℤ ℝ = ℝn /V −→ 0,

following the notations introduced in Example 7.56. Then, consider the diagram of complexes with the exact sequence ℍℤn (V ) represented as horizontal and the exact sequences 𝕍(ℤn )V , 𝕍ℤn , 𝕍ℤn /(ℤn )V (in this order) represented as vertical. Keeping track of the commutativity of diagrams, one may raise for such a configuration the exact sequence ℍℤn (V ) to the exact sequence of ℤ-modules (7.168) 0 −→ Homℤ (((ℤn )V ) , 𝕊1 ) −→ Homℤ (((ℤn )) , 𝕊1 ) = 𝕊n1 Γ

V −→ Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ) −→ 0.

Thus, we have the following technical result. Proposition 7.62. For any affine subset A of ℝn with direction V and any a = (a1 , . . . , an ) ∈ A, the map (7.169)   ΠA : z ∈ trop−1 (A) −→ ΓV arg(ea1 z1 , . . . , ean zn ) ∈ Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ) does not depend on the choice of a and realizes a submersion from trop−1 (A) to the multiplicative compact group Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ). Moreover, each fiber n n  1 Π−1 A ({θ}), where θ ∈ Homℤ ((ℤ /(ℤ )V ) , 𝕊 ), is an (n − m)-dimensional complex ∗ n torus in (ℂ ) . It is invariant under the action of the group Homℤ (((ℤn )V ) , ℂ∗ ) and is the common zero set of a finite family of binomials. Proof. One can see that the kernel of the map   (7.170) z ∈ trop−1 (V ) −→ ΓV arg(z) ∈ Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ) is the (n − m)-dimensional torus Homℤ (((ℤn )V ) , ℂ∗ ). Moreover, any of the fibers of (7.170) is of the form eiξ • Homℤ (((ℤn )V ) , ℂ∗ ), where ξ ∈ 𝕊n1 and 𝕊n1 acts on (ℂ∗ )n by pointwise multiplication. Such pointwise action is denoted here by •. A straightforward argument leads then to the result.  Example 7.63. Let Δ be a generalized (n − m)-dimensional ℤn -rational polyhedron with relative interior Δ◦ in the (n − m)-dimensional subspace AΔ (with direction VΔ ) which it spans in ℝn . The restriction of ΠAΔ to trop−1 (Δ◦ ) still realizes a submersion from trop−1 (Δ◦ ) to the compact group Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ). Each fiber of (ΠAΔ )|trop−1 (Δ◦ ) is the intersection of a (n − m)-dimensional complex torus with trop−1 (Δ◦ ). The closure (7.171)

(ΠAΔ )−1 |trop−1 (Δ◦ ) ({θ})

378

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

in 𝕋n of such an orbit, where θ ∈ Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ), is an (n−m)-dimensional complex manifold with piecewise smooth boundary. To the complex manifold (7.171) one attaches the (m, m)-integration current 

(ΠAΔ )−1 |trop−1 (Δ◦ ) ({θ}) . This current is positive but fails to be closed since (7.171) is a complex manifold with a boundary. For 1 ≤ m < n, let Z be an (n − m)-tropical effective cycle in ℝn = ℤn ⊗ℤ ℝ according to Definition 7.60. This cycle admits as representative the weighted, balanced, effective and purely (n − m)-dimensional ℤn -rational complex Z = (P, ν). Following Farhad Babaee and June Huh [BabH17, §2 and §3], we may attach to Z , hence to Z , a positive closed current trop∗ Z ∈  D(m,m) ((ℂ∗ )n ), in such a way that our construction will not depend on the choice of its representative Z = (P, ν). Let us make this precise with the following statement. Theorem 7.64. Let the (n − m)-tropical effective cycle Z in ℝn be as above with representative Z = (P, ν), a purely (n − m)-dimensional ℤn -rational complex both balanced and effective. Then the (m, m)-current defined on (ℂ∗ )n by (7.172) 

  ∗ ν(Δ) (ΠAΔ )−1 trop Z = |trop−1 (Δ◦ ) ({θ}) dμhaar , Δ∈P(n−m)

Homℤ ((ℤn /(ℤn )V ) , 𝕊1 )

where μhaar denotes the normalized Haar measure on Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ), is well defined since it is independent of the representative (P, ν) of Z . Moreover, the current (7.64) is d-closed. Proof. The proof relies on Fourier analysis techniques.37 We assume this result here and refer instead the reader to [BabH17, §2 and §3].  Definition 7.65. The (m, m) closed positive current attached to the effective (n − m)-tropical cycle Z by (7.172) is called the (n − m)-tropical current of H . Theorem 7.64 bridges tropical geometry, whose intrinsic, combinatorial nature has as effective objects (n − m)-tropical cycles, to the world of closed, (m, m)positive closed currents of specified averaged type, which is of intrinsic complex analytic nature. Example 7.66. If H is the effective (n − 1)-tropical cycle represented by the tropical Laurent polynomial p (with evaluation function p) as in Example 7.61, then the corresponding (1, 1)-tropical current is the positive closed (1, 1)-current defined as ddc (p ◦ trop); see [Bab14, Theorem 5.2]. 7.4.3. Tropicalization and (p, q)-supercurrents on ℝn . Our approach displays in many places the key role played by the pairing of local holomorphic coordinates ζ1 , . . . , ζN to their “shadow” companions ζ 1 , . . . , ζ N in the current presentation of multidimensional residue theory on a complex N -dimensional manifold X (for example, ℂN or ℙN ℂ ) or an embedded, purely n-dimensional complex 37 We shall present briefly in the next subsection Fourier analysis within the context of spaces of distributions or currents on the complex torus 𝕋n .

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379

analytic space V ⊂ X (for example, an affine or projective algebraic subvariety N 𝕏 ⊂ ℂN = 𝔸N ℂ or 𝕏 ⊂ ℙℂ ). In particular, observe that ∂ ζj = 0 ∂ζ k  for 1 ≤ j, k ≤ N and that the sum ζ2 = N k=1 ζ k ζk ≥ 0 is heuristically considered as a linear form in ζ with constant coefficients. Our object of interest in this subsection is the differential manifold ℝn = ℤn ⊗ℤ ℝ, which prevents us from profiting from such combined roles between local coordinates and their shadow companions. Nevertheless, we consider instead the n-dimensional complex manifold (7.173)

(7.174)

Homℤ ((ℤn ) , ℂ∗ ) = (ℂ∗ )n = −trop−1 (ℝn ) = Log−1 (ℝn ),

where Log z = (log |z1 |, . . . , log |zn |) for z ∈ (ℂ∗ )n , following the definition of the tropicalization morphism; see (7.163). Since Fourier analysis plays a major role in the proof of Theorem 7.64 (see [BabH17, §3]), let us describe it in the context of the space  D(p,q) ((ℂ∗ )n ) for 0 ≤ p, q ≤ n. We follow for that purpose the presentation in [Os19, §2] and introduce first the concept of (p, q) supercurrent on ℝn in the tropical sense.38 Definition 7.67. A ℂ-valued (p, q)-current T on (ℂ∗ )n is said to induce a ℂ-valued (p, q)-supercurrent on ℝn = trop ((ℂ∗ )n ) if and only if T is invariant under the action of (𝕊1 )n on (ℂ∗ )n by pointwise multiplication. That is, for each θ ∈ (𝕊1 )n ,   (7.175) z −→ eiθ z ∗ T = T. We denote by  Dtrop (ℝn ) the space of such ℂ-valued (p, q)-currents on (ℂ∗ )n . (p,q)

Example 7.68. Let 0 ≤ m < n. If Z is an (n − m)-tropical effective cycle on ℝn = ℤn ⊗ℤ ℝ, then its associated closed, positive, (m, m)-tropical current trop∗ Z (introduced in Definition 7.65) is the archetypal example of an (m, m)-current on (ℂ∗ )n inducing an (m, m)-supercurrent on ℝn . One has   (7.176) Supp trop∗ Z = trop−1 (|P|), where the weighted ℤn -rational polyhedral complex (P, ν) represents Z . Fourier analysis allows us to attach to any current T ∈  D((ℂ∗ )n ) a multisequence

ℤn   (p,q) T& = T&κ κ∈ℤn ∈  Dtrop (ℝn ) , which will be naturally called the spectrum of T . Definition 7.69. Let T =

  |J|=p |K|=q

T J,K

dzJ dz K ∧ , zJ zK

where J and K are ordered subsets of {1, . . . , n} in increasing order and the T J,K ’s are ℂ-valued distributions on (ℂ∗ )n . The spectrum of T is the multisequence of 38 The (formal) definition of (p, q)-supercurrents on ℝn was introduced in [Lag12]. It became a fundamental tool in nonarchimedean analytic geometry; see [ChD12, §1.2]. The definition suggested below is a concrete realization of such a concept in the tropical setting.

380

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

elements T&κ ∈  Dtrop (ℝn ) indexed by κ ∈ ℤn defined as follows: for every κ ∈ ℤn the corresponding term of the spectrum is given by (p,q)

(7.177) T&κ := n  

 zj κj J,K Log∗ (x) (dx + idθ)J ∧ (dx − idθ)K exp∗ T |zj | j=1 |J|=p |K|=q

=

 

exp∗

|J|=p |K|=q

n  zj κj J,K

dzJ dz K T ∧ , Log∗ (x) |zj | zJ zK j=1

where exp : (x, θ) ∈ ℝn × (𝕊1 )n −→ ex+iθ ∈ (ℂ∗ )n . Remark 7.70. Any T ∈  D(p,q) ((ℂ∗ )n ) splits as n   zj κj & Tκ , T = |zj | n j=1

(7.178)

κ∈ℤ

justifying the terminology used in Definition 7.69. Let T ∈  D((ℂn )∗ ). Its 0-Fourier coefficient T&0 , whose action on any smooth (n − p, n − q)-form ϕ is given by    ∗ dθ & T, z −→ eiθ z ϕ (7.179) T0 , ϕ = (2π)n (𝕊1 )n belongs to  Dtrop (ℝn ). The construction of the (n − m)-tropical current trop∗ H attached to an effective (n − m)-tropical cycle (see Definition 7.65 and Example 7.68) suggest other constructions of (p, q)-supercurrents on ℝn that could be of interest. Suppose, for example, that the effective (n − m)-tropical cycle H , with 0 < m < n, is represented by a weighted balanced ℤn -polyhedral complex (P, ν). Moreover, assume that for any Δ ∈ P(n − m) the (n − m)-dimensional complex torus Homℤ (((ℤn )VΔ ) , ℂ∗ ) is defined as a smooth complete intersection in (ℂ∗ )n . Namely, assume that there are exactly m binomials z aΔ,j −z bΔ,j , where aΔ,j , bΔ,j ∈ (ℕ∗ )n satisfying39 2 4 (7.180) Homℤ (((ℤn )VΔ ) , ℂ∗ ) = z ∈ (ℂ∗ )n : z aΔ,j − z bΔ,j = 0 for j = 1, . . . , m (p,q)

and such that z ∈ Homℤ (((ℤn )VΔ ) , ℂ∗ ) =⇒

(7.181)

m '   d z aΔ,j − z bΔ,j = 0. j=1

Then every fiber of any map40

  z ∈ trop−1 (VΔ ) −→ ΓVΔ arg(z) ∈ Homℤ ((ℤn /(ℤn )VΔ ) , 𝕊1 )

is also defined as the smooth complete intersection 4 2 z ∈ (ℂ∗ )n : (eiξ • z)aΔ,j − (eiξ • z)bΔ,j = 0 for j = 1, . . . , m 39 Such 40 We

is the case when m = 1. refer here to (7.168) and Proposition 7.62 for the notations used here.

7.4. ALGEBRAIC RESIDUES AND TROPICAL CONSIDERATIONS

381

for some ξ ∈ (𝕊1 )n , where • denotes the pointwise multiplication. Let a be an arbitrary point in AΔ . It follows Proposition 7.62 that for any θ in the compact group Homℤ ((ℤn /(ℤn )V ) , 𝕊1 ), (7.182) Π−1 AΔ ({θ}) 4 2 = z ∈ (ℂ∗ )n : (ea+i ξθ • z)aΔ,j − (ea+i ξθ • z)bΔ,j = 0 for j = 1, . . . , m for some ξθ ∈ (𝕊1 )n , in addition to (7.183) z ∈ Π−1 AΔ ({θ}) =⇒ ωΔ,θ (z) =

m '   d (ea+i ξθ • z)aΔ,j − (ea+i ξθ • z)bΔ,j = 0. j=1

One can state under the hypothesis specified above Proposition 7.71. Proposition 7.71. For 0 ≤ m < n, let Z be an effective (n − m)-tropical cycle represented by an effective, balanced weighted complex (P, ν) such that the hypotheses above are fulfilled. Let T be the (0, m)-current on (ℂ∗ )n , whose action on test forms ϕ ∈ D(n,n−m) ((ℂ∗ )n ) is described by  (7.184) T, ϕ = 

Δ∈P(n−m)

ν(Δ) Homℤ ((ℤn /(ℤn )V ) , 𝕊1 )



 (ΠAΔ )−1 |trop−1 (Δ◦ ) ({θ}) ,

ϕ  ωΔ,θ

dνhaar (θ),

where ϕ/ωΔ,θ is the (n − m, n − m)-form obtained in a neighborhood of Π−1 AΔ ({θ}) when dividing the (n, n − m)-form ϕ by the nonvanishing form (m, 0)-form ωΔ,θ . (0,m) Then the current T ∈  Dtrop (ℝn ). Proof. The result is an immediate consequence of the averaging process involved in the construction itself, as was already the case for the current trop∗ Z , whose realization inspired the construction of T .  Remark 7.72. When f1 , . . . , fm define a complete intersection in the complex manifold (ℂ∗ )n , one has (7.185) m  1

' |f 2λj m+1−j | ∧ df1 ∧ · · · ∧ dfm ∧ df1 ∧ · · · ∧ dfm = ∂ ∂ f 2iπ fm+1−j λ1 =···=λm =0 j=1 = R|f | ∧ df1 ∧ · · · ∧ dfm = 1{f =0} · lim+ ddc log(f 2 + ) ε→0  = div(f1 )] ∧ · · · ∧ [div(fm )]; see Proposition 2.54 and the Lelong—Poincar´e factorization formula (2.76). Observe that this is trivial when f = 0 =⇒ df1 ∧ · · · ∧ dfm = 0, which corresponds to the situation we have here. Therefore, the (0, m)-supercurrent T , whose action on test forms is described by (7.184), is the natural candidate for a residue supercurrent attached to the tropical cycle Z . The property a priori missing for such residue supercurrent is its ∂-closedness. To prove that balancing conditions imply that the residue current the T attached to the (n − m)-tropical cycle (P, ν), as in (7.184), is ∂-closed looks like an interesting question. The particular case in which m = 1 seems to be easier to handle, since the current trop∗ (P, ν) admits in this case another representation; see Example 7.66.

382

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

Treating compact algebraic situations requires extending the concept of supercurrent on ℝn , considered from the tropical point of view as the tropical n-dimensional torus (Trop \ {−∞})n ) (as introduced in Definition 7.67) to that of supercurrent on the compactification with corners 𝕏Σ (ℝ≥0 ) of ℝn with respect to a complete ℤn -rational fan Σ in ℝn = ℤn ⊗ℤ ℝ. We refer to [AKS13, §4.1] for the use of notations and for exact references, as well as for a sketchy presentation of the construction of such an object 𝕏Σ (ℝ≥0 ) inspired by the work of W. Fulton. We refer in particular to the realization of the proper continuous mapping ρΣ : (𝕏Σ )ℂ −→ 𝕏Σ (ℝ≥0 ) extending Log : (ℂ∗ )n = Homℤ ((ℤn ) , ℂ∗ ) −→ ℝn = Homℤ ((ℤn ) , ℝ) = (Trop \ {−∞})n , where (𝕏Σ )ℂ is the complete toric complex variety attached to the fan Σ, with Orb{0}  𝕋N = (ℂ∗ )n as dense orbit acting on it; see Remark 3.9. Denote by • the action of the dense orbit Orb{0}  (ℂ∗ )n on (𝕏Σ )ℂ . This construction suggests the following definition, a companion to Definition 7.67. Definition 7.73. A ℂ-valued (p, q)-current T on the complete toric complex variety (𝕏Σ )ℂ is said to induce a ℂ-valued (p, q)-supercurrent on the compactification with corners 𝕏Σ (ℝ≥0 ) = ρΣ ((𝕏Σ )ℂ ) in the tropical sense if and only if T is invariant under the action of the real torus (Orb{0} )ℝ  (𝕊1 )n of the dense orbit Orb{0}  (ℂ∗ )n on (𝕏Σ )ℂ . Namely, T induces a supercurrent on 𝕏Σ (ℝ≥0 ) if and only if for each u ∈ (Orb{0} )ℝ ,   (7.186) x −→ u • x ∗ T = T. We denote by  Dtrop (𝕏Σ (ℝ≥0 )) the space of such ℂ-valued currents T on (𝕏Σ )ℂ . (p,q)

From now on, let us focus on the algebraic situation, where  ca X a F ∈ 𝕂[X1±1 , . . . , Xn±1 ] = a∈(ℤn )

is a Laurent polynomial with coefficients in a subfield 𝕂 of ℂ, whose Newton polyhedron Δ, namely the convex hull of {a ∈ (ℤn ) : ca ∈ 𝕂∗ }, is n-dimensional. Lev Ronkin introduced and studied an important example of a (0, 0)-supercurrent on ℝn in his last paper [Ro00]. Namely, he considered the current 

dθ  ◊ Log |F (ex+iθ )| (7.187) ([log |F |])0 = Log∗ x −→ . (2π)n (𝕊1 )n The convex real function (7.188)



RF : x ∈ ℝ −→

log |F (ex+iθ )|

n

(𝕊1 )n

dθ (2π)n

is called the Ronkin function of the algebraic (n − 1)-cycle div(ℂ∗ )n (F ) in the affine complex variety (ℂ∗ )n . One can alternatively consider the real concave function  dθ log |F (e−x+iθ )| , (7.189) ρF : x ∈ ℝn −→ − n (2π) 1 n (𝕊 ) (see §D.5.2) and the corresponding (0, 0)-supercurrent on ℝn , if one prefers to use in the constructions the tropicalization morphism trop = −Log instead of Log from Homℤ ((ℤn ) , ℂ∗ ) = (ℂ∗ )n to Homℤ ((ℤn ) , ℝ) = ℝn . The convex (respectively

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383

concave) functions RF (respectively ρF ) play an important role in the so-called croftonization process; see Proposition D.49. In the particular arithmetic setting, where 𝕂 = ℚ, Roberto Gualdi pointed in [Gua18, Remark 2.8] that the Ronkin function RF admits ultrametric companions corresponding to nonarchimedean places on ℚ. For each such place v = vp , where p = 2, 3, 5, . . . is a prime integer, such an ultrametric companion of RF is the tropical polynomial (7.190)

RF,p (X) =

{a∈(ℤ )

: ca ∈ℚ∗ }

n 

(−vp (ca ))  X1

a1

 · · ·  Xn

an

,

where vp (ca ) is the exponent of p in the prime decomposition of the strictly positive rational number |ca |. That is, −vp (ca ) = log |ca |p , where | |p is the p-adic ultrametric absolute value on ℚ. The evaluation function RF,p : x ∈ ℝn = (Trop \ {−∞})n −→

max

{a∈((ℤn ))

: ca

∈ℚ∗ }



 log |ca |p + a , x ∈ ℝ = Trop \ {−∞}

is such that the (0, 0)-supercurrent [trop∗ RF,p ] satisfies in (ℂ∗ )n the Green equation −ddc [trop∗ RF,p ] + trop∗ Zp (F ) = 0.

(7.191)

In (7.191), the (n − 1)-tropical current trop∗ Zp (F ) is attached (as described in Theorem 7.64) to the effective (n − 1)-dimensional tropical cycle Zp (F ) = div(Trop\{−∞})n (RF,p ); see Example 7.61 and Example 7.66. On the other hand, one has, given (7.187), that ◊ |F |])0 (1/z), [trop∗ RF ](z) = ([log since trop = −Log. From these observations, we conclude that the following Green equations −ddc [trop∗ RF,p ] + [trop∗ div(Trop\{−∞})n (RF,p )] = 0 for p prime integer, (7.192)

−ddc [trop∗ RF ] + [div¤ (ℂn )∗ (F (1/z))]0 = 0

hold jointly in (ℂ∗ )n . Now, let Σ be a complete, ℤn -rational fan in ℝn , which is compatible with the n-dimensional Newton polyhedron Δ ⊂ (ℤn ) ⊗ℤ ℝ = (ℝn ) of F . It means that the Laurent polynomial F defines a global holomorphic section of a toric Cartier divisor 𝔻 with a concave virtual function Ψ𝔻 ; see Proposition D.21 and Example D.40. Suppose first that F has rational coefficients and that it is irreducible in the polynomial ring ℚ[X1±1 , . . . , Xn±n ]. Given the product formula  ∀ c ∈ ℚ>0 , log |c| + log |c|p = 0, p prime

the pairing of the convex function RF with its ultrametric companions RF,p , once combined with potential theory within the archimedean and nonarchimedean setting (as sketched in §6.6.2), leads to closed expressions for the total logarithmic height of the Zariski closure F −1 ({0}) in 𝕏Σ with respect to metrized toric divisors

384

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

[Gua18, Theorem 1]. The current Green identities (7.192), involving jointly the algebraic hypersurface ([div¤ (ℂ∗ )n (F (1/z))])0 and its ultrametric companions, namely the tropical currents attached to the (n − 1)-tropical cycles div(Trop\{−∞})n (RF,p ), p prime, reveal themselves to be of high importance. Exploring the possibilities of bringing multivariate residue theory into the picture (as we modestly tried to do in Proposition 7.71) deserves to be considered as concerns arithmetic division instead of intersection questions. Let us now return to the general setting, where the coefficients of the Laurent polynomial F considered here are only assumed to be complex. If Δ𝔻 = Stab(Ψ𝔻 ) = {ξ ∈ (ℝn ) : ξ , · − Ψ𝔻 ≥ 0} is the ℤn -integral polyhedron of stability for Ψ𝔻 , then let (7.193)

p𝔻 : x ∈ ℝn −→

max

a∈Δ𝔻 ∩ (ℤn )

a, x =

max

a vertex of Δ𝔻

a, x.

The (0, 0)-supercurrent on ℝn    ◊ ([log |F |])0 − Log∗ p𝔻 = Log∗ RF,p − Log∗ p𝔻 , as well as the currents

  Log∗ RF,p − Log∗ p𝔻 for p prime in the arithmetic setting where F has rational coefficients, extend as (0, 0)-supercurrents on the compactification with corners 𝕏Σ (ℝ≥0 ) of ℝn , thus providing archetypal examples for such objects. For further considerations about similar constructions, including the concept of Ronkin current instead of Ronkin function, attached this time to a sequence (F1 , . . . , Fm ) of elements in ℂ[X1±1 , . . . , Xn±1 ], where m > 1, we refer the reader to [Os19, §6]. Let us conclude this subsection by observing that the results presented here are far from being satisfactory at this stage. Our modest purpose was to address the following point. As we pointed out in Remark 7.13, Theorem 7.47, and its subsequent Remark 7.48, multivariate residue calculus is a most powerful tool for proving the effective Brian¸con–Skoda’s theorem in ℂ[X1 , . . . , Xn ], possibly combining then with the use of Crofton’s averaging formula. In addition, it allows us to formulate and prove an effective version of Hilbert’s nullstellensatz by solving B´ezout identity in the arithmetic setting, where the polynomial entries have integer coefficients; see Theorem 7.27 with Example 7.28. The main point is that one views Hilbert’s nullstellensatz as a particular case of Brian¸con–Skoda’s theorem; see §7.1.2–§7.1.4. Unfortunately, as we already mentioned in Remark 7.48, nothing is known presently concerning the possibility of an arithmetic version of Brian¸con–Skoda’s theorem in ℤ[X1 , . . . , Xn ]. By arithmetic Brian¸con–Skoda’s theorem we mean a reformulation of Theorem 7.47, when the polynomial entries pj lie in ℤ[X1 , . . . , Xn ]. But a major difference is that the solution should be provided this time in ℚ[X1 , . . . , Xn ] with (possibly not sharp) estimates fitting with B´ezout’s arithmetic intersection theorem both for numerators and denominators.41 For these reasons, it is tempting to transpose the concept of residue current and multivariate algebraic residue calculus to the setting of nonarchimedean geometry. We only attempted to do that here in 41 This is the case in the discrete setting, that is, with notations as in Theorem 7.47, when the pj ’s, j = 1, . . . , m define a zero-dimensional variety. M. Elkadi proved this result in [El94].

7.5. JACOBIAN AND SOCLE, RADICAL AND TOP RADICAL

385

a very naive way within the tropical setting using Babaee and Huh’s realization of tropical currents [BabH17]. We aim to suggest from the closed expression for such currents the construction (still very primitive) of tropical residues as in Proposition 7.71, under the additional hypothesis which includes, as we observed, the hypersurface case m = 1. Another interesting direction to explore, given the role devoted to analytic continuation throughout this monograph, would be that of Igusa calculus [Igu]. All these questions remain indeed a challenge for the future. 7.5. Jacobian and socle, radical and top radical Let 𝕀 be a proper ideal in the local ring Oℂn ,0 = O0 and let {Pι }ι be the uniquely determined finite list of distinct prime ideals Pι in O0 so that 3 √ (7.194) 𝕀= Pι ; ι

see decomposition (A.12) in §A.3.1. Let m = m(𝕀) be the minimal cardinal of a possible set for generators of 𝕀 in O0 . We set 𝕀 = 𝕀f1 ,...,fm = 𝕀f , whenever f is a set of generators for 𝕀 whose cardinal number |f | equals the minimal number m(𝕀). The positive integer dev(𝕀f ) = |f | − codim0 V (𝕀f )

(7.195)

is called the deviation of 𝕀f in O0 . For any r = 1, . . . , n, let 3 √ Pι , (7.196) ( 𝕀)r := {ι : codim O0 /Pι =r}

where the distinct prime ideals Pι are involved in the unique (up to the ordering) representation (7.194). Let us recall the terminology introduced by Wolmer Vercosa Vasconcelos in [Va92].42 Definition 7.74. The top radical of 𝕀f , whenever |f | = m, is defined as the radical ideal / / / (7.197) top 𝕀f := ( 𝕀f )|f | = ( 𝕀f )codim0 V (𝕀f )+dev (𝕀f ) . Remark 7.75. If 𝕀f is such that |f | > n, then one has immediately that / top 𝕀f = {0}. When codim 𝕀f = |f |, which is equivalent to saying that f is a / / regular sequence in O0 , then one has the equality top 𝕀f = 𝕀f . In this section we are interested in the ideals 𝕀f whose generators are few, i.e., |f | ≤  nn.  In this case we also introduce the nonzero ideal J (𝕀f ; f ) generated by the m coefficients JK of the germ of the (m, 0) holomorphic differential form (7.198)

m '



dfj (z) =

j=1

Namely, we consider the ideal (7.199)

JK (𝕀f ; f )(z) dzK .

1≤k1 0, M ∈ ℕ∗ , there exist a Hefer form ℍμ,ε and an explicit smooth kernel Kε,R,M (z, ζ), depending on the supports and the orders of the μ &j ’s, with bidegree (0, 0) in z and (n − m, n − m) in ζ, holomorphic in z, such that the following assertion holds: any compactly supported distribution u in ℝn with Supp u ⊂ 𝔹ℝn (0, R) and order less than M is such that the function (7.232)  (ℍμ,ε (z, ζ))m 1 ∗  ∧ · · · ∧ ∗m u : μ &](z) = u &(ζ) Kε,R,M (z, ζ) ∂ z −→ Trε,R,M [& m! μ & 1 ℂn is a well defined entire function on ℂn belonging to Aφ (ℂn ). Moreover, for any such u, this entire function satisfies the growth estimates (7.233)

sup z∈ℂn

  Trε,R,M [& u : μ &](z) (1 +

z)[Bφ (∂(1/μ))]+N (μ,M )

e−(Bφ (∂(1/μ))+B(μ,R)+ε)Im(z) < +∞,

m  where N (μ, M ) := M + 2( m 1 ord(μj )) + 1, B(μ, R) = R + 1 Rj . The exponent Bφ (∂(1/& μ)) is attached to the φ-tempered residue current ∂(1/& μ) as in (7.214). Furthermore, most important is the fact that one has the following trace formula: u & − Trε,R,M [& u : μ &] ∈

(7.234)

m 

μ &j O(ℂn ).

j=1

Proof. For any 0 < ε < 1, one can represent each distribution μj with support in 𝔹ℝn (0, Rj ) as a sum of derivatives of measures μj,ε,ι supported by the euclidean ball 𝔹ℝn (0, Rj + ε/(3m)). The order of these derivatives is at most ord(μj ). Given such measure μj,ε,ι , one has  μ &j,ε,ι (ζ) − μ &j,ε,ι (z) = =

(e−iζ,t − e−iz,t ) dμj,ε,ι (t)

n   k=1

ℝn



1

e−iz+(ζ−z) , t d (−itk ) dμj,ε,ι (t) (ζk − zk ).

0

As a consequence, one can construct a Hefer form (z, ζ) −→ ℍμ,ε (z, ζ) for μ & as in (7.220) satisfying (7.235) 1m m ℍμ,ε (z, ζ)1  sup  e−( j=1 Rj +ε/3) ((Im∗ρ)(z)+(Im∗ρ)(ζ)) < +∞.  m ord(μ ) j 1 z,ζ 1 + z + ζ Moreover, if   m m  u & ℍμ,ε (z, ζ) u & ℍμ,ε (z, ζ) = #K=m

dz K

dz K ,

396

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

then the same argument shows that for any multi-index K ⊂ {1, . . . , n} such that |K| = m and for any multi-index α ∈ ℕn for any (ζ, z) ∈ ℂn × ℂn , (7.236)

     m   ∂/∂ζ α u  & ℍμ,ε (z, ζ) dz K  sup m   1 ord(μj ) z,ζ∈ℂn (1 + ζ)M 1 + ζ + z × e−(R+

m

j=1

Rj +2 ε/3)((Im∗ρ)(z)+(Im∗ρ)(ζ)))

< +∞.

f

Let us decompose the Bochner–Martinelli weight W K• from (7.224). Fix for the & to 𝔹ℂn (0, ), time being N ∈ ℕ∗ and C > 0. Then represent the restriction of u where  ∈ ℕ∗ , using the first integral representation formula in (7.231) as (7.237)

  

∂ ζ 1 − χ(ζ/) q(z, ζ) n+1 u &(z) = u &(ζ) χ(ζ/) + ΩC,N (z, ζ) 2iπ n−m,n−m ℂn m m    (ℍμ,ε (z, ζ)) 1 ∗ × ∂ μ %j O 𝔹ℂn (0, ) .  ∧ · · · ∧ ∗m modulo m! μ & 1 j=1 

In (7.237) one has that    ∂ (Im ∗ ρ) N Wℂn (z, ζ) (7.238) ΩC,N (z, ζ) = exp 2 C , z−ζ ∂ζ r n   2 C ddc (Im ∗ ρ) (ζ) × , r! r=0 using (7.228) and (7.226). As pointed out in Remark 4.22, the plurisubharmonicity of Im ∗ ρ implies the convexity inequality (7.239)   ∂ (Im ∗ ρ)    ∂ (Im ∗ ρ) 

  , z − ζ  = exp 2 C Re , z−ζ  exp 2 C ∂ζ ∂ζ   exp C(Im ∗ ρ)(z)  for any (z, ζ) ∈ ℂn × ℂn . ≤ exp C(Im ∗ ρ)(ζ) Also, the fact that (∂(1/& μ) is (Im z + log(1 + z2 )/2)-tempered, the definition of μ)) (see (7.214)) once combined with (7.215), imply that for any ε > 0 any Bφ (∂(1/& μ) along the basis of distribution coordinate coefficient τ of the (0, m)-current ∂(1/& differential forms {dz K : |K| = m} is of the form (7.240)

τ=

∂ α ∂ β (dμ), ∂ζ ∂ζ

where α, β ∈ ℕn and θ ∈ C 0 (ℂn ) is such that  |dμ(ζ)| (7.241) e−(Bφ (∂(1/μ))+/3) (Im∗ρ)(ζ)) < +∞. B (∂(1/ μ ))−|α|−|β|+ε/3 φ n ℂ (1 + ζ)

7.6. MULTIVARIATE RESIDUE CALCULUS AND THE EXPONENTIAL FUNCTION

397

Let us now specify N ∈ ℕ∗ and C = Cε > 0 as N =M+ (7.242)

m 

   ord(μj ) + Bφ ∂(1/& μ) + 1,

1

C = Cε = R +

m 

  Rj + Bφ ∂(1/& μ) + ε.

j=1

Using the explicit representation (7.240) of all distributions τ ∈ B(∂(1/& μ)), together with the growth estimates (7.236) and (7.241), the expressions (7.238) of the smooth form ΩCε ,Nε , combined with that for the Bochner–Martinelli weight Wℂn (7.227), the convexity inequality (7.239), and, finally, the fact that the derivatives up to a prescribed order M of Im∗ρ are uniformly bounded in ℂn by C(M ), allow us to show that the integral    (ℍμ,ε (z, ζ))m 1 ∗  ∧ · · · ∧ ∗m u &(ζ) ΩC ,N (z, ζ) n−m,n−m ∂ m! μ & 1 ℂn is absolutely convergent. Moreover, it defines, as a function of z, an entire function Trε,R,M [& u : μ &] belonging to Aφ (ℂn ). The function Tr[& u : μ &] satisfies the growth estimates     (ℍμ,ε (z, ζ))m 1 ∗   u &(ζ) ΩC ,N (z, ζ) n−m,n−m ∂ 1 ∧ · · · ∧ ∗m   m! μ & ℂn ≤ Aε (1 + z)[Bφ (∂(1/μ))]+N (μ,M ) e(Bφ (∂(1/μ))+B(μ,R)+ε)Im(z)) . When z < 0 , the sequence of holomorphic functions z ∈ 𝔹ℂn (0, 0 ) −→ 

n+1   (ℍμ,ε (z, ζ))m 1 ∗ ζ u &(ζ) ΩC ,N (z, ζ) n−m,n−m ∂ χ  ∧ · · · ∧ ∗m ,  m! μ & 1 ℂn for  ≥ 0 converges uniformly on any compact subset of Bℂn (0, 0 ) towards the u : μ &] to Bℂn (0, 0 ). Moreover, the same reasoning shows restriction of Trε,R,M [& that the sequence of functions z ∈ 𝔹ℂn (0, 0 ) −→    

∂ ζ 1 − χ(ζ/) q(z, ζ) n+1 u &(ζ) χ(ζ/) + ΩCε ,N (z, ζ) 2iπ n−m,n−m ℂn m (ℍμ,ε (z, ζ)) 1 ∗  ∧ · · · ∧ ∗m × ∂ m! μ & 1 

n+1   (ℍμ,ε (z, ζ))m 1 ∗ ζ − u &(ζ) ΩC ,N (z, ζ) n−m,n−m ∂ χ  ∧ · · · ∧ ∗m  m! μ & 1 ℂn converges uniformly towards 0 on any compact subset of 𝔹ℂn (0, 0 ). The fact that the entire function u & admits in each Bℂn (0, ), where  ∈ ℕ∗ , the integral representation formula (7.237) (with N and C = Cε as in (7.242)) modulo the ideal  m %j O(𝔹ℂn (0, )) then concludes the proof of the theorem.  j=1 μ μ 

Remark 7.88. Let the principal value current P0K• be also φ-tempered, where φ(z) = Imz + log(1 + z2 )/2. Then for C > 0, N ∈ ℕ∗ and 0 < ε < 1 , one

398

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

could also consider, by analogy to (7.228), the (ℂm )∗ valued current μ 

◦ WIm,C,N [& μ] = WℂNn eC(WIm −1) H1 (z, ζ) P0K• (ζ) m 

ℍr−1 μ  μ ,ε K• , P0,r = WℂNn eC(WIm −1) 𝕀 (r − 1)! r=1

(7.243)

where 𝕀 is from (7.223). One can reinforce the choice of C, N in terms of μ &, R, M μ 

and the associated constants Bφ (∂(1/& μ)) and Bφ (P0K• ) in order to construct for (1:r) each r = 1, . . . , m a smooth form Kε,R,M with bidegree (0, 0) in the variables z and (0:m)

(n − (r − 1), n − (r − 1)) in ζ, holomorphic in z, and a smooth form Kε,R,M with bidegree (0, 0) in the variables z and (n − m, n − m) in ζ such that the complete division formula  m 

μ  ℍr−1 μ ,ε (z, ζ) (1:r) K• P0,r u &(ζ) Kε,R,M (z, ζ) 𝕀 (ζ) (7.244) u &(z) = 2iπ μ(z) (r − 1)! ℂn r=1  m ℍμ,ε (z, ζ) 1 ∗ (0:m) 1 ∧ · · · ∧ ∗m u &(ζ) Kε,R,M (z, ζ) ∂ + m! μ & n ℂ (1:r)

(0:m)

holds in Aφ (ℂn ). Note also that the smooth forms Kε,R,M and Kε,R,M depend on μ &. It leads to an explicit division formula (7.245)

 (0:m)

u &(z) = ℂn

u &(ζ) Kε,R,M (z, ζ)

u : μ &](z) + = Tr◦ε,R,M [&

m 1  ℍm μ ,ε (z, ζ) ∂ u &j (z) μ &j (z) ∗1 ∧ · · · ∧ ∗m + m! μ & j=1

m 

u &j (z) μ &j (z),

j=1

but this time with estimates   max1≤j≤m Tr◦ uj (z)| u : μ &](z), |& ε,R,M [& (7.246) sup  K• μ z∈ℂn (1 + z)[Bφ (∂(1/μ),P0 ))]+N (μ,M )  K• μ

× e−(Bφ ((∂(1/μ),P0

))+B(μ,R)+ε)Im(z)

< +∞.

The following important result follows from Theorem 7.87 and concerns mean periodicity.50 Corollary 7.89. Let φ, μ1 , . . . , μm be as in Theorem 7.87, with in addition ord(μ) = max1≤j≤m ord(μj ) and R(μ) = max1≤j≤m Rj . Furthermore, assume that51 m m    n n (7.247) Aφ (ℂ ) ∩ μ &j O(ℂ ) = μ %j Aφ (ℂn ). j=1

j=1

50 The notation used in the statement, as well as in the proof, refers to one introduced in Theorem 7.87 or in Remark 7.88. 51 Observe that this condition is satisfied under the additional assumption suggested in the previous Remark 7.88.

7.6. MULTIVARIATE RESIDUE CALCULUS AND THE EXPONENTIAL FUNCTION

399

Let θ ∈ C ∞ (ℝn ) be mean periodic for the μj ’s acting by convolution. Namely, assume that the equations  θ(t − ξ) dμj (ξ) = (μj ∗ θ)(t) = 0 for any t ∈ ℝn (7.248) ℝn

hold for any j = 1, . . . , m. Let ε, R, R be such that 0 < ε < 1,

R ≥ R + R(μ)

and ηε,R ∈ D(ℝn , [0, 1]) be a radial cut-off function identically equal to 1 in a neighborhood of the euclidean ball 4 2 μ)) + R + R(μ) + ε . (7.249) t ∈ ℝn : t ≤ Bφ (∂(1/& If M ≥ ord(μ), then the mean-periodic function θ is reproduced in 𝔹ℝn (0, R) by (7.250) θ(t) =  eiζ,t ℂn ∗

where  =

∗1

1 (2π)n

∧ ···∧

∗m



η÷ ε,R θ(ω) Kε,R ,M (ω, ζ) ℝm

(ℍμ,ε (ω, ζ))m 1 ∗ dω ∂  , m! μ &

is dual of the canonical basis {1 , . . . , m } for ℂm .

Proof. It follows from Theorem 7.87 and the Paley–Wiener theorem [Hor, Theorem 7.3.1] that, for any t ∈ 𝔹ℝn (0, R), the entire function &] : Trε,R ,M [δ&t : μ



z −→

e−iζ , t Kε,R ,M (z, ζ)

ℂn

(ℍμ,ε (z, ζ))m 1 ∗  ∧ · · · ∧ ∗m ∂ m! μ & 1

is the analytic continuation to the complex plane of the Fourier transform of a distribution uε,R ,M [t] on ℝn with compact support in the euclidean ball (7.249). The hypothesis (7.247), once combined with (7.234), implies that there are distributions uε,R ,M,j [t], j = 1, . . . , m, in ℝn , with compact support, such that δt − uε,R ,M [t] =

m 

uε,R ,M,j [t] ∗ μj .

j=1

The above fact implies (δt ∗ θ)(ξ) = θ(ξ − t) = (uε,R ,M [t] ∗ θ)(ξ) for all ξ ∈ ℝn , since θ is mean periodic (7.248). Applying this at ξ = 0, then changing t into −t, one gets, thanks to the hypothesis on the cut-off function ηε,R and to Fourier inversion formula, that   θ(t) = uε,R ,M [−t](ξ) , θ(−ξ) = uε,R ,M [−t](ξ) , ηε,R (ξ)θ(−ξ)   1  −i,ξ  u = [−t](ξ) , d η÷ ε,R ,M ε,R θ()e n (2π) ℝn 1  ¤ ÷ uε,R ,M [−t] , ηε,R θ = (2π)n   (ℍμ,ε (ω, ζ))m 1 ∗ ÷ 1 iζ,t  ,M (ω, ζ)  ηε,R θ(ω) dω e K ∂ = ε,R (2π)n ℝn m! μ & ℂn   1 (ℍμ,ε (ω, ζ))m 1 ∗ ÷  θ(ω) Kε,R ,M (ω, ζ) eiζ,t η = dω ∂  , ε,R (2π)n ℝm m! μ & ℂn where ∗ = ∗1 ∧ · · · ∧ ∗m . It is precisely the desired integral representation formula. 

400

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

Remark 7.90. One could have chosen R = R and M = 0 according to Theorem 7.87. However, since  eiζ,t−ξ dμj (ξ) = μ %j (ζ) eiζ , t ℝn

for t ∈ ℝn and since μ %j ∂(1/& μ) = 0 for j = 1, . . . , m in the sense of currents (see Theorem 2.79) the choices of R = R + R(μ) in place of R and M in place of 0 are justified by the fact that we want the representation formula (7.250) to reproduce the mean-periodicity of θ with respect to the μj ’s acting by convolution in 𝔹ℝn (0, R). In any case, it follows from the local structure of the Coleff–Herrera ¯ μ) (namely Proposition 2.37, see also §5.1.2) that the right-hand side current ∂(1/& of (7.250) can be understood as follows. Locally, it is a superposition   ωι (ζ) , (7.251) lim qι (ζ, ∂/∂ζ)(eiζ,t ) ε→0 V ( hι (ζ) μ) ∩ {|hι |≥ε} ι where ωι is an (n − m, n − m)-smooth differential form in ζ, qι is a holomorphic differential operator in ζ, and hι is a holomorphic function in ζ, whose zero set intersects properly V (& μ); see §2.2.2 or §5.1.2. Moreover, the superposition (7.251) is such that it realizes, as a function of t, a mean-periodic function for the μj ’s acting by convolution. Let us comment on the results presented above by recalling some facts about the origin of the problem. In their pioneering paper [BT80, Definition 5.1], Carlos Berenstein and Alan Taylor introduced, given f1 , . . . , fm in some weighted algebra Aφ (ℂn ) such that the fj ’s define a quasi-regular sequence, the concept of slowly decreasing family of entire functions with respect to a family of affine (n − m)dimensional subspaces of ℂn . More specifically, let L be a family of (n − m)dimensional affine subspaces of ℂn , whose union covers V (f ). One says that the family f1 , . . . , fm belonging to Aφ (ℂn ) is slowly decreasing with respect to L if and only if there exist positive constants cf , cf , Cf , Cf , independent of any affine subspace l ∈ L so that for any l ∈ L, the connected components of 2  4 (7.252) O(l, cf , cf ) := z ∈ l : f (z) ≤ e−cf φ(z)−cf are all bounded and such that φ(ζ) ≤ Cf φ(z) + Cf when both ζ, z ∈ l belong to the same bounded connected component of (7.252). Going further, the authors introduced a concept of almost parallelism that the family of linear subspaces L is required to satisfy. Under such a hypothesis, they were able to show, using Jacobi– Lagrange interpolation through Cauchy–Weil’s formula, that the ideal generated by the fj ’s in Aφ is closed since it coincides then with Aφ (ℂn ) ∩ (f1 , . . . , fm ) O(ℂn ); see [BT80, Theorem 5.6]. So the additional hypothesis (7.247) needed to ensure that Corollary 7.89 is valid holds under the slowly decreasing (SD) (for an almost parallel family L) requirements. Under the further assumption that the Coleff– ¯ μ) is Aφ -tempered, we obtain, when φ is the Paley– Herrera residue current ∂(1/& Wiener weight, a more explicit version, in terms of currents, of [BT80, Theorem 10.1]. Corollary 7.89 also holds when the conditions in Remark 7.88 are fulfilled. We refer also the reader to [BY95, Mat18] for further examples involving PDEs with commensurate time lags in a fixed direction. Remark 7.91. In this subsection we present the trace results with the Paley– Wiener weight since it appeals to questions in harmonic analysis. The analogue of

7.6. MULTIVARIATE RESIDUE CALCULUS AND THE EXPONENTIAL FUNCTION

401

Theorem 7.87 can be formulated, with almost the same proof, for any weight  γ with γ ≥ 1. The cases γ = 1 and γ = 2 are special. The case γ = 1 corresponds, through the Fourier–Borel transform, to the setting of analytic functionals, while the case γ = 2, where the weight is smooth, profits from the fact that φ is attached to a scalar product. 7.6.3. Ehrenpreis–Palamodov’s fundamental principle revisited. Here is a well known result going back to Leonhard Euler. Let p ∈ ℂ[X] be a polynomial with deg p > 0 and distinct complex roots α1 , . . . , α . Then, the deg p-dimensional ℂ-vector space of solutions of the homogeneous differential equation with constant coefficients d (y) = 0 (7.253) p i dt in C ∞ ([a, b]), where −∞ ≤ a < b ≤ +∞, admits as a basis the elementary solutions t −→ tκ e−iαν t , ν = 1, . . . , , κ = 0, . . . , μαν (p) − 1, where μα (p) denotes the order of α as a zero of p. In this subsection, we consider the semilocal n-dimensional analogue of the above result, where U, in place of ]a, b[, is an open convex subset of ℝn . For technical reasons, we will assume first that U is bounded and that its closure K (playing the role of [a, b]) is a strictly convex compact subset of ℝn with smooth boundary containing the origin. Then, the positive support function52 HK : ω ∈ (ℝn ) −→ sup ω, t

(7.254)

t∈K

is smooth in ℝ \ {0} and its gradient n

(7.255)

∇ HK : ω ∈ ℝn \ {0} −→

∂H

K

∂ω1

(ω), . . . ,

∂HK (ω) ∈ ∂ K ∂ωn

is surjective. We let also (7.256)

C ∞ (K) :=

lim −→ n

C ∞ (U).

{U open in ℝ : U⊃K}

In order to overcome the nonsmoothness of HK at the origin, we will frequently regularize HK by convolution HK ∗ ρℝ with a smooth positive bump function ρℝ , whose support is contained in the closed ball 𝔹ℝn (0, 1) and whose total mass equals 1 with respect to the n-dimensional Lebesgue measure dω. Our choice for ρℝ is irrelevant throughout this subsection, as the choice of ρ was in the previous one. Observe that one has (∇(HK ∗ ρℝ ))(ℝn ) ⊂ K, since ρℝ is positive with mass 1 and K is convex. Instead of the space of solutions of (7.253), we consider the ℂvector subspace of solutions y ∈ C ∞ (K) of a system of m homogeneous PDEs with constant coefficients. Namely, we consider the ℂ-vector space (7.257) ∂ ∂

∂ , . . . , pm i = KerC ∞ (K) p i KerC ∞ (K) p1 i ∂t ∂t ∂t ∂ 4 2 ∞ (y) ≡ 0 on K , = y ∈ C (K) : p i ∂t where pj ∈ ℂ[X1 , . . . , Xn ] for j = 1, . . . , m. 52 Since Fourier analysis is very much present in this subsection (as it was in the previous one), we make a distinction between the time domain ℝn = ℝn t (with variables denoted as usual by t = (t1 , . . . , tn )) and its dual copy (ℝn ) , namely the frequency domain, where the variables will be ω = (ω1 , . . . , ωn ).

402

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

It follows from Hilbert’s syzygy theorem [Hil1890] for ℂ[X1 , . . . , Xn ]-finitely generated modules that there exists a generically exact complex, with length L ≤ n and polynomial morphisms, F

FL−1

F

L 2 (7.258) (Fp• ) : 0 −→ ℂmL −→ ℂmL−1 −→ · · · −→ ℂm =

m 

ℂ ∗j

m p 1

j j

=p(z)

−→

ℂ

1

such that the associated complex of sheaves of sections 0 −→ O⊕

mL

−→ O⊕

mL−1

−→ · · · −→ O⊕

m

 (h1 ,...,hm ) → n 1 hj pj

−→

O −→ 0 O −→ m 1 pj O p

p

is exact. It follows from Proposition 7.84 that the currents P0F• and R0F• attached to the metrization of Fp• (with the canonical metric on each trivial bundle ℂm ) by Proposition 3.28, are log(1 + z2 )/2-tempered. We let p p   BFp = Blog(1+z2 )/2 (P0F• , R0F• )

(7.259)

•

according to (7.214). Now, Proposition 4.43 and Remark 4.45 imply that the Fp• Hefer problem (with respect to the Koszul complex K• D attached to (E , Δ) as in p p }≥0,r≥−1 , where the H,r are polynomial. Proposi(4.100)) admits a solution {H,r tion 4.29, applied on ℂn × ℂn \ {(z, ζ) : p(z) = 0}, shows that the global Bochner– Martinelli weight attached to such a solution of the Hefer problem is described by p

p

p

(7.260) W F• (z, ζ) = 2iπp(z) H1p (z, ζ) P0F• (ζ) + H0p (z, ζ) R0F• (ζ) 

min(m,n+1)

= 2iπm pj (z)j 1



min(m,n) p

F• p H1,r (z, ζ) P0,r (ζ) +

r=1

p

F• p H0,r (z, ζ) R0,r (ζ).

r=1

p We define the degree of the solution {H,r }≥0,r≥−1 to the Fp• -Hefer problem by

(7.261)

p deg≤1 Hp = max max degz,ζ H,r (z, ζ), ≤1 r≥1





where the degree of a polynomial map (z, ζ) −→ Φ(z, ζ) from ℂn ×ℂm to ℂn ×ℂm denotes the maximum of the degrees of its polynomial entries. As in (7.228), we introduce for N ∈ ℕ∗ the Bochner–Martinelli global weight on the open subset {(z, ζ) ∈ ℂn × ℂn : p(z) = 0} defined by p

WK,N [p] = WℂNn eWHK ∗ρℝ −1 W F• ,

(7.262) where

WHK ∗ρℝ (z, ζ)   n

   ∂ (HK ∗ ρℝ )(Im ζ) = 1+2 (zk − ζk ) + 2 ddc (HK ∗ ρℝ )(Im ζ) , ∂ζk k=1

7.6. MULTIVARIATE RESIDUE CALCULUS AND THE EXPONENTIAL FUNCTION

403

so that eWHK ∗ρℝ −1 (z, ζ)  r n     ∂ (H ∗ρ )(Im ζ) 2 ddc (HK ∗ρℝ )(Im ζ) K ℝ , z−ζ = exp 2 ∂ζ r! r=0 = e−iz , (∇ (HK ∗ρℝ ))(Im ζ)  r n   2 ddc (HK ∗ρℝ )(Im ζ)

iζ , (∇ (HK ∗ρℝ ))(Im ζ) . × e r! r=0 Then the convexity inequality (7.263)    −iz , (∇ (H ∗ρ ))(Im ζ) iζ , (∇ (H ∗ρ ))(Im ζ)  exp (HK ∗ ρℝ )(Im z) K ℝ K ℝ e ≤   e exp (HK ∗ ρℝ )(Im ζ)   2 supK ζ exp HK (Im z) , ≤e exp HK (Im ζ) valid for all (z, ζ) ∈ ℂn ×ℂn , plays in this subsection the role played by the convexity inequality (7.239). We can now state the following multivariate, semilocal analogue of Euler’s result; see [AndW07, Theorem 8.2] or also [BerP89]. Theorem 7.92. Let the pj’s, K, HK , ρℝ be as above. Let deg p = maxj deg pj . There is an integer N ∈ ℕ∗ , independent of K, and for each ν ∈ ℕn such that |ν| ≤ N , there exists a distribution TN,ν ∈ D (ℂn ), also independent on K, which is a sum of distributions of the form (∂/∂ζ)α (∂/∂ζ)β [dμ], where α, β ∈ ℕn and (1 + ζ)deg p |dμ| is a positive measure with finite total mass on ℂn , such that TN,ν is annihilated by each pj . Furthermore, any ∂ ∂

, . . . , pm i y ∈ KerC ∞ (K) p1 i ∂t ∂t is reproduced in K as (7.264) ∂ ν     y(t) = [y] ∇(HK ∗ ρℝ )(Im ζ) . TN,ν (ζ) , eiζ,∇(HK ∗ρℝ )(Im ζ)−t ∂τ |ν|≤N

Proof. Let  ∈ ℕ∗ . We set (7.265)

N  = BFp + deg≤1 Hp + deg p + 1. •

Let also (z, ζ) −→ q(z, ζ) be as defined in (7.229) and χ ∈ D(ℂn , [0, 1]) be a cut-off function identically equal to one in a neighborhood of 𝔹ℂn (0, 1). For any t ∈ K and z ∈ 𝔹ℂn (0, ), one has, according to Proposition 4.33, that  (7.266) e−iz,t = δ&t (z) = e−iζ,t ℂn   

∂ ζ 1 − χ(ζ/) q(z, ζ) n+1  N  WH ∗ρ −1 Fp•  W ℂn e K ℝ W (z, ζ) × χ(ζ/) + . 2iπ n,n It follows from the choice of N  in (7.265) (according to the definition of BFp , • see (7.214)), from the convexity inequality (7.263), together with the fact that all derivatives up to a finite order of χ and HK ∗ ρℝ are uniformly bounded, that the

404

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

following assertion holds:53 for each integer r  = 1, . . . , min(m, n), and for each integer r  = 1, . . . , min(m, n + 1), for each ν ∈ ℕn with |ν| ≤ N = N  + deg≤1 Hp , one can find smooth integral kernels (0:r  )

(1:r  )

∞ n mr  ∗ ΩN,ν ∈ Cn,n−r ) ),  (ℂ , (ℂ

ΩN,ν

∞ n mr ∈ Cn,n−(r , (ℂm )∗ ))  −1) (ℂ , Homℂ (ℂ

such that (7.266) leads, when  tends to infinity, to the explicit full division formula (7.267) e−iz,t =



(−iz)ν

min(m,n)  

|ν|≤N

p

ℂn

r=1 min(m,n+1) 

p iζ,∇(HK ∗ρℝ )(Im ζ)−t (1:r) F• ¤ (z) e Ω P δ∇(H 0,r , K ∗ρℝ )(Im ζ) N,ν



+ 2iπ p(z)

ℂn

r=1

(0:r) F• ¤ (z) eiζ,∇(HK ∗ρℝ )(Im ζ)−t ΩN,ν R0,r δ∇(H K ∗ρℝ )(Im ζ)

(0,r  )

(1:r  )

where the vector-valued distributions TN,ν and TN,ν , which are defined for each r  = 1, . . . , min(m, n), r  = 1, . . . , min(m, n + 1) and ν ∈ ℕn with |ν| ≤ N by (7.268)

(0:r  )

(0:r  )

p

F• ΩN,ν R0,r (ddc ζ2 )n ,  = TN,ν

(1:r  )

ΩN,ν

p

(1:r  )

F• P0,r  = TN,ν

(ddc ζ2 )n ,

are sums of distributions of the form (∂/∂ζ)α (∂/∂ζ)β (dμ), where α, β ∈ ℕn and the positive measure (1 + ζ)deg p+|α|+|β| |dμ(ζ)| is a complex measure with finite total mass on ℂn . The fact that all derivatives up to a finite order of HK ∗ρℝ are bounded, together with the convexity inequality (7.263), ensures the absolute convergence of the integrals over ℂn on the right-hand side of (7.267), once the action of the distributions in (7.268) is described in terms of integration by parts. It also ensures that for any ν ∈ ℕn with |ν| ≤ N , one has, since (∇(HK ∗ ρℝ ))(ℝn ) ⊂ K, that min(m,n+1) 

(7.269) 2iπp(z)



ℂn

r=1

=

(1:r) F• ¤ δ∇(H (z) eiζ,∇(HK ∗ρℝ )(Im ζ)−t ΩN,ν P0,r K ∗ρℝ )(Im ζ)

m  j=1

p

m ¤  ’ t,j pj (z) dμt,j (z) = dμ N,ν N,ν ∗ pj (−i∂/∂t)(δ0 )(z), j=1

μt,j N,ν ’s

are measures supported by K. On the other hand, for each such where the ν ∈ ℕn with |ν| ≤ N , the entire function min(m,n) 

z −→

 r=1

ℂn

(0:r) F• ¤ (z) eiζ,∇(HK ∗ρℝ )(Im ζ)−t ΩN,ν R0,r δ∇(H K ∗ρℝ )(Im ζ) p

is the analytic continuation to ℂn of the Fourier transform of the averaged measure  (0:r) Fp • δ∇(HK ∗ρℝ )(Im ζ) eiζ,∇(HK ∗ρℝ )(Im ζ)−t ΩN,ν R0,r (7.270) dμtN,ν := ℂn

supported by K. Taking the antecedents through Fourier transform (acting on the space of compactly supported distributions on ℝn ) of both sides in (7.267) leads 53 One needs to repeat the argument used in the proof of Theorem 7.87 more precisely here: the proof of the full division formula (7.244) in Remark 7.88 with ε = 0 and M = 0.

7.6. MULTIVARIATE RESIDUE CALCULUS AND THE EXPONENTIAL FUNCTION

then to (7.271) δt =

405

m

   ν  ν − ∂/∂t (dμt,j dμtN,ν ∗ − ∂/∂t (δ0 ) + N,ν ) ∗ pj (−i∂/∂t)(δ0 ) .

|ν|≤N 

j=1

It follows from (7.271) that    (7.272) y(τ )dδt (τ ) = y(t) = ℝn

|ν|≤N

+

∂ ν [y](τ ) dμtN,ν (τ ) ℝn ∂τ

m   

|ν|≤N j=1

∂ ν ∂  pj i [y] (τ ) dμt,j N,ν (τ ). ∂t ℝn ∂τ

Since

∂ ∂

y ∈ KerC ∞ (K) p1 i , . . . , pm i , ∂t ∂t one concludes from (7.272) that  

min(m,n) ∂ ν   t [y](τ ) dμN,ν (τ ) = y(t) = ∂τ n r=1 |ν|≤N ℝ |ν|≤N  ∂ ν    (0:r) Fp•  (ζ), eiζ,∇(HK ∗ρℝ )(Im ζ)−t [y] ∇(HK ∗ ρℝ )(Im ζ) ΩN,ν R0,r ∂τ ℂn

which is the integral representation formula (7.264). The structure of the distribu(0:r) tions TN,ν in (7.268), for every r = 1, . . . , min(m, n), shows that the integrals on the right-hand side of (7.264) can be differentiated at least deg p times by Lebesgue’s differentiation theorem as functions of t in the integrand. It follows then from The orem 3.38 that (7.264) reproduces the fact that y ∈ KerC ∞ (K) (p(i∂/∂t)). Let us now come to the more classical (global) formulation of the Ehrenpreis– Palamodov fundamental principle. Theorem 5.5 admits the following transposition to the global setting. Let p1 , . . . , pm ∈ ℂ[X1 , . . . , Xn ] and (7.273)

m 

pj ℂ[X1 , . . . , Xn ] = Q1 ∩ · · · ∩ QM

j=1

m be a reduced primary decomposition of the ideal j=1 pj ℂ[X1 , . . . , Xn ] in the polynomial algebra ℂ[X1 , . . . , Xn ]. It means that / / ι = ι =⇒ Qι = Qι 3 (7.274) Q ⊃ Q  for any ι. ι

ι

ι =ι

√ Recall now that the list of associated primes Pι = Qι is uniquely determined up to order. The associated primes, minimal for ordering by inclusion, are the isolated ones. The remaining elements of the list are the embedded ones. Nœther’s preparation lemma, together with the existence of primitive elements for algebraic extensions (see [Bj74, Chapter 8]) implies that for any ι ∈ {1, . . . , M } there is a list ∂   : Nι,ν ∈ ℂ[X1 , . . . , Xn ] d/dX1 , . . . , d/dXn  , ν = 1, . . . , νι Nι,ν ζ, ∂ζ

406

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

of algebraic nœtherian operators such that the following statement holds for any h ∈ O(Ω), where Ω is an open subset of ℂn : (7.275) h ∈

m 

pj O(Ω) ⇐⇒

j=1

∂ [h] = 0 on V (Pι ) ∩ Ω for ν = 1, . . . , νι . Nι,ν ζ, ∂ζ The existence of such a collection of polynomial differential operators with polynomial coefficients goes back to Macaulay [Mac]. In addition to quoting [Bj74, Chapter 8], we refer the reader also to the pioneering monograph by V. Palamodov [Pal], as well as to [ChHKL22] for an updated bibliography and an algorithmic approach. To formulate the fundamental principle of Ehrenpreis and Palamodov [Ehr, Pal] in its classical (nonexplicit) global form, we need to attach to any convex domain U ⊂ ℝn the family K (U) of all nonvanishing positive continuous functions ψ : ℂn −→ ]0, +∞[ such that |& u| (7.276) ∀ u ∈ D  (ℝn ), Supp u  U =⇒ sup < +∞. ψ ℂn ∀ ι = 1, . . . , M,

Here is the classical formulation of such a principle.54 Theorem 7.93. Let p = (p1 , . . . , pm ) be a2 collection ofpolynomials in the  p ζ, ∂/∂ζ , ι = 1, . . . , M, ν = , . . . , X ] and let N = N polynomial algebra ℂ[X 1 n ι,ν 4 subordinate to a reduced primary 1, . . . , νι be a list of algebraic nœtherian operators m decomposition (7.273) with length M of p ℂ[X , . . . , X ] in ℂ[X1 , . . . , Xn ]. j 1 n 1 Given any convex subset U and any ∂ ∂

, . . . , pm i y ∈ KerC ∞ (U) p1 i ∂t ∂t y there exist ψ ∈ K (U) and complex measures dμι,ν on ℂn , whose support satisfies Supp dμyι,ν ⊂ V (Pι ) such that

(7.277)

νι  M   ι=1 ν=1

y(t) =

∀ ι = 1, . . . , M, ∀ ν = 1, . . . , νι ,

ψ(ζ) |dμyι,ν (ζ)| < ∞,

V (Pι )

νι  M   ι=1 ν=1

∂ Nι,ν ζ, [e−iζ , t ](ζ) dμyι,ν (ζ) ∂ζ ℂn

∀t ∈ U.

We want to reformulate Theorem 7.93 in a form appealing to Bochner–Martinelli residue currents instead of nœtherian operators. For that purpose, let us assume that U is a convex open subset in ℝn admitting the exhaustion  U= U , 0 ∈ U ⊂ K = U ⊂ U+1 for all  ∈ ℕ∗ , ≥1

where the closure K is bounded, strictly convex, and has a smooth boundary. Let ∂ ∂

y ∈ KerC ∞ (U) p1 i , . . . , pm i . ∂t ∂t 54 See

[Han81, §3] for its extension through a functional analysis point of view.

7.6. MULTIVARIATE RESIDUE CALCULUS AND THE EXPONENTIAL FUNCTION

407

Let us observe that, given any sequence of positive real numbers (γ )≥1 such that γ ≥ supK HK for any  ∈ ℕ∗ , the convex function ω ∈ ℝn −→ sup(HK (ω) − γ ) ≥1

is finite on ℝn . Actually, HK (ω) − γ ≤ 0 as soon as ω ∈ K , which occurs when  ≥ (ω) for each ω ∈ U since the K exhaust U. We can then state the following alternative form of the Ehrenpreis–Palamodov fundamental principle in the global setting. Theorem 7.94. Let U ⊂ ℝn be open and convex. Let also the exhausting sequence (K )≥1 and the pj’s be as above. There exists an integer M = M (p) and for each y ∈ KerC ∞ (U) (p(i∂/∂t)), there exists a sequence (γy, )≥1 of positive numbers satisfying γy, ≥ supK HK for any  ∈ ℕ∗ , so that y is reproduced in U as  (7.278) y(t) = Ty (ζ), e−iζ,t . The distribution Ty is annihilated by the pj ’s. Moreover, it is a finite sum of distributions of the form ∂ α ∂ β (dμα,β y ) ∂ζ ∂ζ depending on y such that with |α| + |β| ≤ M and complex measures dμα,β y  (7.279) (1 + ζ)deg p elim sup →∞ (HK (Im ζ)−γy, ) |dμα,β y | < +∞. ℂn

The representation (7.278) reproduces the fact that y ∈ KerC ∞ (U) (p(i∂/∂t)). Proof. For each  ∈ ℕ∗ , the following representation formulae hold for y in K (see Theorem 7.92). For any t ∈ K and  ≥ , (7.280) y(t) =    iζ,∇(H ∗ρ )(Im ζ) ∂ ν   K  ℝ

[y] ∇(HK  ∗ ρℝ )(Im ζ) TN,ν (ζ) , e−iζ,t , e ∂τ |ν|≤N

where N = N (p) ∈ ℕ∗ is conveniently chosen independently of y. Furthermore, the distributions TN,ν involved in the representation (7.280) are independent of y and  and such that ∂ α ∂ β  dμα,β TN,ν = N,ν , ∂ζ ∂ζ {α,β∈ℕn : |α|+|β|≤M } N

where the complex measures

dμα,β N,ν



(7.281) {α,β∈ℕn

: |α|+|β|≤MN }

 ℂn

(1 + ζ)deg p+|α|+|β| |dμα,β N,ν | < +∞.

As a consequence, for each  ≥  and ν ∈ ℕn with |ν| ≤ N , the distribution ∂ ν   [y] ∇(HK  ∗ ρℝ )(Im ζ) TN,ν (ζ), TN,ν : y, := eiζ,∇(HK  ∗ρℝ )(Im ζ) ∂τ

408

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

acting on ζ → e−iζ,t in the bracket (7.280) and remaining annihilated by each polynomial map ζ → pj (ζ), splits as ∂ α ∂ β  TN,ν : y, = dμα,β N,ν : y, . ∂ζ ∂ζ n {α,β∈ℕ : |α|+|β|≤MN }

Here the complex measures dμα,β N,ν : y, are such that    (7.282) (1 + ζ)deg p eHK  (Im ζ)−γy,  |dμα,β N,ν : y, | ≤ 1, |ν|≤N |α|+|β|≤MN

ℂn

where γy, ≥ supK  HK  is a constant depending on K , on the estimates for the derivatives of ρℝ with order less than MN as well as on those of the derivatives of y with order less than MN + |ν| on K , and on the positive constant (7.281). The convexity inequality (7.263) with K = K and z = 0, plays here a crucial role. One has then, for any t ∈ K and  ≥ ,  ∂ α ∂ β    −iζ,t (dμα,β )(ζ) , e (7.283) y(t) =  N,ν ; y, ∂ζ ∂ζ |ν|≤N |α|+|β|≤M N

with estimates (7.282) for the positive measures |dμα,β N,ν ; y, | and ∂ α ∂ β   (7.284) pj (ζ) (dμα,β N,ν ; y, ) = 0 ∂ζ ∂ζ |ν|≤N |α|+|β|≤M N

for j = 1, . . . , m. An argument of the Montel type shows that for any  ∈ ℕ∗ , one can extract from the sequence (indexed by  = ,  + 1,  + 2, . . . ) of vectors (indexed by ν, α, β) of measures   α,β (dμα,β N,ν ; y, )ν  ≥ α,β a subsequence, which converges weakly on ℂn towards (dμα,β N,ν : y,≥ )ν . Thus, for any t ∈ K ,  ∂ α ∂ β    −iζ,t (dμα,β )(ζ) , e (7.285) y(t) = N,ν : y,≥ ∂ζ ∂ζ |ν|≤N |α|+|β|≤MN

with (7.286)







|ν|≤N |α|+|β|≤MN

and (7.287)

pj (ζ)



ℂn

(1 + ζ)deg p esup  ≥ (HK  (Im ζ)−γy,  ) |dμα,β N,ν ; y,≥ | ≤ 1 

|ν|≤N |α|+|β|≤MN

∂ α ∂ β (dμα,β N,ν ; y,≥ ) = 0 ∂ζ ∂ζ

for j = 1, . . . , m. We now repeat the Montel argument and extract from the sequence (indexed this time by  = 1, 2, . . . ) of vectors (indexed by ν, α, β) of measures   α,β (dμα,β N,ν;≥,y )ν ≥1 α,β a subsequence converging weakly on ℂn towards (dμα,β N,ν : y )ν . One has then for any t ∈ U that  ∂ α ∂ β    −iζ,t (dμα,β y(t) = N,ν : y )(ζ) , e ∂ζ ∂ζ |ν|≤N |α|+|β|≤MN

7.6. MULTIVARIATE RESIDUE CALCULUS AND THE EXPONENTIAL FUNCTION

with





|ν|≤N |α|+|β|≤MN

and pj (ζ)

409

 ℂn

(1 + ζ)deg p elim sup →∞ (HK (Im ζ)−γy, ) |dμα,β N,ν : y | ≤ 1





|ν|≤N |α|+|β|≤MN

∂ α ∂ β (dμα,β N,ν : y ) = 0 ∂ζ ∂ζ

for j = 1, . . . , m. The last assertion follows from the fact that the estimates (7.279) authorize the differentiation deg p times of the right-hand side of (7.278) as a function of t under the bracket, combined with the fact that the distribution Ty is annihilated by the polynomial maps ζ −→ pj (ζ) for j = 1, . . . , m. The proof of the theorem is now complete.  The major difference between the representation formulae (7.277) and (7.278) (with constraints (7.279)) is that in the first case, the Ehrenpreis–Palamodov nœtherian operators are involved, while in the second case the Bochner–Martinelli residue p current R0F• is. The action of the residue current is realized through the action (0:r) of the distributions TN,ν introduced in (7.268), on which the materialization of Ty in (7.278) relies, as the proof of Theorem 7.94 shows. The Bochner–Martinelli p current T = R0F• involved there depends in general on the choice of metrics on the complex Fp• and fails in most cases to be a Coleff–Herrera current; see its extrinsic description in §3.4.2 or its intrinsic description in case the ideal generated by the pj ’s is radical in §3.4.3. Nevertheless, there are two significant cases where such a current is Coleff–Herrera: (1) The most important case is the case where m = codim V (p) ≤ n and Fp• = Kp• ,55 thus T = ∂(1/p) ∗1 ∧ · · · ∧ ∗m , if p = p1 1 + · · · + pm m . (2) The other m case is where V (p) is purely dimensional and the quotient sheaf Oℙnℂ / 1 Pj Oℙnℂ , where the Pj ’s denote the homogenizations of the pj ’s, is Cohen–Macaulay in ℂn . In this case, one can take the complex Fp• as (7.280) with length codim V (p) so that the corresponding Bochner– Martinelli current T is Coleff–Herrera, with pure bidegree (0, codim V (p)), but of course matricial.56 Even in such particular cases, where the residue current is Coleff–Herrera, hence can be locally described by Proposition 2.37, the two global formulations (7.277) and (7.278) (with constraints (7.279)) of the Ehrenpreis–Palamodov fundamental principle in the global setting remain different. The main reason for that (see also §5.1.2) is the polar part 1/δ appearing in the local presentations of Coleff–Herrera residue currents as  |δ|2λ

[V (p)] Q∗ δ λ=0 (see Proposition 2.37), which is not the case in representation (7.277) involving nœtherian operators. Thus formulations (7.277) and (7.278) (with constraints (7.279)) remain intrinsically different. The main advantage of the current formulation of the fundamental principle is that semilocally speaking, it can be made explicit in strongly convex relatively compact domains with smooth boundaries 55 This 56 Such

particular case was for example studied in [BGVY, Chapter 5, §3] or in [BerP89]. is the case for example when codim V (p) = n.

410

7. MISCELLANEOUS APPLICATIONS: INTERSECTION, DIVISION

such as in Theorem 7.92. It is not the case for the representation (7.93), since its proof relies on its final step on a functional analysis argument. Moreover, it is not the case for the current formulation in Theorem 7.94 either, since its proof relies on a Montel-type argument too. As seen in Corollary 7.89, another advantage of the current approach is that sometimes it carries through in the transcendental context for the representation of mean-periodic solutions, where compactly supported distributions replace differential operators with constant coefficients. 7.7. Residue calculus outside the commutative setting Although we did not explicitly mention that commutativity remained crucial to settle univariate or multidimensional residue theory, this is what we have seen until now throughout this monograph. Therefore, it would be a fascinating challenge to move towards the noncommutative world some of the concepts that the presented residue theory carries, and some of the tools it provides. As examples, being able to move towards the operator theory setting the strategy used to solve B´ezout identity explicitly, hence Hilbert’s nullstellensatz (as presented in §7.2.1) or Brian¸con–Skoda’s problem in ℂ[X1 , . . . , Xn ] (see Theorem 7.47) would indeed be very exciting.57 We only present in this last section a few modest attempts towards such a goal. 7.7.1. Univariate quaternionic setting and relevant regularity concepts. Consider the quaternionic algebra ℍ as a model in this subsection. We begin by recalling some basic facts about various presentations, as well as various notions of regularity (Fueter, Fueter modified, slice regularity), since our goal is to analyze in §7.7.2 how the concepts of principal value or residue current could get a chance to be transposed in this noncommutative (but fundamental and well explored) setting. The most classical representation of the generic element in ℍ is Z = x1 1 + y1 i + x2 j + y2 k

(7.288)

where (x1 , y1 , x2 , y2 ) ∈ ℝ and the generators i, j, k follow the rules 4

(7.289)

i2 = j 2 = −1, i j = −j i = k,

governing the univariate (noncommutative) quaternionic calculus. There exist several alternatives to such a presentation. A few of them follow. (1) The quaternionic algebra ℍ is the four-dimensional real Clifford algebra Cl0,2 (ℝ) = Cl2 (ℝ) = ℝ + ℝ 1 + ℝ 2 + ℝ 1 · 2 , where {1 , 2 } denotes the canonical basis of ℝ2 and the rules (7.290)

j · j  + j  · j = −2 δjj



govern the noncommutative calculus. The ℝ-isomorphism consisting of the identifications 1 ←→ i, 2 ←→ j, 1 · 2 ←→ k realizes the equivalence with the classical presentation given initially by (7.288) and (7.289); see for example [CSSS04, Example 4.1.2 (2)]. 57 We should mention here that pertinent questions of such nature were frequently posed to us by Nikolai Nikolski.

7.7. RESIDUE CALCULUS OUTSIDE THE COMMUTATIVE SETTING

411

(2) The quaternionic algebra ℍ is the even subalgebra of the eight-dimensional real Clifford algebra Cl0,3 (ℝ) = Cl3 (ℝ) = ℝ+ℝ 1 +ℝ 2 +ℝ 3 +ℝ 1 ·2 +ℝ 2 ·3 +ℝ 3 ·1 +ℝ 1 ·2 ·3 , where {1 , 2 , 3 } is the canonical basis of ℝ3 and the rules (7.290) still govern noncommutative calculus. The ℝ-isomorphism consisting of the identifications 1 · 2 ←→ i, 3 · 1 ←→ j, 1 · 2 = k realizes then the equivalence with the classical presentation given initially by (7.288) and (7.289). (3) More interesting for us is the embedding   x1 − iy1 −x2 + iy2 (7.291) Z = x1 1+ y1 i + x2 j + y2 k −→ ∈ Homℂ (ℂ2 , ℂ2 ); x2 + iy2 x1 + iy1 see, for example, [WaW17] for an updated bibliography since it brings us to the complex ℂ2 world. It is well known that multivariate residue theory in ℂ2 profits extensively from the pairing between holomorphic coordinates (z1 , z2 ) and their ghost companions (z1 , z2 ). In view of such a presentation, we will use the notation (7.292)

(7.293)

z 1 = x1 1 + y1 i,

z 2 = x2 1 + y2 i,

so that the embedding (7.291) is expressed as   z −z2 Z = z 1 + z 2 j −→ 1 ∈ Homℂ (ℂ2 , ℂ2 ), z2 z1

where naturally z1 := x1 + iy1 and z2 := x2 + iy2 . (4) Besides the three above presentations of ℍ, one should mention, inspired by the tropical considerations in §7.4.2 and §7.4.3, that ℍ \ ℝ is diffeomorphic to ℝ × ℝ>0 × 𝕊2  ℝ × ℝ>0 × ∂ Bℝ3 (0, 1) as a four-dimensional differentiable manifold. The corresponding diffeomorphism is (7.294)

(x, r, ξ) ∈ ℝ × ℝ>0 × ∂ 𝔹ℝ3 (0, 1) −→ x1 + r(ξ1 i + ξ2 j + ξ3 k), N (y1 , x2 , y2 ) x1 , y12 + |z2 |2 , / 2 ←− x1 1 + y1 i + x2 j + z2 k ∈ ℍ \ ℝ. y1 + |z2 |2 The involuting conjugation

(7.295)

(7.296)

Z = z 1 + z 2 j −→ (x1 1 − y1 i) − z 2 j = x1 1 − (y1 i + x2 j + y2 k), where z 1 = x1 1 + y1 i and z 2 = x2 1 + y2 j are defined as in (7.292), together with the definition of the operator 1 ∂ ∂ ∂ = +i , ∂z 2 ∂x ∂y which plays a central role in complex analysis and in multivariate residue theory as well, suggests the construction in the quaternionic setting of substitutes for the operator (7.296). It was done by Grigore Constantin Moisil [Moi31] and pursued by Rudolf Fueter [Fu35, Fu36]. It leads to the introduction of the so-called Cauchy–Riemann–Moisil–Fueter left

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(resp., right) differential operators acting on the left (resp.,right, depending on how ϕ is presented, see below) on smooth functions ϕ : Z = x1 1 + y1 i + x2 j + y2 k ∈ 𝕌 −→ ϕ(Z) = X1 (x, y) 1 + Y1 (x, y) i + X2 (x, y) j + Y2 (x, y) k = 1 X1 (x, y) + i Y1 (x, y) i + j X2 (x, y) + k Y2 (x, y) ∈ ℍ from an open subset 𝕌 of the four-dimensional real manifold ℍ to ℍ, with X1 , X2 , Y1 , Y2 smooth real functions on 𝕌. Definition 7.95. The Cauchy–Riemann-Moisil-Fueter left and right operators are respectively defined by →

(7.297)

∂ ∂ ∂ ∂ ∂ , + i +j +k =1 ∂x1 ∂y1 ∂x2 ∂y2 ∂Z ← ∂ ∂ ∂ ∂ ∂ = 1+ i+ j+ k . ∂x1 ∂y1 ∂x2 ∂y2 ∂Z

The subsequent notions of the left (resp., right) regularity in the sense of Fueter follow. We will consider only left regularity here since the concept of right regularity is analogous. Definition 7.96. A differentiable function ϕ : 𝕌 → ℍ is said to be (left) regular in the sense of Fueter if and only one has in 𝕌 that ⎞ ⎛ ⎛ ⎞ → Z (z) −Z (z) ∂ ∂ ⎝ 1 ⎠ ∂ ⎝ 2 ⎠ (7.298) ϕ ≡ 0 ⇐⇒ = ∂z2 Z (z) ∂z1 ∂Z Z1 (z) 2 when ϕ(z) := X1 (x, y) 1 + Y1 (x, y) i + X2 (x, y) j + Y2 (x, y) k and z = (x1 + iy1 , x2 + iy2 ), Z1 (z) = (X1 + iY1 )(x, y), Z2 (z) = (X2 + iY2 )(x, y) according to the notations (7.292). The role played by the Cauchy kernel defined U = U × U by 1 (z, ζ) ∈ U \ D −→ ζ −z in univariate complex analysis, where U is an open subset of ℂ and D ⊂ U = {(z, ζ) : ζ = z}, if one keeps track of the notations introduced in §4.1, is now played by the archetypal example of a (both left and right) regular function in the sense of Fueter outside the diagonal Dℍ = {(z, ζ) ∈ ℍ × ℍ : ζ = z}. Namely, the Cauchy kernel becomes now the Cauchy–Fueter kernel, denoted following the notations from §4.2 and defined by (7.299) ζ−Z (ζ − z 1 ) − (ζ 2 − z 2 ) j 1 = 1 , (ζ − Z)−1 = Kℍ (Z, ζ) = 4 ζ − z ζ − z4 (Z − ζ)(Z − ζ)) where Z = z 1 + z 2 j, ζ = ζ 1 + ζ 2 j, z = (z1 , z2 ) ∈ ℂ2 , ζ = (ζ1 , ζ2 ) ∈ ℂ2 according to (7.292). The analogy between the Cauchy–Fueter kernel in (ℍ × ℍ) \ Dℍ and the Cauchy kernel in (ℂ × ℂ) \ D is reinforced by the fact that functions in C 1 (𝕌, ℍ),

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413

where 𝕌 is a bounded open subset of ℍ with piecewise smooth boundary, can be reproduced in 𝕌 thanks to the Cauchy–Fueter–Pompeiu formula; see for example [Su79], [CSSS04, Theorem 3.1.7] and [Pe07, §2.5].58 Theorem 7.97. Let 𝕌 ⊂ ℍ be as above. Any ϕ ∈ C 1 (𝕌, ℍ) admits in 𝕌 the following Cauchy–Pompeiu type integral representation formula (7.300) ϕ(Z) = →  

1 ∂ c 2 2 c 2 2 . 𝕂ℍ (ζ − Z)  → (dd ζ ) ϕ(ζ) − 𝕂 (ζ − Z) ζ ) ϕ(ζ) (dd ℍ ∂ 4 ∂𝕌 ∂ζ 𝕌 ∂ζ As a consequence, if ϕ ∈ C 0 (𝕌, ℍ) ∩ C 1 (𝕌, ℍ) is (left) regular in 𝕌 in the sense of Fueter, then the Cauchy–Fueter integral representation formula  1 𝕂ℍ (ζ − Z)  → (ddc ζ2 )2 ϕ(ζ) (7.301) ϕ(Z) = ∂ 4 ∂𝕌 ∂ζ holds for any Z ∈ 𝕌. Unfortunately, such an essential function as the identity Idℍ : Z −→ Z fails to be (left) regular in the sense of Fueter in ℍ, since →

∂ Z = 1 + i2 + j 2 + k2 = −2. ∂ Z¯ Any monomial Z k (or Laurent monomial Z −k in ℍ \ 0) for k ≥ 1 fails also to be left or right regular in the sense of Fueter. In the paper [Fu39] devoted to Hartogs’s theorem, Fueter came up with the brilliant suggestion to modify the Moisil operators (7.297) in order that, given a holomorphic map f = (f1 , f2 ) from U ⊂ ℂ2 to ℂ2 , the ℍ-valued map (7.302)

f : Z ∈ {z 1 + z 2 j : (z1 , z2 ) ∈ U } −→ f1 (z) + f2 (z) j ∈ ℍ

inherits regularity with respect to left or right-modified operators. Definition 7.98. The modified Cauchy–Fueter left and right operators are respectively defined by

(7.303)

→ ∂ ∂  ∂ ∂ ∂ , =1 + i +j −k ∂x1 ∂y1 ∂x2 ∂y2 ∂Z ∂ ← ∂  ∂ ∂ ∂ = 1+ i+ j− k . ∂x1 ∂y1 ∂x2 ∂y2 ∂Z

The subsequent notions of the left (resp., right) regularity in the sense of Fueter follow. We will consider again only left regularity here since the notion of right regularity is analogous. We recall now the terminology introduced in [CLSSS07, Dol13]. 58 Observe that (ddc ζ2 )2 equals 2/π 2 times the volume form dx ∧ dy ∧ dx ∧ dy on 1 1 2 2 ℍ  ℝ4  ℂ2 as a four-dimensional differential manifold.

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Definition 7.99. A differentiable function ϕ : 𝕌 → ℍ is said to be (left) hyperholomorphic if and only if one has in 𝕌 that ⎞ ⎞ ⎛ ⎛ Z (z) −Z (z) → ∂ ⎝ 2 ⎠ ∂  ∂ ⎝ 1 ⎠ = ϕ ≡ 0 ⇐⇒ (7.304) ∂z2 ∂z 1 ∂Z Z1 (z) Z2 (z) when ϕ(z) := X1 (x, y) 1 + Y1 (x, y) i + X2 (x, y) j + Y2 (x, y) k and z = (x1 + iy1 , x2 + iy2 ), Z1 (z) = (X1 + iY1 )(x, y), Z2 (z) = (X2 + iY2 )(x, y) according to (7.292). Example 7.100. Here is the major example motivating the modification of the Cauchy–Fueter–Moisil operators. As we already mentioned, given U ⊂ ℂ2 and the map f = (f1 , f2 ) holomorphic from U to ℂ2 , then the map ff ∈ C ∞ (𝕌, ℍ) defined as in (7.302) is (left) hyperholomorphic in 𝕌 = {z 1 + z 2 j : (z1 , z2 ) ∈ U }. Actually, it is enough to observe that the four entries to compare in the matrix equality on the right-hand side of (7.304) equal zero in view of the formal independence of holomorphic coordinates with respect to their antiholomorphic ghost companions; see (7.173). The role devoted to the Cauchy–Fueter kernel Kℍ is played now by the archetypal example of left or right hyperholomorphic function. Namely, it is played by the modified Cauchy–Fueter kernel, denoted again by the notations compatible to those in §4.2 and defined by (7.305)

Kℍ (Z, ζ) =

(ζ 1 − z 1 ) − (ζ 2 − z 2 ) j , ζ − z4

where Z = z 1 + z 2 j, ζ = ζ 1 + ζ 2 j, z = (z1 , z2 ) ∈ ℂ2 , ζ = (ζ1 , ζ2 ) ∈ ℂ2 according to (7.292). The analogy between the modified Cauchy–Fueter kernel in (ℍ×ℍ)\Dℍ and that of the Cauchy kernel in (ℂ×ℂ)\D is reinforced once more by the fact that functions in C 1 (𝕌, ℍ), where 𝕌 is a bounded open subset of ℍ with piecewise smooth boundary, can be reproduced in 𝕌 thanks to the modified Cauchy–Fueter–Pompeiu formula; see for example [CSSS04, Theorem 3.1.7]. Theorem 7.101. Let 𝕌 ⊂ ℍ be as above. Any ϕ ∈ C 1 (𝕌, ℍ) admits in 𝕌 the following Cauchy–Pompeiu type integral representation formula (7.306) ϕ(Z) =   →

1 ∂   c 2 2  c 2 2  (dd ζ ) ϕ(ζ) − 𝕂ℍ (ζ − Z) 𝕂ℍ (ζ − Z) → ϕ(ζ) (dd ζ ) . ∂ 4 ∂𝕌 ∂ζ 𝕌 ∂ζ As a consequence, if ϕ ∈ C 0 (𝕌, ℍ) ∩ C 1 (𝕌, ℍ) is hyperholomorphic in 𝕌, then the modified Cauchy–Fueter integral representation formula  1  (ddc ζ2 )2 ϕ(ζ) 𝕂 (ζ − Z)  → (7.307) ϕ(Z) = ∂ 4 ∂𝕌 ℍ ∂ζ holds for any Z ∈ 𝕌. Remark 7.102. The clever idea of Rudolf Fueter in [Fu39], namely the idea to relate the theory of holomorphic functions in two (resp., more) complex variables to that of hyperholomorphic ones in one quaternionic variable Z (resp., in some

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415

more general hypercomplex variable) inspired Enzo Martinelli. He introduced the Bochner–Martinelli kernel Kℂn in [Mar43]. This kernel Kℂn (see §4.2.1) is a central object in the present monograph. The identity Idℍ : ℍ → ℍ is hyperholomorphic but Z −1 : ℍ\{0} → ℍ\{0} fails to be so. More generally, when h = h 1 + h 2 j, with h 1 (Z) = X1 (x, y) + Y1 (x, y) i and h 2 (Z) = X2 (x, y) + Y2 (x, y) i, is hyperholomorphic in 𝕌, h−1 : Z −→

(X1 (x, y) 1 − Y1 (x, y) i) − (X2 (x, y) 1 + Y2 (x, y) i) j h (Z) h (Z)

fails in general to be hyperholomorphic in 𝕌 \ h−1 ({0}). Similarly, ζ → Kℍ (0, ζ) is hyperholomorphic in ℍ \ {0}, while its inverse ζ → ζ2 (ζ 1 + j ζ 2 ) fails to be hyperholomorphic in ℍ. It is a major stumbling block in view of univariate principal value or residue theory in a quaternionic context. Let us set the following definition from [Dol13, Definition 4.2]. Definition 7.103. Let 𝕌 be an open set of ℍ. A function h defined in 𝕌 is called hypermeromorphic if it is a hyperholomorphic function defined almost everywhere in 𝕌, whose inverse h −1 (see above) shares the same properties. The following proposition is due to Pierre Dolbeault [Dol13, Proposition 3.1]. It emphasizes that hypermeromorphicity (when hyperholomorphicity of the inverse is satisfied) requires undeniably many constraints. For example, these constraints exclude in most cases 1/f , when f = f 1 + f 2 j with f = (f1 , f2 ) : U → ℂ2 being holomorphic. Proposition 7.104. Let f ∈ C 1 (𝕌, ℍ), where 𝕌 is an open subset of ℍ and V (f ) be its zero set in 𝕌. Then f and 1/f are both hyperholomorphic in 𝕌 and 𝕌 \ V (f ), respectively, if and only if ⎞ ⎞ ⎛ ⎛ ⎛ ⎞ f (z) f (z) −f (z) ∂ ⎝ 1 ⎠ ∂ ⎝ 2 ⎠ ∂ ⎝ 1 ⎠ (7.308) 2i Im f1 (z) = f2 (z) , − f2 (z) ∂z1 ∂z1 ∂z 2 f1 (z) f2 (z) f2 (z) where f1 (z) = X1 (z) + iY1 (z), f2 (z) = X2 (z) + iY2 (z), z = (x1 + iy1 , x2 + iy2 ), when f (x1 1 + y1 i + (x2 1 + y2 i) j) = X1 (z) 1 + Y1 (z) i + (X2 (z) 1 + Y2 (z) i) j. Example 7.105. If f = f 1 + f 2 j, where z = (z1 , z2 ) −→ f (z) = (f1 (z), f2 (z)) is holomorphic from U = {z : z 1 + z 2 j ∈ 𝕌} to ℂ2 , conditions (7.308) reduce to ⎞ ⎞ ⎛ ⎛ −f2 (z) f1 (z) ∂ ⎝ ⎠ − f2 (z) ∂ ⎝ ⎠ = 0. f2 (z) (7.309) ∂z1 ∂z 2 f1 (z) f2 (z) Suppose U is connected. The conditions (7.308) are only fulfilled when f2 ≡ 0 or f = constant in U . Example 7.106. If f = f 1 + f 2 j with f1 and f2 real, then the conditions (7.308) reduce to ⎞ ⎞ ⎛ ⎛ −f2 (z) f1 (z) ∂ ⎠− ∂ ⎝ ⎠ = 0. ⎝ (7.310) f2 (z) ∂z1 ∂z 2 f1 (z) f2 (z)

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Note that such conditions are satisfied when f2 ≡ 0 or f1 (z) = x1 + x2 + γ1 and f2 (z) = x2 − x1 + γ2 for real constants γ1 , γ2 ; see [Dol13, Proposition 4]. As we have observed, important quaternionic functions, such as polynomial or Laurent polynomial functions (7.311)

Z ∈ ℍ −→

d 

Z d−ν γ ν ,

Z ∈ ℍ \ 0 −→

ν=0

d 

Z ν−d γ ν ,

ν=0

where γ ν ∈ ℍ, fail to be either left regular in the sense of Fueter or (left) hyperholomorphic. For such reasons, Graziano Gentili and Daniele C. Struppa introduced in [GeS06, GeS07] (see also [CSS16] for an up-to-date bibliography) the concept of (left) slice regularity for quaternionic valued real-differentiable functions on some open subset 𝕌 ⊂ ℍ. Definition 7.107. Let f : 𝕌 −→ ℝ be a real differential function. The function f is said to be (left) slice regular in 𝕌 ⊂ ℍ if and only if for any element I in the unit sphere of purely imaginary quaternions y1 i + x2 j + y2 k with y12 + |z2 |2 = 1, one has ∂ ∂ (7.312) +I f (u + v I) ≡ 0 ∂u ∂v in ΩI := {w = u + iv ∈ ℂ : u 1 + v I ∈ 𝕌}. Remark 7.108. Let us consider the action of the group SO(3, R) on the unit quaternionic sphere 𝕊ℍ = {y1 i + x2 j + y2 k : (y1 , x2 , y2 ) ∈ ∂ 𝔹ℝ3 (0, 1)}. If f is a (left) slice regular function in 𝕌, then for any σ ∈ SO(3, ℝ) and any u + iv ∈ Ωσ(i) , the function u + iv ∈ Ωσ(i) −→ f (u + σ(i)v) decomposes as f (u + v σ(i)) = (Uσ (u, v) + Vσ (u, v) σ(i)) + (Uσ (u, v) + Vσ (u, v) σ(i)) σ(j), where the two complex-valued functions u + iv ∈ Ωσ(i) −→ Uσ + iVσ ,

u + iv ∈ Ωσ(i) −→ Uσ + iVσ

are holomorphic in Ωσ(i) ; see [CSS16, Lemma 2.1]. Example 7.109. Polynomial functions or Laurent polynomial functions as in (7.311) are (left) slice regular. It is the case also for convergent power series ∞ ∞   Z ∈ 𝔹ℍ (0, r) −→ Z κ γ κ , Z ∈ ℍ \ 𝔹ℍ (0, R) −→ Z −κ γ κ . κ=0

κ=1

Let us introduce the commutative multiplication ⍟ for quaternionic functions defined on a common domain as



   (7.313) Z κ γ κ ⍟ Z κ γ κ := Zκ γ κ γ κ . κ

κ

κ

{(κ ,κ ) : κ +κ =κ}

The introduction of multiplication (7.313) allows us to formulate within the algebraic setting of a quaternionic polynomial with coefficients on the right as in (7.311) the following decomposition theorem; see [LavR98, AlB19] or [CSS16, Theorem 3.9].

7.7. RESIDUE CALCULUS OUTSIDE THE COMMUTATIVE SETTING

Theorem 7.110. A quaternionic polynomial map Z −→ p(Z) = with γ ν ∈ ℍ and γ 0 = 0 admits the unique ⍟ decomposition (7.314)

d ν=0

417

Z d−ν γ ν

p(Z) = p𝕊 (Z) ⍟ p0 (Z) = p0 (Z) ⍟ p𝕊 (Z) = p𝕊 (Z) p0 (Z) = p0 (Z) p𝕊 (Z).

d−2m d−2m−ν γ 0,ν The polynomials p𝕊 ∈ ℝ[Z] with degree 2m ≤ d, p0 (Z) = ν=0 Z with γ 0,ν ∈ ℍ and γ 0,0 = 0 are such that V (p0 ) is a discrete or empty subset of ℍ, V (p𝕊 ) is the finite union of at most m ≤ m distinct spheres xι + rι 𝕊ℍ with xι ∈ ℝ and rι > 0 for ι = 1, . . . , m . Moreover, one has the Lelong–Poincar´e type formula m  log |p (Z)|2  𝕊 μι δxι +rι 𝕊ℍ (7.315) Δ2 − = 6 ι=1 in the sense of real-valued distributions on ℍ  ℝ4 , where Δ2 is the bilaplacian59 real differential operator with order 4 ∂2 ∂ 2 2 1 ∂ 2 ∂ 2 ∂ 2 ∂ 2 2 Δ2 = + = + + + ∂z 1 ∂z1 ∂z 2 ∂z2 16 ∂x1 ∂y1 ∂x2 ∂y2   m and μι ∈ ℕ∗ with 1 μι = m. Remark 7.111. Moreover, one can attach to each (isolated) zero of p0 in (7.314) a (nonlocal) multiplicity in terms of the ⍟ factorization. Namely, for each such zero α ∈ x(α) + r(α) 𝕊ℍ = 𝕊ℍ (α) according to the polar representation of quaternions (7.294), one can find the algorithmically unique sequence [α1 = α, . . . , αμα (p0 ) ] with entries in 𝕊ℍ (α) and maximal length μα (p0 ) such that p0 (Z) = (Z − α1 ) ⍟ · · · ⍟ (Z − αμα(p0 ) ) ⍟ qα (Z), where αj+1 = αj for j = 1, . . . , μα (p0 ) − 1 and qα does not vanish on 𝕊ℍ (α); see [CSS16, Definition 3.4]. Furthermore, one has  μα (p0 ) = d − 2m = deg p − deg p𝕊 α∈V (p0 )

in terms of decomposition (7.314). 7.7.2. Principal value and residue currents in the ℍ-setting. Let us first introduce the sheaf of ℍ-valued distributions on ℍ through the representation of quaternions as z 1 + z 2 j with (z1 , z2 ) = (x1 + iy1 , x2 + iy2 ). Definition 7.112. Let 𝕌 be an open subset of ℍ and let U = {(z1 , z2 ) ∈ ℂ2 : z 1 1 + z 2 j ∈ 𝕌}. A distribution T on 𝕌 corresponds to a pair of ℂ-valued distributions T1 = Re T1 + i Im T1 , T2 = Re T2 + i Im T2 on U . The (left) distribution T 1 + j T 2 = (Re T1 + Im T1 i) + j (Re T2 + Im T2 i) 59 Observe that the bilaplacian operator 16 Δ plays here a central role, as it does also in 2 [Fu35].

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acts on smooth ℍ-valued functions ϕ + ψ j on U , where ϕ = Re ϕ + Im ϕ i and ψ = Re ψ + Im ψ i, ϕ and ψ being ℂ-valued test forms on U , as     T 1 + j T 2 , ϕ + ψ j  = T1 , ϕ − T2 , ψ i→i + T1 , ψ + T2 , ϕ i→i j. Remark 7.113. One can as well interpret T 1 + j T 2 as a (0, 0)-current acting on the left on the 4-form (π 2 (ddc z2 )/2) (ϕ + ψ j). 0,0 (·, ℍ). The sheaf of ℍ-valued (0, 0)-currents acting on the left is denoted by  Dleft Given a (left) hyperholomorphic function f : 𝕌 −→ ℍ of the form f = f 1 + f 2 j with z → (f1 (z), f2 (z)) holomorphic in U (see Example 7.100) one associates to it a principal value current P f as follows, according to what we did in one-dimensional complex setting; see Proposition 1.12.

Definition 7.114. Let 𝕌 and f = f 1 + f 2 j be as above. Then (7.316)

λ −→ f 2λ

f1 − j f2 2iπ f 2

0,0 extends from {λ ∈ ℂ : Re λ  1} to ℂ as a  Dleft (𝕌, ℍ)-valued meromorphic map + with poles in −ℚ . Its value at λ = 0 is called the principal value current P f .

For what concerns the approach to the notion of residue current in such a modified Fueter setting, with f : 𝕌 −→ ℍ as above, the noncommutativity generates a lot of complications. We will only write down a few computations below and then conclude with a suggestion. One has for Re λ  1 that (7.317) → ∂  f − j f2 f 2λ 1 2iπ f 2 ∂Z → ∂ 

→ ∂  f 1 − j f 2 2λ f 1 − j f 2 2λ = f  + f  2iπ f 2 2iπ f 2 ∂Z ∂Z ∂ f − j f2 ∂ f1 − j f2 = λf 2(λ−1) (F 1 + j F 2 ) 1 + f 2λ +j 2 iπ f  ∂z1 ∂z2 i→i 2iπ f 2  f 2(λ−2)  =λ f1 F1 + f2 F2 + j (f1 F2 − f2 F1 ) i→i iπ ∂ f1 f2

f 2λ ∂ f1 ∂ f2 ∂ + + +j − , 2 2 2 iπ ∂z1 f  ∂z2 f  ∂z2 f  ∂z1 f 2 i→i where F1 =

∂ ∂ f 2 , F2 = f 2 . ∂ζ 1 ∂ζ 2

Since the (2, 2)-complex valued pseudo-meromorphic currents

f 2λ ∂ f  ∂ f2 1 c 2 2 (dd + z ) , iπ ∂z1 f 2 ∂z2 f 2 λ=0

f 2λ ∂  f1 f2 ∂ c 2 2 (dd − z ) iπ ∂z2 f 2 ∂z1 f 2 λ=0

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419

are clearly of standard extension on U = {z ∈ ℂ2 : z 1 1 + z 2 j ∈ 𝕌} with respect to their support U (see Definition 2.48) the (2, 2)-current 

→ ∂  f ,λ (7.318) R := 1V (f ) · P ∧ dx1 ∧ dy1 ∧ dx2 ∧ dy2 λ=0 ∂Z 

f 2(λ−2)   f1 F1 + f2 F2 + j (f1 F2 − f2 F1 ) i→i dx1 ∧ dy1 ∧ dx2 ∧ dy2 = λ iπ λ=0 could be a natural candidate for a so-called residue current attached to such hyperholomorphic function f : 𝕌 −→ ℍ. Of course, such a candidate would be of a very particular type since f1 and f2 are both holomorphic in U . Observe in particular that     (7.319) R f , (f 1 + f 2 j) ϕ(Z) =

    λf 2(λ−1 (F 1 + j F 2 ) dx1 ∧ dy1 ∧ dx2 ∧ dy2 , ϕ(Z) =0 f

λ=0

for any ϕ ∈ D(ℍ, ℝ). To mimic constructions realized in the one-dimensional complex setting instead of the bivariate complex one in relation with hyperholomorphicity, let us finally consider the (basically univariate) concept of (left) slice regularity and profit from Remark 7.108. Suppose that f : 𝕌 → ℍ is slice regular in 𝕌 as described by Definition 7.107. Given any σ ∈ SO(3, ℝ) (we assume that this group acts on the quaternionic unit sphere 𝕊ℍ ), there is a holomorphic section of the trivial bundle ℂ2 over Ωσ(i) := {u + iv ∈ ℂ : u 1 + v σ(i) ∈ 𝕌} defined by sσ = u + iv −→ (Uσ + iVσ , Uσ + iVσ ) such that f (u 1 + v σ(i)) = (Uσ + iVσ )i→σ(i) + (Uσ + iVσ )i→σ(i) σ(j) for any u + iv ∈ ℂ. The currents Psσ and Rsσ constructed as in Proposition 1.27 are principal value and residue current attached to the restriction of f to the slice 𝕌 ∩ (ℝ + ℝ σ(i)); see also §1.2.1 for the particular case of trivial bundles (here ℂ2 ) with the canonical metric. Any smooth ℍ-valued function in 𝕌 splits on such slice as [2] ϕ(u + v σ(i)) = ϕ[1] σ (u + iv) + ϕσ (u + iv) σ(j), [1]

[2]

where ϕσ (u + iv), ϕσ (u + iv) ∈ ℝ + ℝ σ(i) for any u + iv ∈ Ωσ(i) . A natural f attached to f in the slice regular context would principal value distribution Pslice be then  f (7.320) Pslice , ϕ      [2] Psσ (u+iv) , 1 ϕ[1] = σ (u+iv)+2 ϕσ (u+iv) σ(j) du∧dv dνhaar (σ), SO(3,ℝ)

where (1 , 2 ) is the canonical basis of the trivial bundle ℂ2 . Any smooth ℍ-valued 1-form splits on the slice 𝕌 ∩ (ℝ + ℝ σ(i)) as [2] ω(u + v σ(i)) = ω [1] σ (u + iv) + ω σ (u + iv) σ(j),

420

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[2]

where this time ω σ (u + iv), ω σ (u + iv) are ℝ + ℝ σ(i) valued 1-forms. A natural f attached to f in such slice-regular context would be then residue current Rslice  f (7.321) Rslice , ϕ    = Rsσ (u + iv) , 1 ωσ[1] (u + iv) + 2 ωσ[2] (u + iv) σ(j) dνhaar (σ). SO(3,ℝ)

APPENDIX A

Complex manifolds and analytic spaces Throughout this monograph, we work on either a complex manifold and its structural sheaf1 (X , OX ) or on a complex reduced analytic space and its structural sheaf, denoted also by (X , OX ). Furthermore, we assume that analytic subsets V ⊂ X are equipped with the structural sheaves OV = ι∗ OX inherited from OX through the embedding ι : V → X . Thus Appendix A is devoted to a brief outline of these notions. For a detailed presentation of this material, we invite the reader to consult the extensive monographs by J. P. Demailly [De] and R. O. Wells [W]. A.1. Complex manifold, structural sheaf OX , sheaves of OX -modules A.1.1. Definitions. A complex manifold X of dimension N consists of: (1) a topological space X , which is countable at infinity;2 (2) a collection of charts {(Uι , ζι )ι } on X ", where the family (Uι )ι realizes an open covering of X , namely X = ι Uι , and, for any ι, the mapping ζι : Uι ↔ U ι ⊂ ℂN is a homeomorphism so that, for every pair of indices (ι0 , ι1 ) satisfying Uι0 ∩ Uι1 = ∅, the map ζι0 ,ι1 = ζι0 ◦ (ζι1 )−1 realizes a biholomorphism between the open subsets ζι1 (U ι0 ∩ U ι1 ) and ζι0 (U ι0 ∩ U ι1 ). Such a collection of charts {(Uι , ζι )ι } is called an atlas for the complex manifold X . Given an atlas for the complex manifold X , the homeomorphism ζι : Uι ←→ U ι ⊂ ℂN is called a local system of coordinates in the open subset Uι . Moreover, if 0 ∈ U ι and zι ∈ Uι is such that ζι (zι ) = 0, then the local system of coordinates ζι is said to be centered at zι ∈ Uι . Given a complex analytic manifold X equipped with an atlas {(Uι , ζι )ι }, let us denote by ζι = ξι + iηι , where ξι = Re ζι and ηι = Im ζι , are such that τι = (ζι , ηι ) realizes an homeomorphism between Uι and U ι,ℝ . Here Uι,ℝ denotes the open subset of ℝ2N defined by U ι,ℝ =: {(ξ, η) ∈ ℝ2N : ξ + iη ∈ U ι }. is a C ∞ diffeomorphism For every pair of indices (ι0 , ι1 ), the map τι0 ,ι1 = τι0 ◦ τι−1 1 (which is additionally real-analytic) between the open subsets τι1 (U ι0 ,ℝ ∩U ι1 ,ℝ ) and τι0 (U ι0 ,ℝ ∩ U ι1 ,ℝ ). The collection {(Uι , τι )ι } defines a structure of a differentiable 1 For those readers who are interested in knowing more about sheaf theory, they can consult the books by R. Godement [God] and by H. Grauert and R. Remmert [GrtR] (see also [GuR] and [De, II, §1]). Here, we present the elements from sheaf theory needed within this monograph. 2 The existence of continuous (or of C ∞ ) partitions of unity on the underlying differentiable manifold Xℝ is for us a major ingredient.

421

422

A. COMPLEX MANIFOLDS AND ANALYTIC SPACES

(in fact real-analytic) 2N -dimensional manifold Xℝ , which is called the underlying differentiable manifold of X . Given an open subset U ⊂ X , one denotes as OX (U ) the commutative ring of holomorphic functions in U . Namely, f ∈ OX (U ) if f : U → ℂ is such that for any atlas {(Uι , ζι )ι } and any index ι satisfying U ∩ Uι = 0 the composed function f ◦ (ζι−1 )|ζι (U ∩ Uι ) : ζι (U ∩ Uι ) → ℂ is holomorphic. An element f ∈ OX (U ) is called regular if it is not a zero-divisor in the ring OX (U ). Regular elements in OX (U ) constitute a multiplicative subset SX (U ) and the commutative ring (SX (U ))−1 OX (U ) = MX (U ) (which is canonically isomorphic as a commutative ring to OX (U )[(SX (U ))−1 ]) is called the commutative ring of meromorphic functions in U . Let us recall briefly the definition of sheaves on a topological space X ; see for example [De, II, §1], or for more complete discussion [God,GrtR,GuR]. To define a presheaf F of commutative rings on a topological space X , one needs: (1) first to set a collection of commutative rings {F(U )}U⊂X , where U is an arbitrary open subset of X . The ring F(U ) is called then the ring of sections of the sheaf F over the open set U ; (2) second to associate to every pair of open sets U, U  satisfying U ⊂ U  , a restriction map ρU,U  : F(U  ) → F(U ), which is a homomorphism of rings, so that ρU,U  ◦ ρU  ,U  = ρU,U  whenever U ⊂ U  ⊂ U  and ρU,U = IdF (U) . A presheaf F of commutative rings on the topological space X is said to be a sheaf of commutative rings on X if it satisfies in addition the following two " gluing axioms. Given any open covering {Uι } of an arbitrary open set U , i.e., U = ι Uι then: (A) whenever F, G ∈ F(U ) and ρUι ,U (F ) = ρUι ,U (G) for any index ι, then F = G; (B) given any collection (Fι )ι with Fι ∈ F(Uι ) such that ρUι0 ∩Uι1 ,Uι0 (Fι0 ) = ρUι0 ∩Uι1 ,Uι1 (Fι1 )

∀ ι0 , ι1 ,

then there is an element F ∈ F(U ) such that ρUι ,U (F ) = Fι for any index ι. For any z ∈ X , one defines the stalk Fz of a sheaf of commutative rings at z as the direct limit Fz = lim F(U ). zU

This means that an element of Fz can be represented as an element in F(U ) for some neighborhood U of z, taking into account that two such sections in F(U  ) and F(U  ) are identified if they coincide on a neighborhood U of z lying in U  ∩ U  . Elements of the stalk Fz are called germs at z of sections of the sheaf F, while elements of F(U ) are called sections of F on U ⊂ X , whenever U is an open subset of X . The support of a sheaf of commutative rings on X is the set of points z ∈ X where Fz = 0. Given two sheaves of commutative rings F and G on a topological space X with G  restriction morphisms ρF U,U  and ρU,U  respectively whenever U ⊂ U , a morphism θ : F → G of sheaves of commutative rings from F to G consists of a family {θU }U⊂X of morphisms θU : F(U ) → G(U ) of commutative rings, defined for any G   open subset U ⊂ X , satisfying θU ◦ ρF U,U  = ρU,U  ◦ θU  on F(U ) for any U ⊂ U .

A.1. COMPLEX MANIFOLD, STRUCTURAL SHEAF OX , SHEAVES OF OX -MODULES 423

Given a sheaf of commutative rings F on a topological space X and U an open subset of X , there is a natural restriction operation F → F|U as a sheaf on the topological space U equipped with the topology induced by that on X . Definition A.1 (The structural sheaf OX and the sheaf MX ). Let X be a complex N -dimensional manifold. The collection {F(U )}U⊂X , where F(U ) = OX (U ) (respectively MX (U )), indexed by open subsets of the underlying topological space X , together with restriction maps ρU,U  : f ∈ F(U  ) → f|U ∈ F(U ) whenever U ⊂ U  , defines a sheaf of commutative rings OX (respectively MX ) on the topological space X . The sheaf of commutative rings OX is called the structural sheaf of the complex manifold X , while the sheaf of commutative rings MX is called the sheaf of meromorphic functions on X . A sheaf F = {F(U )}U⊂X with restriction maps ρU,U  (U ⊂ U  ) is said to be a sheaf of OX -modules if, in addition to being a commutative ring, each F(U ) inherits a structure of OX (U )-module and each restriction map ρU,U  realizes then a homomorphism of OX -modules from F(U ) to F(U  ), whenever U ⊂ U  ⊂ X . Each stalk FX,z inherits then a structure of OX,z -module. In the particular case where F = I is a sheaf of OX -modules on X such that F(U ) = I(U ) is an ideal of the commutative ring OX (U ) for any open subset U ⊂ X , the sheaf of commutative rings is said to be an ideal sheaf on X . Given a morphism f of sheaves of OX -modules from F to G, one defines naturally Ker(f) as a subsheaf of F, Im(f) as a subsheaf of G, and the quotient sheaf as coker(f) = G/Im(f). Definition A.2 (Locally free sheaves of OX -modules with constant rank). A sheaf F of OX modules over X is said to be locally free with (constant) rank r over some open subset U ⊂ X if for any z ∈ X , there is a neighborhood Uz of z in X r such that F|U  ((OX )⊕ )|Uz . A.1.2. Examples (ℂN , ℙN ℂ , X (Σ)). N (1) The affine space ℂ is the prototype of a complex manifold. (2) Besides ℂN , the most common example of a complex manifold is the projective space ℙN ℂ , which is realized as the geometric quotient of the open subset ℂN +1 \ {(0, . . . , 0)} ⊂ ℂN +1 by the equivalence relation of colinearity between vectors in ℂN +1 \ {(0, . . . , 0)}. The homogeneous coordinates [z0 : · · · : zN ] represent points of the projective space. Open (N ) := {[z0 : · · · : zN ] : zj = 0}, j = 0, . . . , N , are the charts for sets Uj this compact complex manifold with transition functions ζi,j = zj /zi on (N ) (N ) Ui ∩ Uj . It is important to think here of the projective space ℙN ℂ as a hyperplane at infinity (N +1)

U0

+1 := {[t : z0 : · · · : zN ] : t = 0} ⊂ ℙN ℂ

in the following sense: if (t, z0 , . . . , zN ) is a point in ℂN +2 \ {(0, . . . , 0)}, the complex line {(λ t, λξ0 , . . . , λξN ) : λ ∈ ℂ} intersects the hyperplane at (N +1) +1 infinity U0 of the projective space ℙN exactly at the point identified ℂ with the homogeneous coordinates [z0 : · · · : zN ] ∈ ℙN ℂ . This point of view presides over the concept of perspective as developed by Italian Renaissance artists and appears to be a particular case of a more general fact; see Example 2.7 in the main body of the monograph.

424

A. COMPLEX MANIFOLDS AND ANALYTIC SPACES

(3) Let Σ = {σι }ι be a complete ℤN -rational primitive fan in ℝN . That" is, Σ is a finite collection of strict3 closed cones σι ⊂ ℝN satisfying ℝN = ι σι and having the following properties: • each cone σι is generated by a set of nonzero vectors in ℤN with coprime coordinates (a set of primitive vectors), which can be completed as a basis of the lattice ℤN ; • any face of any σι ∈ Σ still belongs to Σ; • any intersection σι0 ∩ σι1 of two elements of Σ is a common face to the cones σι0 and σι1 . Let σ0 , . . . , σN  be the distinct N -dimensional cones in Σ. The interiors of the cones σj , j = 0, . . . , N  , in ℝN are pairwise disjoint and one has " "N  N 1 N  j=0 σj = ι σι = ℝ . Let {ξj , . . . , ξj }, j = 0, . . . , N , be a set of    N primitive vectors ξj = (ξj,1 , . . . , ξj,N ) ∈ ℤ ,  = 1, . . . , N , which on one hand generate σj as an N -dimensional cone and on the other hand form a basis of ℤN as a lattice. For each j = 0, . . . , N  , consider the invertible monomial map from 𝕋N = (ℂ∗ )N into itself πj : ζ ∈ 𝕋N −→

N  =1

ξ

ζ j,1 , . . . ,

N 

ξ

ζ j,N

∈ 𝕋N .

=1

Let Uj , j = 0, . . . , N  , be the open dense subsets of ℂN defined as 2 4  Uj = z ∈ ℂN : z = 0, whenever ξj,k < 0 for some k ∈ {1, . . . , N } . Each monomial map πj realizes a birational map between Uj and its image U j = πj (Uj ) in ℂN . One can construct a compact complex manifold >  X (Σ) starting from the disjoint union N j=0 Uj , then make the necessary identifications to glue the open subsets Uj (which were initially forced >N  to be disjoint in j=0 Uj ) along their open subsets πj−1 (U j0 ∩ U j1 ) and 0 −1  (U ∩ U ) through the monomial maps π ◦ π , πj−1 j0 j1 j1 0 ≤ j0 < j1 ≤ N j0 1 (see [Ehl75, Da78, Ew] for such construction). The projective space ℙN ℂ can be constructed in this way, starting with the ℤN -rational primitive fan, whose one-dimensional cones are ℝ+ 1 , . . . , ℝ+ N , −ℝ+ (1 + · · · + N ), where {1 , . . . , N } is the canonical basis of ℝN . More generally, given a complete ℤN -rational primitive fan Σ in ℝN , the complex manifold X(Σ), realized as above, inherits a group action on itself by the complex torus 𝕋N . The complex torus 𝕋N stands as a dense orbit for this action. The N -dimensional complex manifold X (Σ) is called the toric manifold subordinate to the primitive fan Σ. Since the transition functions are rational maps (even monomial maps), the complex structure of X (Σ) is algebraic and X (Σ) (such as ℂN or ℙN ℂ ) is said to be an algebraic manifold. One can consider it either as being defined based on algebraically closed field ℂ or based on the field ℚ or its algebraic closure ℚ. 3 For

more details about toric manifolds; see §3.1, which this example illustrates.

A.2. COHERENCE, STEIN MANIFOLDS, FREE RESOLUTIONS

425

A.2. Coherence, Stein manifolds, free resolutions A.2.1. Coherent sheaves of OX -modules. Among the category of sheaves of OX -modules on a complex manifold X , coherent sheaves play an essential role in this monograph. Definition A.3 (Coherence of a sheaf). A sheaf F of OX -modules on an N dimensional complex manifold X is said to be coherent (or OX -coherent) if and only if it fulfills the two following conditions. (1) For any z in X , there exists an open neighborhood Uz of z and qz elements s1 , . . . , sqz in OX (Uz ) such that, for every z  ∈ Uz , the OX ,z -module Fz is generated by the germs sj,z , j = 1, . . . , qz , of the sections sj ∈ F(Uz ) at the point z  ∈ Uz (one phrases this condition saying that F is locally finitely generated). (2) For any open subset U  of X , for any s1 , . . . , sq ∈ F(U  ), the (OX )|U  ⊕q subsheaf of (OX )|U  on U  called the sheaf of relations RU  (s1 , . . . , sq ) on  ⊕q U  , that is, the kernel of the sheaf homomorphism θ U : (OX )|U  → F|U  , where, for any U ⊂ U  , 

U : (g1 , . . . , gq ) ∈ (OX (U ))⊕ −→ θU q

q 

gj sj |U ∈ F(U )

j=1

is also locally finitely generated. The important fact that the structural sheaf OX is coherent is known as the Oka coherence theorem. Given a coherent sheaf of ideals I on X , the sheaf OX /I is also coherent on X since one has the following short exact sequence of sheaves 0 −→ I −→ OX −→ OX /I −→ 0. Any finite locally free sheaf of OX -modules on X , in particular any locally free sheaf of OX -modules with constant rank, inherits the coherence of the structural sheaf OX of the complex manifold X . A.2.2. Stein manifolds and Cartan Theorems A and B. We now turn to a natural class of complex manifolds known as Stein manifolds. Stein manifolds are natural in the sense that they carry enough nonconstant holomorphic functions. For example, this is not the case for the compact complex manifold ℙN ℂ . More specifically, a complex manifold X of dimension N , with structural sheaf OX , is a Stein manifold if satisfies the following two properties. (1) It is holomorphically convex, that is, the holomorphic convex hull & =K & O := {z ∈ X , |f (z)| ≤ sup |f | , ∀ f ∈ OX (X )} K X K

of any compact subset K  X remains compact. (2) It carries enough holomorphic functions to separate points. That is, for any points z, z  ∈ X , z = z  , there exists an element f ∈ OX (X ) such that f (z) = f (z  ). Example A.4. The affine space ℂN is the prototype of a Stein manifold. Any noncompact Riemann surface is a Stein manifold as well.

426

A. COMPLEX MANIFOLDS AND ANALYTIC SPACES

Remark A.5. Any Stein manifold with dimension N admits an exhaustion {Kν }ν≥1 by holomorphically convex compact subsets Kν ⊂ X satisfying, in addi" & ν = Kν . Here tion to properties X = ν≥1 Kν and Kν ⊂ (Kν+1 )0 , the property K 0 (Kν+1 ) denotes the interior of the set. It is known that any Stein manifold X of dimension N can be holomorphically embedded into ℂ2N +1 . One has also two extremely powerful results, due to Henri Cartan. Theorem A.6 (Cartan Theorem A). Let X be a Stein manifold of dimension N . Any coherent sheaf of OX -modules on X is spanned by its global sections. That is, for any open set U ⊂ X , ρU,X (F(X )) generates F(U ) as a OX (U )-module. In particular, for each z ∈ X , one can find global sections fz,1 , . . . , fz,N in OX (X ) such that (fz,1 , . . . , fz,N ) defines a local system of coordinates in a neighborhood of z. Theorem A.7 (Cartan Theorem B). Let X be a Stein manifold of dimension ˇ q (X , F) = 0 N . If F is a coherent sheaf of OX -modules on X , then one has that H ˇ q (X , F) denotes the Cech ˇ for each q > 0, where H cohomology group4 with values in the sheaf F. One of the most remarkable consequences of Theorem A.6 for us will be the following: if g, f1 , . . . , fm are m + 1 holomorphic functions on a Stein manifold X , such that locally, at every point z ∈ X, gz ∈ (f1,z , . . . , fm,z ) OX ,z at the level of germs,  then there exist holomorphic functions a1 , . . . , am on X such thatNone has N g≡ m j=1 aj fj in X . An important example occurs when X = Ω × Ω ⊂ ℂ × ℂ , N N where Ω is a pseudo-convex domain of ℂ (that is a Stein open subset of ℂ ) and fj (z, ζ) := ζj − zj , j = 1, . . . , N . If g is a holomorphic function in Ω, then there exist N holomorphic functions h1 , . . . , hN on Ω × Ω, such that (A.1)

g(ζ) − f (z) =

N 

hj (z, ζ) (ζj − zj )

∀ (z, w) ∈ Ω × Ω.

j=1

This formula is called the Hefer division formula. The Hefer formula can easily be obtained when U is convex. Actually in this case one can use the Taylor formula with integral remainder, namely (A.2)  1  1 N  d ∂f f (ζ) − f (z) = [f (tζ + (1 − t)z)] dt = (ζj − zj ) (tζ + (1 − t)zj ) dt. dt ∂ζ j 0 0 j=1 Note that such a simple argument does not go through in the case of a pseudoconvex domain Ω. 4 Let F be a presheaf of abelian groups on a topological space X and let U = {U } be a ι ι ˇ ˇ p (X , U, F ) = Ker δq /Im δq−1 , q ∈ ℕ, covering family of X by open subsets. The Cech groups H materialize the obstruction for the exactness of the complex δq 0 −→ · · · −→ Cˇq (X , U, F ) −→ Cˇq+1 (X , U, F ) −→ · · · ,

where Cˇq (X , U, F ) is the group of q-cochains of U with values in F and δq is the coboundary ˇ cohomology group operator. δq -closed q-cocycles are called q-cocycles with values in F . The Cech ˇ q (X , U, F ) along the refinements of covering ˇ q (X , F ) is realized as the direct limit of the groups H H families U. See also §D.2.3 for more details, in particular for more details for the definition of the coboundary morphisms δ0 and δ1 in relation with the notion of Cartier divisor.

A.2. COHERENCE, STEIN MANIFOLDS, FREE RESOLUTIONS

427

A.2.3. Free resolutions of coherent sheaves. A.2.3.1. Local aspects. Let X be an N -dimensional complex manifold and let F be a coherent sheaf of OX -modules. The following useful concept arises naturally from the two properties stated in Definition A.3. Definition A.8 (Free finite resolution for Fz ). A finite free resolution for Fz at a point z ∈ X is an exact5 OX ,z -complex (A.3)

rL ,z

fLz ,z

⊕ z 0 −→ OX ,z

rL −1,z

⊕ z −→ OX ,z

f2,z

r1,z

⊕ · · · −→ OX ,z

f1,z

r0,z

⊕ −→ OX ,z ,

with length Lz ∈ ℕ∗ , such that r0,z

⊕ OX ,z

= Fz . Im(f1,z ) Such a finite free resolution for Fz is said to be minimal if and only if coker(f1,z ) :=

(A.4)

(A.5)

r −1,z

⊕ Im(f,z ) ⊂ MX ,z OX ,z

for any  ≥ 1,

where MX ,z denotes the maximal ideal in OX . Given a coherent sheaf of OX -modules over a complex manifold X and z ∈ X , there is a unique minimal finite free resolution for Fz up to an isomorphism of OX ,z -complexes of OX ,z -modules. More precisely, one has the following result [Eis, Theorem 20.2]. Theorem A.9 (Uniqueness of minimal finite free resolutions for Fz ). Let F be a coherent sheaf over X and z ∈ X . Given a minimal finite free resolution Rz,min for Fz , any finite free resolution Rz for Fz is, as a complex of OX ,z -modules, isomorphic to the direct sum of Rz,min with a trivial complex of OX ,z -modules. As a consequence, one introduces the following definition. Definition A.10 (Depth of F at z ∈ X ). Let F be a coherent sheaf of OX modules over a complex N -dimensional manifold X . The depth of F at z ∈ X is defined by N − depthz (F) = Lz ,

(A.6)

where Lz denotes the length of the minimal finite free resolution of Fz . A.2.3.2. Global aspects. When X is additionally a Stein manifold, Theorem B implies that every coherent sheaf F of OX -modules admits, in a convenient open neighborhood UK of a given holomorphically convex compact subset K  X , a (global) free resolution. That is, there exists an exact complex of coherent sheaves of OX -modules over UK : (A.7)

f +1,K

r

f ,K

r

f2,K

r

f1,K

r

,K

,K 1,K 0,K · · · −→ LUK −→ LUK · · · −→ LUK −→ LUK ,

where coker(f1,K ) = F|UK and

,K ⊕ = (OX LUK

r

r ,K

)|UK for any  ∈ ℕ∗ .

Observe that the sequence above may be infinite to the left. The existence of the exact sequence (A.7) follows from the iterated use of [Hor73, Theorem 7.2.1] about the existence of generating global sections for F about a holomorphically convex 5 Each f ,z is a homomorphism of OX ,z -modules such that fLz ,z is injective and one has Ker f,z = Im(f+1,z ) for any  ≥ 1.

428

A. COMPLEX MANIFOLDS AND ANALYTIC SPACES

compact subset K. On the other hand, any holomorphically convex compact subset K can be covered with a finite number of open subsets Uzι above which there exists a locally free finite free resolution6 ⊕ 0 −→ (OX

rL ,z ι ι

fL

ι ι ⊕ )|Uzι −→ (OX ,z

rL −1,z ι ι

f2,z

⊕ )|Uzι · · · −→ι (OX

r1,z ι

f1,z

⊕ )|Uzι −→ι (OX

r0,z ι

)|Uzι ,

with coker(f1,zι ) = F|Uzι and Lι = N − depthzι F. Then, if one takes L  maxι Lι and invokes Theorem A.9, one can see that the coherent sheaf Ker(fL,K ) in (A.7) is locally free with rank rL+1,K over UK . Hence one can truncate the locally free resolution (A.7) to the left as fL,K

r

fL−1,K

r

f2,K

r

f1,K

r

L+1,K L−1,K 1,K 0,K −→ LUK −→ · · · −→ LUK −→ LUK , (A.8) 0 −→ Ker(fL,K ) = LUK

where coker(f1,K ) = F|UK . Thus, one can formulate the following definition. Definition A.11 (Finite locally free resolution for a coherent sheaf). A finite locally free resolution with length L for a coherent sheaf F over a complex manifold X in an open subset U ⊂ X is an exact complex of locally free coherent sheaves with constant rank (A.9)

0 −→ LUrL −→ LUrL −1 · · · −→ LUr1 −→ LUr0 f2

fL

f1

such that coker(f1 ) = F|U .

(A.10)

This definition leads to the following proposition holding in the context of Stein manifolds. Proposition A.12. Let X be an N -dimensional Stein manifold and let F be a coherent sheaf of OX -modules. There exists an exhaustion of X by an increasing sequence of open subsets Uν such that F admits in each Uν a finite, locally free resolution with length Lν as in (A.9) with ⊕ LUνν, = (OX r

rν,

)|Uν

for all  ≥ 1, except at  = Lν . ⊕ Example A.13. Let X be a Stein manifold and let J be a subsheaf of OX . There exists an exhausting sequence of X by open subsets Uν , such that for each ⊕r ν, F = OX /J admits in Uν a locally free resolution as in (A.9) with r

⊕ )|Uν and Im(f1 ) = J . LUrν0 = (OX r

⊕ /J in Uν is sometimes called by extension Such a finite, locally free resolution of OX a free resolution of J in Uν . The search for such a free resolution (locally in a neighborhood of a given point z ∈ X or, when X is Stein, globally in an open subset of X ) is known as the determination of syzygies problem for the quotient sheaf ⊕r OX /J or, by extension, for the subsheaf J . In general, there is no algorithmic process that allows computing such a free resolution, even locally, within polynomial time. We refer the reader to [Eis05] for a complete overview of the important notion of syzygy going back to D. Hilbert. The fact that there exist in the local or global context-free resolutions an ideal sheaf I is known as Hilbert’s syzygy theorem [Hil1890]. In particular, this theorem implies in the local setting of the regular r

6 One takes the minimal finite free resolution (A.3) at z by taking U ι zι sufficiently small around zι and then choose representatives in Uzι for the germs f,zι .

A.3. CLOSED ANALYTIC SUBSETS OF A COMPLEX MANIFOLD

429

local ring OℂN ,0 that, given any ideal I in OℂN ,0 , OℂN ,0 /I admits always a free resolution with length equal to N . A.3. Closed analytic subsets of a complex manifold Definition A.14 (Closed analytic subset). Let X be a complex manifold of dimension N with OX as its structural sheaf. A subset V ⊂ X is called a closed analytic subset of X if the closed subset V ⊂ X is expressed locally in an open neighborhood Uz around any z ∈ X as the set of common zeros of a finite family of elements in OX (Uz ). An analytic subset V ⊂ X is said to be irreducible if it cannot be decomposed into a union of two closed analytic subsets V  and V  such that V  = V and V  = V . Example A.15. Given a coherent sheaf of OX -modules F over a complex manifold X , the support of F is defined as the subset of points z ∈ X such that Fz = 0. It is a closed analytic subset of X . Remark A.16. Suppose that a germ Vz of the closed analytic subset at a point z ∈ X is embedded in two ways through the embeddings ι : V ∩ Uz → U ⊂ ℂNι ,

ι : V ∩ Uz → U ⊂ ℂNι .

Here U and U are open subsets of ℂNι and ℂNι correspondingly. Since the minimal embeddings ιmin , ιmin are equivalent, and any embedding ι factorizes at the level of germs at z in a simple way over a minimal one ιmin , one can always take ι = ιmin minimal (hence unique up to equivalence) and ι = ι, with U = U × ℂν , ν = Nι − Nιmin and ι(z) = (ιmin (z), 0). A.3.1. Local dimension. If V ⊂ X is a closed analytic subset of X , one can associate with it a coherent sheaf of ideals IV ⊂ OX , where (A.11)

IV (U ) := {f ∈ OX (U ) : f (z) = 0,

∀ z ∈ V ∩ U}

for any open subset U ⊂ X . For each z ∈ X , the stalk IV,z is a radical ideal of the stalk OX ,z . Since OX ,z is a nœtherian regular local ring of dimension N , the stalk IV,z admits a unique representation (up to the order of factors) as 3 Pz,ι . (A.12) IV,z = ι

The ideals Pz,ι in (A.12) are prime ideals in OX ,z , which are not contained pairwise in each other since IV,z is radical. The finite set Ass(OX ,z /IV,z ) = {Pz,ι }ι is called the set of associated primes of the finitely generated OX ,z -module OX ,z /IV,z , which is also a reduced (in general nonregular) local ring. For each Pz,ι ∈ Ass(OX ,z /IV,z ), the Krull dimension of the finitely generated OX ,z -module OX ,z /Pz,ι is defined as the maximal length (possibly 0) of strictly increasing chains of distinct prime ideals Pz,ι  P1  · · ·  Pκ  OX ,z . The local dimension of V at the point z, denoted as dimz V, is defined as the maximum of the Krull dimensions of the quotient rings OX ,z /Pz,ι for all associated primes Pz,ι in Ass(OX ,z /IV,z ). To conclude the present section, let us mention the classical box principle. Assume that I1 , . . . , Ir are r ideals in OX ,z and assume that h1 , . . . , hm are m elements in the same local ring such that, for any k = 1, . . . , r, there exists at least one hj

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 so that hj ∈ / Ik . Then, for generic complex coefficients λ1 , . . . , λm , m / Ik j=1 λj hj ∈ for any k = 1, . . . , r. The same result holds in the global setting (polynomial ideals in 𝕂[X1 , . . . , XN ] or homogeneous polynomial ideals in 𝕂[X0 , . . . , XN ]) provided 𝕂 is an infinite commutative field, for example, an algebraically closed field. A.3.2. Splitting V = Vsing ∪ Vreg , global dimension. Let V be a closed irreducible analytic subset of a complex manifold X . Then the set Vsing of singular points of V , that is, the set of points z in V in a neighborhood Uz of which V cannot be described as a submanifold of Uz , is a proper closed analytic subset of V such that, for any z ∈ Vsing , dim(Vsing )z < dim Vz . It implies that V \ Vsing = Vreg is dense in V . Nonsingular points of the set V are called regular points of V . Since V is irreducible, the function z → dim Vz is constant on Vreg . Its constant value on Vreg is defined as the global dimension of the irreducible closed analytic subset V . It takes values between 0 and dim X . The global dimension of a closed analytic subset V ⊂ X is the maximum of the dimensions of its irreducible components Vι . A closed analytic subset V ⊂ X is said to be of pure dimension (or purely dimensional) if all its irreducible components have the same global dimension. A.3.3. Free resolutions of sheaves and codimension of supports. Let X be an N -dimensional complex manifold and let F be a coherent sheaf over X with support the closed analytic subset VF . Let us recall in this section two important results related to the concept of finite locally free resolution introduced in §A.2.3. The first result is a local one. It concerns the notion of depth introduced in Definition A.10. Proposition A.17. One has for any z ∈ X that depthz F ≤ dim VF .

(A.13)

In other words, the length of the minimal finite resolution of Fz equals at least the codimension of VF . We formulate the second result in a global context since we need it within such a context in this monograph. We refer here to [BuE74] or [Eis, Theorem 20.9] for a formulation in the local context, from which our statement can be derived thanks to Theorem A.9. Theorem A.18. Let R be a finite locally free resolution of the coherent sheaf of OX -modules F in X as in (A.9) with U = X and (A.10) fulfilled. Let for any  ∈ ℕ∗  R = (−1)ν r+ν , where r = 0 if  > LR = length(R). ν≥0

Then, one has that (A.14)

codim{z ∈ X : rank(f (z)) < R } ≥ 

for all  ∈ ℕ∗ .

Moreover, conditions (A.14) are equivalent to the exactness of the complex R. Remark A.19. The sequence of closed analytic subsets (A.15)

V (R) := {z ∈ X : rank(f (z)) < R },

 ∈ ℕ∗ ,

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is a decreasing sequences of closed analytic subsets such that V1 (R) = VF when f1 is surjective. This follows from the following observation. One has rank(f+1 (z)) < R+1 =⇒ r − R+1 = R > r − rank(f+1 (z)) = r − dim(Ker(f (z))) = rank(f (z)) and V1 (R) equals in general the closed analytic subset of points z ∈ X in a neighborhood in which F fails to be locally free (that is VF as soon as f1 is surjective). As a consequence of Theorem A.9, the sequence of closed analytic subsets (V (R))≥1 does not depend on the finite locally free resolution R of F. Definition A.20. If the coherent sheaf of OX -modules F admits a finite locally free resolution R in X , then the decreasing sequence (V (R))≥1 = (VF , )≥1 will be called because of [BuE74] the Buchsbaum–Eisenbud–Fitting sequence of the coherent sheaf F. Proposition A.17 suggests we introduce the following concept. Definition A.21 (Cohen–Macaulay sheaves and closed analytic subsets). A coherent sheaf of OX -modules F over a complex manifold X is said to be Cohen– Macaulay if and only if for any z in the support of F, the length Lz of the minimal free resolution of the sheaf F at z equals the codimension of the support of F at this point, which is the minimal value it can take. If I is a coherent sheaf of OX ideals such that OX /I is Cohen–Macaulay, then we say that the coherent sheaf of ideals I is Cohen–Macaulay. Remark A.22. When the coherent sheaf of ideals I is Cohen–Macaulay at a point z ∈ Supp (OX /I), all associated primes of OX ,z /Iz are isolated.7 That is, they are minimal elements in Ass(OX ,z /Iz ) for the ordering induced by inclusion. Example A.23 (Zero-dimensional sheaves of ideals are Cohen–Macaulay). Let X be an N -dimensional complex manifold. Any coherent sheaf of ideals I such that OX /I has zero-dimensional (that is discrete) support is Cohen–Macaulay. A.3.4. Local presentation in the purely dimensional case. Let V be a purely dimensional closed analytic subset of N -dimensional complex manifold X such that dim V = N − M , where M ∈ {1, . . . , N }. Let z0 ∈ V , ζ : Uz0 → ℂN be a local system of coordinates in a neighborhood of z0 , centered at z0 , and U z0 = ζ(Uz0 ). The following proposition ([GrtR, page 72]) plays an important role since it allows us to introduce geometrically the concept of local (Hilbert–Samuel) multiplicity for a purely dimensional closed analytic subset at a given point z0 ∈ V . Proposition A.24 (Presentation of closed analytic sets). There exist an open neighborhood of the origin U ⊂ U z0 in ℂN and holomorphic functions fj : U → ℂ 5 −1 such that M analytic subset j=1 fj ({0}) is a purely (N − M )-dimensional closed " of U, and the set ζz0 (V ) can be described in U as the finite union ι Wι of some 5 −1 irreducible components Wι of the closed analytic subset M j=1 fj ({0}), with df1 ∧ · · · ∧ dfM ≡ 0 on any Wι . Let us make this presentation of V in a neighborhood of z0 more precise and thus introduce geometrically the concept of local multiplicity, known also as Hilbert– Samuel multiplicity. It follows from the normalization lemma of E. Nœther that 7 In

general, the associated primes that are not isolated are called embedded.

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   there is a linear change of coordinates z ↔ (z1 , . . . , zN −M , z1 , . . . , zM ) in a neighborhood of the origin in ℂN (that one chooses here to be a generic point) such that −M   if Δ ⊂ U ⊂ ℂN × ℂM z  is a convenient open poly-cylinder Δ × Δ centered at z (0, 0), the projection

π : (z  , z  ) ∈ ζz0 (V ) ∩ (Δ × Δ ) −→ z  ∈ Δ is a proper map. Since df1 ∧ · · · ∧ dfM ≡ 0 on any Wι , the jacobian determinant  J(z  , z  ) = ∂(f1 , . . . , fM )/∂(z1 , . . . , zM ) is not identically zero on each Wι ∩ Δ for  ∈ Δ : σ(z  ) = 0} of the projection π equals any index ι. The discriminant locus {z      π ζz0 (V )∩{(z , z ) ∈ Δ : J(z , z ) = 0} . Above Δ \σ −1 ({0}), ζz0 (V )∩Δ consists then in μ disjoint leaves (μ ∈ ℕ∗ ), which are (N − M )-dimensional submanifolds parameterized by the coordinates z  → w(ι) (z  ) for ι = 1, . . . , μ. One has μ ≤ μ ˜, where μ ˜ denotes the number of disjoint (N − M )-dimensional leaves z  → w(ι) (z  ) 5M ˜ −1 ({0})) × Δ ), where of the proper projection π ˜ : j=1 fj−1 ({0}) ∩ ((Δ \ σ ˜ (z  ) = 0} ⊃ {z  ∈ Δ : σ(z  ) = 0} {z  ∈ Δ : σ denotes its discriminant locus. The integers μ ≤ μ ˜ are independent of the choice of the generic change of coordinates as well as independent of the choice of the local centered coordinate system in a neighborhood of z0 . The integer μ defines the local Hilbert–Samuel multiplicity multz0 (V ) of V at z0 . In the particular case where M = 1 and ζz0 (V ) = f −1 ({0}), multz0 (V ) equals the degree of the homogeneous part of the lowest degree in the  Taylor development in z in a neighborhood of the origin of f red , where f red = ι fι is thereduced (e.g., square-free) holomorphic function deduced from the splitting f = ι fινι of f in irreducible factors. Given a purely (N − M )-dimensional closed analytic subset V ⊂ X of a complex analytic manifold, the local Hilbert–Samuel multiplicity multz0 (V ) at a point z0 ∈ V can also be interpreted algebraically instead of geometrically as follows. Let IV be the coherent ideal sheaf attached to V and let MV,z0 be the maximal ideal of the local ring OX ,z0 /IV,z0 . The Hilbert function of the local ring OX ,z0 /IV,z0 with maximal ideal MV,z0 is the function from ℕ to ℕ defined as   ν ∈ ℕ −→ HV,z0 (ν) = dimOX ,z0 /IV,z0 (OX ,z0 /IV,z0 )/MνV,z0 . It coincides for ν  1 with the Hilbert polynomial function of the local ring OX ,z0 /IV,z0 with respect to its maximal ideal MV,z0 ν ∈ ℕ −→ H V,z0 (ν) = multz0 (V )

N −M ν N −M + hV,z0 ,k ν N −M −k (N − M )!

(hV,z0 ,k ∈ ℚ).

k=1

We will later complete the definition of the Hilbert–Samuel local multiplicity of a closed analytic set at a point z0 with that of the list of local Segre numbers; see Proposition C.25 and Example D.16. A.3.5. Weakly holomorphic functions and Oka universal denominator. Let V be a closed analytic subset of an N -dimensional complex manifold X . We introduce below three notions of holomorphicity for functions on V . It is not restrictive to assume that V is purely (N − M )-dimensional with 0 ≤ M ≤ N − 1, which we do. Definition A.25. Let V ⊂ X be a closed purely (N −M )-dimensional analytic subset in a N -dimensional complex manifold and h be a function from V to ℂ.

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(1) The function h is said to be a weakly holomorphic function on V if and only if h|Vreg is holomorphic (as a function defined on Vreg equipped with its structure of (N − M )-complex manifold (Vreg , (OX )|Vreg )) and h is locally bounded around any point z ∈ V . (2) The function h is c-holomorphic ([Wh, Ch, Lo]) on V if and only if h|Vreg is holomorphic (see above) and h is continuous on V . (3) The function h is strongly holomorphic on V if and only if it is the restriction to V of a holomorphic function H in some open neighborhood of V in the ambient manifold X . One has the following useful characterizations for the two first concepts. Proposition A.26. Let X be an N -dimensional complex manifold and let V ⊂ X be a closed analytic subset with pure codimension M . • A holomorphic map h : Vreg → ℂ extends as a weakly holomorphic function to the whole of V if and only if the closure in X × ℂ of its graph ΓVreg (h) = {(z, h(z)) : z ∈ Vreg } over Vreg is a closed analytic subset in X × ℂ. • A function h : V → ℂ is c-holomorphic on V if and only if it is continuous on V and its graph ΓV (h) = {(z, h(z)) : z ∈ V } is a closed analytic subset in X × ℂ. Weakly holomorphic functions, as well as c-holomorphic functions, on a closed analytic subset V ⊂ X are restrictions to V of meromorphic functions in a neighborhood of V in the ambient manifold X . In this direction, we have a more precise and important result due to Kiyoshi Oka. This result is known as the universal denominator theorem. Theorem A.27. Let X be a complex manifold, let V ⊂ X be a closed analytic subset with pure codimension M , and let h be a weakly holomorphic function on V . Then, for any z0 ∈ V , h can be expressed in a neighborhood of z0 as the restriction to V of a meromorphic function in a neighborhood of z0 in the ambient manifold X . For z0 ∈ V fixed, the denominator of this meromorphic extension can be chosen to be independent of h. Proof. Since the proof of Oka’s theorem depends on the local Nœther’s presentation given in §A.3.4 and involves ideas appearing in many places in the present monograph, we outline its proof here. We take X = Ω as a neighborhood of the origin in ℂN . We assume that V has pure codimension M ∈ {0, . . . , N − 1} and that the coordinates (z  , z  ) are taken such that the projection map π : (z  , z  ) ∈ V ∩ (Δ × Δ ) → z  is proper; see Proposition A.17 in §A.3.4. We respect such presentations and keep the notation from Proposition A.17 and the whole of §A.3.4 which follows it. There exist holomorphic functions hjk : Δ × Δ × Δ → ℂ such that for any z  =    (z1 , . . . , zN −M ) ∈ Δ , u = (u1 , . . . , uM ), v = (v1 , . . . , vM ) ∈ Δ , fj (z  , u) − fj (z  , v) =

M  k=1

hjk (z  , u, v) (uj − vj ), j = 1, . . . , M.

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This comes from Hefer division formulae for polycylinders (see (A.1) in §A.2.2) 5 −1 since poly-cylinders are Stein manifolds. Let V6 = M j=1 fj ({0}) ⊃ V . If h is a   holomorphic function in the submanifold Vreg ∩ (Δ × Δ ), then it can be extended holomorphically to a holomorphic function in the submanifold V6reg × (Δ × Δ ) by setting h = 0 on (V6reg \ Vreg ) ∩ (Δ × Δ ). For z  ∈ Δ , π −1 ({z  }) ∩ V6 = {w(1) (z  ), . . . , w(˜μ) (z  )} for some μ ˜ ∈ ℕ∗ independent of z  , namely the Hilbert–Samuel local multiplicity at z0 (see §A.3.4) of the closed analytic (N − M )-dimensional subset M 3

fj−1 ({0}) ⊃ V.

j=1

One can see then that the function H : (z  , z  ) ∈ Δ × Δ −→

μ  

h(z, w(ι) (z  )) det[hjk ] (z  , z  , w(ι) (z  ))

ι=1

is holomorphic in both variables (z  , z  ) ∈ (Δ \σ −1 ({0}))×Δ , where σ defines the discriminant locus of the proper projection π. The restriction of this holomorphic map to V ∩ ((Δ \ σ −1 ({0})) × Δ ) equals hJ|V . If h is bounded in Vreg , then Riemann’s theorem shows that the function H has an analytic continuation to a holomorphic function in Δ × Δ and that ∀ (z  , z  ) ∈ V ∩ (Δ × Δ ) , h(z  , z  ) =

H(z  , z  ) . J(z  , z  )

The last equality proves that h coincides with the restriction to V ∩ (Δ × Δ ) of a meromorphic function in Δ × Δ , with J (independent of h) as denominator.  A.3.6. Example: closed analytic subsets of ℙN ℂ , Chow’s theorem. ∗ Since compact complex manifolds ℙN , N ∈ ℕ , play a major role in the present ℂ monograph, we recall here the important theorem of Chow, which can be viewed as a consequence of the Remmert–Stein theorem [RS53]: if V is a purely dimensional closed analytic subset of the open affine subset ℂN +1 \ {(0, . . . , 0)} with dim V > 0, then its closure V in ℂN +1 is a closed analytic subset of ℂN +1 . Chow’s theorem reflects the GAGA principle formulated by J. P. Serre [Se56]. Theorem A.28. Any closed analytic subset V of ℙN ℂ is the zero set of a homogeneous polynomial ideal in ℂ[X0 , . . . , XN ]. In particular, any n-dimensional submanifold X of ℙN ℂ is such that one can find homogeneous polynomials P1 , . . . , Pm in ℂ[X0 , . . . , XN ] so that X = {z ∈ ℙN ℂ , ; P1 (z) = · · · = Pm (z) = 0},  with

z∈X :

N' −n  m  N j=1 k=0

 ∂Pj (z) k = 0 = ∅ , ∂zk

where {0 , . . . , N } denotes the canonical basis of ℂN +1 . Remark A.29. Since any projective algebraic manifold X admits an embedding  N ι : X → ℙN ℂ of complex manifolds in some projective space ℙℂ , the final assertion  in Theorem A.28 holds for ι(X ) ⊂ ℙN ℂ . If X is the reduced complex analytic space

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underlying a projective algebraic variety defined over ℂ, then there is an embedding  N ι : X → ℙN ℂ of X in some projective space ℙℂ and the first assertion in Theorem A.28 applies to ι(X ) as well. A.4. Complex analytic spaces, normalization, and log resolutions A.4.1. Complex analytic spaces. Let 

V  ⊂ Ω ⊂ ℂN ,

V  ⊂ Ω ⊂ ℂN

 



be two closed analytic subsets of the open sets Ω and Ω of ℂN and ℂN , respectively. A continuous function f : V  → V  is called a morphism of analytic sets from V  into V  if and only if, for every z ∈ V  , there exists a neighborhood Uz of z in  Ω , a holomorphic function Fz from Uz into ℂN , such that (Fz )|V  ∩ Uz = f|V  ∩ Uz . If such is the case, then for every z ∈ V  , one can define the mapping OΩ ,f (z) OΩ ,z (A.16) fz∗ : OV  ,f (z) := −→ OV  ,z := IV  ,f (z) IV  ,z as fz∗ (gz ) = (g ◦ Fz )z . This map is called the comorphism of f at the point z. For a  closed analytic subset V  ⊂ Ω ⊂ ℂN , denote by OV  the OΩ -coherent sheaf (see §A.2.1) defined as the quotient sheaf OV  := OΩ /IV  . Complex analytic spaces are realized on the model of complex manifolds (see §A.1.1), except that one needs to glue together closed analytic subsets Vι ⊂ Ωι ⊂ ℂNι instead of copies of open subsets of some fixed ℂN ; see also [De, II, §5]. Definition A.30 (Complex analytic space). A complex analytic space (or, to be more precise, reduced),8 (X , OX ) consists of the following data: (1) a separable and locally compact topological space X , which is countable at infinity; (2) a structural sheaf of rings of holomorphic functions OX , together with a covering (Uι , τι )ι of X , where τι realizes a homeomorphism between Uι and some closed analytic subset Vι = τι (Uι ) ⊂ Ωι ⊂ ℂNι (for some integer Nι ∈ ℕ), such that, for any index ι, the comorphism O Ωι −→ (g ◦ τι )z ∈ (O|Uι )z , z ∈ Uι , τι∗ : g ∈ (OVι )τι (z) = IVι τι (z) is an isomorphism between sheaves of rings. Remark A.31. Suppose that X is an N -dimensional complex manifold with V ⊂ X a purely (N −M )-dimensional closed analytic subset. Given z ∈ V , suppose that, besides the embedding Vz → Xz , Vz is embedded in two ways through the embeddings ι : V ∩ Uz → U ⊂ ℂNι ,

ι : V ∩ Uz → U ⊂ ℂNι

as in Remark A.16. If ι = ιmin is supposed to be minimal, hence ι = ι with U = U × ℂνw with ν = Nι − Nιmin , and Iιmin (V ),U is a radical defining sheaf for ιmin (V ), then Iι(V ),U = Iιmin (V ),U OU + (w1 , . . . , wν )OU is a radical defining sheaf for ι (V ) = ι(V ) in U and the two isomorphic quotient sheaves OU /Iιmin (V ),U  OU /Iι(V ),U 8 In

this monograph we avoid discussing the theory of schemes [EGA, Ha].

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represent at the level of germs at z the structure sheaf OV,z of the reduced analytic space with underlying space V around z. If X is a complex analytic space, then the subset Xreg of regular points of X (these are points in a neighborhood of which (X , OX ) is a complex manifold with its structural sheaf of holomorphic functions) is dense in X (isolated points of X are considered as regular). One denotes by Xsing = X \ Xreg the set of singular points in X . The closures of the connected components of Xreg are called irreducible components of X . The dimension of X at z ∈ X , denoted as dimz X , is by definition dimτι (z) (Vι ), whenever (Uι , τι ) is a local chart containing z . The function z → dimz X is constant on the irreducible components of X and thus it is enough to evaluate it at regular points in X . The complex space X is of pure dimension N if all its irreducible components are of dimension N . 6X of functions Given a complex analytic space X , one defines on X the sheaf O h : X → ℂ such that h|Xreg is holomorphic on Xreg and h is locally bounded on X . Here, Xreg is considered as a complex manifold equipped with the structural sheaf (OX )|Xreg and h being locally bounded on X means that, for any z ∈ X , one can find an open neighborhood Uz of z such that |h(ζ)| ≤ C(z) for any ζ ∈ Uz ∩ Xreg . Such functions are also called weakly holomorphic functions on the complex analytic space X referring to the notion of weakly holomorphic function on a closed analytic subset of a complex manifold introduced previously (Definition A.25 in §A.3.5). If X is irreducible (i.e., the set of regular points has only one connected component), then the local rings OX ,z are integral domains and one may consider their fraction fields MX ,z , z ∈ X . The corresponding sheaf MX is then called the sheaf of meromorphic functions (sometimes called regular functions) over X . If X is not irreducible anymore, then the sheaf MX is a sheaf of rings. The stalk MX ,z is the ring of fractions of OX ,z , that is, the quotient of OX ,z by the ideal of elements that are not zero-divisors. Oka’s theorem (Theorem A.27 in §A.3.5) implies that 6X ⊂ MX . In fact, for z ∈ X , O 6X ,z is the integral closure of the subring OX ,z in O 6X ,z is the set of elements hz in the OX ,z -algebra MX ,z , that is, for every z ∈ X , O MX ,z satisfying a monic integral dependency relation: hνz + a1 hzν−1 + · · · + aν = 0, with a1 , . . . , aν ∈ OX ,z . A.4.2. Normalization and blowup of complex analytic spaces. A.4.2.1. Normal complex analytic spaces, normalization. The notion of weakly holomorphic function on a complex analytic space X introduced in §A.4.1 appeals to Riemann’s analytic continuation theorem: if X is a complex manifold of dimension N and V is a closed analytic subset of X with codimension 1, any function f from X \ V to ℂ, which is holomorphic in X \ V and locally bounded on X , that is, (A.17)

∀z ∈ X , ∃ Uz + z,

sup |f (ζ)| < +∞ ,

ζ∈Uz \V

extends holomorphically over the whole manifold X . In case codim V ≥ 2, the additional hypothesis (A.17) is redundant and the existence of the analytic continuation of f follows from Hartogs’s theorem; see, e.g., [GuR]. When X is an irreducible complex analytic space of dimension N equipped with its structure sheaf OX , Riemann analytic continuation theorem fails to be

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true in general. One may have holomorphic functions in the dense subset Xreg which are bounded in a neighborhood of a singular point z ∈ Xsing , but which do not define elements of OX ,z . Here is an example, namely that of the cusp: let ϕ : t ∈ D(0, ε) → (t3 , t2 ) be an injective (since 2 and 3 are coprime) holomorphic parametrization in a neighborhood of the origin in ℂ2 of the analytic space X defined (as embedded in ℂ2 ) by the equation z12 − z23 = 0. The function h : z → ϕ−1 (z) is holomorphic in Xreg = X \ {(0, 0)} and locally bounded in X , but there is no holomorphic function 6 h in a neighborhood of the origin z = (0, 0) in X such that 6 h(t3 , t2 ) = t. This proves that ϕ−1 cannot be defined at the level of germs at (0, 0) as an element of OX ,(0,0) . The fact that Riemann’s analytic continuation theorem fails in general on a complex analytic space (X , OX ) leads naturally to the introduction of the concept of normality at a given point z ∈ Xsing and of normality at any point z ∈ Xsing . Namely, we have the following. Definition A.32. A complex analytic space (X , OX ) is said to be normal at 6X ,z = OX ,z . The space X is said to be normal if a point z ∈ Xsing if and only if O and only if it is normal at any point z of Xsing . Any complex analytic space (X , OX ) can be normalized in the sense described by the following Oka’s normalization theorem. Theorem A.33. Let X be a complex analytic space. There exists a normal complex analytic space (X&, OX ), together with a proper projection π : X& → X , where |(π −1 ({z})| < +∞ for any z, such that X& \ π −1 (Xsing ) is dense in X& and π realizes an analytic isomorphism between X& \ π −1 (Xsing ) and Xreg . Such a pair ((X&, OX ), π) (or, in short, (X&, π)), is said to be a normalization of the complex %2 , π2 ) are %1 , π1 ) and (X analytic space X . It is unique in the following sense: if (X % % two normalizations of complex analytic space X , then (X1 , OX1 ) and (X2 , OX2 ) are isomorphic as complex analytic spaces. A.4.2.2. Normalized blowup of a complex manifold along a coherent sheaf of ideals. The geometric operation consisting of blowing up plays an important role in analytic or algebraic geometry. In particular, given a OX -coherent sheaf of ideals I over an N -dimensional complex manifold (X , OX ), one will use extensively in this monograph the so-called normalized blowup of X along the sheaf of ideals I (or also with the center of the coherent ideal sheaf I). This is, up to an isomorphism between (reduced) complex analytic spaces, an intrinsic object (depending on X and I); see Proposition A.34. Here are the two operations comprising the normalized blow-up procedure, once X , its structural sheaf OX and the OX -coherent sheaf of ideals I are given. (1) The first one is the blowup of the complex manifold X along the coherent sheaf I (or with the center of the coherent sheaf I). It leads, as described below, to an N -dimensional complex analytic space (XI , OXI ), together with a holomorphic proper surjective projection π : XI → X , such that the inverse image sheaf I · OXI of I by π is invertible. More precisely, the support of OXI /I · OXI is a closed analytic hypersurface HI of XI , and π realizes a biholomorphism between XI \ HI and X \ Supp (OX /I).

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(2) The second one consists of the realization of a normalization πnorm X&I −→ XI

of the complex analytic space XI according to Theorem A.33. The normalized blowup of X along I is then πnorm π (A.18) πI,norm : X&I −→ XI −→ X . The inverse image sheaf I · OXI via π ◦ πnorm is also invertible,9 which means that its support O XI (A.19) H = Supp ⊂ X&I I · OXI is a closed hypersurface. Since X&I is normal, its structural sheaf is regular in codimension one. This means that the generic point of any irreducible closed hypersurface is regular, or, which is equivalent, that codim (X&I )sing ≥ 2. To any irreducible component Hγ of such hypersurface H (defined by (A.19)), whose generic point is denoted by zˆγ , one can then attach a positive integer mγ defined by     (A.20) mγ = length OXI zˆγ / I · OXI zˆγ . The locally finite formal combination (A.21)

EI =



m γ Hγ

γ

is then a Weil divisor over X (see §D.3.1) called the Weil exceptional divisor of the normalized blowup πI,norm . For a deeper understanding of these notions one can refer to [Hir64] or [T73, T82]. The reason why we denoted as EI the exceptional divisor of the normalized blowup follows from the next important proposition, that ensures the uniqueness, up to isomorphism, of the normalized blowup. Proposition A.34. Let X be a complex manifold and let I be a OX -coherent sheaf of ideals on X . The normalized blowup of X along I defined by (A.18) has the following universality property: if τ : X  −→ X is any morphism of complex analytic spaces such that X  is normal and the inverse image sheaf I · OX  is invertible, then τ factorizes in a unique manner as τ = πI,norm ◦ τI , where τI is a morphism of complex analytic spaces from X  into the normalized blowup of X along I. It follows from the universality property of the normalized blowup that the following definition taken from ([Ful, Ful84]) describes intrinsically closed analytic subsets contained in Supp (OX /I). Definition A.35. Let X be a complex N -dimensional manifold and let I be a OX coherent sheaf of ideals. Let also {Hγ }γ be the collection of all irreducible hypersurfaces of the normalized blowup of X along I, whose union is the hypersurface H defined by (A.19). Then the Zγ = πI,norm (Hγ ) are closed irreducible analytic subsets10 of X lying in Supp (OX /I). For codimX (OX /I) ≤ r ≤ N = dim X , the 9 It is the sheaf of sections of a Cartier divisor on X , which is called the exceptional divisor of the normalized blowup. 10 The fact that Z is a closed analytic subset of X follows from the properness of the map γ πI,norm . The irreducibility of Zγ is a consequence of the irreducibility of the hypersurface Hγ .

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irreducible closed analytic subsets Zγr , forming the (highly redundant in general) list {Zγ : codimX Zγ = r}, are called distinguished irreducible varieties of codimension r of Supp (OX /I). The same definition holds if X is an N -dimensional complex analytic space and the notion of normalized blowup is extended to such a context too; see Remark A.38. Let us make explicit the blowup of a complex analytic manifold X along a coherent sheaf of ideals I first in the case I = IY , where Y is a closed submanifold in X . Then we will make it explicit in the general case of a coherent OX -sheaf I on X when the support of OX /I is a submanifold; see, e.g., [SmiKKT] or [De, VII, §12]. To do so, it is convenient to invoke the theory of holomorphic bundles over a complex manifold; see Appendix B, in particular §B.1.3 to §B.1.5. • Suppose first that I = IY , where Y is a closed submanifold in X . The blowup of X along IY is an holomorphic vector bundle with rank equal to codimX Y above Y. Its fiber above y ∈ Y is the projectivization ℙ(Ny ) of the space Ny of all directions orthogonal to Y at the point y. That is, the ℂ-vector space obtained as the quotient of Ny \ {0} by the colinearity relation. The holomorphic vector bundle as constructed above Y is the quotient bundle (TX )|Y /TY , where TX → X is the holomorphic complex tangent bundle to X (see Example B.5 in §B.1.3) that one restricts above Y as a bundle above Y, and TY → Y is the holomorphic tangent complex to Y considered here as a subbundle of (TX )|Y → Y. Suppose the submanifold Y (here with dimension N − M ) is defined in a local chart U around one of its points y as Y := {z ∈ X : f1 (z) = · · · = fM (z) = 0} , where f1 , . . . , fM are holomorphic functions in U , with df1 ∧ · · · ∧ dfM ≡ 0 over Y ∩ U . The blowup of U along IY ∩ U can be expressed, up to an −1 of the isomorphism of complex manifolds, as the closure in U × ℙM ℂ graph of the mapping z ∈ U \ Y → [f1 (z) : · · · : fM (z)]. This closure is denoted by ΓY,U and is defined as the set of points (z, [w1 : · · · : wM ]) −1 satisfying wj fj+1 (ζ) − wj+1 fj (ζ) = 0 in the product manifold U × ℙM ℂ for j = 1, . . . , M − 1. Then the proper projection πU : ΓY,U → U is given by the projection

(z, [w1 : · · · : wM ]) → z ∈ U. −1 Note that the projection map πU is proper, since ℙM is compact. Furℂ −1 −1 thermore, the inverse image HY,U := πU (Y) =: (Y ∩ U ) × ℙM is a ℂ smooth hypersurface in ΓY,U , and the projection πU realizes a biholomorphism between the open subsets ΓY,U \ HY,U and U \ Y. • Consider now the general case where I is an arbitrary OX -coherent sheaf on X . It is enough to blow up (along I|U ) a local chart U around a point y in the support of OX /I. Let (f1 , . . . , fM ) be M elements in O(U ), such

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that Iz = ((f1 )z , . . . , (fM )z ) for every z ∈ U . We introduce, as previously, the closure ΓI,U of the graph of the map z ∈ U \ Supp (OU /I|U ) → [f1 (z) : · · · : fM (z)] in U × ℙM −1 (ℂ). The projection π : ΓI,U → U is described again by (z, [w1 : · · · : wM ]) → z ∈ U . The complex analytic space ΓI,U depends only (up to isomorphisms of complex analytic spaces) on the coherent sheaf I|U and not on the system of generators (f1 , . . . , fM ) chosen in OX (U ). Hartogs’s theorem implies that the inverse image by π of the support of the quotient sheaf OU /IU (i.e., the inverse image by π of the locus of the common zeroes of fj , j = 1, . . . , M , in U ) cannot have components with a codimension larger than or equal to 2. Such an inverse image is therefore a hypersurface HI,U in ΓI,U and π realizes a biholomorphism 5 −1 between ΓI,U \ HI,U and U \ M 1 fj (0) = U \ Supp ((OX )|U /IU ). This geometric description of the blowup UI|U of U along the coherent sheaf I|U is sufficient for our needs in the core of the monograph. A.4.2.3. Integral closure and normalized blowup of a complex analytic space along a coherent sheaf of ideals. The notion of integral closure11 of an ideal in a commutative ring is closely related to the normalized blowup. We recall this algebraic definition in this subsection. Definition A.36. Let 𝔸 be a commutative ring and let a be an ideal in 𝔸. We call integral closure of a in 𝔸 the ideal a in 𝔸 consisting of elements h of 𝔸 satisfying, for some ν ∈ ℕ∗ , a homogeneous integral dependence relation hν + u1 hν−1 + · · · + hν = 0, with uk ∈ ak , k = 1, 2, . . . , ν. An equivalent formulation is to say that hT , considered as an element of the graded algebra 𝔸[T ], is integral over the Rees algebra & a := 𝔸 ⊕ aT ⊕ a T ⊕ · · · = 2

2

∞ (

ak T k ⊂ 𝔸[T ],

k=0

i.e., it satisfies a monic integral dependence relation (hT )ν +

ν 

(hT )ν−j u &j = 0 , u &1 , . . . , u &ν ∈ & a.

j=1

If X is a complex analytic manifold and I is a OX -coherent sheaf of ideals, then one can consider the coherent sheaf I, where, for every z ∈ X , I z denotes the integral closure of Iz in OX ,z . The coherence of the sheaf of ideals I follows from the theorem of Grauert on the transport of coherence through direct image [Grt60] and Proposition A.37. Proposition A.37. Let X be an N -dimensional complex manifold, let I be a coherent sheaf of ideals, let U ⊂ X be an open subset, and let f ∈ I(U ). If πI,norm −→ X denotes the normalized blowup of X along I, then for any irreducible 11 One should not confuse the notion of the integral closure of a ring 𝔸 in an 𝔸-algebra (as introduced at the end of §A.4.1) with the subring OX ,z of the OX ,z -algebra MX ,z .

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−1 component Hγ of the hypersurface H defined by (A.19) intersecting πI,norm (U ) and for a generic point zˆγ in H ∩ U , one has that   (OXI )|U zˆ γ  ≥ mγ , (A.22) length  (πI,norm )∗ f ) OXI )|U zˆ γ

where mγ is defined by (A.20). (A.23)

∗ (πI,norm f)

12

The above claim is also equivalent to the inclusion   · (OXI )|π−1 (U) ⊂ I · (OXI ) |π−1 (U) .

Conversely, if f ∈ OX (U ) is such that either condition (A.22) or condition (A.23) is fulfilled, then f ∈ I(U ). Remark A.38. One can as well introduce the blowup of an N -dimensional complex analytic space (X , OX ) (instead of an N -dimensional complex manifold) along a coherent OX -sheaf of OX ideals I, and then, normalize it to construct the normalized blowup of (X , OX ) along IX . Results analogous to Propositions A.34 and A.37 still hold. From the point of view of complex analytic scheme theory (see [Ha]) one should point out that the blowup (XI , OXI ) of (X , OX ) along the OX -coherent sheaf I corresponds to the complex analytic scheme13 ∞

( (XI , OXI ) = Proj Ik . k=0

The normalized blowup (XI , OXI ) corresponds to the complex analytic scheme (X&I , OXI ) = Proj

∞ ( k=0

∞

(

I k = Proj Ik , k=0

see for example to [LejT08]. A.4.3. Log resolution and the Hironaka theorem. Among closed singular hypersurfaces H in an N -dimensional complex manifold X , hypersurfaces with local normal crossings play a major role, since their singularities can be explicitly described. Definition A.39. Let X be an N -dimensional manifold. A closed hypersurface H ⊂ X is said to be locally with normal crossings if and only if, for any point z ∈ H, there is a centered system of local coordinates in some open chart U around z such that H ∩ U = {ζ ∈ U : ζiν1 · · · ζiνz = 0 , {iν1 , . . . , iνx } ⊂ {1, . . . , N }}, i.e., H can be locally described in centered local coordinates ζ as the zero set of a monomial in ζ. The important concept of log resolution is closely linked with that of closed hypersurface with normal crossings. 12 In other words, if one refers to the notions of Cartier and Weil divisors on a complex analytic space (see respectively §D.2.1 and §D.3.1), one has div((πI,norm )∗ f ) ≥ EI , where div((πI,norm )∗ f ) denotes the Weil divisor which is attached to the principal Cartier divisor (π −1 (U ), f ◦ πI,norm ). 13 We are forced here to introduce the concept of scheme and just refer the reader to [Ha]. We quote this result for the sake of completeness, but we do not invoke it within our monograph, except for the fact that it is lying behind the statement of Theorem 7.8 which we quoted (without detailed proof) from [Hi01, Theorem 5.1].

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Definition A.40 (Log resolution). Let (X , OX ) be an N -dimensional complex irreducible analytic space and let Y be a closed analytic subset in X . A log resolution π X6 → X for Y consists of: (1) a complex manifold (X6, OX ), where dim X6 = N ; (2) a proper surjective holomorphic map π : X6 → X , such that the analytic subset π −1 (Y) ∪ crit(π) is a closed analytic hypersurface with normal crossings in X6. Here crit(π) denotes the set of critical points of π, that is, the points of X6 in a neighborhood of which π does not realize a local biholomorphism. One of the main tools in developing multidimensional residue theory is the theorem on the resolution of singularities due to H. Hironaka [Hir64]. It justifies the existence of a log resolution for any analytic subset Y in an irreducible analytic complex space (X , OX ). Theorem A.41 (Resolution of singularities). Let X be an irreducible complex analytic space of dimension N and let Y be a complex analytic subset of X containing Xsing . There exists a complex manifold X6Y of dimension N (sometimes called a smooth modification of X ) and a proper holomorphic mapping πY : X6Y → X , such 6 = π −1 (Y): that, if H Y 6 and X \ Y; • the mapping πY realizes a biholomorphism between X6Y \ H • the sheaf of ideals IY · OXY (the inverse image by πY of the sheaf of ideals IY on X ) is an invertible sheaf whose support is a hypersurface with locally normal crossings. Remark A.42. The universality property of the normalized blowup (Proposition A.34 in §A.4.2) implies that if I is a coherent sheaf of ideals on a complex manifold X (or even a complex analytic space, see Remark A.38 in §A.4.2) and Y denotes the support of OX /I, then any resolution of singularities π Y X6Y → X

as in Theorem A.41 factorizes in a unique way as πI,norm θ X6Y −→ XI −→ X ,

where θ denotes a morphism of complex analytic spaces (depending on the log resolution).

APPENDIX B

Holomorphic bundles over complex analytic spaces B.1. Analytic cocycles, holomorphic bundles, isomorphism classes B.1.1. Cocycles on a complex analytic space, examples. All through this first subsection, X will be a (purely) N -dimensional complex analytic space with OX as its structural sheaf. Complex analytic 1-cocycles (or also, shortly, OX -cocycles) with values in GL(•, ℂ) on X are defined as follows.1 Definition B.1. Let m ∈ ℕ∗ and let U = {Uι }ι be an open covering of X . A complex analytic 1-cocycle with values in GL(m, ℂ) on X subordinate to the covering U is a collection of matrix-valued maps {gι1 ,ι0 }ι0 ,ι1 , where gι1 ,ι0 is a morphism of complex analytic spaces between Uι0 ∩ Uι1 (equipped with the restriction of the structure sheaf OX ) and GL(m, ℂ) (equipped with its structure of complex 2 submanifold of ℂm ) such that the cocycle conditions (B.1)

∀ z ∈ Uι0 ∩ Uι1 ∩ Uι2 , gι2 ,ι0 (z) = gι2 ,ι1 (z) ◦ gι1 ,ι0 (z)

hold for any triplet of indices ι0 , ι1 , ι2 . Example B.2. Let {Uι }ι be a open covering of X and, for each index ι, let also fι ∈ MX (Uι ) be a nonidentically zero meromorphic function in Uι . Suppose that any pair of indices ι0 , ι1 , fι1 /fι0 takes its values in ℂ∗ . Then {gι1 ,ι0 = fι1 /fι0 }ι0 ,ι1 is a complex analytic 1-cocycle with values in GL(1, ℂ) = ℂ∗ on X which is called a Cartier divisor; see also §D.2.1. Let for example X = ℙN ℂ and, for any k = 0, . . . , n, Uk = {[z0 : · · · : zN ] ∈ X : zj = 0}; for any d ∈ ℤ, {[z0 : · · · : zN ] ∈ Uk −→ z0d /zkd }k is a Cartier divisor on ℙN ℂ . The representation of a Cartier divisor as {(Uι , fι }ι is not unique. Nevertheless, given two open coverings U and U6, one identifies 6˜ι , f˜˜ι }˜ι when the meromorphic section fι /f˜˜ι two Cartier divisors {(Uι , fι )}ι and {U 6ι ), whenever Uι ∩ U 6˜ι is nonempty. Such defines an invertible element in OX (Uι ∩ U ι identification allows us to call Cartier divisors {(X , f )} principal Cartier divisors, when f ∈ MX (X ). Example B.3. Let X be an N -dimensional complex manifold with an atlas {(Uι , ζι )}ι as in §A.1.1. Then   4 2 ) (ζι0 (z)) ∈ GL(N, ℂ) ι ,ι , (B.2) gι1 ,ι0 : z ∈ Uι0 ∩ Uι1 −→ d(ζι1 ◦ ζι−1 0 0

1

is a complex analytic 1-cocycle with values in GL(N, ℂ) on X . Such a complex analytic 1-cocycle allows us to preside over the construction of the holomorphic tangent bundle; see Example B.5. 1 The terminology refers here to Cech ˇ cohomology (with respect to a covering U) with values in the (multiplicative) sheaf of groups F• defined by F• (U ) = O(U, GL(•, ℂ)); see the footnote in the statement of Theorem A.7 or also §D.2.3 for more details.

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B.1.2. Linear algebra and construction of cocycles. Let E1 and E2 be two ℂ-vector spaces with dimension m1 , m2 , respectively. The ℂ-vector spaces E1 and E2 play naturally the role of elementary bricks to construct many other ones, # # E1 ⊕ E2 , E1 ⊗ E2 , Homℂ (E1 , E2 ) = E2 ⊗ E1∗ , E1 or E2 such as E1∗ and E2∗ , # m1 for  ∈ ℕ, det E1 = E1 or det E2 , etc. Let us transpose such constructions in the world of cocycles. • If g = {gι1 ,ι0 : Uι0 ∩ Uι1 −→ GL(m, ℂ)}ι0 ,ι1 is a complex analytic 1cocycle with values in GL(m, ℂ) subordinate to the open covering U, the complex analytic 1-cocycle 4 2 ∗ (B.3) gι1 ,ι0 : z ∈ Uι0 ∩ Uι1 −→ (t gι1 ,ι0 (z))−1 , where t (·) denotes the transposition, is also with values in GL(m, ℂ) and is called the dual cocycle of g. • When g 1 is a complex analytic 1-cocycle with values in GL(m1 , ℂ) subordinate to an open covering U 1 and g 2 is a complex analytic 1-cocycle with values in GL(m2 , ℂ) subordinate to an open covering U 2 , then g 1 ⊕ g 2 (respectively g 1 ⊗ g 2 ) is the complex analytic 1-cocycle with values in GL(m1 + m2 , ℂ) (respectively GL(m1 m2 , ℂ)) subordinate to the covering U 1 ∩ U 2 = {Uι11 ∩ Uι22 }ι1 ,ι2 = {Uι1,2 }ι defined by   (g 1 ⊕ g 2 )ι0 ,ι1 = gι10 ,ι1 ⊕ gι20 ,ι1 |U 1,2 ∩ U 1,2 ι ι1 (B.4)   0   1 1 2 2 resp., (g ⊗ g )ι0 ,ι1 = gι0 ,ι1 ⊗ gι0 ,ι1 |U 1,2 ∩ U 1,2 , ι0

where

 G1 ⊕ G2 =

G1 0

ι1

 0 , G2

(G1 ⊗ G2 )(1ν1 ⊗ 2ν2 ) = G1 (1ν1 ) ⊗ G2 (2ν2 ), when Gj ∈ G(mj , ℂ) and j = {j1 , . . . , jmj } denotes the canonical basis of ℂmj . In particular, the complex analytic 1-cocycle Homℂ (g 1 , g 2 ) subordinate to the open covering U 1,2 is defined as g 2 ⊗ (g 1 )∗ up to the canonical isomorphism Hom(ℂm1 , ℂm2 )  ℂm2 ⊗ (ℂm1 )∗ . • If g is a complex analytic 1-cocycle with values in GL(m, ℂ) subordinate to the open covering U and  0 ≤  ≤ m, the complex analytic 1-cocycle # g with values in GL( m  , ℂ) subordinate to the same open covering is defined by  

' '   gι1 ,ι0 (z) (ν1 ∧ · · · ∧ ν ) = (B.5) gι1 ,ι0 (z)(νj ) j=1

for all 1 ≤ ν1 < · · · < ν ≤ m, where  = {1 , . . . , m } is the canonical m , and z ∈ Uι0 ∩ Uι1 . In particular, the complex analytic basis of ℂ# m 1-cocycle g with values in ℂ∗ subordinate to U is denoted by det g. B.1.3. Holomorphic bundles over a complex analytic space. Let X be a (purely) N -dimensional complex analytic space, m ∈ ℕ, and let (B.6)

Em = {Ez : z ∈ X ,

dimℂ Ez = m}

be a collection of complex m-dimensional vector spaces indexed with z ∈ X . The ℂ-vector space Ez labeled by z is called the stalk at the point z.

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Definition B.4. A holomorphic bundle E → X with rank m over X is a structure (denoted by (𝕏Em , O𝕏Em )) of (N > + m)-purely dimensional complex analytic space on the disjoint union 𝕏Em := z∈X ({z} × Ez ) if Em is as in (B.6), such that • the projection map πEm : (z, ξz ) ∈ 𝕏Em → z ∈ X , where ξz denotes the current vector in Ez , is a morphism between complex analytic spaces; • X admits an open covering {Uι }ι such that for each index ι, there exists (Uι ) an analytic isomorphism of complex analytic spaces θι between πE−1 m and Uι × ℂm such that, for each z ∈ Uι , (B.7)

θι (z, ξz ) = (z, ι,z (ξz )) ∈ {z} × ℂm with ι,z ∈ Isomℂ (Ez , ℂm ). (Uι ) and Uι × ℂm is called Such an analytic isomorphism θι between πE−1 m a trivialization morphism for the bundle E → X over the open subset Uι .

Holomorphic bundles with rank 1 over X are called holomorphic line bundles. Example B.5 (Holomorphic tangent and cotangent bundle over a complex manifold). Let z ∈ X , where X is a complex N -dimensional manifold. Let 𝔻 be the unit disc in ℂ. A centered complex disk γ at z on X is a holomorphic mapping γ from ε𝔻 (for some ε > 0 depending on γ) to X such that γ(0) = z. Two such discs γ, γ˜ are considered to be equivalent if d(ζz ◦ γ)(0) = d(ζz ◦ γ˜ )(0), where ζz is a local centered coordinate in a neighborhood of z (that is such that ζz (z) = 0), the choice of such a centered local coordinate in a neighborhood of z being irrelevant. The set of equivalence classes γ¯ of centered complex discs at z modulo such an equivalence relation inherits a structure of N -dimensional ℂ-vector (1,0) space and is called the holomorphic tangent space2 at the point z. If TX ,z denotes the holomorphic complex tangent space at z ∈ X , then the family EN = {TX1,0,z : z ∈ X } (1,0)

inherits a structure of the N -holomorphic bundle TX , which is called the holomorphic tangent bundle over X . Moreover, for an atlas {(Uι , ζι )} of X , the trivialization morphism θι over Uι is defined by   (B.8) ∀ z ∈ Uι , ∀¯ γ ∈ TX1,0,z , θι (z, γ¯ ) = z, (ζι ◦ γ) (0) ∈ Uι × ℂN . Given a holomorphic bundle E → X with rank m, where X is a complex analytic space and {Uι }ι is an open covering of X such that E admits a trivialization morphism θι over Uι , then the family {gι1 ,ι0 : Uι0 ∩ Uι1 → GL(m, ℂ)}ι0 ,ι1 , defined by   ) (z, ξ) = z, (gι1 ,ι0 (z))(ξ) (B.9) ∀ z ∈ Uι0 ∩ Uι1 , (θι1 ◦ θι−1 0 is a complex analytic 1-cocycle on X with values in GL(m, ℂ). Example B.6. When X is an N -dimensional complex manifold, the complex analytic 1-cocycle on X with values in GL(N, ℂ) attached to the holomorphic tangent bundle (see Example B.5) has been described in Example B.3. 2 The holomorphic tangent space at z admits two alternative presentations besides the geometric one we propose here: it is isomorphic in a canonical way to the ℂ-vector space of ℂ-derivations of the local ring OX ,z , that is ℂ-linear maps D : OX ,z → ℂ such that D(f g) = f (z) Dg + g(z) Df ; it is also isomorphic in a canonical way to the ℂ-dual of the ℂ-vector space MX ,z /M2X ,z , where MX ,z is the maximal ideal in OX ,z .

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B.1.4. Isomorphism classes of holomorphic bundles. Definition B.7. Let E1 → X and E2 → X be two holomorphic bundles with respective ranks m1 and m2 over a complex analytic space X . Assume that they are attached to Ej,mj = {Ej,z : z ∈ X } with dim Ej,z = mj for j = 1, 2. A morphism of holomorphic bundles over X from E1 → X to E2 → X is an analytic morphism between complex analytic spaces F : 𝕏E1 → 𝕏E2 (with complex dimensions dim X + m1 and dim X + m2 , respectively) satisfying (B.10) ∀ z ∈ X , ∀ ξ1,z ∈ E1,z ,

  1,2 F (z, ξ1,z ) = (πE2 (F (z, 0)), L1,2 z (ξ1,z )) with Lz ∈ Homℂ E1,z , E2,πE2 (F (z,0)) .

Two holomorphic bundles E1 → X and E2 → X with the same rank are said to be isomorphic as holomorphic bundles if there are morphisms of holomorphic bundles F 1,2 : 𝕏E1 → 𝕏E2 and F 2,1 : 𝕏E2 → 𝕏E1 satisfying F 2,1 ◦ F 1,2 = Id𝕏E1 ,

F 1,2 ◦ F 2,1 = Id𝕏E2 .

As seen in §B.1.3, one can associate to any holomorphic bundle with rank m over an X a complex analytic 1-cocycle with values in GL(m, ℂ). Conversely, given a complex analytic 1-cocycle g = {gι1 ,ι0 }ι0 ,ι1 with values in GL(m, ℂ) subordinate to the open covering U, one can realize a holomorphic bundle with rank m as follows. Consider the disjoint union I (Uι × ℂm ). 𝕏(U) = ι

Then, whenever Uι0 ∩Uι1 = ∅, an element (zι0 , ξι0 ) ∈ Uι0 ×ℂm is identified with the element (zι1 , ξι1 ) ∈ Uι1 × ℂm if and only if zι0 = zι1 = z as elements in X and one has that ξι1 = (gι1 ,ι0 (z)) (ξι0 ). The quotient of 𝕏(U) modulo such identifications provides a structure of holomorphic m-bundle Eg → X . Before stating Proposition B.8, one needs to formulate an identification rule between complex analytic 1-cocycles with values in GL(m, ℂ); see, e.g., [Colon, Definition 3.4.3]. Any such 1-cocycle g (with values in GL(m, ℂ)) consists in a family U(g) of open subsets {Uι }ι , together with a family of morphisms F(g) : {gι1 ,ι0 : Uι0 ∩ Uι1 → GL(m, ℂ)}ι0 ,ι1 . Two such cocycles g and g˜ are then identified3 if and only if there exists a complex analytic 1-cocycle G (with valued in GL(m, ℂ)) such that E(g) ∪ E(˜ g) ⊂ E(G) and F(g) ∪ F(˜ g ) ⊂ F(G). Proposition B.8. There is a one-to-one correspondence between complex analytic 1-cocycles with values in GL(m, ℂ) on X (with identifications as above) and equivalence classes of holomorphic m-bundles modulo isomorphisms between holomorphic bundles. Example B.9. Let X = ℙN ℂ and d ∈ ℤ. The complex analytic 1-cocycle with ∗ values in ℂ defined by g([z0 : · · · : zN ]) = (z0 /zk )d for each z = [z0 : · · · : zN ], such that zk = 0, defines a class of isomorphic line bundles denoted by O(d). The class O(−1) admits as representing the tautological bundle corresponding to the family of ℂ-vector spaces E = {(z0 , . . . , zN ) ℂ : [z0 : · · · : zN ] ∈ ℙN ℂ } organized in such a N way that it realizes a holomorphic line bundle over ℙℂ . 3 Observe that in this formulation, redundancy in the sets E(G), and therefore F (G), occurs in general in the definition of complex analytic 1-cocycles with values in GL(m, ℂ).

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¯ then form a multiplicative group through Equivalence classes of line bundles L the tensorial multiplication of complex analytic 1-cocycles with values in ℂ∗ . Definition B.10. The group of equivalence classes of holomorphic 1-bundles over a complex analytic space X is called the Picard group of X and is denoted by Pic(X ). It is isomorphic to the quotient group of Cartier divisors modulo the subgroup of principal Cartier divisors, provided identifications between various representations of the same Cartier divisor are performed; see Example B.2. Example B.11. The Picard group of the projective space ℙN ℂ is isomorphic to ℤ. Any isomorphism class of holomorphic 1-bundles equals Oℙnℂ (d) for some unique ¯ ∈ (ℤ, +) realizes such an isomorphism. ¯ ∈ (Pic(ℙN ), ⊗) −→ d(L) d ∈ ℤ and L ℂ Such construction suggests the projectivization of a holomorphic m-bundle. Definition B.12. Let E → X be holomorphic with rank m over a complex analytic space X and let g be a complex analytic 1-cocycle with values in GL(m, ℂ) attached to E and subordinate to an open covering U of X . The projectivization Proj E of E is the (dim X + m − 1)-dimensional complex analytic space defined (up to isomorphism of complex analytic spaces) as follows. One considers the > m−1 ) in which two points (zι0 , [ξι0 ]) ∈ Uι0 × ℙm−1 and disjoint union ι (Uι × ℙℂ ℂ m−1 (zι1 , [ξι1 ]) ∈ Uι1 ×ℙℂ are identified whenever Uι0 ∩ Uι1 = ∅ and zι0 , zι1 represent the same point z = zι0 = zι1 of X in the nonempty intersection Uι0 ∩ Uι1 , and ξι1 = gι1 ,ι0 (z) (ξι0 ) in the open subset ℂm \ {(0, . . . , 0)}. Such a construction comes with a projection π : Proj E → X . All operations on complex analytic 1-cocycles described in §B.1.2 induce operations on holomorphic bundles (up to isomorphism) because of Proposition B.8. Given two holomorphic bundles E1 → X and E2 → X , with respective ranks m1 and m2 , one may then define the holomorphic dual bundle E1∗ → X (with rank m1 ), the holomorphic bundle E1 ⊕ E2 with rank m1 + m2 , the holomorphic bundles E1 ⊗ E2 and Homℂ (E1 , E2 )  E2 ⊗ E1∗ , both with rank m1 m2 , as well as, for  ∈ ℕ,   # the holomorphic bundle E1 with rank m1 . Example B.13 (Holomorphic cotangent bundle over a complex manifold). Let X be an N -dimensional complex manifold. The dual bundle of the holomorphic tangent bundle (see Example B.5) is a holomorphic bundle with rank N which is (1,0)∗ called the holomorphic cotangent bundle and denoted by TX . The line bundle (B.11)

(1,0)∗

can(X ) = det TX

=

N '

(1,0)∗

TX

is called the canonical bundle of X . In the particular case where X = ℙN ℂ , one has N can(ℙN )  O (−N − 1). ℙℂ ℂ B.1.5. Local holomorphic frames and sheaves of holomorphic sections. Definition B.14. Let E → X be a holomorphic bundle with rank m over a complex analytic space and let U be an open subset of X . A morphism s of complex analytic spaces from U to E such that s(z) ∈ Ez for any z ∈ U is called a holomorphic section of E over U .

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Suppose in particular that U = Uι is an open subset of X over which E admits a trivialization morphism θι = θ : π −1 (U ) ←→ U × ℂm . Let {1 , . . . , m } be the canonical basis of ℂm . Definition B.15. The family of holomorphic sections eU = {z ∈ U → θ −1 (z, j ) : j = 1, . . . , m} of E over U is said to be a local holomorphic frame for E in the open subset U over which E admits a trivialization morphism. Given a complex analytic space (X , OX ) and a holomorphic bundle E → X with rank m, the ℂ-vector space OX (U, E) of holomorphic sections of E over U is a OX (U )-module and {OX (U, E) : U ⊂ X } inherits the structure of a locally free sheaf of OX -modules with constant rank m on the complex analytic space X . B.1.6. The concepts of local versus global complete intersection. Given a holomorphic section s of a holomorphic bundle with rank m over a (purely) N -dimensional complex analytic space X , the subset s−1 ({0}) is a closed analytic subset of X . The concept of complete intersection (globally or locally) generalizes in a higher codimension than that of a (globally or locally defined) closed analytic hypersurface in the reduced complex analytic space X . Definition B.16. Let 1 ≤ m ≤ N = dim X . A closed analytic subset V ⊂ X with pure codimension m is said to be globally defined as a complete intersection in X if and only one can find a holomorphic bundle E → X with rank m, together with a holomorphic global section s of E over X , such that V = s−1 ({0}). It is said to be locally defined as a complete intersection in X if and only if, for any point z ∈ V , one can find fz,1 , . . . , fz,m ∈ OX ,z such that the germ Vz of V at z can be 5m −1 realized as Vz = j=1 fj,z ({0}). A sequence (a1 , . . . , am ) in a commutative ring 𝔸 is said to be regular in 𝔸 if and only if a1 = 0 and, for any j = 1, . . . , m − 1, aj+1 is not a zero divisor in the quotient ring 𝔸/(a1 , . . . , aj ). If 1 ≤ m ≤ N and f1,z , . . . , fm,z belong to the maximal ideal of the local ring OX ,z , it is equivalent to saying that (f1,z , . . . , fm,z ) 5m −1 is a regular sequence in OX ,z and that dim j=1 fz,j ({0}) = N − m. Therefore a closed analytic subset V ⊂ X with (pure) codimension m is locally defined in X as a complete intersection if and only if, for any z ∈ V , one can find a regular sequence 5m −1 ({0}). But one of noninvertible elements (f1,z , . . . , fm,z ) such that Vz = j=1 fj,z should be careful that the notions of complete intersection and regularity are not connected anymore in the global context; see Remark B.17. Remark B.17. The fact that a closed analytic subset V with codimension m in a complex analytic space X with dimension N is globally defined as a complete intersection, that is, as the zero set of a holomorphic section s of a m-holomorphic  vector bundle E → X , does not imply that (σ1,z , . . . , σm,z ) (where s = m 1 σj ⊗ j when expressed in a local holomorphic frame in a neighborhood of z ∈ X ), is a regular sequence in OX ,z , unless z ∈ s−1 (0). Take E for example as the trivial bundle over ℂ3 and V = s−1 (0) = {(0, 0, 0)}, where   s(z) = z1 (z3 + 1), z2 (z3 + 1), z3 . The sequence (s1,z , s2,z , s3,z ) is not regular (in this order) in a neighborhood of any point in the complex plane {z3 + 1 = 0}.

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Example B.18. Any closed analytic subset in ℙN ℂ is algebraic projective, which is defined as the zero set of a finite number of homogeneous polynomials in z0 , . . . , zN ; see Theorem A.28 in §A.3.6. If P1 , . . . , Pm are m homogeneous polynomials in z0 , . . . , zN with respective total degrees d1 , . . . , dm , then the section (d1 ) ⊕ · · · ⊕ OℙN (dm ) (with rank m) defines (P1 , . . . , Pm ) of the vector bundle OℙN ℂ ℂ N a globally complete intersection in ℙℂ if and only if {z ∈ ℙN ℂ : P1 (z) = · · · = Pm (z) = 0} is purely (N − m)-dimensional. We conclude this section with a heuristic remark. It is easier to describe the notion of a complete intersection in geometric terms than in algebraic ones, as the following equivalence shows. If f1 , . . . , fm are m holomorphic functions in an open pseudo-convex domain X of ℂN (or more generally, m holomorphic functions over a Stein manifold X , see §A.2.2), then f1 , . . . , fm either define an (N −m)-dimensional globally complete intersection in X or have no common zero in X if and only if, for every ν ∈ ℕ∗ , for every homogeneous relation

ν+1  κm aκ f1κ1 f2κ2 · · · fm ∈ (f1 , . . . , fm ) OX (X ) , κ1 +···+κm =ν

where aκ ∈ OX (X ) are holomorphic functions in X , one has that ∀ κ ∈ ℕm , aκ ∈ (f1 , . . . , fm ) OX (X ) . Thus one can see that the rather simple geometric concept of complete intersection (here global) corresponds, expressed in algebraic terms, to a more involved asymptotic characterization (“for every ν ∈ ℕ∗ , for every relation”, etc.). Compare also the two approaches (geometric or algebraic) to the notion of a gap sheaf, where the same phenomenon occurs (see §D.1.1, in particular (D.4) and (D.5) where the cutting-out operation is introduced on complex analytic cycles): a simple geometric formulation on one side and an asymptotic more-involved algebraic formulation on the other side. ∗(p,q)

over a complex manifold X . B.1.7. The bundles TX B.1.7.1. Complex bundles over Xℝ and their isomorphism classes. Let X be an N -dimensional complex manifold with underlying 2N -dimensional real manifold Xℝ ; see §A.1.1. Definition B.19. A ℂ-bundle E → X with rank m ∈ ℕ over Xℝ is a structure (denoted by (𝕏Em ,ℝ , C𝕏∞Em ,ℝ )) of 2(N + m)-dimensional differentiable manifold on > the disjoint union 𝕏Em := z∈X {z} × Ez , where Em is as in (B.6), such that • the projection map π Em : (z, ξz ) ∈ 𝕏Em → z ∈ Xℝ is a morphism between differentiable manifolds; • X admits an open covering {Uι }ι such that for each index ι, there exists an isomorphism of differential manifolds θ ι between the differentiable manifolds π −1 ((Uι )ℝ ) and (Uι × ℂm )ℝ such that, for each z ∈ Uι , (B.12)

θ ι (z, ξz ) = (z, ι,z (ξz )) ∈ {z} × ℂm with ι,z ∈ Isomℂ (Ez , ℂm ). Such an isomorphism θ ι of differentiable manifolds between π −1 Em ((Uι )ℝ ) and (Uι × ℂm )ℝ is called a trivialization morphism for the bundle E → X over the open subset Uι .

The notion of a local frame extends that of a holomorphic local frame already introduced in Definition B.15.

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Definition B.20. Assume that U = Uι is an open subset of X over which E admits a trivialization morphism θι = θ : π −1 (Uℝ ) ←→ Uℝ × ℂm . Let {1 , . . . , m } be the canonical basis of ℂm . Then the family of sections {z ∈ U → ej (z) = θ−1 (z, j ) : j = 1, . . . , m} of E over U is said to be a local frame for E in the open subset U . The concept of a complex bundle with rank m over the differentiable underlying manifold Xℝ to a complex manifold X is closely linked with that of 1-complex differentiable cocycle with values in GL(m, ℂ) over Xℝ . Definition B.21. Let m ∈ ℕ∗ and U = {Uι }ι be an open covering of X . A complex differentiable 1-cocycle on Xℝ with values in GL(m, ℂ) subordinate to the covering U is a collection of matrix valued maps {gι1 ,ι0 }ι0 ,ι1 , where gι1 ,ι0 is a morphism of differential manifolds between (Uι0 ∩Uι1 )ℝ and (GL(m, ℂ))ℝ such that the cocycle conditions ∀ z ∈ Uι0 ∩ Uι1 ∩ Uι2 , gι2 ,ι0 (z) = gι2 ,ι1 (z) ◦ gι1 ,ι0 (z), hold for any triplet of indices ι0 , ι1 , ι2 . In a similar way to Definition B.7 (see §B.1.4) one may introduce a notion of isomorphism between complex bundles with the same rank over Xℝ , in place of that of an isomorphism between holomorphic bundles over a complex analytic space X . In the same vein as in Proposition B.8 (see §B.1.4) there is a one-to-one correspondence between complex differentiable 1-cocycles on Xℝ with values in GL(m, ℂ) and isomorphism classes of complex bundles with rank m over Xℝ . Operations on complex differentiable 1-cocycles over Xℝ , such as described in §B.1.2, induce in this new context the realization the the correspondence of isomorphism classes of complex bundles over Xℝ . Observe that all such operations provide as output entries holomorphic bundles as soon as the input entries are holomorphic bundles; see §B.1.4. ∗(1,0) ∗(0,1) and TX over Xℝ and their exterior powers. B.1.7.2. The ℂ-bundles TX Let X be an N -dimensional complex manifold and let Xℝ be its 2N -dimensional underlying manifold. Let {(Uι , ζι )}ι be an atlas for X and τι : Uι ←→ U ι,ℝ as in §A.1.1. Then 2   4 g ι1 ,ι0 : z ∈ Uι0 ∩ Uι1 −→ d τι1 ◦ τι−1 ) ∈ GL(2N, ℝ) ι0 ,ι1 0 is a C ∞ (even real analytic) 1-cocycle on the 2N -dimensional underlying real manifold Xℝ , with values in GL(2N, ℝ). For each z ∈ X , let ζz be a local centered system of coordinates ζz : U ←→ U in a neighborhood of z and τz : U ←→ U ℝ be the corresponding map from U to ℝ2N as in §A.1.1. The tangent real vector space TXℝ ,z is defined (independent of the choice of ζz , hence of τz ) as the set of equivalence classes γ˙ ℝ of germs of C ∞ maps γℝ : ] − ε, ε[ → Xℝ such that γℝ (0) = z modulo the identification between γℝ and γ˜ℝ as soon as d(τz ◦γℝ )(0) = d(τz ◦ γ˜ℝ )(0). It inherits a structure of 2N -dimensional ℝ-vector space. Let ℂ ⊗ℝ TXℝ ,z = TXℝ ,z ⊕ i TXℝ ,z = TX ,z ⊕ TX ,z := TX1,0,z ⊕ TX0,1,z , where TX ,z = TX1,0,z is the holomorphic complex tangent space at z; see Example B.5. If ∂/∂ζz,1 , . . . , ∂/∂ζz,N is a holomorphic local frame for the holomorphic tangent

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451

complex bundle in a neighborhood of z, one has TX0,1,z =

N 

∂ ℂ ¯ (z). ∂ ζ k,z k=1

Both the holomorphic bundle (with rank N ) TX1,0 and its conjugate TX0,1 can be considered as complex bundles over Xℝ , according to Definition B.19 in §B.1.7.1. Such are their duals, which we denote respectively as4 ∗(1,0)

TX

= (TX1,0 )∗ = (TX1,0 )∗ , ℝ

∗(0,1)

TXℝ

= (TX0,1 )∗ . ℝ

Definition B.22. Let E be a complex bundle with rank m ∈ ℕ over Xℝ .5 For any 0 ≤ p, q ≤ N , the one-to-one correspondence between complex bundles and complex differentiable 1-cocycles on Xℝ with values in GL(•, ℂ) allows us to define (up to isomorphisms between complex bundles over Xℝ with the same rank, see §B.1.7.1), the complex bundle over Xℝ : ∗(p,q)

(B.13) TXℝ

⊗E =

p '

∗(1,0)

TX

⊗

q '

∗(0,1)

TXℝ

with

⊗ E, ∗(p,q) rank(TXℝ

   N N ⊗ E) = m . p q

B.2. Sheaves of bundle-valued differential forms or currents B.2.1. Differential forms and currents on a complex manifold. In this subsection X is an N -dimensional complex manifold with underlying differentiable manifold Xℝ . B.2.1.1. Sheaves of complex bundle-valued forms on a complex manifold. Definition B.23. Let E be a complex bundle with rank m over Xℝ and let ∗(p,q) 0 ≤ p, q ≤ N , 0 ≤ r ≤ 2N . Smooth sections of the bundle TXℝ ⊗ E are called E-valued differential (p, q)-forms on Xℝ , or complex valued differential (p, q)-forms on Xℝ in case E is the trivial (holomorphic) bundle X × ℂ. For any open subset U ⊂ X , one denotes as ∞ ∞ (Cp,q )X (U, E) := (Cp,q )Xℝ (U, E), DX (U, E) := DXℝ (U, E),   (p,q) ∞ r (Cp,q )X (U, E), DX (U, E) := DX (U, E), (Cr∞ )X (U, E) := (p,q)

(p,q)

p+q=r

p+q=r

max(p,q)≤N

max(p,q)≤N

respectively, the ℂ-vector space of E-valued (p, q)-differential forms in U , its subspace of E-valued (p, q)-differential forms with compact support in U , the ℂ-vector space of E-valued r-differential forms in U , its subspace of E-valued r-differential forms with compact support in U . We will denote shortly the corresponding sheaves ∞ ∞ by Cp,q (·, E), D(p,q) (·, E), Cr∞ (·, E), Dr (·, E) and by Cp,q , D (p,q) , Cr∞ , Dr in the particular case where E = ℂ. 4 Since the holomorphic cotangent complex bundle is holomorphic, we keep X instead of X ℝ as an index in the notation. 5 It could be in particular a holomorphic bundle with rank m over X .

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If e = {ez,1 , . . . , ez,m } is a local frame6 for E in a neighborhood of a trivial∞ (U, E) expresses in local ization open subset U , see Definition B.20, any ϕ ∈ Cp,q coordinates (ζ1 , . . . , ζN ) in a neighborhood of z ∈ U as (B.14) ϕ=

m  j=1

=

m  j=1





ϕjI,J

1 0.

It is said to be Griffiths semipositive, which is denoted by (E, | |) ≥G 0, if and only if (C.32)

∀z ∈ X , ∀ (ξ, ez ) ∈ TX ,z × Ez ,

θE,| | (ξ ⊗ ez , ξ ⊗ ez ) ≥ 0.

When > and ≥ are replaced respectively by < and ≤ in (C.31) or (C.32), (E, | |) is said to be Griffiths negative, (which is denoted by (E, | |) N 0, if and only if the hermitian form θE,| | defines a scalar product on each fiber (TX ⊗ E)z , z ∈ X . It is said to be Nakano semipositive and denoted by (E, | |) ≥N 0, if and only if θE,| | defines a positive form on each fiber (TX ⊗ E)z , z ∈ X . The hermitian holomorphic bundle (E, | |) is said to be Nakano negative (respectively Nakano seminegative) and denoted by (E, | |)