Instituto Nacional de Matemática Pura e...

Hossein Movasati

A course in Hodge Theory

with emphasis on multiple integrals

May 25, 2015

Publisher

Preface

The main objective of the present book is to give an introduction to Hodge the-ory and its main conjecture, the so-called Hodge conjecture. We aim to explore theorigins of Hodge theory much before the introduction of Hodge decomposition ofthe de Rham cohomology of projective varieties. This is namely the study of el-liptic, abelian and multiple integrals originated from the works of Abel, Riemann,Poincare, Picard and Simart among others. The development of Hodge theory dur-ing the last decades has put it far from its origin and the introduction of mirrorsymmetry by string theorists and the period manipulations of the B-model Calabi-Yau varieties, have risen the need for a text in Hodge theory with more emphasis onperiods and multiple integrals. We aim to cover materials which are not covered in J.Lewis’s book ”A survey of the Hodge conjecture” and C. Voisin’s two volume book”Hodge theory and complex algebraic geometry, I, II”. We have tried to keep thetext self-sufficient, however, a basic knowledge of Complex Analysis, DifferentialEquations, Algebraic Topology and Algebraic Geometry will make the reading ofthe text smoother. The text is mainly written for two primary target audiences: grad-uate students who want to learn Hodge theory and get a flavor of why the Hodgeconjecture is hard to deal with, mathematicians who uses periods and multiple inte-grals in their research and would like to put it in a Hodge theoretic framework. Wehope that our text, together with those mentioned above, makes Hodge theory moreaccessible to a broader public.

Hossein MovasatiJanuary 2017

Rio de Janeiro, RJ, Brazil

v

Acknowledgements

The present text is written during many years that I thought Hodge theory at IMPA,Rio de Janeiro.

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Elliptic and abelian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Lefschetz’s Puzzle and the origin of Hodge conjecture . . . . . . . . . . . . 5

2 Homology theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Eilenberg-Steenrod axioms of homology . . . . . . . . . . . . . . . . . . . . . . . 72.2 Singular homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Some consequences of the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Leray-Thom-Gysin isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Intersection map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Lefschetz theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Some consequences of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Lefschetz theorem on hyperplane sections . . . . . . . . . . . . . . . . . . . . . . 193.4 Topology of complete intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Some remarks on hard Lefschetz theorem . . . . . . . . . . . . . . . . . . . . . . 233.6 Lefschetz decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.7 Lefschetz theorems in cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Picard-Lefschetz Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1 Ehresmann’s fibration theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 The case of isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5 Vanishing Cycles as Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.6 Lefschetz pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.7 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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5 Topology of tame polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1 Vanishing cycles and orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Tame polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Picard-Lefschetz theory of tame polynomials . . . . . . . . . . . . . . . . . . . 425.4 Monodromy group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5 Distinguished set of vanishing cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 455.6 Join of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.7 Direct sum of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Hodge conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.1 De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 Hodge decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.4 Hodge conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.5 Real Hodge cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.6 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.7 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 De Rham cohomology of affine hypersurfaces . . . . . . . . . . . . . . . . . . . . . 597.1 The base ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2 Homogeneous tame polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607.3 De Rham Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.4 Tame polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.5 De Rham Lemma for tame polynomials . . . . . . . . . . . . . . . . . . . . . . . . 677.6 The discriminant of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.7 The double discriminant of a tame polynomial . . . . . . . . . . . . . . . . . . 717.8 De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.9 Proof of Theorem 16 for a homogeneous tame polynomial . . . . . . . . 747.10 Proof of Theorem 16 for an arbitrary tame polynomial . . . . . . . . . . . . 76

8 Hodge filtrations and Mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . 798.1 Gauss-Manin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.2 Mixed Hodge structure of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.3 Homogeneous tame polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.4 Weighted projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

9 Fermat varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879.1 De Rham cohomology of affine Fermat varieties . . . . . . . . . . . . . . . . . 879.2 Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899.3 Intersection form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919.4 No Hodge cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.5 Algebraic cycles in a join of two tame polynomial . . . . . . . . . . . . . . . 93

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10 Gauss-Manin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9510.1 Gauss-Manin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9510.2 Picard-Fuchs equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9710.3 Gauss-Manin connection matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9810.4 Calculating Gauss-Manin connection . . . . . . . . . . . . . . . . . . . . . . . . . . 9810.5 R[θ ] structure of H ′′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.6 Gauss-Manin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.7 Griffiths transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

11 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10511.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10511.2 Integrals and Gauss-Manin connections . . . . . . . . . . . . . . . . . . . . . . . . 10611.3 Period matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10711.4 Picard-Fuchs equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10811.5 Modular foliations and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.6 Homogeneous polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.7 Integration over joint cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11011.8 Reduction of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.9 Residue map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11311.10Geometric interpretation of Theorem 19. . . . . . . . . . . . . . . . . . . . . . . . 114

12 Intersection of topological cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

13 Infinitesimal variation of Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . 12313.1 Griffiths-Dwork method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

14 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12514.1 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12514.2 How to compute Cech cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.3 Acyclic sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.4 Resolution of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.5 Cech cohomology and Eilenberg-Steenrod axioms . . . . . . . . . . . . . . . 12914.6 Dolbeault cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

15 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13115.1 Objects and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13115.2 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13215.3 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13315.4 Additive categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13415.5 Kernel and cokernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13415.6 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13515.7 Additive functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13615.8 Injective and projective objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13615.9 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13615.10Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13715.11Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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15.12Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13915.13Acyclic objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13915.14Cohomology of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14015.15Derived functors for complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14015.16Hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14215.17De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14215.18How to calculate hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 14215.19Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

16 Algebraic de Rham cohomology and Hodge filtration . . . . . . . . . . . . . . 14716.1 De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14716.2 Atiyah-Hodge theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14816.3 Proof of Theorem 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14816.4 Hodge filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

17 Lefschetz (1,1) theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15117.1 Lefschetz (1,1) theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15117.2 Some consequences on integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

18 Deformation of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15318.1 Reconstructing the period matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

19 Mixed Hodge structure of affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . 15519.1 Logarithmic differential forms and mixed Hodge structures of

affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15519.2 Pole order filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15619.3 Another pole order filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15719.4 Mixed Hodge structure of affine varieties . . . . . . . . . . . . . . . . . . . . . . . 158

20 Global invariant cycle theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15920.1 Some doubt about Hodge conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 16020.2 A consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

21 Lixo de Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Chapter 1Introduction

The study of multiple integrals of dimension n, that is the integration of differentialn-forms over topological cycles of dimension n and lying in algebraic varieties, goesback to 19th century. Abel, Cauchy, Riemann, Poincare were among many mathe-maticians who studied the one dimensional integrals, nowadays known as ellipticand abelian integrals. Picard was the first person who systematically studied the twodimensional integrals and, jointly with Simart, wrote a two-volume treatise on thissubject1. Between 1911 and 1924, Lefschetz motivated by such an study and withAnalysis Situs of Poincare in hand , started a complete investigation of the topologyof algebraic varieties2, in his own words, he wanted to plant the harpoon of alge-braic topology into the body of the whale of algebraic geometry. It is a remarkablefact that at the time Lefschetz’s work was being done, while the study of algebraictopology was getting under way, the topology tools available were still primitive. Ittook some decades for the precise formulations and proofs of Lefschetz ideas us-ing harmonic integrals and sheaf theory. However, these methods only obtain thehomology groups with complex or real coefficients, whereas the direct method ofLefschetz enables us to use integer coefficient. Nowadays, few mathematics studentsand university professors know that many achievements in algebraic geometry andalgebraic topology were originated by Lefschetz study of the topology of algebraicvarieties, which in turn, originated by the works of Picard for understanding doubleintegrals.

In his mathematical autobiography3, Lefschetz writes: From the ρ0-formula ofPicard, applied to a hyperelliptic surface Φ (topologically the product of four cir-cles) I had come to believe that the second Betti number R2(Φ)= 5, where as clearlyR2(Φ) = 6. What was wrong? After considerable time it dawned upon me that Pi-card only dealt with finite 2-cycles, the only useful cycles for calculating periods ofcertain double integrals. Missing link? The cycle at infinity, that is the plane section

1 Picard E., Simart G. Theorie des fonctions algebriques de 2 variables independentes. Tome 11897, Tome 2 1906, Paris.2 Lefschetz, S., L’analysis situs et la geometrie algebrique, Gauthier Villars, 1924, Paris.3 Lefschetz, S., A page of mathematical autobiography, Bull. Amer. Math. Soc. 74 (1968), 854–879.

1

2 1 Introduction

of the surface at infinity. This drew my attention to cycles carried by an algebraiccurve, that is to algebraic cycles, and ... the harpoon was in. Therefore, the desireof classifying cycles carried by algebraic varieties goes back to the early state ofboth Algebraic Geometry and Algebraic Topology. Lefschetz himself formulated acriterion for cycles carried by an algebraic curve which can be generalized to realcodimension two cycles on algebraic varieties and nowadays it is known as Lef-schetz (1,1) theorem. He states his result in the following form: On an algebraicsurface V a 2-dimensional homology class contains the carrier cycle of a virtualalgebraic curve if and only if all the algebraic double integrals of the first kind havezero periods with respect to it. I was surprised when I found that no modern book inHodge theory states the Lefschetz (1,1) theorem in its original form, namely, usingintegrals. Complains in this direction is expressed by V. Arnold4: ... students whohave sat through courses on differential and algebraic geometry (read by respectedmathematicians) turned out to be acquainted neither with the Riemann surface ofan elliptic curve y2 = x3 +ax+b nor, in fact, with the topological classification ofsurfaces (not even mentioning elliptic integrals of first kind and the group propertyof an elliptic curve, that is, the Euler-Abel addition theorem). They were only taughtHodge structures and Jacobi varieties!

A criterion, called nowadays the Hodge conjecture, to distinguish topologicalcycles carried by algebraic varieties was formulated after the discovery of the Hodgedecomposition for Kahler manifolds5. The objective of the present text is to collectrecent and old developments on the Hodge conjecture, with emphasis on multipleintegrals. In the same time we want to rebuilt the higher dimensional integral theoryof Picard, that is, the study of multiple integrals.

1.1 Elliptic and abelian integrals

In order to trace back the origins of the Hodge conjecture, one must goes back tothe study of elliptic integrals of the form∫ b

a

dy√f (x)

,

where f (x) is a polynomial of degree 3 and with three distinct real roots, and a,bare two consequent elements among the roots of f and ±∞. It is an easy exerciseto show that all the above integrals can be calculated in terms of only two of them.One way to see this is to consider the integration in the complex domain x ∈ C, inwhich we may discard the assumption that f has only real roots. We may go evenfurther and reduce our integrals to integrals of the form

4 V.I. Arnold, On teaching mathematics, Palais de Decouverte in Paris, 7 March 1997.5 Hodge, W. V. D., The Theory and Applications of Harmonic Integrals. Cambridge UniversityPress, Cambridge, 1941.

1.2 Multiple integrals 3∫δ

dxy, δ ∈ H1(E,Z)

whereE : y2 = f (x)

We may define H1(E,Z) as the abelization of the fundamental group of E. We knowthe topology of E; it is a punctured torus and so H1(E,Z) has only two linearlyindependent generators.

The above discussion can be made for f an arbitrary polynomial of degree d andwith distinct roots. In this case the curve E is a Riemann surface of genus g := [ d−1

2 ]

and with d− 2[ d−12 ] removed points. Therefore, H1(E,Z) is of rank d− 1 and we

have d−1 linearly independent integrals.

1.2 Multiple integrals

As we saw, the study of linear independent one dimensional integrals naturally leadsto the study of the topology of curves. Therefore, for the study of higher dimen-sional integrals we need a better understanding of the topology of algebraic va-rieties. It took more than half a decade to conclude the precise definition of the(co)homologies:

X a smooth variety of dimension n → Hq(X ,Z),Hq(X ,Z), q = 0,1, . . . ,n

The foundation of algebraic topology was started by Poincare in his treatise ”Anal-ysis Situs” with a series of addenda. What really was studied in these works werethe rank and torsion elements of a homology and not the homology itself. From nowon the reader can think of an element δ ∈ Hq(X ,Z) in the same style of 19th cen-tury, that is, as the image of a smooth generically one to one map from a compact,oriented q-dimensional manifold to X , or even, a collection of such maps. One ofthe varieties for which Poincare applies his theory is the affine variety

V : z2 = f (x,y), (1.1)

where f (x,y) is a polynomial in two variables and f = 0 is a smooth curve. Poincarestudied the topology of this variety by cutting it with the hyperplanes y = c, where cis a constant.6 This idea was later used by Lefschetz. A generic pencil of hyperplanesnowadays is called a Lefschetz pencil. Poincare needed this in order to study thedouble integral ∫ ∫ dxdy√

f (x,y)

6 Henri Poincare, Sur certaines surfaces algebriques; troisieme complement a l’Analysis Situs,Bulletin de la Societe mathematique de France, 30 (1902), pages 49-70.

4 1 Introduction

In an article7 he associates to Cauchy the interest to such an study C’est a Cauchyque revient la gloire d’avoire fonde la theorie des integrales prises entre des limitesimaginaire ....

E. Picard, together with Simart, in 1900 and 1904 published two books on mul-tiple integrals with emphasis on double integrals. The main tools in his books arethe algebraic geometry of Noether, Severi, Castelnuevo and others, and the basics ofalgebraic topology after Betti and Poincare. Lefschetz after reading these two booksfelt the need for a systematic study of the topology of algebraic varieties and aftereleven years of labor and isolation he published his treatise in 1924. After Lefschetzthe study of multiple integrals were forgotten, until string theorists around eightiesstarted to use triple integrals and produce miraculous predictions on the number ofcurves on Calabi-Yau threefolds.

1.3 An example

For the variety (1.1), let us take the polynomial f (x,y) =−xd−yd + · · · , where · · · isa linear combinations of monomials of degree ≤ d−1 in x,y. The compactificationC3⊂P3 with the coordinates (x,y,z,w), or in other words P3 =C3∪P2

∞, gives us theprojective variety V such that V∞ := V\V is given by [x;y;z] ∈ P2

∞ | xd + yd = 0.This is a union of d curves isomorphic to P1. Let C be any curve in V . We have thetopological classes

[V∞], [C] ∈ H2(V ,Z)

Any Z-linear combination of these cycles is called an algebraic cycle. This is inthe same way that we associate to any real one dimensional curve in a Riemannsurface a homological class. The curve C intersects V∞ in a finite number of points.We define n := 〈[C], [V∞]〉= n, m := 〈[V∞], [V∞]〉 and δ := m[C]−n[V∞] and we have〈δ , [V∞]〉= 0. The conclusion is that the support of δ is in V . For simplicity we writeδ ∈ H2(V,Z). The study of the topology of algebraic varieties leads us to the factthat H2(V,Z) is a free Z-module of rank (d−1)2. The cycle δ is a very special one.It follows that∫

δ

xiy jdx∧dyz

= 0, ∀i, j ∈ N∪0, 12+

i+1d

+j+1

d< 1 (1.2)

Following the terminology in higher dimensions, we call δ ∈ H2(V,Z) with theproperty (1.2) a Hodge cycle. Note that for n = 2,3,4 any δ ∈ H2(V,Z) is a Hodgecycle. Lefschetz (1,1) theorem implies any Hodge cycle δ is a Z-linear combinationof cycles constructed from algebraic cycles as above, that is, any any Hodge cycleis an algebraic cycle. A consequence of δ being an algebraic cycle is the followingstatement ∫

δ

p(x,y)dx∧dyz

∈ Q ·π (1.3)

7 Henri Poincare, Sur les rsidus des intgrales doubles. Acta Math. 9 (1887), no. 1, 321380.

1.4 Lefschetz’s Puzzle and the origin of Hodge conjecture 5

for any polynomial p with algebraic coefficients. This follows from a theorem ofDeligne. These properties show the importance of algebraic cycles in the study ofmultiple integrals. In order to have more concrete and elementary problems, let ustake δ to be

δ := (x,y,z) ∈ R3 | z2 + xd + yd + · · ·= 0.

and with an arbitrary orientation. If we take the coefficients of the monomials in· · · very small, then the above set is a compact two dimensional subvariety of R3

isomorphic to the two dimensional sphere. It is a challenging problem to find anexample of this cycle with (1.2). Assuming that such a cycle exists, to prove thestatement (1.3) is another challenging problem.

In general, for a smooth projective variety X of even dimension n and any subva-riety, not necessarily smooth, Y of X of dimension q we have

[Y ] ∈ H2q(X ,Z)

and any Z-linear combination of such classes is called an algebraic cycle. TheHodge conjecture claims to give a criterion to distinguish algebraic cycles fromother cycles. This criterion can be formulated in terms of vanishing of a bunch ofintegrals.

1.4 Lefschetz’s Puzzle and the origin of Hodge conjecture

In this section we roughly explain Lefschetz’s ρ0 = 5 puzzle and we explain whatdawned upon him so that he started to study algebraic cycles. Our history beginsfrom page 448 of the second volume of Picard’s treatise: A hyperelliptic curve ofgenus two is given by the equation

S : y2 = f (x)

in C2, where f is a degree 5 polynomial and it has not repeated roots. All integrals onthis curve reduces to the integrals

∫ xidxy ,y= 0,1,2,3. Let us set Φ = S×S/∼, where

∼ is defined by ((x1,y1),(x2,y2)) ∼ ((x2,y2),(x1,y1)). We call Φ the hyperellipticsurface. Adding the point at infinity to S, that is S := S∪∞, we get a compactsurface of genus two and in a similar way we can define Φ which is toplogically theset of two points (without order) in S. The image of S×S under the map given by

(x1,y1,x2,y2) 7→ (x,y,z), x := x1 + x2, y = x1x2, z = y1 + y2

8 gives Φ in the affine coordinate (x,y,z). The double integrals∫ ∫(xp

1xq2− xq

1xp2)

dx1dx2

y1y2, p,q = 0,1,2,3, p < q

8 Note that y1y2 =12 (z

2− f (x1)− f (x2)) and f (x1)+ f (x2) can be written in terms of x,y.

6 1 Introduction

gives us essentially six double integrals on Φ of the form∫R(x,y,z)dxdy.

In this example the second Betti number of Φ is 6 but its affine part in (x,y,z)-coordinates has the Betti number 5. The difference 6− 5 = 1 is due to the twodimensional toplogical cycle obtained by Φ\Φ . This is a copy of S embedded in Φ .

Exercises1. Let S be a compact, orientable surface of genus 2. Lefschetz in his autobiography claims that

the set of two points in S is a four dimensional manifold isomorphic to the self product of acircle four times. Prove or disprove this.

Chapter 2Homology theory

In mathematics an object may be constructed during the decades and by many math-ematicians and finally one finds that such an object satisfies an enumerable numberof axioms, previously stated as theorems, and these axioms determine the objectuniquely. An outstanding example to this is the homology theory of topologicalspaces. It was founded by Henri Poincare under the name Analysis Situs1, furtherdeveloped by Solomon Lefschetz and many others and finally, it was axiomatizedby Samuel Eilenberg and Norman Steenrod around 1950 (see [ES52]). A fascinat-ing fact is the study of the topology of algebraic varieties by Lefschetz right at thebegining of homology theory. This goes back even further, to the study of integralsstarted by Abel and pursued by Picard.

In this chapter we present the axiomatic approach to Homology theory introducedby Eilenberg and Steenrod in [ES52]. After many years of using homology theoryin my own research my impression is that instead of spending time and effort toconstruct the homology theory, one has to get the feeling that one uses it correctly,even without knowing its precise definition.

2.1 Eilenberg-Steenrod axioms of homology

An admissible category A of pairs (X ,A) of topological spaces X and A with A⊂ Xand the maps between them f : (X ,A)→ (Y,B) with f : X→Y and f (A)⊂B satisfiesthe following conditions:

1. If (X ,A) is in A then all pairs and inclusion maps in the lattice of (X ,A)

1 Henri Poincare, Analysis Situs, Journal de l’Ecole Polytechnique ser 2, 1 (1895) pages 1-123.See also a series of addenda after this paper

7

8 2 Homology theory

(X , /0)

( /0, /0)→ (A, /0) (X ,A)→ (X ,X)

(A,A)

are in A .2. If f : (X ,A)→ (Y,B) is in A then (X ,A),(Y,B) are in A together with all maps

that f defines of members of the lattice of (X ,A) into corresponding membersof the lattice of (Y,B). If A and B are empty sets then for simplicity we writef : X → Y .

3. If f1 and f2 are in A and their composition is defined then f1 f2 ∈A .4. If I = [0,1] is the closed unit interval and (X ,A) ∈A then the cartesian product

(X ,A)× I := (X× I,A× I)

is in A and the maps given by

g0,g1 : (X ,A)→ (X ,A)× I

g0(x) = (x,0), g1(x) = (x,1)

are in A .5. There is in A a space consisting of a single point. If X ,P ∈A and P is a single

point space and f : P→ X is any map then f ∈A .

The category of all topological pairs and the category of polyhedra are admissible. Inthis text we will not need these general categories. The reader is referred to [ES52]for examples of admissible categories. What we need in this text is the category ofreal differentiable manifolds, possibly with boundaries, which is admissible and it isa sub category of the category of polyhedra. A polyhedra is also called a triangulablespace.

An axiomatic homology theory is a collection of functions

(X ,A) 7→ Hq(X ,A), q = 0,1,2,3, . . .

from an admissible category of pairs (X ,A) of topological spaces X and A with A⊂X to the category of abelian groups such that to each continuous map f : (X ,A)→(Y,B) in the category it is associated a homomorphism:

fq : Hq(X ,A)→ Hq(Y,B), q = 0,1,2, . . .

We denote by H∗(X ,A) the disjoint union of Hq(X ,A) and by f∗ : H∗(X ,A) →H∗(Y,B) the corresponding map constructed from fq’s. We have the following ax-ioms of Eilenberg and Steenrod.

1. If f is the identity map then f ∗ is also the identity map.

2. For (X ,A)f→ (Y,B)

g→ (Z,C) we have (g f )∗ = g∗ f∗.

2.1 Eilenberg-Steenrod axioms of homology 9

3. There are connecting homomorphisms

∂ : Hq(X ,A)→ Hq−1(A)

such that such that for any (X ,A)→ (Y,B) in A the following diagram com-mutes:

Hq(X ,A)→ Hq(Y,B)↓ ↓

Hq−1(A) → Hq−1(B), q = 0,1,2, . . .

Here A means (A, /0).4. The exactness axiom: The homology sequence

· · · → Hq+1(X ,A) ∂→ Hq(A)i∗→ Hq(X)

j∗→ Hq(X ,A)→ ··· → H0(X ,A)

where i∗ and j∗ are induced by inclusions, is exact. This means that the kernel ofevery homomorphism coincides with the image of the previous one.

5. The homotopy axiom: for homotopic maps f ,g : (X ,A)→ (Y,B) we have f∗= g∗.6. The excision axiom: If the closure of a subset U2 in X is contained in the interior

of A and the inclusion(X\U,A\U)

i→ (X ,A)

belongs to the category, then i∗ is an isomorphism.7. The dimension axiom: For a single point set X = p we have Hq(X) = 0 for

q > 0.

The coefficient group of a homology theory is defined to be H0(X) for a single pointset X . From the first and second axioms it follows that for any two single point setsX1 and X2 we have an isomorphism H0(X1)∼= H0(X2).

Axiomatic cohomology theories are dually defined, i.e for (X ,A)f→ (Y,B) we

have Hq(Y,B)f ∗→Hq(X ,A) and the coboundary maps δ : Hq−1(A)→Hq(X ,A) with

similar axioms as listed above. We just change the direction of arrows and insted ofsubscript q we use superscript q.

In [Mil62] we find the Milnor’s additivity axiom which does not follow from theprevious ones if the admissible category of topological spaces has topological setswhich are disjoint union of infinite number of other topological sets:

8. Milnor additivity Axiom. If X is the disjoint union of open subsets Xα with inclu-sion maps iα : Xα → X , all belonging to the category, the the homomorphisims

(iα)q : Hq(Xα)→ Hq(X)

must provide an injective representation of Hq(X) as a direct sum.

The amazing point of the above axioms is the following:

2 In [ES52] it is asumed that U is open. However, in the page 200 of the same book, the authorsshow that for singular homology or cohomology the openness condition is not necessary.

10 2 Homology theory

Theorem 1 In the category of polyhedra the homology (cohomology) theory existsand it is unique for a given coefficient group.

The singular homology (cohomology) is the first explicit example of the homol-ogy (cohomology) theory. Its precise construction took more than sixty years in thehistory of mathematics. For more details. The reader is referred to [Mas67]. Theuniqueness is a fascinating observation of Eilenberg and Steenrod. For a proof ofuniqueness the reader is referred to [ES52] page 100 and [Mil62].

In order to stress the role of the coefficient group G we sometimes write:

Hq(X ,A,G) = Hq(X ,A) and so on.

Using the above axiom we can show that

H0(S1,Z)∼= H1(S1,Z)∼= Z.

Later, we will need this in order to calculate the homology of a torus.

2.2 Singular homology

As we mentioned before, one of the reasons for the development of algebraic topol-ogy was a systematic study of multiple integrals. For this reason, the first example ofhomology theory is the singular homology constructed from simplicial complexes,where our integrations take place.

Fix an abelian group G. Let X be a C∞ manifold and

∆n =

(x0, · · · ,xn) ∈ Rn+1 |∑

ixi = 1 and xi ≥ 0, i = 0,1,2, . . . ,n

be the standard n-simplex. A C∞ map f : ∆ n → M is called a singular n-simplex.The map f need to be neigther surjective nor injective. Therefore, its image may notbe so nice as ∆ n. Let Cn(X) the set of all formal and finite sums

∑i

ni fi, ni ∈ G, fi a singular n-complex.

Cn(X) has a natural structure of an abelian group. For the simplex ∆ n we denote by

Ik : ∆n−1→ ∆

n, Ik(x1,x2, . . . ,xn) := (x1,x2, . . . ,xk,0,xk+1, · · · ,xn)

k = 0,1,2, . . . ,n

the canonical inclusion for which the image is the k-th face of ∆ n. For f a singularn-complex we define

2.3 Some consequences of the axioms 11

∂n f =n

∑k=0

(−1)k f Ik ∈Cn−1(X)

and by linearity we extend it to the homomorphism of abelian groups:

∂ = ∂n : Cn(X)→Cn−1(X).

which we call it the boundary map. The kernel of the boundary map is

Zn(X) = ker(∂n)

and is called the group of singular n-cycles. The image of the boundary map is

Bn(X) = Im(∂n+1)

and is called the group of singular n-boundaries. It is an easy exercise to show that∂n∂n+1 = 0 and so Bn(X)⊂ Zn(X). The n-th homology group of X with coefficientsin G is defined to be

Hn(X ,G) := Hn(C•(X),∂ ) =Zn(X)

Bn(X).

The elements of Hn(X ,G) are called homology classes with coefficients in G.In order to construct relative homologies we proceed as follows: For the pair

(X ,A) of topological spaces with A⊂ X we define

Hn(X ,A,G) := Hn(C•(X)

C•(A),∂ ).

The boundary mapδ : Hn(X ,A,G)→ Hn−1(A,G)

is given by the boundary map ∂n (prove that it is well-dfined).

2.3 Some consequences of the axioms

From the axioms it is possible to prove the following classical theorems in singularhomology theory. For proofs see [ES52].

1. Universal coefficient theorem for homology: For a polyhedra X there is a naturalshort exact sequence

0→ Hq(X ,Z)⊗G→ Hq(X ,G)→ Tor(Hq−1(X ,Z),G)→ 0

For two abelian group A and B, Tor(A,B) := TorZ1 (A,B) is the Tor functor andwe will define it later. For the moment, it is sufficient to know the following:


• Let 0→ F1h→ F0

k→ A→ 0 be a short exact sequence with F0 a free abeliangroup (it follows that F1 is free too). Then there is an exact sequence as fol-lows:

0→ Tor(A,B)→ F1⊗B h⊗1→ F0⊗B k⊗1→ A⊗B→ 0

One can use this property to define or calculate Tor(A,B). It is recommandedto an student to prove the bellow properties using only this one.

• Tor(A,B) and Tor(B,A) are isomorphic.• If either A or B is torsion free then Tor(A,B) = 0.• For n ∈ N we have

Tor(ZnZ

,A)∼= x ∈ A | nx = 0

and so Tor( ZnZ ,

ZmZ ) =

Zgcd(n,m)Z .

For further properties of Tor see [Mas67], p. 270.2. Universal coefficient theorem for cohomology: For a polyhedra X there is a nat-

ural short exact sequence

0→ Ext(Hq−1(X ,Z),G)→ Hq(X ,G)→ Hom(Hq(X ,Z),G)→ 0

For two abelian group A and B, Ext(A,B) is the Ext functor and we will define itlater. For the moment, it is sufficient to know the following:

• Let 0→ F1h→ F0

k→ A→ 0 be a short exact sequence with F0 a free abeliangroup (it follows that F1 is free too). Then there is an exact sequence as fol-lows:

0← Ext(A,B)← hom(F1,B)hom(h,1)← hom(F0,B)

hom(k,1)← hom(A,B)← 0

One can use this property to define or calculate Ext(A,B). It is recommandedto an student to prove the bellow properties using only this one.

• If A is a free abelian group then Ext(A,B) = 0 for any abelian group B.• If B is a divisible group then Ext(A,B) = 0 for any abelian group A.• For n ∈ N we have

Ext(Z

nZ,B)∼=

BnB

.

For further properties of Ext see [Mas67], p. 313.Note that Hq(X ,G)→ Hom(Hq(X ,Z),G) gives us a natural pairing

Hq(X ,G)×Hq(X ,Z)→ G, (α,β ) 7→∫

β

α. (2.1)

3. For a triple Z ⊂ Y ⊂ X of polyhedras we have the long exact sequence:

· · · → Hq(Y,Z;Z)→ Hq(X ,Z;Z)→ Hq(X ,Y ;Z)→ Hq−1(Y,Z;Z)→ ·· ·

2.3 Some consequences of the axioms 13

4. For X and Y two polyhedra we have a cross poduct maps

H p(X ,G1)×Hq(Y,G2)→ H p+q(X×Y,G1⊗G2), (ω1,ω2) 7→ ω1×ω2

For a list of its properties see [Mas67], 174.5. Kuneth theorem for homology: Let X and Y be two polyhedra. Then we have a

natural exact sequence

0→⊕

i+ j=q

Hi(X ,G)⊗G H j(Y,G)→Hq(X×Y,G)→⊕

i+ j=q−1

Tor(Hi(X ,G),H j(Y,G))→ 0.

6. Kuneth theorem for cohomology: Let X and Y be two polyhedra. Let us assumethat all the cohomologies of X with coefficients in Z are finitely generated andat least one of the two spaces X and Y has all cohomology groups torsion free.Then we have a canonical isomorphism⊕

i+ j=q

H i(X ,Z)⊗H j(Y,Z)∼= Hq(X×Y,Z)

given by the cross product. see [Mas67], p. 196.7. For X ,Y as in the previous item and δ1 ∈Hi(X ,Z),δ2 ∈H j(Y,Z), ω1 ∈H i(X ,Z), ω2 ∈

H j(X ,Z) we have ∫δ1⊗δ2

ω1⊗ω2 =∫

δ1

ω1

∫δ2

ω2

8. There are natural cup and cap products:

H p(X ,G1)×Hq(X ,G2)→ H p+q(X ,G1⊗Z G2), (α,β ) 7→ α ∪β

H p(X ,G1)×Hq(X ,G2)→ Hq−p(X ,G1⊗Z G2),(α,β ) 7→ α ∩β

For some properties which ∪ and ∩ satisfy see [Mas67], p. 329, for instance wehave

α ∩ (β ∩ γ) = (α ∪β )∩ γ, α ∈ H p(X ,G1), β ∈ Hq(X ,G2), γ ∈ Hr(X ,G3).

The cup product is defined using the cross product. Let d : X → X ×X , d(x) =(x,x) be the diagonal map. We define

ω1∪ω2 = d∗(ω1×ω2).

9. The cap product for p = q, G1 = G, G2 = Z and X a connected space generalizesthe integration map (2.1):

Hq(X ,G1)×Hq(X ,G2)→ G1⊗G2, (α,β ) 7→∫

β

α := α ∩β . (2.2)

10. Top (co)homology: Let X be a compact connected oriented manifold of dimen-sion n. We have Hn(X ,Z) ∼= Z, Hn(X ,Z) ∼= Z. The choice of a generator of


Hn(X ,Z) or Hn(X ,Z) corresponds to the choice of an orientation and so wesometimes refer to it as a choice of an orientation for X . We denote by [X ] agenerator of Hn(X ,Z) and write∫

Xα :=

∫[X ]

α, α ∈ Hn(X ,Z).

Let Y be a compact connected oriented manifold of dimension m. For a C∞ mapf : X → Y we have the map f∗ : Hn(X ;Z)→ Hn(Y ;Z). If there is no danger ofconfusion then the image of [X ] in Hn(Y ;Z) is denoted again by [X ]. In manycases X is a submanifold of Y and f is the inclusion.

11. Intersection map: Let X be an oriented manifold. There is a natural intersectionmap

Hp(X ,Z)×Hq(X ,Z)→ Hp+q−n(X ,Z),(α,β ) 7→ α ·β

If X is connected for q = n− p this gives us

Hp(X ,Z)×Hn−p(X ,Z)→ Z (2.3)

12. Poincare duality theorem: Let X be a compact oriented manifold of dimensionn. The intersection map (2.3) is unimodular. This means that any linear functionHn−q(X ,Z)→ Z is expressible as intersection with some element in Hq(X ,Z)and any class in Hq(X ,Z) having intersection number zero with all classes inHn−q(X ,Z) is a torsion class. This is equivalent to say that

P : Hq(X ,Z)→ Hn−q(X ,Z), α 7→ α ∩ [X ]

is an isomorphism. For α ∈Hq(X ,Z) we say that α and P(α) are Poincare duals.We have the equality∫

P(α)ω =

∫X

ω ∪α, ω ∈ Hn−q(X ,Z), α ∈ Hq(X ,Z)

By Poincare duality the intersection map in homology is dual to cup product mapand in particular (2.3) is dual to

Hn−q(X ,Z)×Hq(X ,Z)→ Z, (ω1,ω2) 7→∫

Xω1∧ω2.

The first and main example of homology theory with Z-coefficients is the singularhomology

Hsingk (X ,Z), k = 0,1,2, . . .

see for instance [Mas67]. For the interpretation of integration we will need this inter-pretation of homology theory. For cohomology theory there are in fact different con-structions. First, it was constructed singular cohomology and then it appeared Cechcohomology of constant sheaves (see [BT82]). The introduction of the de Rhamcohomology was a significant step toward the formulation of Hodge theory.

2.5 Intersection map 15

2.4 Leray-Thom-Gysin isomorphism

Let us be given a closed oriented submanifold N of real codimension c in an orientedmanifold M. One can define a map

τ : Hq−c(N,Z)→ Hq(M,M\N,Z) (2.4)

for any q, with the convention that Hq(N) = 0 for q < 0, in the following way: Letus be given a cycle δ in Hq−c(N). Its image by this τ is obtained by thickening acycle representing δ , each point of it growing into a closed c-disk transverse to N inM (see for instance [Che91] p. 392).

Theorem 2 (Leray-Thom-Gysin ismorphism) The map (2.4) is an isomorphism.

Proof. Recall that a tubular neighborhood of N in M is a C∞ embedding f : E→M,where E is the normal bundle of N in M, such that

1. f restricted to the zero section of E induces the identity map in N.2. f (E) is an open neighborhood of N in M.

We know that a tubular neighborhood of N in M exists ([Hir76], Theorem 5.2). Nowusing Excision property, it is enough to prove the theorem for a M a vector bundleand N its zero section.

Let M and N as above. Writing the long exact sequence of the pair (M,M\N)and using (2.4) we obtain:

· · · → Hq(M,Z) τ→ Hq−c(N,Z) σ→ Hq−1(M\N,Z) i→ Hq−1(M,Z)→ ··· (2.5)

The map τ is the intersection with N. The map σ is the composition of the boundaryoperator with (2.4).

2.5 Intersection map

Let us be given a closed submanifold N of real codimension c in a manifold M.Using Leray-Thom-Gysin ismorphisim we defined the intersection map

Hq(M)→ Hq−c(N).

If M is another submanifold of M which intersects N transversely then it is left tothe reader the construction of the relative intersection map

Hq(M,N)→ Hq−c(M,N∩ M). (2.6)

Exercises1. Using the Eilenberg-Steenrod axioms, calculate the homology and cohomology groups of


• the n-dimensional sphere Sn,• the projective spaces RPn(n),CP(n).

2. List all the axioms of a cohomology theory.3. Show that the Milnor additivity axiom for finite disjoint union of topological space follows

from the axioms 1 till 7.4. Let us be given a homology theory Hq(X ,Z) with Z-coefficients. Does Hom(Hq(X ,Z),Z) is a

cohomology theory? If no, which axiom fails?5. Try to prove some of the consequences 1 till 7 of homology theory by yourself. For this purpose

you can consult [ES52].6. Show that

∂n ∂n+1 = 0.

and that δ : Hn(X ,A,G)→ Hn−1(A,G) is well-defined.7. Prove all the properties of Tor mentioned in this text using just the first property in the list.8. Construct the relative intersection map (2.6).9. Show that any point in Rn is a deformation retract of Rn.

Chapter 3Lefschetz theorems

Here’s to LefschetzWho’s as argumentative as hell,When he’s at last beneath the sodThen he’ll start to heckle God1

As I see it at last it was my lot to plant the harpoon of algebraic topologyinto the body of the whale of algebraic geomery, Solomon Lefschetz in[Lef68].

In 1924 Lefschetz published his treatise on the topology of algebraic varieties. Whenit was written knowledge of topology was still primitive and Lefschetz ”made usemost uncritically of early topology a la Poincare and even of his own later develop-ments” (see [Lef68]). Later, Lefschetz theorems were proved using harmonic formsor Morse theory or sheaf theory and spectral sequences. But non of these very ele-gant methods yields Lefschetz’s full geometric insight. Two temtations to give pre-cise proofs for Lefschetz theorems are due to A. Wallace 1958 and K. Lamotke1981, see [Lam81]. In this chapter we use the later source and we present Lef-schetz’s theorems on hyperplane sections. Unfortunately, up to the time of writingthe present text there is no topological proof for the so called ”hard Lefschetz the-orem”. In this chapter if the coefficient ring of the homology or cohomology is notmentioned then it is supposed to be the ring of integers Z.

3.1 Main Theorem

Let X be a smooth projective variety of dimension n. By definition X is embedded insome projective space PN and it is the zero set of a finite collection of homogeneouspolynomials. Let also Y,Z be two smooth codimension one hyperplane sections of

1 Hodge, William Solomon Lefschetz. Contemp. Math., 58, Amer. Math. Soc., Providence, RI,1986.

17

18 3 Lefschetz theorems

X . 2. We assume that Y and Z intersect each other transversely at X ′ :=Y ∩Z. We donot assume that Y and Z are hyperplane sections associated to the same embeddingX ⊂ PN . However, we assume that for some k ∈ N the divisor Y − kZ is principal,i.e. for some meromorphic function f on X , Y is the zero divisor of order one of fand Z is the pole divisor of order k of f . We will need the following Theorem:

Theorem 3 We have

Hq(X\Z,Y\X ′) =

0 if q 6= nFree Z-module of finite rank i f q = n , n := dim(X).

where n := dim(X).

Later we will see how to calculate the rank of Hq(X\Z,Y\X ′) by means of algebraicmethods. A proof and further generalizations of Theorem 3 will be presented inChapter 4 in which we develope the Picard-Lefschetz theory. A reader who wantsto get a feeling of this theorem using a basic Algebraic Topology is invited to thinkon Exercise 3 at the end of this chapter.

3.2 Some consequences of Theorem 3

Let us state some consequence of Theorem 3. Let

U := X\Z, V := Y\X ′.

U is an affine variety and V is an affine subvariety of codimension one.

Corollary 1 Let X be smooth projective space and Z be a smooth hyperplane sec-tion of X. We have

Hq(U ;Z) = 0, for q > dimU, U := X\Z.

The homology group Hn(U ;Z) is free of finite rank.

Proof. We write the long exact sequence of the pair V ⊂U and we get a five termexact sequence and the isomorphisms

Hq(U)∼= Hq(V ), q 6= n,n−1. (3.1)

Now our result follows by induction on n. For n = 1 it it trivial because U is notcompact. Let us assume that it is true for n. We know that X ′ is also a smoothhyperplane section of Y and dim(Y ) = n−1. Therefore, the corollary in dimensionn−1 implies the corollary in dimension n.

Applying the first part of the corollary to V we conclude that Hn(V ) = 0. The fiveterm exact sequence mentioned above reduces to the four term exact sequence:

2 A modern terminology is to say that Y,Z are two smooth very ample divisors of X

3.3 Lefschetz theorem on hyperplane sections 19

0→ Hn(U)→ Hn(U,V )→ Hn−1(V )→ Hn−1(U)→ 0. (3.2)

This implies that Hn(U) is a subset of Hn(U,V ) and so by theorem 3 it is free.

Remark 1 In the exact sequence (3.2), Hn(U),Hn(U,V ) and Hn−1(V ) are free Z-modules and Hn−1(U) may have torsions. Later we will see that for complete inter-section affine varieties Hn−1(U) = 0 and so (3.2) reduces to three terms which areall free Z-modules.

Corollary 2 The intersection mappings

Hn+q(X)→ Hn+q−2(Y ), x 7→ [Y ] · x, q = 2,3, . . .

are isomorphism

Proof. We write the long exact sequence of the pair X\Y ⊂ X and use the Leray-Thom-Gysin isomorphism and obtain

· · · → Hn+q(X\Y )→ Hn+q(X)→ Hn+q−2(Y )→ Hn+q−1(X\Y )→ ··· (3.3)

Now, our statement follows from Corollary (1).

Later we will use the dual of the intersection mapping in Corollary 2

Hn+q−2(Y )→ Hn+q(X).

This is an isomorphim of Hodge structures of weight (1,1), that is, if we use thede Rham cohomology instead of singular cohomology then it sends (p,q)-forms to(p+1,q+1)-forms. Despite the fact that so far we have not discussed these topics,it is possible to derive a consequence of the Hodge conjecture:

Conjecture 1 Let us assume that n + q is even. For any algebraic cycle Z =

∑ri=1 ni[Zi], ni ∈Z,dim(Zi) =

n+q2 −1 in Y , there is some m∈N, such that mZ is ho-

molog to another algebraic cycle Z in Y with the property that Z is the intersectionof an algebraic cycle of dimension n+q

2 in X with Y .

The above conjecture follows from the Hodge conjecture for the (n+ q)-th coho-mology of X . At this point, we cannot expect that Z itself is obtained by intersectionof some algebraic cycle with X .

3.3 Lefschetz theorem on hyperplane sections

In this section we are going to prove the following theorem:

Theorem 4 Let X be a smooth projective variety of dimension n and Y ⊂ X be asmooth hyperplane section. Then

Hq(X ,Y ;Z) = 0, 0≤ q≤ n−1,


In other words, the inclusion Y → X induces isomorphisims of the homology groupsin all dimensions stricktly less than n−1 and a surjective map in Hn−1.

Proof. The proof is essentially based on the long exact sequence of triples

V ⊂U ⊂ X ,

V ⊂ Y ⊂ X ,

the Leray-Thom-Gysin isomorphisms for the pairs (V,Y ) and (U,X) and Theorem3.

The long exact sequence of the first triple together with Theorem 3 gives usisomorphisms

Hq(X ,V )∼= Hq(X ,U), q 6= n,n+1

induced by inclusions, and the five term exact sequence:

0→ Hn+1(X ,V )→ Hn+1(X ,U)→ Hn(U,V )→ Hn(X ,V )→ Hn(X ,U)→ 0

Now, we use Thom-Leray-Gysin isomorphism for the pair (U,X) and obtainHq(X ,U)∼= Hq−2(Z). Combining these two isomorphisms we get

Hq(X ,V )∼= Hq−2(Z),q 6= n,n+1 (3.4)

We write the long exact sequence of the second triple and (X ′,Z) in the followingway:

· · · → Hq(Y,V ) → Hq(X ,V )→ Hq(X ,Y ) → Hq−1(Y,V )→ ·· ·↓ ↓ ↓ ↓

· · · → Hq−2(X ′)→ Hq−2(Z) → Hq−2(Z,X ′)→ Hq−3(X ′) → ·· ·

Some words must be said about the down arrows: The first and fourth down arrowsare the Leray-Thom-Gysin isomorphism for the pair (V,Y ). The second down arrowis the isomorphism (3.4). The third morphism is obtained by intersecting the cycleswith Z. It is left to the reader to show that the above diagram commutes and so byfive lemma, there is an isomorphism

Hq(X ,Y )∼= Hq−2(Z,X ′), q 6= n,n+1,n+2.

Now the theorem is proved by induction on n.

Again, we can introduce a consequence of the Hodge conjecture, even though, wehave not stated it so far.

Conjecture 2 Let q be an even natural number. For any algebraic cycle Z =

∑ri=1 ni[Zi], ni ∈ Z,dim(Zi) =

q2 , q < n−1 in X there are m ∈ N and an algebraic

cycle Z in Y such that mZ is homolog to Z.

This follows from the Hodge conjecture for the q-the cohomology of Y .

3.4 Topology of complete intersections 21

3.4 Topology of complete intersections

We start this section by studying the topology of projective spaces:

Proposition 1 For i ∈ N0 we have

Hi(Pn) =

0 if i is odd

〈[P i2 ]〉 if i is even

where P i2 is any linear projective subspace of Pn.

Proof. We take a linear suspace Pn−1 ⊂ Pn, write the long exact sequence ofPn\Pn−1 ⊂ Pn and use the equalities

Hi(Pn\Pn−1) = 0, i 6= 0.

Note that Pn\Pn−1 ∼= Cn can be retracted to a point. We conclude that the intersec-tion with Pn−1 mappings

Hi(Pn)→ Hi−2(Pn−2)

are isomorphism and H1(Pn) = 0. Now the proposition follows by induction on n.

Iterate the sequence X ⊃ Y ⊃ X ′ to

X = X0 ⊃ X1 = Y ⊃ X2 = X ′ ⊃ X3 ⊃ ·· · ⊃ Xn ⊃ Xn+1 = /0 (3.5)

so that Xq is a smooth hyperplane section of Xq−1 and hence dimXq = n−q.

Proposition 2 We have

Hq(X ,Xi) = 0, q≤ dim(Xi) = n− i

Proof. From theorem 4 it follows that

Hq(Xi,Xi+1) = 0, i≤ dim(Xi+1) = n− (i+1) (3.6)

Now, we prove the proposition by induction on i. For i = 1 it is Theorem 4. As-sume that Hq(X ,Xi) = 0, q ≤ dim(Xi) = n− i. We write the long exact sequenceof the triple Xi+1 ⊂ Xi ⊂ X and use the induction hypothesis and (3.6) to obtain theproposition for i+1.

Proposition 3 Let X ⊂ Pn+1 be a smooth hypersurface of dimension n. Then

Hq(Pn+1,X) = 0, q≤ n

In particular, we have

Hq(X) =

0 if q is oddZ if q is even for q≤ n−1


Proof. We can use the Veronese embedding of Pn+1 such that X becomes a smoothhyperplane section of Pn+1.

We have seen that a hypersurface in PN is a smooth hyperplane section.

Definition 1 A smooth projective variety X ⊂ PN of dimension n is called a com-plete intersection if it is given by N−n homogeneous polynomials f1, f2, . . . , fN−n ∈C[x0,x1, . . . ,xN ] such that the matrix

[∂ fi

∂x j]i=1,2,...,N−n, j=0,1,...,N

has the maximum rank N−n for all points x ∈ PN .

Proposition 4 If X ⊂ PN is a complete intersection of dimension n. We have

Hq(PN ,X) = 0, q≤ n

In particular,

Hq(X) =

0 if q is oddZ if q is even for q≤ n−1

Proof. There is a sequence of projective varieties

PN = X0 ⊃ X1 ⊃ X2 ⊃ ·· · ⊃ XN−n = X (3.7)

such that Xi is a hyperplane section of Xi−1 for i = 1,2 . . . ,N − n. In fact for i =1,2, . . . ,N − n, Xi is induced by the zero set of f1, f2, . . . , fi. Now, our assertionfollows from Proposition 2.

Let X be a complete intersection in PN and Z be a smooth hyperplane section corre-sponding to X ⊂ PN . We call U := X\Z an affine complete intersection.

Proposition 5 Let U be an affine complete intersection variety as above. We have

1. For q≤ n−1, Hq(U) = 0 and so by Corollary 1 all Hq(U) is are zero except forq = n.

2. The intersection mappings

Hn−q(X)→ Hn−q−2(Y ), x 7→ [Y ] · x, q = 1,2, . . .

are isomorphism.3. The four term exact sequence (3.2) reduces to three term exact sequence of free

finitely generated Z-modules.

Proof. Let H be the hyperplane in PN such that Z =H∩X . We have a sequence U =UN−n ⊂UN−n−1 ⊂ ·· · ⊂U1 ⊂U0 = CN which is obtained from (3.7) by removingH from Xi’s. We use the the isomorphism in (3.1) to each consecutive inclusionsof Ui’s, we compose them and we get the isomorphism Hq(U) ∼= Hq(CN) for q 6=

3.5 Some remarks on hard Lefschetz theorem 23

n,n+ 1, · · · ,N. In particular for 0 < q ≤ n− 1 we have Hq(U) = 0. The secondpart follows from the long exact sequence of (X ,U) and the Leray-Thom-Gysinisomorphism. This is similar to the proof of Corollary (2). The thir part followsfrom the first part.

3.5 Some remarks on hard Lefschetz theorem

Let us consider the sequence (3.5).

Theorem 5 (Hard Lefschetz theorem) For every q = 1,2, . . . ,n the intersectionwith Xq

Hn+q(X ,Q)→ Hn−q(X ,Q), x 7→ x · [Xq]

is an isomorphism.

First of all note that the Hard Lefschetz theorem is stated with rational coefficientsand hence it is valid for any field of characteristic zero. It is false with Z-coefficientsfor trivial reasons. Take for instance X a Riemann surface and Y a hyperplane sectionwhich consists of m points with m > 1. We use canonical isomorphisms H2(X ;Z)∼=Z, H0(X ;Z)∼=Z and obtain the map Z→Z, x 7→mx which is not surjective. For themoment I do not have any counterexample with the non injective map in Theorem 5with Z coefficients. Since Theorem 5 is valid with Q coefficients, it is natural to looka counterexample which is a two dimensional projective variety X with dim(X) = 2and a torsion α ∈ H3(X ;Z) with zero intersection with Y .

There is no topological proof of hard Lefschetz theorem in the literature. This isin some sense natural because it is a theorem with coefficients in Q. The only preciseproof availble in the literature is due to Hodge by means of harmonic integrals. Thereis also an arithmetic version due to Deligne. We will give a precise proof of Theorem5 after introducing the de Rham cohomology of algebraic varieteis.

First we remark that it is enough to prove Theorem 5 for q = 1. For an arbitraryq the theorem follows from the case q = 1, the diagram

Hn+q(X ;Q)·[Xq]→ Hn−q(X ;Q)

↓ ·[X1] ↑

H(n−1)+q−1(X1;Q)·[Xq+1]→ H(n−1)−(q−1)(X1;Q)

and induction on q, where the up arrow is induced by the inclsion.We write the long exact sequence of the pairs X\Y ⊂ X and Y ⊂ X and we have


Hn(X)↓

Hn(X ,Y )↓

0→ Hn+1(X)→ Hn−1(Y ) → Hn(X\Y )→ Hn(X)→ Hn−2(Y )→ 0↓

Hn−1(X)↓0

(3.8)

Hard-Lefschetz theorem and following statements are equivalent:

1.

Hn−1(Y ;Q)= Im(Hn+1(X ;Q)→Hn−1(Y ;Q))⊕ker(Hn−1(Y ;Q)→Hn−1(X ;Q)).

2.

Im(Hn+1(X ;Q)→ Hn−1(Y ;Q))∩ker(Hn−1(Y ;Q)→ Hn−1(X ;Q)) = 0.

Note that by Poincare duality Hn+1(X ;Q) and Hn−1(X ;Q) have the same dimension.For more equivalent versions of hard Lefschetz theorem see [Lam81].

Proposition 6 Let Y be a smooth hyperplane section of a projective variety X. Theintersection with [Y ] induces an isomorphism

Hn+1(X)tors ∼= Hn−1(Y )tors

In particular if dim(X) = 2 then H3(X) is torsion free.

Proof. This follows from the horizontal line of the diagram (3.8) and the fact thatHn(X\Y ) has no torsion.

Remark 2 Let n+q be even. If the Hodge conjecture is true for cycles in Hn+q(X ,Q)then it is true for cycles in Hn−q(X ,Q) but not necessarily vice versa. It also impliesthat any algebraic cycle in Hn−q(X ,Q) is homologous in Y to an intersection of analgebraic cycle in Hn+q(X ,Q) with Y .

3.6 Lefschetz decomposition

Let us consider the sequence (3.5). We assume that all hyperplane sections in this se-quence are associated to the same embedding X ⊂ PN . Each Xq gives us a homologyclass

[Xq] ∈ H2n−2q(X ;Z)

It is left to the reader to check the equalities:

3.7 Lefschetz theorems in cohomology 25

[Xq] · [Xq′ ] = [Xq+q′ ]. (3.9)

An element x ∈ Hn+q(X), q = 0,1,2, . . . ,n is called primitive if

[Xq+1] · x = 0

Recall that by hard Lefschetz theorem if [Xq] · x = 0 then x = 0. Let us define

Hn+q(X ;Q)prim := x ∈ Hn+q(X ;Q) | [Xq+1] · x = 0.

Theorem 6 Every element x ∈ Hn+q(X) can be written uniquely as

x = x0 +[X1] · x1 +[X2] · x2 + · · · (3.10)

and every element x ∈ Hn−q(X) as

x = [Xq] · x0 +[Xq+1] · x1 +[Xq+2] · x2 + · · · (3.11)

where xi ∈ Hn+q+2i(X) are primitive and q≥ 0.

Proof. We use the fact that intersection with [Xq] induces an isomorphism Hn+q(X)→Hn−q(X) which transforms (3.10) into (3.11). Therefore, it is enough to prove (3.10).For this we use decreasing induction on q starting from q = n,n− 1, where everyelement is primitive. For the induction step from n+q+2 to n+q it suffices to showthat every x ∈ Hn+q(X) can be written uniquely as

x = x0 +[X1] · y, x0 primitive (3.12)

because the induction hypothesis applied to y then yields the decomposition (3.10).In order to prove (3.12) consider [Xq+1] · x according to Theorem 5 there is exactlyone y ∈ Hn+q+2(X) with [Xq+2] · y = [Xq+1] · x and thus

x0 := x− [X1] · y

is primitive. In order to show the uniqueness assume that 0 = x0 + [X1] · y with x0primitive. Then [Xq+1] · x0 = 0, hence [Xq+2] · y = 0 and Theorem 5 implies y = 0,and hence x0 = 0.

3.7 Lefschetz theorems in cohomology

Let u∈H2(X ,Z) denote the Poincare dual of of the algebraic cycle [Y ]∈H2n−2(X ,Z),i.e.

u∩ [X ] = [Y ].

Let also


uq = u∪u∪·· ·∪u︸︷︷︸q-times

∈ H2q(X ,Z).

which is the Poincare dual of Xq. We have

uq∩ x = [Xq] · x

and so Theorem 5 says that for every q = 1,2, . . . ,n the cap product with the q-thpower uq

Hn+q(X ,Q)→ Hn−q(X ,Q), α 7→ uq∩α

is an isomorphism. Poincare dual to the Hard Lefschetz theorem is the following:For every q = 1,2, . . . ,n the cup product with the q-th power of u ∈ H2(X ,Z) is anisomorphism

Lq : Hn−q(X ,Q)→ Hn+q(X ,Q), α 7→ uq∪α

Define the primitive cohomology in the follwong way:

Hn−q(X ,Q)prim := ker(Lq+1 : Hn−q(X ,Q)→ Hn+q+2(X ,Q)

)The Poincare dual to the decomposition theorem 6 is:

Theorem 7 (Lefschetz decomposition) The natural map

⊕qLq :⊕qHm−2q(X ,Q)prim→ Hm(X ,Q)

is an isomorphism.

Exercises1. Prove the equality (3.9).2. Let Y be a hypersurface of dimension n−1. Does Hn−1(Y ) has torsion? If the answer is yes then

by diagram 3.8 and Proposition 6 we have a hypersurface X of dimension n such that Hn+1(X)has torsion and so the surjectivity of hard Lefschetz theorem with Z coefficients fails.

3. Let m1,m2, . . . ,mn+1 be positive integers bigger than one. Consider the following affine varietyin Cn+1:

U : xm11 + xm2

2 + · · ·+ xmn+1n+1 = 1.

Let

Γ := (t1, t2, . . . , tn+1) ∈ Rn+1 | ti ≥ 0,n+1

∑i=1

ti = 1.

be the standard n-simplex, Gmi := 0,1,2, . . . ,mi− 2 and let ζmi be a mi-th primitive root ofunity. For β ∈ G1×G2×·· ·×Gn+1 and a ∈ 0,1n+1 let

Γβ ,a : Γ →U

Γβ ,a(t) = (t1

m11 ζ

β1+a1m1

, t1

m22 ζ

β2+a2m2

, . . . , t1

mn+1n+1 ζ

βn+1+an+1mn+1 ),

where for a positive number r and a natural number s, r1s is the unique positive s-th root of r.

Prove thatδβ := ∑

a(−1)∑

n+1i=1 (1−ai)Γβ ,a

3.7 Lefschetz theorems in cohomology 27

induces a non-zero element in Hn(U,Z). In fact, they from a basis of the Z-module Hn(U,Z)as it is expected in Theorem 3.

4. For β ,β ′ ∈ I, we have〈δβ ,δβ ′ 〉= (−1)n〈δβ ′ ,δβ 〉,

and if β 6= β ′ and βk ≤ β ′k ≤ βk +1, k = 1,2, . . .n+1, β 6= β ′ then

〈δβ ,δβ ′ 〉= (−1)n(n+1)

2 (−1)Σn+1k=1 β ′k−βk

We have also〈δβ ,δβ 〉= (−1)

n(n+1)2 (1+(−1)n), β ∈ I.

In the remaining cases, except those arising from the previous ones by a permutation, we have〈δβ ,δβ ′ 〉= 0. See [Mov11] page 111 and [AGZV88] page 66.

5. Discuss in more details the content of Exercises 3, 4 in the one dimensional case n = 1. In thiscase U is a Riemann surface with some removed points. How many? Determine the genus ofU . Can you determine a Z-linear combination of δβ ’s which are homolog to cycles around theremoved points? Can you do it at least for the particular cases m1 = m2 equals to 3,4,5?

6. Using Proposition 1 and Propoisition (4), we know that for a hypersurface X ⊂ Pn+1 and q < nan even number Hq(X) ∼= Hq(Pn+1) and Hq(Pn+1) is generated by the homology class of alinear projective subspace P

q2 ⊂ Pn+1. However, there might be no such P

q2 inside X . This is

the first indication that integral Hodge conjecture is wrong. A canonical element [Z] of Hq(X)

is obtained by a transverse intersection of a linear space Pq2 +1 ⊂ Pn+1 with X . Compute [Z] in

terms of Pq2 .

7. Prove or disprove the first part of Proposition 5 in the case of case of varieties which arenot affine complete intersections. In other words, can you find a smooth affine variety U ofdimension n such that for some 0 < q < n, Hq(U) is non-zero.

8. Prove that the dimension of the n-th primitive cohomology of a hypersurface X of degree d inPn+1 is given by

dimHn(X ,Q)prim = (d−1)n+1− (d−1)n +(d−1)n−1−·· ·+(−1)n(d−1)

Can you also give a formula in the cases of X a complete intersection of type (d1,d2, . . . ,ds).9. Write a report on Hard Lefschetz theorem. Use Mathscinet with the keyword Hard Lefschetz

theorem and write a report based on the reviw of the articles. You may also use google.10. Describe the topology of the algebraic curve

xy(x+ y−1) =1

100

over R and C and compare them.

Chapter 4Picard-Lefschetz Theory

In 1924 Solomon Lefschetz published his treatise [Lef50] on the topology of al-gebraic varieties. He considered a pencil of hyperplanes in general position withrespect to that variety in order to study its topology. In this way he founded the socalled Picard-Lefschetz theory. In fact, the idea of taking a pencil goes back furtherto Poincare and Picard. In the this chapter we introduce basic concepts of Picard-Lefschetz theory and we prove Theorem 3 in Chapter 3. Concrete examples andcomputations in this theory will be presented in Chapter 5, and hence, it might beuseful to read both the present chapter and Chapter 5 simultaneously.

4.1 Ehresmann’s fibration theorem

Many of the Lefschetz intuitive arguments are made precise by appearance of a fiberbundle which is the basic stone of the so called Picard-Lefschetz theory. Despite thefact that the theorem bellow is proved two decades after Lefschetz treatise, it is thestarting point of Picard-Lefschetz theory.

Theorem 8 (Ehresmann’s Fibration Theorem [Ehr47]). Let f : Y → B be a propersubmersion between the C∞ manifolds Y and B. Then f fibers Y locally triviallyi.e., for every point b ∈ B there is a neighborhood U of b and a C∞-diffeomorphismφ : U× f−1(b)→ f−1(U) such that

f φ = π1 = the first projection.

Moreover if N ⊂ Y is a closed submanifold such that f |N is still a submersion thenf fibers Y locally trivially over N i.e., the diffeomorphism φ above can be chosen tocarry U× ( f−1(b)∩N) onto f−1(U)∩N.

29

30 4 Picard-Lefschetz Theory

The map φ is called the fiber bundle trivialization map. Ehresmann’s theorem canbe rewritten for manifolds with boundary1 and also for stratified analytic sets. In thelast case the result is known as the Thom-Mather theorem.

In Theorem 8 assume that f is not a submersion. Because of this, we define C′ tobe the union of critical values of f and critical values of f |N , and C be the closureof C′ in B. By a critical point of the map f we mean the point in which f is notsubmersion. Now, we can apply the theorem to the function

f : Y\ f−1(C)→ B\C = B′

For any set K ⊂ B, we use the following notations

YK = f−1(K), Y ′K = YK ∩N, LK = YK\Y ′K

and for any point c ∈ B, by Yc we mean the set Yc. By f : (Y,N)→ B we mean themap f together with f |N . It is called the critical fiber bundle map.

Definition 2 Let A⊂ R⊂ S be topological spaces. R is called a strong deformationretract of S over A if there is a continuous map r : [0,1]×S→ S such that

1. r(0, ·) = id,2. ∀x ∈ S, y ∈ R, r(1,x) ∈ R ,r(1,y) = y,3. ∀t ∈ [0,1], x ∈ A, r(t,x) = x.

Here, r is called the contraction map. In a similar way we can do this definition forthe pairs of spaces (R1,R2)⊂ (S1,S2), where R2 ⊂ R1 and R1,S2 ⊂ S1.

We use the following theorem to define a generalized vanishing cycle and also tofind relations between the homology groups of Y\N and the generic fiber Lc of f .

Theorem 9 Let f : Y → B and C′ as before, A ⊂ R ⊂ S ⊂ B and S∩C be a subsetof the interior of A in S, then every retraction from S to R over A can be lifted to aretraction from LS to LR over LA.

Proof. According to Ehresmann’s fibration theorem f : LS\C→ S\C is a C∞ locallytrivial fiber bundle. The homotopy covering theorem, [Ste51], §11.3, implies thatthe contraction of S\C to R\C over A\C can be lifted so that LR\C becomes a strongdeformation retract of LS\C over LA\C. Since C∩S is a subset of the interior of A inS, the singular fibers can be filled in such a way that LR is a deformation retract ofLS over LA.

1 In fact one needs this version of Ehresmann’s fibration theorem in the local Picard-Lefschetztheory developed for singularities, see [AGZV88] and §4.4.

4.2 Monodromy 31

4.2 Monodromy

Let λ be a path in B′ = B\C with the initial and end points b0 and b1. In the sequelby λ we will mean both the path λ : [0,1]→ B and the image of λ ; the meaningbeing clear from the text.

Proposition 7 There is an isotopy

H : Lb0 × [0,1]→ Lλ

such that for all x ∈ Lb0 , t ∈ [0,1] and y ∈ N

H(x,0) = x, H(x, t) ∈ Lλ (t), H(y, t) ∈ N. (4.1)

For every t ∈ [0,1] the map ht = H(., t) is a homeomorphism between Lb0 and Lλ (t).The different choices of H and paths homotopic to λ would give the class of homo-topic maps

hλ : Lb0 → Lb1,

where hλ = H(·,1).

Proof. The interval [0,1] is compact and the local trivializations of Lλ can be fittedtogether along γ to yield an isotopy H.

The class hλ : Lb0 → Lb1 defines the maps

hλ : π∗(Lb0)→ π∗(Lb1),

hλ : H∗(Lb0)→ H∗(Lb1).

In what follows we will consider the homology class of cycles, but many of thearguments can be rewritten for their homotopy class.

Definition 3 For any regular value b of f , we can define

h : π1(B′,b)×H∗(Lb)→ H∗(Lb),

h(λ , ·) = hλ (·).

The image of π1(B′,b) in Aut(H∗(Lb)) is called the monodromy group and its actionh on H∗(Lb) is called the action of monodromy on the homology groups of Lb.

Following the article [Che96], we give the generalized definition of a vanishingcycle.

Definition 4 Let K be a subset of B and b be a point in K\C. Any relative k-cycle∆ ∈ Hk(LK ,Lb) of LK with a boundary in Lb is called a k-thimble above (K,b) andits boundary δ ∈ Hk−1(Lb) is called a vanishing (k−1)-cycle above K.


bci b

i

~λ i

D i

Fig. 4.1

4.3 Vanishing cycles

What we studied in the previous section is developed first in the complex context.Let Y be a complex compact manifold, N be a submanifold of Y of codimensionone, B = P1 and f be a holomorphic function. The set C of critical values of f isfinite and so each point in C is isolated in P1. We write

C := c1,c2, . . . ,cs.

Let ci ∈ C, Di be a small disk around ci and λi be a path in B′ which connectsb ∈ B′ to bi ∈ ∂Di. Put λi the path λi plus the path which connects bi to ci in Di (seeFigure 4.1). Define the set K in the three ways as follows:

Ks =

λi s = 1λi∪Di s = 2λi∪∂Di s = 3

. (4.2)

We can now define the vanishing cycle in Lb above Ks for s = 1,2,3. Since K1 andK3 are subsets of K2, the vanishing cycle above K1 or K3 is also vanishing aboveK2. In the case of K1, we have the intuitional concept of a vanishing cycle. If ciis a critical point of f |N we can see that the vanishing cycle above K2 may notbe vanishing above K1, see Exercise 5. The case K3 gives us the vanishing cyclesobtained by a monodromy around ci. In this case we have the Wang isomorphism

v : Hk−1(Lb)→Hk(LK3 ,Lb)

see [Che91]. Roughly speaking, the image of the cycle α by v is the footprint ofα , taking the monodromy around ci. Let γi be the closed path which parametrizeK3, that is, γi starts from b, goes along λi until bi, turns around ci anti clockwise on∂Di and finally comes back to b along λi

−1. Let also hγi : Hk(Lbi)→ Hk(Lbi) be the

monodromy around the critical value ci. We have

∂ v = hλi − id,

where ∂ is the boundary operator. Therefore the cycle α is a vanishing cycle aboveK3 if and only if it is in the image of hλi − id. For more information about thegeneralized vanishing cycle the reader is referred to [Che96].

4.4 The case of isolated singularities 33

Remark 3 In general by vanishing along the path λi we mean vanishing above K2.

4.4 The case of isolated singularities

First, let us recall some definitions from local theory of vanishing cycles. For allmissing proofs the reader is referred to [AGZV88]. Let f : (Cn,0)→ (C,0) be aholomorphic function with an isolated critical point at 0 ∈Cn. We can take an smallclosed disc D with center 0 ∈ C and a closed ball B with center 0 ∈ Cn such that

f : ( f−1(D)∩B, f−1(D)∩∂B)→ D

is a C∞ fiber bundle over D\0. Here, fibers are real manifolds with boundarieswhich lie in ∂B. In what follows we consider the mentioned domain and image forf . The Milnor number of f is defined to be

dim(OCn,0

〈 d fdx1

, d fdx2

, · · · , d fdxn〉)

where (x1,x2, · · · ,xn) is a local coordinate system around 0 and OCn,0 is the ring ofholomorphic functions in a neighborhood of 0 in Cn.

Proposition 8 For b∈ ∂D the relative homology group Hk( f−1(D), f−1(b)) is zerofor k 6= n and it is a free Z-module of rank µ , where µ is the Milnor number of f . Abasis of Hn( f−1(D), f−1(b)) is given by hemispherical homology classes.

By definition a hemispherical homology class is the image of a generator of infinitecyclic group Hn(Bn,Sn−1)∼= Z, where

Sn−1t := (x1, · · · ,xn) ∈ Rn |∑x2

j = t, 0≤ t ≤ 1, Sn−1 := Sn−11

andBn := (x1, · · · ,xn) ∈ Rn |∑x2

j ≤ 1= ∪t∈[0,1]Sn−1t

under the homeomorphism induced by a continuous mapping of the closed n-ballBn into f−1(λ ) which sends the (n−1)-sphere Sn−1

t , to Lλ (t). Here λ : [0,1]→D isa straight path which connects 0 to b.

Definition 5 We denote the image of Bn in B by ∆ and call it a Lefschetz thimble.Its boundary is denoted by δ and it is called the Lefschetz vanishing cycle.

Let us consider the simplest case

f (x1,x2, · · · ,xn) = x21 + x2

2 + · · ·+ x2n

and assume that b = 1. The Milnor number of f is one and Hn( f−1(D), f−1(b)) isgenerated by the image of the generator of Hn(Bn,Sn−1) under the inclusion Rn ⊂Cn.


We are now ready to consider the global context of the previous section.

Proposition 9 Assume that c ∈ P1 is not a critical point of f restricted to N andf has only isolated critical points p1, p2, . . . , pk in Lc and these are all the criticalpoints of f : (Y,N)→ P1 within Yc. Let also K = K2. The following statements aretrue:

1. For all k 6= n we have Hk(LK ,Lb) = 0. This means that there is no (k− 1)-vanishing cycle along λi for k 6= n;

2. Hn(LK ,Lb) is freely generated by hemispherical homology classes. It is a freeZ-module of rank

µc :=k

∑i=1

µpi .

The proof is a modification of a similar argument in [Lam81], paragraph 5.4.1, andit is left to the reader. In the above example vanishing above K1 and K2 are the same.Also by the Picard-Lefschetz formula the reader can verify that three types of thedefinition of a vanishing cycle coincide.

The monodromy hi around the critical value ci is given by the Picard-Lefschetzformula

h(δ ) = δ +(−1)n(n+1)

2 〈δ ,δi〉δi, δ ∈ Hn−1(Lb)

where 〈·, ·〉 denotes the intersection number of two cycles in Lb.

4.5 Vanishing Cycles as Generators

Let c1,c2, . . . ,cs be a subset of the set of critical values of f : (Y,N)→ P1, andb ∈ P1\C. Consider a system of s paths λ1, . . . ,λs starting from b and ending atc1,c2, . . . ,cs, respectively, and such that:

1. each path λi has no self intersection points,2. two distinct path λi and λ j meet only at their common origin λi(0) = λ j(0) = b

(see Figure 4.2).

This system of paths is called a distinguished system of paths. The set of vanishingcycles along the paths λi, i= 1, . . . ,s is called a distinguished set of vanishing cyclesrelated to the critical points c1,c2, . . . ,cs.

Fix a point b∞ ∈ P1 which may be the critical value of f . Assume that Y is acompact complex manifold, f : (Y,N)→ P1 restricted to N has no critical values,except probably b∞, and f has only isolated critical points in Y\Yb∞

.

Theorem 10 The relative homology group Hk(LP1\b∞,Lb) is zero for k 6= n and itis a freely generated Z-module of rank r for k = n.

Proof. Our proof is a slight modification of arguments in [Lam81] Section 5.We consider our system of distinguished paths inside a large disk D+ so thatb∞ ∈ P1\D+, the point b is in the boundary of D+ and all critical values ci’s in

4.6 Lefschetz pencil 35

D+

b

c

c2

c1

λ

λc3

44

3

λ2

λ1

Fig. 4.2

C\b∞ are interior points of D+. Small disks Di with centers ci i = 1, · · · ,r arechosen so that they are mutually disjoint and contained in D+. Put

Ki = λi∪Di, K = ∪ri=1Ki

The pair (K,b) is a strong deformation retract of (D+,b) and (D+,b) is a strongdeformation retract of (P1\b∞,b). Therefore, by Theorem 9 (LK ,Lb) is a strongdeformation retract of (LP1\b∞,Lb). The set λ = ∪λi can be retract within itselfto the point b and so (LK ,Lb) and (LK ,Lλ

) have the same homotopy type. By theexcision theorem we conclude that

Hk(LD+ ,Lb)'r

∑i=1

Hk(LKi ,Lb)'r

∑i=1

Hk(LDi ,Lbi)

and the so Proposition 9 finishes the proof.

Corollary 3 Suppose that Hn−1(LP1\b∞) = 0 for some b∞ ∈ P1, which may be acritical value. Then a distinguished set of vanishing (n− 1)-cycles related to thecritical points in the set C\b∞= c1,c2, . . . ,cr generates Hn−1(Lb).

Proof. Write the long exact sequence of the pair (L(LP1\b∞,Lb):

. . .→ Hn(LP1\b∞)→ Hn(LP1\b∞,Lb)σ→ Hn−1(Lb)→ Hn−1(LP1\b∞)→ . . .

(4.3)Knowing this long exact sequence, the assertion follows from the hypothesis.

4.6 Lefschetz pencil

Let PN be the projective space of dimension n. The hyperplanes of PN are the pointsof the dual projective space PN and we use the notation


Hy ⊂ PN , y ∈ PN

Let X ⊂ PN be a smooth projective variety of dimension n. By definition X is aconnected complex manifold. The dual variety of X is defined to be:

X := y ∈ PN | Hy is not transverse to X

One can show that X is an irreducible variety of dimension at most N−1. Any lineG in PN gives us in PN a pencil of hyperplanes Htt∈G which is the collection ofall hyperplanes containing a projective subspace A of dimension N− 2 of PN . A iscalled the axis of the pencil.

Let G be a line in PN which intersects X transversely. If dim(X) < N− 1 thismeans that G does not intersects X and if dim(X) = N−1 this means that G inter-sects X transversely in smooth points of X . We define

Xt := X ∩Ht , t ∈ G, X ′ := X ∩A

One can prove that

1. A intersects X transversely and so X ′ is a smooth codimension two subvariety ofX .

2. For t ∈ G∩ X the hyperplane section Xt has a unique singularity which lies inXt\X ′ and in a local holomorphic coordinates is given by

x21 + x2

2 + · · ·+ x2n = 0.

We fix an isomorphism G→ P1 such that t = 0,∞ 6∈ G∩ X . Let L0 and L∞ be thelinear polynomials such that L0 = 0 = H0 and L0 = 0 = H∞. Our pencil ofhyperplanes is now given by

bL0 +aL∞ = 0, [a;b] ∈ P1.

We definef =

L0

L∞

|X

which is a meromorphic function on X with indeterminacy set X ′.Let us now see how Ehresmann’s theorem applies to a Lefschetz pencil Xtt∈G.

LetC := G∩ X .

Proposition 10 The map f : X\X ′→ G is a C∞ fiber bundle over G\C.

Proof. We consider the blow-up of X along the indeterminacy points of f , i.e.

Y := (x, t) ∈ X×G | x ∈ Xt.

There are two projections

4.7 Proof of Theorem 3 37

Xp← Y

g→ G

We haveY ′ := p−1(X ′) = X ′×G

and p maps Y\Y ′ isomorphically to X\X ′. Under this map f is identified with g.We use Ehresmann’s theorem and we conclude that g is a fiber bundle over G\C.Since it is regular restricted to Y ′, we conclude that g restricted to Y\Y ′ is a C∞ fiberbundle over G\C.

4.7 Proof of Theorem 3

Theorem 3 follows from Theorem 10. By our hypothesis there is a meromorphicfunction f on X such that Y is the zero divisor of order one of f and Z is the poledivisor of order k of f . Since Y intersects Z transversely, a similar blow up argumentas in Proposition 10 implies that f is a C∞ fiber bundle map over C\C, where C is theset of critical values of f restricted to X\Z. Now, we have to prove that f in X\Z hasisolated singularities. If this is not the case then we take an irreducible componentS of the locus of singularities of f restricted to X\Z which is of dimension biggerthan one. The variety S necessarily intersects Y in some point p. The point p doesnot lie in Y\Z because Y\Z is smooth. It does not lie in X ′ because Y intersectsZ transversely and hence in some coordinate system (x,y, . . .) around p ∈ X ′, themeromorphic function f can be written as x

yk .

Exercises1. In local context prove that three different definition of a vanishing cycle and thimble coincide.2. X is an irreducible variety of dimension at most N−1 and the set of lines in PN transverse to a

variety form an non-empty Zariski open subset of lines in PN .3. Prove the properties 1 and 2 of the Lefschetz pencil.4. There is version of Picard-Lefschetz theory in which fibers are symplectic manifolds, see

[Sei08]. Write a report on this.5. Following the notations in §4.3, write down an explicit example of a vanishing cycle above K2

which is not vanishing above K1.6. Rewrite the proof of Theorem 10 using the fibrations

f : C→ C, f (x) = P(x)

f : C2→ C, f (x) = y2−P(x)

where P is a polynomial of degree d and with distinct roots. For instance, take P(x) = xd −dx.Draw also a picture in both cases.

Chapter 5Topology of tame polynomials

It was Solomon Lefschetz who for the first time systematically studied the topologyof smooth projective varieties. Later, his theorems were translated into the languageof modern Algebraic Geometry, using Hodge theory, sheaf theory and spectral se-quences. ”But none of these very elegant methods yields Lefschetz’s full geometricinsight, e.g. they do not show us the famous vanishing cycles”1. A direction in whichLefschetz’s topological ideas were developed was in the study of the topology of hy-persurface singularities. The objective of this chapter is to study the topology of thefibers of tame polynomials following the local context [AGZV88] and the globalcontext [Lam81]. This is an essential step in presenting Hodge theory in a waywhich connects it to the study of multiple integrals due to Poincare and Picard. Tomake this chapter self sufficient, we have put many well-known materials from thementioned references. We mainly use a tame polynomial f ∈ C[x] in the sense thatthe topology of the fibers of f does not get wild when one approaches the infinity.

5.1 Vanishing cycles and orientation

We consider in C the canonical orientation

1−2√−1

dx∧dx = d(Re(x))∧d(Im(x)).

This corresponds to the anti-clockwise direction in the complex plane. In this way,every complex manifold carries an orientation obtained by the orientation of C,which we call it the canonical orientation. For a complex manifold of dimension nand an holomorphic nowhere vanishing differential n-form ω on it, the orientation

obtained from 1(−2√−1)n ω ∧ω differs from the canonical one by (−1)

n(n−1)2 , see

Exercise 1 in this chapter. Holomorphic maps between complex manifolds preserve

1 Klaus Lamotke in [Lam81].

39

40 5 Topology of tame polynomials

the canonical orientation. For a zero dimensional manifold an orientation is just amap which associates ±1 to each point of the manifold.

Let f = x21 + x2

2 + · · ·+ x2n+1. For a real positive number t, the n-th homology of

the complex manifold

Lt := (x1,x2, . . . ,xn+1) ∈ Cn+1 | f (x) = t

is generated by the so called vanishing cycle

δt = Sn(t) := Lt ∩Rn+1.

It vanishes along the vanishing path γ which connects t to 0 in the real line. The(Lefschetz) thimble

∆t := ∪0≤s≤tδs = x ∈ Rn+1 | f (x)≤ t

is a real (n+ 1)-dimensional manifold which generates the relative (n+ 1)-th ho-mology Hn+1(Cn+1,Lt ,Z). We consider for Sn(t) the orientation η such that

η ∧Re(d f ) = Re(dx1)∧Re(dx2)∧·· ·∧Re(dxn+1).

(up to multiplication with positive-valued functions). This is an orientation for ∆t .

Proposition 11 The orientation of (Cn+1,0) obtained by the intersection of two

thimbles is (−1)n(n+1)

2 times the orientation of (C,0) obtained by the intersection oftheir vanishing paths (see Figure 5.1).

Proof. Let α be a complex number near to 1 with Im(α)> 0, |α|= 1 and

h : Lt → Lα2t , x 7→ α · x.

The oriented cycle h∗δt is obtained by the monodromy of δt along the shortest pathwhich connects t to α2t. Now the orientation of ∆t wedge with the orientation ofh∗∆t is:

= Re(dx1)∧Re(dx2)∧·· ·∧Re(dxn+1)∧Re(α−1dx1)∧Re(α−1dx2)∧·· ·∧Re(α−1dxn+1)

= (−1)n2+n

2 Im(α)n+1Re(dx1)∧ Im(dx1)∧Re(dx2)∧ Im(dx2)∧·· ·∧Re(dxn+1)∧ Im(dxn+1)

= (−1)n2+n

2 the canonical orientation of Cn+1.

This does not depend on the orientation η that we chose for δt . The assumptionIm(α)> 0 is equivalent to the fact that Re(dt)∧h∗Re(dt) is the canonical orienta-tion of C.

5.2 Tame polynomials 41

C n+1 C

tf

∆

∆tα

2

t

α t

2

Fig. 5.1 Intersection of thimbles

5.2 Tame polynomials

Let n ∈ N0 andα := (α1,α2, . . . ,αn+1) ∈ Nn+1.

Let also R be a localization of Q[t], t = (t1, t2, . . . , ts) a multi parameter, over amultiplicative subgroup of Q[t]. For simplicity one can take R to be the field ofrational functions in t. We regard the entries of t as parameters, that is, we frequentlysubstitute t with a value and work with such a special case. We denote by

R[x] := R[x1,x2, . . . ,xn+1]

the polynomial ring with coefficients in R and variables x1,x2, . . . ,xn+1. It is consid-ered as a graded algebra with

deg(xi) := αi.

For n = 0 (resp. n = 2 and n = 3) we use the notations x (resp. x,y and x,y,z).

Definition 6 A polynomial f ∈ R[x] is called a homogeneous polynomial of degreed with respect to the grading α if f is a linear combination of monomials of the type

xβ := xβ11 xβ2

2 · · ·xβn+1n+1 , deg(xβ ) = α ·β :=

n+1

∑i=1

αiβi = d.

For an arbitrary polynomial f ∈R[x] one can write in a unique way f =∑di=0 fi, fd 6=

0, where fi is a homogeneous polynomial of degree i. The number d is called thedegree of f .

Definition 7 A polynomial f ∈ R[x] is called a tame polynomial if there exist natu-ral numbers α1,α2, . . . ,αn+1 ∈ N such that for g := fd , the last homogeneous pieceof f in the graded algebra R[x], deg(xi) = αi, the R-module

Vg :=R[x]

jacob(g),

is finitely generated. Here,


jacob(g) := 〈 ∂g∂x1

,∂g∂x2

, · · · , ∂g∂xn+1

〉 ⊂ R[x]

is the Jacobian ideal of g. In this way we say that g has an isolated singularity at theorigin.

For simplicity, we assume that the coefficients of f are polynomials in t, and hence,we can evaluate f for any t ∈ Cs. A detailed study of tame polynomials, togetherwith many examples, will be discussed in Chapter 7. In particular we will introducea polynomial ∆(t) such that, roughly speaking, f = 0 is a singular variety if andonly if ∆(t) = 0.

5.3 Picard-Lefschetz theory of tame polynomials

Let us consider a tame polynomial f in the ring R[x], where R is a localization ofQ[t], t = (t1, t2, . . . , ts) a multi parameter, over a multiplicative group generated byai ∈ R, i = 1,2, . . . ,r. Let also

U0 := Cs\(∪ri=1t ∈ Cs | ai(t) = 0),

U1 := (x, t) ∈ Cn+1×U0 | f (x, t) = 0,

T := U0\t ∈ U0 | ∆(t) = 0,

where ∆ is the discriminant of f . We have a canonical projection π : U1→ U0 andwe define:

Lt := π−1(t) = x ∈ Cn+1 | ft(x) = 0,

where ft is the poynomial obtained by fixing the value of t. Let g be the last ho-mogeneous piece of f and Nn+1 = 1,2, . . . ,n+ 1, S = i ∈ Nn+1 | αi = 1 andSc = Nn+1\S.

Definition 8 The homogeneous polynomial g has a strongly isolated singularity atthe origin if g has an isolated singularity at the origin and for all R⊂ 1,2,3 . . . ,n+1 with S ⊂ R, g restricted to ∩i∈Rxi = 0 has also an isolated singularity at theorigin.

If α1 = α2 = · · · = αn+1 = 1 then the condition ’strongly isolated’ is the same as’isolated’. The Picard-Lefschetz theory of tame polynomials is based on the follow-ing statement:

Theorem 11 If the last homogeneous piece of a tame polynomial f is either inde-pendent of any parameter in R or it has a strongly isolated singularity at the originthen the projection π : U1→ U0 is a locally trivial C∞ fibration over T .

Proof. We give only a sketch of the proof. First, assume that the last homogeneouspiece of f , namely g, has a strongly isolated singularity at the origin. Let us add the

5.3 Picard-Lefschetz theory of tame polynomials 43

new variable x0 to R[x] and consider the homogenization F(x0,x) ∈ R[x0,x] of f .Let Ft be the specialization of F at t ∈ T . Define

U1 := ([x0 : x], t) ∈ P1,α ×T | Ft(x0,x) = 0,

where P1,α is the weighted projective space of type

(1,α) = (1,α1,α2, . . . ,αn+1).

Let π : U1 → U0 be the projection in U0. If all the weights αi are equal to 1 thenD := U1\U1 is a smooth submanifold of U1 and π and π |D are proper regular (i.e.the derivative is surjective). For this case one can use directly Ehresmann’s fibrationtheorem (see Theorem 8 in Chapter 4 or the articles [Ehr47, Lam81]). For arbitraryweights we use the generalization of Ehresmann’s theorem for stratified varieties.In P1,α we consider the following stratification

(P1,α\Pα)⋃(Pα\PSc

)⋃∪I⊂Sc(PI\P<I),

where for a subset I of Nn+1, PI denotes the sub projective space of the weightedprojective space Pα given by xi = 0 | i ∈ Nn+1\I and

P<I := ∪J⊂I, J 6=IPJ .

Now in T consider the one piece stratification and in P1,α×T the product stratifica-tion. This gives us a stratification of U1. The morphism π is proper and the fact thatg has a strongly isolated singularity at the origin implies that π restricted to eachstrata is regular. We use Verdier Theorem ([Ver76], Theorem 4.14, Remark 4.15)and obtain the local trivialization of π on a small neighborhood of t ∈ T and com-patible with the stratification of U1. This yields to a local trivialization of π aroundt. If g is independent of any parameter in R then U1\U1 = G×U0, where G is thevariety induced in g = 0 in Pα . We choose an arbitrary stratification in G and theproduct stratification in G×U0 and apply again Verdier Theorem.

The hypothesis of Theorem 11 is not the best one. For instance, the homogeneouspolynomial

g := x3 + tzy+ tz2 ∈ R[x,y,z],

R := C[t,s,1t], deg(x) = 2, deg(y) = deg(z) = 3

depends on the parameter t and g(x,y,0) has not an isolated singularity at the origin.However for f := g− s, π is a C∞ locally trivial fibration over T = C2\t = 0∪s = 0. I do not know any theorem describing explicitly the atypical values of themorphism π . Such theorems must be based either on a precise desingularizationof U1 and Ehresmann’s theorem or various types of stratifications depending onthe polynomial g. For more information in this direction the reader is referred tothe works of J. Mather, R. Thom and J. L. Verdier around 1970 (see [Mat73] and


the references there). Theorem 11 (in the general context of morphism of algebraicvarieties) is also known as the second theorem of isotopy (see [Ver76] Remark 4.15).

5.4 Monodromy group

Let b0 and b1 two points in T and λ be a path in T connecting b0 to b1 and definedup to homotopy. Theorem 11 gives us a unique map

hλ : Lb0 → Lb1

defined up to homotopy. In particular, for b := b0 = b1 we have the action ofπ1(U0,b) on the homology group Hn(Lb,Z). The image of π1(U0,b) in Aut(Hn(Lb,Z))is called the monodromy group.

Example 1 We consider the one variable tame polynomial f = ft = xd +td−1xd−1+· · ·+ t1x+ t0 in R[x], where R = C[t0, t1, . . . , td−1]. The homology H0( ft = 0,Z)is the set of all finite sums ∑i ri[xi], where ri ∈ Z,∑i ri = 0 and xi’s are the roots offt . The monodromy is defined by the continuation of the roots of f along a path inπ1(T,b). To calculate the monodromy we proceed as follows:

The polynomial f = (x−1)(x−2) · · ·(x−d) has µ := d−1 distinct real criticalvalues, namely c1,c2, . . . ,cµ . Let b the point in T corresponding to f . We consider fas a function from C to itself and take a distinguished set of paths λi, i = 1,2, . . . ,µin C which connects 0 to the critical values of f . This mean that the paths λi donot intersect each other except at 0 and the order λ1,λ2, . . . ,λµ around 0 is anti-clockwise. The cycle δi = [i+1]− [i], i = 1,2, . . . ,µ vanishes along the path λi andδ = (δ1,δ2, . . . ,δµ) is called a distinguished set of vanishing cycles in H0(Lb,Z).Now, the monodromy around the critical value ci is given by

δ j 7→

δ j j 6= i−1, i, i+1−δ j j = iδ j +δi j = i−1, i+1

.

In H0(Lb,Z) we have the intersection form induced by

〈x,y〉=

1 if x = y0 otherwise x,y ∈ Lb.

By definition 〈·, ·〉 is a symmetric form in H0(Lb,Z), i.e. for all δ1,δ2 ∈ H0(Lb,Z)we have 〈δ1,δ2〉= 〈δ2,δ1〉. Let Ψ0 be the intersection matrix in the basis δ :

5.5 Distinguished set of vanishing cycles 45

Ψ0 :=

2 −1 0 0 · · · 0−1 2 −1 0 · · · 00 −1 2 −1 · · · 0...

......

......

...0 0 0 0 −1 2

. (5.1)

The monodromy group keeps the intersection form in H0(Lb,Z). In other words:

ΓZ ⊂ A ∈ GL(µ,Z) | AΨ0Atr =Ψ0. (5.2)

Consider the case d = 3. We choose the basis δ1 = [2]− [1], δ2 = [3]− [2] forH0(Lb,Z). In this basis the intersection matrix is given by

Ψ0 :=(

2 −1−1 2

).

There are two critical points for f for which the monodromy is given by:

δ1 7→ −δ1, δ2 7→ δ2 +δ1,

δ2 7→ −δ2, δ1 7→ δ2 +δ1.

Let g1 =

(−1 01 1

), g2 =

(1 10 −1

). The monodromy group satisfies the equalities:

ΓZ = 〈g1,g2 | g21 = g3

2 = I,g1g2g1 = g2g1g2〉= I,g1,g2,g1g2g1,g2g1,g1g2

=

(1 00 1

),

(−1 01 1

),

(1 10 −1

),

(0 −1−1 0

),(

0 1−1 −1

),

(−1 −11 0

).

For this example (5.2) turns out to be an equality (one obtains equations like (a−b)2 +a2 +b2 = 2 for the entries of the matrix A and the calculation is explicit). SeeExercise 4 in this chapter.

5.5 Distinguished set of vanishing cycles

First, let us recall some definitions from the local theory of vanishing cycles, see forinstance [AGZV88]. Let f : (Cn+1,0)→ (C,0) be a holomorphic function with anisolated critical point at 0∈Cn+1. We take convenient neighborhoods U of 0∈Cn+1

and V of 0 ∈ C such that f : U → V is a C∞ fiber bundle over V\0. Let ti ∈V\0, i = 1,2, · · · ,s (not necessarily distinct) and λi be a path which connects 0 toti in V . We assume that λi’s do not intersect each other except at their start/end points


and at 0 they intersect each other transversally. We also assume that the embeddedoriented sphere δi ⊂ f−1(ti) vanishes along λi. The sphere δi is called a vanishingcycle and is defined up to homotopy.

Definition 9 The ordered set of vanishing cycles δ1,δ2, · · · ,δs is called distin-guished if

1. (λ1,λ2, · · · ,λs) near 0 is the clockwise direction;2. for a versal deformation f of f with µ distinguished critical values, where µ is

the Milnor number of f , the deformed paths λi do not intersect each other exceptpossibly at their end points ti’s.

Historically, one is interested in the full distinguished set of vanishing cycles, i.e.the one with µ elements and with b := t1 = t2 = · · ·= tµ . From now on by a distin-guished set of vanishing cycles we mean the full one. It is well-known that a full dis-tinguished set of vanishing cycles form a basis of Hn( f−1(b),Z) (see [AGZV88]).

Example 2 For f := xd the point 0 ∈ C is the only critical value of f . Let λ (s) =s, 0≤ s≤ 1. The set

δi := [ζ i+1d ]− [ζ i

d ], i = 0, . . . ,d−2

is a distinguished set of vanishing cycles for H0( f = 1,Z). The vanishing takesplace along λ (see [AGZV88] Theorem 2.15).

Let f ∈C[x] be a tame polynomial. We fix a regular value b ∈C\C of f and con-sider a system of paths λi, i = 1,2, . . . ,µ connecting the points of C to the point b.Again, we assume that λi’s do not intersect each other except at their start/end pointsand at the points of C they intersect each other transversally. We call λi’s a distin-guished set of paths. In a similar way as in Definition 9 we define a distinguishedset of vanishing cycles δi ⊂ f−1(b), i = 1,2, . . . ,µ (defined up to homotopy). Foreach singularity p of f we use a separate versal deformation which is defined ina neighborhood of p. If the completion of f has a non zero double discriminantthen we can deform f and obtain another tame polynomial f with the same Milnornumber in a such a way that f and f have C∞ isomorphic regular fibers and f hasdistinct µ critical values. For the notions of completion and double discriminant seeDefinition 15 and §7.7. In this case we can use f for the definition of a distinguishedset of vanishing cycles.

Fix an embedded sphere in f−1(b) representing the vanishing cycle δi. For sim-plicity we denote it again by δi.

Theorem 12 For a tame polynomial f ∈ C[x] and a regular value b of f , the com-plex manifold f−1(b) has the homotopy type of ∪µ

i=1δi. In particular, a distinguishedset of vanishing cycles generates the homology Hn( f−1(b),Z).

Proof. The proof of this theorem is a well-known argument in Picard-Lefschetztheory, see for instance [Lam81] §5, [Bro88] Theorem 1.2, [Mov00] Theorem 2.2.1and [DN01]. We have reproduced this argument in the proof of Theorem 14

5.5 Distinguished set of vanishing cycles 47

In the literature the union ∪µ

i=1δi is known as the bouquet of µ spheres.

Theorem 13 If the tame polynomial f ∈ C[x] has µ distinct critical values and thediscriminant of its completion is irreducible then for two vanishing cycles δ0,δ1 in aregular fiber of f , there is a homotopy class γ ∈ π1(C\C,b) such that hγ(δ0) =±δ1,where C is the set of critical values of f .

Similar theorems are stated in [Lam81] 7.3.5 for generic Lefschetz pencils, in[Mov00] Theorem 2.3.2, Corollary 3.1.2 for generic pencils of type F p

Gq in Pn and in[AGZV88] Theorem 3.4 for a versal deformation of a singularity. Note that in theabove theorem we are still talking about the homotopy classes of vanishing cycles.I believe that the discriminant of complete tame polynomials is always irreducible.This can be checked easily for α1 = α2 = · · ·= αn+1 = 1 and many particular casesof weights.

Proof. Let F ∈ R[x] be the completion of f , where R is some localization of C[t],and ∆0 := t ∈U0 | ∆F(t) = 0. We consider f − s, s ∈C as a line Gc0 in U0 whichintersects ∆0 transversally in µ points. If there is no confusion we denote by b thepoint in U0 corresponding to f −b. Let D be the locus of points t ∈ ∆0 such that theline Gt through b and t intersects ∆0 at µ distinct points. Let also δ0 and δ1 vanishalong the paths λ0 and λ1 which connect b to c0,c1 ∈ Gc0 ∩∆0, respectively. Sincethe set D is a proper algebraic subset of ∆0 and ∆0 is an irreducible variety andc0,c1 ∈ ∆0\D, there is a path w in ∆0\D from c0 to c1. After a blow up at the pointb and using the Ehresmann’s theorem, we conclude that: There is an isotopy

H : [0,1]×Gc0 →∪t∈[0,1]Gw(t)

such that

1. H(0, ·) is the identity map;2. for all a∈ [0,1], H(a, .) is a C∞ isomorphism between Gc0 and Gw(a) which sends

points of ∆0 to ∆0;3. For all a ∈ [0,1], H(a,b) = b and H(a,c0) = w(a)

Let λ ′a = H(a,λ0). In each line Gw(a) the cycle δ0 vanishes along the path λ ′a in theunique critical point of Fw(s) = 0. Therefore δ0 vanishes along λ ′1 in c1 = w(1).Consider λ1 and λ ′1 as the paths which start from b and end in a point b1 near c1 andput λ = λ ′1−λ1. By uniqueness (up to sign) of the Lefschetz vanishing cycle alonga fixed path we can see that the path λ is the desired path.

Let f ∈C[x] be a tame polynomial and λ be a path in C which connects a regularvalue b ∈ C\C to a point c ∈ C and do not cross C except at the mentioned pointc. To λ one can associate an element in λ ∈ π1(C\C,b) as follows: The path λ

starts from b goes along λ until a point near c, turns around c anti clockwise andreturns to b along λ . By the monodromy along the path and around c we mean themonodromy associated to λ . The associated monodromy is given by the Picard-Lefschetz formula/mapping:

a 7→ a+∑δ

(−1)(n+1)(n+2)

2 〈a,δ 〉δ , (5.3)


where δ runs through a basis of distinguished vanishing cycles which vanish in thecritical points of the fiber f−1(c). The above mapping keeps the intersection form〈·, ·〉 invariant, i.e.⟨

a+(−1)(n+1)(n+2)

2 〈a,δ 〉δ ,b+(−1)(n+1)(n+2)

2 〈b,δ 〉δ⟩= 〈a,b〉,

∀a,b ∈ Hn( f = 0,Z).

This follows from (12.1) and the fact that 〈·, ·〉 is (−1)n-symmetric. In general a〈·, ·〉-preserving map from Hn( f = 0,Z) to itself is not a composition of somePicard-Lefschetz mappings. A nice example comes from mirror symmetry and thework of Candelas et al in [CdlOGP91]. They compute explicitly the monodromygroup of at theso called mirror quintic family of Calabi-Yau threefolds. Later, in[BT14] Thomas and Brav prove that such a monodromy group has an infinite indexin Sp(4,Z).

Definition 10 A cycle δ ∈Hn( f = 0,Z), f a tame polynomial, is called the cycleat infinity if its intersection with all other cycles in Hn( f = 0,Z) (including itself)is zero.

5.6 Join of topological spaces

We start with a definition.

Definition 11 The join X ∗Y of two topological spaces X and Y is the quotient spaceof the direct product X× I×Y , where I = [0,1], by the equivalence relation:

(x,0,y1)∼ (x,0,y2) ∀ y1,y2 ∈ Y, x ∈ X ,

(x1,1,y)∼ (x2,1,y) ∀ x1,x2 ∈ X , y ∈ Y.

(x,y’)0 r

x

y’

b

x’

y

(x,y)

(x’,y)

(x,y’)

(x’,y’)

(x,y)(x’,y)

(x’,y’)

Fig. 5.2 Join of zero dimensional cycles

Let X and Y be compact oriented real manifolds and π : X ∗Y → I be the projectionon the second coordinate. The real manifold X ∗Y\π−1(0,1) has a canonical ori-entation obtained by the wedge product of the orientations of X , I and Y . Does X ∗Yhave a structure of a real oriented manifold? It does not seem to me that the answer

5.7 Direct sum of polynomials 49

is positive for arbitrary X and Y . In the present text we only need the followingproposition which gives partially a positive answer to our question. Let

Sn := (x1,x2, . . . ,xn+1) ∈ Rn+1 | x21 + x2

2 + · · ·+ x2n+1 = 1

be the n-dimensional sphere with the orientation α such that the vector x togetherwith α give us the orientation of Rn given by the coordinates system (x1,x2, . . . ,xn).In terms of differential forms this is given by dx

d(x21+x2

2+···+x2n+1)

.


Sn ∗SmC0∼= Sn+m+1, n,m ∈ N0,

which is an isomorphisim of oriented manifolds outside π−1(0,1).

Proof. For the proof of the above diffeomorphism we write Sn+m+1 as the set of all(x,y) ∈ Rn+m+2 such that

x21 + · · ·+ x2

n+1 = 1− (y21 + · · ·+ y2

m+1)

Now, let t be the above number and let it varies from 0 to 1. We have the followingisomorphism of topological spaces:

Sn+m+1→ Sn ∗Sm, (x,y) 7→

( x√

t , t,y√1−t

) t 6= 0,1(0,0,y) t = 0(x,1,0) t = 1

The Figure (5.2) shows a geometric construction of S0×S0. The proof of the state-ment about orientations is left to the reader.

5.7 Direct sum of polynomials

Let f ∈ C[x] and g ∈ C[y] be two polynomials in variables x := (x1,x2, . . . ,xn+1)and y := (y1,y2 . . . ,ym+1) respectively. In this section we study the topology of thevariety

X := (x,y) ∈ Cn+1×Cm+1 | f (x) = g(y)

in terms of the topology of the fibrations f : Cn+1 → C and g : Cm+1 → C. Letalso C1 (resp. C2) denotes the set of critical values of f (resp. g). We assume thatC1 ∩C2 = /0, which implies that the variety X is smooth. Fix a regular value b ∈C\(C1∪C2) of both f and g. Let δ1b ∈ Hn( f−1(b),Z) and δ2b ∈ Hm(g−1(b),Z) betwo vanishing cycles and ts, s ∈ [0,1] be a path in C such that it starts from a pointin C1, crosses b and ends in a point of C2 and never crosses C1 ∪C2 except at thementioned cases. We assume that δ1b vanishes along t−1

. when s tends to 0 and δ2bvanishes along t. when s tends to 1. Now


δ1b ∗δ2b ∼= δ1b ∗t. δ2b := ∪s∈[0,1]δ1ts ×δ2ts ∈ Hn+m+1(X ,Z)

is an oriented cycle. Note that its orientation changes when we change the directionof the path t.. We call the triple (ts,δ1,δ2) = (ts,δ1t. ,δ2t.) an admissible triple.

Let b ∈ C\(C1 ∪C2). We take a system of distinguished paths λc, c ∈C1 ∪C2,where λc starts from b and ends at c. Let

δ11 ,δ

21 , · · · ,δ

µ

1 ∈ Hn( f−1(b),Z)

andδ

12 ,δ

22 , · · · ,δ

µ ′

2 ∈ Hm(g−1(b),Z)

be the corresponding distinguished basis of vanishing cycles. Note that many van-ishing cycles may vanish along a path in one singularity.

Theorem 14 The Z-module Hn+m+1(X ,Z) is freely generated by

γ := δi1 ∗δ

j2 , i = 1,2, . . . ,µ, j = 1,2, · · · ,µ ′,

where we have taken the admissible triples

(λc j λ−1ci

,δ i1,δ

j2 ), ci ∈C1, c j ∈C2.

Proof. The proof which we present for this theorem is similar to the well-knownargument in Picard-Lefschetz theory that we have already used it in §4.5, see also[Lam81] or Theorem 2.2.1 of [Mov00]. The homologies bellow are with Z coeffi-cients.

The fibration π : X→C, (x,y) 7→ f (x) = f (y) is toplogically trivial over C\(C1∪C2). Let Y = f−1(b)×g−1(b). We have

0 = Hn+m+1(Y )→ Hn+m+1(X)→ Hn+m+1(X ,Y ) (5.4)

∂→ Hn+m(Y )→ Hn+m(X)→ ·· ·

We take small open disks Dc around each point c ∈C1∪C2. Let bc be a point near cin Dc and Xc = π−1(λc∪Dc) . We have

Hn+m(Y )∼= Hn( f−1(b))⊗Z Hn(g−1(b))

and

Hn+m+1(X ,Y ) ∼= ⊕c∈C1∪C2Hn+m+1(Xc,Y )∼= ⊕c∈C1∪C2Hn+m+1(Xc,Ybc)

∼= ⊕c∈C1 Hn+1( f−1(Dc), f−1(bc))

⊕⊕c∈C2 Hm+1(g−1(Dc),g−1(bc)).

5.7 Direct sum of polynomials 51

We look Hn+m+1(X) as the kernel of the boundary map ∂ in (5.4). Let us take twocycles δ1 and δ2 form the pieces of the last direct sum in the above equation andassume that ∂δ = 0, where δ = δ1− δ2. If δ1 and δ2 belongs to different classes,according to c ∈C1 or c ∈C2, then δ is the join of two vanishing cycles. Otherwise,δ = 0 in Hn+m+1(X ,Z).

It is sometimes useful to take g = b′−g′, where b′ is a fixed complex number andg′ is a tame polynomial . The set of critical values of g′ is denoted by C′2 and hencethe set of critical values of g is C2 = b′−C′2. We define t = F(x,y) := f (x)+g′(y)and so X = F−1(b′). The set of critical values of F is C1 +C′2 and the assumptionthat C1 ∩ (b′−C′2) is empty implies that b′ is a regular value of F . Let (ts,δ1b,δ2b)be an admissible triple and ts starts from c1 and ends in b′− c′2.

Proposition 13 The topological cycle δ1b ∗ δ2b is a vanishing cycle along the patht.+ c2 with respect to the fibration F = t.

Proof. See Figure 5.3.

b’−C’

A

B

b

b’

b’

1

1 2

1C + C’

C + C’

C

2

1C

2b+C

2

b’−C’2

Fig. 5.3 A system of distinguished paths

Remark 4 Let b ∈ C\(C1 ∪C2). We take a system of distinguished paths λc c ∈C1 ∪C2, where λc starts from b and ends at c (see Figure 5.3). If the points of theset C1 (resp. C2) are enough near (resp. far from) each other then the collectionof translations given in Proposition 13 gives us a system of paths, which is distin-guished after performing a proper homotopy, starting from the points of C1+C′2 andending in b′. This together with Theorem 12 gives an alternative proof to Theorem14.

Example 3 Let us assume that all the critical values of f and g′ = b′− g are real.Moreover, assume that f (resp. g) has non-degenerated critical points with distinctimages. For instance, in the case n = m = 0 take


f := (x−1)(x−2) · · ·(x−m1), g′ := (x−m1−1)(x+2) · · ·(x−m1−m2).

Take b′ ∈ C with Im(b′) > 0. We take direct segment of lines which connects thepoints of C1 to the points of b′−C′2. The set of joint cycles constructed in this way,is a basis of vanishing cycles associated the direct segment of paths which connectb′ to the points of C1 +C′2 (see Figure 5.3, B).

Example 4 Using the machinery introduced in this section, we can find a distin-guished basis of vanishing cycles for Hn(g = 1,Z), where g = xm1

1 + xm22 + · · ·+

xmn+1n+1 , 2≤ mi ∈ N is discussed in Example 7. Let

Γ := (t1, t2, . . . , tn+1) ∈ Rn+1 | ti ≥ 0,n+1

∑i=1

ti = 1.

For i = 1,2, . . . ,n+ 1 we take the distinguished set of vanishing cycles δi,βi , βi =0,1, . . . ,mi−2 given in Example 2 and define the joint cycles

δβ = δm1,β1 ∗δm2,β2 ∗ · · · ∗δmn+1,βn+1 :=

∪t∈Γ δm1,β1,t1 ×δm2,β2,t2 ×·· ·×δmn+1,βn+1,tn+1 ∈ Hn(g = 1,Z), β ∈ I,

where I := (β1, . . . ,βn+1) | 0 ≤ βi ≤ mi− 2. They are ordered lexicographicallyand form a distinguished set of vanishing cycles in Hn(g = 1,Z). Another de-scription of δβ ’s is as follows: For β ∈ I and ai = 0,1, where i = 1,2, . . . ,n+ 1,let

Γβ ,a : Γ →g = 1,

Γβ ,a(t) = (t1

m11 ζ

β1+a1m1

, t1

m22 ζ

β2+a2m2

, . . . , t1

mn+1n+1 ζ

βn+1+an+1mn+1 ),

where for a positive number r and a natural number s, r1s is the unique positive s-th

root of r. We haveδβ := ∑

a(−1)∑

n+1i=1 (1−ai)Γβ ,a.

Chapter 6Hodge conjecture

In this chapter we present the Hodge conjecture in a form which appears before theinvention of the Hodge decomposition, that is, using integrals of differential formsover topological cycles. The Hodge conjecture, true or false, is a desire to classifytopological cycles supported in algebraic subvarieties. Even if it is false the desireto obtain other criterions to distinguish such cycles will not end.

6.1 De Rham cohomology

Let M be a C∞ manifold and Ω iM∞ , i = 0,1,2, . . . be the sheaf of C∞ differentiable

i-forms on M. By definition OM∞ = Ω 0M∞ is the sheaf of C∞-functions on M. We

have the de Rham complex

Ω0M∞

d→Ω1M∞

d→ ·· · d→Ωi−1M∞

d→ΩiM∞

d→ ···

and the de Rham cohomology of X is defined to be

H idR(M) = Hn(Γ (M,Ω i

M∞),d) :=global closed i-forms on Mglobal exact i-forms on M

.

By Poincare Lemma we know that if M is a unit ball then

H idR(M) =

R if i = 00 if i = 0

It follows that R→ Ω •M is the resolution of the constant sheaf R on a C∞ manifoldM. Since the sheaves Ω i

M∞ , i = 0,1,2, . . . are fine we conclude that

H i(M,R)∼= H idR(M), i = 0,1,2, . . .

53

54 6 Hodge conjecture

where H i(M,R) can be interpreted as the Cech cohomology of the constant sheafR on M. All these will be explained in details in Cahpter 14. For the purpose ofthis chapter we only need to know that an element of H i

dR(M) is represented by adifferential i-form ω in M which is closed, that is, dω = 0. Further, ω is the zeroelement of H i

dR(M) if it is exact, that is, there is a differential (i− 1)-form η in Msuch that ω = dη .

6.2 Integration

Let Hi(M,Z) be i-th singular homology of of M. We have the integration map

Hi(M,Z)×H idR(M)→ R, (δ ,ω) 7→

∫δ

ω

which is defined as follows: let δ ∈ Hi(M,Z) be a homology class which is repre-sented by a piecewise smooth p-chain

∑ai fi, ai ∈ Z

and fi is a C∞ map from a neighborhood of the standard p-simplex ∆ ⊂ Rp to M.Let also ω a C∞ global differential form on M. Then∫

δ

ω := ∑ai

∫∆

f ∗i ω

By Stokes theorem this definition is well-defined and does not depend on the classof both δ and ω in Hi(M,Z), respectively H i

dR(M).

6.3 Hodge decomposition

Now, let M be a complex manifold. All the discussion in the previous chapter isvalid replacing the R coefficients with C-coefficients. Let Ω

p,qM∞ (resp. Zp,q

M∞ ) be thesheaf of C∞ differential (p,q)-forms (resp. closed (p,q)-forms) on M. We define

H p,q =Γ (M,Zp,q

M∞)

dΓ (Ω p+q−1M∞ )∩Γ (M,Zp,q

M∞)

We have the canonical inclusion:

H p,q→ HmdR(M)

Theorem 15 Let M be a projective smooth variety . We have

6.4 Hodge conjecture 55

HmdR(M) = Hm,0⊕Hm−1,1⊕·· ·⊕H1,m−1⊕H0,m, (6.1)

which is called the Hodge decomposition.

One can prove the above theorem using harmonic forms, see for instance Voisin’sbook [Voi02b] page 115 or Green’s lectures [GMV94], page 14. The Hodge theory,as we learn it from the literature, starts from the above theorem. Surprisingly, wewill not need the above theorem and so we will not prove it.

We have the conjugation mapping

HmdR(M)→ Hm

dR(M), ω 7→ ω (6.2)

which leaves Hm(M,R) invariant and maps H p,q isomorphically to Hq,p. In order toprove the Hodge decomposition it is enough to prove that:

HmdR(M) = Hm,0 +Hm−1,1 + · · ·+H1,m−1 +H0,m. (6.3)

Let αp,m−p ∈H p,m−p, p = 0,1,2, . . . ,m and ∑mp=0 αp,m−p = 0. This equality implies

that the wedge product of αp,m−p with every element in HmdR(M) is zero and hence

αp,m−p = 0.

6.4 Hodge conjecture

One of the central conjectures in Hodge theory is the so called Hodge conjecture. Letm be an even natural number and M a fixed complex compact manifold. Consider aholomorphic map f : Z→M from a complex compact manifold Z of dimension m

2to M. We have then the homology class

[Z] ∈ Hm(M,Z)

which is the image of the generator of Hm(Z,Z) (corresponding to the canonicalorientation of Z) in Hm(X ,Z). The image Z of f in X is a subvariety (probablysingular) of X and if dim(Z)< m

2 then using resolution of singularities we can showthat [Z] = 0. Let us now

Z =s

∑i=1

riZi

where Zi, i = 1,2, . . . ,s is a complex compact manifold of dimension m2 , ri ∈ Z

and the sum is just a formal way of writting. Let also fi : Z→ X , i = 1,2, . . . ,s beholomorphic maps. We have then the homology class

s

∑i=1

ri[Zi] ∈ Hm(M,Z)


which is called an algebraic cycle with Z-coefficients (see [BH61]). We denote byFalgHm(M,Z) the Z-module of algebraic cycles of set Hm(M,Z).

Proposition 14 For an algebraic cycle δ ∈ Hm(M,Z) we have∫δ

ω = 0, (6.4)

For all C∞ (p,q)-form ω on M with p+q = m, p 6= m2.

Proof. The pull-back of a (p,q)-form with p+q = m and p 6= m2 by fi is identically

zero because at least one of p or q is bigger than m2 .

Any torsion element δ ∈ Hm(M,Z) satisfies the property (6.4) and so sometimes itis convenient to consider Hm(M,Z) up to torsions.

Definition 12 A cycle δ ∈ Hm(M,Z) with the property (6.4) is called a Hodge cy-cle. We denote by FHodgeHm(M,Z) the Z-module of Hodge cycles in Hm(M,Z). Bydefinition it contains all the torsion elements of Hm(M,Z).

Conjecture 3 (Hodge conjecture) For any Hodge cycle δ ∈ Hm(M,Z) there is aninteger a ∈ N such that a ·δ is an algebraic cycle.

Note that if δ is a torsion then there is a∈N such that aδ = 0 and in the way that wehave introduced the Hodge conjecture, torsions do not violate the Hodge conjecture.

Remark 5 Let δ ∈Hm(M,Z) be a Hodge cycle and let P−1(δ ) be its Poincae dual.We have ∫

δ

ω =∫

Xω ∧P−1

δ , ω ∈ Hm(X ,C)

and so ω ∧P−1(δ ) = 0 for all (p,q)-form ω with p+ q = m, p 6= q. By Hodgedecomposition and its relation with the intersection form (6.5) we see that P−1(δ )∈Hn−m

2 ,n−m2 and so

P−1(δ ) ∈ Hn−m2 ,n−

m2 ∩H2n−m(M,Z).

where we have identified H2n−m(M,Z) with its image in H2n−m(M,C) and hencewe have killed the torsions. An element of the above set is called a Hodge class. ThePoincare duality gives a bijection between the Q-vector space of Hodge cycles andthe Q-vector space of Hodge classes. In the literature one usually call an element ofHn−m

2 ,n−m2 ∩H2n−m(M,Q) to be a Hodge class. Note that

Hn−m2 ,n−

m2 ∩H2n−m(M,Z) = Fn−m

2 (H2n−m(M,C))∩H2n−m(M,Z).

6.5 Real Hodge cycles

Let k be a field with Q ⊂ k ⊂ C. We can define the set FHodgeHm(M,k) of Hodgecycles defined over k.

6.7 Polarization 57


dimRFHodgeHm(M,R) = hm2 ,

m2

dimCFHodgeHm(M,C) = hm2 ,

m2

where hm2 ,

m2 = dimH

m2 ,

m2 .

Proof. The proof is an easy linear algebra. Note that if k ⊂ R then in (6.4) we canassume that p < m

2 .

Therefore, in order to find counterexamples for the Hodge conjecture over real num-bers it is sufficient to give a variety such that

dimQFAlgHm(M,Q)< hm2 ,

m2 .

For instance, the product of two elliptic curves have this property.

6.6 Counterexamples

The Hodge conjecture is false when it formulated with a = 1. This means that aHodge cycle δ ∈ Hm(M,Z) may not be an algebraic cycle. It is natural to look fora counterexample δ which is a torsion. The first example of torsion non-algebraichomology elements was obtained by Atiyah and Hirzebruch in [AH62]. A new pointof view is presented by Totaro in [Tot97]. In [kol92] Kollar and van Geemen showsthat if X ⊂ P4 is a very general threefold of degree d and p ≥ 5 is a prime numbersuch that p3 divides d, the degree of every curve C contained in X is divisible byp. This provides a counterexample to the Hodge conjecture over Z not involvingtorsion classes, since it implies that the generator α of H4(X ,Z) is not algebraicwhereas dα is algebraic. In [SV05] Voisin and Soule remarks that the methods ofAtiyah-Hirzebruch and Totaro cannot produce non-algebraic p-torsion classes forprime numbers p > dimC(X). They then show that for every prime number p ≥ 3there exist a fivefold Y and a non-algebraic p-torsion class in H6(Y,Z).

The Hodge decomposition is also valid for Kahler manifolds, however, the Hodgeconjecture is not valid in this case, see [Zuc77, Voi02a]

6.7 Polarization

Let Y be a hyperplane section of X and let u ∈ H2(X ,Z) denote the Poincare dualof of the algebraic cycle [Y ] ∈ H2n−2(X ,Z), i.e.

u∩ [X ] = [Y ].


In the de Rham cohomology H2dR(M), u is of type (1,1) and so the Lefschetz de-

compoisitionHm(X ,Q)∼=⊕qHm−2q(X ,Q)prim

is compatible with the Hodge decomposition. We have the wedge product map:

ψ : HmdR(M)×Hm

dR(M)→ C, ψ(ω1,ω2) =1

(2πi)m

∫M

un−m∧ω1∧ω2,

m = 0,1,2, . . .

where n = dimCM. This form is symmetric for m even and alternating otherwise.


ψ(H i,m−i,Hm− j, j) = 0 unless i = j, (6.5)

(−1)m(m−1)

2 +pψ(ω, ω)> 0, ∀ω ∈ H p,m−p∩Hm(M,C)prim, ω 6= 0. (6.6)

Proof. The first part follows from the fact for ωk, k = 1,2 of type (i,m− i) and( j,m− j), respectively, the 2n-forma un−m∧ω1∧ω2 is identically zero for i 6= j.

The proof of the second part uses Harmonic forms and can be found in [Voi02b].The proof for an ω which is locally of the form f dzp ∧ dzq, where p+ q = m andf is a local C∞ function and dzp (resp. dzq) is a wedge product of p (resp. q) dzi’s(resp. dzi)( this is the case for instance for p = n = m). Up to positive numbers wehave:

ω ∧ ω = | f |2dzp∧dzq∧dzp∧dzq

= (−1)(p+q)q+pqdzp∧dzq∧dzp∧dzq

= (−1)q(−1)m(m−1)

2 dzi1 ∧dzi1 ∧·· ·

= (−1)q(−1)m(m−1)

2 (−2i)mdRe(zi1)∧dIm(zi1)∧·· ·

Exercises1. Show that the wedge product in the de Rham cohomology corresponds to the cup product in

the singular cohomology.2. Show that the top de Rham cohomology of an oriented compact manifold is one dimensional.3. Show that (6.3) is equivalent to the Hodge decomposition.

Chapter 7De Rham cohomology of affine hypersurfaces

The objective of the present chapter is to develop one of the basic ingredients inthe study of multiple integrals. This is the algebraic de Rham cohomology of affinevarieties which is essentially the study of the integrand of multiple integrals. Wework with a tame polynomial f with coefficients in a general ring, instead of thefield of complex numbers. We have tried to keep as much as possible the algebraiclanguage and meantime to explain the theorems and examples by their topologicalinterpretations. When one works with affine varieties in an algebraic context, onedoes not need the whole algebraic geometry of schemes and one needs only a basictheory of commutative algebra. This is also the case in this chapter and so fromalgebraic geometry of schemes we only use some standard notations.

Hodge conjecture is not part of the classical algebraic geometry over fields asit uses the singular homology and cohomology. From a different point of view, deRham cohomology and Hodge filtration, after Atiyah, Hodge, Grothendieck andDeligne, can be introduced in the framework of algebraic geometry over fields. Inthis chapter we explain this in the case of hypersurfaces. This is mainly due toAtiyah and Hodge in [HA55] and Griffiths in [Gri69]. We do not need the sheaf orcategory theory in this case and so the exposition of the material is elementary.

7.1 The base ring

We consider a commutative ring R with multiplicative identity element 1. We as-sume that R is without zero divisors, i.e. if for some a,b ∈ R, ab = 0 then a = 0 orb = 0. We also assume that R is Noetherian, i.e. it does not contain an infinite as-cending chain of ideals (equivalently every ideal of R is finitely generated/every setof ideals contains a maximal element). A multiplicative system in a ring R is a sub-set S of R containing 1 and closed under multiplication. The localization MS of anR-module M is defined to be the R-module formed by the quotients a

s , a∈M, s∈ S.If S = 1,a,a2, . . . for some a ∈ R, a 6= 0 then the corresponding localization isdenoted by Ma. Note that by this notation Za, a ∈ Z,a 6= 0 is no more the set of

59

60 7 De Rham cohomology of affine hypersurfaces

integers modulo a ∈ N. By M we mean the dual of the R-module

M := a : M→ R, a is R linear .

Usually we denote a basis or set of generators of M as a column matrix with entriesin M. We denote by k the field obtained by the localization of R over R\0 and byk the algebraic closure of k. In many arguments we need that the characteristic of kto be zero. If this is the case then we mention it explicitly.

For the purpose of the present text we will only use a localization of a polynomialring Q[t1, t2, . . . , tn] instead of the general ring R. Therefore, the reader may followthe content of this and next chapter only for this ring.

7.2 Homogeneous tame polynomials

Let n ∈ N0 and α = (α1,α2, . . . ,αn+1) ∈ Nn+1. For a ring R we denote by R[x] thepolynomial ring with coefficients in R and the variable x := (x1,x2, . . . ,xn+1). Weconsider

R[x] := R[x1,x2, . . . ,xn+1]

as a graded algebra with deg(xi) = αi. For n = 0 (resp. n = 2 and n = 3) we use thenotations x (resp. x,y and x,y,z).

A polynomial f ∈ R[x] is called a homogeneous polynomial of degree d withrespect to the grading α if f is a linear combination of monomials of the type

xβ := xβ11 xβ2

2 · · ·xβn+1n+1 , deg(xβ ) = α ·β :=

n+1

∑i=1

αiβi = d.

For an arbitrary polynomial f ∈R[x] one can write in a unique way f =∑di=0 fi, fd 6=

0, where fi is a homogeneous polynomial of degree i. The number d is called thedegree of f . The Jacobian ideal of f is defined to be:

jacob( f ) := 〈 ∂ f∂x1

,∂ f∂x2

, · · · , ∂ f∂xn+1

〉 ⊂ R[x].

The Tjurina ideal is

tjurina( f ) := jacob( f )+ 〈 f 〉 ⊂ R[x].

We define also the R-modules

V f :=R[x]

jacob( f ), W f :=

R[x]tjurina( f )

.

These modules may be called the Milnor module and Tjurina module of f , analogto the objects with the same name in singularity theroy (see [Bri70]).

7.2 Homogeneous tame polynomials 61

Remark 6 In practice one considers V f as an R[ f ]-module. If we introduce the newparameter s and define

f := f − s ∈ R[x], R := R[s]

then W f as R-module is isomorphic to V f as R[ f ]-module. We have introduced V fbecause the main machinaries are first developed for f −s, f ∈C[x] in the literatureof singularities (see [Mov07b]).

Definition 13 A homogeneous polynomial g∈R[x] in the weighted ring R[x], deg(xi)=αi, i = 1,2, . . . ,n+1 has an isolated singularity at the origin if the R-module Vg isfreely generated of finite rank. We also say that g is a (homogenous) tame polyno-mial in R[x].

In the case R = C, a homogeneous polynomial g has an isolated singularity at theorigin if Z( ∂g

∂x1, . . . , ∂g

∂xn+1) = 0. This justfies the definition geometrically. If the

homogeneous polynomial g ∈ C[x] is tame then the projective variety induced byg = 0 in Pα is a V -manifold/quasi-smooth variety (see Steenbrink [Ste77]). Forthe case α1 =α2 = · · ·=αn+1 = 1 the notions of a V -manifold and smooth manifoldare equivalent.

Example 5 The two variable polynomial f (x) = x2 +y2 is not tame when it is con-sidered in the ring Z[x,y] and it is tame in the ring Z[ 1

2 ][x,y]. In a similar wayf (x,y) = t2x2 + y2 is tame in Q[t, 1

t2 ][x,y] but not in Q[t][x].

Example 6 Consider the case n = 0, deg(x) = 1. For g = xd we have

Vg =⊕d−2i=0 R · x

i⊕⊕∞i=d−1R/(d ·R) · xi

and so g is tame if and only if d is invertible in R. For instance take R= Z[ 1d ], Q,C.

A basis of the R-module Vg is given by I = 1,x,x2, . . . ,xd−2.

Example 7 In the weighted ring R[x], deg(xi) = αi ∈ N for a given degree d ∈ N,we would like to have at least one tame polynomial of degree d. For instance, if

mi :=dαi∈ N, i = 1,2, . . . ,n+1

and all mi’s are invertible in R then the homogeneous polynomial

g := xm11 + xm2

2 + · · ·+ xmn+1n+1

is tame. A basis of the R-module Vg is given by

I = xβ | 0≤ βi ≤ mi−2, i = 1,2, . . . ,n+1.

For other d’s we do not have yet a general method which produces a tame polyno-mial of degree d.


Example 8 For n = 1 and R = C, a homogeneous polynomial has an isolated sin-gularity at the origin if and only if in its irreducible decomposition there is no factorof multiplicity greater than one.

Throughout the present text we will work with a fixed homogeneous tame poly-nomial g and we assume that the degree d of g is invertible in R. We use the follow-ing notations related to a homogeneous tame polynomial g ∈ R[x]: We fix a basis

xI := xβ | β ∈ I

of monomials for the R-module Vg. We also define

wi :=αi

d, 1≤ i≤ n+1, (7.1)

Aβ :=n+1

∑i=1

(βi +1)wi, µ := #I = rankVg

η := (n+1

∑i=1

(−1)i−1wixidxi),

ηβ := xβη , ωβ = xβ dx β ∈ I,

wheredx := dx1∧dx2∧·· ·∧dxn+1,

dxi = dx1∧·· ·∧dxi−1∧dxi+1∧·· ·∧dxn+1.

One may call µ the Milnor number of g.1 To make our notation simpler, we define

U0 := Spec(R), U1 := Spec(R[x])

and denote by π :U1→U0 the canonical morphism. The set of (relative) differentiali-forms in U1 is:

ΩiU1/U0

:= ∑ fk1,k2,...,kidxk1 ∧dxk2 ∧·· ·∧dxki | fk1,k2,...,ki ∈ R[x].

The adjective relative is used with respect to the morphism π . The set Ω iU j, j = 0,1

of differential i-forms and the differential maps

d : ΩiU j→Ω

i+1U j

, i = 0,1, . . .

can be defined in an algebraic context (see [Har77], p.17). The set DU0 of vectorfields in U0 is by definition the dual of the R-module Ω 1

U0. Therefore, we have the

1 J. Milnor in [Mil68] proves that in the case R= C there are small neighborhoods U ⊂ Cn+1 andS⊂C of the origins such that g : U→ S is a C∞ fiber bundle over S\0whose fiber is of homotopytype of a bouquet of µ n-spheres.We will see a similar statement for tame polynomials in Chapter5.

7.3 De Rham Lemma 63

R-bilinear mapDU0 ×Ω

1U0→ R, (v,η) 7→ η(v) := v(η)

and the mapDU0 ×R→ R, (v, p) 7→ d p(v)

We define

deg(dx j) = α j, deg(ω1∧ω2) = deg(ω1)+deg(ω2),

j = 1,2, . . . ,n+1, ω1,ω2 ∈ΩiU1/U0

.

With the above rules, Ω iU1/U0

turns into a graded R[x]-module and we can talkeabout homogeneous differential forms and decomposition of a differential form intohomogeneous pieces. A geometric way to look at this is the following: The multi-plicative group R∗ = R\0 acts on U1 by:

(x1,x2, . . . ,xn+1)→ (λ α1x1,λα2x2, . . . ,λ

αn+1xn+1), λ ∈ R∗.

We also denote the above map by λ : U1→ U1. The polynomial form ω ∈ Ω iU1/U0

is weighted homogeneous of degree m if

λ∗(ω) = λ

mω, λ ∈ R∗.

For the homogeneous polynomial g of degree d this means that

g(λ α1x1,λα2x2, . . . ,λ

αn+1xn+1) = λdg(x1,x2, . . . ,xn+1), ∀λ ∈ R∗.

Remark 7 The reader who wants to follow the present text in a geometric contextmay assume that R=C[t1, t2, . . . , ts] and hence identify Ui, i = 0,1 with its geomet-ric points, i.e.

U0 = Cs, U1 = Cn+1×Cs.

The map π is now the projection on the last s coordinates.

7.3 De Rham Lemma

In this section we state the de Rham lemma for a homogeneous tame polynomial.Originally, a similar Lemma was stated for a germ of holomorphic function f :(Cn+1,0)→ (C,0) in [Bri70], p.110. To make the section self sufficient we recallsome facts from commutative algebra. The page numbers in the bellow paragraphrefer to the book [Eis95].

Let R be a commutative Noetherin ring with the multiplicative identity 1. Thedimension of R is the supremum s of the lengths of chains 0 6= I0 $ I1 $ · · · $


Is of prime ideals in R. For a prime ideal I ⊂ R we define dim(I) = dim( RI ) andcodim(I) = dim(RI) (p. 225), where RI is the localization of R over the complementof I in R.

A sequence of elements a1,a2, . . . ,an+1 ∈ R is called a regular sequence if

〈a1,a2, . . . ,an+1〉 6= R

and for i = 1,2, . . . ,n+1, ai is a non-zero divisor on R〈a1,a2,...,ai−1〉

(p. 17). For I 6= R,the depth of the ideal I is the length of a (indeed any) maximal regular sequence inI.

The ring R is called Cohen-Macaulay if the codimension and the depth of anyproper ideal of R coincide (p. 452). If R is a domain, i.e. it is finitely generated overa field, then we have

dim(I)+ codim(I) = dim(R) (7.2)

(this follows from Theorem A, p. 221) but in general the equality does not hold. IfR is a Cohen-Macaulay ring then R[x] = R[x1,x2, . . . ,xn+1] is also Cohen-Macaulay(p. 452 Proposition. 18.9). In particular, any polynomial ring with coefficients in afield and its localizations are Cohen-Macaulay.

Proposition 17 Let R be a Cohen-Macaulay ring and g be a homogeneous tamepolynomial in R[x]. The depth of the Jacobian ideal jacob(g)⊂ R[x] of g is n+1.

Proof. Let I := jacob(g)⊂ R[x] we have:

codim(I) := dimR[x]I = dimk[x]I = dimk[x]−dimI = n+1.

Here I is the Jacobian ideal of g in k[x], where k is the quotient field of R. In thesecond and last equalities we have used the fact that g is tame and hence I does nocontain any non-zero element of R and dimI := dim( k[x]I ) = 0. We have also useddim(k[x]) = n+ 1 (Theorem A, p.221). We conclude that the depth of jacob(g) ⊂R[x] is n+1.

The purpose of all what we said above is:

Proposition 18 (de Rham Lemma) Let R be a Cohen-Macaulay ring and g be ahomogeneous tame polynomial in R[x]. An element ω ∈Ω i

U1/U0, i≤ n is of the form

dg∧η , η ∈Ωi−1U1/U0

if and only if dg∧ω = 0. This means that the following sequnceis exact

0→Ω0U1/U0

dg∧·→ Ω1U1/U0

dg∧·→ ··· dg∧·→ ΩnU1/U0

dg∧·→ Ωn+1U1/U0

. (7.3)

In other wordsH i(Ω •U1/U0

,dg∧·) = 0, i = 0,1, . . . ,n.

Proof. We have proved the depth of jacob(g)⊂R[x] is n+1. Knowing this the aboveproposition follows from the main theorem of [Sai76]. See also [Eis95] Corollary17.5 p. 424, Crollary 17.7 p. 426 for similar topics.

7.4 Tame polynomials 65

The sequence in (7.3) is also called the Koszul complex.

Proposition 19 The following sequence is exact

0 d→Ω0U1/U0

d→Ω1U1/U0

d→ ··· d→ΩnU1/U0

d→Ωn+1U1/U0

d→ 0.

In other words

H idR(U1/U0) := H i(Ω •U1/U0

,d) = 0, i = 1,2, . . . ,n+1.

Proof. This is [Eis95], Exercise 16.15 c, p. 414.

Note that in the above proposition we do not need R to be Cohen-Macaulay. Later,we will need the following proposition.

Proposition 20 Let R be a Cohen-Macaulay ring. If for ω ∈Ω iU1/U0

, 1≤ i≤ n−1we have

dω = dg∧ω1, for some ω1 ∈ΩiU1/U0

(7.4)

then there is an ω ′ ∈Ωi−1U1/U0

such that

dω = dg∧dω′.

Proof. Since g is homogeneous, in (7.4) we can assume that

degx(ω1) = degx(dω)−d and so degx(ω1)< degx(dω)≤ degx(ω).

We take differential of (7.4) and use Proposition 18. Then we have dω1 = dg∧ω2, and again we can assume that degx(ω2) < degx(ω1). We obtain a sequenceof differential forms ωk, k = 0,1,2,3, . . . , ω0 = ω with decreasing degrees anddωk−1 = dg∧ωk. Therefore, for some k ∈ N we have ωk = 0. We claim that for all0 ≤ j ≤ k we have dω j = dg∧dω ′j for some ω ′j ∈ Ω

i−1U1/U0

. We prove our claim bydecreasing induction on j. For j = k it is already proved. Assume that it is true forj. Then by Proposition 19 we have

ω j = dg∧ω′j +dω

′j−1, ω

′j−1 ∈Ω

i−1U1/U0

.

Putting this in dω j−1 = dg∧ω j, our claim is proved for j−1.

7.4 Tame polynomials

We start this section with the definition of a tame polynomial.

Definition 14 A polynomial f ∈ R[x] is called a tame polynomial if there exist nat-ural numbers α1,α2, . . . ,αn+1 ∈ N such that the R-module Vg is freely generatedR-module of finite rank (g has an isolated singularity at the origin), where g = fd isthe last homogeneous piece of f in the graded algebra R[x], deg(xi) = αi.


In practice, we fix up a weighted ring R[x], deg(xi) = αi ∈ N and a homogeneoustame polynomial g∈ R[x]. The perturbations g+g1, deg(g1)< deg(g) of g are tamepolynomials.

Example 9 For n = 0, if d is invertible in R then

f = xd + td−1xd−1 + · · ·+ t1x+ t0, ti ∈ R

is a tame polynomial in R[x] (we have used for simplicity x = x1).

Example 10 One of the most important class of tame polynomials are the so calledhyperelliptic polynomials

f = y2 + tdxd + td−1xd−1 + · · ·+ t1x+ t0 ∈ R[x,y],

deg(x) = 2, deg(y) = d,

with g = y2 + tdxd . We assume that td and 2d are invertible in R. A R-basis of theVg-module (and hence of V f ) is given by

I := 1,x,x2, . . . ,xd−2.

In this example we have:

Ai =12+

i+1d

, η :=1d

ydy− 12

ydx,

xidxy

=−2xidx∧dy

d f=−2Ai

∇ ∂

∂ t0(xi

η). (7.5)

The last equalities will be explained in §10.1.

Definition 15 The polynomial

f = ∑deg(xα )≤d

tα xα ∈ R[x],

whereR=Q[tα | deg(xα)≤ d]

is called a complete polynomial.

Let R ⊂ R be the polynomial ring generated by the coefficients of the last homo-geneous piece g of f . Let also k be the field obtained by the localization of R overR\0. Assume that the polynomial g ∈ k[x] has an isolated singularity at the originand so it has an isolated singularity at the origin as a polynomial in a localizationRa of R for some a ∈ R. The variety a = 0 contains the locus of parameters forwhich g has not an isolated zero at the origin. It may contains more points. To findsuch an a we choose a monomial basis xβ , β ∈ I of k[x]/jacob(g) and write allxixβ , β ∈ I, i = 1,2, . . . ,n+ 1 as a k-linear combination of xβ ’s and a residue in

7.5 De Rham Lemma for tame polynomials 67

jacob(g). The product of the denominators of all the coefficients (in k) used in thementioned equalities is a candidate for a. The obtained a depends on the choice ofthe monomial basis.

Now, a complete polynomial is tame over RR\0[x]. An arbitrary tame polyno-mial f ∈ R[x] is a specialization of a unique complete tame polynomial, called thecompletion of f .

Remark 8 In the context of the article [Bro88] the polynomial mapping f :Cn+1→C is tame if there is a compact neighborhood U of the critical points of f such thatthe norm of the Jacobian vector of f is bounded away from zero on Cn\U . It hasbeen proved in the same article (Proposition 3.1) that f is tame if and only if theMilnor number of f is finite and the Milnor numbers of f w := f − (w1x1 + · · ·+wn+1xn+1) and f coincide for all sufficiently small (w1, · · · ,wn+1) ∈Cn+1. This andProposition 22 imply that every tame polynomial in the sense of this article is alsotame in the sense of [Bro88]. However, the inverse may not be true (for instancetake f = x2 + y2 + x2y2, see [Sch05] for other examples).

7.5 De Rham Lemma for tame polynomials

Proposition 21 (de Rham lemma for tame polynomials) Proposition 18 is valid re-placing g with a tame polynomial f .

Proof. If there is ω ∈Ω iU1/U0

, i≤ n such that d f ∧ω = 0 then dg∧ω ′= 0, where ω ′

is the last homogeneous piece of ω . We apply Proposition 18 and conclude that ω =d f ∧ω1 +ω2 for some ω1 ∈Ω

i−1U1/U0

and ω2 ∈Ω iU1/U0

with deg(ω2)< deg(ω) andd f ∧ω2 = 0. We repeat this argument for ω2. Since the degree of ω2 is decreasing,at some point we will get ω2 = 0 and then the desired form of ω .

Recall that in §7.2 we fixed a monomial basis xI for the R-module Vg.

Proposition 22 For a tame polynomial f , the R-module V f is freely generated byxI .

Proof. Let f = f0 + f1 + f2 + · · ·+ fd−1 + fd be the homogeneous decompositionof f in the graded ring R[x], deg(xi) = αi and g := fd be the last homogeneouspiece of f . Let also F = f0xd

0 + f1xd−10 + · · ·+ fd−1x0 + g be the homogenization

of f . We claim that the set xI generates freely the R[x0]-module V := R[x0,x]/〈 ∂F∂xi|

i = 1,2, . . . ,n+1〉. More precisely, we prove that every element P ∈ R[x0,x] can bewritten in the form

P = ∑β∈I

Cβ xβ +R, R :=n+1

∑i=1

Qi∂F∂xi

, (7.6)

degx(R)≤ degx(P), Cβ ∈ R[x0], Qi ∈ R[x0,x]. (7.7)

Since xI is a basis of Vg, we can write


P = ∑β∈I

cβ xβ +R′, (7.8)

R′ =n+1

∑i=1

qi∂g∂xi

, cβ ∈ R[x0], qi ∈ R[x0,x].

We can choose qi’s so that

degx(R′)≤ degx(P). (7.9)

If this is not the case then we write the non-trivial homogeneous equation of highestdegree obtained from (7.8). Note that ∂g

∂xiis homogeneous. If some terms of P occur

in this new equation then we have already (7.9). If not we subtract this new equationfrom (7.8). We repeat this until getting the first case and so the desired inequality.Now we have

∂g∂xi

=∂F∂xi− x0

d−1

∑j=0

∂ f j

∂xixd− j−1

0 ,

and soP = ∑

β∈Icβ xβ +R1−P1, (7.10)

where

R1 :=n+1

∑i=1

qi∂F∂xi

, P1 := x0(n+1

∑i=1

d−1

∑j=0

qi∂ f j

∂xixd− j−1

0 ).

From (7.9) we have

degx(P1)≤ degx(P)−1, degx(R1)≤ degx(P).

We write again qi∂ f j∂xi

in the form (7.8) and substitute it in (7.10). By degree condi-tions this process stops and at the end we get the equation (7.6) with the conditions(7.7).

Now let us prove that xI generates the R[x0]-module V freely. If the elements ofxI are not R[x0]-independent then we have

∑β∈I

Cβ xβ = 0

in V for some Cβ ∈ R[x0] or equivalently

∑β∈I

Cβ xβ = dF ∧ω (7.11)

for some ω = ∑n+1i=1 Qi[x,x0]dxi, Qi ∈ R[x,x0], where d is the differnetial with re-

spect to xi, i = 1,2, . . . ,n+1 and hence dx0 = 0. Since F is homogenous in (x,x0),we can assume that in the equality (7.11) the deg(x,x0)

of the left hand side isd + deg(x,x0)

(ω). Let ω = ω0 + x0ω1 and ω0 does not contain the variable x0. In

7.6 The discriminant of a polynomial 69

the equation obtained from (7.11) by putting x0 = 0, the right hand side side mustbe zero otherwise we have a nontrivial relation between the elements of xI in Vg.Therefore, we have dg∧ω0 = 0 and so by de Rham lemma (Proposition 18)

ω0 = dg∧ω′ = dF ∧ω

′+ x0(g−F

x0)∧ω

′,

with degx(ω0) = d+deg(ω ′). Substituting this in ω and then ω in (7.11) we obtaina new ω with the property (7.11) and stricktly less degx.

Proposition 22 implies that f and its last homogeneous piece have the same Milnornumber.

7.6 The discriminant of a polynomial

Definition 16 Let A be the R-linear map in V f induced by multiplication by f .According to (22), V f is freely generated by xI and so we can talk about the matrixAI of A in the basis xI . For a new parameter s define

S(s) := det(A− s · Iµ×µ),

where Iµ×µ is the identity µ times µ matrix and µ = #I. It has the property S( f )V f =0. We define the discriminant of f to be

∆ = ∆ f := S(0) ∈ R.

The discriminant has the following property

∆ f ·W f = 0. (7.12)

In general ∆ f is not the the simplest element in R with the property (7.12).

Remark 9 In the zero dimensional case n = 0 the discriminant of a monic polyno-mial f = xd + td−1xd−1 + · · ·+ t1x+ t0 ∈ R[x] is defined as follows:

∆′f := ∏

1≤i 6= j≤d(xi− x j) =

d

∏i=1

f ′(xi) ∈ R,

where f ′ = ∂ f∂x is the derivative of f . It is an easy exercise to see that the multiplica-

tion of ∆ ′f with the number dd is equal to ∆ f .

Proposition 23 Let R be a closed algebraic field. We have ∆ f = 0 if and only if theaffine variety f = 0 ⊂ Rn+1 is singular.


Proof. ⇐: If ∆ f 6= 0 then A is surjective and 1 ∈ R[x] can be written in the form1 = ∑

n+1i=1

∂ f∂xi

qi + q f . This implies that the variety Z := ∂ f∂xi

= 0, i = 1,2, . . . ,n+1, f = 0 is empty.⇒: If f = 0 is smooth then the variety Z is empty and so by Hilbert Nullste-

lensatz there exists f ∈ R[x] such that f f = 1 in V f . This means that A is invertibleand so ∆ f 6= 0.

The above Proposition implies that in the case of R equal to C[t1, t2, . . . , ts], the affinevariety ∆ f (t) = 0 ⊂ Cs is the locus of parameters t such that the affine variety f = 0 ⊂ Cn+1 is singular.

Definition 17 For a tame polynomial f we say that the affine variety f = 0 issmooth if the discriminant ∆ f of f is not zero.

Proposition 24 Assume that f is a tame polynomial and ∆ f 6= 0. If

d f ∧ω2 = f ω1,

for some ω2 ∈ΩnU1/U0

, ω1 ∈Ωn+1U1/U0

thenω2 = f ω3 +d f ∧ω4, ω1 = d f ∧ω3,

for some ω3 ∈ΩnU1/U0

, ω4 ∈Ωn−1U1/U0

.

Proof. If ω1 is not zero in W then the multiplication by f R-linear map in V f hasa non trivial kernel and so ∆ f = 0 which contradicts the hypothesis. Now let ω1 =d f ∧ω3 and so d f ∧ ( f ω3−ω2) = 0. The de Rham lemma for f (Proposition 21)finishes the proof.

The example bellow shows that the above proposition is not true for singular affinevarieties.

Example 11 For a homogeneous polynomial g in the graded ring R[x], deg(xi) =αiwe have

g =n+1

∑i=1

wixi∂g∂xi

equivalentely gdx = dg∧η

and so the matrix A in the definition of the discriminat of g is the zero matrix. Inparticular, the discriminant of g− s ∈ R[s][x] is (−s)µ .

Example 12 Assume that 2d is invertible in R. For the hypergeometric polynomialf := y2− p(x)∈ R[x,y], deg(p) = d we have V f ∼=Vp and under this isomorphy themultiplication by f linear map in V f coincide with the multiplication by p map inVp. Therefore,

∆ f = ∆p.

7.8 De Rham cohomology 71

7.7 The double discriminant of a tame polynomial

Let f ∈ R[x] be a tame polynomial. We consider a new parameter s and the tamepolynomial f − s ∈ R[s][x]. The discriminant ∆ f−s of f − s as a polynomial in s hasdegree µ and its coefficients are in R. Its leading coefficient is (−1)µ and so if µ isinvertible in R then it is tame (as a polynomial in s) in R[s]. Now, we take again thediscriminant of ∆ f−s with respect to the parameter s and obtain

∆ = ∆ f := ∆∆ f−s ∈ R

which is called the double discriminant of f . We consider a tame polynomial f asa function from kn+1 to k. The set of critical values of f is defined to be P = Pf :=Z(jacob( f )) and the set of critical values of f is C = C f := f (Pf ). It is easy to seethat:

Proposition 25 The tame polynomial f has µ distinct critical values (and hencedistinct critical points) if and only of its double discriminant is not zero.

Note that that µ is the maximum possible number for #C f .


Let f ∈ R[x] be a tame polynomial as a in §7.4. The following quotients

H′ = H′f := (7.13)

Ω nU1

f Ω nU1

+d f ∧Ωn−1U1

+π−1Ω 1U0∧Ω

n−1U1

+dΩn−1U1

,

∼=Ω n

U1/U0

f Ω nU1/U0

+d f ∧Ωn−1U1/U0

+dΩn−1U1/U0

H′′ = H′′f := (7.14)

Ωn+1U1

f Ωn+1U1

+d f ∧dΩn−1U1

+π−1Ω 1U0∧Ω n

U1

∼=Ω

n+1U1/U0

f Ωn+1U1/U0

+d f ∧dΩn−1U1/U0

are R-modules and play the role of de Rham cohomology of the affine variety

f = 0 := Spec(R[x]

f ·R[x]).


Here π : U1 → U0 is the projection corresponding to R ⊂ R[x]. We have assumedthat n > 0. In the case n = 0 we define:

H′ = H′f :=R[x]

f ·R[x]+R

and

H′′ =Ω 1

U1/U0

f ·Ω 1U1/U0

+R ·d f.

Remark 10 We will use H or HndR( f = 0) to denote one of the modules H′ or H′′.

We note that for an arbitrary polynomial f such modules may not coincide with thede Rham cohomology of the affine variety f = 0 defined by Grothendieck, Atiyahand Hodge (see [Gro66]). For instance, for f = x(1+ xy)− t ∈ R[x,y], R= C[t] thedifferential forms yk+1dx+ xykdy, k > 0 are not zero in the corresponding H′ butthey are relatively exact and so zero in H1

dR( f = 0) (see [Bon03]).

One may call H′ and H′′ the Brieskorn modules associated to f in analogy to thelocal modules introduced by Brieskorn in 1970. In fact, the classical Brieskorn mod-ules for n > 0 are

H ′ = H ′f =Ω n

U1/U0

d f ∧Ωn−1U1/U0

+dΩn−1U1/U0

,

H ′′ = H ′′f :=Ω

n+1U1/U0

d f ∧dΩn−1U1/U0

.

and for n = 0

H ′ :=R[x]R[ f ]

,

H ′′ =Ω 1

U1/U0

R[ f ] ·d f.

We consider them as R[ f ]-modules. In [Mov07b] we have worked with the classicalones.

Remark 11 The R[ f ]-module H ′f is isomorphic to the R[s]-module H′f , where f =f − s ∈ R[s][x] and s is a new parameter. A similar statement is true for the otherBrieskorn module.

Remark 12 We have the following well-defined R-linear map

H′→ H′′, ω 7→ d f ∧ω

which is an inclusion by Proposition 24. When we write H′ ⊂ H′′ then we mean theinclusion obtained by the above map. We have

7.8 De Rham cohomology 73

H′′

H′=W f .

For ω ∈ H′′ we define the Gelfand-Leray form

ω

d f:=

ω ′

∆∈ H′∆ , where ∆ ·ω = d f ∧ω

′.

Recall the definition of ωβ and ηβ from §7.2. Let us first state the main results ofthis section.

Theorem 16 Let R be of characteristic zero and Q ⊂ R. If f is a tame polynomialin R[x] then the R[ f ]-modules H ′′ and H ′ are free and ωβ ,β ∈ I (resp. ηβ , β ∈ I)form a basis of H ′′ (resp. H ′). More precisely, in the case n > 0 every ω ∈ Ω

n+1U1/U0

(resp. ω ∈Ω nU1/U0

) can be written

ω = ∑β∈I

pβ ( f )ωβ +d f ∧dξ , (7.15)

pβ ∈ R[ f ], ξ ∈Ωn−1U1/U0

, deg(pβ )≤deg(ω)

d−Aβ

(resp.ω = ∑

β∈Ipβ ( f )ηβ +d f ∧ξ +dξ1, (7.16)

pβ ∈ R[t], ξ ,ξ1 ∈Ωn−1U1/U0

, deg(pβ )≤deg(ω)

d−Aβ

).

A similar statemennt holds for the case n = 0. We leave its formulation and proof tothe reader. We will prove the above theorem in §7.9 and §7.10.

Corollary 1 Let R be of characteristic zero and Q ⊂ R. If f is a tame polynomialin R[x] then the R-modules H′ and H′′ are free and ηβ ,β ∈ I (resp. ωβ , β ∈ I) forma basis of H′ (resp. H′′).

Note that in the above corollary f = 0 may be singular. We call ηβ ,β ∈ I (resp.ωβ , β ∈ I) the canonical basis of H′ (resp. H′′).

Proof. We prove the corollary for H′. The proof for H′′ is similar. We consider thefollowing canonical exact sequence

0→ f H ′→ H ′→ H′→ 0

Using this, Theorem 16 implies that H′ is generated by ηβ , β ∈ I. It remains toprove that ηβ ’s are R-linear independent. If a := Σβ∈Irβ ηβ = 0, rβ ∈ R in H′ thena = f b, b ∈ H ′. We write b as a R[ f ]-linear combination of ηβ ’s and we obtainrβ = f cβ ( f ) for some cβ ( f ) ∈ R[ f ]. This implies that for all β ∈ I, rβ = 0.


Theorem 16 is proved first for the case R = C in [Mov07b]. In this article we haveused a topological argument to prove that the forms ωβ ,β ∈ I (resp. ηβ , β ∈ I)are R[ f ]-linear independent. It is based on the following facts: 1. ηβ ’s generatesthe C[ f ]-module H ′, 2. #I = µ is the dimension of Hn

dR( f = c) for a regularvalue c ∈ C−C, 3. H ′ restricted to f = 0 is isomorphic to Hn

dR( f = c). In theforthcomming sections we present an algebraic proof.

7.9 Proof of Theorem 16 for a homogeneous tame polynomial

Let f = g be a homogeneous tame polynomial with an isolated singularity at origin.We explain the algorithm which writes every element of H ′′ of g as a R[g]-linearcombination of ωβ ’s. Recall that

dg∧d(Pdxi,dx j) = (−1)i+ j+εi, j(∂g∂x j

∂P∂xi− ∂g

∂xi

∂P∂x j

)dx,

where εi, j = 0 if i < j and = 1 if i > j and dxi,dx j is dx without dxi and dx j (wehave not changed the order of dx1,dx2, . . . in dx).

Proposition 26 In the case n > 0, for a monomial P = xβ we have

∂g∂xi·Pdx = (7.17)

dd ·Aβ −αi

∂P∂xi

gdx+dg∧d(∑j 6=i

(−1)i+ j+1+εi, j α j

d ·Aβ −αix jPdxi,dx j).

Proof. The proof is a straightforward calculation:

∑j 6=i

(−1)i+ j+1+εi, j α j

d ·Aβ −αidg∧d(x jPdxi,dx j) =

−1d ·Aβ −αi

∑j 6=i

(α j∂g∂x j

∂ (x jP)∂xi

−α j∂g∂xi

∂ (x jP)∂x j

)dx =

−1d ·Aβ −αi

((d ·g−αixi∂g∂xi

)∂P∂xi−P

∂g∂xi

∑j 6=i

α j(β j +1))dx =

−1d ·Aβ −αi

(d ·g ∂P∂xi−αiβiP

∂g∂xi−P

∂g∂xi

∑j 6=i

α j(β j +1))dx

The only case in which dAβ −αi = 0 is when n = 0 and P = 1. In the case n = 0 forP 6= 1 we have

∂g∂x·Pdx =

dd ·Aβ −α

∂P∂x

gdx =dα

xβ−1gdx

7.9 Proof of Theorem 16 for a homogeneous tame polynomial 75

and if P = 1 then ∂g∂xi·Pdx is zero in H ′′. Based on this observation the following

works also for n = 0.We use the above Proposition to write every Pdx ∈Ω

n+1U1/U0

in the form

Pdx = ∑β∈I

pβ (g)ωβ +dg∧dξ , (7.18)

pβ ∈ R[g], ξ ∈Ωn−1U1/U0

, deg(pβ (g)ωβ ), deg(dg∧dξ )≤ deg(Pdx).

• Input: The homogeneous tame polynomial g and P ∈ R[x] representing [Pdx] ∈H ′′. Output: pβ ,β ∈ I and ξ satisfying (7.18)We write

Pdx = ∑β∈I

cβ xβ ·dx+dg∧η , deg(dg∧η)≤ deg(Pdx). (7.19)

Then we apply (7.17) to each monomial component P ∂g∂xi

of dg∧η and then we

write each ∂ P∂xi

dx in the form (7.19). The degree of the components which makePdx not to be of the form (7.18) always decreases and finally we get the desiredform.

To find a similar algorithm for H ′ we note that if η ∈Ω nU1/U0

is written in the form

η = ∑β∈I

pβ (g)ηβ +dg∧ξ +dξ1, (7.20)

pβ ∈ R[g], ξ ,ξ1 ∈Ωn−1U1/U0

,

where each piece in the right hand side of the above equality has degree less thandeg(η), then

dη = ∑β∈I

(pβ (g)Aβ + p′β(g)g)ωβ −dg∧dξ (7.21)

and the inverse of the map

R[g]→ R[g], p(g) 7→ Aβ .p(g)+ p′(g) ·g

is given byk

∑i=0

aigi→k

∑i=1

ai

Aβ + igi. (7.22)

Now let us prove that there is no R[g]-relation between ωβ ’s in H ′′g . This implies alsothat there is no R[g] relation between ηβ ’s in H ′g. If such a relation exists then wetake its differential and since dg∧ηβ = gωβ and dηβ =Aβ ωβ we obtain a nontrivialrelation in H ′′g .

Since g = dg∧η and xβ are R-linear independent in Vg, the existence of a nontrivoal R[g]-relation between ωβ ’s in H ′′g implies that there is a 0 6= ω ∈ H ′′g suchthat gω = 0 in H ′′g . Therefore, we have to prove that H ′′g has no torsion. Let a ∈ R[x]


andg ·a ·dx = dg∧dω1, for some ω1 ∈Ω

n−1U1/U0

. (7.23)

Since g is homogeneous, we can assume that a is also homogeneous. Now, the aboveequality, Proposition 18 and

dg∧ (aη−dω1) = 0

imply thataη = dω1 +dg∧ω2, for some ω2 ∈Ω

n−1U1/U0

.

We take differential of the above equality and we conclude that

(n+1

∑i=1

wi +deg(a)

d)a ·dx = 0 in H ′′g .

Since Q⊂ R, we conclude that adx = 0 in H ′′g .

Remark 13 The reader may have already noticed that Theorem 16 is not at all trueif R has characteristic different form from zero. In the formulas (7.22) and (7.17)we need to divide over d ·Aβ −αi and Aβ + i. Also, to prove that H ′′g has no torsion

we must be able to divide on ∑n+1i=1 wi +

deg(a)d .

7.10 Proof of Theorem 16 for an arbitrary tame polynomial

For simplicity we assume that n > 0. We explain the algorithm which writes everyelement of H ′′ of f as a R[ f ]-linear combination of ωβ ’s. We write an elementω ∈Ω

n+1U1/U0

,deg(ω) = m in the form

ω = ∑β∈I

pβ (g)ωβ +dg∧dψ,

pβ ∈ R[g], ψ ∈Ωn−1U1/U0

, deg(pβ (g)ωβ )≤ m, deg(dψ)≤ m−d.

This is possible because g is homogeneous. Now, we write the above equality in theform

ω = ∑β∈I

pβ ( f )ωβ +d f ∧dψ +ω′,

withω′ = ∑

β∈I(pβ (g)− pβ ( f ))ωβ +d(g− f )∧dψ.

The degree of ω ′ is strictly less than m and so we repeat what we have done at thebeginning and finally we write ω as a R[ f ]-linear combination of ωβ ’s.

7.10 Proof of Theorem 16 for an arbitrary tame polynomial 77

The algorithm for H ′ is similar and uses the fact that for η ∈ Ω nU1/U0

one canwrite

η = ∑β∈I

pβ (g)ηβ +dg∧ψ1 +dψ2 (7.24)

and each piece in the right hand side of the above equality has degree less thandeg(η).

Let us now prove that the forms ωβ ,β ∈ I (resp. ηβ , β ∈ I) are R[ f ]-linear inde-pendent. If there is a R[ f ]-relation between ωβ ’s in H ′′f , namely

∑β∈I

pβ ( f )ωβ = d f ∧dω, ω ∈Ωn−1U1/U0

, (7.25)

then by taking the last homogeneous piece of the relation, we obtain a nontrivialR[g]-relations between ωβ ’s in H ′′g or

dg∧dω1 = 0, ω1 ∈Ωn−1U1/U0

,

where ω = ω1 +ω ′1 with deg(ω ′1) < deg(ω1) = deg(ω). The first case does nothappen by the proof of our theorem in the f = g case (see §7.9). In the second casewe use Proposition 7.11 and its Proposition 20 and obtain

dω1 = dg∧dω2,

ω2 ∈Ωn−2U1/U0

, deg(dω1) = d +deg(dω2).

Nowd f ∧dω = d f ∧d(ω1 +ω

′1) = d f ∧ (d(g− f )∧dω1 +dω

′1).

This means that we can substitute ω with another one and with less degx. Taking ω

the one with the smallest degree and with the property (7.25), we get a contradiction.In the case of H ′f the proof is similar and is left to the reader.

Exercises1. Let us consider the case R=C. Show that the canonical map from the Brieskorn module H′ to

the classical de Rham cohomology of f = 0 is an isomorphism.2. Show that we can always take a monomial basis of the Milnor module Vg.3. Let t1, t2, t3 ∈C with 27t3

2 − t23 6= 0 and let E be the elliptic curve y2 = 4(x− t1)3− t2x− t3. This

is actually a torus with a removed point. Verify the following equalities in H1dR(E):

x2dxy

= (2t1)xdx

y+(−t2

1 +112

t2)dxy,

x3dxy

= (3t21 +

320

t2)xdx

y+(−2t3

1 +110

t1t2 +110

t3)dxy,

x4dxy

= (4t31 +

35

t1t2 +17

t3)xdx

y+(−3t4

1 −110

t21 t2 +

935

t1t3 +5

336t22 )

dxy,

x5dxy

= (5t41 +

32

t21 t2 +

57

t1t3 +7

240t22 )

xdxy

+(−4t51 −

23

t31 t2 +

27

t21 t3 +

19420

t1t22 +

130

t2t3)dxy.


For similar exercises see [Mov12].

Chapter 8Hodge filtrations and Mixed Hodge structures

So far, we have discussed the algebraic de Rham cohomology of the affine variety f = 0 for a tame polynomial. Its elements are constructed by polynomials andit is natural to make some distinctions between such elements. In this chapter weexplain the mixed Hodge structure of the algebraic de Rham cohomology of f =0 which is going to do this job. In the case of curves this is essentially the oldstory of differential forms of the first kind (holomorphic everywhere), of the secondkind (with poles but no residues) and the third kind (with poles which might haveresidues).

We define the Gauss-Manin system M associated to f which plays the samerole as H and it has the advantage that the Hodge and weight filtrations in M aredefined explicitly. Our approach is by looking at differential forms with poles along f = 0 in U1/U0 which is a proper way when one deals with the tame polynomialsin the sense of present text. The main role of the Hodge and weight filtrations inthe present text is to distinguish between differential forms. We state the Griffithstransversality theorem which is a direct consequence of our definitions. We will lateruse the material of this section for residue of such differential forms along the pole f = 0.

8.1 Gauss-Manin system

The Gauss-Manin system for a tame polynomial f is defined to be:

M f =M :=Ω

n+1U1/U0

[ 1f ]

Ωn+1U1/U0

+d(Ω nU1/U0

[ 1f ])

= ∼=Ω

n+1U1

[ 1f ]

Ωn+1U1

+d(Ω nU1[ 1

f ])+π−1Ω 1U0∧Ω n

U1[ 1

f ],

79

80 8 Hodge filtrations and Mixed Hodge structures

where Ω iU1[ 1

f ] is the set of polynomials in 1f with coefficients in Ω i

U1and etc.. It

has a natural filtration given by the pole order along f = 0, namely

Mi := [ωf i ] ∈M | ω ∈Ω

n+1U1/U0

,

M1 ⊂M2 ⊂ ·· · ⊂Mi ⊂ ·· · ⊂M∞ :=M.

It is useful to identify H′ by its image under d f ∧· in H′′ and define M0 :=H′. Notethat in M we have

[dω

f i−1 ] = [(i−1)d f ∧ω

f i ], ω ∈ΩnU1/U0

, i = 2,3, . . . (8.1)

[d f ∧dω

f i ] = [d(d f ∧ω

f i )] = 0, ω ∈Ωn−1U1/U0

, i = 1,2, . . . (8.2)

Proposition 27 If the discriminant of the tame polynomial f is not zero then thedifferential form ω

f i , i ∈ N, ω ∈Ωn+1U1/U0

is zero in M if and only if ω is of the form

f dω1− (i−1)d f ∧ω1 +d f ∧dω2 + f iω3,

ω1 ∈ΩnU1/U0

, ω2 ∈Ωn−1U1/U0

, ω3 ∈Ωn+1U1/U0

.

Proof. Letω

f i = d(ω1

f s ) mod Ωn+1U1/U0

. (8.3)

If s = i−1 then ω has the desired form. If s≥ i then d f ∧ω1 ∈ f Ωn+1U1/U0

and so byProposition 24 we have ω1 = f ω3 +d f ∧ω2 and so

ω

f i = d(f ω3 +d f ∧ω2

f s ), mod Ωn+1U1/U0

. (8.4)

If s = i then we obtain the desired form for ω . If s > i we get d f ∧ dω2 + (s−1)d f ∧ω3 ∈ f Ω

n+1U1/U0

and so again by Proposition 24 we have dω2 +(s− 1)ω3 =

f ω4 + d f ∧ω5. We calculate ω3 from this equality and substitute it in (8.4) andobtain

ω

f i =1

s−1d(

f ω4 +d f ∧ω5

f s−1 ) mod Ωn+1U1/U0

.

We repeat this until getting the situation s = i.

The structure of M and its relation with H is described in the following proposition.

Proposition 28 We have the well-defined canonical maps

H′′→M1,ω 7→ [ω

f],

8.2 Mixed Hodge structure of M 81

W→Mi/Mi−1, ω 7→ [ω

f i ], i = 1,2, . . . .

If the discriminant of the tame polynomial f is not zero then they are isomorphimsof R-modules.

Proof. The fact that they are well-defined follows from the equalities (8.1) and (8.2).The non-trivial part of the second part is that they are injective. This follows fromProposition 27.

8.2 Mixed Hodge structure of M

Recall that xβ | β ∈ I is a monomial basis of the R-module Vg and ωβ , β ∈ I is abasis of the R-module H′′.

Definition 18 We define the degree of ω

f k , k ∈ N,ω ∈ Ωn+1U1/U0

to be degx(ω)−degx( f k). By definition we have deg(

ωβ

f k ) = d(Aβ − k). The degree of α ∈ M isdefined to be the minimum of the degrees of ω

f k ∈ α .

In order to define the mixed Hodge structure of M we need the following proposi-tion.

Proposition 29 Every element of degree s of M can be written as an R-linear sumof the elements

ωβ

f k , β ∈ I, 1≤ k, Aβ ≤ k, (8.5)

deg(ωβ

f k )≤ s.

Proof. Let us be given an element ω

f k of degree s in M. According to Corollary 1,we write ω = ∑β∈I aβ ωβ +d f ∧dω2 + f ω1 and so

ω

f k = ∑β∈I

aβ

ωβ

f k +ω1

f k−1 in M.

We repeat this argument for ω1. At the end we get ω

f k as a R-linear combination ofωβ

f i , β ∈ I, k ∈N. An alternative way is to say that ω can be written as an R[ f ]-linear

combinations of ωβ , β ∈ I modulo d f ∧ dΩn−1U1/U0

(see Theorem 16). The degree

conditions (7.15) implies that the we have used onlyωβ

f i with deg(ωβ

f i )≤ deg( ω

f k ).

Now, we have to get rid of elements of typeωβ

f k , Aβ > k. Given such an element,in M we have:


ωβ

f k =1

Aβ

dηβ

f k =k

Aβ

d f ∧ηβ

f k+1

=k

Aβ

f ωβ +(g− f )ωβ +d( f −g)∧ηβ

f k+1

and soωβ

f k =k

Aβ − k(g− f )ωβ +d( f −g)∧ηβ

f k+1 . (8.6)

The degree of the right hand side of (8.6) is less than d(Aβ −k), which is the degree

of the left hand side. We write the right hand side in terms ofω

β ′f s , β ′ ∈ I,s ∈ N and

repeat (8.6) for these new terms. Since each time the degree of the new elementsω ′

β

f s

decrease, at some pont we get the desired form forωβ

f k .

By definition of ∇v in (10.9) and Proposition 8.5, we have

deg(∇v(α))≤ deg(α), α ∈M. (8.7)

Now, we can define two natural filtration on M.

Definition 19 We define Wn = WnM∆ to be the R∆ -submodule of M∆ generatedby

ωβ

f k , β ∈ I, Aβ < k

and call0 =: Wn−1 ⊂Wn ⊂Wn+1 :=M∆

the weight filtration of M∆ . We also define Fi = FiM∆ to be the R∆ -submodule ofof M∆ generated by

ωβ

f k , β ∈ I, Aβ ≤ k ≤ n+1− i (8.8)

and call0 = Fn+1 ⊂ Fn ⊂ Fn−1 ⊂ ·· · ⊂ F0

the Hodge filtration of M∆ . The pair (F•,W•) is called the mixed Hodge structureof M∆ .

Since for j = 0,1,2, . . . ,∞ we have the the inclusion H :=M j ⊂M∆ , we definethe mixed Hodge structure of H to be the intersection of the (pieces) of the mixedHodge structure of M∆ with H:

WiH :=WiM∆ ∩H, F jH := F jM∆ ∩H,

i = n−1,n,n+1, j = 0,1,2, . . . ,n+1.

The Hodge filtration induces a filtration on GrWi := Wi/Wi−1, i = n,n+ 1 andwe set

8.3 Homogeneous tame polynomials 83

Gr jFGrWi := F jGrWi /F j+1GrWi

=(F j ∩Wi)+Wi−1

(F j+1∩Wi)+Wi−1.

for j = 0,1,2, . . . ,n+1.

For the original definition of the mixed Hodge structure in the complex context R=C or C(t)(the field of rational functions in t = (t1, t2, . . . , ts)) see [Voi02b, Voi03].In fact we have used Griffiths-Steenbrink theorem (see [Ste77] and §??) in order toformulate the above definition. In particular, in this context we have F0H= H.

Remark 14 Since R is a principal ideal domain and H :=H′, H′′ is a free R-module(Corollary 1), any R-sub-module of H is also free and in particular the pieces ofmixed Hodge structure of H are free R-modules.

Definition 20 A set B = ∪nk=0Bk

n ∪∪nk=1Bk

n+1 ⊂ H is a basis of H compatible withthe mixed Hodge structure if it is a basis of the R-module H and moreover each Bk

mform a basis of Grk

FGrWm H.

8.3 Homogeneous tame polynomials

Bellow for simplicity we use d to denote the differential operator with respect tothe variables x1,x2, . . . ,xn+1. Let us consider a homogeneous polynomial g in thegraded ring R[x], deg(xi) = αi. We have the equality

g =n+1

∑i=1

wixi∂g∂xi

which is equivalent togdx = dg∧η .

We have also

gωβ = dg∧ηβ , dη = (w ·1)dx, dηβ = Aβ ωβ . (8.9)

The discriminant of the polynomial g is zero. We define f := g− t ∈ R[t][x] whichis tame and its discriminant is (−t)µ . The above qualities imply that

∇ ∂

∂ tηβ =

Aβ

tηβ ,

∇ ∂

∂ t(ωβ ) =

(Aβ −1)t

ωβ .

We have


tωβ

f k =− f ωβ +dg∧ηβ

f k = (−1+Aβ

k−1)

ωβ

f k−1

in M. Therefore

ωβ

f k =1

tk−1 (−1+Aβ

k−1)(−1+

Aβ

k−2) · · ·(−1+

Aβ

1)

ωβ

f(8.10)

in Mt . Note that under the canonical inclusion H′ ⊂H′′ of the Brieskorn modules off we have

tωβ = ηβ .

Theorem 17 For a weighted homogeneous polynomial g ∈ R[x] with an isolatedsingularity at the origin, the set

B = ∪nk=1Bk

n+1∪∪nk=0Bk

n

withBk

n+1 = ηβ | Aβ = n− k+1,

Bkn = ηβ | n− k < Aβ < n− k+1,

is a basis of the R-module H′ associated to g− t ∈ R[t][x] compatible with the mixedHodge structure. The same is true for H′′ replacing ηβ with ωβ .

Proof. This theorem with the classical definition of the mixed Hodge structuresis proved by Steenbrink in [Ste77]. In our context it is a direct consequence ofDefinition 19 and the equality (8.10).

8.4 Weighted projective spaces

In this section we recall some terminology on weighted projective spaces. Thereare two main reasons why we use weighted projective spaces instead of the usualprojective spaces. First, most of the historical examples which we have discussed inthe Introduction fits into this general context. The second reason is just for the sakeof completeness, as our main theorem in this chapter, Theorem 18, is stated in sucha general context. For a first reading, the reader may follow this chapter with theclassical projective space of dimension n+ 1. We have used [Dol82, Ste77] as ourmain source on weighted projective spaces. The material of the present section isnecessary in order to say that the Hodge filtration used in Definition 19 is the sameone as in Chapter 6.

Let n be a natural number and α := (α1,α2, . . . ,αn+1) be a vector of naturalnumbers whose greatest common divisor is one. The multiplicative group C∗ actson Cn+1 in the following way:

(X1,X2, . . . ,Xn+1)→ (λ α1X1,λα2X2, . . . ,λ

αn+1 Xn+1), λ ∈ C∗.

8.4 Weighted projective spaces 85

We also denote the above map by λ . The quotient space

Pα := Cn+1/C∗

is called the projective space of weight α . If α1 = α2 = · · ·= αn+1 = 1 then Pα isthe usual projective space Pn (Since n is a natural number, Pn will not mean a zerodimensional weighted projective space). One can give another interpretation of Pα

as follow: Let Gαi := e2π√−1m

αi | m ∈ Z. The group Πn+1i=1 Gαi acts discretely on the

usual projective space Pn as follows:

(ε1,ε2, . . . ,εn+1), [X1 : X2 : · · · : Xn+1]→ [ε1X1 : ε2X2 : · · · : εn+1Xn+1].

The quotient space Pn/Πn+1i=1 Gαi is canonically isomorphic to Pα . This canonical

isomorphism is given by

[X1 : X2 : · · · : Xn+1] ∈ Pn/Πn+1i=1 Gαi → [Xα1

1 : Xα22 : · · · : Xαn+1

n+1 ] ∈ Pα .

Let d be a natural number. The polynomial (resp. the polynomial form) ω inCn+1 is weighted homogeneous of degree d if

λ∗(ω) = λ

dω, λ ∈ C∗.

For a polynomial g this means that

g(λ α1X1,λα2X2, . . . ,λ

αn+1Xn+1) = λdg(X1,X2, . . . ,Xn+1), ∀λ ∈ C∗.

Let g be an irreducible polynomial of (weighted) degree d. The set g = 0 inducesa hypersurface D in Pα , α = (α1,α2, . . . ,αn+1). If g has an isolated singularity at0∈Cn+1 then Steenbrink has proved that D is a V-manifold/quasi-smooth variety. AV -manifold may be singular but it has many common features with smooth varieties(see [Ste77, Dol82]).

For a polynomial form ω of degree dk, k ∈ N in Cn+1 we have λ ∗ ω

gk = ω

gk forall λ ∈ C∗. Therefore, ω

gk induce a meromorphic form on Pα with poles of order kalong D. If there is no confusion we denote it again by ω

gk . The polynomial form

ηα =n+1

∑i=1

(−1)i−1αiXidXi, (8.11)

where dXi = dX1∧·· ·∧dXi−1∧dXi+1∧·· ·∧dXn+1, is of degree ∑n+1i=1 αi.

Let P(1,α) = [X0 : X1 : · · · : Xn+1] | (X0,X1, · · · ,Xn+1) ∈ Cn+2 be the projectivespace of weight (1,α), α = (α1, . . . ,αn+1). One can consider P(1,α) as a compacti-fication of Cn+1 = (x1,x2, . . . ,xn+1) by putting

xi =Xi

Xαi0, i = 1,2, · · · ,n+1 (8.12)


The projective space at infinity Pα∞ =P(1,α)−Cn+1 is of weight α :=(α1,α2, . . . ,αn+1).

Let f be a tame polynomial of degree d in §7 and g be its last quasi-homogeneouspart. We take the homogenization F = Xd

0 f ( X1X

α10, X2

Xα20, . . . ,

Xn+1

Xαn+10

) of f and so we can

regard f = 0 as an affine subvariety in F = 0 ⊂ P(1,α).Now, we are ready to state a classic theorem of Griffiths in [Gri69]. Its general-

ization for quasi-homogeneous spaces is due to Steenbrink in [Ste77]. Since in thepresent text we only deal with the de Rham cohomology of smooth manifolds, thereader may consider all the weights αi equal to one and so Pα is the usual projectivespace Pn+1.

Theorem 18 Let g(X1,X2, · · · ,Xn+1) be a weighted homogeneous polynomial ofdegree d, weight α = (α1,α2, . . . ,αn+1) and with an isolated singularity at 0 ∈Cn+1(and so D = g = 0 is a V -manifold). We have

Hn(Pα −D,C)∼=H0(Pα ,Ω n(∗D))

dH0(Pα ,Ω n−1(∗D))

and under the above isomorphism

Grn+1−kF GrW

n+1Hn+1(Pα −D,C) := Fn−k+1/Fn−k+2 ∼= (8.13)

H0(Pα ,Ω n(kD))

dH0(Pα ,Ω n−1((k−1)D))+H0(Pα ,Ω n((k−1)D))

where 0 = Fn+1 ⊂ Fn ⊂ ·· · ⊂ F1 ⊂ F0 = Hn(Pα −D,C) is the Hodge filtrationof Hn(Pα −D,C). Let Xβ | β ∈ I be a basis of monomials for the Milnor vectorspace

C[X1,X2, · · · ,Xn+1]/ <∂g∂Xi| i = 1,2, . . . ,n+1 >

A basis of (8.13) is given by

Xβ ηα

gk , β ∈ I, Aβ = k (8.14)

where

ηα =n+1

∑i=1

(−1)i−1αiXidXi

In the situation of the above theorem F0 = F1. The essential ingredient in the proofis Bott’s vanishing theorem for quasi-homogeneous spaces and Proposition 64.

Chapter 9Fermat varieties

In this chapter we are going to discuss Hodge and algebraic cycles for Fermat vari-eties

xd0 + xd

1 + · · ·+ xdn+1 = 0.

in the projective space Pn+1. Since the main emphasis of the present text is multipleintegrals and these are usually written using affine varieties, we will take the affinechart x0 = 1 of the Fermat variety, and we will consider a more general affine varietygiven by

V =V (m1,m2, . . . ,mn+1) : xm11 + xm2

2 + · · ·+ xmn+1n+1 −1 = 0.

We will still call V the Fermat variety because it is the generalization of the classicalFermat curve xd + yd = 1.

The whole discussion of the present section is based on the following simpleintegral computation.

Proposition 30 Let δβ , β ∈ I be the basis of the free Z-module described in Exer-cise 3 in Chapter 3. We have

∫δβ

ηβ ′ =1

m1m2 · · ·mn+1·B(β1 +1

m1,

β2 +1m2

, · · · , βn+1 +1mn+1

)·n+1

∏i=1

(ζ(βi+1)(β ′1+1)mi −ζ

βi(β′i +1)

mi )

9.1 De Rham cohomology of affine Fermat varieties

We are going to consider the weighted polynomial ring C[x] with deg(xi) = αi ∈N. For a given degree d ∈ N, we would like to have at least one homogeneouspolynomial g ∈ C[x] with an isolated singularity at the origin and of degree d. Forinstance for α1,α2, . . . ,αn+1 | d we have the polynomial

g = xm11 + xm2

2 + · · ·+ xmn+1n+1 , mi :=

dαi

.

87

88 9 Fermat varieties

For other d’s we do not have yet a general method which produces a tame polyno-mial of degree d. The vector space Vg = C[x]/jacob( f ) has the following basis ofmonomials

xβ , β ∈ I

whereI := β ∈ Zn+1 | 0≤ βi ≤ mi−2

We also defineµ = #I = Π

n+1i=1 (mi−1).

Aβ :=n+1

∑i=1

(βi +1)mi

.

The number µ is the Milnor number of the singularity g = 0, see for instance[AGZV88].

Proposition 31 Let us consider the affine variety

V =V (m1, . . . ,mn+1) := g = 1 ⊂ Cn+1

The set of differential forms ωβ , β ∈ I form a basis of the n-th algebraic de RhamHn

dR(V ) of V . Furthermore, it is compatible with both weight and Hodge filtrationof Hn

dR(V ), more precisely,ωβ ∈ I, Aβ = k, (9.1)

form a basis of Grn+1−kF GrW

n+1HndR(V ) and

ωβ ∈ I, k−1 < Aβ < k. (9.2)

form a basis of Grn+1−kF GrW

n HndR(V ).

The above proposition follows from Theorem 17. For β ∈ I we have

Aβ = Am−β−2,

wherem−β −2 := (m1−β1−2,m2−β2−2, · · ·).

We have the symmetric sequence of numbers

(hk−1,n−k0 , k = 1,2, . . . ,n), (hk−1,n−k+1

0 , k = 1,2, . . . ,n+1)

wherehk−1,n−k

0 := #β ∈ I | Aβ = k,

hk−1,n−k+10 := #β ∈ I | k−1 < Aβ < k

which correspond to the classical Hodge numbers of the primitive cohomologies ofthe weighted projective varieties:

9.2 Vanishing cycles 89

V∞ =V∞(m1, . . . ,mn+1) := g = 0 ⊂ Pα1,α2,...,αn+1 ,

V =V ∪V∞ ⊂ P1,α1,...,αn+1

respectively. Here are some tables of Hodge numbers of weighted hypersurfacesobtained by Proposition 31.

n = 2,α0 = α1 = α2 = α3 = 1d 1 2 3 4 5 6 7 8 9 10

h0,20 0 0 0 1 4 10 20 35 56 84

h1,10 0 1 6 19 44 85 146 231 344 489

h2,00 0 0 0 1 4 10 20 35 56 84

n = 3,α0 = α1 = α2 = α3 = α4 = 1d 1 2 3 4 5 6 7 8 9 10

h0,30 0 0 0 0 1 5 15 35 70 126

h1,20 0 0 5 30 101 255 540 1015 1750 2826

h2,10 0 0 5 30 101 255 540 1015 1750 2826

h3,00 0 0 0 0 1 5 15 35 70 126

n = 4,α0 = α1 = α2 = α3 = α4 = α5 = 1d 1 2 3 4 5 6 7 8 9 10

h0,40 0 0 0 0 0 1 6 21 56 126

h1,30 0 0 1 21 120 426 1161 2667 5432 10116

h2,20 0 1 20 141 580 1751 4332 9331 18152 32661

h3,10 0 0 1 21 120 426 1161 2667 5432 10116

h4,00 0 0 0 0 0 1 6 21 56 126

n = 2,α0 = α1 = α2 = 1, α3 = 3d 3 6 9 12 15 18 21 24 27 30

h0,20 0 1 11 39 94 185 321 511 764 1089

h1,10 0 19 92 255 544 995 1644 2527 3680 5139

h2,00 0 1 11 39 94 185 321 511 764 1089

9.2 Vanishing cycles

We are going to analyze the Hodge cycles of the variety V in more details. We useTheorem 17 and Proposition 45 and obtain an arithmetic interpretation of Hodgecycles which does not involve any topological argument.

For each natural number m let

Im := 0,1,2, . . . ,m−2,∆m := δ0,δ1,δ2, . . . ,δm−2,Ωm := ω0,ω1, . . . ,ωm−2

be three sets with m−1 elements and define:

90 9 Fermat varieties∫δβ

ωβ ′ := ζ(β+1)(β ′+1)m −ζ

β (β ′+1)m , β ,β ′ ∈ Im

Pm(ωβ ) := [∫

δ0

ωβ ,∫

δ1

ωβ , . . . ,∫

δm−2

ωβ ]tr =

[ζ β+1m −1,ζ 2(β+1)

m −ζ(β+1)m , · · · ,ζ (m−1)(β+1)

m −ζ(m−2)(β+1)m ]tr, β ∈ Im.

For a set M let Z[M] be the free Z-module generated by the elements of M. Forarbitrary β ∈ Z we define δi ∈ Z[∆m] using the rules:

δi := δi mod m, (9.3)

δm−1 :=−m−2

∑i=0

δi.

Equivalentlyδi +δi+1 + · · ·+δi+m−1 = 0, ∀i ∈ Z.

Let m = (m1,m2, . . . ,mn+1), 2≤ mi ∈ N and

Im := Im1 × Im2 ×·· ·× Imn+1 ,

∆m := ∆m1 ×∆m2 ×·· ·×∆mn+1 ,

Ωm := Ωm1 ×Ωm2 ×·· ·×Ωmn+1 .

We denote the elements of Im by β = (β1,β2, · · · ,βn+1). We also denote an elementof ∆m (resp. Ωm) by δβ (resp. ωβ ) with β ∈ Im. We define

∫δβ

ωβ ′ :=n+1

∏i=1

∫δβi

ωβ ′i, β ,β ′ ∈ Im,

Pm(ωβ ) = Pm1(ωβ1)∗Pm2(ωβ2)∗ · · · ∗Pmn+1(ωβn+1),

β = (β1,β2, . . . ,βn+1) ∈ Im.

Here for two matrices A and B by A ∗B we mean the coordinate wise product of Aand B ordered lexicographically.

By Z-linearity we define∫δ

ω, δ ∈ Z[∆m], ω ∈ Z[Ωm].

The elements of ∆m are called vanishing cycles and ∆m is called to be a distinguishedset of vanishing cycles.

Definition 21 For ω ∈Z[Ωm], an ω-cycle is an element δ ∈Z[∆m] such that∫

δω =

0.

An ω-cycle written in the canonical basis of Z[∆m] is a 1×µ matrix δ with coeffi-cients in Z such that δ ·Pm(ω) = 0. Recall that

9.3 Intersection form 91

Aβ :=n+1

∑i=1

βi +1mi

, β ∈ Im.

Definition 22 An element δ ∈ Z[∆m] which is an ω-cycle for all

ωβ ∈Ωm,Aβ 6∈ Z, Aβ <n2

is called a Hodge cycle.

Definition 23 An element δ ∈ Z[∆m] is called a cycle at infinity if δ is an ωβ -cyclefor all β ∈ Im with Aβ 6∈ N, i.e.∫

δ

ωβ = 0, ∀(β ∈ Im, Aβ 6∈ N).

9.3 Intersection form

Definition 24 In the freely generated Z-module Z[∆m] we consider the bilinearform 〈·, ·〉 which satisfies

〈δβ ,δβ ′〉= (−1)n〈δβ ′ ,δβ 〉,β ,β ∈ Im,

〈δ(β1,β2,...,βn+1),δ(β ′1,β′2,...,β

′n+1)〉= (−1)

n(n+1)2 (−1)Σ

n+1k=1 β ′k−βk

for βk ≤ β ′k ≤ βk +1, k = 1,2, . . . ,n+1, β 6= β ′, and

〈δβ ,δβ 〉= (−1)n(n−1)

2 (1+(−1)n), β ∈ Im.

In the remaining cases, except those arising from the previous ones by a permuta-tion, we have 〈δβ ,δβ ′〉= 0.

The above bilinear map corresponds to the intersection map of Hn(g = 1,Z), seeExample 6.8 [Mov11].

Using the geometric interpretation of cycles at infinity, one can see that:

Proposition 32 An element δ ∈ Z[∆m] is a cycle at infinity if and only if

〈δ ,δβ 〉= 0, ∀β ∈ Im.

Definition 25 To each vanishing cycle δ ∈ ∆m we associate the monodromy map

hδ : Z[∆m]→ Z[∆m], hδ (a) = a+(−1)(n+1)(n+2)

2 〈a,δ 〉.

and call it the Picard-Lefschetz monodromy map. The full mondromy group M is thesubgroup of the group of Z-linear isomorphisms of Z[∆m] generated by all hδ , δ ∈

92 9 Fermat varieties

∆m. We enlarge the class of vanishing cycles in the following way. A cycle δ ′ ∈Z[∆m] is called a vanishing cycle if there is an element h ∈M and δ ∈ ∆m such thath(δ ) =±δ ′.

Using the geometric interpretation of vanishing cycles and Theorem 13 we have:

Proposition 33 For any two vanishing cycle δ1,δ2 there is a monodromy h ∈ Msuch that h(δ1) =±δ2.

For a decomposition 1,2, . . . ,n+1= A∪B, A∪B = /0, we have a canonical map

∆mA ×∆mB → ∆m,

(δ1,δ2) 7→ δ1 ∗δ2

mA := (mi)i∈A, mB := (mi)i∈B

which is obtained by shuffling δ1 ∈ ∆mA and δ2 ∈ ∆mB according to the mentioneddecomposition. By Z-linearity it extends to

Z[∆A]×Z[∆B]→ Z[∆m].

Definition 26 A cycle δ ∈ Z[∆m] is called a joint cycle if it has the following prop-erty: There exists a decomposition 1,2, . . . ,m= A∪B into disjoint non empty setssuch that δ = δ1 ∗δ2, δ1 ∈ Z[∆mA ], δ2 ∈ Z[∆mB ].

By the definition, if δ1 ∈ Z[∆mA ] is a ωβ1 cycle then for all β2 ∈ IB and δ2 ∈ Z[ΩB],δ1 ∗δ2 is a (β1,β2)-cycle.

Let m and m′ be (n+1)-tuple as before and assume that m′i |mi, i= 1,2, . . . ,n+1.We have a Z-linear map

a∗ : Z[Ωm′ ]→ Z[Ωm]

which is induced by

(β ′1,β′2, . . . ,β

′n+1) 7→

m1

m′1

m2

m′2· · · mn+1

m′n+1

((β ′1 +1)

m1

m′1−1,

(β ′2 +1)m2

m′2−1, . . . ,(β ′n+1 +1)

mn+1

m′n+1−1)

We have also the mapa∗ : Z[∆m]→ Z[∆m′ ]

δ(β1,β2,...,βn+1) 7→ δ(β1 mod m′1,β2 mod m′2,...,βn+1 mod m′n+1)

where we have used the rules (9.3). Again, using the geometric interpretation:

Proposition 34 We have∫a∗δ

ω =∫

δ

a∗ω, δ ∈ Z[∆m], ω ∈ Z[Ωm].

9.4 No Hodge cycles 93

9.4 No Hodge cycles

Proposition 35 Let ωβ ∈Ωm.

1. For natural numbers m1,m2, . . . ,mn+1 the condition

[Q(ζm1 ,ζm2 , . . . ,ζmn+1),Q] = (m1−1)(m2−1) · · ·(mn+1−1) (9.4)

is satisfied if and only if all mi’s are prime numbers.2. If (9.4) is satisfied then there does not exist a non zero ωβ -cycle.3. In particular, there does not exist a non zero Hodge cycle, and also, there does

not exist a cycle at infinity and so

∀β ′ ∈ Im,Aβ ′ 6∈ N.

Proof. Let ki =Q(ζm1 ,ζm2 , . . . ,ζmi), i = 1,2, . . . ,n+1. Since

[kn+1,Q] = [kn+1 : kn] · · · [k2 : k1][k1 : Q], [ki,ki−1]≤ mi−1,

the condition (9.4) implies that [ki,ki−1] = mi−1 and so mi is a prime number. If allmi’s are prime the condition (9.4) is trivially true.

For the proof of the second statement of the theorem, we prove that the entries ofP(ωβ ) form a Q-basis of Q(ζm1 ,ζm2 , . . . ,ζmn+1). This statement can be easily provedby induction on n (since mi’s are prime, we can assume that β = (0,0, . . . ,0)).

Chapter 10Gauss-Manin connection

The objective of the present chapter is to introduce the Gauss-Manin connection ofthe fibration induced by a tame polynomial f . From this one can derive Picard-Fuchsequations and the calculation of the later is abundant in the literature. However, acomplete computation of the Gauss-Manin connection itself in the case of ellipticcurves elliptic curves goes back to Griffiths and Sasai, see [Sas74, Gri66]. For thetwo variable polynomial f (x1,x2)− s with the parameter s such a calculation orparts of it is done by many people in the context of planar differential equations,see for instance [Gav01] and the references within there. The multi variable case ofsuch a calculation can be interesting from the Hodge theory point of view and it isdiscussed in [Mov07b] and [Mov07a] for a tame polynomial in the sense of Chapter7. Our arguments in the present chapter work for a polynomial f defined on a generalring as it is described in Chapter 7. We have tried to keep as much as possible thealgebraic language and meantime to explain the theorems and examples by theirtopological interpretations. When one works with affine varieties in an algebraiccontext, one does not need the whole algebraic geometry of schemes and one needsonly a basic theory of commutative algebra. This is also the case in this chapter andso from algebraic geometry of schemes we only use some standard notations.

10.1 Gauss-Manin connection

In this section we define the so called Gauss-Manin connection of the R-moduleH. Its geometric interpretation in terms of a connection on a vector bundle will beexplained in Chapter 11.

The Tjurina module of f can be rewritten in the form

95

96 10 Gauss-Manin connection

W f :=Ω

n+1U1

d f ∧Ω nU1

+ f Ωn+1U1

+π−1Ω 1U0∧Ω n

U1

∼=Ω

n+1U1/U0

d f ∧Ω nU1/U0

+ f Ωn+1U1/U0

.

Looking in this way, we have the well defined differential map

d : H′→W f .

Let ∆ be the discriminant of f . We define the Gauss-Manin connection on H′ asfollows:

∇ : H′→Ω1T ⊗RH

′

∇ω =1∆

∑i

αi⊗βi,

where

∆dω−∑i

αi∧βi ∈ f Ωn+1U1

+d f ∧ΩnU1, αi ∈Ω

1U0, βi ∈Ω

nU1,

and Ω 1T is the localization of Ω 1

U0on the multiplicative set

1,∆ ,∆ 2, . . ..

From scheme theory point of view this is the set of differential forms defined in

T := Spec(R∆ ) = U0 \∆ = 0.

To define the Gauss-Manin connection on H′′ we use the fact that H′′H′ =W f and

define∇ : H′′→Ω

1T ⊗RH

′′,

∇(ω) = ∇(∆ ·ω

∆) =

∇(∆ ·ω)−d∆ ⊗ω

∆, (10.1)

where ∆ ·ω = d f ∧η , η ∈ H′.The operator ∇ satisfies the Leibniz rule, i.e.

∇(p ·ω) = p ·∇(ω)+d p⊗ω, p ∈ R, ω ∈ H

and so it is a connection on the module H. It defines the operators

∇i = ∇ : ΩiT ⊗RH→Ω

i+1T ⊗RH.

by the rules

10.2 Picard-Fuchs equations 97

∇i(α⊗ω) = dα⊗ω +(−1)iα ∧∇ω, α ∈Ω

iT , ω ∈ H.

If there is no danger of confusion we will use the symbol ∇ for these operators too.

Proposition 36 The connection ∇ is an integrable connection, i.e. ∇∇ = 0.

Proof. We havedω = ∑

iαi∧βi, αi ∈Ω

1T , βi ∈Ω

nU1

modulo f Ωn+1U1

+d f ∧Ω nU1

. We take the differential of this equality and so we have

∑i

dαi∧βi−αi∧dβi = 0

again modulo f Ωn+1U1

+d f ∧Ω nU1

. This implies that

∇∇(ω) = ∇(∑i

αi⊗βi)

= ∑i

dαi⊗βi−αi∧∇βi

= 0.

10.2 Picard-Fuchs equations

It is is useful to look at the Guass-Manin connection in the following way: We havethe operator

DU0 → Lei(H∆ ), v 7→ ∇v,

where DU0 is the set of vector fields in U0, ∇v is the composition

H∆

∇→Ω1T ⊗R∆

H∆

v⊗1→ R∆ ⊗R∆H∆∼= H∆ ,

and Lei(H∆ ) is the set of additive maps ∇v from H∆ to itself which satisfy

∇v(rω) = r∇v(ω)+dr(v) ·ω, v ∈DU0 , ω ∈ H∆ , r ∈ R∆ .

In this way H∆ is a (left) D-module (differential module):

v ·ω := ∇v(ω), v ∈D , ω ∈ H∆ .

Note that we can now iterate ∇v, i.e. ∇sv = ∇v ∇v · · · ∇v s-times, and this is

different from ∇∇ introduced before.For a given vector field v ∈DU0 and ω ∈ H consider

ω,∇v(ω),∇2v(ω), · · · ∈ H⊗R k.


Since the k-vector space H⊗R k is of dimension µ , there exists a positive integerm≤ µ and pi ∈ R, i = 0,1,2, . . . ,m such that

p0ω + p1∇v(ω)+ p2∇2v(ω)+ · · ·+ pm∇

mv (ω) = 0 (10.2)

This is called the Picard-Fuchs equation of ω along the vector field v. Since R is aunique factorization domain, we assume that there is no common factor between pi.

10.3 Gauss-Manin connection matrix

Let ω1,ω2, . . . ,ωµ be a basis of H and define ω = [ω1,ω2, . . . ,ωµ ]tr. The Gauss-

Manin connection in this basis can be written in the following way:

∇ω = A⊗ω, A ∈ 1∆

Matµ×µ(Ω 1T ) (10.3)

The integrability condition translates into dA = A∧A.

10.4 Calculating Gauss-Manin connection

Letd : Ω

•U1→Ω

•+1U1

be the differential map with respect to variable x, i.e. dr = 0 for all r ∈ R, and

d : Ω•U1→Ω

•+1U1

be the differential map with respect to the elements of R. It is the pull-back of thedifferential in U0. We have

d = d + d,

where d is the total differential mapping. Let s be a new parameter and S(s) be thediscriminant of f − s. We have

S( f ) =n+1

∑i=1

pi∂ f∂xi

, pi ∈ k[x]

or equivalently

S( f )dx = d f ∧η f , η f =n+1

∑i=1

(−1)i−1 pidxi. (10.4)

To calculate ∇ of

10.4 Calculating Gauss-Manin connection 99

ω =n+1

∑i=1

Pidxi ∈ H′

we assume that ω has no dr, r ∈ R, but the ingredient polynomials of ω may havecoeficients in R. Let ∆ = S(0) and

dω = P ·dx.

We have

S( f )dω = S( f )dω +S( f )n+1

∑i=1

dPi∧ dxi

= d f ∧ (P ·η f )+S( f )n+1

∑i=1

dPi∧ dxi.

This implies that

∆dω = (∆ −S( f ))(dω−n+1

∑i=1

dPi∧ dxi)+

d f ∧ (P ·η f )+(∆n+1

∑i=1

dPi∧ dxi)− d f ∧ (P ·η f )

= (∆n+1

∑i=1

dPi∧ dxi)− d f ∧ (P ·η f )

= ∑j

dt j ∧

(∆(

n+1

∑i=1

∂Pi

∂ t jdxi)−

∂ f∂ t j·P ·η f

).

all the equalities are in Ω 1U0⊗H′. We conclude that

∇(ω) = (10.5)

1∆

(∑

jdt j⊗

(n+1

∑i=1

(∆

∂Pi

∂ t j− (−1)i−1 ∂ f

∂ t j·P · pi

)dxi

)),

where

P =n+1

∑i=1

(−1)i−1 ∂Pi

∂xi.

It is useful to define∂ω

∂ t j=

n+1

∑i=1

∂Pi

∂ t jdxi.

Then

∇(ω) =1∆

(∑

jdt j⊗

(∆

∂ω

∂ t j− ∂ f

∂ t j·P ·η f

)). (10.6)


The calculation of ∇ in H′′ can be done using

∇(P ·dx) =d f ∧∇(Pη f )−d∆ ⊗Pdx

∆, Pdx ∈ H′′

which is derived from (10.1). Note that we calculate ∇(P ·η f ) from (10.5). We leadto the following explicit formula

∇(P ·dx) = (10.7)

1∆

(∑

jdt j⊗

(d f ∧

∂ (Pη f )

∂ t j− ∂ f

∂ t jQP−

∂∆

∂ t jP))

,

where

QP =n+1

∑i=1

(∂P∂xi

pi +P∂ pi

∂xi).

To be able to calculate the iterations of the Gauss-Manin connection along a vectorfield v in U0, it is useful to introduce the operators:

∇v,k : H→ H, k = 0,1,2, . . .

∇v,k(ω) = ∇v(ω

∆ k )∆k+1 = ∆ ·∇v(ω)− k ·d∆(v) ·ω.

It is easy to show by induction on k that

∇kv =

∇v,k−1 ∇v,k−2 · · · ∇v,0

∆ k . (10.8)

Remark 15 The formulas (10.7) and (10.6) for the Gauss-Manin connection usu-ally produce polynomials of huge size, even for simple examples. Specially whenwe want to iterate the Gauss-Manin connection along a vector field, the size of poly-nomials is so huge that even with a computer (of the time of writing this text) we getthe lack of memory problem. However, if we write the result of the Gauss-Maninconnection, in the canonical basis of the R-module H, and hence reduce it moduloto those differential forms which are zero in H, we get polynomials of reasonablesize.

10.5 R[θ ] structure of H ′′

In this section we consider the R[s]-modules H ′′ and H ′, where sω := f ω . We havethe following well-defined map:

θ : H ′′→ H ′, θω = η , where ω = dη .

10.6 Gauss-Manin system 101

We have used the fact that HndR(U1/U0) = 0 (see Proposition 19). It is well-defined

because:

d f ∧dη1 = dη2⇒ η2 = d f ∧η1 +dη3, for some η3 ∈Ωn−1U1/U0

.

Using the inclusion H ′ → H ′′, ω 7→ d f ∧ω , both H ′ and H ′′ are now R[s,θ ]-modules. The relation between R[s] and R[θ ] structures is given by:

Proposition 37 We have:θ · s = s ·θ −θ ·θ

and for n ∈ Nθ

ns = sθn−nθ

n+1.

Proof. The map d : H ′→ H ′′ satisfies

d · s = s ·d +d f ,

where s stands for the mapping ω 7→ sω and d f stands for the mapping ω 7→ d f ∧ω, ω ∈ H ′. Composing the both sides of the above equality by θ we get the firststatement. The second statement is proved by induction.

For a homogeneous polynomial with an isolated singularity at the origin we havedηβ = Aβ ωβ and so

θωβ =s

Aβ

ωβ .

Remark 16 The action of θ on H ′′ is inverse to to the action of the Gauss-Maninconnection with respect to the parameter s in f − s = 0 (we have composed theGauss-Manin connection with ∂

∂ s ). This arises the following question: Is it possibleto construct similar structures for H′ and H′′?

10.6 Gauss-Manin system

The Gauss-Manin connection on M is the map

∇ : M→Ω1U0⊗RM

which is obtained by derivation with respect to the elements of R (the derivation ofxi is zero). By definition it maps Mi to Ω 1

U0⊗RMi+1 For any vector field in U0, ∇v

is given by

∇v : M→M, ∇v([Pdx

f i ]) = [v(P) · f − iP · v( f )

f i+1 dx], P ∈ R[x]. (10.9)

where v(P) is the differential of P with respect to elements in R and along the vectorfield v (v(·) : R→ R, p 7→ d p(v)). In the case i = 0 it is given by


∇v([d f ∧ω

f]) = [

f · v(d f ∧ω)− v( f ) ·d f ∧ω

f 2 ]

= [v(d f ∧ω)+d(v( f ) ·ω)

f]

and so ∇v maps M0 to M1. The operator ∇v is also called the Gauss-Manin con-nection along the vector field v. To see the relation of the Gauss-Manin connectionof this section with the Gauss-Manin connection of §10.1 we need the followingproposition:

Proposition 38 Suppose that the discriminant ∆ of the tame polynomial f is notzero. Then the multiplication by ∆ in M maps Mi to Mi−1 for all i ∈ N.

Proof. The multiplication by ∆ in W is zero and so for a given ω

f i we can write

∆ω

f i =f ω1 +d f ∧ω2

f i =ω1

f i−1 +1

i−1(

dω2

f i−1 −d(ω2

f i−1 ))

which is equal toω1+

1i−1 dω2

f i−1 in M.

Now, it is easy to see that ∆ ·∇v : H→ H, H = H′,H′′ of this section and §10.1coincide. Recall that for a R-module M and a∈M, Ma denotes the localization of Mover the multiplicative set 1,a,a2, · · ·. As a corollary of Proposition 38 we have:

Proposition 39 The inclusion H→M induces an isomorphism of R-modules M∆∼=

H∆ .

10.7 Griffiths transversality

In the free module H we have introduced the mixed Hodge structure and the Gauss-Manin connection. It is natural to ask whether there is any relation between thesetwo concepts or not. The answer is given by the next theorem. First, we give adefinition

Definition 27 A vector field v in U0 is called a basic vector field if for any p ∈ Rthere is k ∈N such that vk(p) = 0, where vk is the k-th iteration of v(·) : R→R, p 7→d p(v).

For R= C[t1, t2, . . . , ts], the vector fields ∂

∂ ti, i = 1,2, . . . ,s are basic.

Theorem 19 Let (W•,F•) be the mixed Hodge structure of H. The Gauss-Maninconnection on H satisfies:

1. Griffiths transversality:

∇(Fi)⊂Ω1T ⊗R F

i−1, i = 1,2, . . . ,n.

10.7 Griffiths transversality 103

2. No residue at infinity: We have

∇(Wn)⊂Ω1T ⊗RWn.

3. Residue killer: For a tame polynomial f of degree d, ω ∈ H and a basic vectorfield v∈DU0 such that deg(v( f ))< d there exists k ∈N such that ∇k

vω ∈WnM∆ .

Griffiths transversality has been proved in [Gri68b, Gri68a] for Hodge structures.For a recent text see also [Voi02b]. The proof for mixed Hodge structures is similarand can be found in [Zuc84, Zuc87].

Proof. It is enough to prove the theorem for the Gauss-Manin connection ∇v alonga vector field v ∈DU0 and the mixed Hodge of M∆ .

For the Griffiths transversality, we have to prove that ∇v maps FiM∆ to Fi−1M∆ .By Leibniz rule, it is enough to take an element ω =

ωβ

f k , Aβ ≤ k in the set (8.8) and

prove that ∇vω is in Fi−1M. This follows from (10.9) and:

∇vωβ

f k =v( f )ωβ

f k+1 ,

deg(v( f )ωβ

f k+1 )≤ deg(ωβ

f k ) = d(Aβ − k)≤ 0

For the second part of the theorem we have to prove that ∇v maps WnM∆ to WnM∆ .This follows follows form (8.7) and the fact that

ωβ

f k , Aβ < k generate WnM∆

For the third part of the theorem we proceed as follows: For ω ∈ M we useProposition 8.5 and write ω as a R-linear combination of

ωβ

f k , β ∈ I, Aβ ≤ k. By the

second part of the theorem, it is enough to prove that forωβ

f k , Aβ = k and p ∈ R,there exists some s ∈ N such that

∇sv(p[

ωβ

f k ]) ∈WnM∆

Since deg(v( f ))< d we have

deg∇v[ωβ

f k ]< deg(ωβ

f k ) = 0

and so ∇v[ωβ

f k ] ∈WnM∆ . Now modulo WnM∆ we have

∇kv(p[

ωβ

f k ]) = vk(p) · [ωβ

f k ]

and the affirmation follows from the fact that v is a basic vector field.

Definition 28 We say that a polynomial g ∈ R[x] does not depend on R (or param-eters in R) if all the coefficients of g lies in the kernel of the map d : R→ Ω 1

U0. In

other words, v(g) = 0 for all vector field v ∈DU0 .


In the case R := Q[t1, t2, . . . , ts] the above definition simply means that in g the pa-rameters ti, i = 1,2, . . . ,s do not appear. In this case, for a tame polynomial f over Rsuch that its last homogeneous piece g does not depend on R, all v = ∂

∂ ti’s are basic

and deg(v( f ))≤ deg( f ). In practice, we use this as an example for the third part ofTheorem 19.

Chapter 11Integrals

In this chapter we unify the material of Chapters 7, 5 and 10 in order to studyintegrals of algebraic differential forms over topological cycles. We will also discusssome methods for reducing higher dimensional integrals to lower dimensional ones.

11.1 Notations

We fix a finite number of elements ai, i = 1,2, . . . ,r of the polynomial ring Q[t],t = (t1, t2, . . . , ts) a multi parameter, and we assume that R is the localization of Q[t]over the multiplicative group generated by ai’s. As before, f is a tame polynomialin R[x] and we will freely use the notations related to f introduced in §7.2. We have

U0 := Cs\(∪ri=1t ∈ Cs | ai(t) = 0)

andT := U0\t ∈ U0 | ∆(t) = 0,

where ∆ is the discriminant of f . In particular, Ω iT is the set of algebraic i-forms in

T .For a fixed value c ∈ U0 of t, we denote by fc the polynomial obtained by re-

placing c instead of t in f . By a topological cycle δ ∈ Hn( f = 0,Z) we meana continuous family of cycles δtt∈U , δt ∈ Hn( ft = 0,Z), where U is a smallneighborhood in T .

The integral∫δ

ω :=∫

δt

(ω | ft=0

), ω ∈ H′, δ ∈ Hn( f = 0,Z)

is well-defined, i.e. it does not depend on the choice of the differential form (resp.cycle) in the class ω (resp. in the homology class δ ). In the case ω ∈H′′ by

∫δ

ω wemean

∫δ

ω

d f , where the Gelfand-Leray form ω

d f is defined in §7.3. The integral∫

δω

105

106 11 Integrals

is a holomorphic function in U and it can be extended to a multi-valued holomorphicfunction in T .

In the zero dimensional case n = 0, recall that H0( f = 0,Z) is the set of allfinite sums ∑i ri[xi], where ri ∈ Z,∑i ri = 0 and xi’s are the roots of f . We define∫

δ

ω := ∑i

riω(xi),

whereω ∈ H′, δ = ∑

irixi ∈ H0( f = 0,Z),

and call them (zero dimensional Abelian) integrals/periods.

11.2 Integrals and Gauss-Manin connections

The following proposition gives us the most important property of the Gauss-Maninconnection related to integrals.

Proposition 40 Let U be a small open set in T and δtt∈U ,δt ∈ Hn( ft = 0,Z)be a continuous family of topological n-dimensional cycles. Then

d(∫

δt

ω) =µ

∑i=1

αi

∫δt

ωi, ω ∈ H, (11.1)

where

∇ω =µ

∑i=1

αi⊗ωi, αi ∈Ω1T , ωi ∈ H,

and ωi’s form a R-basis of H.

See [AGZV88] for similar statements in the local context and their proof.

Proof. By Theorem 12 a distinguished set of vanishing cycles generate the n-thcohomology of f = 0 and so we assume that δt is a vanishing cycle in a smoothpoint c of the variety ∆ = 0. Therefore, there exists an n+ 1-dimensional realthimble

Dt = ∪s∈[0,1]δγt (s)×γt(s) ⊂ Cn+1×Cs

such that γt is a path in U0 connecting t to c and δγt (s) is the trace of δt when itvanishes along γt . In order to define the Gauss-Manin connection of ω ∈Ω

n+1U1/U0

wewrote

dω−∑i

αi∧ωi ∈ f Ωn+1U1

+d f ∧ΩnU1, αi ∈Ω

1T , ωi ∈Ω

nU1.

Since f |Dt= 0, the integral of the elements of f Ωn+1U1/U0

+d f ∧Ω nU1/U0

on Dt is zeroand we have

11.3 Period matrix 107∫δt

ω =∫

Dt

dω

= ∑i

∫Dt

αi∧ωi

=∫

γt (s)

(∑

iαi

∫δγt (s)

ωi

).

In the first equality we have used Stokes Lemma and in the last equality we haveused integration by parts. Taking the differential of the above equality we get thedesired equality.

Remark 17 From (11.1) it follows that

v(∫

δt

ω) =∫

δt

∇vω, ∀ω ∈ H, v ∈DU0 (11.2)

for any continuous family of cycles δt in a small neighborhood in T . For a fixedv, the operator ∇v : H→ H∆ with the above property is unique. This follows fromthe fact that if ω ∈ H restricted to all regular fibers of f is exact then ω is zero inH (a consequence of Corollary 1). If we want to prove an equality for the Gauss-Manin connection of a tame polynomial f over the function field introduced at thebeginning of this chapter then we may use (11.2). The proof of the same equalityfor an arbitrary R of Chapter 7 demands only algebraic methods.

11.3 Period matrix

Let ω = (ω1,ω2, . . . ,ωµ)tr be a basis of of the free R-module H. In this basis we

can write the matrix of the Gauss-Manin connection ∇:

∇ω = A⊗ω, A ∈Matµ×µ(Ω 1T ).

A fundamental matrix of solutions for the linear differential equation

dY = A ·Y (11.3)

(with Y a µ×1 unknown matrix function defined in an small open neighborhood inU0\∆ = 0 ) is given by Y = Ptr, where

P(t) = [∫

δ

ωtr] =

∫

δ1ω1

∫δ1

ω2 · · ·∫

δ1ωµ∫

δ2ω1

∫δ2

ω2 · · ·∫

δ2ωµ

......

......∫

δµω1∫

δµω2 · · ·

∫δµ

ωµ

, (11.4)

108 11 Integrals

and δ = (δ1,δ2, . . . ,δµ)tr is a basis of the Z-module H0( f = 0,Z). This follows

from Proposition 40. The matrix P is called the period matrix of f (in the basis δ andω). Looking P as a function matrix in t, it is also called the period map. By Theorem12 we know that δ can be chosen as a distinguished set of vanishing cycles.

Proposition 41 Let ∆i, i= 1,2, . . . ,m be the irreducible components of the discrim-inant of a tame polynomial in R[x]. We have

det(P)2 = c ·∆ k11 ∆

k22 · · ·∆

kmm

for some non-zero constant c and k1,k2, . . . ,km ∈ Z.

Proof. First, we prove that det(P)2 is a one-valued function in T . If δ ′ is anotherbasis of Hn( f = 0,Z) obtained by the monodromy of δ then

δ′ = Aδ , AΨ0Atr =Ψ0,

where Ψ0 is the intersection matrix of Hn( f = 0,Z) in the basis δ . This impliesthat det(A)2 = 1 and so det(P)2 is a one-valued function in T . Since our integralshave a finite growth at infinity and ∆ = 0 we conclude that det(P)2 is rationalfunction in U0 with poles along ∆ = 0. It does not have zeros outside ∆ = 0and so it must be of the desired form.

11.4 Picard-Fuchs equation

We saw in §10.2 that for ω ∈H and v ∈DU0 a vector field in U0, there exists m≤ µ

and pi ∈R, i= 0,1,2, . . . ,m such that we have the Picard-Fuchs equation of ω alongv:

p0ω + p1∇v(ω)+ p2∇2v(ω)+ · · ·+ pm∇

mv (ω) = 0 (11.5)

For δ ∈ Hn( f = 0,Z) we take∫

δof the above equality, we use the equality

v(∫

δ·) =

∫δ

∇v(·) and finally we conclude that the analytic functions∫δ

ω, δ ∈ Hn( f = 0,Z) (11.6)

satisfy the linear differential equation:

p0(t)y+ p1(t)y′+ p2(t)y′′+ · · ·+ pm(t)y(m) = 0, (11.7)

y′ := dy(v)

In fact, they span the µ-dimensional vector space of the solutions of (11.7). Thisfollows from the fact that the period matrix (11.4) is a fundamental system for thelinear differential equation (11.3). The number m is called the order of the differen-tial equation (11.7). If m = µ then the integrals (11.6) form a basis of the solutionspace of (11.7).

11.6 Homogeneous polynomials 109

Remark 18 Note that if v = ∂

∂ ti, i = 1,2, . . . ,s then y′ means the derivation with

respect to the parameter ti. Almost all the examples of Picard-Fuchs equations inthe literature are obtained by such vector fields.

11.5 Modular foliations and integrals

As a corollary of Proposition 40 we have:

Proposition 42 The leaves of the modular foliation Fω , ω ∈ H are the loci ofparameters in which the integrals

∫δ

ω, δ ∈ Hn( f = 0,Z) are constant.

Proof. Let ∆ =(δ1,δ2, . . . ,δµ)tr be a Z-basis of Hn( f = 0,Z) and ω =(ω1,ω2, . . . ,ωµ)

tr

be a basis of the free R-module H. Let also P be the corresponding period matrix.By Proposition 40 we have

[d(∫

δ1

ω),d(∫

δ2

ω), · · · ,d(∫

δµ

ω)]tr = P · [α1,α2, . . . ,αµ ]tr,

where ∇ω = ∑µ

i=1 αi⊗ωi, αi ∈ Ω 1T , ωi ∈ H. By Proposition 41, the period ma-

trix has a non-zero determinant outside of ∆ = 0 and the foliation induced byd(∫

δiω), i = 1,2, . . . ,µ and αi, i = 1,2, . . . ,µ are the same.

11.6 Homogeneous polynomials

For a homogeneous polynomial g(x) = g(x1,x2, . . . ,xn+1) let us define:

p(β ,δ ) :=∫

δ

ωβ ∈ C, (11.8)

where ωβ := xβ dx, xβ a monomial in x, and δ ∈ Hn( f = 1,Z). We define f :=g− t ∈ R[t][x] which is tame and its discriminant is (−t)µ . We have

∇ ∂

∂ t(ωβ ) =

(Aβ −1)t

ωβ

and so∂

∂ t

∫δt

ωβ =Aβ −1

t

∫δt

ωβ .

Therefore ∫δt

ωβ = p(β ,δ )tAβ−1. (11.9)

Here we have chosen a branch of tAβ whose evaluation on 1 is 1. Using ηβ = tωβ

in H′′ of f − t we can obtain similar formulas for ηβ .

110 11 Integrals

11.7 Integration over joint cycles

The objective of this and the next section is to introduce techniques for simplifyingintegrals and in the best case to calculate them. For simplicity, we take the tamepolynomials over C but the whole discussion is valid for the tame polynomials de-pending on parameters as it is explained at the beginning of the present chapter.

Let f ∈ C[x] and g ∈ C[y] be two tame polynomials in n+1, respectively m+1,variables. Recall the definition of an admissible triple from §5.7.

Proposition 43 Let ω1 (resp. ω2) be an (n+1)-form (resp. (m+1)-form) in Cn+1

(resp. Cm+1). Let also (ts, s ∈ [0,1],δ1b,δ2b) be an admissible triple and

I1(ts) =∫

δ1,ts

ω1

d f, I2(ts) =

∫δ2,ts

ω2

dg.

Then ∫δ1b∗t.δ2b

ω1∧ω2

d( f −g)=∫

ts,s∈[0,1]I1(ts)I2(ts)dts

Proof. We have

ω1∧ω2 = d f ∧ ω1

d f∧dg∧ ω2

dg

= d( f −g)∧ ω1

d f∧dg∧ ω2

dg

and so restricted to the variety

X := (x,y) ∈ Cn+1×Cm+1 | f (x)−g(y) = 0

we haveω1∧ω2

d( f −g)=

ω1

d f∧dt ∧ ω2

dg,

where t is the holomorphic function on X defined by t(x,y) := f (x) = g(y). Now,the proposition follows by integration in parts.

Recall the B-function

B(a,b) =Γ (a)Γ (b)Γ (a+b)

=∫ 1

0sa−1(1− s)b−1ds, a,b,∈Q.

and its multi parameter form:

B(a1,a2, · · · ,ar) =Γ (a1)Γ (a2) · · ·Γ (ar)

Γ (a1 +a2 + · · ·+ar).

Proposition 44 Let f (x1,x2, . . . ,xn+1) and g(y1,y2, . . . ,ym+1) be two tame homoge-neous polynomials. Let also (ts, s ∈ [0,1],δ1b,δ2b) be an admissible triple, xβ1 be a

11.8 Reduction of integrals 111

monomial in x and yβ2 be a monomial in y. We have

p( f +g = 1,(β1,β2),δ1 ∗t. δ2) =

p( f = 1,β1,δ1)p(g = 1,β2,δ2)B(Aβ1 ,Aβ2)

Proof. In Proposition 43 let us replace g with −g+1. We use (11.9) and we have∫δ1∗t.δ2

xβ1yβ2dx∧dyd( f +g)

= p( f = 1, β1,δ1)p(g = 1,β2,δ2) ·∫ 1

0sAβ1

−1(1− s)Aβ2−1ds

= p( f = 1, β1,δ1)p(g = 1,β2,δ2) ·B(Aβ1 ,Aβ2).

As a corollary of the above proposition we have:

Proposition 45 For zero dimensional cycles

δi = [ai]− [bi] ∈ H0(xmii −1,Z)

we have

p(xm11 + xm2

2 + · · ·+ xmn+1n+1 = 1,(β1,β2, . . . ,βn+1),δ1 ∗δ2 ∗ · · · ∗δn+1) =

(∫

δ1

xβ11 dx1

dxm11

)(∫

δ2

xβ22 dx2

dxm22

) · · ·(∫

δn+1

xβn+1n+1 dxn+1

dxmn+1n+1

)·

B(β1 +1

m1,

β2 +1m2

, · · · , βn+1 +1mn+1

) =

1m1m2 · · ·mn+1

(aβ1+11 −bβ1+1

1 )(aβ2+12 −bβ2+1

2 ) · · ·(aβn+1+1n+1 −bβn+1+1

n+1 )·

B(β1 +1

m1,

β2 +1m2

, · · · , βn+1 +1mn+1

)

Proof. Successive uses of Proposition 44 will give us the desired equality of theproposition.

11.8 Reduction of integrals

In this section we describe some simple rules for reducing a higher dimensionalintegral to a lower dimensional one.

Proposition 46 Let f (x) = f (x1,x2, . . . ,xn+1) be a tame polynomial and g(y) =g(y1,y2, . . . ,ym+1) be a homogeneous tame polynomial. Let δ1 ∈ Hn( f = 1,Z),

112 11 Integrals

δ2 ∈ Hm(g = 1,Z), xβ1 be a monomial in x and yβ2 be a monomial in y. Let alsots, s ∈ [0,1] is a path in the C-plane which connects a critical value of f to 0 (theunique critical value of g). We assume that δ1 vanishes along t−1

. and δ2 vanishesalong t.. Then we have

∫δ1∗t.δ2

xβ1yβ2dx∧dyd( f −g)

=

p(β2,δ2)p(β3,δ3)

∫δ1∗t.δ3

xβ1 zβ3 dx∧dzd( f−zq) Aβ2 6∈ N

p(β2,δ2)∫

δ1θ( f

Aβ2−1

xβ1 dxd f ) Aβ2 ∈ N

In the first case q and β3 are given by the equality Aβ2 =β3+1

q and δ3 is any cycle in

H0(zq = 1,Z) with p(β3,δ3) 6= 0. In the second case, δ1 ∈ Hn( f = 0,Z) is themonodromy of δ1 along the path ts, s ∈ [0,1] and θ is the operator in §10.5.

Proof. Using Proposition 43 we have:∫δ1∗δ2

xβ1yβ2dx∧dyd( f −g)

= p(β2,δ2)∫

ts, s∈[0,1]tAβ2

−1I1(ts)dts

= p(β2,δ2)∫

ts, s∈[0,1]t

β3+1q −1I1(ts)dts,

where I1(ts) :=∫

δ1,ts

xβ1 dxd f . We consider two cases: If Aβ2 6∈ N then we can choose a

cycle δ3 ∈ H0(zq = 1,Z) such that p(β3,δ3) 6= 0 and so

tβ3+1

q =1

p(β3,δ3)I3(t), I3(t) :=

∫δ3,t.

zβ3dzdzq .

We again use Proposition 43 and get the desired equality.If Aβ2 ∈ N then zβ3 is zero in H′′ of the tame one variable polynomial zq− t and

we cannot repeat the argument of the first part. In this case we have

= p(β2,δ2)∫

ts, s∈[0,1](∫

δ1,ts

f Aβ2−1xβ1dxd f

)dts

= p(β2,δ2)∫

∆

f Aβ2−1xβ1dx

= p(β2,δ2)∫

δ1

θ(f Aβ2

−1xβ1dxd f

),

where∆ := ∪s∈[0,1]δ1,ts ∈ Hn+1(Cn+1, f−1(0),Z)

is the Lefschetz thimble with boundary δ1.

Proposition 47 With the notations of Proposition 46

11.9 Residue map 113

∫δ1∗t.δ2

xβ1yβ2dx∧dy( f −g)k =

p(β2,δ2)p(β3,δ3)

∫δ1∗t.δ3

xβ1 zβ3 dx∧dz( f−zq)k Aβ2 6∈ N

p(β2,δ2)∫

δ1

fA

β2−1

xβ1 dxf k−1 Aβ2 ∈ N

for k ≥ 2.

Proof. We assume that f is of the form f −a and use the equality

1(k−1)!

∂ k−1

∂ak−1

∫δ1∗t.δ2

xβ1yβ2dx∧dy( f −g)

=∫

δ1∗t.δ2

xβ1yβ2dx∧dy( f −g)k

and Proposition 46.

11.9 Residue map

Let us be given a closed submanifold N of real codimension c in a manifold M. TheLeray (or Thom-Gysin) isomorphism is

τ : Hk−c(N,Z)→Hk(M,M−N,Z)

holding for any k, with the convention that Hs(N) = 0 for s < 0. Roughly speaking,given δ ∈Hk−c(N), its image by this isomorphism is obtained by thickening a cyclerepresenting δ , each point of it growing into a closed c-disk transverse to N in M (seefor instance [Che96] p. 537). Let N be a connected codimension one submanifoldof the complex manifold M of dimension n. Writing the long exact sequence of thepair (M,M−N) and using τ we obtain:

· · · → Hn+1(M,Z)→ Hn−1(N,Z) σ→ (11.10)

Hn(M−N,Z) i→ Hn(M,Z)→ ···

where σ is the composition of the boundary operator with τ and i is induced byinclusion. Let ω ∈ Hn(M−N,C) := Hn(M−N,Z)⊗C, where Hn(M−N,Z) isthe dual of Hn(M−N,Z) . The composition ω σ : Hn−1(N,Z) → C defines alinear map and its complexification is an element in Hn−1(N,C). It is denoted byResiN(ω) and called the residue of ω in N. We consider the case in which ω in the n-th de Rham cohomology of M−N is represented by a meromorphic C∞ differentialform ω ′ in M with poles of order at most one along N. Let fα = 0 be the definingequation of N in a neighborhood Uα of a point p ∈ N in M and write ω ′ = ωα ∧ d f

f .For two such neighborhoods Uα and Uβ with non empty intersection we have ωα =ωβ restricted to N. Therefore, we get a (n− 1)-form on N which in the de Rhamcohomology of N represents ResiNω (see [Gri69] for details). This is called thePoincare residue.

114 11 Integrals

Let us be given a tame polynomial f , c ∈ T := U0\∆ = 0 and ω ∈ Ωn+1U1/U0

.We can associate to ω

f k , k ∈ N its residue in Lc which is going to be an elementof Hn(Lc,C) (we first substitute t with c in ω

f and then take the residue as it isexplained in the previous paragraph). This gives us a global section Resi[ ω

f k ] of then-th cohomology bundle of the fibration f over T . It is represented by the element[ ω

f k ] ∈ M, where M is defined in §8. Having Proposition 38 in mind, we regard

Resi( ω

f k ) as an element in the localization of H over 1,∆ ,∆ 2, . . .. In the case k = 1,Resi(ω

f ) = [ω] ∈ H′′.If v is a vector field in U0 then we have

v∫

δ

Resi(ω

f k ) = v∫

σ(δ )

ω

f k

=∫

σ(δ )∇v([

ω

f k ])

=∫

δ

Resi(∇v([ω

f k ]))

and soResi(∇v([

ω

f k ]) = ∇v(Resi([ω

f k ]).

11.10 Geometric interpretation of Theorem 19

LetP(1,α) = [X0 : X1 : · · · : Xn+1] | (X0,X1, · · · ,Xn+1) ∈ Cn+2

be the projective space of weight (1,α), α = (α1, . . . ,αn+1). One can considerP(1,α) as a compactification of Cn+1 with coordinates (x1,x2, . . . ,xn+1) by putting

xi =Xi

Xαi0, i = 1,2, · · · ,n+1. (11.11)

The projective space at infinity Pα∞ =P(1,α)−Cn+1 is of weight α :=(α1,α2, . . . ,αn+1).

Let f ∈C[x] be a tame polynomial of degree d and g be its last quasi-homogeneouspart. We take the homogenization F = Xd

0 f ( X1X

α10, X2

Xα20, . . . ,

Xn+1

Xαn+10

) of f and so we can

regard f = 0 as an affine subvariety in F = 0 ⊂ P(1,α).

Proposition 48 For a monomial xβ with Aβ = k ∈ N, the meromorphic form xβ dxf k

has a pole of order one at infinity and its Poincare residue at infinity is Xβ ηα

gk . IfAβ < k then it has no poles at infinity.

Proof. Let us write the above form in the homogeneous coordinates (11.11). We used( Xi

Xαi0) = X−αi

0 dXi−αiXiX−αi−10 dX0 and

11.10 Geometric interpretation of Theorem 19 115

xβ dxf k =

=

( X1X

α10)β1 · · ·( Xn+1

Xαn+10

)βn+1d( X1X

α10)∧·· ·∧d( Xn+1

Xαn+10

)

f ( X1X

α10, · · · , Xn+1

Xαn+10

)k

=Xβ η(1,α)

X(∑n+1

i=1 βiαi)+(∑n+1i=1 αi)+1−kd

0 (X0F−g(X1,X2, · · · ,Xn+1))k

=Xβ η(1,α)

X0(X0F−g(X1,X2, · · · ,Xn+1))k

=dX0

X0∧ Xβ ηα

(X0F−g)k

The last equality is up to forms without pole at X0 = 0. The restriction of Xβ ηα

(X0F−g)k

to X0 = 0 gives us the desired form.If Aβ < 0 then the second equality above tells us that

ωβ

f k has no poles at infinityX0 = 0.

Proposition 48 shows that for a cycle δ ∈ Hn( f = 0,Z) at infinity we have

∫δ

xβ dxf k = 0, if Aβ < k

and ∫δ

xβ dxf k =

∫δ ′

Xβ ηα

gk , if Aβ = k

for some cycle δ ′ ∈ Hn(Pα\g = 0,Z). In particular, if the last homogeneous partg of f does not depend on any parameter of R then for Aβ = k the integral

∫δ

xβ dxf k is

constant. Now it is evident that Wn is the set of differential forms which do not haveany residue at infinity. This gives another proof of Theorem 19, part 2. The topolog-ical interpretation of part 3 is as follows: For simplicity we take R=Q[t1, t2, . . . , ts]and assume that g does not depend on the parameters in R. We write an ω ∈M inthe form (??) and for a cycle at infinity δ we see that∫

δ

ω = ∑Aβ=k,k∈N,β∈I

aβ ,k

∫δ

ωβ

f k

which is a polynomial in t1, t2, . . . , ts (according to the previous discussion∫

δ

ωβ

f k areconstant numbers and so the above polynomial has complex coefficients). Therefore,the n-th iterative derivation of

∫δ

ω with respect to ti must be zero for n bigger thanthe degree in ti of

∫δ

ω .

116 11 Integrals

Using the equality (??) we see that the first integrals of Fω that we have dis-cussed at the end of section (??) are the integration of ω over cycles at infinity (upto a constant).

Chapter 12Intersection of topological cycles

’!20 ! !1 2

!2

bc

! !1 20 ’ ’

Fig. 12.1 Two paths in C

Let us consider two tame polynomials f ,g∈C[x]. A critical value c of f is callednon-degenerated if the fiber f−1(c) contains only one singularity and the Milnornumber of that singularity is one. Around such a singularity f can be written in theform X2

1 +X22 + · · ·+X2

n+1 + c for certain local coordinate functions Xi.For two oriented paths t., t ′. in C which intersect each other at b transversally the

notation t.×+b t ′. means that t. intersects t ′. in the positive direction, i.e. dt.∧dt ′. is the

canonical orientation of C. In a similar way we define t.×−b t ′. (see Figure 12.1).

Theorem 20 Let (t.,δ1,δ2) and (t ′. ,δ′1,δ′2) be two admissible triples. Assume that t.

and t ′. intersect each other transversally in their common points and the start/endcritical points of t. and t ′. are non-degenerated. Then

〈δ1 ∗δ2,δ′1 ∗δ

′2〉= (−1)nm+n+m

∑b

ε1(b)〈δ1b,δ′1b〉〈δ2b,δ

′2b〉

where b runs through all intersection points of t. and t ′. ,

117

118 12 Intersection of topological cycles

ε1(b) =

1 t.×+b t ′. and b is not a start/end point

−1 t.×−b t ′. and b is not a start/end point

(−1)n(n−1)

2 t.×+b t ′. and b is a start point

(−1)n(n+1)

2 +1 t.×−b t ′. and b is a start point

(−1)m(m−1)

2 t.×+b t ′. and b is an end point

(−1)m(m+1)

2 +1 t.×−b t ′. and b is an end point

,

and by 〈0,0〉 we mean 1.

Proof. Let t. intersect t ′. transversally at a point b. Let also a1,a2,a′1,a′2 be the ori-

entation elements of the cycles δ1,δ2,δ′1,δ′2 and a and a′ be the orientation element

of t. and t ′. . We consider two cases:

1. b is not the end/start point of neither t. nor t ′. : Assume that the cycles δ1 and δ ′1(resp. δ2 and δ ′2) intersect each other at p1 (resp. p2) transversally. The cyclesγ = δ1 ∗ δ2 and γ ′ = δ ′1 ∗ δ2 intersect each other transversally at (p1, p2). Theorientation element of the whole space X obtained by the intersection of γ and γ ′

is:

a1∧a∧a2∧a′1∧a′∧a′2 = (−1)nm+n+m(a1∧a′1)∧ (a∧a′)∧ (a2∧a′2)

This is (−1)nm+n+m times the canonical orientation of X .2. b = c is, for instance, the start point of both t. and t ′. and δ1,δ

′1 vanish in the point

p1 ∈ Cn+1 when t tends to c. Assume that the cycles δ2 and δ ′2 intersect eachother transversally at p2. By assumption, p1 is a non-degenerated critical pointof f and so both cycles γ,γ ′ are smooth around (p1, p2) and intersect each othertransversally at (p1, p2). The orientation element of the whole space X obtainedby the intersection of γ and γ ′ is:

(a1∧a)∧a2∧ (a′1∧a′)∧a′2 = (−1)(n+1)m(a1∧a)∧ (a′1∧a′)∧a2∧a′2.

Note that a1 ∧ a has meaning and is the orientation of the thimble formed bythe vanishing of δ1 at p1. According to Proposition 11, (a1∧a)∧ (a′1∧a′) is the

canonical orientation of Cn+1 multiplied with ε , where ε = (−1)n(n+1)

2 if t.×+b t ′.

and = (−1)n(n+1)

2 +n+1 otherwise.

Remark 19 One can use Theorem 20 to calculate the intersection matrix of Hn(( f +g′)−1(b′),Z) in the basis given by Theorem 14. This calculation in the local case isdone by A. M. Gabrielov (see [AGZV88] Theorem 2.11). To state Gabrielov’s resultin the context of this text take f and g two tame polynomials such that the set C1can be separated from C2 by a real line in C. Then take b a point in that line. Theadvantage of our calculation is that it works in the global context and the vanishingcycles are constructed explicitly.

Remark 20 In Theorem 20 we may discard the assumption on the critical points inthe following way: In the case in which δ1 and δ ′1 (resp. δ2 and δ ′2) vanish on the

12 Intersection of topological cycles 119

same critical point, we assume that they are distinguished (see Definition 9). Notethat if two vanishing cycles vanish along transversal paths in the same singularitythen the corresponding thimbles are not necessarily transversal to each other, exceptwhen the singularity is non-degenerated.

Proposition 49 The self intersection of a vanishing cycle of dimension n is given by

(−1)n(n−1)

2 (1+(−1)n). (12.1)

Proof. By Proposition 13 a joint cycle of two vanishing cycle is also a vanishingcycle. We apply Theorem 20 in the case δ1 = δ ′1 and δ2 = δ ′2 and conclude that theself intersection an of a vanishing cycle of dimension n satisfies

an+m+1 = (−1)nm+n+m((−1)n(n−1)

2 am +(−1)m(m+1)

2 +1an),

a0 = 2, n,m ∈ N0.

It is easy to see that (12.1) is the only function with the above property.

Example 13 (Stabilization) We take g = y21 + y2

2 + · · ·+ y2m+1 and f an arbitrary

tame polynomial. Let δ1,δ2, · · · ,δµ be a distinguished set of vanishing cycles inHn( f−1(0),Z) and δ be the vanishing cycle in Hn( f−1(0),Z) (up to multiplicationby ±1 it is unique). The intersection form in the basis δi = δi ∗δ is given by

〈δi, δ j〉= (−1)nm+n+m+m(m−1)

2 〈δi,δ j〉, i > j,

〈δi, δ j〉= (−1)nm+n+m+m(m+1)

2 +1〈δi,δ j〉, i < j,

(see [AGZV88] Theorem 2.14). Now let us assume that m = 0 and n = 1. Chooseδi’s as in Example 2. In this basis the intersection matrix is:

Ψ0 =

2 −1 0 0 · · · 0 0−1 2 −1 0 · · · 0 00 −1 2 −1 · · · 0 0...

......

......

......

0 0 0 0 · · · 2 −10 0 0 0 · · · −1 2

.

The intersection matrix in the basis δi, i = 1,2, · · · ,µ is of the form:

Ψ0 =

0 1 0 0 · · · 0 0−1 0 1 0 · · · 0 00 −1 0 1 · · · 0 0...

......

......

......

0 0 0 0 · · · 0 10 0 0 0 · · · −1 0

.

120 12 Intersection of topological cycles

As an exercise, construct a symplectic basis of the Riemann surface X using thebasis δi, i = 1,2, . . . ,µ and its intersection matrix.

Fig. 12.2 Dynkin diagram of x5 + y4

Example 14 We consider f and g = b′−g′, where f and g′ are two homogeneoustame polynomials. The point 0∈C (resp. b′ ∈C) is the only critical value of f (resp.g) and so, up to homotopy, there is only one path connecting 0 to b′. We choose thestraight piece of line ts = sb′, 0 ≤ s ≤ 1 as the path for our admissible triples. Fora point b between 0 and b′ in t· we choose a distinguished set of vanishing cyclesδi, i = 1,2 . . . ,µ1 (resp. γ j, j = 1,2 . . . ,µ2) of f (resp. g) in the fiber f−1(b) (resp.g−1(b)). By Theorem 14, the cycles

δi ∗ γ j, i = 1,2, . . . ,µ1, j = 2, . . . ,µ2

generate H1( f +g = b′,Z). The intersection matrix in this basis is given by

〈δi ∗ γ j,δi′ ∗ γ j′〉=sgn( j′− j)n+1(−1)(n+1)(m+1)+ n(n+1)

2 〈γ j,γ j′〉 if i′ = i & j′ 6= j

sgn(i′− i)m+1(−1)(n+1)(m+1)+m(m+1)2 〈δi,δi′〉 if j′ = j & i′ 6= i

sgn(i′− i)(−1)(n+1)(m+1)〈δi,δi′〉〈γ j,γ j′〉 if (i′− i)( j′− j)> 00 if (i′− i)( j′− j)< 0

Example 15 In the case f := xm1 and g := b′− ym2 ,

δi := [ζ i+1m1

b1

m1 ]− [ζ im1

b1

m1 ], i = 0, . . . ,m1−2

(resp.

γ j := [ζ j+1m2

(b′−b)1

m2 ]− [ζ jm1(b′−b)

1m2 ], j = 0, . . . ,m2−2)

is a distinguished set of vanishing cycles for H0( f = b,Z) (resp. H0(g = b,Z)), where we have fixed a value of b

1m1 and b

1m2 . See Figure (5.2) for a tentative

picture of the join cycle δi ∗ γ j with δi = x−y and γ j = x′−y′. The upper triangle ofintersection matrix in this basis is given by:

〈δi ∗ γ j,δi′ ∗ γ j′〉=

12 Intersection of topological cycles 1211 if (i′ = i & j′ = j+1)∨ (i′ = i+1 & j′ = j)−1 if (i′ = i & j′ = j−1)∨ (i′ = i+1 & j′ = j+1)0 otherwise

.

This shows that Figure 12.2 is the associated Dynkin diagram.

Example 16 The calculation of the Dynkin diagram of tame polynomials of thetype g = xm1 + xm2 + · · ·+ xmn+1 is done first by F. Pham (see [AGZV88] p. 66).It follows from Example 14 and by induction on n that the intersection map in thebasis δβ , β ∈ I of Example 4 is given by:

〈δβ ,δβ ′〉= (−1)n(n+1)

2 (−1)Σn+1k=1 β ′k−βk , (12.2)

β = (β1,β2, . . . ,βn+1),β′ = (β ′1,β

′2, . . . ,β

′n+1)

for βk ≤ β ′k ≤ βk +1, k = 1,2, . . . ,n+1,β 6= β ′, and

〈δβ ,δβ 〉= (−1)n(n−1)

2 (1+(−1)n), β ∈ I.

In the remaining cases, except those arising from the previous ones by a permuta-tion, we have 〈δβ ,δβ ′〉= 0.

Exercises1. For a complex manifold of dimension n and an holomorphic nowhere vanishing differential

n-form ω on it, the orientation obtained from 1(−2√−1)n ω ∧ω differs from the canonical one

by (−1)n(n−1)

2 . Compare also the orientation Re(ω)∧ Im(ω) with the canonical one. One canassume that the complex manifold is (Cn,0) and ω = dz1∧dz2∧·· ·∧dzn.

2. For a tame polynomial f , the Gelfand-Leray form dxd f in each regular fiber of f is an holomor-

phic nowhere vanishing differential n-form.3. Recall the notations of Exercise 3 in Chapter 3 for n = 1,m1 = m2 = d, that is, we are deal-

ing with U : xd + yd = 1. In a similar way, we describe the vanishing cycles δβ ,t , t ∈ Cin the curves Ut : xd + yd = t. We let t ∈ [0,1] run from 1 to 0 in the real line and weobserve that δβ ,t vanishes at 0 ∈ C2. We consider an orientation for the Lefschetz thim-ble ∆β := ∪t∈[0,1]δβ ,t as it is described in §5.1. This is a two dimensional real submanifoldof C2, provided that we take smooth representations of δβ ,t ’s. From another side we havexd + yd = (x− ζ2dy)(x− ζ 3

2dy) · · ·(x− ζ2d−12d y), and so, we have complex lines x− ζ2dy = 0

(of real dimension two) which crosses 0 ∈ C2. Compute the intersection number of these lineswith the Lefschetz thimbles.

4. Prove that (5.2) for d = 3 is an equality. Is this true for d = 4? How about arbitrary d?

Chapter 13Infinitesimal variation of Hodge structures

13.1 Griffiths-Dwork method

Let us now assume that the polynomial g depends on the parameter t. We have

∂

∂ tPηα

gk = −k∂g∂ t Pηα

gk+1

and we can write the right hand side of the above equality in terms os the basis givenin Theorem 18. We only need to use(

n+1

∑i=1

Ai∂g∂xi

)ηα

gk+1 =1k

(n+1

∑i=1

∂Ai

∂xi

)ηα

qk + exact terms. (13.1)

We conclude that the map

δk : Fn−k+1/Fn−k+2→ Fn−k/Fn−k+1, k = 1,2, (13.2)

induced by the Gauss-Manin connection ∇ ∂

∂ tis given by multiplication by ∂g

∂ t . Fromthis we get the notion of infinitesimal variation of Hodge structures (IVHS) devel-oped by Carlson, Harris, Griffiths and Green in 1983, see page 183 of this article.

123

Chapter 14Cech cohomology

In chapter 2 we discussed the axiomatic approach to homology and cohomologytheories and we saw that the singular homology and cohomologies are examplesof such theories. In this Appendix we discuss another construction of cohomologytheory, namely Cech cohomology of constant sheaves. It has the advantage that it iseasy to calculate and it generalizes to the cohomology of sheaves. We assume thatthe reader is familiar with sheaves of abelian groups on manifolds.

14.1 Cech cohomology

A sheaf S of abelian groups on a topological space X is a collection of abeliangroups

S (U), U ⊂ X open

with restriction maps which satisfy certain properties, for instance see [Gun90]. Inparticular S (X) is called the set of global sections of S and the following equiva-lent notations

S (X) = Γ (X ,S ) = H0(X ,S ).

is used.It is not difficult to see that for an exact sequence of sheaves of abelian groups

0→S1→S2→S3→ 0.

we have0→S1(X)→S2(X)→S3(X)

and the last map is not necessarily surjective. In this section we want to constructabelian groups H i(X ,S ), i = 0,1,2 . . . , H0(X ,S ) = S (X) such that we have thelong exact sequence

0→H0(X ,S1)→H0(X ,S2)→H0(X ,S3)→H1(X ,S1)→H1(X ,S2)→H1(X ,S3)→

125

126 14 Cech cohomology

H2(X ,S1)→ ···

Let X be a topological space, S a sheaf of abelian groups on X and U = Ui, i∈I a covering of X by open sets. In this paragraph we want to define the Cechcohomology of the covering U with coefficients in the sheaf S . Let U p denotesthe set of p-tuples σ = (Ui0 , . . . ,Uip), i0, . . . , ip ∈ I and for σ ∈ U p define |σ | =∩p

j=0Ui j . A p-cochain f = ( fσ )σ∈U p is an element in

Cp(U ,S ) := ∏σ∈U p

H0(|σ |,S )

Let π be the permutation group of the set 0,1,2, . . . , p. It acts on U p in a canoni-cal way and we say that f ∈Cp(U ,S ) is skew-symmetric if fπσ = sign(π) fσ .Theset of skew-symmetric cochains form an abelian subgroup Cs(U ,S ).

For σ ∈ U p and j = 0,1, . . . , p denote by σ j the element in U p−1 obtained byremoving the j-th entry of σ . We have |σ | ⊂ |σ j| and so the restriction maps fromH0(|σ j|,S ) to H0(|σ |,S ) is well-defined. We define the boundary mapping

δ : Cps (U ,S )→Cp+1

s (U ,S ), (δ f )σ =p+1

∑j=0

(−1) j fσ j ||σ |

It is left to the reader to check that δ is well-defined, δ δ = 0 and so

0→C0s (U ,S )

δ→C1s (U ,S) δ→C2

s (U ,S )δ→C3

s (U ,S )δ→ ·· ·

can be viewed as cochain complexes, i.e. the image of a map in the complex is insidethe kernel of the next map.

Definition 29 The Cech cohomology of the covering U with coefficients in thesheaf S is the cohomology groups

H p(U ,S ) :=Kernel(Cp

s (U ,S) δ→Cp+1s (U ,S ))

Image(Cp−1s (U ,S) δ→Cp

s (U ,S )).

The above definition depends on the covering and we wish to obtain cohomologiesH p(X ,S ) which depends only on X and S . We recall that the set of all coveringsU of X is directed: U1 ≤ U2 if U1 is a refinement of U2, i.e. each open subset ofU1 is contained in some open subset of U2. For two covering U1 and U2 there isanother covering U3 such that U3 ≤U1 and U3 ≤U1. It is not difficult to show thatfor U1 ≤U2 we have a well-defined map

H p(U2,S )→ H p(U1,S ).

Now the Cech cohomology of X with coefficients in S is defined in the followingway:

H p(X ,S ) := dir limU H p(U ,S ).

14.4 Resolution of sheaves 127

14.2 How to compute Cech cohomologies

The covering U is called acyclic with respect to S if U is locally finite, i.e. eachpoint of X has an open neighborhood which intersects a finite number of open setsin U , and H p(Ui1 ∩·· ·∩Uik ,S ) = 0 for all Ui1 , . . . ,Uik ∈U and p≥ 1.

Theorem 21 (Leray lemma) Let U be an acyclic covering of a variety X. There isa natural isomorphism

Hµ(U ,S )∼= Hµ(X ,S ).

For a sheaf of abelian groups S over a topological space X , we will mainly useH1(X ,S ). Recall that for an acyclic covering U of X an element of H1(X ,S ) isrepresented by

fi j ∈S (Ui∩U j), i, j ∈ I

fi j + f jk + fki = 0, fi j = f ji, i, j,k ∈ I

It is zero in H1(X ,S ) if and only if there are fi ∈S (Ui), i∈ I such that fi j = f j− fi.

14.3 Acyclic sheaves

A sheaf S of abelian groups on a topological space X is called acyclic if

Hk(X ,S ) = 0, k = 1,2, . . .

The main examples of fine sheaves that we have in mind are the following: Let Mbe a C∞ manifold and Ω i

M∞ be the sheaf of C∞ differential i-forms on M.

Proposition 50 The sheaves Ω iM∞ , i = 0,1,2, . . . are acyclic.

Proof. The proof is based on the partition of unity and is left to the reader.

A sheaf SS is said to be flasque or fine if for every pair of open sets V ⊂U , therestriction map SS(U)→ SS(V ) is surjective.

Proposition 51 Flasque sheaves are acyclic.

See [Voi02b], p.103, Proposition 4.34.

14.4 Resolution of sheaves

A complex of abelian sheaves is the following data:

S • : S0 d0→ S1 d1→ ···dk−1→ Sk dk→ ···


where S k’s are sheaves of abelian groups and S k → S k+1 are morphisms ofsheaves of abelian groups such that the composition of two consecutive morphismis zero, i.e

dk−1 dk = 0, k = 1,2, . . . .

A complex S k,k ∈ N0 is called a resolution of S if

Im(dk) = ker(dk+1), k = 0,1,2, . . .

and there exists an injective morphism i : S →S 0 such that Im(i) = ker(d0). Wewrite this simply in the form

S →S •

Theorem 22 Let S be a sheaf of abelian groups on a topological space X andS → S• be an acyclic resolution of S , i.e. a resultion for which each S k is acyclic,then

Hk(X ,S )∼= Hk(Γ (X ,S •),d), k = 0,1,2, . . . .

whereΓ (X ,S •) : Γ (S0)

d0→ Γ (S1)d1→ ···

dk−1→ Γ (Sk)dk→ ·· ·

and

Hk(Γ (X ,S •),d) :=ker(dk)

Im(dk−1).

Let us come back to the sheaf of differential forms. Let M be a C∞ manifold. The deRham cohomology of M is defined to be

H idR(M) = Hn(Γ (M,Ω i

M∞),d) :=global closed i-forms on Mglobal exact i-forms on M

.

Theorem 23 (Poincare Lemma) If M is a unit ball then

H idR(M) =

R if i = 00 if i = 0

The Poincare lemma and Proposition 50 imply that

R→Ω•M

is the resolution of the constant sheaf R on the C∞ manifold M. By Proposition 22we conclude that

H i(M,R)∼= H idR(M), i = 0,1,2, . . .

where H i(M,R) is the Cech cohomology of the constant sheaf R on M.

14.6 Dolbeault cohomology 129

14.5 Cech cohomology and Eilenberg-Steenrod axioms

Let G be an abelian group and M be a polyhedra. We can consider G as the sheaf ofconstants on M and hence we have the Cech cohomologies Hk(X ,G), k = 0,1,2, . . ..This notation is already used in Chapter 2 to denote a cohomology theory withcoefficients group G which satisfies the Eilenberg-Steenrod axioms. The followingtheorem justifies the usage of the same notation.

Theorem 24 In the category of polyhedra the Cech cohomology of the sheaf ofconstants in G satisfies the Eilenberg-Steenrod axioms.

Therefore, by uniqueness theorem the Cech cohomology of the sheaf of constantsin G is isomorphic to the singular cohomology with coefficients in G. We presentthis isomorphism in the case G = R or C.

Recall the definition of integration from Chapter ??

Hsingi (M,Z)×H i

dR(M)→ R, (δ ,ω) 7→∫

δ

ω

This gives usH i

dR(M)→ Hsingi (M,R)∼= H i

sing(M,R)

Theorem 25 The integration map gives us an isomorphism

H idR(M)∼= H i

sing(M,R)

Under this isomorphism the cup product corresponds to

H idR(M)×H j

dR(M)→ H i+ jdR (M), (ω1,ω2) 7→ ω1∧ω2, i, j = 0,1,2, . . .

where ∧ is the wedge product of differential forms.

If M is an oriented manifold of dimension n then we have the following bilinearmap

H idR(M)×Hn−i

dR (M)→ R, (ω1,ω2) 7→∫

Mω1∧ω2, i = 0,1,2, . . .

14.6 Dolbeault cohomology

Let M be a complex manifold and Ωp,qM∞ be the sheaf of C∞ (p,q)-forms on M. We

have the complex

Ωp,0M∞

∂→Ωp,1M∞

∂→ ··· ∂→Ωp,qM∞

∂→ ···

and the Dolbeault cohomology of M is defined to be


H p,q∂

(M) := Hq(Γ (M,Ω p,•M∞), ∂ ) =

global ∂ -closed (p,q)-forms on Mglobal ∂ -exact (p,q)-forms on M

Theorem 26 (Dolbeault Lemma) If M is a unit disk or a product of one dimen-sional disks then H p,q

∂(M) = 0

Let Ω p be the sheaf of holomorphic p-forms on M. In a similar way as in Proposition50 one can prove that Ω

p,qM∞ ’s are fine sheaves and so we have the resolution of Ω p:

Ωp→Ω

p,•M∞ .

By Proposition 22 we conclude that:

Theorem 27 (Dolbeault theorem) For M a complex manifold

Hq(M,Ω p)∼= H p,q∂

(M)

Chapter 15Category theory

The category theory is a language which unifies many similar ideas which had beenappearing in mathematics since the invention of singular homology and cohomol-ogy. It tells us how similar arguments in topology, geometry and algebraic geometrycan be unified in the context of a unique language. In this unique language the mor-phisms are not more functions from a set to another set. We have used mainly thebook [KS90] of M. Kashiwara and P. Schapira, [GM03] of S. Gelfand and Y. Manin,[Voi02b] of C. Voisin and [Dim04] of A. Dimca.

15.1 Objects and functions

A category A consists of

1. The set of objects Ob(A ).2. For X ,Y ∈ Ob(A ) a set hom(X ,Y ) = homA (X ,Y ), called the set of morphisms.

Instead of f ∈ hom(X ,Y ) we usually write:

f : X → Y or Xf→ Y.

Note that this is just a way of writting and it does not mean that X and Y aresets and f is a map between them. Of course, our main examples of the categorytheory have this interpretation.

3. For X ,Y,Z ∈ Ob(A ) a map

hom(X ,Y )×hom(Y,Z)→ hom(X ,Z), ( f ,g) 7→ g f

called the composition map.

It satisfies the following axioms:

1. The composition of morphisms is associative, i.e

131

132 15 Category theory

f (gh) = ( f g)h.

for X h→ Yg→ Z

f→W .2. For all objects X , there is idX : X → X such that

f idX = f , idX g = g

for f ,g that the equality is defined. It is easy to see that idX with this property isunique. We define the isomorphism in the category A in the following way: For

X ,Y ∈ Ob(A ) we write X ∼= Y if there are Xf→ Y and Y

g→ X such that

g f = idX , f g = idY . (15.1)

We say that the morphism f is an isomorphism and its inverse is g. The inverseof f is unique and is denoted by f−1.

In some texts one has also the following axiom in the definition of a category:

If X ∼= Y then X = Y.

We have not included this property in the definition of a category. However, when wesay that an object X is unique we mean that it is unique up to the above isomorphism.

15.2 Some definitions

For two categories A1,A2 we say that A1 is a subcategory of A2 if Ob(A1) ⊂Ob(A2) and for X ,Y ∈ Ob(A1) we have homA1(X ,Y ) ⊂ homA2(X ,Y ). We saythat A1 is a full sub category of A2 if for X ,Y ∈ Ob(A1) we have homA1(X ,Y ) =homA2(X ,Y ).

For a category A we can associate the opposite category A which is defined by:Ob(A ) =Ob(A ) and for all X ,Y ∈Ob(A ) we have homA (X ,Y ) = homA (Y,X).In the opposite category we have just changed the direction of arrows and it is easyto see that A is in fact a category.

A morphism f : X → Y is called a monomorphism (or an injective morphism) iffor any Z ∈ Ob(A ) and g,g′ ∈ hom(Z,X) such that f g = f g′, one has g = g′.We usually write

X → Y

to denote a monomorphism. f is called an epimorphism (or surjective morphism) ifit is a monomorphism in the opposite category.

An object P in the category A is called initial if hom(P,X) has exactly oneelement for all Y ∈ Ob(A ). It is called final if it is initial in the opposite category.Up to isomorphism, there is only one initial or final object.

15.3 Functors 133

15.3 Functors

A (covariant) functor F between two categories A1 and A2 is a collection of maps

F : Ob(A1)→ Ob(A2),

F : hom(X ,Y )→ hom(F(X),F(Y )), ∀X ,Y ∈A1

such that

1. F(idX ) = idF(x) for all x ∈A1,

2. F( f g) = F( f )F(g) for all Xg→ Y

f→ Z in A1.

A contravariant functor F : A1→A2 is a covariant functor A 1 →A2. For instance,

for X ∈ Ob(A )hom(X , ·) : A → Set, Z 7→ hom(X ,Z)

is a covariant functor. In a similar way hom(·,X) is a contravariant functor. Here Setis the category of sets and functions between them.

A morphism between two functors F1,F2 : A1 → A2 contains the followingdata. For all X ∈ Ob(A1) we have θ(X) ∈ hom(F1(X),F2(X)) such that for allf ∈ homA1(X ,Y ) the following diagram commutes:

F1(X)θ(X)→ F2(X)

↓ F1( f ) ↓ F2( f )

F1(Y )θ(Y )→ F2(Y )

For any two categories A1,A2 we get a new category called the category of functorsfor which the objects are the functors from A1 to A2 and the morphisms are asabove. It is left to the reader to show that this is a category.

Now, we can talk about an isomorphism between two functors. It is simply anisomorphism in the category of functors. Let us describe this in more details. Wesay that two functors F1,F2 : A1→A2 are isomorphic and write F1 ∼= F2 if for X ∈Ob(A1) we have θ1(X) ∈ hom(F1(X),F2(X)), θ2(X) ∈ hom(F2(X),F1(X)) suchthat

θ2(X)θ1(X) = idF1(X),θ1(X)θ2(X) = idF2(X)

and the following diagrams commute:

F1(X)θ1(X),θ2(X)

F2(X)↓ F1( f ) ↓ F2( f )

F1(Y )θ1(Y ),θ2(Y )

F2(Y )

In another words F1(X)∼= F2(X) and F1(Y )∼= F2(Y ) and under these isomorphismsF1( f ) is identified with F2( f ).


A functor F : A → Set is called representable if there exists X ∈ Ob(A ) suchthat it is isomorphic to the functor hom(X , ·). It is not difficult to see that up toisomorphism X is unique.

15.4 Additive categories

A category A is called additive if

1. The set hom(X ,Y ) has a structure of an abelian group such that the compositionmap is bilinear.

2. There is 0 ∈ Ob(A ) such that hom(0,0) is the zero abelian group 0. It can beproved that hom(X ,0) and hom(0,X) are zero groups.

3. For any two objects X1,X2 ∈ Ob(A ) there exist an object Y ∈ Ob(A ) and mor-phisms

Yi2← X1

π1← Yπ2→ X2

i2→ Y

such that

π1 i1 = idX1 , π2 i2 = idX2 , i1 π1 + i2 π2 = idY , π2 ı1 = 0, π1 i2 = 0.

One can show that Y with this definition is unique. We shall denote such Y byX1⊕X2 and we shall call it the direct sum of X1 and X2. For any two morphismsf : X1→ Z and g : X2→ Z we define h : f π1 +gπ2 which makes the followingdiagram commutative:

X1↓

X1⊕X2 → Z↑

X2

15.5 Kernel and cokernel

The kernel of a morphism Xf→ Y in an additive category A is an object C in A

with the following properties: It is equipped with a morphism C i→ X such that forany other object M in the category A the mapping:

hom(M,C)→ψ ∈ hom(M,X) | f ψ = 0, g 7→ ig

is well-defined and it is a bijective map. The kernel of f , if it exists, is usuallydenoted by ker( f ). One can check that the map i : ker( f )→ X is a monomorophism.A better way to define ker( f ) is to say that the contravariant functor

15.6 Abelian categories 135

F : A → Set, F(M) = hom(M,X) | f ψ = 0

is isomorphic to a representable functor.The cokernel of f , denoted by coker( f ), in the category A is defined to be the

kernel of f in the opposite category A . In explicit words, the cokernel of f isan object C with the morphism j : Y → C such that for any other object M in thecategory A the mapping:

hom(C,M)→ψ ∈ hom(Y,M) | ψ f = 0, g 7→ g j.

is well-defined and it is a bijective map. The image and coimage of a morphism

Xf→ Y are defined by

Im( f ) =: ker( j), Coim( f ) =: coker(i).

15.6 Abelian categories

An additive category A is called an abelian category if it satisfies

1. For any f : X → Y in A the kernel and cokernel of f exist.2. The canonical morphism Coim( f )→ Im( f ) is an isomorphism.

Some words must be said about the canonical map Coim( f )→ Im( f ). Let r : X →Coim( f ) be the map in the definition of Coim( f ) := coker(i). Therefore, we have aone to one map:

hom(Coim( f ),M)→ψ ∈ hom(X ,M) | ψ i = 0, g 7→ g r.

which is well-defined and it is an isomorphism. Put M =Y and ψ = f . We have f i = 0 and by the above universal property for coker(i) we have a map o : Coim( f )→Y such that f = o r. Now, we use the universal property of ker( j): We have a maps : Im( f )→ Y and a one to one map

hom(M, Im( f ))→ψ ∈ hom(M,Y ) | j ψ = 0, g 7→ sg.

We set M = Coim( f ) and ψ = o. We have j o = 0 because

j f = ( j o) r = 0 and r is surjective

We conclude that there is a morphism O : Coim( f )→ Im( f ) such that sO = o.


15.7 Additive functors

Let A1 and A2 be two abelian categories. A functor from A1 to A2 is called additiveif for all X ,Y ∈ Ob(A1) the map hom(X ,Y )→ hom(F(X),F(Y )) is a morphism ofadditive groups. An additive functor F from A1 to A2 is called left (resp. right)exact if for any exact sequence in A1:

0→ X1→ X2→ X3( resp. X1→ X2→ X3→ 0)

the sequence

0→ F(X1)→ F(X2)→ F(X3)( resp. F(X1)→ F(X2)→ F(X3)→ 0)

is exact.

15.8 Injective and projective objects

Let A be an abelian category. An object A ∈ Ob(A ) is injective if for any diagram

0→ X → Y↓A

in A there is Y → A which makes the diagram commutative. The object A is calledprojective if it is injective in the opposite category, i.e. for any diagram

0← X ← Y↑A

in A there is Y ← A which makes the diagram commutative. As an exercise, it isleft to the reader to show that any short exact sequence 0→ X → Y → Z→ 0 withX an injective object is split. In the category of abelian groups the injective objectsare exactly divisible abelian groups, i.e. a group G such that for all g ∈G and n ∈Nthere is g′ ∈ G such that ng′ = g.

15.9 Complexes

Let A be an abelian category. A complex in this category is a sequence of mor-phisms

· · · → Xn−1 dn−1→ Xn dn

→ Xn+1→ ·· · ,

15.10 Homotopy 137

withdn dn−1 = 0.

We usually do not write the name of the morphisms dn. We write, for short, X• orXk, k ∈ Z to denote a complex. When we write Xk,k ∈ N0 we mean a complex forwhich Xk = 0,k =−1,−2, . . ..

The category of complexes C(A ) is a category whose objects are the set of com-plexes in A . A morphism between two complexes X• and Y • is the following com-mutative diagram:

· · · → Xn−1 → Xn → Xn+1 → ·· ·· · · ↓ ↓ ↓ · · ·· · · → Y n−1 → Y n → Y n+1 → ·· ·

We have the following natural functor

Hk : C(A )→A , Hk(X) :=ker(dk)

Im(dk−1), k ∈ Z

For a morphism f : X → Y of complexes we have a canonical morphism Hk( f ) :Hk(X)→ Hk(Y ). A morphism of complexes f is called a quasi-isomorphism if theinduced morphisms Hk( f ), k ∈ Z are isomorphisms.

We have three full subcategories of C(A ). The category of left complexesC+(A ) contains the complexes

· · · → 0→ 0→ Xn→ Xn+1→ ···

and the category of right complexes C−(A ) contains the complexes

· · · → Xn−1→ Xn→ 0→ 0→ ·· ·

the category of bounded complexes is Cb(A ) for which the objects are

Ob(C+(A ))∩Ob(C−(A ))

15.10 Homotopy

Let A be an abelian category. The shift functor of degree k

[k] : C(A )→C(A ), k ∈ Z.

is defined in the following way. The complex [k](X) is denoted by X [k] and it isdefined by

X [k]n := Xn+k, dnX [k] := (−1)kdn+k

X .

A homomorphism f : X → Y in C(A ) is called homotopic to zero if there existsmorphisms sn : Xn→ Y n−1 in A such that


f n = sn+1 dnX +dn−1

Y sn

We say that f : X →Y is homotopic to g : X →Y if f −g is homotopic to zero. Thefollowing easy proposition is left to the reader

Proposition 52 Let X and Y be two complexes in A and f ,g : X → Y be two ho-motopic maps. Then the induced maps

Hn( f ),Hn(g) : Hn(X)→ Hn(Y ), n ∈ Z.

are equal.

15.11 Resolution

Let A be an abelian category. A complex Xk,k ∈ N0 is called a resolution of X ∈Ob(A ) if

Im(dk) = ker(dk+1), k = 0,1,2, . . .

and there exists an injective morphism i : X → X0 such that Im(i) = ker(d0). Wewrite this simply in the form

X → X•

We have a natural injective map Im(dk)→ ker(dk+1) and by the equality above wemean that this injective map is an isomorphism.

An object I of the category A is called injective if for any injective morphismA → B and a morphism A→ I there exits a morphism B→ I such that the followingdiagram is commutative:

A → B↓ I

We say that the category A has enough injectives if for all object A in A there arean injective element I and an injective morphism A → I. For instance the categoryof sheaves of abelian groups over a topological space has enough injectives. For agiven sheaf A, I is the sheaf of all not necessarily continuous sections of A.

Proposition 53 If an abelian category contains enough injectives then any objectX of A admits a resolution.

For a proof see [Voi02b], Lemma 4.26.From this lemma on we follow line by line[Voi02b] page 97-108

Proposition 54 Let A i→ I• and Bj→ J• be resolutions of A and B respectively and

φ : A→ B be a morphism. Then if the second resolution is injective there exists amorphism of complexes φ • : I•→ J• satisfying φ 0 i = j φ . Moreover, if we havetwo such morphisms φ • and ψ•, there exists a homotopy between φ • and ψ•.

See [Voi02b] Proposition 4.27.

15.13 Acyclic objects 139

If both I• and J• are injective resolutions of A then we apply Proposition 54 to thecase where A = B and φ is the identity map. We obtain φ • : I•→ J• and ψ• : J•→ I•

such that φ • ψ• and ψ• φ • are both homotopic to identity. Therefore, injectiveresolutions are unique up to homotopy.

15.12 Derived functors

Now, we are in position to announce the first important theorem of this chapter.

Theorem 28 Let A1,A2 be two abelian categories and F be a left exact functorfrom A1 to A2. Assume that A1 has enough injectives. For any object X of A1 thereare objects RkF(X), k = 0,1,2, . . . with the following properties:

1. R0F(X) = F(X).2. For all exact sequence

0→ A→ B→C→ 0

in A1 one can construct a long exact sequence

0→F(A)→F(B)→F(C)→R1F(A)→R1F(B)→R1F(C)→R2F(A)→R2F(B)→···

3. For any injective object object I of A1 we have RkF(I) = 0, k = 1,2, . . .

The objects RkF(X) with the above properties are unique up to isomorphisms.

For a proof see [Voi02b] Theorem 4.28. The construction of RiF(A) is as follows.Let A→ I• be an injective resolution of A. we define

RiF(A) = H iF(I•).

For a morphism φ : A→ B in A we have canonical morphisms

RiF(φ) : RiF(A)→ RiF(B).

15.13 Acyclic objects

Let A1,A2 be two abelian categories and F be a left exact functor from A1 to A2.An object X of A1 is called F-acyclic, or simply acyclic if one understand F fromthe context, if

RkF(X) = 0, k = 1,2,3 . . .

By definition, injective objects are acyclic but not vice verse.

Proposition 55 Let X be an object of A1 and

X → X•


be an F-acyclic resolution of X, i.e. a resultion for which each X i is acyclic, then

RkF(X)∼= Hk(F(X•)).

See [Voi02b], Proposition 4.32.

15.14 Cohomology of sheaves

We consider the category of sheaves of abelian groups on topological spaces. Fromthis category to the category of abelian groups we have the global section functorΓ which associates to each sheaf SS the set of its global sections. For a sheaf SS ofabelian groups on a topological space X we define:

Hk(X ,SS) := RkΓ (SS)

Cech cohomology of sheaves gives a nice interpretation of Hk(X ,SS) in terms ofcovering of X with open sets (see Chapter 14 and [Gun90, BT82]).

Proposition 56 Let X be a topological space and SS a sheaf of abelian groupson X. Then we have a canonical isomorphism between Hk(X ,SS) and the Cechcohomology of X with coefficients in SS.

Proof. We take a covering U = Uii∈I such that the intersection V of any finitenumber of Ui’s satisfies

Hk(V,SSi) = 0, k = 1,2, . . . , i = 0,1,2, . . .

For any open set V in X let SS |V be the sheaf for which the stalk over x∈V coincideswith the stalk of SS over x and outside of V it is zero. Let Ik−1 be the direct productof all the sheaves SS |Ui1∩Ui2∩···∩Uik

. We have now the acyclic resolution of SS

SS→ I0 δ→ I1 δ→ I2 · · ·

where δ is defined in a similar way as the δ in the Cech cohomology. Now ourassertion follows from Proposition 55.

15.15 Derived functors for complexesWe follow [Voi02b] page 186-194

Let A1 be an abelian category which has enough injective objects.

Proposition 57 For a left-bounded complex M• in A1, there exists a left-boundedcomplex I• in A1 such that each Ik, k ∈ Z is injective object of A1, and a morphismφ • : M• → I• of complexes which is a quasi-isomorphism and for all k ∈ Z, φ k :Mk→ Ik is injective.

15.15 Derived functors for complexes 141

See [Voi02b], Proposition 8.4.It is a natural question to ask which objects of the category C+(A1) are injective.

Note that a left-bounded complex I• with Ik injective objects is not necessarily aninjective object of C+(A1).

Let A1,A2 be two abelian categories, where A1 has enough injective objects. Letalso F be a left exact functor from A1 to A2. Let M• be a left-bounded complex inA1 and φ • : M•→ I• as in Proposition 57. We define the derived functors

RkF(M•) := Hk(F(I•)), k ∈ Z.

Note that for an object A in A1 we have

RkF(· · · → 0→ A→ 0→ ···) = RkF(A),

where A is positioned in the 0-th place.

Proposition 58 If we have two quasi-isomorphisms M• → I• and M• → J• as inProposition 57, then we have canonical isomorphisms

Hk(F(I•))∼= Hk(F(J•)).

See [Voi02b] Proposition 8.6.

Proposition 59 Let φ • : M• → I• be a quasi-isomorphism of left-bounded com-plexes in the category A1 with Ik injective for all k ∈ Z. Then we have an isomor-phism

RkF(M•)∼= Hk(F(I•))

See [Voi02b] Proposition 8.8. The only difference with Proposition 57 is that we donot assume that each φ k is injective.

Proposition 60 Let φ • : M• → N• be a quasi-isomorphism of left bounded com-plexes in A1. Then we have a canonical isomorphisms

RkF(φ •) : RkF(M•)∼= RkF(N•)

See [Voi02b], Corollary 8.9.

Proposition 61 Let φ • : M• → N• be a quasi-isomorphism of left bounded com-plexes in A1. Assume that all Nk’s are acyclic for the functor F. Then we have acanonical isomorphisms

RkF(M•)∼= Hk(F(N•)).

See [Voi02b], Proposition 8.12.


15.16 Hypercohomology

In the case where A1 is the category of of sheaves of abelian groups over a topolog-ical space X and F is the global section functor

SS→ Γ (X ,SS)

we writeHk(X ,SS•) := Rk

Γ (SS•).

The group Hk(X ,SS•) is called the k-the hypercohomology of the complex SS•.


Let M be a complex manifold and Ω •M and Ω •M∞ be the complex of the sheaves ofholomorphic, respectively C∞, differential forms on M. In the category of sheavesof abelian groups on M we have a quasi-isomorphism

Ω•M →Ω

•M∞

induced by inclusion. We also know that the sheaves Ω kM∞ are Γ -acyclic. By Propo-

sition 61 we conclude that

Hk(M,Ω •M)∼= HkdR(M).

We also know thatC→Ω

•M

is the resolution of the constant sheaf C. This is the same to say that the complex· · ·→ 0→C→ 0→··· with C in the 0-th place, is quasi-isomorphic to the complexΩ •M and so by Proposition 60 we have

Hk(M,C)∼=Hk(M,Ω •M).

15.18 How to calculate hypercohomology

Let us be given a complex of sheaves of abelian groups on a topological space X .

SS : SS0 d→ SS1 d→ SS2 d→ ··· d→ SSn d→ ·· · , d d = 0. (15.2)

Let U = Uii∈I be a covering of X . Consider the double complex

15.18 How to calculate hypercohomology 143

↑ ↑ ↑ ↑SS0

n → SS1n → SS2

n → ·· · → SSnn →

↑ ↑ ↑ ↑SS0

n−1 → SS1n−1 → SS2

n−1 → ·· · → SSnn−1 →

↑ ↑ ↑ ↑...

......

...↑ ↑ ↑ ↑

SS02 → SS1

2 → SS22 → ·· · → SSn

2 →↑ ↑ ↑ ↑

SS01 → SS1

1 → SS21 → ·· · → SSn

1 →↑ ↑ ↑ ↑

SS00 → SS1

0 → SS20 → ·· · → SSn

0 →

(15.3)

Here SSij is the disjoint union of global sections of SSi in the open sets ∩i∈I1Ui,

I1 ⊂ I, #I1 = j. The horizontal arrows are usual differential operator d of SSi’s andvertical arrows are differential operators δ in the sense of Cech cohomology (seeChpater 14). The k-th piece of the total chain of (15.3) is

L k :=⊕ki=0SSi

k−i

with the the differential operator

d′ = d +(−1)kδ : L k→L k+1.

Proposition 62 The hypercohomology Hm(M,SS•) is canonically isomorphic tothe direct limit of the total cohomology of the double complex (15.3), i.e.

Hm(M,SS•)∼= dirlimU Hm(L •,d′).

IfHk(Ui1 ∩Ui2 ∩·· ·Uir ,SSi) = 0, k,r = 1,2, . . . , i = 0,1,2, . . . . (15.4)

thenHm(M,SS•)∼= Hm(L •,d′). (15.5)

Proof. The fact that the direct limit dirlimU Hm(L •,d′) is canonically isomorphicto Hm(L •,d′) with U satisfying (15.4) is classic and is left to the reader. Therefore,we just prove the second part of the proposition. The proof is similar to the proof ofProposition 56. Let us redefine Ii

j to be the direct product of

SSi |∩i∈I1Ui , I1 ⊂ I, #I1 = j

andIk :=⊕k

i=0Iik−i


We have a canonical quasi-isomorphism SS•→ I• with all Ik acyclic. Now our as-sertion follows from Proposition 61.

If one takes (15.5) as a definition of the hypercohomology then it can be shownthat this definition is independent of the choice of U .

Let us assume that M is a complex manifold and U is a Stein covering, i.e.each Ui is Stein. Further assume that all S k’s are coherent. It can be shown thatthe intersection of finitely many of them is also Stein, see for instance [CM03]Proposition 1.5. We conclude that for a Stein covering of M we have (15.5).

15.19 Filtrations

For a complex SS and k ∈ Z we define the truncated complexes

SS≤k : · · ·SSk−1→ SSk→ 0→ 0→ ···

andSS≥k : · · · → 0→ 0→ SSk→ SSk+1→ ·· ·

We have canonical morphisms of complexes:

SS≤k→ SS, SS≥k→ SS

Assume that SS is a left-bounded complex,

SS : · · · → 0→ SSk→ SSk+1→ ·· ·

The morphism SS≥i→ SS induces a map in hypercohomologies and we define

F i := Im(Hm(X ,SS≥i)→Hm(X ,SS))

This gives us the filtration

· · · ⊂ F i ⊂ F i−1 ⊂ ·· · ⊂ Fk−1 ⊂ Fk :=Hm(M,SS•).

Proposition 63 If the maps

Ha(M,SSi)→ Ha(M,SSi+1), a ∈ N0, i ∈ Z

induced by SSi→ SSi+1 are zero then we have canonical isomorphisms

F i/F i+1 ∼= Hm−i(M,SSi)

Proof. We use the second part of Proposition 62 (the details are done in the class).

15.19 Filtrations 145

Remark 21 The hypothesis of Proposition 63 is satisfied for the complex of holo-morphic differential forms on a complex manifold, see [Gri69], II. The correspond-ing filtration in this case is called the Hodge filtration.

Proposition 64 If Ha(M,SSi) = 0, a > 0, i ∈ Z then

Hm(M,SS•)∼= Hm(Γ SS•,d).

Proof. We Use Proposition 62. The hypothesis implies that the vertical arrows in15.3 are exact. Every element in L k is reduced to an element in SSk

0 whose δ iszero and so corresponds to a global section of SSk.

Exercises1. Prove that in a category 1. the identity morphism idX is unique 2. the morphism i : ker( f )→ X

is a monomorophism and j : Y → coker( f ) is an epimorphism.2. In a category A prove that up to isomorphism, there is a unique initial or final object.3. Let A1 and A2 be two categories. Show that the set of functors from A1 to A2 and morphisms

between them is a category. A functor F : A → Set is called representable if there exists X ∈Ob(A ) such that it is isomorphic to the functor hom(X , ·). Prove that up to isomorphism X isunique.

4. Prove that for an additive category hom(X ,0) and hom(0,X) are zero group.5. Use the definition of X1⊕X2 and show that it is unique.6. Show that ker( f ) and coker( f ) are unique. Define Im( f ) and Coim( f ) and prove they are

unique (if you have made the right definition).

7. Let us be given two morphisms Xf→Y

g→ Z with f g = 0 in an abelian category. Describe theobject ker(g)

Im( f ) .8. Any short exact sequence 0→X→Y → Z→ 0 with X an injective object is split, i.e. Y ∼=X⊕Z

and Y → Z is the projection in the second coordinate and Y → X is the inclusion map.

Chapter 16Algebraic de Rham cohomology and Hodgefiltration

Inspired by the work of Atiyah and Hodge in [HA55], Grothendieck in [Gro66]introduced the de Rham cohomology in algebraic geometry.


Definition 30 Let X be any prescheme locally of finite type over a field k, andsmooth over k. We consider the complex (Ω •X/k,d) if regular differential forms onX . The (algebraic) de Rham cohomology of X is defined to be the hypercohomology

HqdR(X/k) =Hq(X ,Ω •X/k), q = 0,1,2, . . . .

Let k = C be the field of complex numbers and Xan be the underlying complexmanifold of X . In a similar way we can define the analytic de Rham cohomology ofXan, i.e.

HqdR(X

an) =Hq(Xan,Ω •Xan), q = 0,1,2, . . . .

where Ω •Xan is the complex of holomorphic differential forms on Xan.

The reader must note that in Xan we have considered usual topology of complexmanifolds and in X/C we have considered the Zariski topology. By Poincare lemmaΩ •Xan is the resoultion of the constant sheaf C and hence

HqdR(X

an)∼= Hq(Xan,C), q = 0,1,2, . . . .

From another side we have the following:

Theorem 29 The canonical map

HqdR(X/C)→ Hq

dR(Xan), q = 0,1,2, . . .

is an isomorphism of C-vectorspaces.

147

148 16 Algebraic de Rham cohomology and Hodge filtration

16.2 Atiyah-Hodge theorem

In order to prove Theorem 29 we first consider the case in which X is an affinevariety.

Theorem 30 The complex variety Xan is an Stein variety and so by Cartan’s Btheorem

H i(Xan,Ω jXan) = 0, i = 1,2, . . . , j = 0,1,2, . . . .

By Proposition 64 we conclude that

H idR(X

an) = H i(Γ Ω•Xan ,d)

From the algebraic side we have:

Theorem 31 (Serre vanishing theorem) Let X be an affine smooth variety over thefield C of complex numbers. Then

H i(X ,Ω jX ) = 0, i = 1,2, . . . , j = 0,1,2, . . . .

By Proposition 64 we conclude that

H idR(X/C) = H i(Γ Ω

•X ,d)

We are now going to prove that:

Theorem 32 (Atiyah-Hodge) Let X be an affine smooth variety over the field C ofcomplex numbers. Then the canonical map

Hq(Γ (Ω •X/C),d)→ HqdR(X

an)

is an isomorphism of C-vector spaces.

Proof. See [Nar68], p. 86.

Corollary 4 Let X be an affine smooth variety over the field C of complex numbers.Every holomorphic q-form ω1 on X can be written as ω1 = ω2 +dω3, where ω2 isan algebraic q-form on X and ω3 is a holomorphic (q−1)-form on X.

Proof. Hint: dω1 is zero in the algebraic de Rham cohomology of X .

16.3 Proof of Theorem 29

Let Uii∈I a covering of X with affine varieties. The intersection of any two affinevariety is again an affine variety. This implies that the intersection of any finitenumber of affine varieties in Ui is again affine. We use Theorem 30 and theorem31 and conclude that Uii∈I is a good covering in the sense of the second part of

16.4 Hodge filtration 149

Proposition 62. This means that we can take the covering Uii∈I (resp. Uani i∈I)

to calculate HmdR(X/C) (resp. HmdR(Xan)). From this point on one has to use Atiyah-

Hodge theorem and the diagram (15.3). We have to make precise

If X is a projective variety then Theorem 29 follows also from Serre’s GAGA. Make it precise

16.4 Hodge filtration

For the complex of differential forms Ω •X/k we have the complex of truncated dif-ferential forms:

Ω•≥iX/k : · · · →Ω

iX/k→Ω

i+1X/k→ ·· ·

and a natural mapΩ•≥iX/k→Ω

•X/k

We define the Hodge filtration

0 = Fm+1 ⊂ Fm ⊂ ·· · ⊂ F1 ⊂ F0 = HmdR(X/k)

as follows

Fq = FqHmdR(X/k) = Im

(Hm(X ,Ω •≥i

X/k)→Hm(X ,Ω •X/k))


Fq/Fq+1 ∼= Hm−q(X ,Ω q)

This proposition follows from Proposition 63 and the following:

Proposition 66 The maps

Ha(X ,Ω iX )→ Ha(X ,Ω i+1

X ), a ∈ N0, i ∈ Z

induced by the differential map d : Ω iX →Ω

i+1X are all zero.

Proof. See [Gri69].

Proposition 67 Let k = C. Under the canonical isomorphisms

HqdR(X/C)→ Hq

dR(Xan), q = 0,1,2, . . .

the algebraic Hodge filtration is mapped to the usual Hodge filtration of Hm(Xan,C)

Proof. This follows from the diagram (15.3) for the complex Ω •Xan and the fact thatthe sheaf of C∞ differential (p,q)-forms is acyclic.

Chapter 17Lefschetz (1,1) theorem

Lefschetz theorem is special case of the Hodge conjecture. It is stated and proved inthe present chapter.

17.1 Lefschetz (1,1) theorem

Let Y be a divisor in X . We associate to Y a line bundle LY in X in a natual way.Consider the short exact sequence

0→ Z→ O → O∗→ 0

and the corresponding long exact sequence

· · · → H1(X ,O∗)i∗→ H2(X ,Z) δ→ H2(X ,O)→ ·· ·

Proposition 68 Let Y be a divisor in X. We have

i∗(LY ) = Poincare dual of [Y ].

Proof. See [GH94], p. 141 Proposition 1.

In H2(X ,C) we have the Hodge filtration

0= F3 ⊂ F2 ⊂ F1 ⊂ F0 = H2(X ,C)

By Proposition 65 we have a canonical isomorphism F0/F1→H2(X ,O) and so wehave a canonical projection

δ : H2(X ,C)→ H2(X ,O)

151

152 17 Lefschetz (1,1) theorem

Proposition 69 The map δ and δ are the same.

Proof. See [GH94],p. 163.

Theorem 33 Let X be a projective variety. Every class in

H2(X ,Z)∩F1(H2(X ,C))

is a Poincare dual of an analytic variety. Here, we write H2(X ,Z) to denote itsimage in H2(X ,C).

Proof. This is a direct consequence of Proposition 68 and Proposition 69.

Proposition 70 The Hodge conjecture is true for cycles

H2n−2(X ,Z)∩Fn−1,n−1(H2n−2(X ,C))

Proof. This follows from Lefschetz (1,1)-theorem and hard Lefschetz theoremwhich says that

H2(X ,C)→ H2n−2(X ,C), α 7→ α ∪∪n−2i=1 P([Y ])

is an isomorphism.

17.2 Some consequences on integrals

Chapter 18Deformation of hypersurfaces

For a given smooth hypersurface M of degree d in Pn+1 is there any deformationof M which is not embedded in Pn+1? We need the answer of this question becauseit would be essential to us to know that the fibers of a tame polynomial f form themost effective family of affine hypersurfaces. The answer to our question is givenby Kodaira-Spencer Theorem which we are going to explain it in this section. Forthe proof and more information on deformation of complex manifolds the reader isreferred to [Kod86], Chapter 5.

Let M be a complex manifold and Mt , t ∈ B := (Cs,0), M0 = M be a deformationof M0 which is topologically trivial over B. We say that the parameter space B iseffective if the Kodaira-Spencer map

ρ0 : T0B→ H1(M,Θ)

is injective, where Θ is the sheaf of vector fields on M. It is called complete if anyother family which contain M is obtained from Mt , t ∈ B in a canonical way (see[Kod86], p. 228).

Theorem 34 If ρ0 is surjective at 0 then Mt , t ∈ B is complete.

Let m = dimCH1(M,Θ). If one finds an effective deformation of M with m = dimBthen ρ0 is surjective and so by the above theorem it is complete.

Let us now M be a smooth hypersurface of degree d in the projective space Pn+1.Let T be the projectivization of the coefficient space of smooth hypersurfaces inPn+1. In the definition of M one has already dimT =

(n+1+dd

)−1 parameters, from

which only

m :=(

n+1+dd

)− (n+2)2

are not obtained by linear transformations of Pn+1.

Theorem 35 Assume that n ≥ 2, d ≥ 3 and (n,d) 6= (2,4). There exists a m-dimensional smooth subvariety of T through the parameter of M such that theKodaira-Spencer map is injective and so the corresponding deformation is com-plete.

153

154 18 Deformation of hypersurfaces

For the proof see [Kod86] p. 234. Let us now discuss the exceptional cases. For(n,d) = (2,4) we have 19 effective parameter but dimH1(M,Θ) = 20. The differ-ence comes from a non algebraic deformation of M (see [Kod86] p. 247). In thiscase M is a K3 surface. For n = 1, we are talking about the deformation theory of aRiemann surface. According to Riemann’s well-known formula, the complex struc-ture of a Riemann surface of genus g ≥ 2 depends on 3g− 3 parameters which isagain dimH1(M,Θ) ([Kod86] p. 226).

18.1 Reconstructing the period matrix

We have seen that the period matrix X = pmtr satisfies the differential equationdX = Atr ·X . Fix a point t0 ∈ T and let γ be a path in T which connects t0 to t ∈T . The analytic continuation of the flat section throught Iµ×µ := X(t0)−1X(t0) andalong γ is X(t0)−1X(t). This gives us the equality

pm(t) = (Iµ×µ −∫

γ

A+∫

γ

AA−∫

γ

AAA+ · · ·)pm(t0), (18.1)

where we have used iterated integrals (for further details see [Hai87]). Note thatthe above series is convergent and the sum is homotopy invariant but its pieces arenot homotopy invariants. The equality (18.1) implies that if we know the value ofperiod matrix for just one point t0 then we can construct the period matrix of otherpoints of T using the Gauss-Manin connection. The calculation of period matrix forexamples of tame polynomials in C[x] is done in S11.7.

Chapter 19Mixed Hodge structure of affine varieties

19.1 Logarithmic differential forms and mixed Hodge structuresof affine varieties

Let M be a projective smooth variety and D = D1 + D2 + · · ·+ Ds be a normalcrossing divisor in M, i.e. to each point p ∈ M there are holomorphic coordinatesz1,z2, . . . ,zn around p such that D = z1 = 0+ · · ·+ zi = 0 for some i depend-ing on p. Let also Ω i

M(logD) be the sheaf of meromorphic i-forms ω in M withlogarithmic poles along D, i.e. ω and dω have poles of order at most one along D.This is equivalent to the fact that around each point p ∈M the sheaf Ω k

M(logD) isgenerated by k-times wedge products of dz1

z1, dz2

z2, · · · , dzi

zi,dzi+1, · · · ,dzn, where i is

as above. Let A • be the complex of C∞ differential forms in M\D and j : M\D→Mbe the inclusion. A map between two differential complexes, p : SS•1→ SS•2 is calledto be a quasi-isomorphism if the induced maps

Hk(SS1,x,d)→ Hk(SS2,x,d), k = 0,1, . . . , x ∈M

are isomorphisms, where SSkx is the stalk of SSk over x ∈M.

Proposition 71 The canonical map

Ω•(logD)→ j∗A •

is a quasi isomorphism.

This proposition is proved [Gri69] and [Del70]. See also [Voi02b] Proposition8.18. The above proposition implies that we have an isomorphism

Hk(M\D,C)∼=Hk(M,Ω •M), k = 0,1,2, . . . .

The Hodge filtration on Hm(M\D,C) is given by

FkHm(M\D,C) := Im(Hm(FkΩ•M(log(D))→Hm(Ω •M(log(D))))

155

156 19 Mixed Hodge structure of affine varieties

where for a differential complex SS•

FkSS• := SS•≥k : 0→ 0→ ··· → 0︸︷︷︸k times 0

→ SSk→ SSk+1→ ··· → SSn→ ·· ·

is the bete filtration. By definition we have F0/F1 ∼= Hn(M,OM). The weight filtra-tion on Hm(M\D,C) is given by

WkHm(M\D,C) := Im(Hm(PkΩ•M(log(D))→Hm(Ω •M(log(D)))), k ∈ Z,

where PkΩ •M(log(D)) is the Deligne pole order filtration: A logarithmic differen-tial m-form ω is in PkΩ m

M(log(D)) if in local coordinates, there does not appear awedge product of more than k′-times dzi

zi,k′ > k in ω . The Hodge filtration induces

a filtration on GrWa :=Wa/Wa−1 and we set

GrbF GrW

a := FbGrWa /Fb+1GrW

a =(Fb∩Wa)+Wa−1

(Fb+1∩Wa)+Wa−1,a,b ∈ Z (19.1)

In the next sections we will introduce two other filtrations which are called againpole order filtrations and have completely distinct nature.

19.2 Pole order filtration

For an analytic sheaf SS on M we denote by SS(∗D) the sheaf of meromorphicsections of SS with poles of arbitrary order along D. For k = (k1,k2, . . . ,ks) ∈Ns

0 wedenote by SS(kD) the sheaf of meromorphic sections of SS with poles of order atmost ki along Di, i = 1,2, . . . ,s. For SS(∗D) we have also the pole filtration:

PkSS•(∗D) : 0→ 0→ ·· · → 0︸︷︷︸k times 0

→ SSk0→ SSk+1

1 → ·· · → SSpp−k→ ··· ,

whereSSp

p−k := ∪|n|≤p−kSS j((n+1)D) if p≥ k,

(n+1) = (n1 +1,n2 +1, . . . ,ns +1), |n|= n1 +n2 + · · ·+ns.

Let (SS•1,F) and (SS•2,F) be two filtered differential complexes and p : (SS•1,F)→(SS•2,P) a map between them, i.e. we have a collection of maps pk : FkSS1→ PkSS2such that the following diagram commutes

Fk+1SS•1 → Pk+1SS•2↓ ↓

FkSS•1 → PkSS•2

, k = 0,1, . . .

19.3 Another pole order filtration 157

The map p is called a quasi-isomorphism of filtered complexes if pk, k = 0,1, . . .are quasi-isomorphism.

Theorem 36 The inclusion

(Ω •M(logD),F)⊂ (Ω •M(∗D),P) (19.2)

is a quasi-isomorphisms of filtered differential complexes.

Proof. Note that this is a local statement and so we can suppose that M =(Cn,0), D=z1 = 0+z2 = 0+ · · ·zs = 0. The proof can be found in [Del70] Proposition3.13, [Del71] Proposition 3.1.8. See also [Voi03] Proposition 8.18. According to[DD90] p.647, Deligne has inspired the above theorem from the work of Griffiths .

The above proposition implies that the Hodge filtration on Hm(M−D,C) is alsogiven by

F iHm(M\D,C) = Im(Hm(PiΩ•M(∗D))→Hm(Ω •M(∗D))).

19.3 Another pole order filtration

In this section we assume that D is a positive divisor, i.e. the associated line bundleis positive. From this what we need is the following: For any coherent analytic sheafSS on M we have

Hk(M,SS(∗D)) = 0, k = 1,2, . . . (19.3)

We do not assume that D is a normal crossing divisor.

Theorem 37 (Atiyah-Hodge-Grothendieck) If D is positive then

Hk(M\D,C)∼= Hk(Γ Ω•M(∗D),d), k = 0,1,2, . . . (19.4)

Proof. The proof follows from Proposition 64 and (19.3).

From now on assume that D is irreducible. To each cohomology class α ∈Hm(M\D,C)we can associate P(α) ∈ N which is the minimum number k such that there existsa meromorphic m-form in M with poles of order k along D and represents α in theisomorphism (19.4). We have

P(α +β )≤ maxP(α),P(β ), P(kα) = P(α), α,β ∈ Hm(M\D,C), k ∈ C\0.

Using the above facts for a C-basis of Hm(M\D,C), we can find a number h suchthat for all α ∈ Hm(M\D,C) we have P(α) ≤ h. We take the minimum number hwith the mentioned property. Now we have the filtration

H0 ⊂ H1 ⊂ ·· · ⊂ Hh−1 ⊂ Hh = Hm(M\D,C)

158 19 Mixed Hodge structure of affine varieties

Hi = α ∈ Hm(M\D,C) | P(α)≤ i, i = 0,1, . . . ,h.

We call it the new pole order filtration.

Complementary notes1. As the reader may have noticed, Theorem 18 implies that for the complement of smooth hyper-

surfaces the pole order filtrations in S19.2 and S19.3 are the same up to reindexing the pieces.In this point the following question arises: Can one find the pieces of the Hodge filtration ofHm(M\,C) inside the pieces of the new pole order filtration? Using Riemann-Roch Theoremone can find also positive answers to this question for Riemann surfaces. However, the questionin general, as far as I know, is open.

19.4 Mixed Hodge structure of affine varieties

Our main examples of modular foliations in Chapters 7 and ?? are associated toa family of affine hypersufaces and polynomial differential forms in Cn+1. Suchdifferential forms have poles at infinity and the corresponding pole order gives usthe first numerical invariant to distinguish between differential forms and hence thecorresponding modular foliations. Another way to distinguish between differentialforms is by looking at their classes in the de Rham cohomology and its Hodge filtra-tion. It is believed that there exists a close relation between the mentioned conceptsand the testimonies to this belief are P. Griffiths theorem on the Hodge filtration ofthe complement of a smooth hypersurface (see S??) and some calculations relatedto Riemann surfaces.

Chapter 20Global invariant cycle theorem

Let X ⊂ Y be two smooth projective varieties and assume that there is an openZariski subset of Y such that X is a fiber over a point, say it 0, of a smooth properalgebraic morphism π : U → B. Let also

ρ : π1(B,0)→ Aut(Hm(X ,Q)) (20.1)

be the monodromy map. The aim of the present chapter is to prove the followingtheorem.

Theorem 38 The space of invariant classes

Hm(X ,Q)ρ := δ ∈ Hm(X ,Q) | ρ(γ)(δ ) = δ , ∀γ ∈ π1(B,0)

is equal to the image of the restriction map

i∗ : Hm(Y,Q)→ Hm(X ,Q)

which is a morphism of Hodge structures, where i : X → Y is the inclusion map.Further, for m even, any Hodge class in Hm(X ,Q)ρ is in the image of a Hodge classin Hm(Y,Q)

The above theorem appears in Deligne’s article [Del71], Theorem 4.1.1 under thename ”theorem de la partie fixe”. It has been restated and used by Voisin in [Voi07,Voi13].

Remark 22 If X is a smooth hyperplane section of Y , then theorem 38 is essentiallya version of Hard Lefschetz theorem stated in [Lam81] and reproduced in Section5, see also [Del68]

159

160 20 Global invariant cycle theorem

20.1 Some doubt about Hodge conjecture

Consider X ⊂ Y as in the previous section. The last part of Theorem 38 says that ifthe Hodge conjecture is true then any algebraic cycle of dimension m

2 in Y is homo-logically equivalent to an algebraic cycle in X . This a really very strong affirmation.

20.2 A consequence

We derive a consequence of gloabl invariant cycle theorem. Let X ,Y,π : U → B asbefore, but defined over a field k of characteristic zero. Let

Theorem 39 Let δ be a global section of the relative algebraic de Rham cohomol-ogy Hm

dR(U/B). If ∇δ = 0 then its evaluation δ0 at 0 ∈ B is in the image of themap

HmdR(Y )→ Hm

dR(X).

induced by inclusion X → Y . Conversely, any δ0 in the image of the above mapcomes from a global section δ with ∇δ = 0.

Proof. The proof is done for k = C. By our hypothesis δ is a glonal section ofRmπ∗Q⊗QC and so it is invariant under the monodromy. Therefore,

δ0 ∈ Hm(X ,Q)ρ ⊗QC.

Chapter 21Lixo de Hodge Theory

The truth in mathematics is a state of satisfaction but not vice versa. The classicalway of doing mathematics is to prove, and hence to feel, the truth and then to enjoythe consequent satisfaction. However, with the rapid development of mathematics itseems to be very difficult to transfer to an student all the details leading to a truth.Manytimes we need to use an object and in order to construct it explicitly we spend alot of time so that the student lose all his/her interest on the subject. In this situation,I think, it is crucial to invest on inducing the state of satisfaction in new learnersrather than using the classical methodology of doing mathematics which is definingand proving every thing precisely. In the present text I will try to follow this methodin order to introduce one of the main conjectures in Algebraic Geometry, namelythe Hodge conjecture.

161

162 21 Lixo de Hodge Theory

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