periods of algebraic cycles - instituto nacional de...

Hossein Movasati, Roberto Villaflor Loyola

A Course in Hodge Theory

Periods of Algebraic Cycles

June 6, 2019

Publisher

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Computational Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Some missing details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The organization of the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Coverings and Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Acyclic sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 How to compute Cech cohomology groups . . . . . . . . . . . . . . . . . . . . . 112.6 Short exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Resolution of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 Cech cohomology and Eilenberg-Steenrod axioms . . . . . . . . . . . . . . . 182.9 Dolbeault cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.10 Cohomology of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Hypercohomology of complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 An element of hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Acyclic sheaves and hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Long exact sequence in hypercohomology . . . . . . . . . . . . . . . . . . . . . . 283.6 Quasi-isomorphism and hypercohomology . . . . . . . . . . . . . . . . . . . . . 303.7 A description of an isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.8 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.9 Subsequent quotients and a spectral sequence . . . . . . . . . . . . . . . . . . . 33

4 Atiyah-Hodge theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Algebraic de Rham cohomology on affine varieties . . . . . . . . . . . . . . 37

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vi Contents

4.3 Proof of Atiyah-Hodge theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Algebraic de Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.3 The isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Hodge filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.5 Cup product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.7 Top cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.8 Standard top form in PN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.9 Chern class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.10 The cohomological class of an algebraic cycle . . . . . . . . . . . . . . . . . . 565.11 Pull-back in algebraic de Rham cohomology . . . . . . . . . . . . . . . . . . . . 58

6 Logarithmic differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Logarithmic differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.3 Deligne’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.4 Carlson-Griffiths Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.5 Second proof of Deligne’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.6 Residue map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.7 Hodge filtration for affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Cohomology of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Bott’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.3 Griffiths’ Theorem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.4 Griffiths’ Theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.5 Carlson-Griffiths Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.6 Computing the residue map in algebraic de Rham cohomology . . . . 76

8 Periods of algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2 Periods of top forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.3 Coboundary map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.4 Periods of complete intersection algebraic cycles . . . . . . . . . . . . . . . . 86

9 Gauss-Manin connection: general theory . . . . . . . . . . . . . . . . . . . . . . . . . 939.1 De Rham cohomology of projective varieties over a ring . . . . . . . . . . 939.2 Gauss-Manin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.4 Griffiths transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.5 Geometric Gauss-Manin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.6 Algebraic vs. Analytic Gauss-Manin connection . . . . . . . . . . . . . . . . . 989.7 A consequence of global invariant cycle theorem . . . . . . . . . . . . . . . . 98

Contents vii

9.8 The regularity of Gauss-Manin connection . . . . . . . . . . . . . . . . . . . . . 98

10 Infinitesimal variation of Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . 9910.1 Starting from Gauss-Manin connection . . . . . . . . . . . . . . . . . . . . . . . . 10010.2 Algebraic polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.3 Kodaira-Spencer map I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.4 Kodaira-Spencer map II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.5 A theorem of Griffiths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10510.6 IVHS for hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10510.7 Griffiths-Dwork method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.8 Noether-Lefschetz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10710.9 Algebraic deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

11 Hodge cycles and Artinian Gorenstein rings . . . . . . . . . . . . . . . . . . . . . . . 10911.1 Intoduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.2 Artinian Gorenstein rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.3 Artinian Gorenstein ring attached to Hodge cycles . . . . . . . . . . . . . . . 11111.4 Sum of two algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

12 Some components of Hodge loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.1 Hodge locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.2 Hodge locus for hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11612.3 Complete intersection algebraic cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 11712.4 Sum of two linear cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Chapter 1Introduction

The first draft of the present book was the lecture notes of a second course in Hodgetheory presented by the first author in 2015 to the second author. These notes weredeveloped into the second author’s Ph.D. thesis and the present text is the outcomeof this collaboration.

1.1 Computational Hodge theory

In order to solve a mathematical problem one may generalize it until a solutioncomes out by itself and this method is mainly attributed to A. Grothendieck. A com-pletely different approach must be adopted if one has the feeling that the Hodgeconjecture is wrong. Instead of generalizations, one has to study so many particularexamples, and one has to compute many well-known theoretical data, such that thecounterexample comes out by itself. Once you are in the ocean without compass, alldirections might lead you to a land. If such a counterexample is found then it wouldbe like a Columbus’ egg and there will be a explosion of other counterexamples.Even if the Hodge conjecture is true, the belief that it is false makes us to take amore computational approach, and at least this makes the Hodge theory accessibleto broader class of mathematicians, and in particular those who love computationalmathematics, either by hand or by computer. The first author’s book [Mov17b] isthe first attempt in this direction. Since the main focus of this book is smooth hy-persurfaces, we feel that one has to prepare the computational ground for arbitrarysmooth projective varieties and the present text is the output of this attempt.

1.2 Some missing details

In attempt to make a piece of mathematics more computational, one might think thatall the credit belongs to theory makers and one does only hard exercises of these

1

2 1 Introduction

theories, and therefore, this kind of mathematics might be banned from publicationin “prestigious journals”. The amount of time and effort needed for this purpose isusually higher than if we wanted to contribute for producing more theories. One mayalso find some missing details in the work of theory makers which are as importantas the main body of such theories. Here, we would like to highlight one example.In a personal communication (November 09, 2018) with P. Deligne, the first authorposed following question: “ I got motivated to write this email after seeing your talk‘what do we mean by equal’, and after getting the feeling that in the foundationof Hodge theory not every detail is explained. Let me explain this. For a smoothprojective variety over C we know that there is a concrete canonical isomorphismbetween the usual de Rham cohomology by C∞ forms and the algebraic de Rhamcohomology. Therefore, all the concepts in the topological side, such as cup product,cohomology class of cycles etc. can be transported to the algebraic side and canbe defined in a purely algebraic fashion. However, my impression is that nobodyhas verified concretely that the algebraic objects are the exact transportation of thecorresponding topological objects. For instance, Grothendiecks definition of a classof an algebraic cycle must corresponds to the one defined by integration, but I donot see if it is written somewhere. In particular, when the the algebraic cycle issingular, the only rigorous proof that I see is the resolution of singularities. Anyway,with my student Roberto Villaflor, we are trying to write a book containing themaximum details, however, we are stuck in this kind of issue. I would be gratefulif you clarify this for us.” The answer came few days later. “I have not thoughtabout proving the compatibility between Grothendieck’s definition of the class ofa cycle and integration, because I never used the latter. I like to use cohomology,not homology, and I like definitions which are uniform across cohomology theories(motivic philosophy)...”, (P. Deligne, personal communication November 13, 2018).As an idiom says “the devil is in the detail” and such missing details in the literaturetook more than 4 years of both authors.

1.3 The organization of the text

The emphasis of the first book [Mov17b] was mainly on hypersurfaces and the studyof the Hodge locus through the Fermat variety. This book intends to tell us theHodge theory of smooth projective varieties and their properties inside families.This is the study of Cech cohomology, hypercohomology, Gauss-Manin connection,infinitesimal variation of Hodge structures, Hodge loci etc. Despite in order to doconcrete computations, we will be back to our favorite example of hypersurfaces. Asynopsis of each chapter is explained below.

In Chapter 2 we present a minimum amount of material so that the reader getsfamiliar with Cech cohomology. We need to represent elements of cohomologieswith concrete data and we do this using an acyclic covering.

In Chapter 3 we aim to define hypercohomology of a complex of sheaves relativeto an acyclic covering, and hence, we describe elements of a hypercohomology with

1.3 The organization of the text 3

concrete data. We discuss quasi-isomorphisms, filtrations etc., adopted for compu-tations. This chapter is presented for general sheaves, however, our main examplefor this is the sheaf of differential forms.

In Chapter 4 we prove the Atiyah-Hodge theorem which says that the elementsof the de Rham cohomology of an affine variety is given by algebraic differentialforms. This paves the road for definition of the algebraic de Rham cohomology inthe next chapter.

Chapter 5 is fully dedicated to algebraic de Rham cohomology and the fact thatit is isomorphic to the classical de Rham cohomology. We need to describe this iso-morphism as explicitly as possible because we want to transport the integration ofC∞ forms to the algebraic side, where integration becomes a pure algebraic opera-tion. The objective of this chapter is to collect all necessary material for computingthe integration of elements of algebraic de Rham cohomologies over algebraic cy-cles.

Chapter 9 is devoted to the description of Gauss-Manin connection of familiesof algebraic varieties. In this chapter we work on the general context of arbitraryfamilies of projective varieties, whereas [Mov17b] focused on the computation ofGauss-Manin connection for tame polynomials, and in particular families of hyper-surfaces. One of the main theorems proved in this chapter is Griffiths transversality.It relates the Gauss-Manin connection to the underlying Hodge filtrations.

We do not give concrete applications of the Gauss-Manin connection in Al-gebraic Geometry, however, its partial data, namely the infinitesimal variation ofHodge structures (IVHS) has successful applications. This includes the famousNoether-Lefschetz theorem which says that a generic surface in the projective spaceof dimension three has Picard rank one. Chapter 10 is dedicated to this topic.

Chapter 2Cech cohomology

Il faut faisceautiser. (The motto of french revolution in algebraic and complex ge-ometry, see [Rem95], page 6).

2.1 Introduction

In Chapter 4 [Mov17b] we discussed the axiomatic approach to singular homologyand cohomology. These are the first examples of cohomology theories constructedin the first half of 20th century. Almost in the same time, other cohomolgy the-ories, such as De Rham and Cech cohomologies, and their properties were underconstruction and intensive investigation.

In this chapter we aim to introduce sheaf cohomologies and its explicit construc-tion using Cech cohomologies. Similar to the case of singular homology and coho-mology, the categorical approach to sheaf cohomology and the way that it is usedin mathematics, shows that in most of occasions we only need to know a bunch ofproperties of the sheaf cohomology and not its concrete construction. However, insome other occasions, mainly when we want to formulate some obstructions, weobtain elements in some sheaf cohomologies and so just axioms of Cech cohomol-ogy would not work. Therefore, we introduce some properties of sheaf chomologywhich can be taken as axioms and we also explain its explicit construction usingCech cohomology. We assume that the reader is familiar with sheaves of abeliangroups on topological spaces. The reader who is interested in a more elaboratedversion of this section may consult other books like [BT82], Section 10, [Voi02]Section 4.3.

For a complex manifold X , our main examples of sheaves are, OX the sheaf ofholomorphic functions, Ω

pX the sheaf of holomorphic p-forms in X , Ω

p,qX , the sheaf

of C∞ (p,q)-forms in X , ΩpX∞ the sheaf of C∞ p-forms in X , IY the ideal sheaf of a

subvariety Y of X and O∗X the sheaf of invertible holomorphic functions. Note thatthe group structure in all these examples is given by addition, except in the last onewhich is given by multiplication. The main examples of short exact sequences are

5

6 2 Cech cohomology

0→ Z ·2πi−−→ OXexp−−→ O∗X → 0, (2.1)

0→IY → OX → OY → 0. (2.2)

2.2 Cech cohomology

Recall that a sheaf S of abelian groups on a topological space X is a collection ofabelian groups

S (U), U ⊂ X open,

called the sections of S defined over U , together with restriction maps of sections.These restriction maps satisfy that every section is uniquely determined by its re-strictions to any open cover of its domain. In particular, S (X) is called the set ofglobal sections of S . Some other equivalent notations for this are

S (X) = Γ (X ,S ) = H0(X ,S ).

For every point x ∈ X , the stalk of S at x is denoted Sx and corresponds to thegroup of germs of sections of S defined over some neighbourhood of x ∈ X . Ashort sequence of sheaves

0→S1→S2→S3→ 0,

is exact if and only if it is exact at its stalks

0→S1,x→S2,x→S3,x→ 0,

for all x ∈ X . It is not difficult to see that for an exact sequence of sheaves of abeliangroups

0→S1→S2→S3→ 0 (2.3)

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)→

H2(X ,S1)→ ··· .

Let us explain the basic idea behind H1(X ,S1). The elements of H1(X ,S1) areconsidered as obstructions to the surjectivity of H0(X ,S2)→H0(X ,S3). This mapis not surjective, however, we can look at an element f ∈S3(X) locally and use thesurjectivity of S2→S3. Suppose there exist an open covering U = Ui, i ∈ I of

2.3 Coverings and Cech cohomology 7

X such that the exact sequence (2.3) remains exact when taking the sections definedover Ui and Ui∩U j for every i, j ∈ I. For instance, we have

0→S1(Ui)→S2(Ui)→S3(Ui)→ 0.

Remark 2.2.1 In general such covering might not exist. For instance, the surjec-tivity of S2,x→S3,x does not imply the surjectivity of S2(U)→S3(U) for someopen neighborhood of x. However, this will be the case in all our examples in thisbook and in particular for (2.1) and (2.2).

We take fi ∈ S2(Ui), i ∈ I such that fi is mapped to f |Ui under S2(Ui)→S3(Ui). This means that the elements f j− fi, which are defined in the intersectionsUi∩U j’s, are mapped to zero and so there are elements fi j ∈S1(Ui∩U j) such thatfi j is mapped to f j− fi. If we consider a different choice fi mapped to f |Ui underS2(Ui)→S3(Ui), then defining

gi := fi− fi ∈S2(Ui),

we see that it is mapped to zero in S3(Ui) and so there exist gi ∈ S1(Ui) that ismapped to gi. In consequence

g j− gi− fi j + fi j = 0, (2.4)

since it is mapped to zero in S2(Ui ∩U j). This lead us to define H1(U ,S1) tobe the set of ( fi j, i, j ∈ I) modulo the equivalence relation given by (2.4), that is,( fi j, i, j ∈ I) and ( fi j, i, j ∈ I) are equivalent if there exist gi ∈S1(Ui) such that(2.4) holds.

2.3 Coverings and Cech cohomology

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+1)-tuples σ = (Ui0 , . . . ,Uip), i0, . . . , ip ∈ I and for σ ∈U p define

|σ | :=Ui0···ip := ∩pj=0Ui j .

A p-cochain f = ( fσ )σ∈U p is an element in

Cp(U ,S ) := ∏σ∈U p

H0(|σ |,S )

Definition 2.3.1 Let π be an element in the permutation group of the set 0,1,2, . . . , p.It acts on U p in a canonical way: for σ =(Ui0 , . . . ,Uip) we have πσ =(Uiπ(0) , . . . ,Uiπ(p)).

8 2 Cech cohomology

We say that f ∈Cp(U ,S ) is skew-symmetric if fπσ = sign(π) fσ for all σ ∈U p.The set of skew-symmetric cochains form an abelian subgroup Cp

s (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 map as

δ : Cps (U ,S )→Cp+1

s (U ,S ), (δ f )σ :=p+1

∑j=0

(−1) j fσ j ||σ | .

We have to check that

Proposition 2.3.1 The above map is well-defined, that is, if f ∈Cp(U ,S ) is skew-symmetric then δ f is also skew-symmetric.

The proof of Proposition 2.3.1 can be easily seen from the following simplificationof our notations. From now on we identify σ with i0i1 · · · ip and write a p-cochainas f = ( fi0i1···ip , i j ∈ I). For simplicity we also write

(δ f )i0i1···ip+1 :=p+1

∑j=0

(−1) j fi0i1···i j−1 i j i j+1···ip+1,

where i j means that i j is removed.

Proposition 2.3.2 We haveδ δ = 0.

Proof. Let f ∈Cps (U ,S ). We have

(δ 2 f )i0i1·ip+2 =p+2

∑j=0

(−1) j(δ f )i0i1···i j ·ip+2

=p+2

∑j=0

p+2

∑k=0, k 6= j

(−1) j(−1)k fi0i1···ik···i j ···ip+2= 0,

where k = k if k < j and k = k−1 if k > j. See [BT82] Proposition 8.3. utNow

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 2.3.2 The p-th Cech cohomology of the covering U with coefficients inthe sheaf S is the cohomology group

H p(U ,S ) :=Kernel(Cp

s (U ,S )δ→Cp+1

s (U ,S ))

Image(Cp−1s (U ,S )

δ→Cps (U ,S ))

.

2.3 Coverings and Cech cohomology 9

The above definition depends on the covering and we wish to obtain cohomologiesH p(X ,S ) which depend only on X and S . We recall that the set of all coveringsU of X is directed:

Definition 2.3.3 For two coverings Ui = Ui, j, j ∈ Ii, i = 1,2 we write U1 ≤U2and say that U1 is a refinement of U2, if there is a map from φ : I1 → I2 such thatU1,i ⊂U2,φ(i) for all i ∈ I1.

For two coverings U1 and U2 there is another covering U3 such that U3 ≤U1 andU3 ≤U2. It is not difficult to show that for U1 ≤U2 we have a well-defined map

H p(U2,S )→ H p(U1,S ).

In fact, given any refinement map φ : I1→ I2 it induces a map

Φ : Cps (U2,S )→Cp

s (U1,S )

given byΦ(s)i0···ip := sφ(i0)···φ(ip)|Ui0 ···ip

.

This map induces the desired map in cohomology

Φ : H p(U2,S )→ H p(U1,S ).

We just need to check that this map does not depend on the refinement map φ . Ifψ : I1→ I2 is another refinement, the induced map

Ψ : Cps (U2,S )→Cp

s (U1,S )

is homotopic to Φ . In fact, the homotopy H : Cp+1s (U2,S )→Cp

s (U1,S ) is givenby

H(s)i0···ip :=p

∑l=0

(−1)lsψ(i0)···ψ(il)φ(il)···φ(ip)|Ui0 ···ip. (2.5)

It is left as an exercise to the reader to verify that

Φ−Ψ = δ H +H δ .

Definition 2.3.4 The Cech cohomology of X with coefficients in S is defined inthe following way:

H p(X ,S ) := dir limU H p(U ,S ).

We may view H p(X ,S ) as the union of all H p(U ,S ) for all coverings U , quo-tiented by the following equivalence relation. Two elements α ∈ H p(U1,S ) andβ ∈ H p(U2,S ) are equivalent if there is a covering U3 ≤ U1 and U3 ≤ U2 suchthat α and β are mapped to the same element in H p(U3,S ).

10 2 Cech cohomology

2.4 Acyclic sheaves

In practice, we never use the direct limit in Definition 2.3.4. This has to do withsome good coverings which are called acyclic.

Definition 2.4.1 A sheaf S of abelian groups on a topological space X is calledacyclic if

Hk(X ,S ) = 0, k = 1,2, . . .

Definition 2.4.2 Let A be a sheaf of rings over a topological space X . Every sheafF of A -modules is called fine if A satisfies the following condition: For everylocally finite open covering U = Uii∈I of X , there exist a partition of unity fi ∈A (X) with Supp fi ⊆Ui and ∑i∈I fi = 1 (where this last sum is well defined locallyand define a global section since A is a sheaf).

Example 2.1. The main examples of fine sheaves that we have in mind are the fol-lowing: Let X be a C∞ manifold, and denote C∞

X the sheaf of C∞ functions with realor complex values. Then all the C∞

X -modules are fine, such as Ω iX of C∞ differen-

tial i-forms on X . If further X is a complex manifold then the sheaves Ωp,qX of C∞

differential (p,q)-forms on X are fine.

Proposition 2.4.1 Let S be any fine sheaf over a topological space X. Then forevery open covering U of X and every k ≥ 1

Hk(U ,S ) = 0.

In particular, every fine sheaf is acyclic.

Proof. Let U = Uii∈I be the open covering of X . Then for every k ≥ 1 and σ ∈Ck

s (U ,S ) with δσ = 0 let us define τ ∈Ck−1s (U ,S ) as

τi0···ik−1 := ∑i∈I

fiσii0···ik−1 .

Then

(δτ)i0···ik =k

∑l=0

(−1)l∑i∈I

fiσii0···il ···ik= ∑∈I

fi

k

∑l=0

(−1)lσii0···il ···ik

= σi0···ik .

In other words σ = δτ , and so Hk(U ,S ) = 0, ∀k ≥ 1 and every cover U . ut

Definition 2.4.3 A sheaf S is said to be flasque if for every pair of open sets V ⊂U ,the restriction map S (U)→S (V ) is surjective.

Proposition 2.4.2 Let S be any flasque sheaf over a topological space X. Then forevery open covering U of X and every k ≥ 1

Hk(U ,S ) = 0.

In particular, every flasque sheaf is acyclic.

2.5 How to compute Cech cohomology groups 11

Proof. Let S be a flasque sheaf and σ ∈Cks (U ,S ) such that δσ = 0. Let us de-

fine τ ∈Ck−1s (U ,S ). For simplicity let us assume that the covering U = Uii∈I

is finite (this assumption might be removed by a transfinite induction argument).Inductively on the set of multi-indexes (i0, ..., ik) with i0 < · · · < ik ordered lexico-graphically, we define

τi1···ik :=

any element of S (Ui1···ik) if i1 = 0any extension of σi1···ik if i1 > 0

Where σi1···ik ∈S (∪i1−1j=0 U ji1···ik) is locally defined as

σi1···ik |U ji1 ···ik:= σ ji1···ik −

k

∑l=1

(−1)lτ ji1···il ik

|U ji1 ···ik.

It is routine to check τ ∈Ck−1s (U ,S ) is well defined and δτ = σ . ut

Example 2.2. Let X be a topological space and G any abelian group. The skyscrapersheaf supported in a point x ∈ X with values in G is the sheaf ix(G) given by

ix(G)(U) :=

G if x ∈U0 if x /∈U

with restrictions the identity or the zero map. This an example of flasque sheaf.

Example 2.3. Consider now the constant sheaf G, which corresponds to the sheafof locally constant functions from a topological space X to an abelian group G.This sheaf is not flasque in general. In fact, when X is an irreducible topologicalspace, G will be flasque (for instance if X is an irreducible algebraic variety with itsZariski topology). But as we will see later, when X is a smooth manifold and G =R,H i(X ,R) = H i

dR(X) and so R is not acyclic.

Example 2.4. Every sheaf S over a topological space X is a subsheaf of a flasquesheaf. In fact, it is enough to define F as the sheaf of discontinuous sections of S ,i.e.

F (U) = ∏x∈U

Sx.

This construction was used by Godement in order to produce acyclic resolutions ofany sheaf of abelian groups.

2.5 How to compute Cech cohomology groups

Definition 2.5.1 The covering U is called acyclic with respect to S if U is locallyfinite, i.e. each point of X has an open neighborhood which intersects a finite numberof open sets in U , and H p(Ui1 ∩·· ·∩Uik ,S ) = 0 for all Ui1 , . . . ,Uik ∈U and p≥ 1.


Theorem 2.5.1 (Leray’s lemma) Let U be an acyclic covering of a variety X withrespect to a sheaf S . There is a natural isomorphism

Hk(U ,S )∼= Hk(X ,S ).

The proof of this theorem will be given in the next section. For a sheaf of abeliangroups S over a topological space X , we will mainly use H1(X ,S ). Recall that foran acyclic covering U of X an element of H1(X ,S ) is represented 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.

Remark 2.5.1 For sheaves of abelian groups Si, i = 1,2, . . . ,k over a variety X itis easy to see that

H p(U ,⊕i≥1

Si) =⊕i≥1

H p(U ,Si), p = 0,1, . . .

for every open cover U of X . In particular,

H p(X ,⊕i≥1

Si) =⊕i≥1

H p(X ,Si), p = 0,1, . . .

2.6 Short exact sequences

In this section we return to one of the main motivations of sheaf cohomology,namely, for an exact sequence of sheaves of abelian groups

0→S1→S2→S3→ 0

we have a long exact sequence

· · · → H i(X ,S1)→ H i(X ,S2)→ H i(X ,S3)→ H i+1(X ,S1)→ ··· (2.6)

that is, in each step the image and kernel of two consecutive maps are equal. Allthe maps in the above sequence are canonical except those from i-dimensional co-homology to (i+ 1)-dimensional cohomology. In this section we explain how thismap is constructed in Cech cohomology.

Proposition 2.6.1 Let X be a topological space and

0→S1f−→S2

g−→S3→ 0

be a short exact sequence of sheaves. If H1(X ,S1) = 0, then

2.6 Short exact sequences 13

0→S1(X)Γ ( f )−−−→S2(X)

Γ (g)−−→S3(X)→ 0

is exact.

Proof. We want to show the surjectivity of Γ (g). For any ω ∈S3(X), there existan open covering U = Uii∈I of X such that for every i ∈ I there exist some µi ∈S2(Ui) such that gUi(µi) = ω|Ui . In particular, for every i, j ∈ I

δ (µ)i j = µ j−µi ∈ Ker gUi j .

In consequence, there exist ηi j ∈S1(Ui j) such that fUi j(ηi j) = δ (µ)i j. Then

fUi jk((δη)i jk) = 0,

and since fUi jk is injective it follows that δη = 0. Since H1(X ,S1) = 0, there existsome refinement U ′ ≤ U and ζ ∈ C0

s (U′,S1) such that η |U ′ = δζ . In conse-

quence µ|U ′ − f (ζ ) ∈C0s (U

′,S2) and δ (µ|U ′ − f (ζ )) = 0, so it defines a globalsection µ ∈S2(X) such that Γ (g)(µ) = ω . ut

Corollary 2.6.1 Consider over a topological space X a short exact sequence ofsheaves

0→S1f−→S2

g−→S3→ 0

and an open covering U = Uii∈I such that H1(∩kl=1Uil ,S1) = 0 for every i0 <

· · ·< ik ⊆ I finite not empty. Then there exist a long exact sequence

· · · → H i(U ,S1)→ H i(U ,S2)→ H i(U ,S3)→ H i+1(U ,S1)→ ·· · . (2.7)

Proof. By Proposition 2.6.1 we have a short exact sequence of complexes of abeliangroups

0→C•s (U ,S1)→C•s (U ,S2)→C•s (U ,S3)→ 0

which induces the long exact sequence in cohomology (2.7). ut

After Corollary 2.6.1, we can give an explicit description of the coboundary map

H i(U ,S3)→ H i+1(U ,S1),

for U an acyclic open cover with respect to S1. In fact, given any ω ∈Cis(U ,S3),

there exist some µ ∈Cis(U ,S2) given by Proposition 2.6.1 such that ω = g(µ). If

δω = 0, then g(δ µ) = 0 and so δ µ = f (η) for some η ∈Ci+1s (U ,S1). Since f is

injective and f (δη) = 0, it follows that δη = 0 and so the image of ω ∈H i(U ,S3)under the coboundary map is η ∈ H i+1(U ,S1).

Now it is easy to describe the coboundary map

H i(X ,S3)→ H i+1(X ,S1),

in general. Take any open cover U of X and any ω ∈Cis(U ,S3). There exist some

refinement U ′ ≤ U such that there exist µ ∈Cis(U

′,S2) such that g(µ) = ω|U ′ .


Imitating the construction of the coboundary map explained above, we produce theimage of ω ∈ H i(U ,S3) under the coboundary map as η ∈ H i+1(U ′,S1). It isroutine to check that this construction is independent of the choices made, and iscompatible with restrictions to refinements, thus it determines the coboundary mapin Cech cohomology. Now it is an exercise to prove that the long sequence inducedis exact.

Proposition 2.6.2 For every short exact sequence of sheaves over a topologicalspace X

0→S1f−→S2

g−→S3→ 0

There exist a long exact sequence in Cech cohomology

· · · → H i(X ,S1)→ H i(X ,S2)→ H i(X ,S3)→ H i+1(X ,S1)→ ·· · . (2.8)

Now we are in position to prove Leray’s lemma:

Proof. (of Theorem 2.5.1) Let S be a sheaf over X , and U an acyclic open coverof X with respect to S . Let F be the Godement’s flasque sheaf associated to Sgiven in Example 2.4. And let G := F/S . We will show by induction on k that thenatural map

Hk(U ,S )→ Hk(X ,S )

is an isomorphism. This is clear for k = 0. For k ≥ 1, it follows from Proposition2.4.2 that Hm(U ,F ) = Hm(X ,F ) = 0 for all m≥ 1. In order to use the inductionhypothesis for G , we need to show that U is an acyclic cover with respect to G .Considering the long exact sequence in Cech cohomology associated to the shortexact sequence

0→S →F → G → 0

restricted to each intersection Ui0···ip = ∩pl=0Uil , and using the fact that F |Ui0···ip is

flasque, we get that

Hm(Ui0···ip ,G )' Hm+1(Ui0···ip ,S ) = 0, ∀m≥ 1,∀p≥ 0,

and so U is an acyclic cover with respect to G . By induction hypothesis we havethe following commutative diagram with exact rows (2.7) and (2.8)

Hk−1(X ,F ) Hk−1(X ,G ) Hk(X ,S ) Hk(X ,F ) = 0 0

Hk−1(U ,F ) Hk−1(U ,G ) Hk(U ,S ) Hk(U ,F ) = 0 0

'' ''

and the result follows from the five lemma. ut

2.7 Resolution of sheaves 15

2.7 Resolution of sheaves

A complex of abelian sheaves is the following data:

S • : S 0 d0→S 1 d1→ ·· ·dk−1→ S k 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 •

For simplicity we will write d = dk, being clear in the text the domain of the map d.

Definition 2.7.1 A resolution S →S • is called acyclic if all S k are acyclic.

Proposition 2.7.1 If 0→S • is an acyclic resolution, then

0→ Γ (S 0)d0→ Γ (S 1)

d1→ ···dk−1→ Γ (S k)

dk→ ···

is exact.

Proof. For every i ≥ 0 define the sheaf F i := Im di. We get short exact sequencesfor every i≥ 1

0→F i−1→S i→F i→ 0.

We claim that F i is acyclic for every i ≥ 0. In fact, F 0 = S 0 is acyclic. And fori≥ 1, it follows from the long exact sequence in Cech cohomology and the fact thatS i is acyclic, that

Hk(X ,F i)' Hk+1(X ,F i−1) = 0, ∀k ≥ 1.

Then, by Proposition 2.6.1 we obtain that

0→ Γ (F i−1)→ Γ (S i)→ Γ (F i)→ 0

is exact, and the result follows. ut

Theorem 2.7.1 Let S be a sheaf of abelian groups on a topological space X andS →S • be an acyclic resolution of S then

Hk(X ,S )∼= Hk(Γ (X ,S •),d), k = 0,1,2, . . . . (2.9)


whereΓ (X ,S •) : Γ (S 0)

d0→ Γ (S 1)d1→ ·· ·

dk−1→ Γ (S k)dk→ ·· ·

and

Hk(Γ (X ,S •),d) :=ker(dk)

Im(dk−1).

Proof. We will prove it assuming there exist an open cover U = Uii∈I of X whichis acyclic with respect to S and all S i, i≥ 1. Let

S ij :=C j

s (U ,S i), S j :=C js (U ,S ), Γ (S i) := Γ (X ,S i).

Consider the double complex

↑ ↑ ↑ ↑0→ Sn → S 0

n → S 1n → S 2

n → ·· · → S nn →

↑ ↑ ↑ ↑ ↑0→ Sn−1 → S 0

n−1 → S 1n−1 → S 2

n−1 → ·· · → S nn−1 →

↑ ↑ ↑ ↑ ↑...

......

......

↑ ↑ ↑ ↑ ↑0→ S2 → S 0

2 → S 12 → S 2

2 → ·· · → S n2 →

↑ ↑ ↑ ↑ ↑0→ S1 → S 0

1 → S 11 → S 2

1 → ·· · → S n1 →

↑ ↑ ↑ ↑ ↑0→ S0 → S 0

0 → S 10 → S 2

0 → ·· · → S n0 →

↑ ↑ ↑ ↑Γ (S 0)→ Γ (S 1)→ Γ (S 2)→ ·· · → Γ (S n)→↑ ↑ ↑ ↑0 0 0 0

(2.10)

The up arrows are δ and the left arrow at S pq is (−1)qd, that is, we have multiplied

the map d with (−1)q, where q denotes the index related to Cech cohomology. Letus define the map A : Hk(Γ (X ,S •),d)→ Hk(X ,S ). It sends a d-closed globalsection ω of S n to a δ -closed cocycle α ∈ Cn

s (U ,S ) and the recipe is sketchedhere: Using Proposition 2.7.1 we can construct

2.7 Resolution of sheaves 17

0↑

α → ω0 → 0↑ ↑

η0 → ω1 → 0↑

η1 . . . . . .. . . ωn−1 → 0

↑ ↑ηn−1 → ωn → 0

↑ω

(2.11)

Where an arrow a→ b means that a is mapped to b under the corresponding map in(3.2), and ωn is the restriction of ω to opens sets Ui’s. The same diagram 2.11 canbe used to explain the map B : Hk(X ,S )→ Hk(Γ (X ,S •),d). In this case we startfrom α and we reach ω . We have to check that

Exercise 1 Show that

1. A and B are well-defined.2. AB and BA are identity maps.

In the general case we cannot apply Proposition 2.7.1 anymore in order to con-struct the maps A and B. Nevertheless, we can proceed in the same way in order toconstruct A and B taking care of passing to adequate refinements at each step. It isstill an exercise to check that these constructions are well-defined, and that one isthe inverse of the other.

If we do not care about using d or (−1)qd we will still get isomorphisms A and B,however, they are defined up to multiplication by −1. The mines sign in (−1)qd isinserted so that D := δ +(−1)qd becomes a differential operator, that is, DD = 0.For further details, see [BT82] Chapter 2. Another way to justify (−1)q is to see itin the double complex of differential (p,q)-forms in a complex manifold. ut

Let us come back to the sheaf of differential forms. Let M be a C∞ manifold. Thede Rham 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 2.7.2 (Poincare Lemma) If M is a unit ball then

H idR(M) =

R if i = 00 if i 6= 0

The Poincare lemma and Example 2.1 imply that


R→Ω•M∞

is an acyclic resolution of the constant sheaf R on the C∞ manifold M. By Theorem2.7.1 we 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.

2.8 Cech cohomology and Eilenberg-Steenrod axioms

Let G be an abelian group and M be a polyhedra. We can consider G as the sheafof constants on M and hence we have the Cech cohomology groups Hk(X ,G), k =0,1,2, . . .. This notation is already used in Chapter 4 of [Mov17b] to denote a co-homology theory with coefficients group G which satisfies the Eilenberg-Steenrodaxioms. The following theorem justifies the usage of the same notation.

Theorem 2.8.1 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

Hsingi (M,Z)×H i

dR(M)→ R, (δ ,ω) 7→∫

δ

ω

This gives us

H idR(M)→ ˇHsing

i (M,R)∼= H ising(M,R)

whereˇmeans dual of vector space.

Theorem 2.8.2 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, . . .

2.10 Cohomology of manifolds 19

2.9 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 2.9.1 (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. By Example 2.1 we know thatΩ

p,qM∞ ’s are fine sheaves and so we have the resolution of Ω p:

Ωp→Ω

p,•M∞ .

By Theorem 2.7.1 we conclude that:

Theorem 2.9.2 (Dolbeault theorem) For M a complex manifold

Hq(M,Ω p)∼= H p,q∂

(M).

There are examples of domains D in Cn such that H0,1∂

(D) 6= 0. See [Gun90a],the end of Chapter E.

2.10 Cohomology of manifolds

The first natural sheaves are constant sheaves. For an abelian group G, the sheaf ofconstants on X with coefficients in G is a sheaf such that for any connected openset it associates G and the restriction maps are the identity. We also denote by G thecorresponding sheaf. Our main examples are (k,+), k = Z,Q,R,C. For a smoothmanifold X , the cohomologies H i(X ,G) are isomorphic in a natural way to singularcohomologies and de Rham cohomologies, see respectively [Voi02] Theorem 4.47and [BT82] Proposition 10.6. We will need the following topological statements.

Proposition 2.10.1 Let X be a topological space which is contractible to a point.Then H p(X ,G) = 0 for all p > 0.

This statement follows from another statement which says that two homotopic mapsinduce the same map in cohomology.


Proposition 2.10.2 Let X be a manifold of dimension n. Then X has a coveringU = Ui, i ∈ I such that

1. all Ui’s and their intersections are contractible to points.2. The intersection of any n+2 open sets Ui is empty.

Using both propositions we get an acyclic covering of a manifold and we prove.

Proposition 2.10.3 Let M be an orientable manifold of dimension m.

1. We have H i(M,Z) = 0 for i > m.2. If M is not compact then the top cohomology Hm(M,Z) is zero.3. If M is compact then we have a canonical isomorphism Hm(M,Z)∼= Z given by

the orientation of M.

If M is a complex manifold of dimension n, then it has a canonical orientation givenby the orientation of C and so we can apply the above proposition in this case. Notethat M is of real dimension m = 2n.

Chapter 3Hypercohomology

My mathematics work is proceeding beyond my wildest hopes, and I am even a bitworried - if it’s only in prison that I work so well, will I have to arrange to spendtwo or three months locked up every year? (Andre Weil writes from Rouen prison,[OR16]).

3.1 Introduction

After a fairly complete understanding of singular homology and de Rham cohomol-ogy of manifolds and the invention of Cech cohomology, a new wave of abstrac-tion in mathematics started. H. Cartan and S. Eilenberg in their foundational book[CE56] called Homological Algebra took many ideas from topology and replaced itwith categories and functors. A. Grothendieck in [Gro57] took this into a new levelof abstraction and the by-product of his effort was the creation of many cohomol-ogy theories, such as Etale and algebraic de Rham cohomology. Etale cohomologywere mainly created in order to solve Weil conjectures, however, the relevant one tointegrals is the algebraic de Rham cohomology. Its main ingredient is the conceptof hypercohomology of complexes of sheaves which soon after its creation was re-placed with derived functors and derived categories. This has made it an abstractconcept far beyond concrete computations and the situation is so that the introduc-tion of C. A. Weibel’s book [Wei94] starts with: “Homological algebra is a tool usedto prove nonconstructive existence theorems in algebra (and in algebraic topology).”In this chapter we aim to introduce hypercohomology without going into categoricalapproach, and the main reason for this is that we would like to emphasize that itselements can be computed and in the case of complex of differential forms, they canbe integrated over topological cycles. For further information the reader is referredto [Voi03, Bry08].

21

22 3 Hypercohomology

3.2 Hypercohomology of complexes

In this section we assume that the reader is familiar with basic concepts of Cechcohomology (see Chapter 2).

Let us be given a complex of sheaves of abelian groups on a topological space X :

S • : S 0 d→S 1 d→S 2 d→ ·· · d→S n d→ ··· , d d = 0. (3.1)

We would like to associate to S • a cohomology which encodes all the Cech coho-mologies of individual S i together with the differential operators d. We first explainthis cohomology using a covering U = Uii∈I of X .

Before proceeding further, let us mention our main example in this chapter. Wetake a smooth projective variety X ⊂ PN of dimension n over an algebraically closedfield k. We take the complex Ω •X of differential forms on X .

Proposition 3.2.1 There is a covering of X with n+1 affine Zariski open subsets.

Proof. The covering is going to be

Ui := gi 6= 0, i = 0,1,2, . . . ,n,

where gi’s are linear homogeneous polynomials in x0,x1, ...,xN and we have as-sumed that the projective space PN−n−1 ⊂ PN given by g0 = g1 = · · · = gn+1 = 0does not intersect X . This happens for a generic PN−n−1. For instance for a genericlinear PN−n intersects X in deg(X) distinct points. This is the definition of the de-gree of a projective variety. Now we can take PN−n−1 ⊂ PN−n such that it does notcross the mentioned deg(X) points. ut

For a smooth hypersurface X ⊂ Pn+1 given by the homogeneous degree d polyno-mial f (x0,x1, ...,xn+1), we have also another useful covering given by

Ui :=

∂ f∂xi6= 0, i = 0,1,2, . . . ,n+1

which is called the Jacobian covering of X . Note that for this covering we use thefact that X is smooth. It has n+2 open sets. For a fixed k = 0,1,2, . . . ,n+1 the opensets U0,U1, ...,Uk−1,Uk+1, ...,Un+1 cover X if X is smooth and xk = 0 intersects Xtransversely.

Consider the double complex

3.2 Hypercohomology of complexes 23

↑ ↑ ↑ ↑S 0

n → S 1n → S 2

n → ·· · → S nn →

↑ ↑ ↑ ↑S 0

n−1 → S 1n−1 → S 2

n−1 → ·· · → S nn−1 →

↑ ↑ ↑ ↑...

......

...↑ ↑ ↑ ↑

S 02 → S 1

2 → S 22 → ·· · → S n

2 →↑ ↑ ↑ ↑

S 01 → S 1

1 → S 21 → ·· · → S n

1 →↑ ↑ ↑ ↑

S 00 → S 1

0 → S 20 → ·· · → S n

0 →

(3.2)

whereS i

j :=C js (U ,S i).

The horizontal arrows are the usual differential operators d of S i’s and the verticalarrows are the differential operators δ in the sense of Cech cohomology. The m-thpiece of the total chain of (3.2) is

L m :=⊕mi=0S

im−i

with the differential operator D which is defined on S ij by:

D = δ +(−1) jd. (3.3)

Remark 3.2.1 Our convention for D is compatible with the one used in [BT82]page 90 and [Bry08] page 14. In other references such as [Voi03], we also findD′ = (−1)iδ +d. This difference produces a sign ambiguity. In order to correct thisambiguity it is necessary to identify both hipercohomologies via the isomorphismHk(L •,D)' Hk(L •,D′) induced by the map

ϕ : L k→L k,

defined over each ω = ∑ki=0 ω i ∈ ⊕k

i=0Sij as

ϕ(ω i) := (−1)i(k−i)ω

i.

Exercise 2 For the operator D in (3.3) of the double complex (3.2) show that D D = 0.

This also justifies the appearance of the sign (−1) j in the definition of D.

Definition 3.2.1 The hypercohomology of the complex S • relative to the coveringU is defined as


Hm(U ,S •) := Hm(L •,D) =ker(L m→L m+1)

Im(L m−1→L m).

Remark 3.2.2 As in the case of Cech cohomology, given U1 ≤U2, there is a well-defined map

Hm(U2,S•)→Hm(U1,S

•).

In fact, if Ui = Ui, j, j ∈ Ii for i = 1,2, given any refinement map φ : I1→ I2, wehave an induced map

Φ :⊕mi=0Ci

s(U2,Sm−i)→⊕m

i=0Cis(U1,S

m−i).

This map induces a map in hypercohomology

Φ : Hm(U2,S•)→Hm(U1,S

•)

which does not depend on the refinement map. In fact, given other refinement mapψ : I1→ I2, there exist an homotopy

H :⊕m+1i=0 Ci

s(U2,Sm+1−i)→⊕m

i=0Cis(U1,S

m−i)

such thatΦ−Ψ = DH +H D.

It is an exercise left to the reader to check that this homotopy map is the one inducedby the homotopy map (2.5) described for Cech cohomology (it is enough to see thatthe homotopy commutes with the differentials d).

Definition 3.2.2 The hypercohomology Hm(X ,S •) is defined to be the direct limitof the total cohomology of the double complex (3.2), i.e.

Hm(X ,S •)∼= dirlimU Hm(U ,S •).

From a computational point of view this definition is not very useful. We have tosearch for coverings U such that Hm(U ,S •) becomes the hypercohomology itself.

Theorem 3.2.1 If the covering U is acyclic with respect to all abelian sheavesS i’s, that is, U is locally finite and

Hk(Ui1 ∩Ui2 ∩·· ·∩Uir ,Si) = 0, k,r = 1,2, . . . , i = 0,1,2, . . . . (3.4)

thenHm(X ,S •)∼=Hm(U ,S •). (3.5)

Definition 3.2.3 A covering U is said to be a good covering with respect to a com-plex of sheaves S •, if it is locally finite and satisfies (3.4).

By definition of the direct limit, we have already a map

Hm(U ,S •)→Hm(X ,S •)

3.3 An element of hypercohomology 25

which sends an α to its equivalence class. We have to show that it is a bijection. Theproof of this result will be given in § 3.5.

3.3 An element of hypercohomology

The reader who is mainly interested in computational aspects of hypercohomologymay take (3.5) as the definition of hypercohomology. In this way we can describeits elements explicitly. An element of Hm(X ,S •) is represented by

ω = ω0 +ω

1 + · · ·+ωm, ω

j ∈Cm− js (U ,S j),

where U is a good covering with respect to S •. Each ω j itself is the followingdata:

ωj

i0i1···im− j∈S j(Ui0 ∩Ui1 ∩·· ·∩Uim− j)

for all i0, i1, · · · , im− j ∈ I. Such an ω is D-closed, that is, D(ω) = 0, if and only if,the following equalities hold

0 = δω0,

(−1)m−1dω0 = δ (ω1),

(−1)m−2dω1 = δ (ω2),

... (3.6)dω

m−1 = δ (ωm),

dωm = 0.

Such an ω is D-exact, or equivalently it is zero in Hm(X ,S •), if and only if there isη = ∑

m−1j=0 η j, η j ∈Cm−1− j

s (U ,S j) such that

ω0 = δ (η0),

ω1 = (−1)m−1d(η0)+δ (η1),

... (3.7)ω

m−1 = −d(ηm−2)+δ (ηm−1),

ωm = dηm−1.

In order to memorize better these equalities the diagram below can be helpful


0ω0 0η0 ω1 0

η1 . . . . . .. . . ωm−1 0

ηm−1 ωm 0

(3.8)

Recall that one must use (−1) jd for horizontal map and δ for vertical maps.It is instructive to consider the cases m = 0,1,2 separately.

• (m = 0) We have

H0(X ,S •)∼=

ω ∈S 0(X) | dω = 0.

• (m = 1) The first hypercohomology H1(X ,S •) is the set of pairs (ω0,ω1),where ω0 consists of ω0

i0i1 ∈ S 0(Ui0 ∩Ui1), i0, i1 ∈ I and ω1 consists ofω1

i0 ∈S 1(Ui0), i0 ∈ I which satisfy the relation

ω1i1 −ω

1i0 = dω

0i0i1 .

Such an ω is taken modulo those of the form ( fi1 − fi0 ,d fi0).• (m = 2) The second hypercohomology H2(X ,S •) is the set of triples ω =

(ω0,ω1,ω2), where ω0 consists of ω0i0i1i2 ∈ S 0(Ui0 ∩Ui1 ∩Ui2), i0, i1, i2 ∈ I,

ω1 consists of ω1i0i1 ∈ S 1(Ui0 ∩Ui1), i0, i1 ∈ I and ω2 consists of ω2

i0 ∈S 2(Ui0), i0 ∈ I. They satisfy the relations

ω0i1i2i3 −ω

0i0i2i3 +ω

0i0i1i3 −ω

0i0i1i2 = 0,

ω1i1i2 −ω

1i0i2 +ω

1i0i1 =−dω

0i0i1i2 ,

ω2i1 −ω

2i0 = dω

1i1i0 .

Such an ω is zero in H2(X ,S •) if

ω0i0i1i2 = η

0i1i2 −η

0i0i2 +η

0i0i1 , (3.9)

ω1i0i1 = −dη

0i0i1 +η

1i1 −η

1i0 , (3.10)

ω2i0 = dη

1i0 , (3.11)

for some η’s whose type can be determined by their indices.

When the covering U := U0,U1, · · · ,Un consists of n open sets and S m = 0, m>n then by definition

Hm(X ,S •) = 0, m > 2n

3.4 Acyclic sheaves and hypercohomology 27

Moreover, if we define Ui :=U0∩U1∩·· ·∩Ui−1∩Ui+1∩·· ·∩Un then

H2n(X ,S •)∼=S n(U0∩U1∩·· ·∩Un)

dS n−1(U0∩U1∩·· ·∩Un)+S n(U0)+S n(U1)+ · · ·+S n(Un).

(3.12)For simplicity, we have not written the restriction maps. Note that the last n+ 1terms in the denominator of (3.12) form the set δηn = ηn

0 −ηn1 + · · ·+(−1)nηn

n |ηn

i ∈S n(Ui). For n = 1 we get

H2(X ,S •)∼=S 1(U0∩U1)

dS 0(U0∩U1)+S 1(U1)+S 1(U0).

3.4 Acyclic sheaves and hypercohomology

Proposition 3.4.1 If all the sheaves S i are acyclic, that is, H j(X ,S i) = 0, j =1,2, . . . then for every good covering U with respect to S •

Hm(U ,S •)∼= Hm(H0(X ,S •),d) :=ker(H0(X ,S m)→ H0(X ,S m+1))

Im(H0(X ,S m−1)→ H0(X ,S m)).

Proof. We have already a map

f : Hm(H0(X ,S •),d)→Hm(U ,S •) (3.13)

which is obtained by restricting a global section of S m to the open sets of thecovering U . We have to define its inverse

f−1 : Hm(U ,S •)→ Hm(H0(X ,S •),d). (3.14)

Let us take an element ω = ∑mj=0 ω j ∈ L m which is D-closed. We have written

the ingredient equalities derived from this in (3.6). In particular, δω0 = 0 and byour hypothesis ω0 = δη0 for some η0 ∈S 0

m−1 (here we are using Leray’s lemma,since U is an acyclic covering with respect to S 0, and that Hm(X ,S 0) = 0). Theelements ω and ω−Dη0 represent the same object in Hm(U ,S •), and so we canassume that ω0 = 0. This process continues and finally we get an element in S m

0which is is equivalent to ω in Hm(U ,S •), and moreover, it is both δ and d-closed.This gives us a global section f−1(ω) ∈ H0(X ,S m). We have to check that f−1 iswell-defined, and it is the inverse of f . These details are left to the reader. ut

Using the same argument, but taking care of passing through the correspondingrefinements it is an exercise to prove the following result:

Proposition 3.4.2 If S • is a complex of acyclic sheaves, then

Hm(X ,S •)∼= Hm(H0(X ,S •),d).


3.5 Long exact sequence in hypercohomology

Proposition 3.5.1 Consider

0→S •1

f−→S •2

g−→S •3 → 0

a short exact sequence of complexes of sheaves. If U = Uii∈I is an open cover-ing such that H1(Ui0···ik ,S

p1 ) = 0 for every k, p ≥ 0, then there exist a long exact

sequence in hypercohomology

· · · →Hm(U ,S •1 )→Hm(U ,S •

2 )→Hm(U ,S •3 )→Hm+1(U ,S •

1 )→ ··· .(3.15)

Proof. By the hypothesis, for every p≥ 0, we have a short exact sequence of com-plexes

0→C•s (U ,S p1 )→C•s (U ,S p

2 )→C•s (U ,S p3 )→ 0.

This induces a short exact sequence of complexes of abelian groups

0→⊕p+q=•Cqs (U ,S p

1 )→⊕p+q=•Cqs (U ,S p

2 )→⊕p+q=•Cqs (U ,S p

3 )→ 0

whose induced long exact sequence corresponds to (3.15). ut

We can give an explicit description of the coboundary map in hypercohomology

Hm(U ,S •3 )→Hm+1(U ,S •

1 ),

for U a good cover with respect to S •1 . In fact, given any ω ∈⊕p+q=mCq

s (U ,S p3 ),

there exist some µ ∈ ⊕p+q=mCqs (U ,S p

2 ) such that ω = g(µ). If Dω = 0, theng(Dµ) = 0 and so Dµ = f (η) for some η ∈ ⊕p+q=m+1Cq

s (U ,S p1 ). Since f is in-

jective and f (Dη) = 0, it follows that Dη = 0 and so the image of ω ∈Hm(U ,S •3 )

under the coboundary map is η ∈Hm+1(U ,S •1 ).

Now it is easy to describe the coboundary map

Hm(X ,S •3 )→Hm+1(X ,S •

1 ),

in general. Take any open cover U of X and any ω ∈ ⊕p+q=mCqs (U ,S p

3 ). Thereexist some refinement U ′ ≤U such that there exist µ ∈ ⊕p+q=mCq

s (U ,S p2 ) such

that g(µ) = ω|U ′ . Imitating the construction of the coboundary map explainedabove, we produce the image of ω ∈ Hm(U ,S •

3 ) under the coboundary map asη ∈ Hm+1(U ,S •

1 ). It is routine to check that this construction is independent ofthe choices made, and is compatible with restrictions to refinements, thus it deter-mines the coboundary map in hypercohomology. Now it is an exercise to prove thatthe long sequence induced is exact.

Proposition 3.5.2 Consider

0→S •1

f−→S •2

g−→S •3 → 0

3.5 Long exact sequence in hypercohomology 29

a short exact sequence of complexes of sheaves. Then there exist a long exact se-quence in hypercohomology

· · · →Hm(X ,S •1 )→Hm(X ,S •

2 )→Hm(X ,S •3 )→Hm+1(X ,S •

1 )→ ··· . (3.16)

Now we are in position to give a proof of Theorem 3.2.1. But first we need analgebraic lemma about sheaves.

Lemma 3.1. Let S • be a bounded complex of sheaves, i.e. S m = 0 for m >>0. Then we can embed S • in a complex of sheaves I • such that H k−1 :=Ker (I k−1→I k) is flasque and Hk(I •) = 0 for every k > 0.

Exercise 3 Prove Lemma 3.1 using Godement’s construction of Example 2.4.

Proof (of Theorem 3.2.1). Let U be a good covering with respect to S •. we wantto show that for every m≥ 0 the natural map

Hm(U ,S •)→Hm(X ,S •)

is an isomorphism. Fixing m, we can truncate the complex at some n >> m and re-duce ourselves to the case S • is bounded. Let us assume then that S • is bounded.Let S • → I • be given by Lemma 3.1 and define G • := I •/S •. We will provethe result by induction on m. For m = 0 it is clear that H0(U ,S •) = H0(U ,S 0) =H0(X ,S 0) =H0(X ,S •) independently of the fact that U is a good covering. Form > 0, it follows from Lemma 3.1 that for every k ≥ 0 we have a short exact se-quence

0→H k→I k→H k+1→ 0. (3.17)

Since every H k is flasque, it follows that every I k is acyclic and that

0→ Γ (H k)→ Γ (I k)→ Γ (H k+1)→ 0 (3.18)

is exact for every k ≥ 0. In consequence Hk(Γ (I •)) = 0 for every k > 0. Fur-thermore (3.17) remains exact when restricting to any open set U ⊆ X , and soI k|U is acyclic for every k ≥ 0. In consequence, every locally finite coveringof X is a good covering with respect to I •. By Proposition 3.4.1 we get thatHk(U ′,I •)'Hk(Γ (I •)) = 0 for every locally finite covering U ′ of X and everyk > 0. In particular Hk(U ,I •) =Hk(X ,I •) = 0 for all k > 0.

In order to apply the induction hypothesis to G •, we need to check that U isa good covering with respect to G •. This follows from the long exact sequence inCech cohomology associated to the short exact sequence

0→S p|Ui0 ···ik→I p|Ui0 ···ik

→ G p|Ui0 ···ik→ 0

together with the fact that U is a good covering with respect to S • and thatI p|Ui0 ···ik

is flasque.Finally, by Propositions 3.5.1 and 3.5.2 we have the following commutative dia-

gram


Hm−1(X ,I •) Hm−1(X ,G •) Hm(X ,S •) Hm(X ,I •) = 0 0

Hm−1(U ,I •) Hm−1(U ,G •) Hm(U ,S •) Hm(U ,I •) = 0 0

'' ''

and the result follows from the five lemma. ut

3.6 Quasi-isomorphism and hypercohomology

A morphism between two complexes S • and S • is the following commutativediagram:

· · · → S n−1 → S n → S n+1 → ·· ·↓ ↓ ↓

· · · → S n−1 → S n → S n+1 → ·· ·

It induces a canonical map in the hypercohomologies

Hm(X ,S •)→Hm(X ,S •).

For a morphism f : S •→ S • of complexes we have a canonical morphism

Hm( f ) : Hm(S •)→ Hm(S •)

where for a complex S • we have defined

Hm(S •) := The sheaf constructed from the presheafker(dm)

Im(dm−1), m ∈ Z

Definition 3.6.1 A morphism f of complexes is called a quasi-isomorphism if theinduced morphisms Hm( f ), m ∈ Z are isomorphisms.

Example 3.1. Let X be a smooth variety over the field of complex numbers and letX∞ be the underlying C∞ manifold. Let Ω •X be the complex of (algebraic) differentialforms in X . We also consider the complex Ω •X∞ of C∞ differential forms in X for theZariski topology of X∞. We will only need to consider Zariski open sets. In Chapter5 we are going to show that the natural inclusion Ω •X→Ω •X∞ is a quasi-isomorphism.

Proposition 3.6.1 Let f : S •→ S • be a quasi-isomorphism then the induced mapin hypercohomology f∗ : Hm(U ,S •)→ Hm(U ,S •) is an isomorphism for everyopen covering U of X. In consequence Hm(U ,S •)'Hm(U ,S •).

Proof. We have to define the inverse map

f−1∗ : Hm(U ,S •)→Hm(U ,S •) (3.19)

3.6 Quasi-isomorphism and hypercohomology 31

Let ω = ∑mj=0 ω j ∈ Hm(U ,S •). We start by looking at ωm. We have d(ωm) = 0

and so there isω

m ∈S m0 ,dω

m = 0

such thatω

m− f∗ωm = dηm−1

for some ηm−1 ∈ S m−10 . We replace ω with ω −Dηm−1 and in this way we can

assume thatω

m = f∗ωm.

Now we look at the (m−1)-level in which we have

dωm−1 = δω

m = δ f∗ωm = f∗ (δωm) . (3.20)

Then δωm is a d-closed element that it is mapped to zero under Hm( f ). Therefore,using that f is a quasi-isomorphism, we conclude that δωm is d-exact, that is, thereexists νm−1 ∈S m−1

1 such that

δωm−dν

m−1 = 0. (3.21)

Combining (3.20) and (3.21) we get

d(ω

m−1− f∗νm−1)= 0. (3.22)

We are now in a similar situation as in the beginning. Since f is a quasi-isomorphismwe have

ωm−1− f∗νm−1− f∗µm−1 = dη

m−2,

for some ηm−2 ∈ S m−21 , µm−1 ∈ S m−1

1 and dµm−1 = 0. Replacing ω by ω +

Dηm−2 and defining ωm−1 := νm−1 +µm−1 ∈S m−11 we obtain

ωm−1 = f∗ωm−1 and δω

m = dωm−1.

At the (m− 2)-the level we get ωm ∈S m0 ,ωm−1 ∈S m−1

1 ,νm−2 ∈S m−22 with the

following identities

ωm = f∗ωm, ω

m−1 = f∗ωm−1, −dωm−1 +δω

m = 0,

andd(ω

m−2− f∗νm−2)= 0. (3.23)

This process stops at (m+ 1)-th step and we get ω = ∑mi=0 ωm which is D-closed

and f∗ω = ω . We define f−1∗ (ω) to be equal to ω . It is left to the reader to prove

that f−1∗ is well-defined, i.e. it is independent of all choices made, and it is inverse

to f∗. ut

Exercise 4 Complete the details of the proof of Proposition 3.6.1.

Proposition 3.6.2 Let S →S • be a resolution of S •. Then


Hk(X ,S •)∼= Hk(X ,S )

Proof. By hypothesis the complex · · · → 0→ S → 0→ ··· with S in the 0-thplace, is quasi-isomorphic to the complex S • and so the result follows from Propo-sition 3.6.1. ut

3.7 A description of an isomorphism

Let us now be given a quasi-isomorphism of complexes S •→ S • and assume thatall the abelian sheaves S m are acyclic. We use Proposition 3.4.1 and Proposition3.6.1 and we get an isomorphism

Hm(X ,S •)∼= Hm(S •(X),d).

For later applications we need to describe the two maps

A : Hm(X ,S •)→ Hm(S •(X),d)

A−1 : Hm(S •(X),d)→Hm(X ,S •).

such that both A A−1 and A−1 A are identity maps. We take an acyclic coveringU = Uii∈I with respect to all sheaves S i and S i. The map A is obtained by thecomposition of the maps

Hm(X ,S •)→Hm(X ,S •)→ Hm(S •(X),d)

The first map is simply induced by the quasi-isomorphism. The second map is themap (3.14). Its explicit description is given in the proof of Proposition 3.4.1. Themap A−1 is the composition of

Hm(S •(X),d)→Hm(X ,S •)→Hm(X ,S •)

The first map is obtained by restricting a global section of S m to the open sets ofthe covering U . The second map is (3.19) and its explicit description is given in theproof of Proposition 3.6.1.

3.8 Filtrations

For a complex S • and k ∈ Z we define the truncated complexes

S ≤k : · · ·S k−1→S k→ 0→ 0→ ···

andS ≥k : · · · → 0→ 0→S k→S k+1→ ···

3.9 Subsequent quotients and a spectral sequence 33

We have canonical morphisms of complexes:

S ≤k→S •, S ≥k→S •.

Assume that S • is a left-bounded complex, that is,

S • : 0→S 0→S 1→ ··· .

The morphism S ≥i→S • induces a map in hypercohomologies and we define

F i := Im(Hm(X ,S ≥i)→Hm(X ,S •)). (3.24)

This gives us the “naive” filtration in hypercohomology

· · · ⊂ F i ⊂ F i−1 ⊂ ·· · ⊂ F1 ⊂ F0 :=Hm(X ,S •).

An element of F i in an acyclic covering U of X is given by

ω = ωi +ω

i+1 + · · ·+ωm, ω

j ∈Cm− js (U ,S j)

where ω j itself is the collection of ωj

i0i1···im− j∈S j(Ui0 ∩Ui1 ∩ ·· · ∩Uim− j), for all

i0, i1, · · · , im− j ∈ I. Note that a priori the element ω might be D-exact in Hm(X ,S •),but as an element of Hm(X ,S ≥i) it might not be D-exact. In other words, the mapin (3.24) might not be injective. This phenomenon is related to the degeneration atE1 of the spectral sequence associated to the naive filtration.

3.9 Subsequent quotients and a spectral sequence

The differential map d : S i→S i+1 induces the maps

d1 : H j(X ,S i)→ H j(X ,S i+1), j ∈ N0.

If all these maps are zero then we can define the maps

d2 : H j(X ,S i)→ H j−1(X ,S i+2), j ∈ N0

which are defined as follows: we take ω i ∈ H j(X ,S i) and since d1 is zero we haveω i+1 ∈S i+1

j−1 such that δω i+1 +(−1) jdω i = 0. Under d2 the element ω i is mappedto dω i+1. In a similar way, we define

dr : H j(X ,S i)→ H j−r+1(X ,S i+r), j ∈ N0.

and if dk, k ≤ r are all zero we define dr+1.

Theorem 3.9.1 Assume that all dr’s constructed above are zero. Then we havecanonical isomorphisms


F i/F i+1 ∼= Hm−i(X ,S i). (3.25)

The reader who is familiar with spectral sequences has noticed that the hypothesisin Theorem 3.9.1 is equivalent to say that the spectral sequence associated to thedouble complex (3.2) with respect to the naive filtration degenerates at E1. In thissection we will recall briefly how to construct this spectral sequence. For furtherdetails, and a treatment of spectral sequences associated to any complex of filteredabelian groups see [Voi02, §8.3].

Let U be a good cover with respect to S •. Define for every p,q,r≥ 0 the abeliangroups

Zp,qr := ω ∈L p+q : ω

0 = · · ·= ωp−1 = (Dω)p = · · ·= (Dω)p+r−1 = 0,

Bp,qr := Zp+1,q−1

r−1 +DZp−r+1,q+r−2r−1 ⊆ Zp,q

r ,

and

E p,qr :=

Zp,qr

Bp,qr

.

Since DZp,qr ⊆ Zp+r,q−r+1

r and DBp,qr ⊆ Bp+r,q−r+1

r , we have the induced map

dr := D : E p,qr → E p+r,q−r+1

r

which turns (E p+r•,q−(r−1)•r ,dr) into a complex. It is clear that E p,q

0 = S pq and d0 =

δ . On the other hand, since Zp,qr+1 ⊆ Zp+1,q−1

r , we have that DZp,qr+1 ⊆ Bp+r,q−r+1

r andso there is a natural map

Zp,qr+1→ ker

(E p,q

rdr→ E p+r,q−r+1

r

).

It is an exercise to check that this map induces the isomorphism

E p,qr+1 '

ker(

E p,qr

dr→ E p+r,q−r+1r

)Im(

E p−r,q+r−1r

dr→ E p,qr

) .In particular E p,q

1 = Hq(X ,S p) and d1 = (−1)qd. Furthermore, if we define

F pL p+q := ω ∈L p+q : ω0 = · · ·= ω

p−1 = 0,

then for r > p+q+1

E p,qr+1 =

ker(D : F pL p+q→ F pL p+q+1

)ker(D : F p+1L p+q→ F p+1L p+q+1)+FPL p+q∩ ImD

' F p/F p+1 = GrFpHp+q(X ,S •) =: E p,q

∞ .

3.9 Subsequent quotients and a spectral sequence 35

Definition 3.9.1 We say that the spectral sequence (E p,qr ,dr) degenerates at Er if

dk = 0 for all k ≥ r. In such case we have

E p,qr = E p,q

k = E p,q∞ = F p/F p+1.

We say that the spectral sequence abuts to Hp+q(X ,S •), and it is denoted by

E p,qr ⇒Hp+q(X ,S •).

The principal question behind the isomorphism (3.25) is under which hypothesiscan we assure that every ω i ∈S i

m−i such that δω i = 0, can be extended to an ω =ω i + · · ·+ωm ∈ F iL m such that Dω = 0.

Definition 3.9.2 Let U be a good covering of X with respect to S •. An elementω i ∈ S i

m−i is said to be extendable in hypercohomology if there exist ω = ω i +· · ·+ωm ∈ F iL m such that

Dω = 0.

For every r ≥ 0 we say that ω i ∈S im−i is r-extendable if there exist ω = ω i + · · ·+

ωm ∈ F iL m such thatDω ∈ F i+rL m+1.

In particular, for r > m+1− i being r-extendable is the same as being extendable inhypercohomology. Note that the hypothesis of Theorem 3.9.1 is the same as sayingthat every element 1-extendable is extendable in hypercohomology.

Proposition 3.9.1 The spectral sequence (E p,qr ,dr) degenerates at Er if and only if

every r-extendable element is extendable in hypercohomology.

Proof. Let ω i ∈S im−i be an element r-extendable. Then there exist ω = ω i + · · ·+

ωm ∈ F iL m such that Dω ∈ F i+rL m+1. Then ω ∈ Zi,m−ir , and since dr = 0 we

conclude that Dω ∈ Bi+r,m−i−r+1r = Zi+r+1,m−i−r

r−1 +DZi+1,m−i−1r−1 . In consequence,

there exist η =η i+1+ · · ·+ηm ∈F i+1L m such that D(ω−η)∈F i+r+1L m+1. Thisimplies that ω i is (r+ 1)-extendable. Inductively we show that ω i is k-extendablefor every k ≥ r and so it is extendable in hypercohomology.

Conversely, since every element k-extendable for k ≥ r is also r-extendable, it isenough to show that dr = 0. Let ω = ω i+ · · ·+ωm ∈ Zi,m−i

r , then Dω ∈ F i+rL m+1,and so ω i is r-extendable. Therefore ω i is extendable in hypercohomology, i.e. thereexist η = ω i+η i+1+ · · ·+ηm ∈ F iL m such that Dη = 0. Then Dω = D(ω−η) ∈DZi+1,m−i−1

r−1 ⊆ Bi+r,m−i−r+1r , and so drω = 0. ut

Proposition 3.9.2 The spectral sequence (E p,qr ,dr) degenerates at E1 if and only if

any of the following equivalent assertions hold:

1. For every p≥ 0 we have the isomorphism (3.25)

F p/F p+1 ' Hq(X ,S p).

2. The natural map Hm(X ,S ≥p)→Hm(X ,S •) is injective.


Furthermore, when S • satisfies Hk(S •) = 0 for k > 0, this is also equivalent to:

3. For all p,q ≥ 0 the map Hq(X ,S p)→ Hq(X ,S p) is surjective, where S p :=ker(d : S p→S p+1), or equivalently d : Hq(X ,S p)→ Hq(X ,S p+1) is zero.

Proof. Suppose the spectral sequence degenerates at E1, then

Hq(X ,S p)' E p,q1 = E p,q

∞ = F p/F p+1,

this proves 1.Consider now the map ω i + · · ·+ωm ∈Hp+q(X ,S ≥p) 7→ ω i ∈Hq(X ,S p). It is

easy to see that this map is well-defined, and its kernel is

F p+1Hp+q(X ,S ≥p) := Im(Hp+q(X ,S ≥p+1)→Hp+q(X ,S ≥p)

).

We have the natural maps

Hp+q(X ,S ≥p)/F p+1Hp+q(X ,S ≥p) F p/F p+1, (3.26)

andHp+q(X ,S ≥p)/F p+1Hp+q(X ,S ≥p) → Hq(X ,S p). (3.27)

Assuming 1. we have that both maps are isomorphisms and so we conclude lookingat the isomorphism (3.26) for decreasing values of p, that Hp+q(X ,S ≥p)' F p forevery p≥ 0. This proves 2.

Suppose now that 2. holds. We will show that the spectral sequence degeneratesat E1 by applying Proposition 3.9.1. Consider ω i ∈S i

m−i that is 1-extendable, i.e.δω i = 0. Then Dω i ∈ F i+1 ' Hm(X ,S ≥i+1). Therefore, there exist some ω i+1 +· · ·+ωm ∈ F i+1L m such that Dω i = D(ω i+1 + · · ·+ωm), and so D(ω i− (ω i+1 +· · ·+ωm)) = 0, i.e. ω i is extendable in hypercohomology.

Finally, it is clear that 3. implies that every 1-extendable ω i ∈S im−i is extendable

in hypercohomology since dω i = δω i+1 for some ω i+1 ∈S i+1m−i−1 with dω i+1 = 0.

And conversely, every ω i ∈ S im−i such that δω i = 0 is 1-extendable, and so it is

extendable in hypercohomology, i.e. there exist ω = ω i + · · ·+ωk ∈ F iL m withDω = 0 for some k ≤ m. We claim we can choose ω such that k = i+ 1. In fact,if we suppose k > i+ 1 we have dωk = 0, and so there exist ηk−1 ∈ S k−1

m−k suchthat (−1)m−kdηk−1 = ωk. Replacing ω by ω −Dηk−1 we reduce the value of k.Repeating this process until k = i+1 we obtain 3. ut

Exercise 5 (Deligne’s cohomology) For the complex S • if the the kernel of d :S 0 → S 1 is non-trivial then we can take any abelian subgroup B of ker(d) andform the new complex B→S • and take its hypercohomology. Discuss this.

Chapter 4Atiyah-Hodge theorem

4.1 Introduction

The algebraic de Rham cohomology was introduced by Grothendieck in [Gro69] af-ter many efforts in order to understand the de Rham cohomology of affine varieties.This is in some sense natural because the integration domain of integrals in Picard,Simart and Poincae’s works [PS06, Pic89, Poi87, Poi95] are usually supported inaffine varieties. The Atiyah-Hodge theorem is the final outcome of all these efforts.

After Hodge decomposition and Dolbeault theorem we are able to recover eachpiece of the de Rham cohomology group of a smooth projective variety using onlyCech (or sheaf) cohomology with coefficients in sheaves of algebraic differentialforms. This is the first manifestation of an algebraic approach to de Rham cohomol-ogy. Unfortunately, this approach does not work for arbitrary smooth varieties, forinstance for affine varieties. In this chapter we will introduce a result due Atiyahand Hodge, which says that we can recover the de Rham cohomology groups in thecase of affine varieties using only global algebraic differential forms. This theoremwill allow us to introduce algebraic de Rham cohomology for any algebraic variety.

4.2 Algebraic de Rham cohomology on affine varieties

Let X be a smooth affine variety over C. Consider the algebraic de Rham complexof X

0→ OXd−→Ω

1X/C

d−→ ·· · d−→ΩnX/C

d−→ 0.

The following classical result due to Serre, implies this is an acyclic complex.

Theorem 4.2.1 (Serre vanishing theorem) Let X be an affine variety over the fieldC of complex numbers. Then

H i(X ,Ω jX/C) = 0, i = 1,2, . . . , j = 0,1,2, . . . .

37

38 4 Atiyah-Hodge theorem

Proof. The Theorem is valid in general for quasi-coherent sheaves, see [Har77, The-orem 3.7]. ut

On the other hand, the analytic de Rham complex of X

0→ OXan∂−→Ω

1Xan

∂−→ ·· · ∂−→ΩnXan

∂−→ 0,

is a resolution of the constant sheaf C, by Dolbeault Lemma (Theorem ??). Andfurthermore, it is acyclic by the fact that every affine variety is Stein, see [Gun90b,page 131, Theorem 2], and the following classical theorem due to Cartan.

Theorem 4.2.2 (Cartan B theorem) Let X be a Stein variety. Then

H i(X ,Ω jXan) = 0, i = 1,2, . . . , j = 0,1,2, . . . .

Proof. The Theorem is valid in general for coherent analytic sheaves, see [Gun90b,page 90]. ut

Finally, considering the C ∞ de Rham cohomology complex of X

0→ OX∞d−→Ω

1X∞

d−→ ·· · d−→ΩnX∞

d−→ 0.

We know this is an acyclic resolution of the constant sheaf C, by Poincare Lemma(Theorem ??) and the fact each sheaf Ω

qX∞ is fine.

Applying Proposition 3.4.1 and Proposition 3.6.2, we obtain the following iso-morphisms

Hq(Γ (Ω •X/C),d)'Hq(X ,Ω •X/C), (4.1)

HqdR(X

an) := Hq(Γ (Ω •Xan),∂ )'Hq(Xan,Ω •Xan)' Hq(Xan,C), (4.2)

HqdR(X

∞) = Hq(Γ (Ω •X∞),d)'Hq(X∞,Ω •X∞)' Hq(X∞,C). (4.3)

So it is clear that (4.2) is isomorphic to (4.3). On the other hand, applying Proposi-tion 3.4.1 to the acyclic complex (Ω •Xan ,∂ ) over the Zariski topology of X we obtainthe isomorphism

HqdR(X

an)'Hq(X ,Ω •Xan).

In view of the previous isomorphisms and Proposition 3.6.1, it is enough to showshow that the natural inclusion (of sheaves over the Zariski topology)

(Ω •X/C,d) → (Ω •Xan ,∂ ) (4.4)

is a quasi-isomorphism to conclude that (4.1) is isomorphic to (4.3), i.e. global alge-braic differential forms are enough to recover the de Rham cohomology groups. Thedifficulty in proving (4.4) is a quasi-isomorphism lies on working over the Zariskitopology, because we do not understand the stalks of the cohomology sheaves asso-ciated to each complex. Despite this fact, we can prove the following result:

Theorem 4.2.3 (Atiyah-Hodge Theorem, [HA55]) Let X be an affine smooth va-riety over the field C of complex numbers. Then the canonical map

4.3 Proof of Atiyah-Hodge theorem 39

Hq(Γ (Ω •X/C),d)→ HqdR(X

∞)

is an isomorphism of C-vector spaces.

4.3 Proof of Atiyah-Hodge theorem

The idea of the proof is to get rid of Zariski topology, and reduce everything to provea quasi-isomorphism between complexes of sheaves over the analytic topology. Forthis we need to complete the variety X to a well behaved projective variety X , trans-port the algebraic sheaf to it and use Serre’s GAGA principle [Ser56, §3] in order towork with sheaves over the analytic topology of X . The following proof was essen-tially taken from [Nar68], except from the lemma that was taken from [Voi03] page160, Lemma 6.6.

By Hironaka’s resolution of singularities theorem [], we can take j : X → Xwhere X is smooth projective variety, j is an open immersion and Y = X \X is anample normal crossing divisor of X (see [Kol07, Theorem 3.35]).

Transporting the sheaf of algebraic p-forms over X to X , we know that

j∗ΩpX/C = lim

k→∞Ω

pX/C⊗OX (Y )

k.

Then, if we take the analytic sheaf

ΩpXan(∗Y ) := lim

k→∞Ω

pXan ⊗OXan(Y )k,

we have (passing to the limit in Serre’s GAGA correspondence) that

Γ (Ω pXan(∗Y ))' Γ ( j∗Ω

pX ) = Γ (Ω p

X ).

Hence, it is enough to show that the cohomology of global sections associated tothe complex (Ω •Xan(∗Y ),∂ ) corresponds to the de Rham cohomology of X . UsingKodaira vanishing theorem (see [GH94, Chapter 1, §2]), we see (Ω •Xan(∗Y ),∂ ) is anacyclic complex, then

Hq(Γ (Ω •Xan(∗Y )),∂ )'Hq(X ,Ω •Xan(∗Y )).

On the other hand, considering the C ∞ de Rham cohomology complex of X trans-ported to X , we have another acyclic complex ( j∗Ω •X∞ ,d), and

HqdR(X

∞) = Hq(Γ ( j∗Ω •X∞),d)'Hq(X , j∗Ω •X∞).

This reduces the proof to show that the natural inclusion of complexes of sheavesover X (over the analytic topology)

ι : (Ω •Xan(∗Y ),∂ ) → ( j∗Ω •X∞ ,d)

40 4 Atiyah-Hodge theorem

is a quasi-isomorphism.

Since polydiscs form a basis of the analytic topology, we can focus on them. Wedivide the analysis on whether the polydisc intersects Y or not. If ∆ is a polydiscnot intersecting Y , we have that (Ω •Xan(∗Y )|∆ ,∂ ) = (Ω •

∆ an ,∂ ) and ( j∗Ω •X∞ |∆ ,d) =(Ω •

∆ ∞ ,d). Since both are resolutions of the constant sheaf C over ∆ , we have thedesired quasi-isomorphism. This means Hq(ι)x is an isomorphism for all q ≥ 0,x ∈ X , i.e. Hq(ι)|X is an isomorphism.

For x ∈Y , we can choose a local chart and a polidisc ∆ such that ∆ \Y is biholo-morphic to (z1, ...,zn) ∈ Dn|z1 · · ·zr 6= 0= (D×)r×Dn−r. These polydiscs form aneighbourhoods system for x, so we can use them to compute the stalk at x of thecohomology sheaves associated to each complex.

We know that ∆ \Y is homotopically equivalent to (D×)r, so we get the isomor-phism

Hq(Γ (∆ , j∗Ω •X∞),d)'HqdR(((D

×)r)∞)=

0 if q > r,⊕

1≤i1<...<iq≤r C[

dzi1∧...∧dziqzi1 ···ziq

]if 1≤ q≤ r,

C if q = 0.

On the other hand, it is clear that H0(Γ (∆ ,Ω •Xan(∗Y )),∂ ) = C, then H0(ι)x is anisomorphism.

For q > 0, we clearly see that Hq(ι)x is surjective (since the representatives ofHq( j∗Ω •X∞ ,d)x belong to Hq(Ω •Xan(∗Y ),∂ )x). So, it is enough to show it is injectivein order to finish the proof. This is exactly what we prove in the following lemma.ut

Lemma 4.1. Let ω be a closed meromorphic q-form on ∆ = Dn, which is holomor-phic in ∆ \Y with poles along Y = (z1, ...zn) ∈ ∆ | z1 · · ·zr = 0. Further, assumethat ω = 0 in Hq

dR((∆ \Y )∞). Then there exist a meromorphic (q− 1)-form η (de-fined on a possibly smaller polydisc) also with poles along Y , such that ω = ∂η .

Proof. We claim there exist a meromorphic (q− 1)-form ν in some (possiblysmaller) polydisc ∆ with poles along Y such that ω − ∂ν does not depend on ziand dzi for i = r+1, ...,n. In fact, assuming the claim for i > m≥ r+1, in a possi-bly smaller polydisc we have

ω−∂ν =µ

(z1 · · ·zr)l ,

for some µ holomorphic q-form not depending on zi and dzi for i = m+ 1, ...,n.Writing

µ = µ1 +dzm∧µ2,

with µ1 and µ2 two holomorphic forms not depending on z j for j = m+1, ...,n, andon dzi for i = m, ...,n. We can formally integrate the power series expansion of µ2to obtain a holomorphic (q− 1)-form δ such that µ2 = ∂δ

∂ zm(here ∂δ

∂ zmis obtained

4.3 Proof of Atiyah-Hodge theorem 41

by applying ∂

∂ zmto every component of δ ) and also does not depend on z j for j =

m+ 1, ...,n, and on dzi for i = m, ...,n. Taking ν = ν + δ

(z1···zr)l , the form ω − ∂ ν

does not depend on z j for j = m+1, ...,n, and on dzi for i = m, ...,n. Furthermore,since it is ∂ -closed, it follows that it does not depend on zm. This proves the claim.Now, we can assume

ω =µ

(z1 · · ·zr)l

with µ holomorphic q-form not depending on zi and dzi for i = r+1, ...,n.We have reduced the lemma to the following assertion:

“For any closed meromorphic q-form ω in ∆ = Dr with poles along Y =z1 · · ·zr = 0 such that ω = 0 in Hq

dR(((D×)r)∞), there exist a meromorphic (q−1)-

form η (in a possibly smaller polydisc) with poles along Y such that ω = ∂η .”

We prove this assertion by induction on r, being the case r = 1 clear. For r > 1,we can write µ = µ1 + dzr ∧ µ2 with µ1 and µ2 not depending on dzr. Taking thepower series expansion of µ2 we can write

ω = ψ1 +dzr ∧ψ2,

where

ψ2 =∞

∑j=−l

φ j(z1, ...,zr−1)z jr ,

with φ j meromorphic form in Dr−1 with poles along z1 · · ·zr−1 = 0. Using formalintegration we obtain a meromorphic (q−1)-form ϕ such that

ψ2−φ−1

zr=

∂ϕ

∂ zr.

Thenω−∂ϕ = ξ1 +ξ2∧

dzr

zr,

with ξ1 meromorphic not depending on dzr, and ξ2 meromorphic not dependingon zr and dzr. Since ω is closed, we have ∂ξ1

∂ zr= ±1

zr∂ξ2. Then zr

∂ξ1∂ zr

does not de-pend on zr, i.e. ξ1 also does not depend on zr. Consequently, ξ1 and ξ2 are closedmeromorphic forms in Dr−1.

Kunneth’s theorem (see [BT82, Chapter 1, §5]) and the exactness of ω implythat ξ1 and ξ2 are exact forms on Dr−1. By induction hypothesis we conclude theassertion. This finishes the proof of the lemma and completes the proof of Atiyah-Hodge’s theorem. ut

Remark 4.3.1 Looking carefully to the proof of Lemma 4.1, we notice that for ω

meromorphic with poles along Y , we can choose η (also with poles along Y ) suchthat its pole order is less or equal than the pole order of ω .

Chapter 5Algebraic de Rham cohomology

I do feel however that while we wrote algebraic GEOMETRY they [Weil, Zariski,Grothendieck] make it ALGEBRAIC geometry with all that it implies, (SolomonLefschetz in [Lef68]).

5.1 Introduction

The main objective in this chapter is to define the algebraic de Rham cohomol-ogy Hm

dR(X/k) of a smooth algebraic veriety X defined over a field k of charac-teristic zero. When k = C we have the underlying C∞ manifold X∞ and we aimto construct explicitly the isomorphism between the algebraic Hm

dR(X/k) and theclassical Hm

dR(X∞) de Rham cohomologies. In this way, we are able to write down

explicit formulas for cup products, Gauss-Manin connection etc, in algebraic deRham cohomology. A. Grothendieck is the main responsible for the definition ofalgebraic de Rham cohomology, however, it must be noted that he was largely in-spired by the work of Atiyah and Hodge in [HA55]. His paper [Gro66] was origi-nally written as a letter to Atiyah and Hodge and it would be fair to call this Atiyah-Hodge-Grothendieck algebraic de Rham cohomology. However, in the literature,and mainly in Algebraic Geometry we find the name Grothendieck’s algebraic deRham cohomology. Algebraic de Rham cohomology for singular schemes has beenstudied by several authors during the seventies, see for example Hartshorne’s work[Har75] and the references therein.

5.2 The definition

The following is the final format of the definition of an algebraic de Rham cohomol-ogy by A. Grothendieck [Gro66] after a long period of dealing with integrals withalgebraic integrand.

43

44 5 Algebraic de Rham cohomology

Definition 5.2.1 Let X be a smooth variety over a field k. We consider the complex(Ω •X/k,d) of regular differential forms on X . The (algebraic) de Rham cohomologyof X is defined to be the hypercohomology

HqdR(X/k) :=Hq(X ,Ω •X/k), q = 0,1,2, . . . .

We take a covering U of X/k by open affine subsets. By Serre’s vanishing Theorem(Theorem 4.2.1) this covering is acyclic with respect to all sheaves Ω i

X/k. Therefore,by Proposition 3.2.1, the hypercohomology in the above definition can be takenrelative to U . Therefore, we can write an element of Hq

dR(X/k) relative to U in theformat described in §3.3.

5.3 The isomorphism

Let X be a smooth affine variety over C. We have

H idR(X/C)∼= H i(Γ (X ,Ω •X/C),d)∼= H i

dR(X∞)

The first isomorphism follows from Serre’s vanishing Theorem (Theorem 4.2.1) andProposition 3.4.1. The second isomorphism is the statement of the Atiyah-Hodgetheorem (Theorem 4.2.3).

For an arbitrary algebraic variety X over complex numbers, the Atiyah-Hodgetheorem implies that the inclusion Ω •X/C→ Ω •X∞ is a quasi-isomorphism (over theZariski topology). In this way, using Proposition 3.4.1 and Proposition 3.6.1 we getthe isomorphism

HmdR(X/k)∼= Hm

dR(X∞).

In §3.7 we have described this isomorphism and its inverse explicitly. Since thiswill be an important tool for later applications, we are going to describe again theexplicit construction of the maps

A : HmdR(X

∞)→ HmdR(X/C), (5.1)

A−1 : HmdR(X/C)→ Hm

dR(X∞). (5.2)

Let us describe first the map (5.1). Take an element in HmdR(X

∞), represented by aclosed differential m-form ω . In what follows, differential forms withˇwill representC∞ differential forms, and those without ˇ will be algebraic differential forms. Werestrict ω to each open set Ui, say ωm

i = ω|Ui . Using Atiyah-Hodge theorem we findalgebraic differential m-forms ωm

i and C∞ (m−1)-forms ηm−1i in Ui such that

ωmi = ω

mi −dη

m−1i , and dω

m = 0.

Applying δ to ωm we obtain

5.3 The isomorphism 45

ωmj −ω

mi = d(ηm−1

j − ηm−1i ) in Ui∩U j.

Again by Atiyah-Hodge theorem, the right hand side of the above equality repre-sents the zero element in the (m−1)-th de Rham cohomology of Ui∩U j. Therefore,there are algebraic differential (m−1)-forms ω

m−1i j in Ui∩U j, such that

ωmj −ω

mi −dω

m−1i j = 0,

and sod(ηm−1

j − ηm−1i −ω

m−1i j ) = 0.

Using Atiyah-Hodge theorem in the intersections Ui∩U j, we can add a closed alge-braic differential form to ω

m−1i j and assume that

ηm−1j − η

m−1i −ω

m−1i j = dη

m−2i j ,

where ηm−2i j are C∞ (m− 2)-forms in Ui ∩U j. Applying δ to ωm−1 we get in Ui ∩

U j ∩Uk the relation

ωm−1jk −ω

m−1ik +ω

m−1i j =−d(ηm−1

jk −ηm−1ik +η

m−1i j ).

Again we have an algebraic (m−1)-form which is exact using C∞ differential formsand so it must be exact using algebraic differential forms. We repeat the process toproduce ωm−2. This process at the k-th step gives us ωm−k such that

δωm−k +(−1)k+1d(δ η

m−k−1) = 0.

Using Atiyah-Hodge theorem we can produce ωm−k−1 such that

δωm−k +(−1)k+1dω

m−k−1 = 0,δ η

m−k−1−ωm−k−1− (−1)k+2dη

m−k−2 = 0.

At the end of the process we get the element Aω = ω = ω0 +ω1 + · · ·+ωm ∈Hm

dR(X/C). Note that the last equality for k = m−1 is just δ η0−ω0 = 0.Let us now describe the map (5.2). Take an element in Hm

dR(X/C), represented byω = ∑

mj=0 ω j such that Dω = 0. In particular, δω0 = 0. Since the Cech cohomology

of the sheaf of C∞ differential forms is zero (except in dimension zero), we have

ω0 = δ η

0,

for some C∞ differential forms η0. Replacing ω by ω −Dη0, we can assume thatω0 = 0. This process continues with δω1 = 0. In the final step we get a closed C∞

m-form in X∞. The complete description can be done in terms of a partition of unity.

Proposition 5.3.1 Let X be a smooth algebraic variety over C. Let U = Uii∈Ibe a locally finite affine cover of X. Given ω = ω0 + · · ·+ωm ∈ Hm

dR(X/C), and apartition of unity aii∈I subordinated to U , we can compute


(A−1ω)i =

m

∑l=0

∑k1,...,kl∈I

dak1 ∧·· ·∧dakl ∧ωm−lik1···kl

∈ H0(U ,Ω mX∞).

Proof. We claim that for every j = 0, ...,m we can represent ω as ω j +ω j+1+ · · ·+ωm, where

ωj

i0···im− j:=

j

∑l=0

∑k1,...,kl∈I

dak1 ∧·· ·∧dakl ∧ωj−l

i0···im− jk1···kl∈Cm− j(U ,Ω j

X∞).

In fact, this is clear for j = 0. Assuming the claim for j≥ 0, define η j ∈Cm− j−1(U ,Ω jX∞)

byη

ji0···im− j−1

:= ∑k∈I

akωj

i0···im− j−1k.

Then δη j = (−1)m− jω j, and so Dη j = (−1)m− j(ω j − dη j). Representing ω byω +(−1)m− j−1Dη j we obtain the claim for j+1. ut

5.4 Hodge filtration

For the complex of differential forms Ω •X/k, we have the truncated complex of dif-ferential forms

Ω•≥iX/k : · · · → 0→ 0→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)).

For the special case of the complex of differential forms, Theorem 3.9.1 becomes:

Theorem 5.4.1 Let k be an algebraically closed field of characteristic zero and Xbe a smooth projective variety over k. We have

Fq/Fq+1 ∼= Hm−q(X ,Ω qX/k)

By Lefschetz principle we can assume that X is defined over complex numbers.We denote by Xan the underlying complex manifold of X . Let Ω i

Xan be the sheaf ofclosed holomorphic differential form on Xan.

5.5 Cup product 47

Theorem 5.4.2 (Dolbeault) Let X be a smooth projective variety over C. The maps

H j(Xan,Ω iXan)→ H j(Xan,Ω i+1

Xan ), i, j ∈ N0,

induced by the differential map d : Ω iXan → Ω

i+1Xan are all zero.

This theorem is taken from Griffiths’ article [Gri69a], II. He uses this property inorder to prove that the Hodge filtration varies holomorphically. He associate thisproposition to Dolbeault, and in the appendix of [Gri69b], he gives a proof usingLaplacian operators. The situation is quite similar to the case of Hard Lefschetztheorem, where the only available proof is done using harmonic forms. Theorem5.4.2 is equivalent to the fact the Frolicher spectral sequence of Xan degeneratesat E1, see for instance Theorem 8.28 in Voisin’s book [Voi02]. In that book thementioned fact is basically derived from the Hodge decomposition.

Proof (of Theorem 5.4.1). We need to check that the hypothesis of Proposition 3.9.1are satisfied in our case. By GAGA principle the natural morphisms

H j(X ,Ω iX )→ H j(Xan,Ω i

Xan)

are isomorphisms of C-vector spaces and under this identifications the algebraic drcoincides with the holomorphic dr. Therefore, it is enough to check that the holo-morphic dr satisfies the hypothesis of Proposition 3.9.1. In the holomorphic context,by Theorem 5.4.2 we have H j(Xan,Ω i

Xan) ∼= H j(Xan,Ω iXan). For this we write the

long exact sequence of the short exact sequence:

0→ ΩiXan →Ω

iXan → Ω

i+1Xan → 0.

In the definition if d1 we can choose a representative of ω i ∈ H j(Xan,Ω iXan) such

that dω i = 0, and so, by definition all dr’s are zero. ut

According to [Voi02] page 207, the algebraic proof of degeneracy at E1 of the com-plex of algebraic differential forms (and hence an algebraic proof of Theorem 5.4.1)is done by Deligne and Illusie in 1987 and Illusie in 1996.

5.5 Cup product

In the usual de Rham cohomology we have the cup/wedge product

HmdR(X

∞)×HndR(X

∞)→ Hn+mdR (X∞), ω, α 7→ ω ∧ α

and it is natural to ask for the corresponding bilinear map in algebraic de Rhamcohomology. For partial result in this direction see [CG80].

Theorem 5.5.1 The cup product of ω ∈ HmdR(X/k) and α ∈ Hn

dR(X/k) is given byγ = ω ∪α , where


γn+mi0 = ω

mi0 ∧α

ni0

γn+m−1i0i1 = (−1)m

ωmi0 ∧α

n−1i0i1 +ω

m−1i0i1 ∧α

ni1

......

γn+m− ji0i1···i j

=j

∑r=0

(−1)m( j−r)+r( j−1)ω

m−ri0···ir ∧α

n− j+rir ···i j

γ0i0···in+m = ω

0i0···im ·α

0im···in+m

Proof. We have to prove that under the canonical isomorphism between the classicaland algebraic de Rham cohomology, the wedge product is transformed in the prod-uct given in the theorem. Let us take ω ∈Hm

dR(X∞) and α ∈Hn

dR(X∞) and construct

the algebraic counterpart of ω, α, ω ∪ α . For this we will use the explicit construc-tion of (5.1). Followng this, it is possible to compute the first two lines in Theorem5.5.1. Once the general formula is guessed the proof is as bellow: First we checkthat Dγ = 0.

(δγm+n− j)i0···i j+1 =

j+1

∑k=0

(−1)kγ

m+n− ji0···ik···i j+1

=j+1

∑k=0

k−1

∑r=0

(−1)k+m( j−r)+r( j−1)ω


n− j+rir ···ik···i j+1

+j+1

∑k=0

j+1

∑r=k+1

(−1)k+m( j−r+1)+(r−1)( j−1)ω

m−r+1i0···ik···ir

∧αn− j+rir ···i j+1

=j

∑r=0

j+1

∑k=r+1

(−1)k+m( j−r)+r( j−1)ω


n− j+rir ···ik···i j+1

+j+1

∑r=1

r−1

∑k=0

(−1)k+m( j−r+1)+(r−1)( j−1)ω

m−r+1i0···ik···ir

∧αn− j+rir ···i j+1

=j

∑r=0

(−1)r+m( j−r)+r( j−1)ω

m−ri0···ir ∧

[(δα

n− j+r)ir ···i j+1 −αn− j+rir+1···i j+1

]+

j+1

∑r=1

(−1)m( j−r+1)+(r−1)( j−1)[(δω

m−r+1)i0···ir − (−1)rω

m−r+1i0···ir−1

]∧α

n− j+rir ···i j+1

=j

∑r=0

(−1) j+m( j−r)+r( j−1)ω

m−ri0···ir ∧dα

n− j+r−1ir ···i j+1

+j+1

∑r=1

(−1)r−1+m( j−r+1)+(r−1)( j−1)(dωm−r)i0···ir ∧α

n− j+rir ···i j+1

5.5 Cup product 49

= (−1) jd

(j+1

∑r=0

(−1)m( j+1−r)+r jω


n− j−1+rir ···i j+1

)= (−1) jdγ

m+n− j−1i0···i j+1

Second, it is easy to verify that the cup product formula announced in Theorem 5.5.1satisfies

ω ∪α = (−1)mnα ∪ω, (5.3)

Dη ∪α = D(η ∪α) for Dα = 0. (5.4)

Therefore, the cup is independent of the choice of representatives.We have to prove that ω ∧ α and γ induce the same element in Hn+m(X∞,Ω •X∞),

for k= C. This follows from the following equalities in Hn+m(X∞,Ω •X∞)

[γ] = γ = ω ∪α = [ω]∪ [α] = [ω ∧ α].

where [·] means the induced class in hypercohomology. ut

Let us assume that m and n are even numbers and ω and α have only the middlepieces ω

m2

i0i1···i m2

and αn2

i0i1···i n2

, respectively. In this particular case, ω ∪α has also

only the middle piece given by

(ω ∪α)n+m

2i0i1···i n+m

2

= (−1)nm4 ω

m2

i0i1···i m2∧α

n2

i m2

i m2 +1···i n+m

2

. (5.5)

Remark 5.5.1 Let U be a convex locally finite open cover of Xan. Since each openconvex set has trivial de Rham cohomology (since it is contractible), an element ofHn

dR(X/C) can be represented by ω0 ∈ Hm(U ,C) and α0 ∈ Hn(U ,C). Let aii∈Ibe a partition of unity subordinated to U , then

ω = ∑k1,...,km∈I

ω0ik1···km

dak1 ∧·· ·∧dakm ,

andα = ∑

l1,...,ln∈Iα

0jl1···lndal1 ∧·· ·∧daln ,

for any i, j ∈ I. On the other hand γ = ω∪α will correspond to γ ∈Hm+ndR (X∞) given

by

γ = ∑k1,...,km,l1,...,ln∈I

dak1 ∧·· ·∧dakm ∧dal1 ∧·· ·∧dalnγ0ik1···kml1···ln = ω ∧ α.

For the description of cup product in Cech cohomology Hm(X ,Z) see [Bry08]. Forsimple applications of cup product in the case of elliptic curves see [Mov12].


5.6 Polarization

Let PN be a projective space of dimension N, with projective coordinates [x0 : x1 :· · · : xN ] and fi j := xi

x j. The rational function fi j has a zero (resp. pole) of order 1 at

xi = 0 (resp. x j = 0). For the algebraic de Rham cohomology of PN we consider thestandard covering and represent its elements in this covering.

Definition 5.6.1 The following

θ = θi j, θi j :=d fi j

fi j=

d( x jxi)

x jxi

=dx j

x j− dxi

xi,

induces an element in H2dR(PN/k) and it is called the polarization. For any subvariety

X of PN over k, its restriction to H2dR(X/k) is also called the polarization.

Note that δθ = 0 and dθ = 0. Therefore, Dθ = 0 and hence θ induces an elementin H2

dR(X/k). Note also that if we write θ = θ 0 +θ 1 +θ 2, then θ 0 and θ 2 are zero.

Proposition 5.6.1 The polarization in the usual de Rham cohomology H2dR(X

∞) isgiven by the (1,1)-form

θ := ∂ ∂ log(N

∑i=0|xi|2). (5.6)

Moreover, for any linear P1 ⊂ PN we have∫P1

θ = 2π√−1.

Proof. The proof of the first part is as follows. We have to follow the recipe in §5.3and construct A−1(θ). We define:

pi : PN\xi = 0→ R+, pi(x) :=N

∑j=0

∣∣∣∣x j

xi

∣∣∣∣2and

η1i :=

∂ pi

pi= ∂ log pi.

We have

p j =

∣∣∣∣x j

xi

∣∣∣∣−2

· pi and so η1j −η

1i =−

d fi j

fi j=−θi j.

Let η1 = η1i , i∈ I. Recall the explicit description of H2 in (3.9),(3.10) and (3.11).

We substitute θ by θ := θ +Dη . This satisfies θ 0 = 0, θ 1 = 0 and

θ2i = dη

1i = ∂ ∂ log pi.

5.6 Polarization 51

The (1,1)-forms θ 2i ’s in the intersection of Ui’s coincide and give us the element

(5.6) in the usual de Rham cohomology.The second statement follows from the fact that 1

2πi θ is the first Chern class ofthe tautological line bundle of PN , see for instance [Mov17a]. An elementary proofis as follows. Since all lines in PN are homologous to each other, we assume thatP1 is given by xN = xN−1 = · · · = x2 = 0, and hence, we only need to prove thecase N = 1. In this case, let us take the chart x1 = 1 and use z := x0. With this newnotation, let x and y be the real and complex part of z, that is, z := x+ iy. We have

∂ ∂ log(1+ |z|2) = 2√−1

dx∧dy(1+ x2 + y2)2

The desired result follows from∫R2

dx∧dy(1+ x2 + y2)2 = π (5.7)

ut

Using Theorem 5.5.1 we know that θ m ∈ H2mdR (PN

k ) has only the middle piece. Thismiddle piece is given by

(θ m)i0i1···im =(−1)(m+1

2 )(

dxi0xi0−

dxi1xi1

)∧(

dxi1xi1−

dxi2xi2

)∧·· ·∧

(dxim−1

xim−1

− dximxim

).

(5.8)In particular, in the top cohomology H2n

dR(PN/k) we have the element θ N given by

(θ N)012···N = (−1)(N+1

2 )(

dx0

x0− dx1

x1

)∧(

dx1

x1− dx2

x2

)∧·· ·∧

(dxN−1

xN−1− dxN

xN

)= (−1)(

N2)∑

Ni=0(−1)ixidxi

x0x1x2 · · ·xN

The summary of our discussion is in the following theorem:

Theorem 5.6.1 Let k be a field of characteristic zero and not necessarily alge-braically closed. The algebraic de Rham cohomology ring H∗dR(PN/k) is generatedby θ .

Proof. We can take k small enough to have the embedding k → C, and since thestatement of the theorem remain valid under field extensions, we can assume thatk = C. By Proposition 5.6.1 1

2πi θ maps to u ∈ H2((PN)∞,Z) under the canonicalisomorphism Hm

dR(PN/C)∼= Hm

dR((PN)∞). Here, u is the cohomology class of a hy-

perplane PN−1 ⊂ PN . Now the the theorem follows from the same theorem in thetopological side. ut

It would be more convenient to give a proof of Theorem 5.6.1 without referring tothe same statement in topology.


5.7 Top cohomology

Let X ⊂ PN be a smooth projective variety of dimension n over k.

Proposition 5.7.1 The top cohomology H2NdR (X) is of dimension 1 and it is gener-

ated by the restriction of θ n ∈ H2ndR(P

N). Moreover, for an embedding k → C wehave ∫

Xanθ

n = deg(X) · (2πi)n.

Proof. The proof follows from a similar statement in topology and the fact thatu = 1

2πi θ ∈ H2(Xan,Z) is the topological polarization. ut

It turns out that we have a canonical isomorphism

Tr : H2ndR(X)∼= k, Tr(ω) := deg(X) · ω

θ n

over the field k. In the complex context k= C, it is given by

ω → 1(2πi)n

∫Xan

ω.

Definition 5.7.1 The element α := θ n

deg(X) is called the volume top form of X . It ischaracterized by the fact that Tr(α) = 1.

Let PN−n−1⊂PN be a projective space such that PN−n−1∩X = /0. We assume thatit is given by linear equations g0 = g1 = · · ·= gn = 0 and consider the correspondingcovering U = Ui, i = 0,1, . . . ,n, Ui := x ∈ X |gi(x) 6= 0. In this covering θ n ∈H2n

dR(X/k) is given by

(θ n)012···n = (−1)(n+1

2 )(

dg0

g0− dg1

g1

)∧(

dg1

g1− dg2

g2

)∧·· ·∧

(dgn−1

gn−1− dgn

gn

).

(5.9)

Remark 5.7.1 The appearance of the sign (−1)(n+1

2 ) in our formulas is natural. In[DMOS82] the pairing between the N dimensional homology and de Rham coho-mology is modified with this sign and it has been remarked: “Our signs differ fromthe usual signs because the standard sign conventions∫

σ

dω =∫

dσ

ω,∫

X×Ypr∗1ω ∧ pr∗2η =

∫X

ω ·∫

Yη , etc.

violate the sign conventions for complexes.”

5.8 Standard top form in PN 53

5.8 Standard top form in PN

All our methods to compute periods reduce at some point to compute the periodof an element in the top cohomology of the projective space. Since H2N

dR (PN) =HN,N(PN)' HN(PN ,Ω N

PN )' C, we just need to know the period of one generator.This will give us the top form of PN in the sense of Definition 5.7.1. We will fix theelement

Ω

x0 · · ·xN=

∑Ni=0(−1)ixidxi

x0 · · ·xN=

(dx1

x1− dx0

x0

)∧·· ·∧

(dxN

xN− dx0

x0

)∈HN(U ,Ω N

PN ),

and compute its period. Here U = UiNi=0 is the standard open covering of PN , i.e.

Ui = xi 6= 0.

Proposition 5.8.1 We have∫PN

Ω

x0 · · ·xN= (−1)(

N2)(2π

√−1)N .

Since we have natural isomorphisms HN(PN ,Ω NPN )'H2N(PN ,Ω •

(PN)∞)'H2NdR (PN).

The element Ω

x0···xN∈ HN(PN ,Ω N

PN ) corresponds to a top form ω ∈ H2NdR (PN). By

abuse of notation we will denote∫PN

Ω

x0 · · ·xN:=∫PN

ω.

We will always use this identification when we talk about periods. From Proposition5.8.1 we get:

Proposition 5.8.2 The top form of PN in the sense of Definition 5.7.1 and associatedto the standard covering U of PN is given by

(−1)(N2)∑

Ni=0(−1)ixidxi

x0 · · ·xN.

Note that this top form is equal to θ N , where θ ∈ H2dR(PN) is the polarization ele-

ment defined in §5.6. Therefore, Proposition 5.8.1 follows from Proposition 5.6.1.We would like to give a different proof of this using a partition of unity subordi-nated to the covering U . The main reason for this is that the isomorphism betweenalgebraic and C∞ de Rham cohomologies described in §5.3 can be explicitely writ-ten down using a partition of unity. However, this resulting formula might be huge.Therefore, the case of PN serves as a nice hard exercise in case one needs a moreconcrete presentation of the material in §5.3.

Proof (of Proposition 5.8.1). We have to determine the image of Ω

x0···xNvia the iso-

morphismsHN(PN ,Ω N

PN )' H2NdR (PN)'H2N(PN ,Ω •(PN)∞).


Let aiNi=0 be a partition of unity subordinated to UiN

i=0. In the affine chartCN ⊂ PN , consider the coordinate system z with zi := xi/x0 for i = 1,2, . . . ,N. With-out loss of generality, we can take the partition of unity such that

ai =

0 if |zi| ≤ 1,1 if |zi| ≥ 2, |z j| ≤ 1∀ j ∈ 1, · · · ,N\i,

anda1 + · · ·+aN = 1 if ∃ j ∈ 1, · · · ,N : |z j| ≥ 2.

Thus

Supp(da1∧·· ·∧daN)⊆z∈CN : 1≤ |zi| ≤ 2,∀i= 1, . . . ,N=(D(0;2)\D(0;1))N .

Using Proposition 5.3.1, we know that the image of Ω

x0···xNin H2N

dR (PN) is repre-sented by the global closed 2N-form ω2N which in the chart Ui is given by

(ω2N)i = N! · (−1)ida0∧·· · dai · · ·∧daN ∧Ω

x0 · · ·xN∈Ω

2N(PN)∞(Ui).

Since Supp(ω2N) ⊂U0,1,...,N . In order to integrate ω2N over PN , is enough to do iton any affine chart:∫

PN

Ω

x0 · · ·xN=∫

U0

(ω2N)0

= N! ·∫CN

da1∧·· ·∧daN ∧dz1

z1∧·· ·∧ dzN

zN

= N! · (−1)(N2)∫CN

d(

a1 ·dz1

z1

)∧·· ·∧d

(aN ·

dzN

zN

)= (−1)(

N2)(2πi)N .

where in the last equality we have used Stokes formula and the Cauchy residueformula. ut

It is easy to see that an element of ZN(U ,Ω NPN ) is of the form

PΩ

xα00 · · ·x

αNN

with α0, ..,αN ∈ N such that α0 + · · ·+αN = deg(P)+N +1. Then it is a C-linearcombination of terms of the form

xβ00 · · ·x

βNN Ω

with β0, . . . ,βN ∈ Z such that β0 + · · ·+ βN = −N− 1. The following propositiontells us how to compute the period of any such form.

5.9 Chern class 55

Proposition 5.8.3 The form

xβ00 · · ·x

βNN Ω ∈ HN(U ,Ω N

PN )

represents an exact top form (when is identified with an element of HN,N(PN)) if andonly if (β0, . . . ,βN) 6= (−1, · · · ,−1). In particular,

∫PN

xβ00 · · ·x

βNN Ω =

0 if (β0, . . . ,βN) 6= (−1, . . . ,−1),(−1)(

N2)(2π

√−1)N if (β0, . . . ,βN) = (−1, . . . ,−1).

Proof. The only thing left to prove is that for (β0, . . . ,βN) 6= (−1, . . . ,−1) the formis D-exact. Using the isomorphism HN(PN ,Ω N

PN ) ' H2N(PN ,Ω •PN ) it is enough toshow that the element ω ∈H2N(PN ,Ω •PN ) given by

ωN = xβ0

0 · · ·xβNN Ω ,

and ω i = 0 for i 6= N, is zero in hypercohomology. Note that if (β0, . . . ,βN) 6=(−1, . . . ,−1), then there exist some βi ≥ 0. Therefore,

xβ00 · · ·x

βNN Ω ∈Ω

NPN (∩ j 6=iU j).

Define η ∈⊕2N−1

j=0 C2N−1− j(U ,Ω jPN ) by

ηNJ := (−1)ixβ0

0 · · ·xβNN Ω ,

for J = (0, . . . , i− 1, i+ 1, . . . ,N), ηNJ′ = 0 for J′ 6= J, and η j = 0 for j 6= N. This

form clearly satisfy Dη = ω as desired. ut

5.9 Chern class

Consider the short exact sequence

0→ 2πiZ→ OXan → O∗Xan → 0 (5.10)

over a complex manifold Xan. The map OXan → O∗Xan is given by the exponentialmap

f 7→ e f .

We write the corresponding long exact sequence

· · · → H1(Xan,O∗Xan)c→ H2(Xan,2πiZ)→ ··· (5.11)

and call c the Chern class map. For a line bundle L ∈ H1(Xan,O∗Xan) on Xan, c(L) iscalled the Chern class of L.


Now, let us consider X over the field k.

Proposition 5.9.1 The algebraic Chern class map c : H1(X ,O∗X/k)→ H2dR(X/k) is

given by

c(L) :=

dhi j

hi j

∈ H2

dR(X/k), L := hi j ∈ H1(X ,O∗X/k).

Proof. For k = C. Take an open covering U = Uii∈I such that there existgi j = loghi j for every i, j ∈ I. Then c(L) = δg ∈ H2(U ,2πiZ). Representingc(L) ∈H2(X ,Ω •Xan) = H2

dR(X/k) we see that D(g) = c(L)−dg. Therefore

c(L)i j =dhi j

hi j,

as claimed. ut

Definition 5.9.1 We define

Pic(X/k) := H1(Xan,O∗X/k)

and call it the Picard group of X/k. We also define

NS(X/k) :=H1(X ,O∗X/k)

Im(H1(X ,OX/k)→ H1(X ,O∗X/k))∼= Im(H1(X ,O∗X/k)

c→ H2dR(X/k))

and call it the Neron-severi group of X/k.

5.10 The cohomological class of an algebraic cycle

Let Z ⊂ X be an algebraic cycle of codimension p in a smooth projective variety X ,both of them are defined over a field k of characteritic 0. We may assume that k issmall enough so that we have an embedding k → C. Since any projective variety Xover k uses a finite number of coefficients in k, the assumption on the embedding ofk in C is not problematic.

In this section we construct the cohomology class [Z]pd ∈ H pdR(X). Note that

the notation [Z] is reserved for the homology class of Z in H2n−2p(Xan,Z) and ournotation [Z]pd is justified by Theorem 5.10.1. According to Deligne in [DMOS82]the construction of cl(Z) = [Z]pd is as follows: “In each cohomology theory thereis a canonical way of attaching a class cl(Z) in H2p(X)(p) to an algebraic cycleZ on X of pure codimension p. Since our cohomology groups are without torsion,we can do this using Chern classes (Grothendieck 1958). Starting with a functorialisomorphism c1 : Pic(X)→ H2(X)(1), one uses the splitting principle to define theChern polynomial

5.10 The cohomological class of an algebraic cycle 57

ct(E) = ∑cp(E)t p, cp(E) ∈ H2p(X)(p),

of a vector bundle E on X . The map E 7→ ct(E) is additive, and therefore factorsthrough the Grothendieck group of the category of vector bundles on X. But, asX is smooth, this group is the same as the Grothendieck group of the category ofcoherent OX -modules, and we can therefore define

cl(Z) :=1

(p−1)!cp(OZ).

” We can also construct [Z]pd using the resolution of singularities in characteristic0. This is as follows. We know that there is a smooth projective variety Z and amorphism Z→ X , all defined over k, whose image is Z. Now we have the inducedmap in de Rham cohomologies composed with the trace map

H2(n−p)dR (X/k)→ H2(n−p)

dR (Z/k) Tr→ k

Since the pairing H2(n−p)dR (X)×H2p

dR(X) → k is non-degenerate we get the class[Z]pd. In the algebric context, it is not clear why [Z]pd is independent of the res-olution map. For this we have to take k small enough in order to embedd it inside Cand use the equalities (5.12) and (5.13).

Theorem 5.10.1 (Poincare duality for algebraic cycles) We have∫[Z]

ω =∫

Xanω ∪ [Z]pd, ∀ω ∈ H2n−2m

dR (Xan). (5.12)

Proof. In the left hand side of (5.12) we can replace the integration over [Z] withthe integration over the smooth part of Z:∫

Zsmooth

ω =∫[Z]

ω, ∀ω ∈ H2n−2mdR (Xan). (5.13)

ut

Corollary 5.10.1 We have

1(2πi)dim(Z)

∫[Z]

ω = Tr(ω ∪ [Z]pd), ∀ω ∈ H2n−2mdR (X/k)

Corollary 5.10.1 tell us that the integration over algebraic cycles is a purely algebraicoperation for which we do not need to embed the base field inside complex numbers.

Proposition 5.10.1 (Deligne, [DMOS82, Proposition 1.5]) Let X be a projectivesmooth variety over an algebraically closed field k⊂ C and let Z be an irreduciblesubvariety in X of dimension m

2 . For any element of the algebraic de Rham coho-mology Hm

dR(X/k), we have


1(2πi)dim(Z)

∫[Z]

ω ∈ k (5.14)

where [Z]⊂ Hn(Xan,Z) is the homololgy class of Z.

Proof. This is a direct consequence of Corollary 5.10.1. ut

5.11 Pull-back in algebraic de Rham cohomology

Given a morphism Y → X of smooth complex algebraic varieties, we have an in-duced homomorphism in algebraic de Rham cohomology

HkdR(X/C)→ Hk

dR(Y/C).

In this section we describe these pull-back homomorphisms.

Proposition 5.11.1 Let X and Y be smooth complex algebraic varieties, and Uan affine open covering of X. Consider an affine morphism ϕ : Y → X (i.e. suchthat ϕ−1(U) is affine, for each open affine U of X), and ω ∈ Hk

dR(X/C). denotingϕ−1(U ) := ϕ−1(U)U∈U , then

ϕ∗ω ∈ Hk

dR(Y/C)

is given by

ϕ∗ω =

k

∑i=0

ϕ∗ω

i ∈k⊕

i=0

Ck−i(ϕ−1(U ),Ω iY ),

with

ω =k

∑i=0

ωi ∈

k⊕i=0

Ck−i(U ,Ω iX ),

where(ϕ∗ω i) j0··· jk−i := ϕ

∗(ω ij0··· jk−i

) ∈ΩiY (ϕ

−1(U j0··· jk−i)),

and

ϕ∗

(∑

IaIdxi1 ∧·· ·∧dxik

):= ∑

Iϕ∗aId(ϕ∗xi1)∧·· ·∧d(ϕ∗xik)

(in particular, ϕ∗ commutes with d and δ , then it also commutes with D).

Proof. It is easy to see that D(ϕ∗ω) = ϕ∗(Dω) = 0. Now, in order to show thatϕ∗ω corresponds to the pull-back of the form ω , we have to show there exist arepresentative of the hypercohomology class of ω of the form

µ = µ0 + · · ·+µ

k ∈k⊕

i=0

Ck−i(U ,Ω iX∞),

5.11 Pull-back in algebraic de Rham cohomology 59

with µ i = 0, ∀i = 0, ...,k−1. And a representative of the hypercohomology class ofϕ∗ω of the form

µ = µ0 + · · ·+ µ

k ∈k⊕

i=0

Ck−i(ϕ−1(U ),Ω iY ∞),

with µ i = 0, ∀i = 0, ...,k− 1. Such that µk ∈ Ω kY ∞(Y ) is the pull-back of µk ∈

Ω kX∞(X) as C∞ differential k-forms. We will in fact show more, we will show induc-

tively that for every l = 0, ...,k there exist a representative of the hypercohomologyclass of ω of the form

µl = µ0l + · · ·+µ

kl ∈

k⊕i=0

Ck−i(U ,Ω iX∞),

with µ il = 0, ∀i = 0, ..., l−1. And a representative of the hypercohomology class of

ϕ∗ω of the form

µl = µ0l + · · ·+ µ

kl ∈

k⊕i=0

Ck−i(ϕ−1(U ),Ω iY ∞),

with µ il = 0, ∀i = 0, ..., l−1. Such that, for every j = l, ...,k, the form (µ j

l )i0···ik− j ∈Ω

jY ∞(ϕ−1(Ui0···ik− j)) is the pull-back of the form (µ j

l )i0···ik− j ∈ Ωj

X∞(Ui0···ik− j). Infact, the claim follows for l = 0 by the definition of ϕ∗ω . Assuming the claim for lwe will show it for l +1. Consider ahN

h=0 a partition of unity subordinated to thecovering U . Define

η = η0 + · · ·+η

k−1 ∈k−1⊕i=0

Ck−1−i(U ,Ω iX∞),

such that

ηi =

0 if i 6= l,λ if i = l,

where

λi0···ik−1−l :=N

∑h=0

ah(µll )i0···ik−1−lh ∈Ω

lX∞(Ui0···ik−1−l ).

And define

η = η0 + · · ·+ η

k−1 ∈k−1⊕i=0

Ck−1−i(ϕ−1(U ),Ω iY ∞),

such that

ηi =

0 if i 6= l,λ if i = l,

where


λi0···ik−1−l :=N

∑h=0

ϕ∗ah(µ

ll )i0···ik−1−lh ∈Ω

lY ∞(ϕ−1(Ui0···ik−1−l )).

Then, using that δ (µ ll ) = 0 we see that δ (λ ) = (−1)k−l µ l

l . Also noticing thatδ (µ l

l )= 0 (and using that ϕ∗ahNh=0 is a partition of unity subordinated to ϕ−1(U ))

we get δ (λ ) = (−1)k−l µ ll . Thus, defining µl+1 := µl +(−1)k+1Dη , and µl+1 :=

µl + (−1)k+1Dη , the claim follows for l + 1 since dλ = ϕ∗(dλ ) (because λ =ϕ∗(λ )). ut

A first application of Proposition 5.11.1 is the description of pull-backs in Cechcohomology for each piece of the Hodge structure.

Corollary 5.11.1 Let X and Y be smooth complex algebraic varieties, and U anaffine open covering of X. Consider an affine morphism ϕ : Y → X, and ω ∈Hq(U ,Ω p

X ), then the pull-back

ϕ∗ω ∈ Hq(ϕ−1(U ),Ω p

Y ),

is given byϕ∗ω j0··· jq = ϕ

∗(ω j0··· jq) ∈ΩpY (ϕ

−1(U j0··· jq)).

Proof. The pull-back morphism in algebraic de Rham cohomology

ϕ∗ : H p+q

dR (X/C)→ H p+qdR (Y/C)

is compatible with the Hodge filtration (i.e. ϕ∗(Fk(X))⊆ Fk(Y )), thus the pull-backin Cech cohomology is induced by

ϕ∗ : Hq(X ,Ω p

X )' Fq(X)/Fq+1(X)→ Fq(Y )/Fq+1(Y )' Hq(Y,Ω pY ).

ut

Chapter 6Logarithmic differential forms

6.1 Introduction

Let X be an affine variety. It follows from Theorem 4.2.1 and Theorem 5.4.1 that theHodge filtration of X defined in (5.4) is trivial, that is, all the pieces of the Hodgefiltration are equal. Therefore, the truncated complexes Ω

•≥iX give us the correct

Hodge filtration for projective varieties and beyond this we may be in trouble todefine the ‘correct’ Hodge filtrations. One has to clarify what the adjective correctmeans. Deligne in [Del71] uses logarithmic differential forms and defines the mixedHodge structure of affine varieties, and in particular its basic ingredient, namely theHodge filtration. In this section we give an exposition of this topic. Even if one isinterested in projective varieties, computations are usually done in affine varieties,and this topic is indispensable for further study of projective varieties. The readermay also consult Voisin’s books [Voi02, Section 8.2.3] and [Voi03, Section 6.1] .

6.2 Logarithmic differential forms

For simplicity, we will restrict ourselves to the following context: Let X be a smoothprojective variety, and Y ⊆ X a smooth hyperplane section. We will consider affinevarieties of the form

U := X \Y.

Definition 6.2.1 Let X be a smooth projective variety and Y ⊆ X be a smooth hy-perplane section. We define the sheaf of rational p-forms with logarithmic polesalong Y as

ΩpX/C(logY ) := Ker(Ω p

X/C(Y )d−→Ω

pX/C(2Y )/Ω

pX/C(Y )).

Analogously, we define the sheaf of meromorphic p-forms with logarithmic polesalong Y as

61

62 6 Logarithmic differential forms

ΩpXan(logY ) := Ker(Ω p

Xan(Y )d−→Ω

pXan(2Y )/Ω

pXan(Y )).

By Serre’s GAGA principle we know ΩpXan(logY ) is the analytification of Ω

pX/C(logY ).

The following proposition describes how logarithmic forms look like in localsystems of coordinates.

Proposition 6.2.1 Let X be a smooth projective variety of dimension n and Y ⊆ Xa smooth hyperplane section. Let z1, ...,zn be local coordinates on an open set Vof X, such that V ∩Y = z1 = 0. Then Ω

pXan(logY )|V is a free OV an -module, for

which dzi1 ∧ ·· · ∧ dzip and dz1z1∧ dz j1 ∧ ·· · ∧ dz jp−1 , ik, jl ∈ 2, ...,n, form a basis.

In particular ΩpXan(logY ) is locally free.

Proof. Let α ∈ Γ (V,Ω pXan(logY )). Since it has pole of order 1 there exist β ∈

Γ (V,Ω pXan) such that α = β

z1. Furthermore, since the same happens to dα , we con-

clude dz1 ∧ β = 0, i.e. β = dz1 ∧ γ , where γ is a holomorphic (p− 1)-form, justdepending on dz2, ....,dzn. ut

6.3 Deligne’s theorem

The following Theorem due to Deligne allows us to define another filtration onalgebraic de Rham cohomology of affine varieties.

Theorem 6.3.1 (Deligne [Del70]) In the context of the previous definition, let i :U = X \Y → X be the inclusion. Then, the natural map

Ω•X/C(logY ) → i∗Ω •U/C =: Ω

•X/C(∗Y ),

induces the isomorphism

Hk(X ,Ω •X/C(logY ))' HkdR(U/C).

We will provide two different proofs of this fact. The first one being more con-ceptual (following the argument of Atiyah-Hodge’s theorem), while the second oneis more computational, and relies on a lemma due to Carlson and Griffiths.

First proof of Theorem 6.3.1 Consider the composition

ι : Ω•Xan(logY ) →Ω

•Xan(∗Y ) → i∗Ω •U∞ ,

of complexes of sheaves over the analytic topology. If we prove that ι is a quasi-isomorphism, we obtain the result by Serre’s GAGA correspondence (since Ω

pX/C(logY )

is coherent).In order to show the quasi-isomorphism we proceed in the same way we did for

Atiyah-Hodge’s theorem. It is clear that for x ∈U , Hk(ι)x is an isomorphism for all

6.4 Carlson-Griffiths Lemma 63

k ≥ 0. And for x ∈ Y , is clear that Hk(ι)x is surjective. Then it remains to show theinjectivity. For this, we use the following logarithmic counterpart of Lemma 4.1:

“If ω is a closed meromorphic k-form on ∆ = Dn, which is holomorphic in ∆ \Y with logarithmic poles along Y = (z1, ...zn) ∈ ∆ |z1 · · ·zr = 0. And ω = 0 inHk

dR((∆ \Y )∞). Then there exist a meromorphic k-form η (on a possibly smallerpolydisc) also with logarithmic poles along Y , such that ω = ∂η . ”

This result is a corollary of the original lemma, since by Remark 4.3.1, we canchoose η with pole of order less or equal than the order of ω , i.e. η has pole orderless or equal than 1 along Y , while ∂η = ω also has pole order less or equal than 1.This implies η has a logarithmic pole along Y , as desired. ut

Remark 6.3.1 Deligne’s Theorem 6.3.1 remains valid for U = X \Y , and Y anample normal crossing divisor of X. In fact, the proof above applies in this contextwithout modifications.

6.4 Carlson-Griffiths Lemma

For the second proof Theorem 6.3.1, we will use the following Lemma due to Carl-son and Griffiths.

Definition 6.4.1 Suppose X is embedded in a projective space PN , and Y = X ∩F = 0 for some homogeneous polynomial F ∈ C[x0, ...,xN ]. Consider U =UiN

i=0 the Jacobian covering of X given by Ui = X ∩Fi 6= 0, where Fi := ∂F∂xi

.For every l ≥ 2 define

Hl : Cq(U ,Ω pX (lY ))→Cq(U ,Ω p−1

X ((l−1)Y )),

(Hlω) j0··· jq :=1

1− lF

Fj0ι ∂

∂x j0

(ω j0··· jq).

Theorem 6.4.1 (Carlson-Griffiths Lemma [CG80]) For every l ≥ 2, letting D =d +(−1)pδ , then

DH +HD :⊕

p+q=k

Cq(U ,Ω pX/C(lY ))

Cq(U ,Ω pX/C((l−1)Y ))

→⊕

p+q=k

Cq(U ,Ω pX/C(lY ))


is the identity map. Note that D is the is the differential map in used to define thehypercohomology groups.

Proof. We claim

dHl +Hl+1d :Cq(U ,Ω p

X/C(lY ))


→Cq(U ,Ω p

X/C(lY ))



is the identity map. In fact, take ωJ =αJF l ∈Cq(U ,Ω p

X/C(lY )), then

(dHl +Hl+1d)ωJ = d

(1

1− lF

Fj0ι ∂

∂x j0

(αJ

F l

))− 1

lF

Fj0ι ∂

∂x j0

(dαJ

F l − ldF ∧αJ

F l+1

)

≡ 11− l

dFFj0∧

ι ∂

∂x j0

(αJ)

F l − l1− l

FFj0

dF ∧ ι ∂

∂x j0

(αJ)

F l+1

+F

Fj0

(Fj0αJ−dF ∧ ι ∂

∂x j0

(αJ))

F l+1

=αJ

F l = ωJ ,

where the congruence is taken mod Cq(U ,Ω pX/C((l− 1)Y )). Using this, take ω =

∑kp=0 ω p ∈

⊕p+q=k Cq(U ,Ω p

X/C(lY )) then

ω−DHω = ∑p+q=k

(ω p−dHωp− (−1)p−1

δHωp)

≡ ∑p+q=k

(Hdωp +(−1)p

δHωp)

≡ ∑p+q=k

H(dωp +(−1)p

δωp) = HDω.

Where in the last congruence we used δHω p ≡ Hδω p. In fact

(δHlωp) j0··· jq+1 =

q+1

∑m=0

(−1)m(Hlωp) j0··· jm··· jq+1

=1

1− l

(F

Fj1ι ∂

∂x j1

(ω pj1··· jq+1

)+F

Fj0ι ∂

∂x j0

(q+1

∑m=1

(−1)mω

pj0··· jm··· jq+1

))

=F

1− l

ι ∂

∂x j1

(ω pj1··· jq+1

)

Fj1−

ι ∂

∂x j0

(ω pj1··· jq+1

)

Fj0

+(Hlδωp) j0··· jq+1 .

ut

6.5 Second proof of Deligne’s Theorem

The idea of the second proof of Deligne’s Theorem 6.3.1 is that thanks to Carlson-Griffiths Lemma (Theorem 6.4.1), we can use the operator H defined in (??) andreduce the pole order of a given algebraic form with poles along Y to transform itinto a logarithmic form with the same hypercohomology class.

6.5 Second proof of Deligne’s Theorem 65

In fact, consider the inclusion ι : Ω •X/C(logY ) →Ω •X/C(∗Y ). It induces the mor-phism

ϕ : Hk(U ,Ω •X/C(logY ))→Hk(U ,Ω •X/C(∗Y )).

Let us prove ϕ is an isomorphism. Given

ω =k

∑p=0

ωp ∈

⊕p+q=k

Cq(U ,Ω pX/C(∗Y ))

such that Dω = 0, there exist l ≥ 1 such that

ω ∈⊕

p+q=k

Cq(U ,Ω pX/C(lY )).

If l = 1, we claim ω ∈⊕

p+q=k Cq(U ,Ω pY/C(logY )). In fact, since Dω = 0, we get

dωp = (−1)p

δωp+1 ∈Cq(U ,Ω p+1

X/C(Y )),

as desired.For l ≥ 2, it follows from the lemma that ω is cohomologous (in hypercohomol-

ogy) to the D-closed element

ν := (1−DH)l−1ω ∈

⊕p+q=k

Cq(U ,Ω pX/C(Y )).

And it follows from the case l = 1 that

ν ∈⊕

p+q=k

Cq(U ,Ω pY/C(logY )),

i.e. ϕ is surjective.Now, for the injectivity, consider

ω ∈⊕

p+q=k

Cq(U ,Ω pX/C(logY )),

such that there existη ∈

⊕p+q=k−1

Cq(U ,Ω pX/C(lY ))

for some l ≥ 1, withDη = ω.

If l = 1, we claim η ∈⊕

p+q=k−1 Cq(U ,Ω pY/C(logY )). In fact, since Dη = ω , we

getdη

k−1 = ωk ∈C0(U ,Ω k

X/C(Y )),


then ηk−1 ∈C0(U ,Ω k−1X/C(logY )). Inductively, if we assume η p+1 ∈Cq−1(U ,Ω p+1

X/C(logY )),then

dηp = ω

p+1 +(−1)pδη

p+1 ∈Cq(U ,Ω p+1X/C(logY )),

in consequence η p ∈Cq(U ,Ω pX/C(logY )).

Finally, for l ≥ 2, if we take

φ := (1−DH)l−1η .

Since (1−DH)η =Hω+µ ∈⊕

p+q=k Cq(U ,Ω pX/C((l−1)Y )), and D((1−DH)η)=

Dη = ω , it is clear that Dφ = ω and

φ ∈⊕

p+q=k−1

Cq(U ,Ω pX/C(Y )),

and by the case l = 1, we actually see that

φ ∈⊕

p+q=k−1

Cq(U ,Ω pX/C(logY )),

as desired. ut

6.6 Residue map

Let X be a smooth projective variety, Y ⊆ X a smooth hyperplane section, and U :=X \Y . Taking the long exact sequence in cohomology of the pair (X ,U), and usingLeray-Thom-Gysin isomorphism (see [Mov17b, Chapter 4, §6]) we obtain the exactsequence

· · · → Hk+1dR (X∞)→ Hk+1

dR (U∞)res−→ Hk

dR(Y∞)

τ−→ Hk+2dR (X∞)→ ·· · (6.1)

where res is the residue map, and τ corresponds to the wedge product with thepolarization θ . In particular, we obtain a surjective map

res : Hk+1dR (U∞)→ Hk

dR(Y∞)0 = Ker τ.

Where HkdR(Y

∞)0 is the primitive part of de Rham cohomology, i.e. is the comple-mentary space to θ

k2 inside Hk

dR(Y∞) (see [Mov17b, Chapter 5, §7]). We want to

determine the algebraic counterpart of this map (together with its long exact se-quence).

Definition 6.6.1 Let X be a smooth projective variety of dimension n and Y ⊆ X asmooth hyperplane section. Let z1, ...,zn be local coordinates on an open set V of X,such that V ∩Y = z1 = 0. For α ∈ Γ (V,Ω p

Xan(logY )) we define its residue at Y tobe

6.7 Hodge filtration for affine varieties 67

Res(α) := β |Y∩V ∈ Γ (V, j∗Ωp−1Y an ),

where α = β ∧ dz1z1

+ γ , and β , γ are holomorphic forms. This definition does notdepend on the choice of the coordinates, and defines the residue map

Res : ΩpXan(logY )→ j∗Ω

p−1Y an .

Which is part of the following exact sequence

0→ΩpXan →Ω

pXan(logY ) Res−−→ j∗Ω

p−1Y an → 0,

called Poincare’s residue sequence. Using Serre’s GAGA principle we have its al-gebraic counterpart

0→ΩpX/C→Ω

pX/C(logY ) Res−−→ j∗Ω

p−1Y/C → 0,

which we also call Poincare’s residue sequence.

This sequence gives rise to a short exact sequence of complexes

0→Ω•X/C→Ω

•X/C(logY ) Res−−→ j∗Ω •−1

Y/C → 0. (6.2)

Taking the long exact sequence in hypercohomology, and applying Deligne’s Theo-rem 6.3.1 we get the exact sequence

· · · → Hk+1dR (X/C)→ Hk+1

dR (U/C) res−→ HkdR(Y/C)

τ−→ Hk+2dR (X/C)→ ··· , (6.3)

Which turns out to be the algebraic counterpart of the sequence (6.1).

6.7 Hodge filtration for affine varieties

Since for every i≥ 0 we have the short exact sequence of complexes

0→Ω•≥iX/C→Ω

•≥iX/C(logY ) Res−−→ j∗Ω •≥i−1

Y/C → 0.

We have the following commutative diagram with exact rows

· · · Hk+1dR (X/C) Hk+1

dR (U/C) HkdR(Y/C) · · ·

· · · Hk+1(X ,Ω •≥iX/C) Hk+1(X ,Ω •≥i

X/C(logY )) Hk(Y,Ω •≥i−1Y/C ) · · ·

f g hτres

The vertical arrows of this diagram are all injective. In fact, the maps f and h areinjective by Proposition ??. The injectivity of g is more delicate and is consequence


of a theorem due to Deligne on the degeneration of the spectral sequence associatedto the naive filtration of Ω •X/C(logY ) (see [Voi02] Theorem 8.35 or [Del71] Corol-lary 3.2.13). In consequence, the exact sequence (6.3) is compatible with Hodgefiltrations, i.e. we have the exact sequence

· · ·→F iHk+1dR (X/C)→F iHk+1

dR (U/C) res−→F i−1HkdR(Y/C)

τ−→F iHk+2dR (X/C)→·· · .

Chapter 7Cohomology of hypersurfaces

7.1 Introduction

In this chapter we will present Griffiths’ results on the cohomology of hypersurfaces[Gri69a]. This work culminates with an explicit basis for the primitive cohomologyof a hypersurface, compatible with the Hodge filtration.

Let X ⊂ Pn+1 be a smooth hypersurface of degree d, and U = Pn+1 \X . To con-struct the basis for Hk

dR(X/C)0 we will give generators for Hk+1dR (U/C), compatible

with the algebraic Hodge filtration F i+1Hk+1dR (U/C). Then, we will obtain the de-

sired basis by applying the algebraic residue map to the generators, and reduce theset of generators to a basis. Furthermore, we will prove a theorem due to Carlsonand Griffiths which describes this basis in each piece of the Hodge structure

F iHkdR(X/C)/F i+1Hk

dR(X/C)' Hk−i(X ,Ω iX ).

With a little more effort we will also describe this basis in all HkdR(X/C).

7.2 Bott’s formula

In this section we will prove a theorem due to Bott that determines the Cech co-homology groups of PN with coefficients in Ω

pPN/C(k). This theorem contains the

cases of Bott’s vanishing theorem. In order to prove this theorem we need to intro-duce Euler’s exact sequence.

Definition 7.2.1 Consider over CN+1 the global vector field

E :=N

∑i=0

xi∂

∂xi.

This field is called Euler’s vector field. We define Euler’s sequence

69

70 7 Cohomology of hypersurfaces

0→Ωp+1PN/C(p+1)→ O

⊕(N+1p+1)

PN →ΩpPN/C(p+1)→ 0, (7.1)

as follows: For every Ui in the standard covering of PN , and every ω =∑|J|=p aJdx j0∧·· ·∧dx jp ∈Ω

p+1PN/C(p+1)(Ui), the first map sends ω to (aJ)|J|=p. While the second

one is sending (bJ)|J|=p, where each bJ ∈OPN (Ui), to the form η = ιE(∑|J|=p bJdx j0∧·· ·∧dx jp).

Proposition 7.2.1 Euler’s sequence (7.1) is an exact sequence.

Proof. To show the exactness, we will show it remains exact in every Ui of the stan-dard covering. Let us do it for U0 = PN \V (x0). The injectivity of the left morphismis clear. In order to show that Euler’s sequence is a complex, it is enough to noticethat ιE(d(

xix0)) = 0. In order to show that the right morphism is well defined, i.e. that

ιE(∑|J|=p bJdx j0 ∧ ·· · ∧ dx jp) ∈ ΩpPN/C(p+ 1)(U0), it is enough to show that any

η = ιE(dx j0 ∧·· ·∧dx jp) ∈ΩpPN/C(p)(U0). This follows from the equality

η = xp0x j0 · · ·x jp

(d(

x j1x0

)1

x j1−d(

x j0x0

)1

x j0

)∧·· ·∧

(d(

x jp

x0

)1

x jp

−d(

x j0x0

)1

x j0

).

The surjectivity of the right morphism follows from taking any η ∈ ΩpPN/C(p +

1)(U0) and write it as η = ιE(x−10 dx0 ∧ η) (notice that ιE(η) = 0). Finally, the

middle exactness follows in the same way. If ω = ∑|J|=p bJdx j0 ∧ ·· ·∧dx jp is suchthat ιE(ω) = 0, then we write ω = ιE(x−1

0 dx0 ∧ω), and we know the image of ιE

determines a form in the projective space, so ω ∈Ωp+1PN/C(p+1)(U0). ut

Theorem 7.2.1 (Bott’s formula [Bot57])

dimCHq(PN ,Ω pPN/C(k)) =

(k+N−p

k

)(k−1p

)if q = 0,0≤ p≤ N,k > p,

1 if k = 0,0≤ p = q≤ N,(−k+p−k

)(−k−1N−p

)if q = N,0≤ p≤ N,k < p−N,

0 otherwise.

Proof. The result follows by induction on p. For p = 0 we know that

H0(PN ,OPN (k)) = C[x0, ...,xN ]k,

andHN(PN ,OPN (k))' C[x0, ...,xN ]−k−N−1, (7.2)

for k<−N. Where the isomorphism (7.2) takes each monomial xI ∈C[x0, ...,xN ]−k−N−1to 1

xIx0···xN∈HN(PN ,OPN (k)). And the rest of groups are zero. This is a well known

fact, see for instance [Har77, page 225, Theorem 5.1]. For the induction step we usethe long exact sequence in cohomology associated to the twist of Euler’s sequence

0→Ωp+1PN/C(k)→ O

⊕(N+1p+1)

PN (k− p−1)→ΩpPN/C(k)→ 0. (7.3)

7.2 Bott’s formula 71

The proof of the induction step is tedious but straightforward if we separate in thecases given by the formula. First of all, (7.3) remains exact in global sections. Infact, since each global section of Ω

pPN/C(k) lift to a global section of Ω

pCN+1/C of the

formη = ∑

|J|=p−1cJdx j0 ∧·· ·∧dx jp−1

with cJ ∈ C[x0, ...,xN ]k−p. We can use Euler’s indentity to write

η = x0η0 + ...+ xNηN .

Then η = ιE(dx0∧η0 + ...+dxN ∧ηN). This gives us the short exact sequence

0→H0(PN ,Ω p+1PN/C(k))→H0(PN ,OPN (k− p−1))(

N+1p+1)→H0(PN ,Ω p

PN/C(k))→ 0,

and we can deduce the inductive step for q = 0. On the other hand, by inductionhypothesis we have the short exact sequence

0→HN(PN ,Ω p+1PN/C(k))→HN(PN ,OPN (k− p−1))(

N+1p+1)→HN(PN ,Ω p

PN/C(k))→ 0,

from which we deduce the inductive step for q = N. For the remaining cases, wehave by the base case p = 0 that Hq(PN ,OPN (k− p−1)) = 0 for 0 < q≤ N−1. Inconsequence

Hq(PN ,Ω pPN/C(k))' Hq+1(PN ,Ω p+1

PN/C(k)),

for 0 < q < N−1, and H1(PN ,Ω p+1PN/C(k)) = 0. These two facts are what we need to

finish the induction for the remaining values of q. ut

Corollary 7.2.1 Every ω ∈ H0(PN ,Ω N−1PN/C(k)) is of the form

ω =N

∑i=0

Tiι ∂

∂xi(Ω),

with Ti ∈ C[x0, ...,xN ]k−N . Where Ω := ιE(dx0 ∧ ·· · ∧ dxN) = ∑Ni=0(−1)ixidxi, and

dxi := dx0∧dx1∧·· ·∧dxi−1∧dxi+1∧·· ·∧dxN .

Proof. Using (7.3) for p = N−1 and Bott’s formula we see that H0(PN ,Ω N−1PN/C(k))

is generated by


ιE(N

∑i=0

Pidxi) = ιE(N

∑i=0

(−1)iPiι ∂

∂xi(dx0∧·· ·∧dxN)),

= ιE(ι∑

Ni=0(−1)iPi

∂

∂xi(dx0∧·· ·∧dxN)),

= −ι∑

Ni=0(−1)iPi

∂

∂xi(ιE(dx0∧·· ·∧dxN)),

=N

∑i=0

(−1)i+1Piι ∂

∂xi(Ω),

where Pi ∈ H0(PN ,OPN (k−N)). Take Ti := (−1)i+1Pi. ut

7.3 Griffiths’ Theorem I

Since the residue mapHk+1

dR (U/C) res−→ HkdR(X/C)0

is an isomorphism (because Hk+1dR (Pn+1)0 = 0). We conclude that Hk

dR(X/C)0 = 0for all k 6= n (because Hk+1

dR (U/C) = 0 for k + 1 6= n+ 1, see [Mov17b] Chapter5, section 5). Thus, we are just interested in determining a basis for the non-trivialprimitive cohomology group of X

HndR(X/C)0.

We start by giving a set of generators.

Theorem 7.3.1 (Griffiths [Gri69a]) For every q = 0, ...,n, the natural map

H0(Pn+1,Ω n+1Pn+1/C((q+1)X))→ Hn+1

dR (U/C)

has image equal to Fn+1−qHn+1dR (U/C). Consequently, every piece of the Hodge

filtrationFn−qHn

dR(X/C)0,

is generated by the residues of global forms with pole of order at most q+1 alongX.

Proof. Consider ωn+1 ∈H0(Pn+1,Ω n+1Pn+1/C((q+1)X)). The natural map sends it to

ω ∈ Hn+1dR (U/C) =Hn+1(Pn+1,Ω •Pn+1/C(∗X)),

by letting ωk = 0 for k = 0, ...,n. To see in which part of Hodge filtration ω is, weneed to write it as an element of Hn+1(Pn+1,ΩPn+1(logX)), i.e. we need to reducethe order of the pole of ωn+1 up to order 1. Thanks to Carlson-Griffiths Lemma6.4.1, we know how to do this applying the operator (1−DH). In order to obtain

7.4 Griffiths’ Theorem II 73

a form with poles of order 1 we need to apply it q times, i.e. the image of ω inHn+1(Pn+1,Ω •Pn+1/C(logX)) is represented by (1−DH)qω . It is clear by the defini-tion of H and D that

(1−DH)lω ∈ Im(Hn+1(Pn+1,Ω •≥n+1−l

Pn+1/C ((q+1−l)X))→Hn+1(Pn+1,Ω •Pn+1/C((q+1−l)X))).

In consequence, for l = q we see that

(1−DH)qω ∈ Fn+1−qHn+1

dR (U/C).

Conversely, let ω ∈ Fn+1−qHn+1dR (U/C). Then we can represent ω = ωn+1−q +

· · ·+ωn+1, where each

ωk ∈Cn+1−k(U ,Ω k

Pn+1/C(logX)),

and ωk = 0 for k = 0, ...,n−q. We claim that for every l = 0, ...,q we can representω ∈Hn+1(Pn+1,Ω •Pn+1/C(∗X)) as ω = η

n+1−q+ll +ωn+2−q+l · · ·+ωn+1 with

ηn+1−q+ll ∈Cq−l(U ,Ω n+1−q+l

Pn+1/C ((l +1)X)).

We prove this claim by induction on l. The case l = 0 is clear taking ηn+1−q0 =

ωn+1−q. Now, for l > 0 suppose ω = ηn−q+ll−1 +ωn+1−q+l + · · ·+ωn+1. Since Dω =

0, we know δηn−q+ll−1 = 0. By Bott’s formula (Theorem 7.2.1)

Hq−l+1(Pn+1,Ω n−q+lPn+1/C(lX)) = 0.

Then, there exist µ ∈ Cq−l(U ,Ω n−q+lPn+1/C(lX)) such that δ µ = η

n−q+ll−1 . Subtracting

from ω the exact form in hypercohomology (−1)n−q+lDµ , we get the claim forl, and we finish the induction. Finally, applying the claim for l = q we can writeω = ηn+1

q withη

n+1q ∈ H0(U ,Ω n+1

Pn+1/C((q+1)X)),

as desired. ut

7.4 Griffiths’ Theorem II

Griffiths’ Theorem 7.3.1 tells us that the elements of the form

res(

PΩ

Fq+1

)∈ Fn−qHn

dR(X/C)0,

where P∈H0(Pn+1,OPn+1(d(q+1)−n−2)) and q= 0, ...,n, generate all HndR(X/C)0.


The following theorem tells us how we can choose a basis from these generators.

Theorem 7.4.1 (Griffiths [Gri69a]) For every q = 0, ...,n the kernel of the map

ϕ : H0(Pn+1,OPn+1(d(q+1)−n−2)) Fn−qHndR(X/C)0/Fn+1−qHn

dR(X/C)0

P 7→ res(

PΩ

Fq+1

)is the degree N = d(q+1)−n−2 part of the Jacobian ideal of F, JF

N ⊆C[x0, ...,xn+1]N .

Definition 7.4.1 Recall that the Jacobian ideal of F is the homogeneous ideal

JF := 〈F0, ...,Fn+1〉 ⊆ C[x0, ...,xn+1],

where, from now on, we denote

Fi :=∂F∂xi

,

for i = 0, ...,n+1. The Jacobian ring of F is

RF := C[x0, ...,xn+1]/JF .

Remark 7.4.1 Theorem 7.4.1 implies that to choose a basis for

Fn−qHndR(X/C)0/Fn+1−qHn

dR(X/C)0 ' Hn−q,q(X)0

it is enough to take the elements of the form res(

PΩ

Fq+1

), for P ∈ C[x0, ...,xn+1]N

forming a basis of RFN . In particular

hn−q,q(X)0 = dimCRFN .

Proof. of Theorem 7.4.1 By Theorem 7.3.1, it is clear that P is in the kernel of ϕ , ifand only if, there exist Q ∈ H0(Pn+1,OPn+1(dq−n−2)) such that

res(

PΩ

Fq+1

)= res

(QΩ

Fq

).

Since the residue map is an isomorphism between Hn+1dR (U/C)'Hn

dR(X/C)0. Thisis equivalent to say

(P−FQ)Ω

Fq+1 = 0 ∈ Hn+1dR (U/C). (7.4)

Since Hn+1dR (U/C)' Hn+1(Γ (Ω •U/C),d), (7.4) is equivalent to

(P−QF)Ω

Fq+1 = dγ,

7.5 Carlson-Griffiths Theorem 75

for some γ ∈ H0(Pn+1,Ω nPn+1/C(qX)). Recall from Corollary 7.2.1 that every γ ∈

H0(Pn+1,Ω nPn+1/C(qX)) is of the form

γ =∑

n+1i=0 Tiι ∂

∂xi(Ω)

Fq ,

for some Ti ∈ C[x0, ...,xn+1]dq−n−1. In consequence, P is in the kernel of ϕ , if andonly if,

PΩ

Fq+1 ≡−q∑

n+1i=0 TiFiΩ

Fq+1 (mod H0(Pn+1,Ω nPn+1/C(qX))),

in other words

P≡−qn+1

∑i=0

TiFi (mod F).

Since F ∈ JF (by Euler’s identity), this is equivalent to P ∈ JF . ut

7.5 Carlson-Griffiths Theorem

In this section we give an explicit description in Cech cohomology of the residuemap for the generators given by Griffiths’ theorem. This was done in [CG80] as aconsequence of Carlson-Griffiths Lemma 6.4.1.

Let X ⊆ Pn+1 a smooth degree d hypersurface given by X = F = 0. Recall thatH0(Pn+1,Ω n+1

Pn+1/C(n+2))' C is generated by

Ω =n+1

∑i=0

(−1)ixidx0∧·· · dxi · · ·∧dxn+1.

Theorem 7.5.1 (Carlson-Griffiths [CG80]) Let q∈0,1, ...,n, P∈H0(Pn+1,OPn+1(d(q+1)−n−2)). Then

res(

PΩ

Fq+1

)=

(−1)n(q+1)

q!

PΩJ

FJ

|J|=q∈ Hq(U ,Ω n−q

X/C). (7.5)

Where ΩJ := ι ∂

∂x jq

(...ι ∂

∂x j0

(Ω)...), FJ :=Fj0 · · ·Fjq and U = Uin+1i=0 is the Jacobian

covering restricted to X, given by Ui = Fi 6= 0∩X.

Proof. Let U := Pn+1 \X . For l = 0, ..,q define

(l)ω := (1−DH)l

(PΩ

Fq+1

)∈ Hn+1

dR (U/C),

where H is the operator defined in Definition 6.4.1. We claim


(l)ω

n+1−l =

(q− l)!(−1)n(l+1)P

q! ·Fq−l

(ΩJ

FJ∧ dF

F+d · (−1)n VJ

FJ

)|J|=l

∈Cl(U ,Ω n+1−lPn+1/C((q−l+1)X)),

where V := dx0 ∧ ·· · ∧ dxn+1, and VJ := ι ∂

∂x jm

(...ι ∂

∂x j0

(V )...). In fact, for l = 0, the

claim follows from the identity

Ω

F=

dFF∧

Ω(i)

Fi+d ·

V(i)

Fi.

(Which is obtained by contracting ι ∂

∂xito the equality dF∧Ω = d ·F ·V .) Assuming

the claim for l ≥ 0, then

Hq−l+1((l)

ωn−l+1)J =

(q− l−1)!(−1)nl+l+1Pq! ·Fq−l

ΩJ

FJ.

In consequence,

(l+1)ω

n−lJ =(−1)n−l+1

δHq−l+1((l)

ωn+1−l)J =

(q− l−1)!q!

l+1

∑m=0

(−1)n(l+1)+m PΩJ\ jmFq−lFJ\ jm

.

Using the following identity

ΩJ ∧dF +(−1)nd ·F ·VJ = (−1)nl+1

∑m=0

(−1)mFjmΩJ\ jm, (7.6)

(this identity is obtained by successive contraction of the identity dF ∧Ω = d ·F ·Vby the ι ∂

∂x jm

, for m = 0, ..., l +1) we obtain the claim for l +1. In conclusion

(q)ω

n+1−q =

(−1)n(q+1)P

q!

(ΩJ

FJ∧ dF

F+d · (−1)n VJ

FJ

)|J|=q

∈Cq(U ,Ω n+1−qPn+1/C(logX)),

the rest is just to apply the residue map. ut

7.6 Computing the residue map in algebraic de Rhamcohomology

Using Carlson-Griffiths’ lemma, it is possible to describe explicitly the residue mapin all algebraic de Rham cohomology, i.e. as an element of

res(

PΩ

Fq+1

)∈ Fn−qHn

dR(X/C).

7.6 Computing the residue map in algebraic de Rham cohomology 77

This is resumed in the following theorem.

Theorem 7.6.1 Let q ∈ 0,1, ...,n, P ∈ H0(Pn+1,OPn+1(d(q+ 1)− n− 2)). Forl = 0, ..,q define

(l)ω := (1−DH)l

(PΩ

Fq+1

).

Then, for each m = 0, ..., l

(l)ω

n+1−m =

(l)αn+1−m

JFq−l

(ΩJ

FJ∧ dF

F+d · (−1)n VJ

FJ

)|J|=m

∈Cm(U ,Ω n+1−mPn+1/C((q+1−l)X)).

Where V := dx0∧·· ·∧dxn+1, VJ := ι ∂

∂x jm

(...ι ∂

∂x j0

(V )...), U is the Jacobian covering

of Pn+1. And (l)αn+1−m ∈Cm(U ,OPn+1(d(q− l +m+1)−n−2)) are such that

(l)α

n+1−mJ =

1q− l +1

((−1)n ·(l−1)

αn+2−mJ\ jk

+FJ\ jk∂

∂x jk

((l−1)αn+1−m

JFJ

)),

(7.7)for any k ∈ 0, ...,m, and (0)αn+1

j = (−1)nP for every j = 0, ...,n+1.

Proof. Let us proceed by induction in l = 0, ...,q. The base case follows directlyfrom the identity

Ω

F=

dFF∧

Ω(i)

Fi+d ·

V(i)

Fi.

(To obtain this identity just apply ι ∂

∂xito the identity dF∧Ω = d ·FV .) Now, assume

the theorem is true for some l ≥ 0. Then

(l+1)ω = (1−DH)(l)ω.

For any m = 0, ..., l +1 we have that

Hq−l+1((l)

ωn+1−mJ ) =

(−1)n+1−m

(q− l)

(l)αn+1−mJ ΩJ

Fq−lFJ.

Using the following identity

ΩJ ∧dQ+(−1)ngQVJ = (−1)nm

∑k=0

(−1)kQ jk ΩJ\ jk, (7.8)

for Q ∈ C(x0, ...,xn+1) homogeneous of degree g (this identity is obtained by suc-cessive contraction of the identity dQ∧Ω = gQV by the ι ∂

∂x jk

, for k = 0, ...,m).

NoticingdΩJ = (−1)m(m−n−1)VJ ,

and applying (7.8) to Q =(−1)n+1−m·(l)αn+1−m

J(q−l)Fq−lFJ

we obtain


dHq−l+1((l)

ωn+1−mJ ) = (−1)m

m

∑k=0

(−1)k ∂

∂x jk

((−1)n+1−m

(q− l)

(l)αn+1−mJ

Fq−lFJ

)ΩJ\ jk.

By the other hand, applying (7.8) for Q = F we get

(l)ω

n+1−m = (−1)nm

∑k=0

(−1)k

((l)αn+1−m

J FjkFq+1−lFJ

)ΩJ\ jk.

Using these relations we conclude

(l+1)ω

n+1−m =m

∑k=0

1

q−l

((l)αn+2−m

J\ jk+(−1)nFJ\ jk

∂

∂x jk

((l)αn+1−m

JFJ

))Fq−lFJ

(−1)kFjk ΩJ\ jk,

and the result follows if we prove that the expression

(l)En+1−mk := (l)

αn+2−mJ\ jk

+(−1)nFJ\ jk∂

∂x jk

((l)αn+1−m

JFJ

)(7.9)

is independent of k (and we conclude that (l+1)αn+1−mJ = (l)En+1−m

k ). If l = 0 this isclear, and for l > 0 we know by induction hypothesis this is true for (l−1)En+2−m

h =(l)αn+2−m

J\ jkand (l−1)En+1−m

h = (l)αn+1−mJ , i.e. we can write

(l)α

n+2−mJ\ jk

= (l−1)α

n+3−mJ\ jk, jh

+(−1)nFJ\ jk, jh∂

∂x jh

((l−1)αn+2−m

J\ jkFJ\ jk

),

and(l)

αn+1−mJ = (l−1)

αn+2−mJ\ jh

+(−1)nFJ\ jh∂

∂x jh

((l−1)αn+1−m

JFJ

),

for h ∈ 0, ...,m\k. Replacing this in (7.9) we obtain

(l)En+1−mk = S(k,h)+(−1)n+1

∂ 2F∂x jk ∂x jh

Fjk Fjh

(l−1)En+1−mh ,

where

S(k,h) := (l−1)αn+3−mJ\ jk, jh

+FJ\ jk, jh

(∂ 2

∂x jk ∂x jh

((l−1)αn+1−m

JFJ

)+(−1)n ∂

∂x jk

((l−1)αn+2−m

J\ jhFJ\ jh

)+(−1)n ∂

∂x jh

((l−1)αn+2−m

J\ jkFJ\ jk

))= S(h,k),

7.6 Computing the residue map in algebraic de Rham cohomology 79

is a symmetric expression in terms of k and h. Since (l−1)En+1−mh = (l−1)En+1−m

k weconclude

(l)En+1−mk = (l)En+1−m

h ,

as desired. ut

Remark 7.6.1 Theorem 7.6.1 is giving us an explicit expression of PΩ/Fq+1 inhypercohomology with meromorphic forms with logarithmic poles along X, namely(q)ω . By (7.7) we know that

(l)α

n+1−lJ =

(−1)n

(q− l +1)(l−1)

αn+2−lJ\ jk

,

for l = 0, ...,q. Recursively, we get the expression

(q)α

n+1−qJ =

(−1)nq

q!(0)

αn+1j =

(−1)n(q+1)Pq!

.

In consequence,

(q)ω

n+1−q =

(−1)n(q+1)P

q!

(ΩJ

FJ∧ dF

F+d · (−1)n VJ

FJ

)|J|=q

∈Cm(U ,Ω n+1−qPn+1 (logX)).

Applying residue to (q)ω , we re-obtain Carlson-Griffiths’ Theorem 7.5.1.

Chapter 8Periods of algebraic cycles

In its early phase (Abel, Riemann, Weierstrass), algebraic geometry was just a chap-ter in analytic function theory, (Solomon Lefschetz in [Lef68] page).

8.1 Introduction

Periods of algebraic cycles play a fundamental role when we look at Hodge conjec-ture in families. In fact, they determine the cohomology class of the algebraic cyclewhich together with the infinitesimal variation of Hodge conjecture gives us the tan-gent space of the underlying Hodge locus. This is our main motivation to computethose periods, and it is the central topic of this chapter. Since non-zero dimensionalalgebraic cycles cannot lie inside affine varieties, their periods do not fit well into themultiple integral context of Simart and Picard in [PS06]. This has actually producedPicard’s ρ0 puzzle, for details see [Mov17b] Chapter 3. Historically, the first useof periods of algebraic cycles in the literature is by Deligne in [DMOS82, Proposi-tion 1.5] which leads him to define the notion of an absolute Hodge cycle. This hasbeen reproduced in Proposition 5.10.1. When algebraic cycles are known exactly,which is the case in this chapter, computation of such periods can be done by the-oretical Cech cohomology manipulations such as in Carlson and Griffiths [CG80],and this eventually leads us to the verification of the alternative Hodge conjecturein explicit examples. This method was suggested in [Mov17c, § 3.5] and carriedout in [MV17] for complete intersection algebraic cycles. The full computation forcomplete intersection algebraic cycles is done by the second author in his thesis,see [Vil18], and this is the main content of this chapter. The applications are left toChapter ?? in which we discuss the variational and alternative Hodge conjecture.For Fermat varieties the periods are known exactly and the problem reduces to find-ing algebraic relations between values of the Gamma function on rational numbers,see [DMOS82, § 7] and [Mov17b, Chapter 16].

81

82 8 Periods of algebraic cycles

In Proposition 5.10.1 We have already seen that integration over algebraic cyclesis a purely algebraic operation. In this chapter we consider the case of an evendimensional smooth hypersurface X = f = 0 ⊆ Pn+1 given by a homogeneouspolynomial F of degree d. Every n

2 -dimensional subvariety Z of X determines analgebraic cycle [Z] ∈ Hn(X ,Z). Recall that by Griffiths’ Theorem 7.3.1, each pieceof the Hodge filtration is generated by the differential forms

ωP := Resi(

PΩ

f q+1

)∈ Fn−qHn

dR(X/C)0,

for P ∈C[x0, . . . ,xn+1]d(q+1)−n−2. Notice that, since Z is a projective variety of pos-itive dimension, it intersects every divisor of X . This means that it is impossible tofind an affine chart of X containing Z, and hence, the computation of periods of Zby calculus methods would be impossible. Our strategy to compute the periods is toreduce the computation to a period of the projective space PN .

8.2 Periods of top forms

In section 5.8 we computed the period of the standard top form relative to the stan-dard covering of PN . Using Corollary 5.11.1 we can compute the periods of topforms in PN described with other open coverings (not just the standard one), forinstance the Jacobian covering associated to a smooth hypersurface.

Proposition 8.2.1 Let f0, . . . , fN ∈ C[x0, . . . ,xN ]d homogeneous polynomials of thesame degree d > 0, such that

f0 = · · ·= fN = 0=∅.

They define the finite morphism F : PN → PN given by

F(x0 : · · · : xN) := ( f0 : · · · : fN).

Let UF = ViNi=0 be the open covering associated, i.e. Vi = fi 6= 0. Then the top

formΩF

f0 · · · fN:=

∑Ni=0(−1)i fid fi

f0 · · · fN∈ HN(UF ,Ω

NPN ),

has period ∫PN

ΩF

f0 · · · fN= dN · (−1)(

N+12 )(2π

√−1)N .

Proof. If Ω is the standard top form associated to the standard covering, apply-ing Corollary 5.11.1 we get F−1(U ) = UF and F∗Ω = ΩF . Then it follows fromtopological degree theory that

8.2 Periods of top forms 83∫PN

ΩF

f0 · · · fN= deg(F) ·

∫PN

Ω

x0 · · ·xN= deg(F) · (−1)(

N+12 )(2π

√−1)N .

Since F is defined by a base point free linear system, the fiber of F is genericallyreduced by Bertini’s theorem (see [Har77] page 179), and corresponds to dN pointsby Bezout’s theorem, i.e. deg(F) = dN .

Before going further we will recall the following theorem due to Macaulay (fora proof see [Voi03] Theorem 6.19):

Theorem 8.2.1 (Macaulay [Mac16]) Given f0, . . . , fN ∈C[x0, . . . ,xN ] homogeneouspolynomials with deg( fi) = di and

f0 = · · ·= fN = 0=∅.

Letting

R :=C[x0, . . . ,xN ]

〈 f0, . . . , fN〉,

then for σ := ∑Ni=0(di−1), we have that

(i) For every 0≤ i≤ σ the product Ri×Rσ−i→ Rσ is a perfect pairing.(ii)dimCRσ = 1.(iii)Re = 0 for e > σ .

Definition 8.2.1 A ring of the form R=C[x0, . . . ,xN ]/I for some homogeneous idealI is called a 0-dimensional Gorenstein ring of socle degree σ , if there exist σ ∈ Nsuch that R satisfies properties (i), (ii) and (iii) of Macaulay’s Theorem 11.2.1. Wealso say that I is a Gorenstein ideal of socle σ .

Remark 8.2.1 It is easy to see (using Euler’s identity) that

ΩF = d−1det(Jac(F))Ω

where Jac(F)=(

∂ fi∂x j

)0≤i, j≤N

is the Jacobian matrix of F . Any element of ZN(UF ,ΩNPN )

is of the formω =

PΩ

f α00 · · · f

αNN

where α0, . . . ,αN ∈Z>0 with d ·(α0+· · ·+αN)= deg(P)+N+1. Using Macaulay’sTheorem 11.2.1, we see that

P = ∑d(β0+···+βN)=deg(P)−d(N+1)

f β00 · · · f

βNN Pβ

with deg(Pβ ) = (d− 1)(N + 1) = σ . This reduce the problem of computation ofperiods, to forms of the form

Pβ Ω

f α00 · · · f

αNN

, (8.1)


with α0, . . . ,αN ∈ Z such that α0 + · · ·+αN = N+1 and deg(Pβ ) = (d−1)(N+1).It is clear that such a form represents an exact top form of PN if some αi is non-positive (in fact, it is equivalent to show that it is zero in hypercohomology and thisis clear because the form extends to a d-closed form on V0···i···N , as in the proof ofProposition 5.8.3) then we reduce the computation to forms of the form

QΩ

f0 · · · fN,

where deg(Q) = σ .

Corollary 8.2.1 If Q ∈ C[x0, . . . ,xN ]σ , then∫PN

QΩ

f0 · · · fN= c ·dN+1 · (−1)(

N+12 )(2π

√−1)N ,

where c ∈ C is the unique number such that

Q≡ c ·det(Jac(F)) (mod 〈 f0, . . . , fN〉).

Proof. To show the existence and uniqueness of c∈C we use item (ii) of Macaulay’sTheorem 11.2.1. So, it is enough to show

det(Jac(F)) /∈ 〈 f0, . . . , fN〉.

This is direct from the previous considerations and the fact

ΩF

f0 · · · fN∈ HN(PN ,Ω N

PN )

does not represent an exact top form by Proposition 8.2.1.

Remark 8.2.2 In summary, the computation of the period reduces to the computa-tion of such constant c that relates Q with det(Jac(F)) in Rσ .

8.3 Coboundary map

In order to compute periods of complete intersection algebraic cycles, we need tocompute periods of smooth hyperplane sections Y of a given projective smooth va-riety X . In other words, for Y → X a smooth hypersurface given by F = 0, weneed an explicit description of the isomorphism

Hn(Y,Ω nY )

τ' Hn+1(X ,Ω n+1X ), (8.2)

ω 7→ ω

together with the relation of periods, that is, the number a ∈ C such that

8.3 Coboundary map 85∫X

ω = a∫

Yω.

For this purpose recall the exact sequence (6.3)

· · · → Hk+1dR (X/C)→ Hk+1

dR (U/C) res−→ HkdR(Y/C)

τ−→ Hk+2dR (X/C)→ ·· · ,

induced by the short exact sequence of complexes

0→Ω•X →Ω

•X (logY ) Res−−→ j∗Ω •−1

Y → 0.

Since H2n+1dR (U) = H2n+2

dR (U) = 0, the coboundary map is an isomorphism

H2ndR(Y/C)

τ' H2n+2dR (X/C).

These vector spaces are one dimensional, and τ preserves Hodge filtration. There-fore, it induces the desired isomorphism in (8.2).

Proposition 8.3.1 Let X ⊆ PN be an smooth complete intersection of dimensionn+ 1, and Y ⊆ X a smooth hyperplane section given by F = 0 ∩ X, for somehomogeneous F ∈ C[x0, . . . ,xN ]d . Let ω ∈ Cn(X ,Ω n

X ) such that ω|Y ∈ Zn(Y,Ω nY ).

For any ω ∈Cn(X ,Ω n+1X (logY )) such that

ω ≡ ω ∧ dFF

(mod Cn(X ,Ω n+1X )),

we haveτ(ω) = ω := (−1)n+1

δ (ω) ∈ Zn(X ,Ω n+1X ).

Furthermore, ω ∈ Hn+1(X ,Ω n+1X ) is uniquely determined by ω|Y ∈ Hn(Y,Ω n

Y ) and

∫X

ω =(−1)n+1 ·2π

√−1

d

∫Y

ω.

Remark 8.3.1 Since any hypersurface section F = 0∩X as above is a hyperplanesection after using a Veronese embedding (see for instance [Mov17b, §5.2]) we willonly use the latter.

Proof. The map described in the proposition is the coboundary map τ , i.e. τ(ω) =ω . Therefore, it is left to prove the period relation. By the fact that τ is an isomor-phism of one dimensional spaces we have a constant aX ,Y ∈ C× such that∫

Xτ(ω) = aX ,Y

∫Y

ω,

for every ω ∈ Hn(Y,Ω nY ). Since X and Y are complete intersections, Lefschetz

hyperplane section theorem (see for instance [Mov17b, Chapter 4]) implies wecan extend ω and τ(ω) to PN . If X is complete intersection of type (d1, ..,dk),then [X ] = d1 · · ·dk[Pn+1] ∈H2n+2(PN ;Z) and [Y ] = dd1 · · ·dk[Pn] ∈H2n(PN ;Z). By


Stokes’ theorem

d1 · · ·dk

∫Pn+1

τ(ω) = aX ,Y dd1 · · ·dk

∫Pn

ω.

In other wordsaX ,Y =

aPn+1,Pn

d.

Finally, to compute aPn+1,Pn we suppose Pn = xn+1 = 0, we take ω ∈Cn(U ,Pn+1),where U is the standard open covering of Pn+1, and

ωJ =∑

ni=0(−1)ix ji dx ji

x j0 · · ·x jn, for |J|= n.

Then

ωJ =

∑

n+1i=0 (−1)ixidxi

x0···xn+1if J = 0, . . . ,n,

0 otherwise.

Consequently

ω0···n =∑

n+1i=0 (−1)ixidxi

x0 · · ·xn+1.

It follows by Proposition 5.8.1 that aPn+1,Pn = (−1)n+1 ·2π√−1. ut

8.4 Periods of complete intersection algebraic cycles

In this section we compute periods of complete intersection algebraic cycles insidea smooth hypersurface X ⊆ Pn+1, of even dimension n. After Carlson-Griffiths’theorem we know that the integrands in these periods are of the forms

ωP = res(

PΩ

Fn2+1

)=

1n2 !

PΩJ

FJ

|J|= n

2

∈ Hn2 (X ,Ω

n2

X ), (8.3)

where P ∈ C[x0, . . . ,xn+1]σ , and σ = (d − 2)( n2 + 1). In order to compute these

periods over a complete intersection subvariety Z of Pn+1 (contained in X), themain ingredient is the explicit description of the coboundary map. For a completeintersection Z ⊆ X of dimension n

2 , we construct a chain of subvarieties

Z = Z0 ⊆ Z1 ⊆ Z2 ⊆ ·· · ⊆ Z n2+1 = Pn+1,

where each Zi is a hypersurface section of Zi+1, and apply inductively the cobound-ary map, to reduce the computation of the period of Z to the computation ofthe integral of a top form in Pn+1. Recall that for functions F : Cn+1 and H =[H0,H1, · · · ,Hn+1] : Cn+1→ Cn+1 the Hessian and Jacobian matrices are defined inthe following way:

8.4 Periods of complete intersection algebraic cycles 87

Hess(F) :=[

∂ 2F∂xi∂x j

](n+1)×(n+1)

, Jac(H) :=[

∂Hi

∂x j

](n+1)×(n+1)

(8.4)

Theorem 8.4.1 Let X ⊆ Pn+1 be a smooth hypersurface given by X = F = 0.Suppose

F = f1g1 + · · ·+ f n2+1g n

2+1,

such that Z := f1 = · · ·= f n2+1 = 0 ⊆ X is a complete intersection (i.e. dim(Z) =

n2 ). Define

H = (h0, . . . ,hn+1) := ( f1,g1, f2,g2, . . . , f n2+1,g n

2+1).

Then ∫ZωP =

(2π√−1)

n2

n2 !

c · (d−1)n+2 ·d1 · · ·d n2+1,

where di = deg fi, ωP is given by (8.3), and c ∈ C is the unique number such that

P ·det(Jac(H))≡ c ·det(Hess(F)) (mod JF).

Remark 8.4.1 To understand the statement of theorem 8.4.1 recall that Macaulay’sTheorem 11.2.1 implies that

RF :=C[x0, . . . ,xn+1]

JF ,

where JF := 〈F0, . . . ,Fn+1〉 is the Jacobian ideal, is a 0-dimensional Gorenstein ringof socle 2σ . In particular dimCRF

2σ= 1. Furthermore, by Proposition 8.2.1 and Re-

mark 8.2.1, RF2σ

is generated by det(Hess(F)). In consequence, for any pair of poly-nomials P,Q ∈ C[x0, . . . ,xn+1]σ there exist a unique c ∈ C such that

P ·Q≡ c ·det(Hess(F)) (mod JF).

Definition 8.4.1 We say that a algebraic cycle δ ∈ Hn(X ,Z) is of complete inter-section type if

δ =k

∑i=1

ni[Zi],

for Z1, . . . ,Zk ⊆ X a set of n2 -dimensional subvarieties that are complete intersection

inside Pn+1, given by

Zi = fi,1 = · · ·= fi, n2+1 = 0, i = 1, . . . ,k,

such that there exist gi,1, . . . ,gi,k ∈ C[x0, . . . ,xn+1] with

F =

n2+1

∑j=1

fi, jgi, j.

For such an algebraic cycle, we define its associated polynomial


Pδ :=k

∑i=1

di ·ni ·det(Jac(Hi)) ∈ RFσ ,

where di := deg Zi, σ := (d−1)( n2 +1) and Hi := ( fi,1,gi,1, . . . , fi, n

2+1,gi, n2+1).

Remark 8.4.2 Theorem 8.4.1 tells us that in order to compute the periods of a com-plete intersection type cycle δ it is enough to know its associated polynomial Pδ . Infact, we are determining the Poincare dual of the cycle δ

δpd = res

(Pδ Ω

Fn2+1

)∈ F

n2+1Hn

dR(X)0,

that is, it satisfies (up to some constant non-zero factor)∫δ

ω =∫

Xω ∧ res

(Pδ Ω

Fn2+1

), ∀ω ∈ Hn

dR(X).

Let ω = ωP be as in (8.3). In order to prove Theorem 8.4.1, we will use Proposi-tion 8.3.1 to construct inductively

ω(0) := ω|Z ∈ H

n2 (Z,Ω

n2

Z ),

ω(l) := ω(l−1) ∈ H

n2+l(Zl ,Ω

n2+l

Zl), l = 1, . . . ,

n2+1

with Zl := fl+1 = · · ·= f n2+1 = 0 ⊆ Pn+1 and Z0 = Z and Z n

2+1 = Pn+1.

Remark 8.4.3 In order to use the coboundary map, we will assume that every Zlis a smooth hyperplane section of Zl−1. We can reduce ourselves to this situationby noticing this will hold for a general pair (Z,X) satisfying the hypothesis of thetheorem, then by continuity the result will extend to every such pair.

Lemma 8.1. For each l = 0, . . . , n2 + 1 and J = j0, . . . , j n

2+l ⊆ 0, . . . ,n+ 1 let|J|= j0 + · · ·+ j n

2+l and K = k0, . . . ,k n2−l= 0,1, . . . ,n+1\ J. We have

(ω(l))J =(−1)(

n2 +2

2 )+ n2 l+(l+1

2 )+σ(J)P ·dl ·d1 ·d2 · · ·dln2 ! ·FJ

· l

∑m=1

(−1)m−1gmdgm

d

n2−l∧r=0

dxkr

l∧t=1

d ftdt

+(−1)l

n2−l

∑p=0

(−1)pxkp

l∧s=1

dgs

d∧ dxkp ∧

l∧t=1

d ftdt

+(−1)n2+l

l

∑q=1

dgq

d∧ dF

d

n2−l∧r=0

dxkr ∧d fq

dq

,Proof. We proceed by induction on l:

l = 0: We have


ΩJ := ι ∂

∂x j n2

(· · · ι ∂

∂x j0

(Ω) · · ·) = (−1)j0+···+ j n

2+(

n2 +2

2 )n2

∑l=0

(−1)lxkl dxkl , (8.5)

This gives us

(ω(0)) j0··· j n2= (ω) j0··· j n

2=

(−1)(n2 +2

2 )+|J|Pn2 ! ·FJ

[ n2

∑p=0

(−1)pxkp dxkp

].

l⇒ l +1:

(ω(l))J ∧d fl+1

fl+1≡ (−1)(

n2 +2

2 )+ n2 l+(l+1

2 )+|J|Pdld1 · · ·dl+1n2 ! ·FJ · fl+1

l

∑m=1

(−1)m−1gmdgm

d

n2−l∧r=0

dxkr

l+1∧t=1

d ftdt

+(−1)l

n2−l

∑p=0

(−1)pxkp

l∧s=1

dgs

d∧ dxkp ∧

l+1∧t=1

d ftdt

+(−1)n2+l

l

∑q=1

dgq

d∧ dF

d

n2−l∧r=0

dxkr ∧d fq

dq∧ d fl+1

dl+1

+(−1)n2+l+1 fl+1

l+1

∑u=1

dgu

du

n2−l∧r=0

dxkr ∧d fu

du

.Applying δ of the Cech cohomology we get

ω(l+1)J =

(−1)(n2 +2

2 )+ n2 (l+1)+(l+2

2 )+|J|dld1 · · ·dl+1n2 ! ·FJ · fl+1

·

l

∑m=1

(−1)m−1gmdgm

d∧

( n2+l+1

∑p=0

Fjpdx jp

) n2−l−1∧

r=0

dxkr

l+1∧t=1

d ftdt

+(−1)l

( n2+l+1

∑p=0

Fjpx jp

)l∧

s=1

dgs

d

n2−l−1∧

q=0

dxkq

l+1∧t=1

d ftdt

+(−1)l+1

n2−l−1

∑p=0

(−1)pxkp

l∧s=1

dgs

d∧

( n2+l+1

∑r=0

Fjr dx jr

)∧ dxkp

l+1∧t=1

d ftdt

+(−1)n2+l

l

∑q=1

dgq

d∧ dF

d∧

( n2+l+1

∑p=0

Fjpdx jp

) n2−l−1∧

r=0

dxkr ∧d fq

dq∧ d fl+1

dl+1

+(−1)n2+l+1 fl+1

l+1

∑u=1

dgu

du∧

( n2+l+1

∑p=0

Fjpdx jp

) n2−l∧r=0

dxkr ∧d fu

du


=(−1)(

n2 +2

2 )+ n2 (l+1)+(l+2

2 )+|J|dld1 · · ·dl+1n2 ! ·FJ · fl+1

l

∑m=1

(−1)m−1gmdgm

d∧ dF

d

n2−l−1∧

r=0

dxkr

l+1∧t=1

d ftdt

+(−1)lFl∧

s=1

dgs

d

n2−l−1∧

q=0

dxkq

l+1∧t=1

d ftdt

+(−1)l+1

n2−l−1

∑p=0

(−1)pxkp

l∧s=1

dgs

d∧ dF

d∧ dxkp

l+1∧t=1

d ftdt

+(−1)n2+l

l

∑q=1

dgq

d∧ dF

d∧ dF

d

n2−l−1∧

r=0

dxkr ∧d fq

dq∧ d fl+1

dl+1

+(−1)n2+l+1 fl+1

l+1

∑u=1

dgu

du∧ dF

d

n2−l∧r=0

dxkr ∧d fu

du

.Replacing F = f1g1 + · · ·+ f n

2+1g n2+1 in the first three expressions we finish the

induction.

Proof (of Theorem 8.4.1). Using Lemma 8.1 for l = n2 +1 we get

(ω( n2+1))0···n+1 =

(−1)|J|Pdn2+1d1 · · ·d n

2+1n2 ! ·F0 · · ·Fn+1

n2+1

∑m=1

(−1)m−1gmdgm

d

n2+1∧t=1

d ftdt

+(−1)n+1

n2+1

∑q=1

dgq

d∧ dF

d∧

d fq

dq

].

Replacing F = f1g1 + · · ·+ f n2+1g n

2+1 we obtain

(ω( n2+1))0···n+1 =

(−1)|J|Pdn2+1d1 · · ·d n

2+1n2 ! ·F0 · · ·Fn+1

n2+1

∑m=1

(−1)m−1(

d−dm

d

)gm

dgm

d

n2+1∧t=1

d ftdt

+(−1)n2

n2+1

∑q=1

(−1)q fq

n2+1∧s=1

dgs

d∧

d fq

dq

,in other words

(ω( n2+1))0···n+1 =

(−1)|J|+(n2 +2

2 )Pe0 · · ·en+1n2 ! ·F0 · · ·Fn+1

n+1

∑k=0

(−1)khkdhk

ek,

where ek = deg(hk). Replacing ei ·hi =n+1

∑j=0

∂hi

∂x j·x j and dhi =

n+1

∑j=0

∂hi

∂x jdx j we obtain

(ω( n

2+1)i )0···n+1 =

(−1)(n2 +1

2 )P ·det(Jac(H))n2 ! ·F0 · · ·Fn+1

n+1

∑k=0

(−1)kxkdxk.


The theorem follows from Proposition 8.3.1, Proposition 5.8.1, and Corollary 8.2.1.ut

Note that by Theorem 8.4.1, all periods are zero if

det(Jac(H)) = 0.

For instance, if some gi is constant. This is consistent with the fact that Z will be thecomplete intersection of X with a codimension n

2 complete intersection of Pn+1, infact in this case

Z = f1 = · · ·= fi−1 = fi+1 = · · ·= f n2+1 = 0∩X .

Furthermore, the formula gives us a characterization of such algebraic cycles:

Corollary 8.4.1 Let X ⊆ Pn+1 be a smooth hypersurface given by

X = F = 0.

If δ ∈ Hn(X ,Z) is a complete intersection type algebraic cycle, then

Pδ ∈ JF if and only if δ = α · [X ∩P n2+1] in Hn(X ,Q), for some α ∈Q.

Proof. By Macaulay’s Theorem 11.2.1, Pδ ∈ JF , if and only if, P ·Pδ ∈ JF for allP∈C[x0, . . . ,xn+1](d−2)( n

2+1). Theorem 8.4.1 says this is equivalent to the vanishingof all periods for ω ∈Hn

dR(X)0. By Poincare’s duality we conclude this is equivalentto δ = α · [X ∩P n

2+1] in Hn(X ,Q), for some α ∈Q.

Remark 8.4.4 This corollary is giving us an algebraic, and hence computable, cri-terion to determine whether a complete intersection algebraic cycle is zero in ho-mology or not. This can be also done by computing the intersection number of al-gebraic cycles which is a more cumbersome process than computing the ingredientsof Theorem 8.4.1.

Remark 8.4.5 Another observation we can derive from Theorem 8.4.1 is that eachperiod is of the form (2π

√−1)

n2 times a number in a field k, where k is the small-

est field such that f1,g1, . . . , f n2+1,g n

2+1 ∈ k[x0, . . . ,xn+1], i.e. the periods belong tothe same field where we can decompose F as f1g1 + · · ·+ f n

2+1g n2+1. This was al-

ready mentioned in Deligne’s work about absolute Hodge cycles (see [DMOS82]Proposition 7.1).

Chapter 9Gauss-Manin connection: general theory

In 1958 Yu. Manin solved the Mordell conjecture over function fields and A.Grothendieck after reading his article invented the name Gauss-Manin connection.I did not find any simple exposition of this subject, the one which could be under-standable by Gauss’s mathematics. I hope that the following explains the presenceof the name of Gauss on this notion. Our story again goes back to integrals. Manytimes an integral depend on some parameter and so the resulting integration is afunction in that parameter. For instance take the elliptic integral∫

δ

Q(x)dx√P(x)

(9.1)

and assume that P and Q depends on the parameter t and the interval δ does notdepend on t. In any course in calculus we learn that the integration and derivationwith respect to t commute:

∂

∂ t

∫δ

Q(x)√P(x)

dx =∫

δ

∂

∂ t(

Q(x)√P(x)

)dx.

As before we know that the right hand side of the above equality can be writtenas a linear combination of two integrals

∫δ

dx√P

and∫

δxdx√P(x)

. This is the historical

origin of the notion of Gauss-Manin connection, that is, derivation of integrals withrespect to parameters and simplifying the result in terms of integrals which cannotbe simplified more.

9.1 De Rham cohomology of projective varieties over a ring

Recall our convention about the ring R in §?? and let T := Spec(R) be the corre-sponding variety over k.

93

94 9 Gauss-Manin connection: general theory

Remark 9.1.1 The usage of T is just for the sake of compatibility of our notationswith those in the literature.

Let X be a projective, smooth, reduced variety over k and let X→ T be a morphismof varieties over k. We also say that X is a variety over the ring R.

We have the complex of sheaves of differential forms (Ω •X/T,d) and we definethe i-th relative de Rham cohomology of X as the i-th hypercohomology of thecomplex (Ω •X/T,d), that is

H idR(X/T) =Hi(Ω •X/T,d).

We explain how the elements of the hypercohomology look like and how to calculateit.

Let U = Uii∈I be any open covering of X by affine subsets, where I is a totallyordered finite set. We have the following double complex

......

...↑ ↑ ↑

(ΩX/T)02 → (ΩX/T)

12 → (ΩX/T)

22 → ·· ·

↑ ↑ ↑(ΩX/T)

01 → (ΩX/T)

11 → (ΩX/T)

21 → ·· ·

↑ ↑ ↑(ΩX/T)

00 → (ΩX/T)

10 → (ΩX/T)

20 → ·· ·

(9.2)

Here (ΩX/T)ij is the product over I1 ⊂ I, #I1 = j+1 of the set of global sections ωσ

of Ω iX/T in the open set σ = ∩i∈I1Ui. The horizontal arrows are usual differential

operator d of (ΩX/T)iX/T ’s and vertical arrows are differential operators δ in the

sense of Cech cohomology, that is,

δ : (ΩX/T)ij→ (ΩX/T)

ij+1, ωσσ 7→

j+1

∑k=0

(−1)kωσk |σσ . (9.3)

Here σk is obtained from σ , neglecting the k-th open set in the definition of σ . Thek-th piece of the total chain of (9.2) is

L k :=⊕ki=0(ΩX/T)

ik−i

with the differential operator

d′ = d +(−1)kδ : L k→L k+1. (9.4)

The relative de Rham cohomology HkdR(X/T) is the total cohomology of the double

complex (9.2), that is,

9.2 Gauss-Manin connection 95

HkdR(X/T) :=Hk(M,Ω •X/T) :=

ker(L k d→L k+1)

Im(L k−1 d→L k).

9.2 Gauss-Manin connection

What we do in this section in the framework of Algebraic Geometry is as follows:Let X be a smooth reduced variety over R. We construct a connection

∇ : H idR(X/T)→Ω

1T⊗R H i

dR(X/T)

where Ω 1T is by definition Ω 1

R/k, that is, the R-module of differential attached to R.By definition of a connection, ∇ is k-linear and satisfies the Leibniz rule

∇(rω) = dr⊗ω + r∇ω.

A vector field v in T is an R-linear map Ω 1T→ R. We define

∇v : H idR(X/T)→ H i

dR(X/T)

to be ∇ composed with

v⊗ Id : Ω1T⊗R H i

dR(X/T)→ R⊗R H idR(X/T) = H i

dR(X/T).

If R is a polynomial ring Q[t1, t2, . . .] then we have vector fields ∂

∂ tiwhich are defined

by the rule∂

∂ ti(dt j) = 1 if i = j and = 0 if i 6= j.

In this case we simply write ∂

∂ tiinstead of ∇ ∂

∂ ti.

Sometimes it is useful to choose a basis ω1,ω2, . . . ,ωh of the R-modular H i(X/T)and write the Gauss-Manin connection in this basis:

∇

ω1ω2...

ωh

= A⊗

ω1ω2...

ωh

(9.5)

where A is a h×h matrix with entries in Ω 1T.


9.3 Construction

Let X be a variety over the ring R and T := Spec(R) Let us take a covering U =Uii∈I of X by affine open sets. In this section we need to distinguish between thedifferential operator relative to R

dR : ΩkX/T→Ω

k+1X/T

and the differential operator dk relative to k:

dk : ΩkX→Ω

k+1X

We also need to consider the double complex similar to (9.2) relative to k:

......

...↑ ↑ ↑

(ΩX)02 → (ΩX)

12 → (ΩX)

22 → ·· ·

↑ ↑ ↑(ΩX)

01 → (ΩX)

11 → (ΩX)

21 → ·· ·

↑ ↑ ↑(ΩX)

00 → (ΩX)

10 → (ΩX)

20 → ·· ·

(9.6)

We have a projection map from the double complex (9.6) to (9.2).Let ω ∈Hk

dR(X/T). By our definition ω is represented by⊕ki=0ωi, ωi ∈ (ΩX/T)

ik−i

and ωi is a collection of i-forms ωi,σσ . We have By definition we have d′Rω = 0.Since the differential map dR used in the definition of d′R is relative to R, that is, bydefinition dr = 0,r ∈ R. Now, let us take any element ω in the double complex (3.8)which is mapped into ω under the canonical projection. We apply dk on ω and theresult is not necessarily zero. However, by our choise of ω we have d′Rω = 0, andso, d′ω maps to zero in the double complex (9.2). This is equivalent to say that

d′kω = η =⊕k+1i=0 ηi, ηi ∈Ω

1T∧ (ΩX)

ik+1−i.

We map η into the wedge product of Ω 1T with the double complex (9.2) and we get

an elementη ∈Ω

1T⊗R (ΩX/T)

ik+1−i.

In this process we have replaced the notation ∧ with the tensor product ⊗R. Sinced′k d′kω = 0

(id⊗R d′R)(η) = 0.

This gives us an element in Ω 1T ⊗R Hk

dR(X/T) which is by definition ∇ω .

9.5 Geometric Gauss-Manin connection 97

9.4 Griffiths transversality

Let X/T be as before. The relative algebraic de Rham cohomology Hm(X/T) carriesa natural filtration which is called then Hodge filtration:

0 = Fm+1 ⊂ Fm ⊂ ·· ·F1 ⊂ F0 = Hm(X/T)

Its ingredients are defined by

F i = F iHmdR(X/T) = Im

(Hm(X ,Ω •≥i

X/T)→Hm(X,Ω •X/T))

Theorem 9.4.1 (Griffiths transversality) The Gauss-Manin connection maps F i

to Ω 1T⊗R F i−1, that is,

∇(F i)⊂Ω1T⊗R F i−1, i = 1,2, . . . ,m.

Proof. This follows from the definition of the Gauss-Manin connection.

9.5 Geometric Gauss-Manin connection

Let us now assume that k=C. The main motivation, which is also the historical one,for defining the Gauss-Manin connection is the following: For any ω ∈H i

dR(X) anda continuous family of cycles δt ∈ Hi(Xt ,Z) we have

d(∫

δt

ω

)=∫

δt

∇ω. (9.7)

Here, by definition ∫δt

α⊗β = α

∫δt

β ,

where β ∈ H idR(X) and α ∈ Ω 1

T . Integrating both side of the equality (9.5) over a abasis δ1,δ2, . . . ,δh ∈ Hi(Xt ,Q) we conclude that

d([∫

δ j

ωi]) = [∫

δ j

ωi] ·A. (9.8)


9.6 Algebraic vs. Analytic Gauss-Manin connection

9.7 A consequence of global invariant cycle theorem

We derive a consequence of gloabl invariant cycle theorem. For the references onthis topic see [Mov17b] §6.11.

Let Y,X ,π : U→ B as before, but defined over a field k of characteristic zero. Let

Theorem 9.7.1 Let δ be a global section of the relative algebraic de Rham coho-mology Hm

dR(U/B). If ∇δ = 0 then its evaluation δ0 at 0 ∈ B is in the image of themap

HmdR(X)→ Hm

dR(Y ).

induced by inclusion Y → X. 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(Y,Q)ρ ⊗QC.

1. Show that the above definition does not depend on the choice of covering, thatis, if U1 and U2 are two open covering of X then the corresponding hypercoho-mologies are isomorphic in a canonical way.

2. For which varieties X , we have H0dR(X) = R.

9.8 The regularity of Gauss-Manin connection

In this section we discuss the regularity of Gauss-Manin connection near the de-generacy locus of families of projective varieties. The main references for this are[Gri70] page 237, [AGZV88] Chapter 13 and [Del70].

Chapter 10Infinitesimal variation of Hodge structures

In its early phase (Abel, Riemann, Weierstrass), algebraic geometry was just a chap-ter in analytic function theory [...] A new current appeared however (1870) underthe powerfull influence of Max Noether who really put ”geometry” and more ”bi-rational geometry” into algebraic geometry [...] The next step in the same directionwas taken by Castelnuovo (1892) and Enriques (1893). They applied analogousmethods to the creation of an entirely new theory of algebraic surfaces. Their ba-sic instrument was the study of linear systems of curves on a surface. Many newbirationally invariant properties were discovered and an entirely new and beauti-ful chapter of geometry was opened. In 1902 the Castelnuovo-Enriques team wasenriched by the brilliant personality of Severi. More than his associates he wasinterested in the contacts with the analytic theory developed since 1882 by EmilePicard. The most important contribution of Severi, his theory of the base was infact obtained by utilizing the Picard number p. The theory of the great Italian ge-ometers was essentially, like Noether’s, of algebraic nature. Curiously enough thisholds in good part regarding the work of Picard. This was natural since in his timePoincare’s creation of algebraic topology was in its infancy. Indeed when I arrivedon the scene (1915) it was hardly further along. [...] I cannot refrain, however, frommention of [....] the systematic algebraic attack on algebraic geometry by OscarZariski and his school, and beyond that of Andre Weil and Grothendieck.

Gauss-Manin connection carries many information of the underlying family ofalgebraic varieties, however, in general it is hard to compute it and verify some of itsproperties. For this reason it is sometimes convenient to break it into smaller peicesand study these by their own. Infinitesimal variation of Hodge structures, IVHS forshort, is one of these pieces of the Gauss-Manin connection, and we explain it in thischapter. It was originated by the articles of Griffiths around sixties [Gri68a, Gri68b,Gri69a], and was introduced by him and his collaborators in the subsequence articles[CG80, CGGH83]. See for instance page 183 of the last article for some historicalcomments on this.

99

100 10 Infinitesimal variation of Hodge structures

10.1 Starting from Gauss-Manin connection

Recall that for a family of projective varieties X→ T we have the Gauss-Maninconnection

∇ : Hm(X/T)→Ω1T⊗OT

Hm(X/T)

We have also the Hodge filtration

0 = Fm+1 ⊂ Fm ⊂ ·· ·F1 ⊂ F0 = Hm(X/T)

and the Gauss-Manin connection satisfies the so called Griffiths transversality:

∇(F i)⊂Ω1T⊗OT

F i−1, i = 1,2, . . . ,m.

Therefore, the Gauss-Manin connection induces well-defined maps

∇i :F i

F i+1 →Ω1T⊗OT

F i−1

F i (10.1)

In Theorem 5.4.1 we have learned that

F i

F i+1∼= Hm−i(X, Ω

iX/T).

Therefore, we get the following R-linear map

∇i : Hm−i(X, ΩiX/T)→Ω

1T⊗R Hm−i+1(X, Ω

i−1X/T) (10.2)

After analysing the definition of the Gauss-Manin connection we get the followingdescription of ∇i. Let ω ∈ Hm−i(X, Ω i

X/T). It is given by a cocycle in

(ΩX/T)im−i =

(ΩX)im−i

Ω 1T∧ (ΩX)

i−1m−i

.

Let ω ∈ (ΩX)im−i such that it maps to ω under the canonical projection. We have

δ (ω) ∈Ω1T∧ (ΩX)

i−1m−i

Projecting this in Ω 1T⊗R (ΩX/T)

i−1m−i we get ∇i(ω).

Sometimes it is usefull to write (10.2) in the following way:

∇i : ΘT→ hom(

Hm−i(X, ΩiX/T),H

m−i+1(X, Ωi−1X/T)

)(10.3)

10.3 Kodaira-Spencer map I 101

10.2 Algebraic polarization

A smooth projective variety comes with an embedding X ⊂ PN , and this gives us thepolarization θ ∈ H2(X ,Z)∩H11 which is Poincare dual [Y ] ⊂ H2n−2(X ,Z), whereY is a smooth hyperplance section of X . We would like to introduce this in theframework of algebraic de Rham cohomology.

Recall that θ is the Chern class of the line bundle O(1) |X :

θ := c(O(1)).

We make the following composition

H1(X ,O∗X )→ H2(X ,Z)→ H2dR(X)

where the first one is the Chern class map and define the resulting map

c : H1(X ,O∗X )→ H2dR(X) (10.4)

in the algebraic context. Its image is in the F1 piece of the de Rham cohomology.Therefore, we will have also the map

c : H1(X ,O∗X )→ H1(X ,Ω 1X ) (10.5)

that we will denote it by the same letter c. We will call c the algebraic Chern classmap.

Proposition 10.2.1 The algebraic Chern class of a line bundle in H1(X ,O∗X ) givenby a cocycle fi ji, j∈I is given by ω = ω2

0 +ω11 +ω0

2 , where ω20 and ω0

2 are zeroand ω1

1 is given by d fi jfi ji, j∈I .

Proof.

10.3 Kodaira-Spencer map I

Let π : X→ T be a family of smooth projective varieties over k as before.

Definition 10.3.1 The sheaf ΘX/T of relative differential forms is by definition thedual of the sheaf of OX-modules Ω 1

X/T. In a small open set U ⊂ T, an elementv ∈ΘX/T(U) is induced by a OX(U)-linear map Ω 1

X/T(U)→ OX (U).

Now we define the Kodaira-Spencer map

K : H0(T,ΘT)→ H1(X,ΘX/T) (10.6)

For a global vector field v in T we define K(v) in the following way. We choose anacyclic covering U = Uii∈I and vector fields vi in Ui such that vi is mapped to v


under X→ T. The vector fields vi− v j are tangent to the fibers of X→ T and theygive us K(v).

The Kodaira-Spencer map K is related to the connecting homomorphism of thelong exact sequence attached to the short exact sequence:

0→ΘX/T→ ΘX→ π∗ΘT→ 0 (10.7)

where ΘX ⊂ΘX is by definition the sheaf of vector fields in X which are mapped tovector fields in T, and π∗ΘT is a sheaf in X such that its section in a Zariski openset X are just the elements of ΘT.

Therefore, we have

H0(X,π∗ΘT) = H0(T,ΘT), H1(X,π∗ΘT) = H1(T,ΘT) (10.8)

The long exact sequence of (10.7) turns out to be

· · · →H0(X,ΘX)→H0(T,ΘT)K→H1(X,ΘX/T)→H1(X,ΘX)→H1(T,ΘT)→ ···

(10.9)We conclude that

Proposition 10.3.1 Assume that H1(T,ΘT) = 0. We have H1(X,ΘX) = 0 if andonly if the Kodaira-Spencer map is surjective.

For the main purpose of the present book, it would be essential to compute H i(X,ΘX), i=0,1.

Let us now consider the parameter space T of smooth hypersurfaces X ⊂ Pm+1.In this case T is the affine space Aa

k mines a discriminant locus ∆ = 0. It has thecoordinate system (tα ,α ∈ I), where

I :=(α0,α1, . . . ,αm+1) | 0≤ αe ≤ d, ∑αe = d

The polynomial expressions of ∆ in terms of the variables tα is in general huge. Thevariety X⊂ Pm+1×T is given by

X : g = 0.

where g := ∑α∈I

tα xα = 0

and X→ T is the projection on T. Let ∂

∂ tα, α ∈ I be canonical vector fields in T.

Since the fibers of X→ T are smooth, we can take the Jacobian covering U =Ui j=0,1,2,...,m+1 of X, where

U j :∂g∂x j6= 0.

In the affine open set Ui a pull-back of ∂

∂ tαis given by

10.4 Kodaira-Spencer map II 103

∂

∂ tα−

∂g∂ tα∂g∂x j

∂

∂x j

Therefore, the Kodaira-Spencer map is given by

K(∂

∂ tα) :=

∂g∂ t

((

∂g∂x j

)−1 ∂

∂x j− (

∂g∂xi

)−1 ∂

∂xi

)i, j=0,1,...,m+1

Proposition 10.3.2 For the universal family of hupersurfaces X → T, we haveH1(X,ΘX) = 0 except for hypersurface of dimension two and degree four. In thisexceptional case it is a one dimensional Q-vector spaces.

This proposition will be proved in the next section.

10.4 Kodaira-Spencer map II

Most of the time we take a point 0 ∈ T, set X := X0 and specialize (10.6) at a point0 ∈ T:

K : TT,0→ H1(X ,ΘX ) (10.10)

In the literature we mainly find this map. The next discussion is take from Voisin’sbook [Voi03] Lemma 6.15. We start with

0→ TX → TX|X → NX⊂X → 0↓ ↓ ↓

0→ TX → TPn+1 |X → NX⊂Pn+1 → 0(10.11)

Here, NA⊂B is the normal bundle of A inside B, the first down arrow map is theidentity, the second is the derivation of the projection X→ Pn+1 and the third is themap induced in the quotient. We note that NX⊂X is the trivial bundle in X . In fact,a trivialization of this bundle is given by the restriction of the derivation of the mapX→ T to the points of X . Therefore, we have canonical identifications

H0(X ,NX⊂X) = TT,0,

H1(X ,NX⊂X) = TT,0⊗k H1(X ,OX ).

Now we write the long exact sequence of (10.11) and we use the above data

H0(X , TX|X ) → TT,0K→ H1(X ,TX ) → H1(X , TX|X ) → TT,0⊗k H1(X ,OX )

↓ ↓ ↓ ↓ ↓H0(X , TPn+1 |X )

b→ H0(X ,NX⊂Pn+1 )c→ H1(X ,TX ) → H1(X , TPn+1 |X ) → H1(X ,NX⊂Pn+1 )

↑ aH0(X ,TPn+1 )

(10.12)


The last up arrow map is just the restriction map that we will need later.Let us now focus on the case of hypersurfaces X in Pn+1 given by the homoge-

neous polynomial g ∈ k[x] of degree d. In this case we have many canonical identi-fications and vanishings so that (10.13) becomes

H0(X , TX|X ) → k[x]dK→ H1(X ,TX ) → H1(X , TX|X ) → 0

↓ ↓ ↓ ↓ ↓H0(X , TPn+1 |X )

b→ k[x]dc→ H1(X ,TX ) → H1(X , TPn+1 |X ) → 0

↑ a〈xi

∂

∂x j, i, j = 1,2, . . . ,n+1〉

(10.13)

Let us explain the details of this. The identification

TT,0 = k[x]d

is done in the following way. For p∈ k[x]d we first consider the curve g+ t p, t ∈Ak

whose derivation at t = 0 gives the corresponding vector in TT,0. We have also

H0(X ,NX⊂Pn+1)∼= H0(Pn+1,OPn+1(d))∼= k[x]d

The map TT,0 → H0(X ,NX⊂Pn+1) turns out to be the identity map. The k-vectorspace H0(X ,TPn+1) is generated by x j

∂

∂xiand the composition ba after these iden-

tifications is

x j∂

∂xi→ x j

∂g∂xi

We will need the following:

Theorem 10.4.1 (Bott, [Bot57]) For a smooth hypersurface X in Pn+1 we have

H1(Pn+1,TPn+1(X)) = 0,

where TPn+1(X) is the sheaf of vector fields in Pn+1 vanishing along X. Further, ifn 6= 2 and the degree of X is not 4 then

H1(X ,TPn+1 |X ) = 0

The first part of Bott’s theorem implies that the maps a and c are surjective. Finallywe get

Theorem 10.4.2 For a fixed hypersurface X ⊂ Pn+1 given by the homogeneouspolynomial g we have the Kodaira-Spencer map

(C[x]/jacob(g))dK→ H1(X ,ΘX )

given by

K(xα) :=

xα

((

∂g∂x j

)−1 ∂

∂x j− (

∂g∂xi

)−1 ∂

∂xi

)i, j=0,1,...,m+1

10.6 IVHS for hypersurfaces 105

for xα ∈ (C[x]/jacob(g))d . For (n,d) 6= (2,4) it is an isomorphism and for (n,d) =(2,4) it is an injection whose image is of codimension 1 in H1(X ,ΘX ).

10.5 A theorem of Griffiths

Under the assumption that H1(X,ΘX) = 0, we are going to define a canonical map

∇i : H1(X,ΘX/T)→ hom(

Hm−i(X, ΩiX/T),H

m−i+1(X, Ωi−1X/T)

)(10.14)

Let v= vi j∈H1(X,ΘX/T) and ω ∈Hm−i(X, Ω iX/T). We want to define ∇i(v)(ω)∈

Hm−i+1(X, Ωi−1X/T). We take an acyclic covering U := Uii∈I and a cocycle

ω ∈ (ΩX)im−i which maps to ω under the canonical projection Ω i

X → Ω iX/T. The

ingredients of δω are sections of the sheaf Ω 1T∧Ω

i−1X .

We have H1(X,ΘX) = 0 and ΘX/T ⊂ΘX. Therefore, we have vector fields vi inUi such that vi j = v j− vi. In any intersections U0∩U1∩·· ·∩Uim ∩Um−i+1 we have

0 = ivi j(δω) = iv j(δω)− ivi(δω)

The first equality is in Ωi−1XT . Therefore, iv j(δω) does not depend on the choice of

j. This gives us the desired element in Hm−i+1(X, Ωi−1X/T).

Theorem 10.5.1 (Griffiths) The map (10.3) factors through the Kodaira-Spencermap (10.6) and one gets the map ∇i, that is,

∇i = ∇i K

Proof. This follows from the definition of ∇i and ∇i.

In many situtation we want to use

∇i : H1(X ,ΘX )→ hom(Hm−i(X , Ω

iX ),H

m−i+1(X , Ωi−1X )

)(10.15)

which is (10.14) over a fixed fiber X =Xt of X→T. Since we have used H1(X,ΘX)=0, we cannot simply take T as a one point variety and proceed the definition of ∇ias before.

10.6 IVHS for hypersurfaces

Let us consider a hyper surface X⊂ Pm+1 given by the homogeneous poynomial g.Griffiths theorem gives a basis of the algebraic de Rham cohomology Hm

dR(Pm+1−

X). Recall that we have a long exact sequence


· · · → Hn+1dR (Pn+1)→ Hn+1

dR (Pn+1−X)→ HndR(X)→ Hn+2(Pn+1)→ ···

For n odd we have HndR(X)0 = Hn

dR(X) and for n even HndR(X) = Hn

dR(X)0⊕ θn2 ,

where θ ∈ H2dR(X) is the polarization of X . Under the residue map Hn+1

dR (Pn+1−X)→ Hn

dR(X)0 is an isomorphism. Since θn2 ∈ F

n2 Hn

dR(X), we get the followingidentifications

Hk(X ,Ω n−kX )0 ∼= (C[X ]/J)(k+1)d−n−2, k = 0,1, . . . ,m. (10.16)

For n even and k = n2 , we have Hk(X ,Ω n−k

X ) = Hk(X ,Ω n−kX )0 +θ

n2 . For all other n

and k, Hk(X ,Ω n−kX )0 = Hk(X ,Ω n−k

X ).Assume that the polynomial g depends on a parameter t. According to the defi-

nition of the Gauss-Manin connection, we have

∇ ∂

∂ t

(Pηα

gk

)=

(−k

∂g∂ t Pηα

gk+1

)⊗dt (10.17)

We have (n+1

∑i=1

Ai∂g∂xi

)ηα

gk+1 =1k

(n+1

∑i=1

∂Ai

∂xi

)ηα

qk + exact terms. (10.18)

We conclude that

Proposition 10.6.1 The infinitesimal variation of Hodge structures

H1(X ,ΘX )×Hm−k(X ,Ω kX )→ Hm−k+1(X ,Ω k−1

X )

for a hypersurface X : g = 0 of degree d and dimension n is given by the multiplica-tion of polynomials

(C[X ]/J)d×(C[X ]/J)(k+1)d−m−2→ (C[X ]/J)(k+2)d−m−2, (F,G) 7→ FG, (10.19)

provided that (n,d) 6= (2,4). In this exceptional case, the same statement is true ifwe replace H1(X ,ΘX ) with the image of the Kodaira-Spencer map.

10.7 Griffiths-Dwork method

We can use Theorem ?? and find a basis of HmdR(P

m+1−X). Applying the Gauss-Manin connection to the elements of this basis, we can write the right hand side of(10.17) in terms of the basis. For this we may use (10.18) in order to reduce the poleorder. This is mainly known as Griffiths-Dwork method.

10.8 Noether-Lefschetz theorem 107

10.8 Noether-Lefschetz theorem

Theorem 10.8.1 (Noether-Lefschetz theorem) For d ≥ 4 a generic surface X ⊂P3 of degree has Picard group Pic(X) ∼= ZO(1), that is, every curve C ⊂ X is acomplete intersction of X with another surface.

Let PN be the projectivization of the parameter space of surfaces in P3. Here by ageneric surface we mean that there is a countable union V of proper subvarieties ofPN such that X is in PN−V .

For the proof of Noether-Lefschez theorem, we need varieties which parameter-ize all curves in P3. This is done using Hilbert schemes. Let us consider the set Hg,nof all curves of genus g and degree d in P3. We know that Hg,n is a projective variety,see for instance [ACG11]. It maynot be irreducible. We call Hg,n the Hilbert schemeof degree n and genus g curves in P3. Let also T be the space of surfaces of degreed surfaces in P3. we have the following incidence variety

Σg,n := (C,X) ∈ Hg,n×T |C ⊂ X

which is again a projective variety. For particular classes of (g,n), we have an irre-ducible component Σg,n which parametrizes the pairs (C,X), where C is a completeintersection of X with another surfaces. The closure Σg,n of Σg,n− Σg,n is still a pro-jective variety and the projection Σg,n→ T is a proper map. The image of this mapis a finite union of irreducible subvarieties of T. The union of all these for all (g,n)is called the Noether-Lefschetz loci in T.

Proof. We give two proofs. The first one is topological and is due to Lefschetz. Thesecond one uses IVHS for surfaces.

First proof: If the theorem is not true then there is a cycle δ ∈ H2(X ,Z) which isinvariant under the monodromy. This is not possible except when δ is the homologyclass of a complete intersection of X with another surface.

Second proof: Let us assume the theorem is not true. This means that there issome g and n and a component Σ of Σg,n such that the projection π : Σ → T issurjective. For each t ∈ T we want to choose a point in π−1(t), and hence, a curveCt ⊂ Xt which varies continously with t. We choose a smooth point t ∈ Σ such thatthe drivative of π at t is surjective. By implicit function theorem, we can choose ananalytic subvariety A ⊂ Σ such that πA is a biholomorphism between A and (Σ , t).This gives us an an analytic family (Xt ,Ct) of hypersurfaces Xt and curves Ct ⊂ Xtfor t in an open set of T. For the rest of our argument, we only need the topologicalclass δt := [Ct ] ∈ H2(Xt ,Z) of Ct . We have find a continous family of topologicalcycles δt , t being in an open subset of T, such that,

∫δt

ω = 0, for all global 2-formsω in Xt . We get ∫

δt

∇ ∂

∂ tω = 0, ∀ ∂

∂ t∈ΘT, ω ∈ H0(X ,ΩX ).

For a fixed X = Xt among this family the map


H1(X ,ΘX )×H0(X ,Ω 2X )→ H1(X ,Ω 1

X )0

is surjective. For d = 4 we use the image of the Kodaira-Spencer map insetad of thefull H1(X ,ΘX ). Therefore, H0(X ,Ω 2

X )+ΘTH0(X ,Ω 2X ) and its complex conjugate

generate the whole primitive cohomology H2dR(X)0. Since we have∫

δt

H0(X ,Ω 2X )+ΘTH0(X ,Ω 2

X )+H0(X ,Ω 2X )+ΘTH0(X ,Ω 2

X ) = 0

we conclude that δt is the Poincare dual to the polarization θ . This is the same asto say that Ct is homolog to an a curve Dt which is a complete intersection. SinceH1(X ,OX ) = 0, we conclude that Ct −Dt is a zero divisor of a function on X .

10.9 Algebraic deformations

Let us now consider the following IVHS

∇ : H1(X ,ΘX )×H1(X ,Ω 1X )→ H2(X ,Ω 0

X ).

The polarization θ is originally defined as an element of H2(X ,Z), therefore, if X ⊂PN depends on a parametr then it is a flat section of the Gauss-Manin connection.Therefore, it is natural to define

H1(X ,ΘX )θ :=

a ∈ H1(X ,ΘX ) | ∇(a,θ) = 0,

(10.20)

In the case of a hypersurface X ⊂ Pn+1, by Lefschetz theorems we know that forn≥ 3 we have H2(X ,Ω 0

X ) = 0. Therefore, in this case we have

H1(X ,ΘX )θ = H1(X ,ΘX ),

For n = 2, the dimension of H2(X ,Ω 0X ) is the Hodge number h20, and so, the

*****A hypersurface X ⊂ P3 of degree 4 is called a K3 surface. Using Serre duality

we haveH1(X ,ΘX )∼= H1(X ,Ω 1

X )

Note that Ω 1X is dual to ΘX and Ω 2

X is the trivial line bundle. We find that the dimen-sion of H1(X ,ΘX ) is the hodge number h11 of X . This is h11 = 20. From another sidedim(C[x]/J)4 = 19. We conclude that the complex moduli space of a K3 surface isof dimension 20. Algebraic deformations correspond to a 19 dimensional subspaceof this space.

Exercises1. Prove that the vector space H0(Pn,ΘPn ) of global vector fields in Pn is generated by xi

∂

∂x j,

where xi’s are coordinates of Pn. Therefore, it is of dimension n2.

Chapter 11Hodge cycles and Artinian Gorenstein rings

11.1 Intoduction

In this chapter to any Hodge cycle of a smooth hypersurface we attach an ArtinianGorenstein ring, and in this way many problems related to Hodge cycles, boils downto problems in commutative algebra. Some of the material of the present text areinspired from [Otw02, Otw03, Voi03, Dan14, MV17].

11.2 Artinian Gorenstein rings

Definition 11.2.1 Let n ∈ N, and I ⊆ C[x0, ...,xn+1] be a homogeneous ideal. Wesay I is Artinian Gorenstein if R := C[x0, ...,xn+1]/I for some σ ∈ N satisfies

(i) dimCRσ = 1.(ii)For every 0≤ i≤ σ the multiplication map

Ri×Rσ−i→ Rσ

is a perfect pairing.(iii)Re = 0 for e > σ .

We also say that R is a Gorestein ring. The number σ is called the socle of I and R.In the literature, Rσ is mainly called the socle of R.

The above definition is mainly inspired from the following example.

Theorem 11.2.1 (Macaulay [Mac16]) Let f0, ..., fn+1 ∈C[x0, ...,xn+1] be homoge-neous polynomials with deg( fi) = di and

f0 = · · ·= fn+1 = 0=∅⊆ Pn+1.

The following

109

110 11 Hodge cycles and Artinian Gorenstein rings

R :=C[x0, ...,xn+1]

〈 f0, ..., fn+1〉,

is an Artinian Gorenstein ring of socle σ := ∑n+1i=0 (di−1).

For a proof see [Voi03, Theorem 6.19].

Example 11.2.1 Let f ∈ C[x0,x1, · · · ,xn+1] be a homogeneous polynomial of de-gree d such that its Jacobian ideal jacob( f ) := 〈 ∂ f

∂x0, ∂ f

∂x1, · · · ∂ f

∂xn+1〉 has an isolated

singularity at the origin 0 ∈ Cn+2. This is our main example of an Artinian Goren-stein ideal which is of socle (n+2)(d−2). For f := xd

0 + xd1 + · · ·+ xd

n+1 this is

jacob( f ) = 〈xd−10 ,xd−1

1 , · · · ,xd−1n+1〉.

Definition 11.2.2 For an ideal I of a ring R then the quotient ideal is defined asfollows:

(I : P) := Q ∈ R : PQ ∈ I.

The following elementary proposition will be used frequently.

Proposition 11.2.1 We have

1. If I is Artinian Gorenstein with socle σ , and P ∈C[x0, ...,xn+1]µ \ Iµ , then (I : P)is Artinian Gorenstein of socle σ −µ .

2. If I1, I2 are two Artinian Gorenstein ideals with the same socle σ and (I1)σ =(I2)σ then I1 = I2.

Proof. The proof of the first statement is as follows. Let R = C[x0, · · · ,xn+1]. Thefirst item in the definition of an Artinian Gorenstein ring for R/(I : P) follows fromthe fact that (R/I)µ × (R/I)σ−µ is a perfect pairing. This also implies that (R/(I :P))a× (R/(I : P))σ−µ−a → (R/(I : P))σ−µ is a perfect pairing. The third item istrivial.

For the second item we first observe that I1 ∩ I2 is also Artinian Gorenstein. Wethen prove the same statement with the additional hypothesis I1 ⊂ I2. We only needto prove that the pairing (C[x]/I1∩ I2)a×(C[x]/I1∩ I2)σ−a→ (C[x]/I1∩ I2)σ is per-fect. If it not then we have P∈ (C[x]a with P 6∈ (I1)a or P 6∈ (I2)a whose product withall Q ∈ C[x]σ−a is in (I1)σ = (I2)σ . If for instance, P 6∈ (I1)a then this contradictsthe same property for I1. Now let us assume that I⊂I2. If Q ∈ I2 and Q 6∈ I1 then Qgives a non-zero element in R/I1 and so there is an element P ∈ R/I1 such that PQis a non-zero element of the socle (R/I1)σ . This is equal to (R/I2)σ which impliesthat PQ is zero, and hence, we get a contradiction.

ut

Remark 11.2.1 The intersection of two Artinian Gorenstein ideal I1, I2 is not nec-essarily Artinian Gorenstein. For instance, consider the case in which I1 and I2 havethe same socle and (I1)σ 6= (I2)σ which implies that dimC(C[x]/I1∩ I2)σ = 2. Notealso that for two polynomials P,Q ∈ C[x]µ and an Artinian Gorenstein ideal I wehave the inclusion

(I : P)∩ (I : Q)⊂ (I : P+Q)

11.3 Artinian Gorenstein ring attached to Hodge cycles 111

which is a strict inclusion if (I : P)σ−µ 6= (I : Q)σ−µ .

11.3 Artinian Gorenstein ring attached to Hodge cycles

Let X ⊂ Pn+1 be a smooth hypersurface of degree d and dimension n, and

σ := (n2+1)(d−2).

Let also Z∞ be the intersection of a linear P n2+1 with X and [Z∞] ∈ Hn(X ,Z) be the

induced element in homology (the polarization). Since the intersection pairing inHn(X ,Z) is non-degenerate, we have isomorphism

Hn(X ,Z)/Z[Z∞]∼= Hn(X ,Z)0 := δ ∈ Hn(X ,Z)|δ · [Z∞] = 0.

It is also well-known that Hn(X ,Z) has no torsions, see for instance [Mov17b, §5.5]

Definition 11.3.1 Any Hodge cycle δ ∈ Hn(X ,Z)/ defines a ring R(δ ) whose a-thgraded piece for a≤ σ is R(δ )a := C[x]a/I(δ )a, where

I(δ )a :=

Q ∈ C[x]a

∣∣∣∣∣∫

δ

QPΩ

fn2+1 = 0, ∀P ∈ C[x]σ−a

.

By definition (I(δ )m = C[x]m for all m≥ σ +1 and so Ra = 0.

Let Pδ ∈ C[x]( n2−2)(d−2) be such that

δpd =

[Pδ Ω

fn2+1

]

Proposition 11.3.1 We have

I(δ ) = [jacob( f ) : Pδ ]

and hence I(δ ) is Artinian Gorenstein of socle σ := ( n2 +1)(d−2).

Proof. We need to show that jacob(δ )⊂ I(δ ). This follows from the equality

dω

f i−1 = (i−1)d f ∧ω

f i−1 +d(

ω

f i−1

)for i = n

2 +1, and the fact that δ is Hodge. ut

Proposition 11.3.2 If δ = [Z] and Z is given by the ideal IZ then

IZ,d ⊂ (Iδ )d . (11.1)


In particular, if the primitive part of the cycles [Z1], [Z2], . . . , [Zk] form a one dimen-sional sybspace of Hn(X ,Q)0 then

k

∑i=1

IZi,d ⊂ (Iδ )d . (11.2)

Proof. This follows from Carlson-Griffiths Theorem used in [MV17]. ut

Definition 11.3.2 An algebraic cycle Z1 is called perfect if there are other algebraiccycles Zi, i = 2, . . . ,k as in the above definition such that (11.2) is an equality.

We have the following natural isomorphism of one dimensional vector spaces:

H(δ ) : Rσ → C, P 7→ 1(2πi)

n2

∫δ

PΩ

fn2+1 (11.3)

and hence we get:H(δ ) : Ra×Rσ−a→ Rσ

∼= C.

Proposition 11.3.3 If δ = [Z] is an algebraic cycle, and both Z and X are definedover a field k⊂ C then I(δ ), R(δ ) and H(δ ) are defined over an algebraic k.

Proof. It follows from Proposition 5.10.1 that Pδ ∈ k[x]. The Jacobian ideal is de-fined over k and so I(δ ),R(δ ) are defined over k. In a similar way, because of Propo-sition 5.10.1 H(δ ) is defined over k.

Knowing Proposition 11.3.3, the following is a consequence of the Hodge conjec-ture.

Conjecture 11.3.1 If f is defined over a field k⊂ C then I(δ ),R(δ ) and H(δ ) aredefined over a finite algebraic extension of k.

It seems that the algebraic extension in conjecture 3.6 highly depends on the under-lying variety X . The following particular case might indicate some general phenom-ena.

Theorem 11.3.1 (P. Deligne, [DMOS82]) If f is the Fermat polynomial, and hencedefined over Q, then I(δ ),R(δ ) and H(δ ) are defined over an abelian extension ofQ.

Proof.

A major problem in our way is that for a generic f there is no no-zero primitiveHodge cycle, and we might be interested to translate this into non-existence of cer-tain Artinian Gorenstein rings of socle degree σ for such polynomials. Note thatConjecture 11.3.1 and Theorem 11.3.1 are the only manifestation of the fact that δ

has coefficients in Z.

11.4 Sum of two algebraic cycles 113

11.4 Sum of two algebraic cycles

Let Z and Z be two complete intersection algebraic cycles given by the idealsIZ = 〈 f1, f2, . . . , f n

2+1〉 and IZ = 〈 f1, f2, . . . , f n2+1〉, respectively. Let also δ :=

r[Z]+ r[Z] ∈ Hn(X ,Z)0 be the corresponding Hodge cycle. We define

IZ+Z :=⊕

Radical(IZ ·IZ

),

where the sum is over all possible replacements of companions of the generators ofIZ and IZ . By Proposition 11.3.2 we know that

IZ+Z ⊂ Iδ .

If the the ring C[x]/IZ+Z is Artinian Gorenstein and has socle degree σ := ( n2 +

1)(d− 2) then we expect that Vr[Z]+r[Z] = VZ ∩VZ and hence the alternative Hodgeconjecture is true for (X ,rZ + rZ). Our hypothesis implies that IZ+Z = Iδ and weneed to check that (IZ+Z)d is the tangent space of VZ ∩VZ . We will analyze this sit-uation in the following particular case. We follow the notations in [Mov17b] §17.9.

Let X ⊂ Pn+1 be a smooth hypersurface given by the homogeneous polynomial

f = f1 fs+1 + f2 fs+1 + · · ·+ fm fs+m + fm+1 fs+m+1g1 + · · ·+ fs f2sgs−m.

with s = n2 +1. We have the algebraic cycles:

Z : f1 = f2 = · · ·= fm = fm+1 = · · ·= fs = 0,Z : f1 = f2 = · · ·= fm = fs+m+1 = · · ·= f2s = 0.

Proposition 11.4.1 The ideal IZ+Z in C[x] contains

fi, 1≤ i≤ m, (11.4)fi, i = s+1≤ i≤ s+m, (11.5)fi f j, m+1≤ i≤ s, s+m+1≤ j ≤ 2s, (11.6)fk fs+m+ jg j, 1≤ j ≤ s−m, s+m+1≤ k ≤ 2s (11.7)fk fm+ jg j, 1≤ j ≤ s−m, m+1≤ k ≤ s (11.8)fig j, 1≤ j ≤ s−m, i ∈ s+m+ j,m+ j (11.9)

Proof. The idea is similar to the case of a single complete intersection algebraiccycle treated in §12.3. We start with the radical I of the ideal of Z+ Z which contains(11.4) and (11.6). As before, we can replace any polynomial in the ideal of Z andZ with its companion and add it the ideal to I. Replacing each element in (11.4)with its companion in both Z and Z we get the elements (11.5). All other companionsubstitution will give us (11.7) and (11.8). There elements among these polynomials


which are of the form a2b. Knowing that each time we take radical of the ideal weget elements (11.9). ut

Let I∗Z+Z be the ideal generated by the polynomials (11.4), (11.5), (11.6) with j− i=s and (11.9). These are exactly the companion of single fi or gi’s in f . We couldassume that fi and gi’s are irreducible and so these are companion of irreducibleingredients of f . The tangent space of VZ+Z at a generic point is given by the d-thhomogeneous piece of I∗Z+Z and we have the inclusions:

J ⊂ I∗Z+Z ⊂ IZ+Z ⊂ I[Z]+[Z]. (11.10)

We reduce the verification of alternative Hodge conjecture in an example to thefollowing purely commutative algebra problem. For simplicity we have only con-sidered the case m = s, that is, the cycles Z and Z do not intersects each other, andit can be formulate easily for arbitrary m.

Problem 11.4.1 Let f ∈ C[x] = C[x0,x1, · · · ,xn+1] be a homogeneous polynomialof degree d and of the following format

f = f1 fs+1 f2s+1 + f2 fs+2 f2s+2 + · · ·+ · · ·+ fs f2s f3s, s :=n2+1

where fi’s are homogeneous and we have assumed that n is even. Let I∗ be the idealin C[x] generated by fi fs+i, fi f2s+i, fs+i f2s+i, i = 1,2, . . . ,s and J be the Jacobianideal of f , that is, it is generated by ∂ f

∂xi, i = 0,1,2, . . . ,n+ 1. We have J ⊂ I∗ and

we assume that f = 0 is smooth and hence by Macaulay theorem C[x]/J is ArtinianGorenstein of socle degree σ := (n+2)(d−2). Show that if there is a third ideal Isuch that J ⊂ I∗ ⊂ I and C[x]/I∗ is Artinian Gorenstein of socle degree σ

2 = ( n2 +

1)(d−2) then for large d, the degree d pieces of I and I∗ are equal.

Chapter 12Some components of Hodge loci

In this chapter we aim to gather all well-known components of the Hodge locus inthe case of hypersurfaces. Some of these components are conjectural and for somewe have proofs.

12.1 Hodge locus

Definition 12.1.1 Let Y be a smooth projective variety. A Hodge class is any el-ement in the intersection of the rational cohomology Hm(Y,Q) ⊂ Hm

dR(Y ) andF

m2 ⊂ Hm

dR(Y ), where Fm2 = F

m2 Hm

dR(Y ) is the m2 -th piece of the Hodge filtration

of HmdR(Y ).

Therefore, the Q-vector space of Hodge classes is simply the intersection Hm(Y,Q)∩H

m2 ,m2 =Hm(Y,Q)∩F

m2 . Now, let Y →V be a family of smooth complex projective

varieties (Y ⊂ PN ×V and Y → V is obtained by projection on the second coordi-nate).

Definition 12.1.2 Let Fm2 Hm

dR(Y/V ) be the vector bundle of Fm2 pieces of the

Hodge filtration of HmdR(Yt), t ∈ V . The locus of Hodge classes is the subset of

Fm2 Hm

dR(Y/V ) containing all Hodge classes.

Note that Fm2 Hm

dR(Y/V ) is an algebraic bundle, however, the locus of Hodge classesis a union of local analytic varieties. Now, we define the Hodge locus in V itself.

Definition 12.1.3 The projection of the locus of Hodge classes under Fm2 Hm

dR(Y/V )→V is called the Hodge locus in V . An irreducible component H of the Hodge locusin a (usual) neighborhood of a point t0 ∈ V is characterized in the following way.It is an irreducible closed analytic subvariety of (V, t0) with a continuous family ofHodge classes δt ∈ Hm(Yt ,Q)∩H

m2 ,m2 in varieties Yt , t ∈ H such that for points t in

a dense open subset of H, the monodromy of δt to a point in a neighborhood (in theusual topology of V ) of t and outside H is no more a Hodge class.

115

116 12 Some components of Hodge loci

The following consequence of the Hodge conjecture is well-known:

Conjecture 12.1.1 Let Y → V be a family of smooth projective varieties definedover a field k ⊂ C. All the components of the locus of Hodge classes are algebraicsubsets of F

m2 Hm

dR(Y/V ) defined over the algebraic closure of k.

In particular, the components of the Hodge locus in V are also algebraic. The alge-braicity statement has been proved by Cattani, Deligne and Kaplan.

Theorem 12.1.1 (Cattani-Deligne-Kaplan, [CDK95]) The irreducible componentsof the locus of Hodge classes in F

m2 Hm

dR(Y/V ) are algebraic sets.

The main ingredient of their proof is Schmid’s nilpotent orbit theorem in [Sch73]together with some results in [CKS86]. All these are purely transcendental methodsin algebraic geometry, and hence, their proof does not give any light into the secondpart of the Conjecture 12.1.1, that is, any component of the locus of Hodge classesis defined over the algebraic closure of the base field k.

The algebraicity statement for the locus of Hodge classes is slightly stronger thanthe same statement for the Hodge locus. Let us explain this. We take an irreduciblecomponent H of the Hodge locus. Above each point t ∈ H we have a Hodge classβ and the above theorem implies that the action of the monodromy representationπ1(H, t)→Aut(Hm(Yt,Q)) on β produces a finite number of cohomological classes(which are again Hodge classes). This topological fact does not follow just fromthe algebraicity of H. We will work with Hodge cycles which lives in homologiesin comparison with Hodge classes which live in cohomologies. Both notions arerelated to each other by Poincare duality.

12.2 Hodge locus for hypersurfaces

Let T ⊂ C[x]d be the parameter space of smooth hypersurface of degree d and di-mension d in Pn+1. For t ∈ T we have the hypersurface Xt given by ft ∈ C[x]d . Wefix 0∈T. Recall the following definition of Hodge locus from [Mov17b, §16.5]. Fora Hodge cycle δ ∈Hodgen(X ,Z)0, let δt ∈Hn(Xt ,Z)0 be the monodromy of δ to thehypersurface Xt . Let OT,0 be the ring of holomorphic functions in a neighborhoodof 0 in T. We have the elements

FP,i(t) :=∫

δt

PΩ

f i ∈ OT,0, P ∈ C[x]σ−( n2−i+1)d , i = 1,2, . . . ,

n2.

Definition 12.2.1 The (analytic) Hodge locus passing through 0 and correspondingto δ is the analytic variety

Vδ :=

t ∈ (T,0)

∣∣∣∣∣ FP,i(t) = 0, ∀P ∈ C[x]σ−( n2−i+1)d , i = 1,2, . . . ,

n2

.

(12.1)

12.3 Complete intersection algebraic cycle 117

We consider it as an analytic scheme with

OVδ:= OT,0

/⟨P ∈ C[x]σ−( n

2−i+1)d , i = 1,2, . . . ,n2

⟩. (12.2)

In the two dimensional case, that is dim(Xt) = 2, the Hodge locus is usually calledNoether-Lefschetz locus.

The main reason for the introduction of Artinian Gorenstein rings in Chapter 11 isthe following theorem.

Theorem 12.2.1 Let Vδ ⊂ (T,0) be the Hodge locus passing through 0 and corre-sponding to δ = δ0. The Zariski tangent space of Vδ at 0 is canonically identifiedwith (Iδ )d .

Proof. This follows by our definition of Hodge locus above. Note that

∂

∂ tα

∫δt

PΩ

f i =

∫δt

QPΩ

f i+1 , Q :=− ∂ f∂ tα

. (12.3)

This implies that the linear part of FP,i for i < n2 is zero and only the linear parts of

FP, n2

contribute to the tangent space of Vδ at 0 ∈ T. ut

Im [Voi02] terminology, t∇(δ pd) = (Iδ )d is the Zariski tangent space of Vδ at 0.

12.3 Complete intersection algebraic cycle

Assume that n≥ 2 is even and f ∈ C[x]d is of the following format:

f = f1 f n2+2 + f2 f n

2+3 + · · ·+ f n2+1 fn+2, fi ∈ C[x]di , f n

2+1+i ∈ C[x]d−di , (12.4)

where 1≤ di < d, i = 1,2, . . . , n2 +1 is a sequence of natural numbers.

Definition 12.3.1 We say that the fi and f n2+1+i are companion of each other. The

notation fi is used to denote either fi or f n2+1+i.

In this section we aim to prove the following.

Theorem 12.3.1 The subvariety Td of T is a component of the Hodge locus. Inother words, there is a Zariski open (and hence dense) subset U of Td such thatfor all t ∈ U and a complete intersection algebraic cycle Z ⊂ X := Xt as above,deformations of Z as an algebraic cycle and Hodge cycle are the same.

This theorem is proved in [Dan14, Theorem 1.1] in which the author assumes d >deg(Z) which is not necessary. The computational proof for particular examples of nand d is done in [MV17]. It has the advantage that it works for other algebraic cycleswhich are not complete intersections, see Chapter 12. The main result in [Otw03]


implies Theorem 12.3.1 for very large degrees, however, the lower bound in thisarticle is not explicit and cannot be applied for a given degree.

Proof. Let X ⊂ Pn+1 be the hypersurface given by f = 0 and Z ⊂ X be the algebraiccycle given by

Z : f1 = f2 = · · ·= f n2+1 = 0.

In Hn(X ,Z) the homology classes of all cycles

g1 = g2 = · · ·= g n2+1 = 0, gi ∈ fi, f n

2+1+i

are equal up to sign and up to Z[Z∞]. Let us denote it by δ . This with Proposition11.3.2 imply that Z is perfect and

Jd := 〈 f1, f2, . . . , fn+2〉d ⊂ (Iδ )d .

Now Macaulay’s theorem (Theorem 11.2.1) implies that the ring (C[x]/J) is alsoGroenstein of socle degree σ := d−d1+d1+ · · ·+d−d n

2+1−d n2+1 = ( n

2 +1)d andso Iδ = J, and in particular, (Iδ )d = Jd . Note that Jd is the tangent space of Td at Xand this proves the theorem. ut

Exercise 6 In the case of Fermat variety Xdn and

xd2i−2−xd

2i−1 = fi f n2+1+i, fi ∈C[x2i−2,x2i−1]di , f n

2+1+i ∈C[x2i−2,x2i−1]d−di , i= 1, . . . ,n2+1

the Macaulay’s theorem is an easy exercise in commutative algebra. Prove it.

12.4 Sum of two linear cycles

Proposition 12.4.1 For d≥ 3 consider the idea I := 〈xd−10 , ...,xd−1

2r−1〉⊂C[x0, ...,x2r−1].Let β1,β2,c1,c2 ∈ C× with β1 6= β2 and

Ri := ci ·r

∏j=1

(xd−12 j−2− (βix2 j−1)

d−1)

(x2 j−2−βix2 j−1).

We have(I : R1)e∩ (I : R2)e = (I : R1 +R2)e, (12.5)

if and only if e 6= (d−2) · r.

Proof. We know that I is Gorestein of socle 2σ , where

σ := r · (d−2),

and deg(R1) = deg(R2) = deg(R1 +R2) = σ and R1,R2,R1 +R2 do not belong toI. Therefore, by Proposition 11.2.1, part 1 we know that (I : R1), (I : R2) and (I :

12.4 Sum of two linear cycles 119

R1 +R2) are Gorenstein ideals of socle σ . We have

(I : R1)σ ∩ (I : R2)σ 6= (I : R1 +R2)σ .

Otherwise, we would have (I : R1 +R2)σ ⊆ (I : Ri)σ , and so by Proposition 11.2.1part 2 we have (I : R1) = (I : R1 +R2) = (I : R2). This implies that R1 = cR2 forsome c ∈ C∗ and modulo I, and hence, β1 = β2.

The equality (12.5) is trivially valid for all e > σ because in this case (I : R1)e =(I : R2)e = (I : R1 +R2)e = C[x]e which follows from part property (iii) of the defi-nition of a Gorenstein ideal applied to (I : R1), (I : R2) and (I : R1 +R2).

If e < σ then we claim that the proof of (12.5) reduces to the case e = σ −1. If(12.5) fails for some e < σ then we choose

p ∈ (I : R1 +R2)e \ (I : R1)e.

Using item (ii) of the definition of a Gorenstein ring in the case of (I : R1), we canfind a monomial xi = xi0

0 · · ·xi2r−12r−1 ∈ C[x0, ...,x2r−1]σ−e such that

xi · p ∈ (I : R1 +R2)σ \ (I : R1)σ .

Since deg(xi)> 0, there exist some i j > 0, and so

xi

x j· p ∈ (I : R1 +R2)σ−1 \ (I : R1)σ−1,

as claimed. Therefore, consider the case e = σ − 1. It is enough to show (I : R1 +R2)e ⊆ (I : R1)e ∩ (I : R2)e. Take p ∈ (I : R1 +R2)e. Since deg(p) = σ −1, we canwrite it as

p = ∑k even

d−3

∑e=0

xekxd−3−e

k+1 pk,e,

where each pk,e does not depend on xk and xk+1, and it is a C-linear combinationof monomials of the form xi0

0 · · ·xik−1k−1xik+2

k+2 · · ·xi2r−12r−1 with i2 j−2 + i2 j−1 = d−2, for all

j ∈ 1, ...,r\ k2 +1. The ideal Ik := 〈xd−1

0 , ...,xd−1k−1 ,x

d−1k+2 , ...,x

d−12r−1〉 is Gorenstein

of socle (d− 2)(2r− 2) and so for any k and e, and i = 1,2, there exist a constantak,e,i ∈ C such that

pk,eRi

(xd−2k + xd−3

k (βixk+1)+ · · ·+(βixk+1)d−2)≡ ak,e,i

(x0 · · ·x2r−1)d−2

(xkxk+1)d−2 ,

modulo Ik. Therefore

pRi ≡ (x0 · · ·x2r−1)d−2

∑k even

(1xk

d−3

∑e=0

ak,e,iβe+1i +

1xk+1

d−3

∑e=0

ak,e,iβei

),

modulo I. Since p · (R1 +R2) ∈ I we conclude that


d−3

∑e=0

ak,e,1βe+11 +

d−3

∑e=0

ak,e,2βe+12 =

d−3

∑e=0

ak,e,1βe1 +

d−3

∑e=0

ak,e,2βe2 = 0.

Since β1 6= β2, this implies

d−3

∑e=0

ak,e,1βe1 =

d−3

∑e=0

ak,e,2βe2 = 0,

and so pRi ∈ I for i = 1,2 which is a contradiction. utProposition 12.4.2 Let −1 ≤ m ≤ n

2 be an integer. In the ring C[x0,x1, · · · ,xn+1]consider the homogeneous polynomials

Q := ∏k≥n−2m even

xd−1k − (β0xk+1)

d−1

xk−β0xk+1,

R1 := c1 · ∏k<n−2m even

xd−1k − (β1xk+1)

d−1)

(xk−β1xk+1

R2 := c2 · ∏k<n−2m even

xd−1k − (β2xk+1)

d−1

xk−β2xk+1,

for some c1,c2,β0,β1,β2 ∈ C×, β1 6= β2 and P1 = R1Q and P2 = R2Q. We have

(JF : P1)e∩ (JF : P2)e = (JF : P1 +P2)e, (12.6)

if and only if e < (d−2)( n2 −m) or e > (d−2)( n

2 +1) .

Proof. For e > (d−2)( n2 +1) the claim follows from the fact that (d−2)( n

2 +1) isthe socle of the three ideals appearing in (12.6). For e < (d− 2)( n

2 −m), considerany q ∈ (JF : P1 +P2)e. Write

q = r+ s,

where r ∈ C[x0, ...,xn−2m−1]e and s ∈ 〈xn−2m, ...,xn+1〉e ⊆ C[x0, ...,xn+1]e. Notingthat

(JF : Q) = 〈xd−10 , ...,xd−1

n−2m−1,xn−2m, ...,xn+1〉,

it is clear that s ∈ (JF : Pi) = ((JF : Q) : Ri) for every i = 1,2. In consequencer ∈ ((JF : Q) : R1 +R2). Since r · (R1 +R2) does not depend on xn−2m, ...,xn+1 weconclude that

r ∈ (I : R1 +R2)e ⊆ C[x0, ...,xn−2m−1]e

for I = 〈xd−10 , ...,xd−1

n−2m−1〉. Using Proposition 12.4.1 for r = n2 −m, we conclude

that r ∈ (I : Ri)e for i = 1,2, and so q ∈ (JF : Pi)e for i = 1,2 as claimed.It remains to prove that for (d − 2)( n

2 −m) ≤ e ≤ (d − 2)( n2 + 1) we have an

inequality in (12.6). We know from Proposition 12.4.1 for r = n2 −m, that there

exists some p ∈ C[x0, ...,xn−2m] such that

p ∈ (JF : R1 +R2)(d−2)( n2−m) \ (JF : R1)(d−2)( n

2−m),

12.4 Sum of two linear cycles 121

Since Q is a polynomial in xn−2m,xn−2m+1, · · · ,xn+1, this implies that

p ∈ (JF : P1 +P2)(d−2)( n2−m) \ (JF : P1)(d−2)( n

2−m).

The ideal (JF : P1) is Gorenstein of socle (d− 2)( n2 + 1). This implies that there

exist some q ∈ C[x0, ...,xn+1]e−(d−2)( n2−m) such that

pq ∈ (JF : P1 +P2)e \ (JF : P1)e,

as desired. ut


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