Three lectures on Hodge structures

Damien Megy∗

June 2, 2014

These are preliminary notes for three lectures given at the spring school “classical and p-adic Hodge theories”in Rennes, France. Latest version available at http://iecl.univ-lorraine.fr/~Damien.Megy/Megy_Hodge.pdf, I plan to upload a new version in a few days (hopefully).Comments are most welcome.

Contents

1 The Hodge Decomposition 31.1 Betti Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Singular cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Comparison with De Rham cohomology . . . . . . . . . . . . . . . . . . 51.1.3 Integral structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Complex structure and Dolbeault double complex . . . . . . . . . . . . . . . . 61.2.1 Dolbeault cohomology and other invariants . . . . . . . . . . . . . . . . 6

1.3 Strong Hodge decomposition and the ∂∂ lemma . . . . . . . . . . . . . . . . . . 81.3.1 What is a (strong) Hodge decomposition ? . . . . . . . . . . . . . . . . . 81.3.2 Cohomology and decomposition of double complexes . . . . . . . . . . 91.3.3 The ∂∂-lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 The category of Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 Objects and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Geometric origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 The Lefschetz Package for Compact Kahler Manifolds 142.1 Strong Hodge decomposition via harmonic theory . . . . . . . . . . . . . . . . . 14

2.1.1 Harmonic theory on compact complex manifolds . . . . . . . . . . . . . 142.1.2 Proportionality and ∂∂ lemma . . . . . . . . . . . . . . . . . . . . . . . 152.1.3 Proportionality (and Hodge decomposition) for compact Kahler manifolds 162.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 The Hard Lefschetz property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Hard Lefschetz property and geometry . . . . . . . . . . . . . . . . . . . 19

2.3 Hodge-Riemann bilinear relations . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Polarizable Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

∗Institut Elie Cartan, universite de Lorraine, France. Email: damien.megy@univ-lorraine.fr

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3 Spectral sequences, holomorphic and algebraic de Rham Cohomology 223.1 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Motivation and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Frolicher spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.3 Effect of a modification on the ∂∂-lemma . . . . . . . . . . . . . . . . . 26

3.2 Holomorphic De Rham complex, Hodge-De Rham spectral sequence . . . . . . 273.3 Algebraic De Rham cohomology and comparison theorem . . . . . . . . . . . . 27

Consider your favorite functorial invariant from a category of geometric objects to someabelian or linear category. For example cohomological invariants. First question : inverseproblem ? For example:Question 1: is there a smooth projective variety X/C with H1(X,Z) ' Z ?

More generally, we can ask if these functors are full, faithfull, essentially surjective... insome cases to get a reasonable answer one must schrink, expand or enrich the categories. Insome other cases there is no known (or reasonable) answer.

What are Hodge structures ?

Hodge structures are a way to package several cohomological invariants and comparisonisomorphisms between them in one object. They form an abelian category endowed with aforgetful functor to abelian groups, and singular cohomology factors throught an enrichedcohomology functor with values in the category of Hodge structures. The question of thefullness of this functor is related to the famous Hodge conjecture (which we won’t discuss atall).

What are they useful for ?

• Better invariant : Hodge structures can distinguish between two non-isomorphic ellipticcurves, for instance. Singular cohomology cannot because the curves are homeomorphic.

• Hodge theory can answer the above question : No, there cannot be any smooth projec-tive variety X/C with H1(X,Z) = Z. Also can answer similar questions.

• Very strong conditions on the cohomology of smooth projective varieties, and in partic-ular: the Lefschetz Package : a set of beautiful properties satisfied by the cohomologyof smooth projective varieties over C. This will be discussed in the second lecture. Therelative version of the Hodge package (in both smooth and singular settings) will beexplained by Migliorini.

In the first lecture we discuss what it means for a variety to admit a (strong) Hodge de-composition on its cohomology, and we introduce the category of Hodge structures. In lecture2 we describe the Hodge-Lefschetz package for compact-Kahler manifolds. The third lectureis about spectral sequences and in particular the Frolicher spectral sequence, holomorphic deRham cohomology, and algebraic de Rham cohomology. All the material is classical and can

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be found in books such as [Dem], [Huy] and particularly [Voi]. The original papers of Deligneoften provide a more conceptual formalism which applies in both the complex and arithmeticsettings.

1 The Hodge Decomposition

1.1 Betti Cohomology

1.1.1 Singular cohomology

From the topological point of view, a cohomology theory with coefficients in a ring A isa contravariant functor H from a (nice) category of topological spaces to the category ofgraded A-modules (or graded-commutative algebras) which statisfies a number of axioms,such as existence of long exact sequences etc. Then, if the category of topological spaces issufficiently nice, it is possible to prove that such a cohomology theory is essentially unique[May, chap. 13 and 18].

Here, we consider complex manifolds, or more generally topological or differentiable mani-folds. These are (very) nice spaces, (for instance, they are CW-complexes) and we can choosethe cohomology theory to be singular (or cellular) cohomology. In the rest of the lectures,will call this the Betti cohomology of a complex variety.

Betti cohomology can have coefficients in Z, Q, R or C and other rings or modules. Theuniversal coefficients theorem says that there is an isomorphism

Hk(X,Q) ' Hk(X,Z)⊗Z Q

and similarly for R and C.We recall a some relevant features of singular cohomology in the geometric context. Betti

cohomology with coefficients in Q is a contravariant functor

H : Vopp → GrVecQ

from the category V of compact orientable smooth manifolds. to the category of finite-dimensional graded Q-vector spaces, such that:

1. for all X,Y ∈ V, there is a canonical isomorphism H(X ∪Y)∼−→ H(X)⊕H(Y);

2. (Kunneth formula) H(X)⊗H(Y)∼−→ H(X×Y); this means that H is a ⊗-functor;

3. Hk(X) = 0 if k 6∈ [0, 2 dim X];

4. for each X ∈ V of dimension d, the choice of an orientation yields an isomorphism1

H2d ∼−→ Q, inducing a perfect pairing

Hk(X)⊗H2d−k(X)→ H2d(X)→ Q.

This looks like part of the general definition of a Weil cohomology theory. Here, we arelacking the Tate object, the cycle class map and compatibility with cup-product and more.Hopefully we will come back to this in the third lecture.

1Note that there is no way to canonically identify the top cohomology to Q; in the arithmetic setting, suchan identification is part of the data, it is called the trace map.

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1.1.1 Definition. For 0 ≤ k ≤ dimR X, the Betti numbers bk are the dimensions of theQ-vector spaces Hk(X,Q). The Euler characteristic is χ(X) =

∑k∈N(−1)kbk(X).

1.1.2 Remark. Poincare duality can fail if X is not a smooth. Take the projective cone overan complex elliptic curve for example: the Betti numbers are not symmetric (exercise). ThusBetti cohomology is probably not a good invariant when working with singular varieties. SeeMigliorini’s course.

1.1.3 Example. All complex elliptic curves are diffeomorphic to S1 × S1 and thereby havethe same Betti cohomology, even though they are in general not algebraically isomorphic. TheBetti numbers are b0 = 1, b1 = 2, b2 = 1.

1.1.4 Example. (Real and complex Tori) A real torus is a real compact manifold diffeomor-phic to a product of circles : M ' (S1)d. Its Betti numbers are given by Kunneth formula :bk =

(dk

). A complex torus is a complex compact manifold obtained by taking the quotient of

a complex vector space by a discrete subgroup Λ ⊂ Cn of maximal rank : T ' Cn/Λ. Theunderlying real manifold is a real torus.

1.1.5 Example. (Hopf surface) The Hopf surface is the quotient of C2 \ (0, 0) by the equiv-alence relation (z1, z2) ' 1

2(z1, z2). It is diffeomorphic to S3 × S1, and its Betti numbers are1, 1, 0, 1, 1. This looks very bad for several reasons. The Hopf surface will be our favoritecounter-example.

1.1.6 Example. Compute the Euler characteristic of smooth degree d complex hypersurfacein Pn(C). All such hypersurfaces are diffeomorphic so take the Fermat hypersurface Xd

0 + ...+Xdn = 0 for example.

1.1.7 Remark. Betti cohomology can be computed as the cohomology of several complexesof abelian groups : cellular complexes, singular complexes, etc. Note that any complex K•

of free abelian groups decomposes as the direct sum of objects supported in only two degrees.For a complex of finite-dimensional Q-vector spaces, there is a splitting into indecomposablesummands (or blocks):

...→ 0→ Q→ 0→ ... and ..→ 0→ Q ∼−→ Q→ 0→ ...

Only the first ones contribute to the cohomology of the complex. For example, the complex

Q2

1 00 0

−−−−−−→ Q2

0 00 1

−−−−−−→ Q2

decomposes as the direct sum of the following blocks:

Q ∼−→ Q→ 0, 0→ Q ∼−→ Q, Q→ 0→ 0, 0→ 0→ Q.

Later we will see a similar decomposition result for double complexes.

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1.1.2 Comparison with De Rham cohomology

Let X be a smooth manifold, and (A•(X), d) its De Rham complex, given by the exteriorderivative acting on global C∞ forms on X. The (real) De Rham cohomology groups are :

HkDR(X,R) = Hk(A•(X), d) =

Ker d : Ak(X)→ Ak+1(X)

d(Ak−1(X)).

De Rham cohomology with complex coefficients is defined as the cohomology of the com-plex A•(X)⊗ C. It is isomorphic to Hk

DR(X,R)⊗ C.Let a ∈ Hk

DR(X,R) be the class of a closed k-form α. For every homology class c ofdegree k, we can choose a smooth representative γ of c. Stokes’ theorem gives a well-definedfunctional Ψa : c 7→

∫γ α.

1.1.8 Theorem. (De Rham) For all k ∈ N, the map

Ψ : HkDR(X,R)→ Hk

sing(X,R), a 7→ Ψa

is an isomorphism.

Note that Ak(X) is the space of global sections of the sheaf AkX of smooth forms. Thecomplex of sheaves

R→ A0X → A1

X → ...

is exact by the Poincare lemma and provides a resolution of the constant sheaf R. On the otherhand, the sheaves AkX are acyclic. Therefore, this resolution computes the sheaf cohomologyof R and we get an isomorphism HDR(X,R) ' Hk(X,R).

1.1.9 Example. Let S be the Hopf surface. Denote by r =√z · z =

√z1z1 + z2z2 the radius

function on C2. The closed 1-form d(r2)r2

= d(z·z)z·z represents a non-trivial cohomology class in

H1(S,R).

1.1.10 Example. (choice of a basis) Consider X = P1(C). Topologically, it is a 2-sphereand we have

H0(X,C) = C, H1(X,C) = 0, H2(X,C) ' C.

What generator should we choose for H2(X,C) ? The form ω = i2π

dz∧dz(1+|z|2)2

(where z is an

affine coordinate) has the advantage that∫

X ω = 1 (exercise), so the cohomology class of ω isactually a generator of H2(X,Z), which is nice from a topological point of view. But from analgebraic point of view, the i

2π factor looks weird and unnecessary (the algebraic variety seems

to have nothing to do with π or i), and we would like to choose ωalg = dz∧dz(1+|z|2)2

as a natural

generator... but then of course ωalg doesn’t belong to H2(X,Z). In fact, there is no way to getrid of 2πi in this situation, and the interplay between the algebraic and topological structureis precisely the interesting thing here. This kind of data is encoded in Hodge structures andperiod matrices.

1.1.3 Integral structure

De Rham Cohomology doesn’t compute rational or integral cohomology. Nevertheless we canrecover rational or non-torsion integral classes as classes c such that the integral over anychain is integral or rational.

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Examples of integral classes are given by cohomology classes of cycles. Consider for in-stance a codimension c closed submanifold N with orientable normal bundle of a n-dimensionalmanifold.

Assume the normal bundle of N is oriented. Then we have Thom’s isomorphism

T : Hc(M,M \N,Z)∼−→ H0(N,Z).

Now, define the cohomology class of N as follows : take the image of 1 ∈ H0(N,Z) under theinverse of Thom’s isomrphism, and then the image under the natural map Hk(M,M\N,Z)→Hk(M,Z) in the long exacte sequence of relative cohomology.

1.1.11 Remark. If the submanifold N is compact and oriented, then the cohomology class[N] of N can be described as follows. Consider the homoloy class of N in Hn−c(M,Z). Thecohomology class of N in Hc(M,Z) is the image of this homology class under the Poincareduality isomorphism Hn−c(M) ' Hc(M,Z).

1.1.12 Remark. The cohomology class of a submanifold can be trivial: consider for exampleS1 ⊂ S2. This can occur for compact complex manifolds, too. We will see in the next lecturethat if M is a compact Kahler manifold and N a complex submanifold, the cohomology class[N] is never trivial. This is a very important feature of compact Kahler manifolds – and inparticular of smooth projective varieties over C.

1.2 Complex structure and Dolbeault double complex

1.2.1 Dolbeault cohomology and other invariants

Let X be a n-dimensional compact complex variety. The (almost) complex structure onX allows to define a decomposition by type on de De Rham complex of (complex-valued)differential forms.

Locally, this decomposition can be described as follows. let T an n-dimensional complexvector space (T should be thought of as the tangent space of the complex manifold), andF = HomR(V,C). The complex vector space F has two natural complex subspaces T′ and T′′

of C-linear and C-antilinear forms, and we have F = T′⊕T′′. Note that T′ is isomorphic to thedual T∨ of T and T′′ = T′. Let k ∈ N, and consider

∧k F. It splits as∧k F =

⊕p+q=k

∧p,q F,where

∧p,q F :=∧p T′ ⊗

∧q T′′.We denote by Ap,qX the C∞X -module of C-valued smooth forms of type (p, q), and by

Ap,q(X) the space of global sections. Then, there are decompositions AkX =⊕p+q=k

Ap,qX and

Ak(X) =⊕p+q=k

Ap,q(X), and we define two operators ∂ and ∂ obtained by composing d with

the projections.

1.2.1 Example. Let S be the Hopf surface. The closed 1-form d(r2)r2

= d(z·z)z·z splits as α+ α,

with α = z·dzz·z of type (1, 0). Note that α and α are not closed.

The manifold X is a complex manifold, which means that the almost complex structureis integrable. As a result, the operator d : Ak(X) → Ak+1(X) splits as ∂ + ∂. The identityd d = 0 implies that the De Rham complex (A•, d) is the total complex associated to the

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double complex (A•,•, ∂, ∂). From this we can build a number of cohomological invariants,for example vertical and horizontal cohomology:

Hp,q

∂(X) =

Ker ∂ ∩Ap,q

∂Ap,q−1Hp,q∂ (X) =

Ker ∂ ∩Ap,q

∂Ap−1,q

And Bott-Chern and Aeppli cohomology :

Hp,qBC(X) =

Ker d ∩Ap,q

∂∂Ap−1,q−1Hp,q

A (X) =Ker ∂∂ ∩Ap,q

∂Ap−1,q + ∂Ap,q−1

More generally, we can consider

H∗∂(X) =Ker ∂

Im ∂, H∗∂(X) =

Ker ∂

Im ∂, H∗BC(X) =

Ker ∂ ∩Ker ∂

Im ∂∂, Hp,q

A (X) =Ker ∂∂

Im ∂ + Im ∂

These groups fit into the commutative diagram

H∗BC

##H∗∂

##

H∗DR(X)

H∗∂

H∗A

(1.2.1)

and there is a similar diagram for (p, q)-components. All this can of course be defined for anydouble complex.

The vertical cohomology group Hp,q

∂(X) is called the (p, q) Dolbeault cohomology group.

In general, we cannot compute it by topological methods, such as singular chains, Morsetheory etc. Note that if X is not compact, Hp,q

∂(X) has no reason to be finite-dimensional. If

it is, then we set hp,q = dim Hp,q

∂(X). The numbers hp,q are called the Hodge numbers of the

manifold X.

1.2.2 Exercise. Let S be the Hopf surface. Prove that h0,1(S) = 1 and h1,0(S) = 0. (Use theNoether-Lefschetz formula [?] for example).

1.2.3 Remark. There is no natural morphism betwen Dolbeault cohmology and De Rhamcohomology, except for a few cases. For example there are arrows Hk(X,C) → H0,k

∂(X) and

Hk(X,C) → Hk,0∂ (X), called edge-homomorphisms. In general, if k > 1, they are neither

injective nor surjective. These morphisms come from spectral sequences.

1.2.4 Remark. (Construction problem)Is it easy to construct manifolds with given Hodge numbers ? As we will see in the nextlectures, Hodge theory gives a lot of restrictions on the possible Betti and Hodge numbers.The question is partially answered in [?].

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1.3 Strong Hodge decomposition and the ∂∂ lemma

1.3.1 What is a (strong) Hodge decomposition ?

Let X be a compact complex manifold of dimension n We have seen that the spaces of k-forms decompose as the direct sum Ak(X,C) =

⊕p+q=k Ap,q(X,C). The question is wether

this decomposition carries over to cohomology in a natural way. This means a decompositionHk(X,C) =

⊕Hp,q(X) as a direct sum of subspaces called the Hodge subspaces, which are yet

to be defined.Let α be a closed k-form on X, representing a cohomology class. Then we have α =∑p,q α

p,q, but the (p, q) components have no reason to be closed. A (p, q) decomposition in

the cohomology group Hk would provide an equality α =∑

p,q αp,q up to an exact k-form, with

closed representatives αp,q of the (p, q)-components. We would also like the decomposition tobe unique, which means that in the previous decomposition, if α is exact, then all componentsαp,q are also exact.

In other words, introducing

Hp,q(X) =α ∈ Ap,q(X), dα = 0

Im(d),

we ask whether the sum of the inclusions Hp,q(X) → Hk(X,C) is an isomorphism onto Hk(X).We could probably also add some additional conditions on the Dolbeault groups, and com-parison conditions with Bott-Chern cohomology.

In general, there is no reason whatsoever for all these conditions to be satisfied.Note that a that Hp,q(X) and Hq,p(X) have the same dimension because they are complex

conjugate to each other. Therefore, a strong Hodge decomposition implies the followingproperty:

b2l+1(X) =∑

p+q=2l+1

dim Hp,q = 2∑

0≤p≤ldim Hp,q(X) ∈ 2N.

It also implies that every cohomology class can be represented by a ∂-closed and ∂-closedsmooth form, because each (p, q) component is closed so by purity it is ∂-closed and ∂-closed.

1.3.1 Example. Here are a few ways to see that the Hopf surface S cannot admit a strongHodge decomposition.

1. We have seen that b1(S) = 1 6∈ 2N.

2. The class of the 1-form d(r2)r2

cannot be represented by a ∂ and ∂-closed form.

3. H0,1 ⊕ H1,0 → H1(X) is not surjective. In fact, H1,0 is trivial, because there are noholomorphic 1-forms on S. This implies that H0,1 is also trivial.

4. H1,2 ⊕H2,1 → H3(X) is not injective: both Hodge subspaces are equal to H3(X).

5. H0,1

∂(X) is one-dimensional, spanned by the Dolbeault class of alpha = z·dz

z·z which is

∂-closed, whereas H1,0

∂(X) is the space of holomorphic 1-forms and is trivial.

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1.3.2 Cohomology and decomposition of double complexes

In these notes a double complex K•,•,∂,∂ is a bigraded object in an abelian category withanti-commuting arrows of ∂, ∂ of degree (1, 0) and (0, 1). We choose this convention so thatthe De Rham double complex of a complex manifold is a double complex. Changing the signsof the differentials gives a complex of complexes.

1.3.2 Example. The following double complexes play a particular role in the rest of thelecture. A double complex is a dot if all terms are zero except for one Kp0,q0 6= 0. Alldifferentials are zero.

0 0 0

0 // Kp,q

OO

// 0

0 0

OO

0

A square is a double complex of the form

Kp,q+1 ∼ // Kp+1,q+1

Kp,q

∼OO

∼// Kp+1,q

∼OO

A zigzag is a bounded double complex of the form

Kp,q ∼ // Kp+1,q

Kp+1,q−1

∼OO

∼ // Kp+2,q−1

Kp+2,q−2

∼OO

∼ // ...

of any length, where the length is the number of non-trivial terms (for instance a dot is azigzag of length one). A zigzag can also start with a vertical arrow:

Kp+1,q

Kp+1,q−1

∼OO

∼ // Kp+2,q−1

...

∼OO

1.3.3 Theorem. (Does anyone know a precise/convenient reference ? Exercise.)Every double complex decomposes as a direct sum of squares, dots, and zigzags.

If ∂∂ is always zero, then this should follow from a theorem of Dieudonne. For eachelement x in some Kp,q such that ∂∂x 6= 0, there is a split square.

It seems that the theorem can be recovered from the representation theory of specialbiserial algebras.

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1.3.3 The ∂∂-lemma

1.3.4 Proposition. Let (K•,•, ∂, ∂) be a bounded double complex in an abelian category, anddenote by (K•, d = ∂+∂) the associated total complex. For each n ∈ Z, the following conditionsare equivalent:

(a)n Ker ∂ ∩Ker ∂ ∩ Im d = Im ∂∂;

(b)n Ker ∂ ∩Ker ∂ ∩ (Im ∂ + Im ∂) = Im ∂∂;

Proof. [DGMS, 5.15]

1.3.5 Definition. We say that the ∂∂ lemma holds for K•,• if the above equivalent conditionshold for every n ∈ Z. We say that it holds for a complex manifold if it holds for its Dolbeaultdouble complex.

If the lemma holds, the natural maps between Dolbeault, De Rham, Bott-Chern andAeppli cohomology in the commutative diagram 1.2.1 are isomorphisms.

1.3.6 Example. The ∂∂-lemma fails for any zigzag.

1.3.7 Exercise. Find a zig-zag in the Dolbeault double complex of the surface. Hint : startwith the 1-form α.

1.3.8 Theorem. ([DGMS, 5.17])Let (K•,•, ∂, ∂) be a bounded double complex in a semisimple abelian category, and denote by(K•, d = ∂ + ∂) the associated total complex. The following conditions are equivalent:

1. The ∂∂ lemma holds for K;

2. K•,• decomposes as a direct sum of squares and dots.

3. (a) The differential d of the total complex K• is strict for the two filtrations on K• (i.e.FpK• ∩ dK = dFpK•) and:

(b) For all n, the two induced filtrations F1 and F2 on Hn(K•), d are n-opposite, i.e.Hn = Fp1Hn ⊕ Fq+1

2 Hn for all p+ q = n.

Proof. (2)⇒ (1) and (2)⇒ (3) : it is enough to prove the results for squares and dots, whichis easy.

(1)⇒ (2) : there can be no zigzags in the decomposition into elementary blocks, becausethe ∂∂-lemma doesn’t hold for them.

(2)⇔(3) : again, decompose into elementary blocks. There are two types of zigzags ofeven-length and each kind the prevents one strictness property. Odd-length zigzags are thereason why the n-oppositeness can fail.

Note that the proof given in [DGMS] doesn’t make use of the decomposition into ele-mentary blocks. The authors prove (1)⇒ (2) by explicitly splitting each Kp,q by hand, andthen prove (3)⇒ (1) by a direct argument. The statement (3.a) can be rephrased in terms ofspectral sequences, see third lecture.

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1.3.9 Corollary. The cohomology of a complex manifold has a strong Hodge decompositionif and only if the ∂∂-lemma holds on X.

Compact Kahler manifolds (in particular, smooth projective varieties over C) satisfy the∂∂-lemma (second lecture).

The condition 3.b of the theorem leads to one of the definitions of Hodge structures.

1.4 The category of Hodge structures

In this section, A is a noetherian subalgebra of R such that A ⊗ Q is a field, for example Z,Q or R.2

1.4.1 Objects and morphisms

Let n ∈ Z. A pure A-Hodge structure of weight n is a triple (VA, (VC,F•), α) where VA is

an A-module of finite type, VC is a finite-dimensional C-vector space with a finite decreasingfiltration F•, and α is an isomorphism VA ⊗ C → VC such that the filtration F• and itscomplex conjugate (with respect to the real structure induced by α) are n-opposite, that is:

VC = FpVC ⊕ Fn−p+1VC.

1.4.1 Remark. This definition can be simplified by setting VC := VA ⊗ C and forgettingabout the comparison isomorphism α. But in practice, the filtration F• is often not definedon VA ⊗ C itself, but on an isomorphic vector space. Keeping track of the isomorphism canbe useful.

Let V and W be pure A-Hodge structures of weight n. A morphism f of Hodge structuresbetween V and W is a pair (fA, fC) where fA : VA → WA is a morphism of A-modules,fC : VC →WC is a filtered morphism of filtered C-vector spaces, and αW fA = fC αV.

If V is a pure A-Hodge structure of weight n, we denote by Vp,q the subspaces FpVC ∩Fn−pVC of VC. These subspaces verify the following properties:

1. Hodge decomposition: VC = ⊕p+q=nVp,q.

2. Hodge symmetry : Vq,p = Vp,q.

Conversely, given complex subspaces of VC satisfying Hodge decomposition and symmetry,we recover the filtration F as follows : FpVC := ⊕p≤r≤nVr,n−r. This filtration and its complexconjugate are n-opposite. The type of a Hodge structure is the minimal subset T ⊂ Z2 suchthat (p, q) 6∈ T⇒ Vp,q = 0. A Hodge structure is effective if T ⊂ N2.

1.4.2 Example. • The trivial Hodge structure A(0) is the pure Hodge structure of weight0 given by A(0)A = A, A(0)C = C, F0 = C, F1 = 0 and α(1) = 1. The only non-trivialHodge subspace is A(0)0,0 = C.

• The Tate A-Hodge structure is the pure A-Hodge structure A(1) of weight −2 withA(1)A = A, A(1)C = C, F−1 = C, F0 = 0, and α(1) = 2iπ. The only non-trivialHodge subspace is A(0)−1,−1 = C.

2There are notions of K-Hodge structures for certain subfields K of C not contained in R. This is implicitor explicit in the work of Deligne–Mostow, Carlson–Toledo and more. Also there is the rather weak notion ofC-Hodge structure. Together with polarizations, C-(V)HS are a very useful, see the work of Simpson.

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• More generally, for n ∈ Z, the Hodge structure A(n) is the weight −2n Hodge structurewith A(n)A = A, A(n)C = C, F−n = C, F−n+1 = 0 and α(1) = (2iπ)n. The onlynon-trivial Hodge subspace is A(0)−n,−n = C.

1.4.3 Example. (H2(P1) is the Lefschetz structure)Let HZ := H2(P1,Z), HC := H2(P1,C), F1HC = HC,F

2HC = 0, and let α be the inverse of thecomparison isomorphism between De Rham and singular cohomology :

H2(P1,C)→ H2(P1,Z)⊗ C = Hom(H2(P1,Z),C), [u] 7→σ 7→

∫σu

.

Then, the Hodge structure (HZ, (HC,F•), α) is isomorphic to the so-called Lefschetz Hodge

structure Z(−1). It has weight 2, and pure type (1, 1).

1.4.4 Example. More generally, we will see later that if X is a compact Kahler manifold,in particular a smooth projective variety over C, then Hk(X,Z) underlies a pure Z-Hodgestructure of weight k.

Pure A-Hodge structures of weight n form a category AHS(n).

1.4.5 Theorem. The category AHS(n) is abelian.

Proof. This is not obvious as the category of filtered objects in an abelian category is notabelian. It follows from the important fact that if f = (fA, fC) is a morphism of pure Hodgestructures, then the filtered morphism fC is strict for the filtrations. See [Del71, 1.2.10].

1.4.6 Exercise. Prove that the category of filtered finite-dimensional vector spaces is notabelian. (find an example such that CoIm(f) is not isomorphic to Im f)

The category of A-Hodge structures, denoted AHS, is the direct sum for n ∈ Z of thecategories AHS(n). This means that the objects of AHS are finite direct sums of pure A-HSof some weight, and morphisms are direct sums of morphisms of pure A-Hodge structures.

1.4.7 Theorem. The category AHS is abelian.

Let VA be a Hodge structure. There is an action ρ of C∗ on VC given by z · v = z−pz−q,for all v ∈ Vp,q. The operator ρ(i) is the endomorphism acting on Vp,q by multiplication byiq−p. It maps VR into itself, which gives an endomorphism C of VR, called the Weil operator.A morphism of real HS is a morphism which commutes with the Weil operators.

The advantage of working with AHS instead of AHS(n) is that the former has a tensorstructure.

Tensor structure If V and V′ are pure A-HS of weight n and n′, then the tensor productV⊗V′ is a pure Hodge structure of weight n+n′, and Hom(V,V′) is a pure Hodge structureof weight n′−n. In particular, the dual V∨ of V is a pure Hodge structure of weight −n. Wealso have the Hodge structures SymkV and

∧k V, both of weight kn. The category AHS isa tensor category.

1.4.8 Exercise. Prove that if n ≥ 1, then A(n) = A(1)⊗n. If n ≤ −1, then A(n) =(A(1)∨)⊗−n.

If V is A-Hodge structure, we write V(n) instead of V⊗A(n). Twisting a Hodge structureby some A(n) is the standard way to modify the weight.

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Differences between Q and R Hodge structures The notion of R-Hodge structure ismuch weaker than that of Q or Z Hodge structure. For example, any R-Hodge structure ofdimension two and weight one is isomorphic to some Hodge structure Vτ with Hodge filtrationC2 = F0 ⊃ F1 = span (1, τ), Im (τ) > 0. Any two such Hodge structures Vτ and Vη areisomorphic over A if the two complex numbers τ and η are in the same GL2(A)-orbit. Thisis always the case over R, but not over Q for example.

Invariants of Hodge structures What tools can we use to distinguish between two Hodgestructures ? Automorphisms can help : even with the same type, some HS have more auto-morphisms than others. For example, for weight one and dimension two Hodge structures,Vi has four automorphisms whereas a generic Vτ has none.

In some situations, the automorphisms are the same, but one Hodge structure has moreendomorphisms than the other. Consider for example the weight one Hodge structure Vi

√2 :

it has no automorphisms, but can be distinguished from a very general Hodge structure Vτ

because it has more endomorphisms than just homothetic transformations.This last example leads to the study of “CM” (complex multiplication) Hodge structures

and Mumford-Tate groups.

1.4.2 Geometric origin

One of the first geometric examples of Hodge structures of weight one is the H1 of a compactRieman surface. We have a map

Riemann surfaces of genus g H1

−−→ effective weight one HS of rank 2g .

It is natural to ask if all weight 1 HS come from Riemann surfaces. This is the case ifH0,1 = 1.

In general, the moduli space of curves of genus g has dimension 3g− 3, and that of Hodgestructures of type (1, 0) + (0, 1) of dimension 2g has dimension g(g + 1)/2. Thus for g > 2,there are Hodge structures not coming from curves3.

In fact at this point it is not clear whether or not all Hodge structures (of any weight)come from (algebraic) geometry4.

In weight one (or minus one), we have the following positive result.

1.4.9 Proposition. The functor

Complex tori −→ Z−Hodge structures of weight -1 and type (−1, 0) + (0,−1)

given by X 7→ H1(,Z), is an equivalence of categories.

A quasi-inverse is obtained as follows. Let V be such a Hodge structure. The Weil operatorC on VR is a complex structure. The complex variety VR/VZ is a complex torus.

Complex tori up to isogeny correspond to Q-Hodge structures.

1.4.10 Corollary. The abelian category Q−HS is not semisimple.

We have seen that weight −1 rationnal Hodge structures correspond to complex tori (upto isogeny). It is easy to construct a two-dimensional torus with a non-split subtorus, evenup to isogeny. Take a so-called Shafarevich extension of two elliptic curves (ref?).

3The image of the map is still not known in detail. The injectivity of the map is exactly the Torelli theoremfor curves. For precise statements and definitions of the objects, see Chris Peters’ lectures.

4Jan Nagel explained in his lectures why it is not the case in general, because of Griffiths transversality

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1.4.3 Exercises

C∗-action.Degree, slope of a filtration. Hodge structures as stable filtrations.Intermediate jacobians.Twistors. Hodge polynomial.

2 The Lefschetz Package for Compact Kahler Manifolds

Introduction : what is the (Hodge-)Lefschetz package ? Hodge structure, Lefschetz structure,Hodge-Riemann bilinear relations.

2.1 Strong Hodge decomposition via harmonic theory

2.1.1 Harmonic theory on compact complex manifolds

Let first M denote a smooth compact oriented n-dimensional riemannian manifold with metricg = 〈−,−〈. This metric induces metrics on all exterior powers of the tangent bundle. For ωa smooth k-form on M, define

||ω||2M :=

∫M||ω||2pdVolg.

This is a positive-definite inner product on the space Ak(M). The idea is to use this metricto find representatives of De Rham cohomology classes.

2.1.1 Definition. A smooth k-form on M is harmonic if it is closed and

||ω||M ≤ ||ω + dβ||M

for all smooth k − 1-forms β. The space of harmonic k-forms is denoted by Hk(M).

This means that a harmonic form has minimum norm inside its cohomology class. Notethat for a small real number t, we have

||ω + tdβ||2M = ||ω||2M + 2t(ω, dβ) + O(t2).

We see that if ω harmonic then we must have (ω, dβ) = 0 for all β ∈ Ak−1. The operator d hasan adjoint d? for (−,−)M. (In an open subset of Rn, it is given by d?u = −

∑I,j

∂uI∂xj

∂∂xj¬dxI.)

Then, (ω, dβ)M = 0 implies (d?ω, β)M = 0, and this holds for all β ∈ Ak−1 if omega isharmonic. Thus, a harmonic form is d?-closed. Conversely, if dω = d?ω = 0, then ||ω+dβ||2M =||ω||2M + ||dβ||M ≥ ||ω||2M and ω is harmonic.

2.1.2 Definition. The Laplace operator ∆ : Ak(M)→ Ak(M) is defined by

∆ = dd? + d?d.

Note that this operator depends on the metric.Any harmonic form satisfies the Laplace equation ∆ω = 0, and conversely, if ∆ω = 0,

then0 = (∆ω, ω)M = (dd?ω, ω)M + (d?dω, ω)M = ||d?ω||2M + ||dω||2M,

so ω is harmonic.

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2.1.3 Theorem. (Hodge theorem for ∆)Let M be a compact oriented riemannian manifold. For all k, the space of smooth k-formsdecomposes as

Ak(M) = Hk(M)⊕ Im d⊕ Im d?.

This sum is orthogonal for the metric on Ak(M), the subspace Hk(M) is finite-dimensionaland maps isomorphically to Hk(M,R).

This shows that a cohomology class contains a unique harmonic representative. If wechange the riemannian metric on M, this representative may not be the same.

Harmonic theory on complex manifolds Let X be a compact complex manifold. Ahermitian metric h on X is a positive-definite hermitian form on the tangent bundle of X.

The operators ∂ and ∂ have adjoints ∂? and ∂? of type (−1, 0) and (0,−1), and we haved? = ∂? + ∂?. Now set ∆′ = ∂∂? + ∂?∂, and ∆′′ = ∂∂? + ∂?∂, and denote by Hp,q∆′ ⊂ Ap,q andHp,q∆′′ ⊂ Ap,q the corresponding subspaces of ∆′-harmonic and ∆′′-harmonic forms. Then wealso have decompositions for these operators:

2.1.4 Theorem. (Hodge theorem for ∆′ and ∆′′)Let X be a compact complex manifold. For all p, q ≥ 0, the harmonic subpaces for ∆′ an ∆′′

are finite-dimensional and we have orthogonal decompositions

Ap,q = Hp,q∆′ ⊕ Im ∂ ⊕ Im ∂?,

Ap,q = Hp,q∆′′ ⊕ Im ∂ ⊕ Im ∂?.

This proves that the Dolbeault cohomology groups of a compact manifold are finite-dimensional, and that any Dolbeault class contains a unique ∆′′-harmonic representative.

2.1.2 Proportionality and ∂∂ lemma

Consider the expansion:

∆ = dd? + d?d = (∂ + ∂)(∂? + ∂∗) + (∂? + ∂∗)(∂ + ∂) = ∆′+ ∆′′+ (∂∂? + ∂?∂) + (∂∂? + ∂?∂).

The cross-terms ∂∂? + ∂?∂ and ∂∂? + ∂?∂ have bidegree (−1, 1) and (1,−1) and do notvanish in general. Thus the Laplace operator is not bihomogeneous of bidegree (0, 0).

2.1.5 Proposition. Let X be a compact complex manifold, and fix a hermitian metric. If∆ = 2∆′ = 2∆′′, then the ∂∂ lemma holds on X.

Proof. The hypothesis implies that the cross-terms vanish and that ∆ is bihomogeneous ofdegree (0, 0). Note that every cohomology class can be represented by a ∂ and ∂ closedform : choose an harmonic representative, then its (p, q) -components are also harmonic andtherefore ∂ and ∂ closed.

Now let us prove the pure ∂∂ lemma. Let ω be a closed (p, q) form. If it is d or ∂ or ∂exact, then it is orthogonal to the harmonic subspace Hp,q. Thus it sufficient to prove that ifω is orthogonal to Hp,q, then it is ∂∂ exact. Apply the Hodge theorem for ∆′′ to ω : we getω = ∂α+ ∂?β. The closed form ω is of pure type so it is ∂-closed. It follows that ∂∂?β = 0,

15

so 0 = (∂∂?β, β) = ||∂?β||2, i.e. ∂?β = 0. As a result, ω = ∂α. The Hodge theorem for ∆′ toα gives a decomposition α = h + ∂γ + ∂?ε. The harmonic component is ∂-closed. Thus, wehave

ω = ∂α = ∂∂γ + ∂∂?ε.

It is enough to prove that the second component is zero. Recall that proportionality implies∂∂? = −∂?∂. Consequently, ω = −∂∂γ−∂?∂ε. Now, ω is also ∂-closed, this forces ∂∂?∂ε = 0and the same argument as before allows to conclude that ∂?∂ε = 0.

In the next section we will see that proportionality holds on a compact Kahler manifold.

2.1.3 Proportionality (and Hodge decomposition) for compact Kahler manifolds

Let (X, h) a complex manifold with a hermitian metric. The associated fundamental form isω = −Im (h). This is a positive (1, 1)-form. The metric h is said to be Kahler if ω is closed.In this case, ω is called the Kahler form. A complex manifolds is Kahler if it admits at leastone Kahler metric.

Both Cn, the complex ball Bn and the projective space Pn are Kahler, with the metrics

ωCn = i∑

dzk ∧ dzk, ωBn = i∂∂ log(1− ||z||2), ωFS =i

2π∂∂ log(1 + ||z||2).

The metric ωFS is called the Fubiny-Study metric, it is not canonical and depends on thechoice of a hermitian scalar product || − ||2 on Cn.

Any complex curve is Kahler, because any (1, 1)-form on a complex curve is closed. Inhigher dimension, there are non-Kahler manifolds, for example the Hopf surface.

Every complex torus T = Cn/Λ is Kahler. Take any scalar product on the Cn = R2n

compatible by the complex structure of Cn. This defines a constant hence Λ-invariant Kahlermetric which descends on T.

Every projective manifold is Kahler. Such a manifold can be embedded in a projectivespace. Restricting the Fubiny-Study metric yields a Kahler metric.

Lefschetz operator Let (X, ω) be a Kahler manifold, and let L be the operator defined onsmooth forms by

L : α 7→ ω ∧ α.

The closedness condition dω = 0 is equivalent to ∂ω = 0 and ∂ω = 0. In terms of theLefschetz operator L, it can be rewritten as

[L, d] = [L, ∂] = [L, ∂] = 0.

The operator L admits a formal adjoint Λ. For all smooth forms u, v, we have (Lu, v)X =(u,Λv)X.

One of the key results in the theory of Kahler manifolds is the following

2.1.6 Proposition. (Kahler identity) Let X be a Kahler manifold. Then

[Λ, ∂] = i∂?.

Several other identities can be obtained by taking the complex conjugate or adjoint of thisexpression.

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Proof. In Cn, this is a computation. The point is precisely that on a Kahler manifold,statements involving the metric up to first derivatives can be checked for the flat metric.See for example [Dem]. There is another proof which doesn’t use local coordinates but ratherthe local Lefschetz decomposition on forms, see [Huy] or [Sab].

The Kahler identity above can be illustrated by the following diagram : one can replace

Ap,q

i∂?

Ap,q−1

by Ap,q ∂ //

Λ

yy

Ap+1,q

ΛyyAp−1,q−1

−∂// Ap,q−1

.

Now we see that this identity implies proportionality for the laplacians ∆, ∆′ and ∆′′ onX. For instance,

∂∂? + ∂?∂ = −i(∂[Λ, ∂] + [Λ, ∂]∂) = −i(∂Λ∂ − ∂∂Λ + Λ∂∂ − ∂Λ∂) = 0,

and the other cross-term ∂∂? + ∂?∂ vanishes too. The equality ∆′′ = ∆′ also follows from a(longer) computation which can be summarized by the following drawing:

∂?

•

∂

EE

∂?

∂

GG

//

−

+

// •

OO OO

//

−

+

//

OO OO

− −//// • ////+

OO

+

OO

OO OO

∂'' •

∂%%

∂?ee

∂?gg

The first step uses the Kahler identity (dotted arrows), the second step is just ∂∂ = −∂∂,and the last step uses the complex conjugate of the Kahler identity: [Λ, ∂] = −i∂?.

2.1.4 Exercises

Non-Kahler aspects, considerations on volume, Wirtinger, more examples. Holonomy ?

2.2 The Hard Lefschetz property

2.2.1 Linear algebra

See [Del80, 1.6] and [Del68].Let V be a vector space, and N a nilpotent endomorphism of V of nilpotency index k, i.e.

k is the first positive integer such that Nk+1 = 0.

2.2.1 Proposition. There exists a unique increasing filtration W = W(N) of V:

0 ⊂W−k ⊂ ... ⊂Wk−1 ⊂Wk = V,

called the weight filtration of N, such that:

1. N(Wk) ⊂Wk−2;

2. For all k ≥ 0, Nk : GrWk V→ GrW

−kV is an isomorphism.

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Proof. The subspaces Wk are defined in terms of Ker(Ni) and Im (Nj).

Exercise : prove the proposition if N has a single Jordan block.

2.2.2 Example. If the nilpotency index is 1, that is, N2 = 0, the filtration is

0 ⊂ Im N ⊂ Ker N ⊂ V.

For nilpotency index 2, the filtration starts with 0 ⊂ Im N2 and ends with Ker N2 ⊂ V.COmpute the rest of the filtration. In fact this is the mechanism of the proof in the generalcase, see Deligne.

2.2.3 Definition. Let V =⊕

l∈Z Vl be a finite-dimensional graded vector space and N andendomorphism. We say that the pair (V,N) satisfies the Hard Lefschetz (HL) property if forall positive integers l, N send Vl to Vl+2 and if

Nl : V−l −→ Vl

is an isomorphism

If (V,N) satisfies (HL), then N is nilpotent. The proposition shows that for any vectorspace V with a nilpotent endomorphism N, denoting by W• the weight filtration of N andW• the opposite filtration, then the pair (GrW•V,N) satisfies (HL).

2.2.4 Definition. (Primitive part) Let (V,N) be a pair satisfying (HL). The primitive partof V−l, denoted by 0V−l, is

0V−l = Ker(

Nl+1 : V−l → Vl).

2.2.5 Proposition. (Lefschetz decomposition) Let (V,N) be a pair satisfying (HL). For alll ≥ 0, the maps⊕

k≥0

Nk :⊕k≥0

0V−l−2k → V−l and⊕k≥0

Nk+l :⊕k≥0

0V−l−2k → Vl

are isomorphisms.

The Lefschetz decomposition implies in turn the Hard Lefschetz property.The basic example comes from the local structure of the sheaf of forms on a complex

manifold :

2.2.6 Example. (Local Lefschetz structure in the geometric setting) Let T an n-dimensionalcomplex vector space and F = HomR(V,C).

In what follows, we use the notations introduced in ?? Let h be an hermitian scalar producton T, and ω = −Im (h). Then ω ∈

∧1,1 F is positive in the sense that ω(x, ix) > 0 for allx. Now let N be the operator ω ∧ −. With the shifted grading Vl =

∧n+l F, the pair (V,N)satisfies (HL). More precisely, let Λ be the adjoint of L with respect to the hermitian metricon V induced by h, and define B = (n− l)Id on Vl. Then, we have

[B,L] = 2L, [B,Λ] = −2Λ, [L,Λ] = B.

This means that V is a sl2(C)-module, and we recover the Lefschetz decomposition from thedecomposition into irreductible submodules.

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2.2.2 Hard Lefschetz property and geometry

2.2.7 Definition. Let X be an n-dimensional complex manifold, u ∈ H2(X,C), and

L : H∗(X,C)→ H∗(X,C), α 7→ u ∪ α.

The pair (X, u) is said to satisfy the Hard Lefschetz property if (H∗(X,C),L) does. Themanifold X satisfies HL if there is at least one class u such that (X, u) satisfies HL.

If a X satisfies the HL property, then the odd and even Betti numbers bk increase fork ≤ n (and decrease for k ≥ n), and b2l 6= 0 for all 0 ≤ l ≤ n.

2.2.8 Example. Consider a smooth projective surface S, and a hyperplane class u ∈ H2(S,C).

Then u2 generates H4(S,C). The non-trivial part of the Lefschetz property says that H1(S,C)∪u−−→

H3(S,C) should be an isomorphism. This is not totally trivial. See exercise 2.5.2 for a possibleapproach.

2.2.9 Theorem. (Hard Lefschetz theorem for Kahler manifolds)Let X be a compact Kahler manifold and ω a Kahler form. Then, the pair (X, [ω]) satifies theHard Lefschetz property.

The idea of the proof is the following. Write L for the operator ω∧ on forms. The firststep is the pointwise calculation stated in 2.2.6. This means that the complex vector bundle∧∗T∨X has a Lefschetz decomposition and that Lk :

∧n−k T∨X →∧n+k T∨X is an isomorphism.

In particular, Lk is injective at the level of smooth (n− k)-forms.The second step is that the morphism L preserves harmonic forms. This is a consequence

of the equality[L,∆] = 0.

By proportionality, it suffices to prove [L,∆′] = 0 and this follows from the Kahler identity.One concludes that Lk : Hn−k → Hn+k is injective, so it is bijective because both spaces

have same dimension.Note that if X is compact Kahler and the (X, u) satisfies the Lefschetz property, the class

u doesn’t have to be a Kahler class. (For example, if u works, so does −u.)

2.2.10 Remark. The Hard Lefschetz property is purely topological and makes sense even ifX is not a complex manifold : for any topological space X together with a fixed integer n anda class in H2(X), we can ask if the maps ∪uk : Hn−k(X,Q) → Hn+k(X) are bijective. TheHard Lefschetz property is intensively studied in symplectic geometry : in that context, it isequivalent to a symplectic version of the ∂∂-lemma.

More generally, the Hard Lefschetz makes sense for algebraic varieties and several co-homology theories. If X is a smooth projective variety of dimension n over a field k, letH := H∗t (Xk,Ql) the l-adic cohomology of X and u ∈ H2(X,Ql) the first Chern class of anample line bundle. Then the pair (H, u) satisfies the Hard Lefschetz property. This is veryhard ([Del80, Theoreme 4.1.1]). There is an analogue of this for crystalline cohomology. Seethe other lectures for the arithmetic side of this story.

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2.3 Hodge-Riemann bilinear relations

Let (X, ω) a compact complex manifold. Part of the Hard Lefschetz property can be rephrasedin the following way. Consider the pairing

(a, b) 7→∫

Xa ∧ b ∧ ωn−k.

This is a (−1)k-symmetric bilinear form and the Hard Lefschetz property implies that it isnon-degenerate.

Now consider the associated sesquilinear pairing

(a, b) 7→ ik∫

Xa ∧ b ∧ ωn−k.

We would like to study the signature of these pairings.Let us first look at one example. If X is a Riemann surface, and consider the pairing

h(a, b) = i∫

X a ∧ b on H1(X). If u ∈ H1,0, it is represented by a holomorphic form which canbe written u = f.dz in local coordinates. Then, iu ∧ u = i|f |2dz ∧ dz is positive, thus

h(u, u) > 0.

Similarly, if u ∈ H0,1, we have h(u, u) < 0.It is also possible to say something about compact Kahler surfaces. Consider this time the

sesquilinear pairing h(a, b) =∫

X a ∧ b on the second cohomology group H2(X). It is positive-definite on H2,0 and H0,2 (just write a (2, 0) form in local coordinates as above), and it canbe shown that if α is a primitive and harmonic (1, 1)-form, then h(α, α) < 0 (exercise : writethe form in local coordinates).

The conclusion is that it seems that these sesquilinear pairings are definite on the primitivesummands 0Hp,q(X). What should be the sign ? The following linear algebra lemma gives aplausible answer:

2.3.1 Lemma. (local Hodge-Riemann relations [Huy, 1.2.36] ) Let (T, h) a hermitian com-plex vector space as in 2.2.6, and ω the associated fundamental form. Recall that V∗ :=∧∗HomR(T,C). The Hodge-Riemann pairing is the bilinear form

Q : Vk ×Vk → C, (a, b) 7→ (−1)k(k−1)/2a ∧ b ∧ ωn−k,

where V2n is identified to C via the volume form. The, the decomposition V = ⊕p,qVp,q isorthogonal for Q, and:

ip−qQ(a, a) > 0

for all primitive elements a of bidegree (p, q).

Proof. See [Huy], p. 39, or [Sab].

This local lemma gives the sign (−1)k(k−1)/2 and implies rather directly the global resultwhich we now state.

2.3.2 Proposition. (Hodge-Riemann bilinear relations) Let (X, ω) be a compact Kahler man-ifold, and let

Q : Hk ×Hk → C, (a, b) 7→ (−1)k(k−1)/2a ∧ b ∧ ωn−k

be the Hodge-Riemann pairing and h(a, b) = ikQ(a, b) the associated sesquilinear pairing.

20

1. The Hodge decomposition on Hk(X,C) is orthogonal with respect to h.

2. For all primitive (p, q) cohomology class u, we have:

ip−qQ(u, u) > 0.

In other words, the restriction of the hermitian form (−1)qh to the primitive Hodgesubspace 0Hp,q(X) is positive-definite.

Proof. The first property holds because if the types of a and b are different, then a∧b∧ωn−k isnot of type (n, n). Let us prove the second assertion. The primitive class u can be representedby an harmonic form α, primitive at each point. The pointwise computation in the preceedinglemma (??) allows to conclude.

In terms of Hodge structures, the Hodge-Riemann bilinear relations say that the bilinearform Q is a polarization of the primitive cohomology.

2.4 Polarizable Hodge structures

2.4.1 Definition. A polarization on V is a morphism of Hodge structures φ : V⊗V→ A(−n)such that Q(x, y) = (2iπ)nφ(Cx, y) is a symmetric positive definite bilinear form on VR.

Since φ is a morphism of Hodge structures, we have

φ(x, y) = φ(Cx,Cy) = φ(C2y, x) = (−1)nφ(y, x)

Hence φ is symmetric if n is even, and alternating if m is odd.A pure Hodge structure is polarizable if it admits a polarization. If V is a polarizable

Hodge structure and W is a sub-Hodge structure, then any polarization of V restricts to apolarization of W.

The category of polarizable A-Hodge is denoted by APHS. If A is a field, it is semisimple.

2.4.2 Theorem. The category of polarizable Q-Hodge structures is semisimple.

Under the equivalence between complex tori and weight −1 Z-Hodge structures of type(−1, 0) + (0,−1), abelian varieties correspond to polarizable Hodge structures, polarizedabelian correspond to polarized Hodge structures, and abelian varieties up to usigeny cor-respond to Q-Hodge structures.

2.5 Exercises

2.5.1 Exercise. Let V be a real finite-dimensional vector space, and Q a symmetric bilinearform on V. Describe the set of effective Hodge structures of weight two on V and hodgenumbers h2,0, h1,1, h0,2 = h2,0, polarized by the form Q, as a subset of Grass(V, h2,0).

2.5.2 Exercise. (Another approach to the Hard Lefschetz theorem)

Let S be a smooth projective surface, C a smooth hyperplane section of S, and u = [C] ∈H2(X,Z) its cohomology class. Prove that

L = u∪ : H1(S,C)→ H3(S,C)

is an isomorphism by using the following tools:

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• Weak Lefschetz : the inclusion i : C → S induces an injective map in cohomology

H1(S)i∗−→ H1(C).

• Hodge-Riemann bilinear relations on the curve C (see example 2.3 : this can be provedwhithout Hard Lefschetz). This means that H1(C) is a polarized Hodge structure ofweight one.

• The Gysin morphism i! : H1(C) → H3(S) (the Poincare dual of i∗, thereby surjective),and the commutative diagram

H1(S,C)L //

i∗

H3(S,C)

H1(C,C)

i!

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First prove that the intersection form on H1(C) restricted to i∗H1(S) is nondegenerate. Showthat this implies the injectivity of i! on the subspace i∗H1(S).

This approach is the one used by Deligne to prove Hard Lefschetz by induction, see [Del80,4.1]. Instead of using the Hodge-Riemann bilinear relations, Deligne puts the hyperplanesection inside a Lefschetz pencil and then applies his global invariant cycle and semisimplicitytheorems to the monodromy representation of the Lefschetz pencil.

3 Spectral sequences, holomorphic and algebraic de Rham Co-homology

3.1 Spectral sequences

3.1.1 Motivation and definitions

Let A be an abelian category.

3.1.1 Definition. A differential object is a pair (X, d) with X ∈ Ob(A) and d is an endo-morphism of X with d2 = 0. A morphism of differential objects f : (X, dX) → (Y, dY) is amorphism f : X→ Y such that the following square commutes:

A

dX

f // Y

dY

Xf

// Y

This defines a category Diff(A). It is abelian.

3.1.2 Definition. Let (X, d) ∈ Diff(A). We define the cycles and boundaries to be Z(X) =

Ker d and B(X) = Im d. The cohomology of X is H(X) := Z(X)B(X) . This defines a functor

H : Diff(A)→ A.

Computing H(X) can be difficult. Additional structure on X may be useful.

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3.1.3 Definition. A filtered differential object is a filtered object (X,F•) with a filtered en-domorphism d : X → X, such that d2 = 0. A filtration on a differential object induces afiltration on its cohomology H(X).

How can filtrations be useful when computing cohomology ? First consider the mostelementary case : a filtration with only two graded pieces. This means that we have afiltration

0 ⊂ Y ⊂ X.

Then, the cohomology of X can be computed with the help of the long exact sequence

...→ H(Y)→ H(X)→ H(X/Y)→ H(Y)→ ...,

which is known to be very useful. Let us now consider a less elementary example, for examplea filtration

0 ⊂ Z ⊂ Y ⊂ X.

Then, there are three short exact sequences :

Z→ Y → Y/Z, Y → X→ X/Y, Z→ X→ X/Z.

These three exact sequences give rise to three long exact sequences in cohomology. This kindof data is what is encoded in a spectral sequence.

More generally, the idea of spectral sequences is to use the filtration on X to compute thecohomology H(X) or at least its associated graded GrH(X). The idea is to approximate thecycles and boundaries Z and B and then compute Z/B as some kind of limit of Zr/Br. Whatdoes it mean to approximate or to take the limit ?

3.1.4 Definition. (Abstract spectral sequence)A bigraded spectral sequence in the abelian category A (starting at r0) is the following data:for all p, q ∈ Z and r ≥ r0:

• Objects Ep,qr .

• Morphisms dp,q : Ep,qr → Ep+r,q−r+1r such that dp+r,q−r+1 dp,q = 0.

• Isomorphisms αp,q : Ep,qr+1 'Ker dp,qr

Im dp−r,q+r−1r

.

Note that if we set Enr =⊕

p+q=n Ep,qr and dr =⊕

p+q=n dp,qr , then dr : Enr → En+1

r ,(E•r , dr) is a complex and its cohomology objects are E∗r+1.

3.1.5 Definition. The spectral sequence converges is for all (p, q), the sequence (Ep,qr )r iseventually constant. The limit is then denoted by Ep,q∞ . The spectral sequence collapses (ordegenerates) at Er if all differentials dp,qs vanish for s ≥ r.

A filtration F• of a complex K• is said to be biregular if for each degree l there are indicespmin and pmax such that FpKl = Kl for p < pmin and FpKl = 0 for p > pmax.

3.1.6 Proposition. (Spectral sequence of a filtered complex, Leray) Let K• be a complex witha biregular filtration F•. There exists a spectral sequence (Ep,qr ), such that

1. Ep,q0 = GrpKp+q.

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2. Ep,q1 = Hp+q(GrpK•, d).

3. The spectral sequence converges and the limit is GrH∗(K•) in the sense that for all (p, q)the sequence Ep,qr is stationary for large r and this limit is GrpHp+q(K•, d).

Proof. Define Zp,qr = x ∈ FpKp+q, dx ∈ Fp+rKp+q+1 and

Ep,qr :=Zp,qr

Zp+1,q−1r−1 + dZp−r+1,q+r−2

r−1

,

and check the remaining properties [Del71, 1.3.1]. See also [Voi].

Let K•,• be a double complex, and denote by K• the associated total complex. The K•

has two natural filtrations F•1 and F•2 given by:

Fp1K• =⊕r≥p,s

Kr,s

Fp2K• =⊕r,s≥p

Kr,s

This yields two different spectral sequences, one for each filtration.We will mostly use the first filtration in our examples and applications. In this case, we

have:Ep,q0 = Kp,q, d0 = ∂; Ep,q1 = Hq(Kp,•, ∂), d1 = ∂.

This follows from the description of the spectral sequence of a filtered complex.

3.1.7 Proposition. (Deligne, [Del71, 1.3.2])Let (K•, d) be a complex with a biregular filtration F. The following two conditions are equiv-alent:

1. The spectral sequence degenerates at E1.

2. The morphisms d : Kl → Kl+1 are strict for the filtration

Proof. (particular case : double complexes)

If K• is the total complex of a double complex K•,•, let us prove that both statements areequivalent to:

3. There are no even-length zigzags ending with an horizontal arrow in the decompositionof K•,•.

(1) ⇔ (3) follows from the following simple remarks.

• The spectral sequence of a dot is constant, i.e. degenerates at E0.

• The spectral sequence of a square collapses to zero at E1.

• The spectral sequence of an odd-length zigzag has only one non-zero term at the E1

stage : the beginning of the zigzag if it begins with a horizontal arrow, the end of thezigzag otherwise. In any case it degenerates at E1.

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• The spectral sequence of an even-length zigzag starting (and thereby also ending) witha vertical arrow collapses to zero at E1.

• The spectral sequence of a zigzag of length 2k starting (and ending) with a horizontalarrow has exactly one nonzero differential dk. It degenerates at Ek+1, not before.

(2) ⇔ (3) is left as an exercise.

The proof in the general case is not very difficult but uses an explicit description of thearrows dr : Ep,qr → Ep+r,q−r+1

r . See Hodge II.

3.1.2 Frolicher spectral sequence

Let X be a compact complex manifold. The Hodge Frolicher (or Frolicher) spectral sequenceof X is the spectral sequence associated to the double complex A•,•(X), ∂, ∂), filtered by

FpA•,•, ∂, ∂) = A•≥p,•, ∂, ∂) =⊕r≥p,s

Ar,s(X).

It starts withEp,q1 = Hp+q(GrpA• = Hq(Ap,•, ∂) = Hp,q

∂(X).

The filtration Fp on the total De Rham complex A•(X) induces a filtration on its cohomol-ogy H∗(X,C). We call it the Hodge-Frolicher filtration. The associated graded GrFH∗(X,C)is the limit of the spectral sequence.

3.1.8 Proposition. (Degeneration)The existence of an isomorphism Hk(X,C) '

⊕p+q=k Hp,q

∂(X) is equivalent to the degeneration

of the Frolicher spectral sequence at E1. In this case, we say that X admits a (weak) Hodgedecomposition.

Proof. The limit terms Ep,q∞ are subquotients of Ep,q1 , thus dim Ep,q∞ ≤ dim Ep,q1 . Existence ofan isomorphism Hk(X,C) '

⊕p+q=k Hp,q

∂(X) thus implies

dim Hk(X,C) =∑p+q=k

dim Ep,q∞ ≤∑p+q=k

dim Ep,q1 = dim Hk(X,C).

This forces the equalities dim Ep,q∞ = dim Ep,q1 for all p, q. As a result, the spectral sequenceis E1-degenerate.

In terms of forms, E1-degeneration means the following. Let a ∈ Hp,q

∂a Dolbeault class.

Then, there exists a closed k-form α = αp,q + αp+1,q−1 + ...+ αk−1,1 + αk,0 in FpAk(X), suchthat the Dolbeault class of the (p, q)-component αp,q is equal to a. This is weaker than strongHodge decomposition (or equivalently the ∂∂-lemma), which says that in this setting, thereexists a closed form α of pure type (p, q) representing a, and that the associated De Rhamclass [α]DR ∈ Hk is uniquely determined by the Dolbeault class a.

Every smooth projective or compact Kahler manifold has of course a strong Hodge decom-position. The Hopf surface has a weak Hodge decomposition, but not a strong one. In fact, the

25

Frolicher spectral sequence of any compact complex surface degenerates at E1 [?]! The non-trivial part consists in proving that the arrow d1 = ∂ : E0,1

1 = H1(S,OS) → E1,11 = H1(S,ΩS)

is zero.In higher dimension, the Frolicher spectral sequence need not degenerate at E1. The

Iwasawa manifold [Dem, VI, 8.10] is a three-dimensional compact manifold with a non-closedholomorphic 1-form (!)5, and the arrow d1 : E1,0

1 → E2,01 is not zero. Recently, Rollenske

constructed examples with arbitrarily non-degenerate Frolicher spectral sequence. Even ifthe spectral sequence degenerates at E1, the groups FpHk ∩ FqHk, where F• denotes theHodge-Frolicher filtration on cohomology, are not necessarily equal to the “Hodge subspaces”we have defined before. For this we really the ∂∂-lemma on the double complex.

3.1.3 Effect of a modification on the ∂∂-lemma

3.1.9 Theorem. [Del68, 4.3, 5.3][DGMS, 5.22] Let X and Y be compact complex manifolds,and f : X → Y a holomorphic (birational) surjective morphism. Id the ∂∂ lemma holds forX, then it holds for Y.

Proof. We first prove that the Frolicher spectral sequence for Y degenerates at E1, and then,that the limit filtration and its conjugate are k-opposite on Hk(Y). By ??, this is equivalentto the ∂∂-lemma.

First step : degeneration

The map Hq(Y,Ωp)f∗−→ Hq(X,Ωp) is injective because there is a map f∗ such that

f∗ f∗ : Hq(Y,Ωp)f∗−→ Hq(X,Ωp)

f∗−→ Hq(Y,Ωp)

is the identity. In [Del68], the map f is a proper map between algebraic varieties and f∗is the relative trace map Trf obtained from (R)f !. The following analytic construction ofsuch a map can be found in [Dem, VI,12.2]. The complex of (p, •) currents is a resolution ofthe sheaf Ωp [Dem, IV,6.17] and can be used to compute Dolbeault cohomology. There is awell-defined push-forward (or direct image) at the level of currents [Dem, I, 2.14]. The directimage commutes with d and satisfies the projection formula : f∗(T∧ f∗α) = f∗T∧α, for anysmooth form α. It induces a direct image f∗ : Hq(X,Ωp

X) → Hq(Y,ΩpY). Now, f∗f

∗ = Id onHq(Y,Ωp

Y) follows from the fact that f is a biholomorphism outside a set of measure zero.This proves that the first page of the Frolicher sequence of X contains that of Y as a direct

summand. An induction shows that Er(Y) = Er+1(Y), f∗ : Er(Y) → Er(X) is injective, andthat dr : Er(Y)→ Er(Y) vanishes for all r ≥ 1, so the spectral sequence degenerates at E1.

Step two : k-opposite filtrations

The ∂∂ lemma holds for X, thus Hk(X) = FpHk(X) ⊕ Fk−p+1Hk(X). The map f∗ sendsFpHk(Y) injectively into FpHk(X) and commutes with complex conjugation. As a result, the

intersection FpHk(Y)∩Fk−p+1Hk(Y) = 0 is zero (the filtrations F and F are k-transverse).This holds for all k, and in particular, for each k it holds for 2n−k. A Serre duality argumentthen implies that FpHk(Y) + Fk−p+1Hk(X) = Hk(Y). See [Del68, 5.3].

5This wouldn’t happen on a Kahler manifold. An holomorphic 1-form is ∂-closed by definition and ∂?-closedfor type reasons. So it is ∆′′-harmonic. Proportionality would imply that the form is harmonic, thus closed.

26

Remark : f∗H∗(Y,C) is a sub-Hodge structure of H∗(X,C). Demailly [Dem, VI, 12.9] givesanother proof, using a sheaf-theoretic description of Bott-Chern cohomology. (Same idea :there are injective pull-backs and surjective push-forward between Bott-Chern (p, q)-groups).

3.2 Holomorphic De Rham complex, Hodge-De Rham spectral sequence

Let X be an n-dimensional complex manifold. The holomorphic De Rham complex Ω•X is thecomplex of sheaves

0→ OX∂−→ Ω1

X∂−→ ...

∂−→ ΩnX → 0.

By the holomorphic Poincare lemma, this complex is a resolution of the constant sheaf Con X. The quasi-isomorphism C → Ω•X induces an isomorphism in hypercohomology :

Hk(X,C)∼−→ Hk(X,Ω•X).

The holomorphic De Rham complex admits the naıve6 filtration FpΩ• = Ω•≥p. It inducesthe filtration

FpHk(X,C) = Im (Hk(X,Ω•≥pX )→ Hk(X,Ω•X)),

and the associated graded is the limit of a spectral sequence which starts with Ep,q1 =Hq(X,GrpΩ•X) = Hq(X,Ωp).

3.3 Algebraic De Rham cohomology and comparison theorem

Let us now consider algebraic varieties. What cohomology theory should we use ? Basically,we would like to consider cohomological invariants that behave like singular cohomology inthe analytic context. A partial list of expected properties has been given in ??.

There are notions of “well-behaved” cohomology theories, for example the so-called Weilcohomology theories, see []. In addition to what has already been discussed, there are require-ments on the multiplicative structure, the cycle class maps, and the way the interact.

A discussion on Weil cohomology theories would lead us too far from the subject of thislectures, but the following remark may be useful. When given a cohomology theory, it can beenlightening to test it on Ank \ 0 and on curves over k. The natural expectation is that weshould get the same result as in the complex analytic setting. For example, the cohomologyof Ank \ 0 should look like the cohomology of topological spheres.

How to construct a good cohomology theory for algebraic varieties over a field k? Thenaıve analogue of the Betti cohomology for complex varieties would be to consider the sheafcohomology of the constant sheaf kX on X, where X is endowed with its Zariski topology. Butthis is not well-behaved. In fact, due to Grothendieck’s vanishing theorem, the cohomologyHl(X,F) of any sheaf F on X vanishes for l > dim X. But one of the requirements of a goodcohmology theory is precisely that the cohomology on degree 2 dim X should be isomorphic tothe field, just as in the analytic case, where the isomorphism is given by integration of forms.This remark rules out sheaf cohomology.

Now, recall that singular cohomology of complex varieties can be computed by holomorphicdifferentials. This remarkable fact shows us how to use the algebraic structure, in the agebraicsetting.

6as opposed to the standard truncation of a complex σ≥pK• = ...→ 0→ Coker dpdp−→ Kp+1 dp+1−−−→ Kp+2 →

... which has much better functorial properties : adjunction etc.

27

Let X be a non-singular n-dimensional quasi-projective variety over a field K of charac-teristic zero, Ω1

X/K the sheaf of Khahler differentials, and ΩlX/K =

∧l Ω1X/K. The algebraic

De Rham complex of X/K is

OXd−→ Ω1

X/Kd−→ ...→ Ωn

X/K.

3.3.1 Definition. The algebraic De Rham cohomology consists of the K-vector spaces

HiDR(X/K) := Hi

(OX

d−→ Ω1X/K

d−→ ...→ ΩnX/K

).

Given a morphism f : X→ Y, we obtain a map of complexe f∗Ω•Y/K → Ω•X/K, which givesrise to a map

f∗ : HiDR(Y/K)→ Hi

DR(X/K).

The inclusion of complexes σp : Ω•≥pX/K → Ω•X/K induces a map in hypercohomology, whose

image in HiDR(X/K) we denote by FpHi

DR(X/K). This defines a decreasing filtration onHi

DR(X/K) by K-vector spaces.At this point, it is not clear whether this cohomology theory is well-behaved or not. Before

proving the general theorem, let us compute a few elementary examples.

3.3.2 Example. (Punctured line)Let X = A1

Q \ 0 = SpecQ[z, z−1]. The de Rham cohomology Hk(X,Ω•X) can be computed usingthe Hodge-to-de Rham spectral sequence Ep,q1 = Hq(X,Ωp)⇒ Hp+q(X,Ω•X).

In this particular case, X has dimension one, so Ep,q1 = 0 for p > 1, and X is affine, soEp,q1 = 0 for q > 0. The first page of the spectral sequence is

0 0 0 0

0 E0,01

d1 // E1,01 0

This means that the spectral sequence has to degenerate at E2, and we have:

H0dR(X/Q) ' E0,0

2 = Ker(d : H0(X,OX)→ H0(X,Ω1

X))' Q,

H1dR(X/Q) ' E0,0

2 = Coker(d : H0(X,OX)→ H0(X,Ω1

X))

= Qdz

z' Q.

After taking the tensor product with C, we get the expected result, that is, the H0 and H1

of the punctured complex plane C∗ are one-dimensional.

3.3.3 Example. (Complete curve)Let X be a smooth projective curve over Q. The Hodge-to-de Rham spectral sequence startswith Ep,q1 = Hq(X,Ωp). We have Ep,q1 = 0 for p > 1 because X is a curve, and Ep,q1 = 0 forq > dim X = 1 by Grothendieck’s vanishing.

0 0 0 0

0 E0,11

d1 // E1,11 0

0 E0,01

d1 // E1,01 0

28

The spectral sequence must degenerate at E2, but this time there can be several graded

pieces. In fact, the spectral sequence degenerates at E1. The differential E0,01

d1−→ E1,01 is given

by H0(X,OX)d−→ H0(X,Ω1

X) which vanishes because the curve is projective, so regular functions

are constant. Serre duality yields E0,11 = H0(X,Ω1

X) = H1(X,OX)∨ and E1,11 = H1(X,ΩX) =

H0(X,OX)∨. The map E0,11

d1−→ E1,11 is dual to E0,0

1d1−→ E1,0

1 , therefore it is also zero.In the end, we have

H0dR(X/Q) ' E0,0

1 = Q,

H2dR(X/Q) ' E1,1

1 = H1(X,ΩX) ' Q,and a short exact sequence :

0→ H0(X,Ω1X)→ H1

dR(X/Q)→ H1(X,OX)→ 0.

We get the expected Betti numbers 1, 2g, 1, with g := dim H0(X,Ω1X).

In both cases, the result is what we expected. This is a consequence of the followinggeneral comparison theorem.

3.3.4 Theorem. (Grothendieck, [Gro])Let X be a smooth quasiprojective variety over C, and denote by Xan the associated complexmanifold. The canonical map of ringed spaces (Xan,OXan)→ (X,OX) induces an isomorphism

HiDR(X/C)→ Hi(Xan,C).

If X is projective, the image of the filtration F• on HiDR(X/C) under this isomorphism is the

Hodge filtration on Betti cohomology.

Proof. We give the proof only for projective varieties.

Extension of scalars and comparison of rational structures Let K ⊂ L be a fieldextension and XL = X×K SpecL, then we have Ω1

X/L ' Ω1X/K ⊗K L, Ωi

X/L ' ΩiX/K ⊗K L, and

thusHi

DR(X/L) ' HiDR(X/K)⊗K L.

Suppose that the projective variety X is defined over Q. Then composing with the basefield extension isomorphism and the universal coefficients theorm, we get an isomorphism

HiDR(X/Q)⊗Q C ∼−→ Hi(Xan,Q)⊗Q C.

It seems a natural question to ask whether or not this isomorphism of C-vector spacescomes from a natural isomorphism

HiDR(X/Q)

???−−→ Hi(Xan,Q)

of Q-vector spaces...The answer is no. For instance, if X = A1

Q\0, the comparison isomorphism H1DR(X/Q)⊗Q

C → H1(Xan,C) maps the generator dtt of H1

DR(X/Q) to 2iπ. This map is not defined overQ. For an example with projective varieties, consider X = P1

Q and H2. It is more or less thesame example.

With respect to basis of the Q-vector spaces HiDR(X/Q) and Hi(Xan,Q), the comparison

isomorphism is given by a matrix with complex coefficients. It is called the period matrix.

29

3.3.5 Definition. Complex numbers that appear in period matrices are called periods.

The basic example of a period is 2πi.We finish by stating an important result :

3.3.6 Theorem. In characteristic zero, algebraic De Rham cohomology is a Weil cohomologytheory.

This means in particular that there is a good notion of cup-product and cycle class mapon algebraic de Rham cohomology.

References

[Del68] P. Deligne, Theoreme de Lefschetz et criteres de degenerescence de suites spectrales,Publ. Math. IHES, 1968

[Del71] P. Deligne, Theorie de Hodge II, Publ. Math. IHES 40 (1971)

[Del80] P. Deligne, La conjecture de Weil, II, Publ. Math. IHES (1980)

[DGMS] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real Homotopy Theory of K ihlerManifolds, Invent. Math. 29 (1975) 245–274

[Dem] J-P. Demailly, Complex analytic and differential geometry, available at http://

www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

[Gro] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ. Math. IHES29 (1966) 95–103

[Huy] D. Huybrechts, Complex geometry, an introduction, Universitext, Springer, 2004

[May] J. P. May, A Concise Course in Algebraic Topology, University of Chicago Press 1999,available at http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

[Sab] C. Sabbah, Theorie de Hodge et theoremes de Lefschetz difficiles, Notes de cours,Strasbourg 2000, Bordeaux 2001, available at www.math.polytechnique.fr/~sabbah/

hodge-str.pdf

[Voi] C. Voisin, Theorie de Hodge et geometrie algebrique complexe, cours specialises 10,SMF (2002)

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