indico.ictp.itindico.ictp.it/event/a09153/material/2/0.pdf · mixed hodge structures fouad el zein...

MIXED HODGE STRUCTURES

FOUAD EL ZEIN

Summer School on Hodge Theory and Related TopicsInternational center for theoretical Physics

14 June 2010 - 2 July 2010Trieste - ITALY

Abstract. These notes start with a summary of cohomology theory (section2), necessary to understand the background to Hodge theory (section 3) andthe polarization (section 4). The Mixed Hodge structure (MHS) is defined asan object of interest in linear algebra (section 5). After developing the spectralsequence of a filtered complex (section 6), canonical MHS are constructed oncohomology of algebraic varieties (sections 7,8).The lectures in previous courses cover the first three sections. The course onMHS will be presented in a more elementary language than these notesLecture 1:Mixed Hodge structure (section 5).Lecture 2:Polarization (section 4, 5).Lecture 3:Logarithmic complex (section 6, 7) .Lecture 4:Spectral sequence and Hodge theory (section 6, 7).Lecture 5: MHS on cohomology of complex algebraic varieties (section 8).

1. Introduction

Cohomology theory attach to a manifold a group, invariant under homeomor-phisms. Introduced first as a topological invariant, it depends only on the topologyof the manifold, but it plays a significant role in the differential study of a manifoldas shown in DeRham theory. Its importance in algebraic and analytic geometries isreflected in Hodge theory as well in the introduction by J.P. Serre of the cohomol-ogy of coherent sheaves. Inspired by the properties of etale cohomology introducedby A. Grothendieck in the study of varieties over fields with positive characteristic,Deligne established the existence of a linear algebraic structure on cohomology ofalgebraic varieties over fields of characteristic zero, called mixed Hodge structure(MHS); it reflects geometric and topological properties of algebraic varieties, whichmeans it depends on the geometry and not only the topology of such varieties.This structure proved to be very useful in the interpretation of various geometric in-variants classically attached to algebraic varieties, as Picard group or the Jacobianvariety. Deformation of geometric structures result into a linear invariant on thecohomlogy of the family of varieties defined by the deformation, called variation ofHodge structure (VHS) by P. Griffiths. More generally a variation of mixed Hodgestructure (VMHS) is attached to an algebraic family of varieties not necessarilyproper or smooth.

1

2 FOUAD EL ZEIN

2. Review of cohomology theories

Many techniques have been used to construct homology and cohomology groups.We mention here :triangulation, singular chains, Cech coverings and finally sheaftheory via injective or acyclic resolutions. In particular De Rham cohomologydefined only on differentiable manifolds, is of special interest in our case. We referto Dieudonne [3] for an historical account of each contribution.All these constructions give, when they are defined and under mild assumptions onthe topological space, isomorphic cohomology groups. Based on their main commonproperties on the class of spaces defined as finite euclidean simplicial complexes, theaxiomatization of the theory occurred in the 1940’s. The use of spectral sequences, introduced by Leray and developed by Serre, appeared to be a powerful tool incomputation, see for example [2]. The most general approach since 1955 is dueto Grothendieck, first with the systematic use of injective resolutions in abeliancategories [10], second with the introduction with Verdier [13] of the derived categoryand the use of hypercohomology, third with the introduction with M. Artin, of anew (Grothendieck) topology and (etale) cohomology theory developed with manycolleagues and students (namely P. Deligne, L. Illusie and J.L. Verdier). However,to introduce MHS we may use only flabby resolutions of Godement [8] to constructthe weight filtration W on cohomology with coefficients Q. Dolbeault resolutionsare of particular interest in the study of the Hodge filtration F , which is an analyticinvariant, on the cohomology with coefficients C.

2.1. Simplicial objects. Let ∆n denote the set of integers 0, 1, . . . , n and Hp,q

the set of increasing (large sense) mappings from ∆p to ∆q for integers p, q ≥ 0. Wedefine in this way the simplicial category ∆ whose objects are ∆n, n ∈ N, with thenatural composition of mappings : Hpq×Hqr → Hpr. The semi-simplicial category∆> is obtained when we consider only the strictly increasing morphisms in ∆.A simplicial (resp. co-simplicial) object of a category C is a contravariant (resp.covariant) functor from ∆ to C. For example a simplicial group consists of a familyof groups Xi for each i ∈ N and of morphisms X(f) : Xq → Xp for each f ∈ Hp,q,compatible with composition. A semi-simplicial (resp. co-semi-simplicial) object ofa category C is a contravariant (resp. covariant) functor from ∆> to C. We definefor 0 ≤ i ≤ n + 1 the i−th face map and denote by

δi : ∆n → ∆n+1, i 6∈ δi(∆n) := Im δi

the unique increasing mapping such that i 6∈ δi(∆n). Such suggestive name cor-responds to the topological realization of the simplex where δi refers to the basicdefinition of the boundary. Its image by a functor is also denoted abusively by thesame symbol δi : Xn+1 → Xn (resp. δi : Xn → Xn+1).

2.1.1. Complexes. A complex (Y ., d) (resp. chain complex (Y., d)) in an abeliancategory is given by a sequence of objects (Y n) and morphisms dn : Y n → Y n+1

(resp. dn : Yn → Yn−1 ) such that dn+1 dn = 0 (resp. dn−1 dn = 0) for allintegers n. It follows that Im dn ⊂ Ker dn+1 (resp. Im dn ⊂ Ker dn−1).Given a complex (Y ., d) , we deduce a chain complex (X., d) if we put Xn = Y −n,with the same differential dn : Xn → Xn−1. Such construction may be inverted.

MIXED HODGE STRUCTURES 3

Definition 2.1. The cohomology of the complex (Y ., d) is defined for each n by

Hn(Y ., d) = Ker(d : Y n → Y n+1)/Im(d : Y n−1 → Y n).

The nth homology of a chain complex (X., d) is defined as

Hn(X., d) = ker(d : Xn → Xn−1)/Im(d : Xn+1 → Xn).

2.1.2. Chain complex defined by a simplicial object of an abelian category. LetXn, n ∈ N be a simplicial object of an abelian category, we define the differen-tial of a chain complex

dn+1 : Xn+1 → Xn, dn+1 = Σn+1i=0 (−1)iδi, then dn dn+1 = 0.

A complex may be defined similarly by a cosimplicial object.

2.2. Triangulation. [8;I.3.2 ], [ 12 ; 3.1] According to Dieudonne [3, p 26 ], theword triangulation is due to Weyl but the concept can be found in the paperAnalysis situs written by Poincare in 1895 and which is considered as a foundationalarticle of Homotopy and Homology theory.

Definition 2.2. A combinatorial complex K consists ofi) a set K of elements v called vertices.ii) a set S of finite non empty subsets s of K called simplices, such that1) any set consisting of exactly one vertex is a simplex,2) any non empty subset of a simplex is a simplex.iii) A simplex s is of dimension n if it contains n + 1 vertices |s| = n + 1.

Definition 2.3. 1) A singular simplex of dimension n of K is a mappingσ : ∆n → K whose image is a simplex of K.2) Let Σn(K) be the free abelian group generated by the set Sn(K) of all singularsimplices of dim n of K (resp. all singular simplices of dim n of K), then Σ.(K) isa simplicial group. We denote also by Σ.(K) the chain complex associated to it.3) The homology of K is defined as the homology of the complex Σ.(K).

Remark 2.4 (3;I.3.8 ). Suppose we are given an order on K such that each simplexis totally ordered, then the set of simplices totally ordered is the same as theset of strictly increasing singular simplices σ : ∆n → K and denote by Σ+

n (K)the subgroup of Σn(K) generated by such subset of simplices s ∈ K. Then thesubcomplex Σ+.(K) of Σ.(K) has the same homology groups as Σ.(K).

2.2.1. Realization. The topological realization of ∆n is the real simplex Σn in Rn+1

defined as the set of points t = (to, . . . , tn) such that ti ≥ 0 and Σni=0ti = 1. As well

a topological space R(K), called its realization, is defined by K; it consists of realsimplices R(s) embedded in some real vector spaces of dimension big enough, onefor each simplex s in K, R(s) and R(s′) being glued along R(s ∩ s′) considered asa face of R(σ) and R(σ′). One way of defining this space is given in [G] as follows

Definition 2.5. The realization R(K) of K is the set of mappings f : K → R suchthati) The set of elements v of K where f(v) 6= 0 is a simplex.ii) f(v) ≥ 0 for all v ∈ K and Σv∈Kf(v) = 1.

The topology on R(K) is defined as follows. For each simplex σ let R(s) denotes theset of elements in R(K) vanishing on K − s. It is isomorphic to a real topologicalsimplex in the finite dimensional real vector subspace generated by R(v) for all v

4 FOUAD EL ZEIN

in s in the vector space of all mappings f : K → R. If v is a vertex of s then R(v) isa point in R(s). A subset U of R(K) is open if it induces an open subset U ∩R(s)in R(s) for each simplex s in K.

Definition 2.6. A triangulation of a topological space E is given by a homeomor-phism of E onto a realization R(K) of a combinatorial complex K.

The homology of a triangulated space E is defined as the homology of its trian-gulation K. This notion becomes interesting when one proves the independence ofthe choice of the triangulation, which is the case [3]. A natural proof is deduced bycomparison of homology with singular homology.

2.3. Singular homology and cohomology. It is based on the use of continu-ous mappings defined on the topological simplex Σn and generally adapted for anintroduction to homology theory, although triangulations are more intuitive.

Definition 2.7. A singular simplex of dimension n of a topological space E is acontinuous mapping σ : Σn → E.

Let Σn(E) denote the set of singular simplices of dimension n, and Sn(E) thefree abelian group generated by Σn(E). Then Σn(E) (resp. Sn(E)) define a sim-plicial set (resp. group) hence a chain complex; while HomZ(Sn(E),Z) define acosimplicial group, hence a complex.

Definition 2.8. i) The homology groups Hi(E,Z) of the chain complex S.(E) de-fined by the simplicial complex are called the singular homology groups of E withcoefficients in Z.ii) The cohomology groups Hi(E,Z) of the cochain complex HomZ(S.(E),Z) de-fined by the cosimplicial group are called the singular cohomology groups of E withcoefficients in Z.

When E is triangulable, and this is the case of differentiable manifolds or al-gebraic varieties, the singular homology coincides with the homology of any trian-gulation. Homology and Cohomology groups have in general torsion elements. InHodge theory we are mainly interested in the image of Hi(E,Z) in Hi(E,Q) whichis torsion free.

Definition 2.9. A piecewise smooth singular simplex of dimension n of a topologi-cal differentiable manifold X is a map σ : Σn → X which extends to a differentiablemap on a neighbourhood of Σn.

We form the chain complex Sps. (E) of piecewise smooth integral chains and define

its homology denoted by Hpsi (X,Z)

Lemma 2.10. The natural morphism Hpsi (X,Z) ∼−→ Hi(X,Z) is an isomorphism.

2.3.1. Poincare duality. A general intersection theory is defined on triangulablespaces, in particular compact differentiable manifolds, on the level of chains oftriangles. It induces an intersection pairing on homology which is used to prove ageneral Poincare duality on oriented topological spaces.

Theorem 2.11 (Poincare duality). On a compact oriented n-manifold, the inter-section pairing

Hi(X,Z)×Hn−i(X,Z) → H0(X,R)degree−−−−→ Z


is unimodular, that is the induced morphism

Hi(X,Z) → Hom(Hn−i(X,Z),Z)

is surjective and its kernel is the torsion subgroup

We need this duality on compact oriented differentiable manifolds where it canbe proved using differential forms. For rational coefficients, it defines a perfectduality

Hi(X,Q) ' Hom(Hn−i(X,Q),Q).

2.4. Cech cohomology. In this case, we can define cohomology groups with co-efficients in any sheaf F of abelian groups.Let U = (Ui)i∈I denotes an open covering of a topological space E and Sp the setof sequences (io, . . . , ip) of length p+1 in the set of indices I such that ∩p

j=0Uij6= ∅

(Remark that this condition defines a structure of combinatorial complex on I asfollows : a finite subset J ∈ I is a simplex if and only if ∩i∈JUi 6= ∅). Define

Us = ∩i∈sUi, s ∈ Sp ; Cp = Cp(U ,F) =∏

s∈Sp

F(Us)

then C . is a cosimplicial group in the sense that to a map f : ∆p → ∆q correspondscovariantly with f ; if we denote an element β of Cp as βi0,...,ip ∈ F(Ui0,...,ip), then

f∗ : Cp → Cq : (f∗α)io...iq = (αif(o)...if(p))|Uio∩...∩Uiq

where we use | for the restriction map of sections on Uif(o) ∩ . . .∩Uif(p) to sectionson Uio ∩ . . .∩Uiq . Hence a structure of complex is defined on C . by the differential

d : Cp → Cp+1, (dα)io...ip+1= Σp+1

j=0(−1)jαio...ij ...ip+1

where we write αio...ij ...ip+1for δj(α). The nth cohomology group of C . is denoted

by Hn(U ,F). To obtain a group independent of the covering, one needs to definean inductive limit on the set of all coverings of E using the notion of refinement ofcoverings, then one put

Hn(X,F) = lim−→UHn(U ,F)

When the space is triangulated, the Cech cohomology groups with coefficients inthe constant sheaf Z coincides with simplicial cohomology groups.

2.5. Hypercohomology. Modern theory of sheaves lead to a general theory ofcohomology [G,Gr]. An abelian category A is said to have enough injective objectsif for any object M there exists a monomorphism M → I into an injective objectI of A. In this case, for any complex X . of objects of A , bounded to the left, andany functor Ψ : A → B to an abelian category B, which is left exact, Grothendieckdefines the hypercohomology H∗(Ψ, X .) of Ψ at X . as follows:i)Construct first an injective resolution of X ., that is a complex I . of injectiveobjects in A and a morphism Φ: X . → I . which induces an isomorphism on coho-mology objects (then Φ is called a quasi-isomorphism and an isomorphism in thederived category D+(A)[D]).ii) Define the hypercohomology as an object in B

H∗(Ψ, X .) = H∗(Ψ(I .))

In particular, the previous cohomology groups of a topological space X with co-efficients in Z are isomorphic to the hypercohomology of the functor Γ of global

6 FOUAD EL ZEIN

sections of sheaves of abelian groups on X applied to the constant sheaf Z definedby the group Z on X. The proof is based on the use of adequate resolutions. A sheafis called acyclic for a functor if its hypercohomology vanishes in degree ≥ 1. Usingspectral sequence theory, one can show that all resolutions by acyclic sheaves forthe functor of global sections, will give isomorphic cohomology groups. Godement[G] uses a particular type of acyclic sheaves called flabby. Any sheaf has naturalresolutions by flabby sheaves and injective sheaves are flabby, so we have the samecohomology objects using either resolutions.

2.6. DeRham cohomology. Now let X be a differentiable manifold and let EX

be the sheaf of C∞ real-valued functions on X, EpX the EX -module of C∞ differential

p-forms on X and consider the complex E∗X with its exterior differential.

Definition 2.12. The cohomology groups of the complex of global sections of E∗Xare called De Rham cohomology

HiDR(X,R) := Hi(Γ(X, E∗X)) := Hi(E∗(X)) :=

Ker(di : E i(X) → E i+1(X)Im(di−1 : E i−1(X) → E i(X)

This definition uses the differentiable structure to introduce the above groupscalled DeRham realization of the cohomology, as they give isomorphic groups withprevious definitions of the cohomology.

2.6.1. DeRham resolution. By Poincare lemma this complex is a resolution of theconstant sheaf R and using ”partition of unity” one can prove this is a resolutionby fine sheaves.

Lemma 2.13. The complex of sheaves of differential forms E∗X is a fine resolutionof the constant sheaf R.

Since fine resolutions are acyclic for the global section functor, we deduce theisomorphisms

Hi(X,R) ' HiDR(X,R)

We define also the cohomology of X with complex coefficients using the globalsections of complex valued differential forms

Hi(X,C) ∼−→ Hi(X,R)⊗R C ∼−→ Hi(E∗(X)⊗R C)

We note that the cohomology spaces are defined naturally over Z,R and C, whichprovides a natural lattice inside the real vector space and a real structure insidethe complex cohomology

Hi(X,Z)⊗ R ' Hi(X,R), Hi(X,R)⊗ C ' Hi(X,C)

2.6.2. Comparison with Singular cohomology. Given a differential form ω ∈ Γ(X, EnX)

and a piecewise smooth singular chain of simplices σ ∈ Spsn (X) one can define the

integral∫

σω in R. This integral can be used to construct a map

Γ(X, EnX) → Sn

ps(X,R) := HomR(Spsn (X),R)

defined by (ω, σ) = (−1)n(n+1)/2∫σω.

By Stoke’s theorem :∫

σdω =

∫∂σ

ω where ∂ denotes the boundary map for chains;from which we deduce that the above map extends to a map of complexes.DeRham’s theorem states that the map defined above induces an isomorphism oncohomology groups for each n ∈ N


Theorem 2.14 (DeRham). Integration over piecewise smooth singular cycles in adifferentiable manifold X, defines an isomorphism of cohomology groups

HnDR(X,R) ∼−→ Hn(S∗ps(X,R)) ∼−→ Hn(X,R)

compatible with wedge product on DeRham cohomology and cup-product on singularcohomology.

2.6.3. Poincare duality. It can be checked directly on DeRham cohomology

Theorem 2.15 (Poincare). The bilinear product

Hn−i(X,R)⊗Hn+i(X,R) → H2n(X,R) → R : (α⊗ β) 7→ α ∧ β 7→∫

X

α ∧ β

defines an isomorphism: Hn−i(X,R) ' Hom(Hn+i(X,R),R).

When we identify above the dual space to cohomology with homology, thePoincare duality is stated as an isomorphism

Hn−i(X,R) ' Hn+i(X,R)

Then the cup-product on cohomology corresponds to the topological Intersectiontheory on the manifold X, such that the duality is stated on homology as follows

Hn−i(X,R)⊗Hn+i(X,R) ∩→H0(X,R)degree−−−−→ R : (α⊗ β) 7→ α ∩ β 7→ deg α ∩ β

3. Harmonic forms

In this section X will denote successively a differentiable , complex analytic, andKahler compact manifold.

3.1. Harmonic forms on a differentiable manifold. A Riemannian manifoldX is endowed with a scalar product on its tangent bundle defining a metric on X. Abasic result in analysis states that the cohomology of a compact smooth Riemannianmanifold is represented by real harmonic global differential forms denoted in degreei by Hi(X). To define harmonic forms, we need to introduce the ?-operator andthe Laplacian.

3.1.1. Riemannian metric. A bilinear form g on X is defined at each point x as aproduct on the tangent space Tx to X at x

gx(, ) : Tx ⊗R Tx → R

where gx vary smoothly with x, that is hij(x) := gx( ∂∂xi

, ∂∂xj

) is differentiable inx, then the product is given as gx =

∑i,j hij(x)dxi ⊗ dxj . It is called a metric if

moreover the matrix of the the product defined by hij(x) is positive definite, thatis: gx(u, u) > 0 for all u 6= 0 ∈ Tx.An induced metric is defined on Ω1

x := T ∗x := Hom(Tx,R) and on Ωpx as follows.

Let e1, . . . , en be an orthonormal basis, then on Ωpx we define the unique metric

such that e∗i1 ∧ . . . ,∧e∗ipis an orthonormal basis where (e∗i )i∈I form a dual basis to

(ei)i∈I .

8 FOUAD EL ZEIN

3.1.2. Volume form, L2 metric. Suppose now that the manifold is oriented, thenan orthonormal positively oriented basis ei define a nowhere vanishing section ofΩn

X called volume section vol satisfying volx := e∗1 ∧ . . . ,∧e∗n.

Exercise 3.1. Let x1, . . . , xn denote a local ordered set of coordinates on an opensubset U of a covering of X, compatible with the orientation. Prove that√

det(hij)dx1 ∧ · · · ∧ dxn

defines the global volume form.

We deduce from the volume form a global pairing called the L2 metric

∀ψ, η ∈ E i(X), (ψ, η)L2 =∫

X

gx(ψ(x), η(x))vol(x)

3.1.3. Laplacian. We prove the existence of an adjoint d∗ : E i+1X → E i

X to thedifferential d : E i

X → E i+1X , satisfying

(dψ, η)L2 = (ψ, d∗η)L2 , ∀ψ ∈ E i(X), η ∈ E i+1(X).

The adjoint operator is defined by constructing first an operator ∗E i

X∗−→ En−i

X

by requiring at each point x ∈ X

(ψ, η)volx = ψx ∧ ∗ηx, ∀ψ, η ∈ E iX,x.

The section vol defines an isomorphism R ' EnX , inducing the isomorphism

Hom(E iX ,R) ' Hom(E i

X , EnX) ' En−i

X . The definition of ∗ is obtained by compo-sition with the isomorphism E i

X → Hom(E iX ,R) defined by the scalar product on

E iX . We have on E i

X

∗2 = (−1)i(n−i)Id, d∗ = (−1)n+in+1 ∗ d ∗.Lemma 3.2. We have for all α, β ∈ E i(X)

(α, β)L2 =∫

X

α ∧ ∗β.

Definition 3.3. The Laplacian ∆ is defined as

∆ = d∗ d + d d∗

Harmonic forms are defined as the solutions of the Laplacian

Hi(X) = ψ ∈ E i(X) : ∆(ψ) = 0The importance of harmonic forms in Hodge theory stems from the representa-

tion of cohomology. However they have the following interesting property.Consider φ ∈ E i(X) satisfying dφ = 0 and its cohomology class φ + dψ ∈ E i(X).The Laplace equation defines exactly the form in this class with minimal L2−norm.First note that the class is an affine subspace of the E i(X). To show the existenceof the solution we introduce the Hilbert space of L2−forms; then the closure of theaffine subspace contains a unique element of minimal distance to 0. A remarkableresult in the theory of elliptic differential equations [Warner, Wells] proves the reg-ularity of the solution that is, such element is in fact in the affine space and notonly in its completion. The link to Laplace equation is as follows. An element φhas minimal norm if

∀t ∈ R, ‖φ‖2L2 ≤ ‖φ + dψ‖2L2


The expansion of the right term in t, ‖φ‖2L2 + 2t(φ, dψ)L2 + O(t2) shows that theinequality holds if and only if (φ, dψ)L2 = 0, and equivalently (d∗φ, ψ)L2 = 0 forall ψ, hence d∗φ = 0. That is dφ = 0 (φ is closed), d∗φ = 0 (φ is co-closed)and ∆(φ) = 0. Reciprocally if ∆(φ) = 0, then 0 = (∆(φ), φ)L2 = (dd∗φ, φ)L2 +(d∗dφ, φ)L2 = ‖d∗φ‖2L2 + ‖dφ‖2L2 , hence φ is closed and co-closed.A basic result admitted here, is

Theorem 3.4. On a compact smooth oriented Riemannian manifold each coho-mology class is represented by a unique real harmonic global differential form

Hi(X) ' Hi(X,R).

3.2. Complex manifolds. An important feature of a complex manifold is theexistence of an almost complex structure J on the real tangent bundle.

3.2.1. Almost complex structure. Let V be a real vector space. A almost complexstructure on V is defined by a linear map J : V → V satisfying J2 = −1; thendim.V is even and the eigenvalues are i and −i with associated eigenspaces inVC = V ⊗R C

V + = x− iJx : x ∈ V ⊂ VC, V − = x + iJx : x ∈ V ⊂ VC.

Example 3.5. 1- The space R2 with the action of J defined on the canonical basisby J(1, 0) := (0, 1).The isomorphism ϕ : C ∼−→ R2, ϕ(1) = (1, 0), ϕ(i) = (0, 1) is compatible with theaction of i on C and the action of J on R2, hence the action of J amounts to definea complex structure for which J is the multiplication by i.2- Let V be a complex vector space, then the isomorphism ϕ : V → VR with thereal underlying space to V transports the action of i to an action J . The image of acomplex basis (ei)i∈[1,n] defines a basis a real basis ϕ(ei), J(ϕ(ei)) = J(iϕ(ei), i ∈[1, n].Reciprocally, a real vector space W with the action of J corresponds to a structureof complex vector space on V as follows. Let (ei)i∈[1,n] be a subset of W suchthat ei, J(ei)i∈[1,n] form a basis of W ( first choose a non zero vector e1 in Wand prove that W1 generated by e1, J(e1) is of dimension two, then continue withe2 /∈ W1 and so on). Then there is a complex structure on W for which ej , j ∈ [1, n]is a complex basis with the action of i ∈ C defined by i.ej := J(ej).

3.2.2. Decomposition into types. Let V be a complex vector space and (VR, J) theunderlying real vector space with its involution J defined by multiplication by i.Let W := HomR(VR,R) and WC := HomR(VR,C) ' W ⊗ C, then HomC(V,C)embeds into WC and its image is denoted W 1,0 while its conjugate with respect toW is the subspace denoted W 0,1, moreover

WC = W 1,0 ⊕W 0,1, ∧2WC = W 2,0 ⊕W 1,1 ⊕W 0,2

where we define resp. the spaces of forms of type (2, 0), (1, 1), (0, 2) as :

W 2,0 = W 1,0 ⊗W 1,0,W 1,1 = W 1,0 ⊗W 0,1,W 0,2 = W 0,1 ⊗W 0,1.

3.2.3. Decomposition into types on the complexified tangent bundle. The existenceof a complex structure on the manifold leads to an almost complex structure on thetangent bundle TX,R and a decomposition into types of the complexified tangentbundle TX,R ⊗R C which splits as a direct sum T 1,0

X ⊕ T 0,1X where T 1,0

X,z = u− iJu :

10 FOUAD EL ZEIN

u ∈ TX,R,z and T 1,0X = T 0,1

X . Precisely, if zj = xj + iyj , j ∈ [1, n] are local complexcoordinates, then J( ∂

∂xj) = ∂

∂yjin TX,R,z. The holomorphic tangent space TX,z

embeds isomorphically into T 1,0X generated by ∂

∂zj:= 1

2 ( ∂∂xj

− i ∂∂yj

) of the formuj − iJ(uj). From now on we identify the two bundles

TX∼−→ T 1,0

X

The dual of the above decomposition of the tangent space is written as E1X ⊗R C '

E1,0X ⊕ E0,1

X which induces a decomposition of the sheaves of differential forms intotypes

E iX ⊗R C ' ⊕p+q=i Ep,q

X

In terms of complex local coordinates on an open set U , φ ∈ Ep,q(U) is written asa linear combination of

dzI ∧ dzJ := dzi1 ∧ · · · ∧ dzip∧ dzj1 ∧ · · · ∧ dzjq

.

where I = i1, · · · , ip, J = j1, cdots, jq . The differential decomposes as as welld = ∂ + ∂, where ∂ : Ep,q

X → Ep,q+1X and ∂ : Ep,q

X → Ep+1,qX are compatible with the

decomposition up to a shift on the bidegree, we deduce from d2 = ∂2 +∂2+∂ ∂ +

∂ ∂ = 0 that the DeRham complex is the simple complex associated to a doublecomplex

(E∗X ⊗R C, d) ' s((E∗,∗X , ∂, ∂)

In particular we recover the notion of a subspace of cohomology type (p, q).

Hp,q(X) := Cohomology classes representable by a closed form of type (p, q) We have Hp,q(X) = Hq,p(X) but it is not always true that Hi

DR(X) = ⊕p+q=iHp,q(X).

Hodge theory apply to a complex manifold when exactly such decomposition exists.

Remark 3.6. Given a bundle of real vector spaces with differentiable action ofJ , the problem to lift this action into a complex structure on the bundle is notstraightforward. An important result in this direction is the Newlander-Nirenbergtheorem.

Example 3.7. 1 - On a compact complex torus X = Cn/Λ, a basis of the coho-mology is given by the translation-invariant forms, hence with constant coefficientssince the tangent space is isomorphic to a direct sum of the structural sheaf E0

X .Hence the decomposition reduces to the case of R2n = Cn ⊕ Cn.2 - The case of a compact Riemann surface , reduces to prove

H1(X,C) ' H1,0(X)⊕H0,1(X).

3.2.4. Holomorphic Poincare lemma. On complex manifolds the sheaf of holomor-phic forms Ω1

X is an important geometric invariant. The holomorphic version ofPoincare lemma resolution lemma shows that the DeRham complex of holomorphicforms Ω∗X is a resolution of the constant sheaf C. However, since the resolutionis not acyclic, cohomology spaces are computed only as hypercohomology of theglobal sections functor, that is after acyclic resolution.

Hi(X,C) ' Hi(X, Ω∗X) := RiΓ(X, Ω∗X) ' Hi(E∗(X)⊗R C)


3.2.5. Dolbeault cohomology. The complex (Er,∗X , ∂) for each r ≥ 0, is called a Dol-

beault resolution since Dolbeault’s result shows it is a resolution of Ω∗X .

Lemma 3.8. The Dolbeault complex (Er,∗X , ∂) for r ≥ 0, is a fine resolution of Ωr

X ,hence

Hi(X, ΩrX) ' Hi(Er,∗

X (X), ∂) := Hr,i

∂(X)

The cohomology of the complex of its global sections is called the ∂ cohomologyof X of type (r, i)

Hr,i

∂(X) := Hi(Er,∗

X (X), ∂) =Ker(∂

i: Er,i(X) → Er,i+1(X))

Im(∂i−1

: Er,i−1(X) → Er,i(X))

A cohomology class of X of type (r, i) defines a ∂ cohomology class of the sametype. It will follows that this map defines an isomorphism whenever Hodge theoryapply.

3.3. Hermitian metric. A Hermitian product h on a complex manifold X isdefined continuously on the tangent space.

3.3.1. Hermitian metric on a complex vector space. Let V be a complex vectorspace and (VR, J) the underlying real vector space with its involution J defined bymultiplication by i. A Hermitian product h on V is a product

h : V × V → C, satisfying h(u, v) = h(v, u)

which is biadditive, linear in the first term and antilinear in the second term.Considering the restriction to VR

h|VR = Reh + iImh

The product defines a Hermitian metric if moreover the real number h(u, u) > 0 ispositive definite for all u 6= 0 ∈ V , then we deduce a Riemannian structure on VRdefined by g := Reh.

Definition 3.9 ( The (1,1) form ω). It is associated to the hermitian metric a real2−form ω := −Imh on VR , satisfying:ω(Ju, v) = −ω(u, Jv), and ω(Ju, Jv) = ω(u, v).

Proof. We have a 2−form since ω(v, u) = −Imh(v, u) = Imh(u, v) = −ω(u, v).We checkω(u, Jv) = −Imh(u, iv) = g(u, v), ω(Ju, v) = −Imh(iu, v) = −g(u, v)ω(Ju, Jv) = −Imh(iu, iv) = −Imh(u, v) = ω(u, v). ¤

Remark 3.10. The hermitian form is entirely determined by g or ω since g(u, v) =ω(u, Jv) and ω(u, v) = −g(u, Jv), as one can deduce from the identity: h(u, iv) =−ih(u, v).

Let W := HomR(VR,R) and WC := HomR(VR,C) ' W ⊗ C, recall the decom-positions

WC = W 1,0 ⊕W 0,1, ∧2WC = W 2,0 ⊕W 1,1 ⊕W 0,2

Considering ∧2RW ⊂ ∧2WC, we introduce the intersection in ∧2WC

W 1,1R := W 1,1 ∩ (∧2

RW ) ⊂ ∧2WR

12 FOUAD EL ZEIN

Lemma 3.11. The correspondence: h → −Imh is a natural isomorphism betweenthe real vector space of Hermitian forms on V ×V and the space W 1,1

R of real formsof type (1, 1).

Definition 3.12. A real form of type (1, 1) is positive if the corresponding Her-mitian form is positive definite.

Let ei of V denote a basis of V . The projections dzj , j ∈ [1, n], denote a dualbasis of the dual space V ∗; then the Hermitian product is written as

h =∑

i,j

hijdzi ⊗ dzj ∈ V ∗ ⊗C V∗ ' W 1,0 ⊗W 0,1

where hij = h(ei, ej) and V is the complex conjugate space to V . Let H = (hij)denotes the matrix with coefficients hij (equal to its conjugate transpose), then

h(u, v) = tV HU

where U, V are the coordinates of u and v written in columns. For a Hermitianmetric, the matrix H is positive definite.We can always suppose the basis orthonormal, that is H = Id. If we write dzj =dxj + idyj , then

h =∑

j

dzj ⊗ dzj =∑

j

(dxj ⊗ dxj + dyj ⊗ dyj)− i∑

j

(dxj ⊗ dyj − dyj ⊗ dxj)

where∑

j(dxj ⊗ dxj + dyj ⊗ dyj) defines the Riemannian metric andω =

∑j dxj ∧ dyj :=

∑j(dxj ⊗ dyj − dyj ⊗ dxj) = i

2

∑j dzj ⊗ dzj (note that we

avoid the factor 12 in some definition of

∑j dxj ∧ dyj).

The volume form vol ∈ ∧2nVR, associated to g, can be computed via ω and is equalto 1

n!ωn. We have

ω =i

2Σjdzj ∧ dzj = Σjdxj ∧ dyj , vol =

ωn

n!= dx1 ∧ dy1 ∧ · · · ∧ dxn ∧ dyn.

3.3.2. Extensions of h to W p,q. Note that the hermitian product h extends toh1,0 on W 1,0 by the condition that the dual basis dzjj∈[1,n] to an orthonormalbasis ejj∈[1,n] of V , is orthonormal for h1,0. On W 0,1 the basis dzjj∈[1,n] isorthonormal for the extension h0,1.Independently, g on V extends by the condition of orthonormality to W and bylinearity to W ⊗ C, then we define h1 on WC by h1(u, v) := g(u, v). Moreover wedefine hp,q on W p,q ∧p W 1,0 ⊗ ∧qW 0,1 and hk on ∧kWC

Lemma 3.13. The extension hk of h is equal to 2k∑

hp,q on ∧kWC '∑

p+q=k W p,q.

In fact, h1,0(dzj , dzj) = 1 while h1(dzj , dzj) = g(dxj + idyj , dxj − idyj)= g(dxj , dxj)+g(dyj , dyj)+i(g(dyj , dxj)−g(dxj , dyj) = 2. Similarly h1(dzj , dzj) =2.

3.3.3. Hermitian metric on X. The Hermitian product h on the complex manifoldX is defined at each point z as a Hermitian product on the holomorphic tangentspace Tz to X at z

hz : Tz × Tz → Cmoreover hz vary smoothly with z, that is hij(z) := hz( ∂

∂zi, ∂

∂zj) is differentiable in

z and hij(z) = hji(z).


The isomorphism Tz∼−→ T 1,0

z . The holomorphic tangent space is generated by ∂∂zi

and the isomorphism is defined by∂

∂zi=

12(

∂

∂xj− i

∂

∂yj)

in T 1,0z ⊂ TX,R,z ⊗ C. Hence it is equivalent to define the Hermitian product as

hz : T 1,0z × T 1,0

z → C, hz =∑

i,j

hij(z)dzi ⊗ dzj ∈ E1,0z ⊗C E0,1

z

The product is called a Hermitian metric if moreover the real number hz(u, u) > 0is positive definite for all u 6= 0 ∈ Tz; equivalently the matrix of the product definedby hij(z) (equal to its conjugate transpose) is positive definite.Considering the R−linear isomorphism

TX,R,z∼−→ T 1,0

z :∂

∂xj7→ ∂

∂zj,

∂

∂yj= J(

∂

∂xj) 7→ i

∂

∂zj

with inverse defined by the real part u of u + iv, and compatible with the action ofJ on TX,R,z and i on T 1,0

z ; the induced product is written as

hz|TX,R,z= Rehz + iImhz

We associate to the metric a Riemannian structure on X defined by gz := Rehz.Then X of dimension n is viewed as a Riemann manifold of dimension 2n withmetric gz. Note that g is defined over the reals since it can be represented by a realmatrix over a basis of the real tangent space.Since hz is hermitian, we have gz(Ju, Jv) = gz(u, v).

3.3.4. The (1,1) form ω. We associate also to the metric a real 2−form ω := −ImH

of type (1, 1) (ω ∈ E1,1X ∩ E2

X). The volume form vol ∈ E2n defined by g on X, canbe defined by ω and is equal to 1

n!ωn. We have locally

ω =i

2Σidzi ∧ dzi, vol =

ωn

n!= dx1 ∧ dy1 ∧ · · · ∧ dxn ∧ dyn.

Reciprocally, the metric can be recovered from ω or from g.

Example 3.14. 1) The Hermitian metric on Cn is defined by h :=∑n

i=1 dzi ⊗ dzj

with induced metric the standard metric on R2n

H(z, w) =n∑

i=1

ziwi

where z = (z1, . . . , zn) and w = (w1, . . . , wn). By writing zk = xk + iyk, it ispossible to consider Cn = R2n, in which case ReH is the Euclidean inner productand ImH is a nondegenerate alternating bilinear form, i.e., a symplectic form.Explicitly, in C2, the standard Hermitian form is expressed as

H((z1,1, z1,2), (z2,1, z2,2)) =x1,1x2,1 + x1,2x2,2 + y1,1y2,1 + y1,2y2,2

+ i(x2,1y1,1 − x1,1y2,1 + x2,2y1,2 − x1,2y2,2).

2) If Λ ⊂ Cn is a full lattice, then the above metric induces a Hermitian metric onthe torus C

n

Λ .3) The Fubini -Study metric on the projective space Pn. Let π : Cn−0 → Pn be theprojection and consider a section Z : U → Cn−0 of π on an open subset U , then theform ω = i

2π ∂∂log||Z||2 is a (1, 1) form, globally defined on Pn, since we can check

14 FOUAD EL ZEIN

that forms on two open subsets glue together on the intersection. For the sectionZ = (1, w1, . . . , wn), ω = i

2π

∑ni=1 dwi ∧ dwi > 0 is a real form (takes real value on

real tangent vectors) and positive that is associated to a positive Hermitian form.Hence the projective space admits a Hermitian metric.

3.4. Harmonic forms on compact complex manifolds. In this section, a studyof the ∂ operator (similar to the previous study of d) will lead to the representa-tion of Dolbeault cohomology by ∂ harmonic forms as stated in Hodge theorem.We choose a Hermitian metric h with associated (1, 1) form ω with volume formvol = 1

n!ωn. Then we define the L2−inner product on Ep,q

X using the underlyingRiemannian metric g and the adjoint operator to ∂

∂∗

: Ep,qX → Ep,q−1

X

defined as ∂∗

= − ∗ ∂∗ satisfying

(∂∗ψ, η)L2 = (ψ, ∂η)L2

so to introduce the ∂−Laplacian

Definition 3.15. The ∂−Laplacian is defined as

∆∂ = ∂ ∂∗

+ ∂∗ ∂

Harmonic forms of type (p, q) are defined as the solutions of the ∂−Laplacian

Hp,q(X) = ψ ∈ Ep,q(X)⊗ C : ∆∂(ψ) = 0.We define similarly ∂∗ = − ∗ ∂∗ and a Laplacian

∆∂ = ∂ ∂∗ + ∂∗ ∂

A basic result is

Theorem 3.16 (Hodge theorem). On a compact complex Hermitian manifold eachcohomology class of type (p, q) is represented by a unique ∂−harmonic global complexdifferential form of type (p, q)

Hp,q(X) ' Hp,q

∂(X)

moreover the space of ∂−harmonic forms is finite dimensional (hence the space of∂−cohomology also).

Corollary 3.17.Hq(X, Ωp

X) ' Hp,q(X)

Corollary 3.18 (Serre duality). The trace map is an isomorphism

Tr : Hn(X, ΩnX) ∼−→ H2n(X,C) ∼−→ C, ω 7→ 1

(2iπ)n

∫

X

ω

and the pairing is

Hp(X, ΩnX)⊗Hn−p(X, Ωn

X) → Hn(X, ΩnX)

is nondegenerate.

Notice that we introduce a factor 1(2iπ)n of the integral to fit later with the fact

that Hn(X, ΩnX) is a Hodge structure of weight 2n.


3.5. Kahler manifolds. In general the exterior product of harmonic forms is notharmonic neither the restriction of harmonic forms to a submanifold is harmonicfor the induced metric. The ∂−Laplacian ∆∂ and the Laplacian ∆d are not relatedin general. The theory will be more natural if we add the Kahler condition onthe metric, most importantly the components of harmonic forms into type will beharmonic.

Definition 3.19. The Hermitian metric is Kahler if its associated (1, 1) form ω isclosed: dω = 0.

Example 3.20. 1) A Hermitian metric on a compact Riemann surface is Kahlersince dω of degree 3 must vanish.2) The Enclidean metric on a compact complex torus is Kahler.3) The projective space with its canonical metric is Kahler.4) The restriction of a Kahler metric on a submanifold is Kahler with associated(1, 1) form induced from the associated (1, 1) form on the ambient manifold.5) The product of two Kahler manifolds is Kahler.

Corollary 3.21. 1) The fundamental class of a subvariety Y ⊂ X is non zero inH∗

DR(X,C). 2) The even Betti numbers b2q(X) are positive. Moreover

H0(X, ΩqX) → Hq

DR(X,C)

We have in this case the following important relations of the Laplacians

Lemma 3.22.∆d = 2∆∂ = 2∆∂

Consider the projections

Πp,q : Er(X)⊗ C→ Ep,q(X)

Corollary 3.23. The projection on the (p, q) component of an harmonic form isharmonic and we have a natural decomposition

Hr(X)⊗ C = ⊕p+q=rHp,q(X), Hp,q(X) = Hq,p(X)

One prove first that the Laplacian ∆d commutes with the projection Πp,q, hencethe (p, q) components of an harmonic form, are harmonic. Since ∆d is real, wededuce the conjugation property.

3.5.1. Hodge decomposition. Recall the definition of the type of cohomology

Hp,q(X) =Zp+q

d (X)dE∗(X) ∩ Zp+q

d (X)where Zp+q

d (X) = Ker d ∩ Ep,q(X)

Theorem 3.24 (Hodge decomposition). Let X be a compact Kahler manifold.There is an isomorphism of cohomology classes of type (p, q) with harmonic formsof the same type

Hp,q(X) ' Hp,q(X).which results into a decomposition of the complex cohomology spaces into a directsum of complex subspaces

Hi(X,C) = ⊕p+q=iHp,q(X), Hp,q(X) = Hq,p(X)

called Hodge decomposition.

16 FOUAD EL ZEIN

Corollary 3.25. There exists a Hodge decomposition on the cohomology of asmooth complex projective variety.

3.5.2. Applications to Hodge theory. We remark first the isomorphisms

Hp,q(X) ' Hp,q

∂(X) ' Hq(X, Ωp

X)

which shows in particular for q = 0

Hp,0(X) ' H0(X, ΩpX)

1) The holomorphic forms are closed and harmonic for any Kahler metric.2) There are no non zero global holomorphic forms on Pn and more generally

Hp,q

∂(Pn) ' Hq(Pn, Ωp

Pn) =

0, if p 6= q

C if p = q

Exercise 3.26 ( Holomorphic invariants in cohomology). Any holomorphic p−formon a compact Kahler manifold is closed and it is 0 iff it is exact. Equivalently, thereis a natural injective morphism: global holomorphic p−forms → Hp

DR with imageHp,0.

Proof. Let φ be holomorphic of type p, 0, then ∂φ = 0 and ∂∗φ of type p,−1 must

vanish, hence the Laplacian ∆∂φ = 0, hence on Kahler manifold ∆dφ = 0 anddφ = 0.Since φ is harmonic, it is orthogonal to the space of exact forms, hence if it is exactit must be 0. ¤

Remark 3.27. i) The form zdw on C2 is not closed.ii) The spaces Hp,q in Hp+q

DR are isomorphic to the holomorphic invariant Hq(X, Ωp),but for a family the embedding of Hp,q(Xt) into DeRham cohomology is not holo-morphic.

Exercise 3.28 (Riemann Surface). Let C be a compact Riemann surface, and letC = C − x1, . . . , xm be the open surface with m points in C deleted. Consider thelong exact sequence

0 → H1(C,Z) → H1(C,Z) → ⊕i=mi=1 Z→ H2(C,Z) ' Z→ H2(C,Z) = 0

then0 → H1(C,Z) → H1(C,Z) → Zm−1 ' Ker(⊕i=m

i=1 Z→ Z)

presents H1(C,Z) as an extension, with weight W1H1(C,Z) = H1(C,Z) and

W2H1(C,Z) = H1(C,Z).

The Hodge filtration is defined by the residue

0 → Ω1C→ Ω1

C(logx1, . . . , xm) → Ox1,...,xm → 0

F 0H1(C,C) = H1(C,C),F 1H1(C,C) = H0(C, Ω1

C(logx1, . . . , xm).

F 2H1(C,C) = 0. That is an extension of pure HS

0 → H1(C) → H1(C) → Ker (⊕i=mi=1 Z(−1) → H2(C) → 0

.


Exercise 3.29 (Algebraic cycles). Let X be a compact Kahler manifold and ω :=−ImH its real 2−form of type (1, 1) (ω ∈ E1,1

X ∩ E2X). The volume form vol ∈ E2n

defined by g on X, can be defined by ω and is equal to 1n!ω

n. We have locally

ω =i

2Σidzi ∧ dzi, vol =

ωn

n!= dx1 ∧ dy1 ∧ · · · ∧ dxn ∧ dyn.

Lemma 3.30. For a manifold X compact and Kahler Hp,p(X,Z) 6= 0 for 0 ≤ p ≤dim.M .

In fact the integral of the volume form∫

Xωn > 0. It follows that the cohomology

class ωn 6= 0 ∈ H2n(X,C), hence the cohomology class ωp 6= 0 ∈ Hp,p(X,C) sinceits cup product with ωn−p is not zero.

3.6. Cohomology class of a subvariety. A compact oriented manifold X of realdimension m is triangulated and the sum of the oriented simplices of dim. m definea cycle class [X] ∈ Hm(X,Z). The Poincare dual of [X] is called a cohomlologyfundamental [X] ∈ H0(X,R). In particular, if X is a complex manifold of dim. n,then it has a class [X] ∈ H2n(X,Z). Let Z be a complex submanifold of codim. pin X and i : Z → X the embedding, then i∗[Z] ∈ H2(n−p)(X,Z) corresponds, byrational Poincare duality, to a class in H2p(X,Z). Let

Hp,p(X,Z) := a ∈ H2p(X,Z), Im (a) ∈ Hp,p(X,C)We define similarly the rational version by taking the image Hp,p(X,Z) in the ratio-nal cohomology and by Poincare duality its homology version the image Hp,p(X,Z)in the rational homology.

Definition 3.31. For a complex compact manifold X the cohomology class [ηZ ] ∈Hp,p(X,Z) of a complex submanifold Z of codim p is defined by the followingrelation

∀ω ∈ En−p,n−p(X),∫

X

ω ∧ ηZ =∫

i∗[Z]

ω.

Lemma 3.32. For a manifold X compact and Kahler the rational cohomology classof a complex submanifold Z of codim p is a non zero element [ηZ ] 6= 0 ∈ Hp,p(X,Z)for 0 ≤ p ≤ dim X.

Proof. For a compact Kahler manifold X, let ω be a Kahler form, then the integralon the homology class [Z] of the restriction i∗ω is positive since it is a Kahler formon Z ∫

i∗[Z]

∧n−pω =∫

[Z]

i∗ ∧n−p ω > 0.

¤

Lemma 3.33. For a projective smooth variety X, the fundamental class of anirreducible subvariety Z of codimension p is well defined and it is a non zero element[ηZ ] 6= 0 ∈ Hp,p(X,Z) for 0 ≤ p ≤ dimX.

The easiest proof is to use a desingularisation π : Z ′ → Z and define [Z] =π∗[Z ′] ∈ Hn−p,n−p(X,Z), then take its Poincare dual in Hp,p(X,Z).

Definition 3.34. An r−cycle of an algebraic variety X is a formal finite linearcombination Σi∈[1,h]miZi of irreducible subvarieties Zi with integer coefficients mi.The group of r−cycles is denoted by Zr(X).

18 FOUAD EL ZEIN

The class of irreducible subvarieties extend into a linear morphism

cl : Zr(X) → Hr,r(X,Z)

defined modulo linear equivalence of cycles. The cohomology cycles in the imageof the cycle class map are called algebraic. Respectively we define the map

clQ : Zr(X)⊗Q→ Hr,r(X,Q)

3.6.1. Hodge conjecture. Is the map clQ surjective?

4. Polarized Hodge structure

4.1. Hodge structure. Having the above properties of cohomology of Kahlermanifolds in mind, it appeared to be rewarding to introduce this notion as a formalstructure in linear algebra

Definition 4.1. A Hodge structure (HS) of weight n is defined by the datai)A free abelian group HZ (the lattice)ii) A decomposition by complex subspaces

HC := HZ ⊗Z C = ⊕p+q=nHp,q satisfying Hp,q = Hq,p

Remark 4.2. The conjugation on HC makes sense with respect to HZ. A subspaceV ⊂ HC := HZ ⊗Z C satisfying V = V has a real structure, that is V = (V ∩HR)⊗R C. In particular Hp,p = (Hp,p ∩HR)⊗R C.

Example 4.3. Tate Hodge structure Z(1) is defined by

HZ = 2iπZ ⊂ C, HC = H−1,−1

The tensor product of Hodge structures is defined in general. In this case then−tensor product Z(1)⊗ · · · ⊗ Z(1) of Z(1) is defined by

Z(m) := (HZ = (2iπ)mZ ⊂ C, HC = H−m,−m)

and the Tate twist of a HS H = (HZ,⊕p+q=nHp,q) is H(m) defined by

(HZ ⊗ (2iπ)mZ,⊕p+q=nHp−m,q−m)

The DeRham cohomology Hi(X,R) of a compact Kahler manifold is a HS ofweight i with lattice defined by the image of integral cohomology Im : Hi(X,Z) →Hi(X,R), in particular if we consider the following trace map

H2n(X,R) ∼−→ R(−n), ω → 1(2iπ)n

∫

X

ω.

then Poincare duality will be compatible with HS

Hn−i(X,R) ' Hom(Hn+i(X,R),R(−n)).

Theorem 4.4 (Deligne). Let X be a smooth complex algebraic variety. There existsa decomposition of the complex cohomology spaces into a direct sum of complexsubspaces

Hi(X,C) = ⊕p+q=iHp,q(X), Hp,q(X) = Hq,p(X)

defining a HS of weight i on cohomology.


4.1.1. The Hodge filtration. To study variations of HS, Griffiths introduced anequivalent structure defined by the Hodge filtration which vary holomorphically.Given a HS: (HZ,Hp,q) of weight i, we define a filtration F ∗ on HC by subspaces

F pHC := ⊕r≥pHr,i−r

the filtration F is decreasing such that the following decomposition is satisfied

HC = F pHC ⊕ F i−p+1HC

The Hodge decomposition is then determined by the filtration as

Hp,q = F pHC ∩ F qHC, p + q = i

4.1.2. The Hodge filtration on the DeRham complex. The Hodge filtration on coho-mology plays an important role since its definition does not use the theory of har-monic forms, although the Hodge result on the filtration is based on such theory. Itis defined by a filtration F of the holomorphic DeRham complex by subcomplexes

F pΩ∗X := 0 · · · 0 → ΩpX → · · ·Ωq

X → · · ·ΩnX

The simple complex, associated to the resolution of each term ΩrX by a Dolbeault

complex (Er,∗X , ∂) for r ≥ p

F pEX = s(Er,∗X , ∂), ∂)r≥p

is a fine resolution of (F pΩ∗X , d = ∂) and define a Hodge filtration by subcomplexesof EX ⊗R C. We have

Hi(F pE∗(X), d) ' F pHi(X,C)

Remark 4.5. We will see that the spectral sequence defined by the Hodge filtrationdegenerates at rank one.

4.2. Lefschetz decomposition. Let [ω] ∈ H2(X,R) denotes the Kahler class ofX. The cup-product with [ω] define morphisms

L : Hq(X,R) → Hq+2(X,R), L : Hq(X,C) → Hq+2(X,C)

Let n = dim X and define the primitive subspaces

Hn−iprim(R) := Ker(Li+1 : Hn−i(X,R) → Hn+i+2(X,R))

and similarly for complex coefficients Hn−iprim(C) ' Hn−i

prim(R) ⊗R C, then we de-duce from the following Lefschetz isomorphism theorem a decomposition of thecohomology in terms of primitive spaces

Theorem 4.6 (Lefschetz ). The iterated linear operator L induces isomorphismsfor each i

Li : Hn−i(X,R) ∼−→ Hn+i(X,R), Li : Hn−i(X,C) ∼−→ Hn+i(X,C)

It gives rise to a decomposition of the cohomology into direct sum of image ofprimitive subspaces by Lr, r ≥ 0

Hq(X,R) = ⊕r≥0LrHq−2r

prim (R), Hq(X,C) = ⊕r≥0LrHq−2r

prim (C)

Since the operator L acting on the HS is of bidegree (1, 1), the decomposition andthe primitive subspaces are compatible with HS.If X is moreover projective, then the action of L is defined with rational coefficientsand the decomposition apply on rational cohomology.

20 FOUAD EL ZEIN

Proof. 1) First we consider the action of L on sheaves, L = ∧ω : ErX → Er+2

X , thenwe introduce its adjoint operator Λ = L∗ : Er

X → Er−2X which can be defined by

Λ = ∗−1 L ∗. Note that the operator

h =2n∑

p=0

(n− p)Πp

where Πp is the projection in degree p on E∗X and n = dim X satisfy the relation

[Λ, L] = h.

from which we can deduce the injectivity of the morphism

Li : En−iX → En+i

X

For this we use the formula

[Lr, Λ] = (r(k − n) + r(r − 1))Lr−1

which is proved by induction on r. Such morphism commutes with the Laplacianand since cohomology classes can be represented by global harmonic sections, itinduces an isomorphism on cohomology vector spaces of finite equal dimension

Li : Hn−i(X,R) ∼−→ Hn+i(X,R)

Moreover the extension of the operator L to complex coefficients is compatible withthe bigrading (p, q) since ω is of type (1, 1). The decomposition of the cohomologyinto direct sum of image of primitive subspaces by Lr, r ≥ 0 follows from theprevious isomorphisms.2) Another proof is based on the representation theory of the Lie algebra sl2. Werepresent L by the action on global sections of the operator L = ∧ω : Ep,q(X) →Ep+1,q+1(X) and its adjoint Λ = L∗ : Ep,q(X) → Ep−1,q−1(X) is defined by Λ =∗−1 L ∗. We admit the relations

[Λ, L] = h

[h,L] = −2L

[h, Λ] = 2Λ

We deduce from the action of the operators L, h, Λ on the space of harmonic forms⊕H∗d(X) ' ⊕H∗(X), a representation of the Lie algebra sl2 by identifying thematrices (

0 10 0

),

(0 01 0

),

(1 00 −1

)resp. withΛ, L, h.

Then the theorem follows from the general structure of such representation. ¤

A geometric interpretation of this theorem in the case of a projective subvarietyX ⊂ PN of dimension n is given via Poincare isomorphism Hn−k(X) ∼−→ Hn−k(X),so that the operator L corresponds to the intersection with an hyperplane sectionV of X so that the result is an isomorphism

Hn+k(X) ∩PN−k

−−−−→ Hn−k(X)

The primitive cohomology Hn−kprim(X) corresponds to a subspace Hf

n−k(X) ⊂ Hn−k(X)equal to the image of Hn−k(X − V ) → Hn−k(X) (f for finite cycles or outside in-finity).


4.3. Polarization of HS. We define in this subsection a scalar product on coho-mology compatible with HS and satisfying relations known as Hodge Riemann re-lations leading to a polarization of the primitive cohomology which is an additionalhighly rich structure on cohomology of proper algebraic varieties. Such product isdefined via the wedge product on DeRham cohomology, or cup-product on singu-lar cohomology and called sometimes an intersection form (referring implicitly tointersection on homology, with which it is compatible via Poincare duality).

4.3.1. Hermitian product on cohomology. Representing cohomology classes by dif-ferential forms, we define a bilinear form

Q(α, β) = (−1)j(j−1)

2

∫

X

α ∧ β ∧ ωn−j , ∀[α], [β] ∈ Hj(X,C)

where ω is the Kahler class. On projective varieties the Kahler class is in the integrallattice in rational cohomology, the product is defined on rational cohomology andpreserves the integral lattice. It is alternating if j is odd, symmetric if j is even andnon degenerate by Lefschetz isomorphism and Poincare duality. By considerationof type, the Hodge and Lefschetz decompositions are orthogonal with respect to Q

Q(Hp,q,Hp′,q′) = 0, unless p = q′, q = p′.

Proposition 4.7 (Hodge-Riemann bilinear relations). On the primitive componentHp,q

prim := Hj(X,C)p,qprim

ip−qQ(α, α) > 0, ∀α 6= 0 ∈ Hp,qprim

the form ip−qQ(α, α) is positive definite.

Proof. We use

Lemma 4.8. Let ω ∈ Ωq,pX,x ⊂ Ej

X,x be a primitive element, then

∗ω = (−1)j(j+1)

2 iq−p Ln−j

(n− j)!ω

We represent a primitive cohomolgy class by a primitive harmonic form α ofdegree j, then since (−1)jiq−p = (−1)q−piq−p = ip−q we deduce from the lemmaLn−jα = iq−p(n− j)!(−1)

j(j−1)2 (∗α) and the product is written via the L2−norm,

asip−qQ(α, α) = (n− j)!

∫

X

α ∧ ∗α = (n− j)!||α||2L2

¤This result suggest to introduce the Weil operator C on cohomology

C(α) = ip−qα, ∀α ∈ Hp,q

Notice that C depends on the decomposition, in particular the action of C in avariation of Hodge structure is differentiable in the parameter space.We deduce from Q a new non degenerate Hermitian product

H(α, β) = Q(Cα, β), ∀[α], [β] ∈ Hj(X,C)

It satisfies for α, β ∈ Hp,q

H(β, α) = H(α, β)sinceH(α, β) = Q(ip−qα, β) = i

p−qQ(α, β) = (−1)2jip−qQ(β, α) = H(β, α).

22 FOUAD EL ZEIN

Remark 4.9. i)When the class [ω] ∈ Hj(X,Z) is integral, which is the case foralgebraic varieties, the product Q is integral (with integral value on integral classes).ii) The integral bilinear form Q extends by linearity to the complex space Hk

Z ⊗C.Its associated form H(α, β) := Q(α, β) is not Hermitian if k is odd. One way toovercome this sign problem is to define H as H(α, β) := ikQ(α, β), still this formwill not be positive definite in general.

Definition 4.10 (Polarization of HS). A Hodge structure ( HZ, (HC, F )) of weightn is polarized if a non degenerate scalar product Q is defined on HZ, alternating ifn is odd and symmetric if n is even, such that the Hermitian form on HC defined asH(α, β) := ip−qQ(α, β) for α, β ∈ Hp,q is orthogonal to the Hodge decompositionand ip−qH(α, α) is positive definite on the primitive component of type (p, q)(satisfyHodge-Riemann bilinear relations).

4.3.2. Polarized subvarieties are algebraic. A theorem of Kodaira states that aKahler variety with an integral class [ω] is projective (which means it can be em-bedded as a closed analytic subvariety in a projective space, hence it is necessarilyan algebraic subvariety.

4.4. Examples.

4.4.1. Projective space. Let H = c1(OPn(1)) denotes the Chern class of the canon-ical line bundle; it is dual to the homology class of an hyperplane.

Theorem 4.11. Hi(Pn,Z) = 0 for i odd and Hi(Pn,Z) = Z for i even with [H]i thecohomology class of an hyperplane to the power i for the cup product, as generator;hence: H2r(Pn,Z) = Z(−r) as HS.

4.4.2. Hodge structures of weight 1 and abelian varieties. Given a HS:(HZ,H1,0 ⊕ H0,1), the projection on H0,1 induces an isomorphism of real vectorspaces

HR → HC = H1,0 ⊕H0,1 → H0,1

since H0,1 = H1,0. Then we deduce that HZ is a lattice in the complex space H0,1,and the quotient T := H0,1/HZ is a complex torus . When HZ is identified withthe image of cohomology spaces Im(H1(X,Z) → H1(X,R)) of a complex manifoldX, resp. H0,1 with H1(X,O), the torus T will be identified with the Picard varietyPic0(X) parameterizing the holomorphic line bundles on X with first Chern classequal to zero as follows. We consider the exact sequence of sheaves defined byf 7→ e2iπf

0 → Z→ OX → O∗X → 1

where 1 is the neutral element of the sheaf of multiplicative groups O∗X , and itsassociated long exact sequence of derived functors of the global sections functor

→ H1(X,Z) → H1(X,OX) → H1(X,O∗X) → H2(X,Z)

where the morphisms can be interpreted geometrically; when the space H1(X,O∗X)is identified with isomorphisms classes of line bundles on X, the last morphismdefines the Chern class of the line bundle. Hence the torus T is identified with theisomorphism classes L with c1(L) = 0

T :=H1(X,OX)ImH1(X,Z)

' Pic0(X) := Ker(H1(X,O∗X) c1−→ H2(X,Z))


It is possible to show that the Picard variety of a smooth projective variety is anabelian variety. (Define a Kahler form with integral class on Pic0(X)).

4.4.3. Hodge structures of weight 2.

(HZ, H2,0 ⊕H1,1 ⊕H0,2; H0,2 = H2,0, H1,1 = H1,1; Q)

the intersection form Q is symmetric and H(α, β) = Q(α, β) is Hermitian. The de-composition is orthogonal for H with H positive definite on H1,1, negative definiteon H2,0 and H2,1.

Lemma 4.12. The HS is completely determined by the subspace H2,0 ⊂ HC s.t.H2,0 is totally isotropic for Q and the associated Hermitian product H is negativedefinite on H2,0. The signature of Q is (h1,1, 2h2,0).

In the lemma H0,2 is determined by conjugation and H1,1 = (H2,0 ⊕H0,2)⊥.

4.4.4. Hodge structures of K3 type.

Lemma 4.13. Let Q be a symmetric non degenerate form on a lattice HZ ofsignature (h1,1, 2). The HS of weight 2 on HC polarized by Q are parameterized bythe complex manifold

D := ω ∈ P(HC)|Q(ω, ω) = 0, Q(ω, ω) < 0Here H2,0 must be a line parameterized by ω ∈ P(HC) where the signature is −1

as well on its conjugate and h1,1 on H1,1.

5. Mixed Hodge Structures (MHS)

Classical Hodge theory apply only for compact Kahler manifolds. In order todescribe specific properties of cohomology of algebraic varieties inspired by etalecohomology and the solution of Weil conjectures, Deligne introduced in [6] a farreaching generalization of HS called mixed Hodge structure (MHS) that can bedefined on the cohomology of all algebraic varieties.We are essentially concerned by filtrations of vector spaces. However it is not moredifficult to describe this notion in the terminology of abelian categories.The formal study of MHS, namely the fact that they form an abelian category, willhelp the reader before he is confronted to their construction.

5.1. Filtrations. Let A denote an abelian category.

Definition 5.1. A decreasing filtration F (resp. increasing) of an object A of A isa family of sub-objects of A, satisfying

∀n,m, n ≤ m =⇒ Fm(A) ⊂ Fn(A)(resp. n ≤ m =⇒ Fn(A) ⊂ Fm(A))

If F is a decreasing filtration (resp. W an increasing filtration), a shifted filtrationF [n] by an integer n is defined as

(F [n])p(A) = Fn+p(A).

Decreasing filtrations F will be considered for a general study. Statements forincreasing filtrations W can be deduced by the change of indices Wn(A) = F−n(A).A filtration is finite if there exists integers n and m such that Fn(A) = A andFm(A) = 0.

A morphism of filtered objects (A,F )f−→ (B, F ) is a morphism A

f−→ B satisfying

24 FOUAD EL ZEIN

f(Fn(A)) ⊂ Fn(B) for all n ∈ Z. Filtered objects (resp. of finite filtration) form anadditive category with existence of kernel and cokernel of a morphism with naturalinduced filtrations as well image and coimage, however the image and coimage willnot be necessarily filtered-isomorphic, which is the main obstruction to obtain anabelian category.In the case of the category of modules over a ring, a morphism f : (A,F ) → (B,F )is strict if

f(Fn(A) = f(A) ∩ Fn(B)so that any element b ∈ Fn(B) ∩ ImA is already in Im Fn(A).More generally a morphism f : (A,F ) → (B, F ) is strict if it induces a filteredisomorphism Coim(f) → Im(f) from the coimage to the image of f .

Definition 5.2. The graded object associated to (A,F ) is defined as

GrF (A) = ⊕n∈ZGrnF (A) where Grn

F (A) = Fn(A)/Fn+1(A)

5.1.1. Induced filtration. A filtered object (A,F ) induces a filtration on a sub-objecti : B → A of A, defined by Fn(B) = B ∩ Fn(A). It is the unique filtration s.t. i isstrict. Dually, the quotient filtration on A/B is defined by

Fn(A/B) = p(Fn(A)) = (B + Fn(A))/B ' Fn(A)/(B ∩ Fn(A))

where p : A → A/B is the projection.

5.1.2. A sequence

Af−→ B

g−→ C

is a 0−sequence if g f = 0; if B is filtered, the cohomology H = Ker g/Im f isendowed with the quotient filtration of the induced filtration on Ker g. It is equal tothe induced filtration on H by the quotient filtration on B/Im f (H ⊂ (B/Im f)).

5.1.3. Two filtrations. Let A be an object of A with two filtrations F and G. Bydefinition, Grn

F (A) is a quotient of a sub-object of A, and as such, it is endowedwith an induced filtration by G. Its associated graded object defines a bigradedobject Grn

GGrmF (A)n,m∈Z. We refer to [6] for

Lemma 5.3 ( Zassenhaus’ lemma). The objects GrnGGrm

F (A) and GrmF Grn

G(A) areisomorphic.

5.1.4. Let H be a third filtration on A. It induces a filtration on GrF (A), henceon GrGGrF (A). It induces also a filtration on GrF GrG(A). These filtrations do notcorrespond in general under the above isomorphism. In the formula GrHGrGGrF (A),G et H have symmetric role, but not F and G.

5.1.5. Hom and tensor functor. If A and B are two filtered objects of A, we definea filtration on the left exact functor Hom

F kHom(A,B) = f : A → B : ∀n, f(Fn(A)) ⊂ Fn+k(B)Hence

Hom((A,F ), (B, F )) = F 0(Hom(A,B)).If A and B are modules on some ring, we define

F k(A⊗B) =∑m

Im(Fm(A)⊗ F k−m(B) → A⊗B)


5.1.6. Multifunctor. In general if H : A1 × . . . × An → B is a right exact multi-additive functor, we define

F k(H(A1, ..., An)) = ΣΣki=kIm(H(F k1A1, ..., FknAn) → H(A1, ..., An))

and dually if H is left exact

F k(H(A1, ..., An) = ∩Σki=kKer(H(A1, ..., An) → H(A1/F k1A1, ..., An/F knAn))

If H is exact, both definitions are equivalent.

5.2. Hodge Structure (HS). For any sub-ring A of R equal to Z, Q or R andany A-module HA, the complex conjugation extends to a conjugation on the spaceHC = HA ⊗A C. A filtration F on HC has a conjugate filtration F s.t. (F )iHC =F iHC.

Definition 5.4 (HS1). An A-Hodge structure H of weight n consists of(1) an A-module of finite type HA (the lattice).(2) a finite filtration F on HC (the Hodge filtration) s.t. F and its conjuguate

F satisfy the relation

GrpF Grq

F(HC) = 0, for p + q 6= n

The HS is real when A = R, rational when A = Q and integral when A = Z.

5.3. Opposite filtrations. Two finite filtrations F and G on an object A of A aren-opposite if

GrpF Grq

G(A) = 0 for p + q 6= n.

hence the Hodge filtration F on a HS of weight n is n-opposite to its conjugate F .The following constructions will define an equivalence of categories between objectsof A with two n-opposite filtrations and bigraded objects of A of the following type.

Example 5.5. Let Ap,q be a bigraded object of A s.t. Ap,q = 0 for all but a finitenumber of couples (p, q) and Ap,q = 0 for p + q 6= n; then we define two n-oppositefiltrations on A =

∑p,qA

p,q:

F p(A) = Σp′≥pAp′,q′ , Gq(A) = Σq′≥qA

p′,q′

We have GrpF Grq

G(A) = Ap,q.

We admit reciprocally [6]

Proposition 5.6. i) Two finite filtrations F and G on an object A are n-opposite,if and only if

∀ p, q, p + q = n + 1 ⇒ F p(A)⊕Gq(A) ' A.

ii) If F and G are n-opposite, and if we put Ap,q = F p(A) ∩Gq(A) for p + q = n,Ap,q = 0 for p + q 6= n, then A is a direct sum of Ap,q, moreover F and G can bededuced from the bigraded object Ap,q of A by the above procedure.

We have an equivalent definition of HS.

Definition 5.7 (HS2). An A−HS on H of weight n is a pair of an A-module HA

and a decomposition into a direct sum on HC = HA ⊗A CHC = ⊕p+q=nHp,q such that H

p,q= Hq,p

The relation with the previous definition is given by Hp,q = F p(HC) ∩ Fq(HC)

for p + q = n

26 FOUAD EL ZEIN

5.4. Morphism of HS. A morphism f : H → H ′ of HS, is an homomorphismf : HA → H ′

A such that fC : HC → H ′C is compatible with the bigrading, orequivalently with Hodge filtration.If H and H ′ are of distinct weights, f is necessarily 0.The HS of weight n form an abelian category. If H is of weight n and H ′ of weightn′, we define a HS, H ⊗H ′ of weight n + n′ as followsi) (H ⊗H ′)A = HA ⊗H ′

A

ii) the bigrading (resp. the Hodge filtration) of (H ⊗H ′)C = HC⊗H ′C is the tensor

product of the bigradings (resp. Hodge filtrations of HC and H ′C).

We define a HS of weight n′ − n on Hom(H, H ′) by the formula

F rHom(HC,H ′C) = f : HC → H ′

C : ∀p, f(F p(HC) ⊂ F p+r(H ′C)

In particular H∗ the dual to H is a HS of weight −n. An homomorphism f : HA →H ′

A of type (r, r) is strictly compatible with F that is

f(F p(HC) = F p+r(H ′C) ∩ Imf.

Similarly the Hodge structure ∧pH is of weight pn.

5.4.1. HomHS. The internal Hom to the category of HS, denoted by HomHS(H,H ′),is the sub-group of HomZ(HZ,H ′

Z) of elements of type (0, 0) in the HS on Hom(H,H ′).

Example 5.8. Tate HS, Z(1) is the HS of weight −2, rank 1, purely bigraded oftype (−1,−1), and of lattice 2πiZ ⊂ C.For n ∈ Z, we define Z(n) as the n tensor product of Z(1): Z(n) is the HS of weight−2n, rank 1, purely bigraded of type (−n,−n), with lattice (2πi)n Z ⊂ C.

5.4.2. Complex HS.

Definition 5.9. A complex HS of weight n on a complex vector space H is givenby a pair of n−opposite filtrations F and F , hence a decomposition into a directsum of subspaces

H = ⊕p+q=nHp,q, where Hp,q = F p ∩ Fq

The two n−opposite filtrations F and F on a complex HS of weight n can berecovered from the decomposition by the formula

F p = ⊕i≥pHi,n−i F

q= ⊕i≤n−qH

i,n−i

Here we don’t assume the existence of conjugation although we keep the notationF . An A−HS of weight n defines a complex HS on H = HC.Let H denotes the complex conjugate space of H and c : H → H the R−linearidentity isomorphism s.t. λ×H c(v) := c(λ×H v).

Definition 5.10. A polarization is a bilinear morphism S : H ⊗H → C s.t.

S(F p, c(Fq) = S(F

p, c(F q)) = 0 for p + q > n

and moreover S(C(H)u, c(v)) is a positive definite Hermitian form on H whereC(H) denotes Weil action on H.

Example 5.11. A complex HS of weight 0 on a complex vector space H is givenby a decomposition into a direct sum of subspaces

H = ⊕p∈ZHp

A polarization is an Hermitian form on H for which the decomposition is orthogonaland whose restriction to Hp is definite for p even and negative definite for odd p.


5.5. Mixed Hodge structure (MHS). This structure has been introduced in[6]. Let A be equal to Z, Q or R, then for an A-module of finite type HA, HA ⊗Qis defined as HA ⊗A Q := HQ, if A is equal to Z, otherwise as HA ⊗Q Q = HA.

Definition 5.12 (Deligne). An A-mixed Hodge structure H consists of1) an A-module of finite type HA

2) a finite increasing filtration W of the A ⊗ Q vector space HA ⊗ Q called theweight filtration3) a finite decreasing filtration F of the C-vector space HC = HA ⊗A C, called theHodge filtration, such that the systems

GrWn H = (GrW

n (HA⊗Q), (GrWn HC, F ))

consist of A⊗Q−HS of weight n.

The MHS is called real if A = R, rational if A = Q and integral if A = Z.

Example 5.13. 1) A HS, H of weight n, is a MHS with weight filtration

Wi(HQ) = 0 for i < n and Wi(HQ) = HQ for i ≥ n.

2) Let (Hi, Fi) be a finite family of A−HS of weight i ∈ Z; then H = ⊕iHi is

endowed with the following MHS

HA = ⊕iHiA, Wn = ⊕i≤nHi

A ⊗Q, F = ⊕iFi.

5.5.1. Morphism of MHS. A morphism f : H → H ′ of MHS is a morphism f :HA → H ′

A whose extension to HQ (resp. HC) is compatible with the filtration W

(resp. F , hence F ).

Definition 5.14 (opposite filtrations). Three finite filtrations W (increasing), Fand G on an object A of A are opposite if

GrpF Grq

GGrWn (A) = 0 for p + q 6= n.

This condition is symmetric in F and G. It means that F and G induce onWn(A)/Wn−1(A) two n-opposite filtrations, then GrW

n (A) is bigraded

Wn(A)/Wn−1(A) = ⊕p+q=nAp,q where Ap,q = GrpF Grq

GGrWp+q(A)

Example 5.15. i) A bigraded object A = ⊕Ap,q of finite bigrading has the follow-ing three opposite filtrations

Wn = ⊕p+q≤nAp,q , F p = ⊕p′≥pAp′q′ , Gq = ⊕q′≥qA

p′q′

ii) In the definition of an A−MHS, the filtration WC on HC deduced from Wby saclar extension, the filtration F and its complex conjugate , form a system(WC, F, F ) of three opposite filtrations.

5.5.2. A canonical decomposition of the weight filtration. Hodge decomposition.The definition of weight is given by a filtration and not a decomposition as a HS.The quotient Wn/Wn−2 is a non split extension of the two pure Hodge structuresGrW

n and GrWn−1, in the category of MHS that we aim to study. The question

here is how far we can lift the bigrading on GrW (A) and at what price do we cansplit the weight filtration W . Deligne constructed a decomposition of the object Awith its three opposite filtrations. For each pair of integers (p, q), he introduces thesubspaces of A

Ip,q = (F p ∩Wp+q) ∩ (F q ∩Wp+q + F q−1 ∩Wp+q−2 + F q−2 ∩Wp+q−3 + · · · )

28 FOUAD EL ZEIN

By construction they are related for p + q = n to the components Ap,q of theHodge decomposition: GrW

n A ' ⊕n=p+qAp,q; however, in general Ip,q 6= Iq,p but

Ip,q ≡ Iq,p modulo Wp+q−2.

Lemma 5.16. The projection Wp+q → GrWp+qA ' ⊕p′+q′=p+qA

p′,q′ induces anisomorphism ϕ : Ip,q ∼−→ Ap,q. Moreover

Wn = ⊕p+q≤nIp,q, F p = ⊕p′≥pIp′,q′

Proof. Injectivity of ϕ. Let m = p + q; modulo Wm−1, the image ϕ(Ip,q) ⊂ Ap,q =(F p∩F q)(GrW

m ). Let v ∈ Ip,q s.t. ϕ(v) = 0, then v ∈ F p∩Wm−1. The class cl(v) in(F p∩F q)(GrW

m−1) = 0 must vanish, since p+q > m−1; so we deduce v ∈ F p∩Wm−2.This is a step in an inductive argument based on the formula F p⊕F q−r+1 ' Wm−r.Suppose v ∈ F p ∩ F q−r+1 ∩Wm−r; since (F p ∩ F q−r+1)GrW

m−r = 0 for r > 0, theclass cl(v) = 0 ∈ GrW

m−r where v ∈ F p ∩Wm−r−1 hence by definition

v ∈ F q ∩Wm−r−1 +∑

r+1≥i≥2

F q−i+1 ∩Wm−r−1 +∑

i>r+1≥2

F q−i+1 ∩Wm−i

hence cl(v) ∈ (F p ∩ F q−r)GrWm−r−1 = 0. Since Wm−r−1 = 0 for large r, we deduce

v = 0.Surjectivity. Let α ∈ Ap,q; there exists v0 ∈ F p ∩Wm (resp. u0 ∈ F q ∩Wm) s.t.ϕ(v0) = α = ϕ(u0), hence v0 = u0 +w0 with w0 ∈ Wm−1. In GrW

m−1 the class of w0

decomposes as cl(w0) = cl(v1)+ cl(u1) with v1 ∈ F p ∩Wm−1 and u1 ∈ F q ∩Wm−1;hence there exists w1 ∈ Wm−2 s.t. v0 = u0+v1+u1+w1, then v0−v1 = u0+u1+w1

with v0 − v1 ∈ F p ∩Wm and u0 + u1 ∈ F q ∩Wm.By an increasing inductive argument on k, we can find vectors

vk ∈ F p ∩Wm, wk ∈ Wm−1−k

uk ∈ F q ∩Wm + F q−1 ∩Wm−2 + F q−2 ∩Wn−3 + ... + F q+1−k ∩Wm−k

s.t. vk represents α and vk = uk + wk, then we decompose the class of wk inGrW

m−k−1 as above in the inductive step. For large k, Wm−1−k = 0 and we representα in Ip,q.Moreover, Wn = Wn−1 ⊕ (⊕p+q=nIp,q), which prove the formula for the weight.Next suppose v ∈ F p and consider the least integer n s.t. v ∈ Wn. The image ofv in GrW

n A decomposes into Hodge components of type (i, n − i) with i ≥ p sincev ∈ F p ∩Wn. By subtracting the representatives of the components in Ii,n−i fori ≥ p we push v in Wn−1 so a decreasing inductive argument proves the formulafor F p. ¤

Theorem 5.17 (Deligne). Let A be an abelian category and A′ the category ofobjects of A endowed with three opposite filtrations W (increasing), F and G. Themorphisms of A′ are morphisms in A compatible with the three filtrations.i) A′ is an abelian category.ii) The kernel (resp. cokernel) of a morphism f : A → B in A′ is the kernel (resp.cokernel ) of f in A, endowed with the three induced filtrations from A (resp. quo-tient of the filtrations in B).iii) Any morphism f : A → B in A′ is strictly compatible with the filtrations W,Fand G; the morphism GrW (f) is compatible with the bigradings of GrW (A) andGrW (B); the morphisms GrF (f) and GrG(f) are strictly compatibles with the in-duced filtration by W .


iv) The forget functors of the filtrations, as well GrW , GrF , GrG, GrW GrF 'GrF GrW , GrF GrGGrW and GrGGrW ' GrW GrG of A′ in A are exact.

Proof. A morphism compatible with the filtrations is necessarily compatible withthe above decomposition into ⊕Ip,q. The proof of (i) follows from the precedingstatements.ii) Let f : A → B in A′ and consider on K = Ker(f) the induced filtrationsfrom A. The morphism GrW K → GrW A is injective, since it is injective on thecorresponding terms Ip,q; moreover, the filtration F (resp. G) of K induces onGrW K the inverse image of the filtration F (resp. G) on GrW A. The sub-objectGrW K of GrW A is compatible with the bigrading of GrW A, that is

GrW K = ⊕p,q((GrW K) ∩Ap,q) where GrpF Grq

GGrWn K → Grp

F GrqGGrW

n A

Hence the filtrations W,F,G on K are opposite and K is a kernel of f in A′. Thisand the dual result, prove (ii). If f is a morphism in A′, the canonical morphismCoim(f) → Im(f) is an isomorphism in A; it follows from (iii), it is an isomorphismin A′, which is therefore abelian. Forget the filtrations is an exact functor after (ii).Hence the exactness of the other functors in (iv) results. ¤

We can deduce now the following

Theorem 5.18 (Deligne). i) The category of MHS is abelian.ii) The kernel (resp. cokernel) of a morphism f : H → H ′ has as integral latticethe kernel ( resp. cokernel) K of f : HA → H ′

A, moreover K ⊗Q and K ⊗ C areendowed with induced filtrations (resp. quotient filtrations) by W and F of HA⊗Qand HC (resp. H ′

A⊗Q and H ′C).iii) Each morphism f : H → H ′ is strictly compatible with the filtrations W onHA⊗Q and H ′

A⊗Q as well the filtrations F on HC and H ′C. It induces morphismsof A⊗Q−HS, GrW

n (f) : GrWn (HA⊗Q) → GrW

n (H ′A⊗Q) and morphisms Grp

F (f) :Grp

F (HC) → GrpF (H ′C) strictly compatible with the induced filtrations WC.

iv) The functor GrWn from the category of MHS to the category A⊗Q−HS of weight

n is exact.v) The functor Grp

F is exact.

5.5.3. Hodge numbers. Let H be a MHS and set Hpq = GrpF Grq

FGrW

p+qHC =(GrW

p+qHC)p,q. The Hodge numbers of H are the integers hpq = dimCHpq, thatis the Hodge numbers hpq of the HS: GrW

p+qH.

5.5.4. Complex MHS. Although the cohomolgy of algebraic varieties carry MHSdefined over Z, we may need to work in analysis without such structure over Z.

Definition 5.19. A complex MHS of weight n on a complex vector space H isgiven by an increasing filtration W and two decreasing filtrations F and G suchthat (GrW

k H, F, G) with the induced filtrations is a complex HS of weight n + k.

For n = 0, we say a complex MHS. The definition of complex HS of weight n isobtained in the particular case when Wn = H and Wn−1 = 0.Let H denotes the complex conjugate space of H and c : H → H the R−linearidentity isomorphism s.t. λ.Hc(v) := c(λ.Hv).

Definition 5.20. i) A complex MHS of weight n is graded polarisable if thereexists a bilinear morphism S : H ⊗H → C s.t.

S(F p, c(Gq) = S(Gp, c(F q) = 0 for p + q > n

30 FOUAD EL ZEIN

and moreover (GrWk H, F, G) is an induced polarized complex HS.

5.6. Variation of complex MHS. The structure which appears in deformationtheory on cohomology of the fibers of a morphism of algebraic varieties leads tointroduce the concept of Variation of MHS.

Definition 5.21. i) A variation (VHS) of complex HS on a complex manifold X ofweight n is given by a data (H, F, F ) where H is a complex local system, F and Fare two decreasing filtrations by sub-bundles of the vector bundle OX ⊗CH s.t. foreach point x ∈ X, the data (H(x), F (x), F (x)) form a HS of weight n. Moreover,the connection ∇ defined by the local system satisfy Griffiths tranversality

(∇F p) ⊂ F p−1, (∇Fp) ⊂ F

p−1

ii) A variation (VMHS) of complex MHS of weight n on X is given by the followingdata

(H,W, F, F )where H is a complex local system, W an increasing filtration by sub-local systems,F and F two decreasing filtrations by sub-bundles of the vector bundle OX ⊗C Hsatisfying Griffiths tranversality

(∇F p) ⊂ F p−1, (∇Fp) ⊂ F

p−1

s.t. (GrWk H, F, F ) with the induced filtrations is a complex VHS of weight n + k.

For n = 0 we just say a complex VMHS. Let H be the conjugate local systemof L. A linear morphism S : H ⊗C H → CX defines a polarization of a VHS if itdefines a polarization at each point x ∈ X. A complex MHS of weight n is gradedpolarisable if (GrW

k H, F, F ) is a polarized VHS.

6. Hypercohomology of a Filtered Complex

The construction of MHS on cohomology groups of algebraic varieties proceedsfirst by attaching to the variety a bifiltered complex of sheaves (Ω∗,W, F )(essentiallya variant of DeRham complex ) satisfying certain conditions to insure that the in-duced filtrations by W and F on cohomology define a MHS. Such bifiltered com-plexes, called mixed Hodge complexes (MHC), are defined in [Deligne]. Since theconstruction will be applied respectively in the case of compact Kahler manifolds,the complement of a normal crossing divisor (NCD) and then a simplicial scheme,it has been necessary to establish in [6] a common language and a set of axiomswhich lead naturally to the construction of a MHS on the hypercohomology of suchcomplexes. For a deep knowledge of sheaf theory, the book by Godement [8] forexample is good for a start.

6.1. Derived categories. The language of derived categories is the most appro-priate at this point. It is based on the fact that a complex is a more rich objectthan merely its cohomology. However since we need acyclic resolutions and thereis in general many choices of such resolutions, one needs to change the structureof the category by changing only the structure of morphisms. Verdier showed thatthe following two constructions lead to a new category satisfactory for our purposeand which proved to be useful in various domains of geometry.1) The objects of the category are the complexes.2) The morphisms of the category are defined after the following modifications


i) Two morphisms are equivalent if they are homotopic.ii) A morphism is invertible if it induces an isomorphism on cohomology (then it iscalled a quasi-isomorphism).These two operations have been carried by Verdier [13]. Although the objectsremain the same, the modifications on morphisms can transform the category sig-nificantly (for example non zero morphisms may become isomorphic to zero andthe inverse ψ of a morphism ϕ need to satisfy ψ ϕ and ϕ ψ are only homotopicto the identity [1]).In practice the reader should apply the following rule : the sign RT in front of afunctor T applied to a complex of sheaves K refers to the complex TK ′ where K ′

is a resolution of K by T−acyclic sheaves .

6.2. Filtered derived category. Let K be a complex of objects of an abeliancategory A, filtered by subcomplexes F i. The filtration is biregular if it inducesa finite filtration on each component of K . A morphism of complexes is called aquasi-isomorphism if it induces an isomorphism on the cohomology. The filtrationF of a bifiltered complex (K,F, W ) induces by restriction a filtration F on theterms W iK, which induces on its own a quotient filtration F on Gri

W K. We definein this way GrF GrW K.

Definition 6.1. (i) A morphism f : (K,F ) → (K ′, F ′) of complexes with biregularfiltrations is a filtered quasi-isomorphism if GrF (f) is a quasi-isomorphism.(ii) A morphism f : (K,F, W ) → (K ′, F ′,W ′) of complexes with biregular filtra-tions is a bifiltered quasi-isomorphism if GrF GrW (f) is a quasi-isomorphism.

6.2.1. For an abelian category A, let FA (resp. F2A) denotes the category of fil-tered objects (resp. bifilered) ofA with finite filtration(s), C+A (resp. C+FA, C+F2A)the category of complexes of A (resp. FA, F2A) bounded on the left ( zero in de-grees near −∞) with morphisms of complexes (resp. morphisms respecting thefiltrations).An homotopy from a morphism u : K → K ′ to u′, is a family of morphismski+1 : Ki+1 → K

′i for i ∈ Z of degree −1, satisfying u− u′ = dK′ h + h dK ; inpresence of filtrations, the homotopy should respect the filtrations.

6.2.2. Let K+A (resp. K+FA,K+F2A) be the category whose objects are com-plexes bounded below of objects of A (resp. filtered, bifiltered complexes), andwhose morphisms are equivalence classes modulo homotopy (resp. homotopy re-specting the filtrations).

6.2.3. Let D+A (resp. D+FA, D+F2A) be the category deduced from K+A (resp.K+FA,K+F2A) by inverting the quasi-isomorphisms (resp. the filtered, bifilteredquasi-isomorphisms). That is the derived category (resp. derived filtered, bifiltered) as defined by Verdier (resp. [Deligne]). What is important to retain is the following: the objects of D+A (resp.D+FA, D+F2A) are complexes of objects of A, hencethe same as those of K+A (resp.K+FA, K+F2A). An isomorphism f : K1 → K2

in D+A (resp.D+FA, D+F2A) is represented by one of the diagrams of morphisms

K1g1≈←−−− K ′

1f1−→ K2 , K1

f2−→ K ′2

g2≈←−−− K2

in K+A (resp.K+FA,K+F2A) where g1 and g2 are quasi-isomorphisms (resp.filtered, bifiltered).

32 FOUAD EL ZEIN

6.2.4. Triangles. The complex TK or K[1] is the complex (K, dK) with shifteddegrees defined as

(TK)i = Ki+1, dTK = −dK

Let u : K → K ′ be a morphism of C+A(resp. C+FA, C+F2A), the cone C(u)is the complex TK ⊕K ′ with the differential

(−dK

u0

dK′

)(resp. endowed with the

sum of the filtrations). The exact sequence associated to C(u) is

0 → K ′ → C(u) → TK → 0

Let h denotes an homotopy (resp. filtered, bifiltered ) from a morphism u : K → K ′

to u′, we define an isomorphism (resp. filtered, bifiltered) Ih : C(u) ∼−→ C(u′) bythe matrix

(Idh

0Id

)acting on TK ⊕K ′, which commute to the injections of K ′ in

C(u) and C(u′), and to the projections on TK.Let h and h′ be two homotopies of u to u′. A second homotopy of h to h′, that is afamily of morphisms ki+2 : Ki+2 → K

′i for i ∈ Z, satisfying h−h′ = dK′ k−kdK ,defines an homotopy of Ih to Ih′ .A distinguished triangle (or exact ) in K+A (resp. K+FA, K+F2A)) is a sequenceof complexes isomorphic to the image of an exact sequence associated to a cone inC+A (resp.C+FA, C+F2A).The derived category (resp. derived filtered, bifiltered ) is a triangulated category(that is endowed with a class of distinguished triangles). A distinguished trianglein D+A (resp.D+FA, D+F2A) is a sequence of complexes isomorphic to the imageof a distinguished triangle in K+A (resp.K+FA, K+F2A).

6.3. Filtered complexes. We follow in this section Deligne’s notations in [6]where all complexes are bounded below. There exist different point of views inthe literature.

6.3.1. Consider a left exact functor T : A → B, (A,F ) a filtered object of finitefiltration, and TF the filtration of TA by its sub-objects TF p(A) (here we use T leftexact). If GrF (A) is T -acyclic, the F p(A) are T -acyclic as successive extensionsof T -acyclic objects. Hence, the image by T of the sequence o → F p+1(A) →F p(A) → Grp

F (A) → o is distinguished, and

GrTF TA ' TGrF A.

6.3.2. Let A be an object with two finite filtrations F and W such that GrF GrW A isT -acyclic. Then the objects GrF A andGrW A are T -acyclic, as well F q(A)∩W p(A).The sequences

o → T (F q ∩W p+1) → T (F q ∩W p) → T ((F q ∩W p)/(F q ∩W p+1)) → o

are exact, and T (F q(GrpW A)) is the image in T (Grp

W (A) of T (F p ∩ W q). Thediagram

T (F q ∩W p) → T (F qGrpW A) → TGrp

W A' ↓ '↓

TF q ∩ TW p → GrpTW TA

shows that the isomorphism relative to W transform the filtration GrTW (TF ) onGrTW TA into the filtration T (GrW (F )) on TGrW A.


6.3.3. Whenever the category A has enough injectives, the following derived func-tors of a left exact functor are defined functorially

RT : D+(A) → D+(B), RT : D+F (A) → D+F (B), RT : D+F2(A) → D+F2(B)

as followsa) Let K be a complex in D+(A) ( resp. D+F (A), D+F2(A)). We choose a quasi-isomorphism i : K

≈→ I (resp. filtered, bifiltered), where the components of I areinjectives in each degree ( resp. with injective filtrations). For example, I is thesimple complex defined by an injective Cartan-Eilenberg resolution of K.b) We define RTK = T (I).c) A morphism f : K → K ′ gives rise to a morphism RT (K) → RT (K ′) functo-rially, since f can be extended on the injective resolutions to a morphism I(f) :T (I) → T (I ′), defined uniquely up to homotopy.The hypercohomology RiTK is equal to the cohomology Hi(RTK). In particular,for a different choice of an injective resolution J of K, we have an isomorphismT (I) ' T (J) in D+(B).

Remark 6.2. A single sheaf F is T−acyclic when RiT (F) = 0 for i > 0. Animportant result shows that for any resolution A of a complex K with acyclicsheaves, TA is isomorphic to the derived complex RTK.

6.3.4. Let F be a biregular filtration of K. A filtered T -acyclic resolution of K isgiven by a filtered quasi-isomorphism i : K → K ′ to a complex with a biregularfiltration s.t. Grp

F (K′n) are acyclic for T . If K ′ is such a resolution, the F p(K

′n)are acyclic for T and the derived functor is defined as followsa) We choose a filtered quasi-isomorphism i : (K,F ) → (K ′, F ′) s.t. the componentsof Grp

F ′(K′) are acyclic for T .

b) We define RT (K, F ) = (TK ′, TF ′). For a different choice (K ′′, F ′′) of (K ′, F ′)we have an isomorphism (TK ′′, TF ′′) ' (TK ′, TF ′) in D+F (B). In fact we have

RT (GrF K) ' GrTF ′T (K ′) ' GrTF ′′T (K ′′)

The complex T (K ′, F ′) is functorial in (K ′, F ′), and a filtered quasi-isomorphism f :(K1, F1) → (K2, F2) induces an isomorphism RT (f) : RT (K1, F1) ' RT (K2, F2).

6.3.5. Let F and W be two biregular filtrations of K. A bifiltered T -acyclic resolu-tion of K is a bifiltered quasi- isomorphism i : K → K ′ of K to a bifiltered complexbiregular for each filtration s.t. Grp

F GrqW (K

′n) is acyclic for T . If K ′ is such aresolution, the terms F pK ′n ∩W qK ′n are acyclic for T , then the derived functoris defined as followsa) We chose a bifiltered quasi-isomorphism i : (K, F, W ) ≈−→ (K ′, F,W ) s.t. thecomponents of Grp

F GrqW K ′ are acyclic for T (such i always exists). In the partic-

ular case where A is the category of sheaves of A-modules on a topological spaceX, and where T is the functor Γ of A to the category of A-modules, an example ofbifiltered T -acyclic resolution of K is the simple complex associated to the doublecomplex defined by Godement resolution G∗ of K, filtered by G∗(F pK) and byG∗(WnK). Since G∗ is exact, we have

GrF GrW (G∗K) ' G∗(GrF GrW K)

b) We define RT (K, F,W ) = (TK ′, TF, TW ).If K ′ is a bifiltered T -acyclic resolution of K, the complex TK ′ is filtered by TF pK ′

and by TW qK ′. Moreover, GrnW K ′ is a filtered T -acyclic resolution for F of Grn

W K,

34 FOUAD EL ZEIN

hence GrnF Grm

W K ′ is a T -acyclic resolution of GrnF Grm

W K, and TGrF K ′ ' GrF TK ′

(as a W -filtered complex), TGrW K ′ ' GrW TK ′ (as a F -filtered complex) andTGrF GrW K ′ ' GrF GrW TK ′. We deduce

RTGrF GrW K ' GrF GrW TK ′

For a different choice (K ′′, F, W ) of (K ′,W, F ), we have an isomorphism

(TK ′′, TF, TW ) ' (TK ′, TF, TW ) in D+F2(B).

The complex T (K ′, F, W ) is functorial in (K,F, W ) and a bifiltered quasi-isomorphismf : K1 → K2 induces an isomorphism RT (f) : RTK1

∼−→ RTK2.

6.4. Spectral sequence of a filtered complex. A filtration F of a complex Kby sub-complexes induces a filtration on its cohomology with associated gradedcohomology . The spectral sequence ([6], [8]) Er(K, F ) associated to F leads,for large r and under mild conditions, to such graded cohomology defined by thefiltration. This chapter is technical and we recommend to follow the proofs at a firstreading on an example, and return later with the study of MHC where the spectralsequence becomes interesting and meaningful since it contains deep geometricalinformation.Let K be a complex of objects of A, with a biregular filtration F .

6.4.1. To construct the terms Epqr (K, F ), or simply Epq

r of the spectral sequencedefined by F , we set for r ≥ 1

Zpqr = Ker(d : F pKp+q → Kp+q+1/F p+rKp+q+1)

and we define dually Bpqr by the formula

Kp+q/Bpqr = coker(d : F p−r+1Kp+q−1 → Kp+q/F p+1(Kp+q).

Such formula still makes sense for r = ∞ if we set, for a filtered object (A,F ),F−∞(A) = A and F∞(A) = 0

Zpq∞ = Ker(d : F pKp+q → Kp+q+1)

Kp+q/Bpq∞ = coker(d : Kp+q+1 → Kp+q/F p+1Kp+q)

The notations here are similar to [6] but different from [Go]. We set by definition

Epqr = Im(Zpq

r → Kp+q/Bpqr ) = Zpq

r /(Bpqr ∩ Zpq

r )

= Ker(Kp+q/Bpqr → Kp+q/(Zpq

r + Bpqr ))

Epq∞ = Zpq

∞/Bpq∞ ∩ Zpq

∞ .

The differential d : Zpqr → Zp+r,q−r+1

r induces a differential dr : Epqr → Ep+r,q−r+1

r .For r < ∞, the terms Er form a complex of objects, graded by the degree p−r(p+q)(or a direct sum of complexes with index p−r(p+q)); moreover Er+1 is isomorphicto the cohomology of this complex:

Epqr+1 = H(Ep−r,q+r−1

rdr−→ Epq

rdr−→ Ep+r,q−r+1

r ).

for r = 0, we setEpq

0 = GrpF (Kp+q).

Consider the filtration F on H∗(K) defined by

F iH∗(K) = Im(H∗(F iK) → H∗(K)).


Then we haveEpq∞ = Grp

F (Hp+q(K)).Epq

1 = Hp+q(GrpF (K)).

The differentials d1 are the connecting morphisms defined by the exact sequences

0 → Grp+1F K → F pK/F p+2K → Grp

F K → 0.

Remark 6.3. The precedent formulas are defined for an increasing filtration on Kby a change of indices.Let W be an increasing filtration on K. We set for all j, n ≤ m and n ≤ i ≤ m

WiHj(WmK/WnK) = Im(Hj(WiK/WnK) → Hj(WmK/WnK)

then we define with these notations, for all r ≥ 1, p and q

Epqr = GrW

−pHp+q(W−p+r−1K/W−p−rK).

This definition coincides with the previous one, written for W , as we can check

Epqr (K, W ) = Zpq

r /Bpqr ∩ Zpq

r =

Ker(d : W−pKp+q → Kp+q+1/W−p−rK

p+q+1)/(W−p−1Kp+q+dW−p+r−1K

p+q−1)∩Zpqr

=Ker(d : W−pK

p+q/W−p−rKp+q → W−p+r−1K

p+q+1/W−p−rKp+q+1)

((W−p−1Kp+q + dW−p+r−1Kp+q−1)/W−p−rKp+q) ∩ Zpq∞

=

Zpq∞

Bpq∞ ∩ Zpq

∞= Epq

∞(W−p+r−1K/W−p−rK) = GrW−pH

pq(W−p+r−1K/W−p−rK)

Consider the exact sequence

0 → W−p−rK/W−p−2rK → W−p+r−1K/W−p−2rK → W−p+r−1K/W−p−rK → 0.

The connecting morphism

Hp+q(W−p+r−1K/W−p−rK) ∂→Hp+q+1(W−p−rK/W−p−2rK).

send W−p to W−p−r. The injection W−p−rK → W−p−1K induces a morphism

Hp+q+1(W−p−rK/W−p−2rK)ϕ−→ Ep+r,q−r+1

r = GrW−p−rH

p+q+1(W−p−1K/W−p−2rK)

Then ϕ ∂ restricted to W−p induces the differential

dr : Epqr → Ep+r,q−r+1

r .

The injection W−p+r−1 → W−p+rK induces the isomorphism

H(Epqr , dr)

∼−→ Epqr+1 = GrW

−pHp+q(W−p+rK/W−p−r−1K).

The spectral sequence Ep,qi (K,W ) degenerates at Er if the differentials di are zero

for i ≥ r. Then we write Er = E∞. Moreover, in this case

Epqr = Epq

i = Epq∞ pour i ≥ r.

Proposition 6.4 (Deligne). Let K be a complex with a biregular filtration F . Thefollowing conditions are equivalent

(1) The spectral sequence defined by F degenerates at E1(E1 = E∞)(2) The morphisms d : Ki → Ki+1 are strictly compatibles to the filtrations.

Proposition 6.5. A filtered quasi-isomorphism f : (K, F ) → (K ′, F ′) induces anisomorphism of spectral sequences Epq

r (f) : Epqr (K, F ) ∼−→ Ep,q

r (K ′, F ′).

Corollary 6.6. Let f : (K, F ) → (K ′, F ′) be a filtered morphism. The followingassertions are equivalent

36 FOUAD EL ZEIN

(1) f is a filtered quasi-isomorphism.(2) Epq

1 (f) : Epq1 (K, F ) → Epq

1 (K ′, F ′) is an isomorphism(3) Epq

r (f) : Epqr (K, F ) → Epq

r (K ′, F ′) is an isomorphism for all r ≥ 1.

6.4.2. Let T : A → B be a functor, and (K, F ) an object of D+FA. The complexRT (K, F ) defines a spectral sequence

Ep,q1 = Rp+qT (Grp

F ) ⇒ GrpF Rp+qT (K)

This is the hypercohomology spectral sequence of the filtered complex K. It iswritten for an increasing filtration W on K as

Ep,q1 = Rp+qT (GrW

−p) ⇒ GrW−pR

p+qT (K)

It depends functorially on K and a filtered quasi-isomorphism induces an isomor-phism of spectral sequences. The differentials d1 of this spectral sequence are theconnecting morphisms defined by the short exact sequences

0 → Grp+1K → F pK/F p+2K → GrpK → 0.

6.4.3. Let K be a complex. We denote by τ≤p(K) the following sub-complex

τ≤p (K)n =

Kn for n < p

Ker(d) for n = p

0 for n > p.

The filtration of K by the τ≤p(K) is called canonical, then

Hi(τ≤p(K)) = Hi(K) if i = p, and 0 if i > p.

A quasi-isomorphism f : K → K ′ is necessarily a filtered quasi-isomorphism forthe canonical filtrations.

6.4.4. The sub-complexes σ≥p(K) of K

σ≥p (K)n =

0 if n < p

Kn if n ≥ p.

define a biregular filtration, the trivial filtration of K (called bete by Deligne).The hypercohomology spectral sequences of a left exact functor attached to thetrivial and canonical filtrations of K are the two natural hypercohomology spectralsequences of K.

Example 6.7. Let f : X → Y be a continued map of topological spaces. LetF be an abelian sheaf on X and F∗ a resolution of F by f∗−acyclic sheaves,then Rif∗F ∼= Hi(f∗F∗). The hypercohomology spectral sequence of the complexRf∗F∗ with its canonical filtration with respect to the global section functor Γ(Y, ∗)is

Epq1 = H2p+q(Y,R−pf∗F∗) ⇒ Grτ

−pHp+q(X,F)

it coincides, up to renumbering Epqr → E2p+q,−p

r+1 , with Leray’s spectral sequencefor f and F

Epq2 = Hp(Y, Rqf∗F).


Example 6.8. On DeRham complex, the trivial filtration is called the Hodgefiltration since it is defined by the subcomplexes

F p(Ω∗X) = 0 7→ · · · 0 7→ ΩpX 7→ · · · 7→ Ωm

X 7→ 0

It induces on DeRham cohomology a filtration by sub-vector spaces

F pHn(X, Ω∗X) = ImHn(X, F pΩ∗X).

The spectral sequence with respect to the global section functor and the filtration F(in different terms of the filtered complex (RΓ(X, Ω∗X),RΓ(X, F pΩ∗X)) degeneratesat rank one

Epq1 = · · · = Epq

r = · · · = GrpFH

p+q(X, Ω∗X)The proof uses Hodge theory from which we deduce Grp

FHp+q(X, Ω∗X) ' Hp,q(X) 'Hq(X, Ωp

X), then it follows that dim Epq1 = · · · = dim Epq

r and dr = 0 for r > 0.In particulae we deduce that the morphism

Hn(X, F pΩ∗X) ∼−→ F pHn(X, Ω∗X)

is an isomorphism. Such result is algebraic and has been obtained directly byDeligne and Illusie. However Hodge theory tells more. Precisely, the conjugateHodge filtration F of F with respect to Hn(X,Q)) satisfy

Hp,q(X) = F pHn(X, Ω∗X) ∩ F qHn(X, Ω∗X), Hp,q = Hq,p.

as well the decomposition

Hn(X,C) = ⊕p+q=nHp,q(X).

6.5. Two filtrations. This section relates results on various induced filtrations onterms of a spectral sequence, contained in [6](lemma on two filtrations). A filtrationF of a complex by sub-complexes define a spectral sequence Er(K,F ). A secondfiltration W induces in various ways filtrations on Er(K, F ), different in general.We discuss here axioms on W and F , at the level of complexes, in order to getcompatibility of the various induced filtrations by W on the spectral sequence of(K,F ). What we have in mind is to find axioms leading to define the MHS withinduced filtrations W and F on cohomology. The proofs are technical and may beassumed for a first reading.

6.5.1. Let (K, F, W ) be a bifiltered complex of objects of an abelian category,bounded below. The filtration F , assumed to be biregular, induces on the termsEpq

r of the spectral sequence E(K, W ) various filtrations as follows

Definition 6.9. The first direct filtration on Er(K,W ) is the filtration Fd definedby images

F pd (Er(K, W )) = Im(Er(F pK, W ) → Er(K, W )).

Dually, the second direct filtration Fd∗ on Er(K,W ) is defined by kernels

F pd∗(Er(K, W )) = Ker(Er(K, W ) → Er(K/F pK,W )).

Lemma 6.10. On E0 and E1, we have Fd = Fd∗ .

For r=0 or 1, we have Bpqr ⊂ Zpq

r .

Definition 6.11. The recurrent filtration Frec on Epqr is defined as follows

i) On Epq0 , Frec = Fd = Fd∗ .

ii) On Epqr+1, as a quotient of a sub-object of Epq

r , the recurrent filtration Frec isinduced by Frec on Epq

r .

38 FOUAD EL ZEIN

6.5.2. The precedent definitions of direct filtrations apply to Epq∞ as well and they

are compatible with the isomorphism Epqr ' Epq

∞ for large r. We deduce, via thisisomorphism a recurrent filtration Frec on Epq

∞ . The filtrations F and W induceeach a filtration on H∗(K). The isomorphism E∗∗

∞ ' Gr∗W H∗(K) leads to a newfiltration F on Epq

∞ .

Proposition 6.12. (i) The morphisms dr are compatible with the first direct fil-tration. If Epq

r+1 is considered as a quotient of a sub-object of Epqr , the first direct

filtration on Epqr+1 is finer then the filtration F ′ induced by the first direct filtration

on Epqr : Fd(E

pqr+1) ⊂ F ′(Epq

r+1).(ii) Dually, the morphisms dr are compatible with the second direct filtration, andthe second direct filtration on Epq

r+1 is less fine then the filtration F ′′ induced fromEpq

r : F ′′(Epqr+1) ⊂ Fd∗(E

pqr+1).

(iii) Fd(Epqr ) ⊂ Frec(Epq

r ) ⊂ Fd∗(Epqr ).

(iv) On Epq∞ , the filtration induced by the filtration F on H∗(K) satisfy

Fd(Epq∞) ⊂ F (Epq

∞) ⊂ Fd∗(Epq∞).

6.5.3. The bifiltered complex (K, F,W ) is said to be F−split if the filtered complex(K,W ) is isomorphic to a sum of filtered complexes (Kn,Wn)n∈Z

(K, W ) = ⊕n(Kn,Wn) with FnK = ⊕n′≥nKn′

6.6. Let r0 be an integer ≥ 0, or +∞ and consider the following condition(2filt, r0): For each non negative integer r < r0, the differentials dr of the gradedcomplex Er(K,W ) are strictly compatible with the recurrent filtration defined byF .

6.6.1. It is clear that the condition (2filt, ∞) is satisfied by F−split (K,F, W ).Reciprocally, if the condition (2filt, ∞) is satisfied, the exact sequence

0 → (F i+1K,W ) → (F iK,W ) → (GriF K,W ) → 0

behaves like it was split. For example we have exact sequences for all i and j

0 → GrW Hj(F i+1K) → GrW Hj(F iK) → GrW Hj(GriF K) → 0,

the term GrF of the spectral sequence E(K, W ) is isomorphic to the spectral se-quence E(GrF K, W ), and the spectral sequence E(K,F ) degenerates at rank 1(E1 = E∞).

Theorem 6.13 (Deligne). Let K be a complex with two filtrations W and F , whereF is biregular.(i) If (K,F, W ) satisfy (2 filt, r0), then for r ≤ r0 the sequence

0 → Er(F pK,W ) → Er(K,W ) → Er(K/F pK, W ) → 0

is exact, and for r = r0 + 1, the sequence

Er(F pK,W ) → Er(K,W ) → Er(K/F pK, W )

is exact. In particular for r ≤ r0 + 1, the two direct and the recurrent filtration onEr(K,W ) coincide Fd = Frec = Fd∗ .(ii) If (K,F, W ) satisfy (2 filt, ∞), then the spectral sequence E(K, F ) degeneratesat E1 (E1 = E∞) and the filtrations Fd, Frec, Fd∗ coincide with the filtration F onE∞ induced by (H∗(K), F ).


Corollary 6.14. (i) In the conditions of case (i) of the theorem the condition (2filt, r0) is satisfied by the complex (F aK/F bK, W,F ) if it is already satisfied by(K,W,F ).(ii) In the conditions of case (ii) of the theorem, we have a strict filtered exactsequence

0 → (H(F pK),W ) → (H(K),W ) → (H(K/F pK), W ) → 0

and an isomorphism of spectral sequences

GrpF (E.(K, W )) ' E.(Grp

F K, W ).

7. Mixed Hodge Complex(MHC)

We consider complexes of abelian sheaves endowed with new structures needed todeduce a MHS on their cohomology. Deligne called them Mixed Hodge complexes(MHC). The technique of mixed cone associates to a morphism of MHC a new MHC,which leads to a MHS on relative cohomology and long exact sequences of MHS. Thefirst example is constructed on the cohomology of a divisor Y with normal crossings(NCD) in a smooth complex complete variety X and the cohomology with compactsupport of X − Y . The dual case is the logarithmic complex constructed definingthe MHS on the cohomology of X − Y . The Hodge filtration F may be deducedalso from the filtration by order of the pole on the sheaf of meromorphic formsholomorphic on X − Y . Other examples are given by a construction of naturalmixed cones, including the case of cohomology with compact support on X − Y .The most general type of MHS is on the cohomology of open NCD.

7.1. Hodge Complex (HC). In this section we consider complexes with addi-tional properties that enables us to obtain a HS on their cohomology, and arrive tothe definition of a Hodge complex. The cohomology of a compact complex smoothalgebraic variety carry a HS whose Hodge filtration is deduced from the filtration onthe algebraic DeRham complex, however the proof of the decomposition is reducedto the case of a projective variety, hence compact Kahler manifold.

7.1.1. Let A = Z,Q or R. A Hodge A-complex K of weight n consists of(1) a complex of A−modules KA ∈ ObD+(A), s.t. Hk(KA) is an A-module of

finite type for all k,(2) a filtered complex (KC, F ) ∈ ObD+F (C)(3) an isomorphism α : KC ' KA ⊗ C in D+(C).

The following axioms must be satisfied(CH 1) the differential d of KC is strictly compatible with the filtration F ; equiva-lently, the spectral sequence defined by (KC, F ) degenerates at E1(E1 = E∞),(CH 2) for all k, the filtration F on Hk(KC) ' Hk(KA)⊗ C define an A−HS ofweight n + k on Hk(KA).Equivalently, the filtration F is (n + k)-opposed to its complex conjugate (whichmakes sense since A ⊂ R).

7.1.2. Let X be a topological space. An A-cohomological Hodge complex (CHC)Kof weight n on X, consists of

(1) a complex KA ∈ ObD+(X,A)(2) a filtered complex (KC, F ) ∈ ObD+F (X,C)(3) an isomorphism α : KC

∼−→ KA ⊗ C in D+(X,C).

40 FOUAD EL ZEIN

Moreover, the following axiom must be satisfied(CHC) The triple (RΓ(KA),RΓ(KC, F ),RΓ(α)) is a HC of weight n.When A = Z, we say simply HC or CHC.

Remark 7.1. If (K, F ) is a HC (resp. CHC) of weight n, then (K[m], F [p]) is a HC(resp. CHC) of weight n + m− 2p.

Theorem 7.2. Let X be a compact Kahler manifold and consider(1) KZ the complex reduced to a constant sheaf Z on X in degree zero.(2) KC the De Rham holomorphic complex Ω∗X with its Hodge filtration F = σ

F pΩ∗X := 0 → · · · → ΩpX → Ωp+1

X · · · → ΩnX → 0

(3) The quasi-isomorphism α : KZ ⊗ C ≈−→ C ≈−→ Ω∗X (Poincare’s lemma).Then (KZ , (KC, F ), α) is a CHC of weight.

We need to prove that the complex RΓ(X, Ω∗X) defined by the derived globalsection functor Γ is a HC. In simple terms, the global section of the filtered resolu-tion (EX , F ) defined by Dolbeault resolutions of F pΩ∗X is a HC. This means exactlythat the induced Hodge filtration F on Hn(X,C) defines a HS, hence the result isjust a reformulation of classical Hodge theory.

7.1.3. Let L be a rational local system with a polarization, rationally defined, onthe associated local system LC = L ⊗Q C, then the spectral sequence defined bythe Hodge filtration on DeRham complex with coefficients Ω∗X ⊗C L degenerates atrank 1

Epq1 = Hq(X, Ωp

X(L)) ⇒ Hp+q(X,LC)and the induced filtration by F on cohomology defines a HS. The proof is similarto the constant case.

7.1.4. A CHC is defined by the Hodge filtration on DeRham complex for anycomplex algebraic manifold X. Using the existence of a projective manifold aboveX and birational to X ( Chow’s lemma), and resolution of singularities, Delignededuces a Hodge decomposition in this way on the cohomology of any compactcomplex algebraic manifold not necessarily Kahler. The HS defined in this way arefunctorial for morphisms of algebraic varieties.

7.2. An A- mixed Hodge complex (MHC) K consists of(1) a complex KA ∈ ObD+(A) s.t. Hk(KA) is an A-module of finite type for

all k,(2) a filtered complex (KA⊗Q,W ) ∈ ObD+F (A⊗Q) with an increasing filtra-

tion W ,(3) an isomorphism KQ

∼−→ KA ⊗Q in D+(A⊗Q),(4) a bifiltered complex (KC,W, F ) ∈ ObD+F2(C) with an increasing (resp.

decreasing) filtration W (resp. F ) and an isomorphism α : (KC,W ) ∼−→(KA⊗Q,W )⊗ C in D+F (C).

Moreover, the following axiom is satisfied(MHC) For all n, the system consisting of the complex GrW

n (KA⊗Q) ∈ ObD+(A⊗Q), the complex (GrW

n (KC, F ) ∈ ObD+F (C) with induced F and the isomorphismGrW

n (α) : GrWn (KA⊗Q)⊗ C ∼−→ GrW

n (KC), is an A⊗Q−HC of weight n.


7.2.1. An A-cohomological mixed Hodge complexe K (CMHC) on a topologicalspace X consists of

(1) a complex KA ∈ ObD+(X,A) s.t. Hk(X, KA) are A-modules of finite type,and H∗(X, KA)⊗Q ∼−→ H∗(X,KA ⊗Q),

(2) a filtered complex (KA⊗Q, W ) ∈ ObD+F (X, A ⊗ Q) with an increasingfiltration W and an isomorphism KA⊗Q ' KA ⊗Q in D+(X, A⊗Q),

(3) a bifilitered complex (KC,W, F ) ∈ ObD+F2(X,C) with an increasing (resp.decreasing) filtration W (resp. F ) and an isomorphism α : (KC,W ) ∼−→(KA⊗Q,W )⊗ C in D+F (X,C).

Moreover, the following axiom is satisfied(CMHC) For all n, the system consisting of the complex GrW

n (KA⊗Q) ∈ ObD+(X, A⊗Q), the complex (GrW

n (KC, F ) ∈ ObD+F (X,C) with induced F and the isomor-phism GrW

n (α) : GrWn (KA⊗Q) ⊗ C ∼−→ GrW

n (KC), is an A ⊗ Q − CHC of weightn.

Proposition 7.3. If K = (KA, KA⊗Q,W ), (KC,W, F )) is an A−CMHC thenRΓK = (RΓKA,RΓ(KA⊗Q,W ),RΓ(KC,W, F )) is an A-MHC.

Remark 7.4. If (K, W,F ) is a MHC (resp. CMHC), then for all m and n ∈Z, (K[m],W [m− 2n], F [n]) is a MHC (resp. CMHC).

7.2.2. MHS on the cohomology of a MHC.

Theorem 7.5 (Deligne). Let K be an A−MHC.(i)The filtration W [n] of Hn(KA⊗Q) ' Hn(KA) ⊗ Q and the filtration F onHn(KC) ' Hn(KA)⊗A C define on Hn(K) an A−MHS i.e.(Hn(KA), (Hn(KA⊗Q), W ), (Hn(KC), W, F ) is an A-MHS.(ii)On the terms W Epq

r of the spectral sequence of (KC,W ), the recurrent filtrationand the two direct filtrations defined by F coincide Fd = Frec = Fd∗ .(iii) The morphisms d1 : W Ep,q

1 → W Ep+1,q1 are strictly compatible with the filtra-

tion F .(iv) The spectral sequence of (KA⊗Q, W) degenerates at E2 (W E2 = W E∞).(v) The spectral sequence of (KC, F ) degenerates at E1(F E1 = F E∞).(vi) The spectral sequence of the complex Grp

F (KC), with the induced filtration W ,degenerates at E2.

Proof. The two direct filtration and the recurrent filtration Frec defined by F co-incide on W Epq

1 = Hp+q(GrW−pK). The axiom on (MHC) shows that the complex

GrW−pK is a HC of weight −p, and the axiom (HC2)) shows that the filtration Frec,

induced by F on GrW−pK, is q-opposed on W Epq

1 to its complex conjugate F rec.The differential d1 is compatible with the direct filtrations, hence with Frec andcommutes with complex conjugation since it is defined on A⊗Q, hence it is com-patible with F rec. Then it is strictly compatible with Hodge filtration F = Frec,which proves (iii).We deduce that the two direct filtrations and the recurrent filtration induced byF on W Epq

2 coincide. The filtration Frec defined in this way is q-opposed to itscomplex conjugate and defines a HS of weight q on W Epq

2 .The proof of the following lemma is by induction on r

Lemma 7.6. For r ≥ 0, the differentials dr of the spectral sequence W E are strictlycompatibles to the recurrent filtration F = Frec. For r ≥ 2, they vanish.

42 FOUAD EL ZEIN

For r = 0 we apply (HC1). We just proved it for r = 1 and for r ≥ 2, it isenough to prove dr = 0. We suppose by induction that the two direct filtrationsand the recurrent filtration coincide on W Es(s ≥ r + 1) : Fd = Frec = Fd∗ andW Er = W E2. Then dr is compatible with the filtration Frec := F . On W Ep,q

2 =W Ep,q

r , la filtration F is q-opposed to its complex conjugate. Hence the morphismdr : W Ep,q

r → W Ep+r,q−r+1r satisfy, for r − 1 > 0

dr(W Ep,qr ) = dr(

∑

a+b=q

F a(W Ep,qr ) ∩ F

b(W Ep,q

r )) ⊂∑

a+b=q

F a(W Ep+r,q−r+1r ) ∩ F

b(W Ep+r,q−r+1

r )) = 0

which proves (ii) et (iv).After the results on two filtrations , the filtration on W Ep,q

∞ induced by the filtrationF on Hp+q(K) coincide with the filtration Frec on W Ep,q

2 . Hence, it is q-oppositeto its complex conjugate, which proves (i). ¤

7.3. MHS of a normal crossing divisor (NCD). Let Y be a normal crossingdivisor in a proper complex smooth algebraic variety. We suppose the irreduciblecomponents (Yi)i∈I of Y smooth and ordered.

7.3.1. Mayer-Vietoris resolution. Let Sq denotes the set of strictly increasing se-quences σ = (σ0, ..., σq) on the ordered set of indices I, Yσ = Yσ0 ∩ ... ∩ Yσq , Yq =∑

σ∈SqYσ and for all j ∈ [0, q], q ≥ 1 let λj,q : Yq → Yq−1 denotes a map induc-

ing for each σ the embedding λj,σ : Yσ → Yσ(j) where σ(j) = (σ0, ..., σj , ..., σq) isobtained by deleting σj . Let Πq : Yq → Y (or simply Π) denotes the canonicalprojection and λ∗j,q : Π∗ZYq−1 → Π∗ZYq the morphism defined by λj,q for j ∈ [0, q].

Definition 7.7 (Mayer-Vietoris resolution of ZY ). It is defined by the followingcomplex of sheaves Π∗ZY .

0 → Π∗ZY0 → Π∗ZY1 → · · · → Π∗ZYq−1

δq−1→ Π∗ZYq→ · · ·

where δq−1 =∑

j∈[0,q](−1)jλ∗j,q.

This resolution is associated to a resolution of Y by topological spaces in thefollowing sense. Consider the diagram of spaces over Y

Y. =

Y0

←← Y1

←←←

· · · Yq−1

...λj,q←−−...

Yq · · ·

Π−→ Y

This diagram is the strict simplicial scheme associated in [7] to the normal cross-ing divisor Y , called here after Mayer-Vietoris. The Mayer-Vietoris complex iscanonically associated as direct image by Π of the sheaf ZY . equal to ZYi on Yi.

7.3.2. The cohomological mixed Hodge complex of a NCD. The weight filtrationW on Π∗QY . is defined by

W−q(Π∗QY .) = σ·≥qΠ∗QY . = Π∗σ·≥qQY ., GrW−q(Π∗QY .) ' Π∗QYq [−q]


We introduce the complexes Ω∗Yiof differential forms on Yi. The simple complex

s(Ω∗Y .) is associated to the double complexΠ∗Ω∗Y . with the exterior differential d offorms and the differential δ. defined by δq−1 =

∑j∈[0,q](−1)jλ∗j,q on Π∗Ω∗Yq−1

. Theweight W , and Hodge F filtrations are defined as

W−q = s(σ·≥qΩ∗Y .) = s(0 → · · · 0 → Π∗Ω∗Yq→ Π∗Ω∗Yq+1

→ · · · )

F p = s(σ∗≥pΩ∗Y .) = s(0 → · · · 0 → Π∗ΩpY . → Π∗Ω

p+1Y . → · · · )

We have a filtered isomorphism in D+F (Y,C)

(GrW−qs(Ω

∗Y .), F ) ' (Π∗Ω∗Yq

[−q], F ) in D+F (Y,C).

inducing isomorphisms in D+(Y,C)

(Π∗QY .,W )⊗ C = (CY .,W ) α∼−−→ (s(Ω∗Y .), W )

GrW−q(Π∗CY .) ' Π∗CYq [−q]∼→Π∗Ω∗Yq

[−q] ' GrW−qs(Ω

∗Y .)

Let K be the system consisting of

(Π∗QY ., W ),QY ' Π∗QY ., (s(Ω∗Y .),W, F ), (Π∗QY .,W )⊗ C ' (s(Ω∗Y .),W )

Proposition 7.8. The system K associated to a normal crossing divisor Y withsmooth proper irreducible components, is a CMHC on Y . It defines a functorialMHS on the cohomology Hi(Y,Q), with weights varying between 0 and i.

In terms of Dolbeault resolutions : (s(C)∗,∗Y . ,W, F ), the statement means that thecomplex of global sections Γ(Y, s(C)∗,∗Y . ,W, F ) := (RΓ(Y,C),W, F ) is a MHC in thefollowing sense

GrW−i(RΓ(Y,C) := (Γ(Y, W−is(C∗,∗Y . )/Γ(Y, W−i−1s(C∗,∗Y . ), F )

' (Γ(Y, GrW−is(C∗,∗Y . ), F ) ' (RΓ(Yi,Ω∗Yi

[−i]), F )

is a HC of weight −i in the sense that

(Hn(GrW−iRΓ(Y,C)), F ) ' (Hn−i(Yi,C), F )

is a HS of weight n− i.The terms of the spectral sequence E1(K,W ) of (K, W ) are written as

W Epq1 = Hp+q(Y,GrW

−p(sΩ∗Y .)) ' Hp+q(Y, Π∗ΩYp [−p]) ' Hq(Yp,C)

They carry the HS of the space Yp. The differential is a combinatorial restrictionmapd1 =

∑j≤p+1(−1)jλ∗j,p+1 : Hq(Yp,C) → Hq(Yp+1,C)

is a morphism of HS. The spectral sequence degenerates at E2 (E2 = E∞).

Corollary 7.9. The HS on GrWq Hp+q(Y,C) is the cohomology of the complex of

HS defined by (Hq(Yp,C), d1):

(GrWq Hp+q(Y,C), F ) ' ((Hp(Hq(Yp,C), d1), F )

44 FOUAD EL ZEIN

7.4. Logarithmic complex. Now we construct the MHS on the cohomology of asmooth complex algebraic variety V . By a result of Nagata, V can be embedded asan open Zariski subset of a complete variety X. After Hironaka’s desingularisationtheorem in characteristic zero, we can suppose X smooth and Y = X − V a NCDwith smooth components. Hence we reduce the construction of the MHS to thecomplement X−Y of a NCD. we show that this construction is independent of thechoice of X and Y s.t. V = X − Y .With the previous notations this case is the Poincare’s dual of the cohomologywith compact support on X − Y which can be constructed by the double DeRhamcomplex Ω∗X → Ω∗Y ..However we introduce here the logarithmic complex, important for its own proper-ties as well to obtain functorial constructions later. Let X be a complex manifoldand Y be a NCD in X. By definition, at each point y ∈ Y , there exist local coor-dinates (zi)i∈[1,n] of X s.t. Y is defined at y by the equation Πi∈I⊂[1,n]zi = 0. LetX∗ = X − Y and j : X∗ → X denotes the embedding of X∗ into X.

Definition 7.10. The logarithmic DeRahm complex of X along Y is a sub-complexΩ∗X(LogY ) of the complex ΩX(∗Y ) ⊂ j∗Ω∗X∗ of meromorphic forms along Y , holo-morphic on X∗. A section of ω ∈ Ωp

X(logY ) is called a differential form withlogarithmic pole along Y and may be written locally as

ω = Σi1,··· ,irωi1,··· ,ir

dzi1

zi1

∧ · · · ∧ dzir

zir

; ωi1,··· ,ir holomorphic

The definition is independent of the choice of coordinates; that is ω is writtenin this form with respect to any set of local coordinates at y. The OX -moduleΩ1

X(LogY ) is locally free with basis (dzi/zi)i∈I and (dzj)j /∈I and ΩpX(LogY ) =

∧pΩ1X(LogY ).

An intrinsic property of the sections ω of the logarithmic complex that can be takenas a definition, is that ω and dω have both a pole of order at most 1 at y.In particular, given a local equation

∏ki=1 zi = 0 of Y at y, a meromorphic section

of j∗O∗X∗ along Y is written locally as f = g∏k

i=1 zkii with g invertible, then

df/f = dg/g +k∑1

kidzi/zi

is the sum of a regular form and a linear combination of the independent vectorsdzi/zi.Let f : X1 → X2 be a morphism of complex manifolds, with normal crossingdivisors Yi in Xi for i = 1, 2, s.t. f−1(Y2) = Y1. Then, the reciprocal morphismf∗ : f∗(j2∗Ω∗X∗

2) → j1∗Ω∗X∗

1induces a morphism on logarithmic complexes

f∗ : f∗Ω∗X2(LogY2) → Ω∗X1

(LogY1).

7.4.1. Weight filtration W . Let Y = ∪i∈IYi be the union of smooth irreducibledivisors. Let Sq denotes the set of strictly increasing sequences σ = (σ1, ..., σq) inthe set of indices I, Yσ = Yσ1...σq = Yσ1 ∩ ... ∩ Yσq , Y

q =∐

σ∈Sq Yσ the disjointunion of Yσ. Set Y 0 = X, Π : Y q → Y the canonical projection. An increasingfiltration W , called the weight, is defined as follows

Wm(ΩpX(LogY )) =

∑

σ∈Sm

Ωp−mX ∧ dzσ1/zσ1 ∧ ... ∧ dzσm/zσm


The sub-OX -module Wm(ΩpX(LogY )) ⊂ Ωp

X(LogY ) is the smallest sub-module sta-ble by exterior multiplication with local sections of Ω∗X and containing the productsdf1/f1 ∧ ...∧ dfk/fk for k ≤ m for local sections fi of j∗O∗X∗ meromorphic along Y .

7.4.2. The Residue isomorphism. The Poincare residue defines an isomorphism

Res : GrWm (Ωp

X(LogY )) → Π∗ΩpY m [−m] : Res(α∧(dzσ1/zσ1∧...∧(dzσm

/zσm)) = α/Yσ

In the case p = 1, it defines

Res : Ω1X(LogY ) → Π∗OY 1

We construct its inverse. Consider for σ ∈ Sm the morphismρσ : Ωp

X → GrWm (Ωp+m

X (LogY )) defined locally as

ρσ(α) = α ∧ dzσ1/zσ1 ∧ ... ∧ dzσm/zσm

It does not depends on the choice of zi since for another choice of coordinates z′i,zi/z′i are holomorphic and the difference (dzi/zi) − (dz′i/z′i) = d(zi/z′i)/(zi/z′i) isholomorphic too.Then ρσ(α)−α∧dz′σ1

/z′σ1∧ ...∧dzσm/zσm ∈ Wm−1Ω

p+mX (LogY ), and successively

ρσ(α) − ρ′σ(α) ∈ Wm−1Ωp+mX (LogY ). We have ρσ(zσi · β) = 0 and ρσ(dzσi ∧

β′) = 0 for sections β of ΩpX and β′ of Ωp−1

X ; hence ρσ factors by ρσ : Π∗ΩpYσ→

GrWm (Ωp+m

X (LogY )) defined locally and glue globally into a morphism of complexeson X

ρ : Π∗Ω∗Y m [−m] → GrWm (Ω∗X(LogY )).

Lemma 7.11. We havei) Hi(GrW

m Ω∗X(LogY )) ' Π∗CY m for i = m and vanishes for i 6= m.ii) Hi(Ω∗X(LogY )) ' Π∗CY i .

Proof. (ii) We deduce from the long exact sequence associated to the short exactsequence

0 → Wr → Wr+1 → GrWr+1 → 0

by induction on r, the statement:Hi(WrΩ∗X(LogY )) ' Π∗CY i for i ≤ r and vanishes for i > r. ¤

7.4.3. Hodge filtration F . It is defined on Ω∗X(LogY ) by the formula F p = σ≥p

including all forms of type (p′, q′) with p′ ≥ p. We have

Res : F p(GrWm Ω∗X(LogY )) ' Π∗F p−mΩ∗Y m [−m]

hence a filtered isomorphism

Res : (GrWm Ω∗X(LogY ), F ) ' (Π∗Ω∗Y m [−m], F [−m]).

Proposition 7.12 (Weight filtration W ). The morphisms of filtered complexes

(Ω∗X(LogY ), W ) α← (Ω∗X(LogY ), τ)β→ (j∗Ω∗X∗ , τ)

are filtered quasi-isomorphisms. They define an isomorphism of the spectral se-quence of hypercohomology of the filtered complex (Ω∗X(LogY ),W ) on X withLeray spectral sequence for j with complex coefficients. The main point here isthat the τ filtration is defined with rational coefficients as (Rj∗QX∗ , τ)⊗ C.

46 FOUAD EL ZEIN

7.4.4. MHS on the hypercohomology of Ω∗X(LogY ). Let j : X∗ = X − Y → Xdenotes the open embedding. The system K

(1) (KQ,W ) = (Rj∗QX∗ , τ) ∈ ObD+F (X,Q)(2) (KC,W, F ) = (Ω∗X(LogY ),W, F ) ∈ ObD+F2(X,C)(3) The isomorphism (KQ,W )⊗ C ' (KC,W ) in D+F (X,C)

form a CMHC on X.

Theorem 7.13 (6). The system K = RΓ(X,K) is a MHC. It endows the coho-mology of V with a canonical MHS.

Proof. The weight spectral sequence at rank 1 is written as

W Epq1 (RΓ(X, Ω∗X(LogY )) = Hp+q(X, GrW

−pΩ∗X(LogY )) ' Hp+q(X, Π∗Ω∗Y −p [p])

' H2p+q(Y −p,C) ⇒ GrWq Hp+q(V,C).

where the double arrow means that the spectral sequence degenerates to the coho-mology graded with respect to W induced by the weight on the complex level. Infact we show that it degenerates at rank 2. The differential d1

d1 =−p∑

j=1

(−1)j+1G(λj,−p) = G : H2p+q(Y −p,C) −→ H2p+q+2(Y −p−1,C)

is equal to an alternate Gysin morphism, Poincare’s dual to the alternate restrictionmorphism

ρ =−p∑

j=1

(−1)j+1λ∗j,−p : H2n−q(Y −p−1,C) → H2n−q(Y −p,C)

hence the first term

(W Epq1 , d1)p∈Z = (H2p+q(Y −p,C), d1)p∈Z

is viewed as a complex in the category of HS of weight q. It follows that the termsW Epq

2 = Hp(W E∗,q1 , d1) are endowed with a HS of weight q. The differential d2

which is compatible with the induced HS, being a morphism from Epq2 a HS of

weight q to Ep+2,q−12 a HS of weight q − 1, is necessarily zero. The proof consists

of a recurrent argument to show in this way that the differentials di for i ≥ 2 arezero. ¤

Exercise 7.14 (Riemann Surface). Let C be a connected compact Riemann sur-face, Y = x1, . . . , xm a subset of m points, and C = C−Y the open surface withm points in C deleted. Consider the long exact sequence

0 → H1(C,Z) → H1(C,Z) → H2Y (C,Z) = ⊕i=m

i=1 Z→ H2(C,Z) ' Z→ H2(C,Z) = 0

then0 → H1(C,Z) → H1(C,Z) → Zm−1 ' Ker(⊕i=m

i=1 Z→ Z)

represents H1(C,Z) as an extension, with weight W1H1(C,Z) = H1(C,Z) and

W2H1(C,Z) = H1(C,Z).

The Hodge filtration is defined by the residue

0 → Ω1C→ Ω1

C(logx1, . . . , xm) → Ox1,...,xm → 0


F 0H1(C,C) = H1(C,C),F 1H1(C,C) = H0(C, Ω1

C(logx1, . . . , xm).

F 2H1(C,C) = 0. That is an extension of two different weights

0 → H1(C) → H1(C) → Ker (⊕i=mi=1 Z(−1) → H2(C)) → 0

.

Exercise 7.15 (Hypersurfaces). Let Y ⊂ P be a smooth hypersurface in a projec-tive space P . To describe Hodge theory on U = P − Y we may use by a result ofGrothendieck on algebraic DeRham cohomology, forms on P meromorphic along Yand holomorphic on U , denoted as a sheaf by ΩP (∗Y ) ⊂ j∗ΩU where j : U → P ,or the logarithmic complex by Deligne’s result.1) In the case of a curve Y in a plane, holomorphic one forms are residues of rational2−forms on P with simple pole along the curve.For example, if the homogeneous equation is F = 0, Ω2

P (LogY ) = Ω2P (Y ) and we

have an exact sequence

0 → Ω2P → Ω2

P (Y ) → Ω1Y → 0

Since h2,0 = h2,1 = 0, H0(P, Ω2P ) = H1(P, Ω2

P ) = 0, so we deduce the isomorphism

H0(P, Ω2P (Y )) Res−−→ H0(Y, Ω1

Y )

In homogeneous coordinates, we take the residue along Y of the rational formA(z0dz1 ∧ dz2 − z1dz0 ∧ dz2 + z2dz0 ∧ dz1)

F

where A is homogeneous of degree d− 3 if F has degree d.2) In general, we consider the exact sequence for relative cohomology (or cohomol-ogy with support)

Hk−1(U) ∂−→ HkY (P ) → Hk(P )

j∗−→ Hk(U)

which reduces via Thom’s isomorphism, to

Hk−1(U) r−→ Hk−2(Y ) i∗−→ Hk(P )j∗−→ Hk(U)

where r is the topological Leray’s residue map dual to the tube over cycle mapτ : Hk−2(Y ) → Hk−1(U) associating to a cycle c the boundary in U of a tube overc, and i∗ is Gysin map, Poincare dual to the map i∗ in cohomology.For P = Pn+1 and n odd, the map r is an isomorphism

Hn−1(Y ) ' Hn+1(P ) → Hn+1(U) r−→ Hn(Y ) i∗−→ Hn+2(P ) = 0j∗−→ Hn+2(U)

and for n even the map r is injective

Hn+1(P ) = 0 → Hn+1(U) r−→ Hn(Y ) i∗−→ Hn+2(P ) = Q j∗−→ Hn+2(U)

thenr : Hn+1(U) ∼−→ Hn

prim(X)If Y is ample in P such U = P − Y is affine

Hn−1(Y ) i∗−→ Hn+1(P )j∗−→ Hn+1(U) r−→ Hn(Y ) i∗−→ Hn+2(P ) → Hn+2(U) = 0

then we define the variable cohomology

Hnvar(Y ) = Ker (Hn(Y ) i∗−→ Hn+2(P ))

48 FOUAD EL ZEIN

equal also to = Im r. The fixed cohomology is

Hnfix(Y ) = i∗Hn(P )

In particular, if Hn+1prim(P ) = 0

Hn+1(U,Q) r∼−−→ Hnvar(Y,Q)

MoreoverHn(Y,Q) = Hn

var(Y,Q)⊕Hnfix(Y,Q)

8. MHS on the cohomology of a complex algebraic variety

The aim of this section is to prove

Theorem 8.1 (6). The cohomology of a complex algebraic variety carry a naturalMHS.

8.1. Simplicial varieties. To construct a natural MHS on the cohomology ofany algebraic variety X, not necessarily smooth or compact, Deligne consider asemi-simplicial variety which is a cohomological resolution in some sense of theoriginal variety (descent theorem). On each term of the simplicial resolution, whichconsists of the complement of a NCD in a smooth compact complex variety, thevarious logarithmic complexes are connected by functorial relations and referred toas a simplicial CMHC giving rise to the CMHC defining the MHS we are lookingfor on the cohomology of X. Although such construction is technically elaborate,the above abstract development of MHC structure leads easily to the result withoutfurther difficulty.

8.1.1. Cohomology of topological simplicial spaces. In what follows we could re-strict the constructions to semi-simplicial spaces which underly the simplicial spacesand work only with semi-simplicial spaces since we use only the face maps.Recall the definition of the simplicial category ∆ whose objects are ∆n, n ∈ N, theset of integers 0, 1, . . . , n and morphisms Hp,q, the set of increasing (large sense)mappings from ∆p to ∆q for integers p, q ≥ 0, with the natural composition ofmappings : Hpq ×Hqr → Hpr. The semi-simplicial category ∆> is obtained whenwe consider only the strictly increasing morphisms in ∆.A simplicial (resp. co-simplicial) object of a category C is a contravariant (resp.covariant) functor from ∆ to C.8.1.2. Sheaves. A sheaf F . on a simplicial space X. is defined by1) A family of sheaves Fn on Xn,2) for each f : ∆n → ∆m with X.(f) : Xm → Xn, an X.(f)−morphism F.(f)from Fn to Fm, that is X.(f)∗Fn → Fm on Xm satisfying for all g : ∆r → ∆n,F.(f g) = F.(f) F.(g).A morphism u : F . → G. is a family of morphisms un : Fn → Gn s.t. for all f :∆n → ∆m, umF .(f) = G.(f)un where the left term is: X.(f)∗Fn → Fm um

−−→ Gm

and the right term is X.(f)∗Fn X.(f)∗(un)−−−−−−−→ X.(f)∗Gn → Gm.We may consider the derived category of cosimplicial A−modules.

8.1.3. A topological space S defines a simplicial constant space S. s.t. Sn = S forall n and S.(f) = Id for all f . An augmented simplicial space π : X. → S is definedby a morphism of simplicial spaces π. : (X.) → (S.).


8.1.4. The structural sheaves OXn of a simplicial analytic space form a simplicialsheaf of rings. Let π : X. → S be an augmentation to an analytic space S, theDeRham complex of sheaves Ω∗Xn/S for various n form a complex of sheaves on X.

denoted Ω∗X./S .A simplicial sheaf F . on a constant simplicial space S. defined by S is a a co-simplicial sheaf on S; hence if F . is abelian, it defines a complex withd =

∑i(−1)iδi : Fn → Fn+1.

A complex of abelian sheaves K on S., denoted by Kn,m with m the cosimplicialdegree, defines a simple complex sK:

(sK)n := ⊕p+q=nKpq; d(xpq = dK(xpq) + (−1)iδixpq.

The filtration L with respect to the second degree will be useful

Lr(sK) = s(Kpq)q≥r.

8.1.5. Direct image on derived category of abelian sheaves. Consider an augmentedsimplicial space a : X. → S, we define a functor denoted Ra∗ for complexes K ofabelian sheaves on (X.). We may view S as a constant simplicial scheme S. anda as a morphism a. : (X.) → (S.). In the first step we construct an injective orflabby complex I quasi-isomorphic to K (K ' I); we can always take Godementresolutions for example, then in each degree p, aq

∗Ip on Sq = S defines for varyingq a cosimplicial sheaf on S denoted a.

∗Ip, and a differential graded complex for

varying p, which is a double complex whose associated simple complex is denotedsa∗I := Ra∗K

(Ra∗K)n := ⊕p+q=naq∗I

p,q; dxpq = dI(xpq)+(−1)pΣq+1i=0 (−1)iδix

pq ∈ (Ra∗K)n+1

where q is the simplicial index ( δi(xpq) ∈ Ip,q+1 and p is the degree. In particularfor S a point we define the hypercohomology of K

RΓ(X.,K) := sRΓ.(X., K); Hi(X., K) := Hi(RΓ(X., K)).

The filtration L on sa∗I := Ra∗K defines a spectral sequence

Epq1 = Rqap∗(K|Xp

) := Hq(ap∗(K|Xp) ⇒ Hp+q(Ra∗K) := Rp+qa∗K

8.1.6. Topological realization. Recall that a morphism of simplices f : ∆n → ∆m

has a geometric realization |f | : |∆n| → |∆m| as the affine map defined whenwe identify a simplex ∆n with the vertices of its affine realization in R∆n . Weconstruct the topological realization of a topological semi-simplicial space X. asthe quotient of the topological space Y =

∐n≥0 Xn × |∆n| by the equivalence

relation R generated by the identifications

∀f : ∆n → ∆m, x ∈ Xm, a ∈ |∆n|, (x, |f |(a)) ≡ (X.(f)(x), a)

The topological realization |X.| is the quotient space of Y , modulo the relation R,with its quotient topology. The construction above of the cohomology amounts tothe computation of the cohomology of the topological space |X.| with coefficient inan abelian group A

Hi(X., A) ' Hi(|X.|, A).

50 FOUAD EL ZEIN

8.1.7. Cohomological descent. Let a : X. → S be an augmented simplicial scheme.Any abelian sheaf F on S, lifts to a sheaf a∗F on X. and we have a natural morphism

ϕ(a) : F → Ra∗a∗F in D+(S)

Definition 8.2 (cohomological descent). The morphism a : X. → S is of cohomo-logical descent if the natural morphism ϕ(a) is an isomorphism in D+(S) for allabelian sheaves F on S.

The definition amounts to the following conditions

F∼−→ Ker(a0∗a∗0F

δ1−δ0−−−−→ a∗1F ); Ria∗a∗F = 0 for i > 0.

In this case for all complexes K in D+(S)

RΓ(S, K) ' RΓ(X., a∗K)

and we have a spectral sequence

Epq1 = Hq(Xp, a

∗pK) ⇒ Hp+q(S, K), d1 =

∑

i

(−1)iδi : Ep,q1 → Ep+1,q

1 .

8.1.8. Simplicial varieties. A complex simplicial variety X. is smooth (resp. proper)if every Xn is smooth (resp. compact).

Definition 8.3 (NCD). A normal crossing divisor is a family Yn ⊂ Xn of normalcrossing divisors such that the family of open subsets Un := Xn − Yn form asimplicial subvariety U. of X. s.t. the family of filtered logarithmic complexes(Ω∗Xn

(LogYn))n≥0,W ) form a filtered complex on (X.).

Theorem 8.4. For each separated complex variety S, there existi) A simplicial variety proper and smooth X. over Cii) A normal crossing divisor Y. in X. and an augmentation a : U. = (X.−Y.) → Ssatisfying the cohomological descent property. Hence for all abelian sheaves F onS, F

∼−→ Ra∗a∗F .Moreover, for each morphism f : S → S′, there exists a morphism f. : X. → X ′. ofsimplicial varieties proper and smooth with normal crossing divisors Y. and Y ′. andaugmented complements a : U. → S and a′ : U ′. → S′ satisfying the cohomologicaldescent property, such that the restriction u : U. → U ′. of f commutes with theaugmentations.

The proof is based on Hironaka’s desingularisation theorem and on a generalcontruction of hypercoverings described briefly by Deligne after the general theoryof hypercoverings. The desingularisation is carried at each step of the contruction.

Remark 8.5. We can (and we will) assume the NCD with smooth irreducible com-ponents.

8.1.9. An A-cohomological mixed Hodge complexe K (CMHC) on a topologicalsimplicial space X. consists of

(1) a complex KA ∈ ObD+(X.,A) s.t. Hk(X.,KA) are A-modules of finitetype, and H∗(X., KA)⊗Q ∼−→ H∗(X., KA ⊗Q),

(2) a filtered complex (KA⊗Q,W ) ∈ ObD+F (X., A ⊗ Q) with an increasingfiltration W and an isomorphism KA⊗Q ' KA ⊗Q in D+(X., A⊗Q),

(3) a bifilitered complex (KC, W, F ) ∈ ObD+F2(X.,C) with an increasing (resp.decreasing) filtration W (resp. F ) and an isomorphism α : (KC,W ) ∼−→(KA⊗Q,W )⊗ C in D+F (X.,C).


Moreover, the following axiom is satisfied(CMH) The restriction of K to each Xn is an A−CMHC.

8.1.10. Let X. be a complex compact smooth simplicial algebraic variety with Y.a NCD in X. s.t. j : U. = (X.− Y.) → X. is an open simplicial subvariety, then

(Rj∗Z, (Rj∗Q, τ≤), (Ω∗X.(LogY.),W, F ))

is a CMHC on (X.).

8.1.11. If we apply the global section functor to an A−CMHC K on X., we getan A−cosimplicial mixed Hodge complex defined as follows1) A cosimplicial complex RΓ.KA in the derived category of cosimplicial A−modules,2 ) A filtered cosimplicial complex RΓ.(KA⊗Q,W ) in the derived category of fil-tered cosimplicial vector spaces,3) A bifiltered cosimplicial complex RΓ.(KC,W, F ) in the derived category of bifil-tered cosimplicial vector spaces,4) An isomorphism RΓ.(KA⊗Q ⊗C, W ⊗C) ' RΓ.(KC,W, F ) in the derived cate-gory of filtered cosimplicial vector spaces.The hypercohomology of a cosimplicial A-CMHC on X. is such a complex.

8.1.12. Diagonal filtration. A cosimplicial mixed Hodge complex K defines a dif-ferential graded complex which is viewed as a double complex whose associatedsimple complex is sK. We deduce on sK a weight filtration by a diagonal process.

Definition 8.6 (Differential graded A−MHC). A differential graded (DG) (resp.DG+) complex ( or a complex of graded objects) has two degrees (resp boundedbelow), the first defined by the complex and the second by the gradings. It can beviewed as a double complex.A graded differential A−MHC is defined by a system of a DG+−complex (resp.filtered, bifiltered)

K, (K, W ), (K, W,F )

s.t. for each degree n of the grading, the component (K .,n,W, F ) is an A-MHC.

A graded differential A−MHC is not a MHC. A cosimplicial mixed Hodge com-plex (K, W,F ) defines a DG+ −A−MHC

sK, (sK, W ), (sK,W,F )

the degree of the grading is the cosimplicial degree.

Definition 8.7 (Diagonal filtration). The diagonal filtration δ(W,L) of W and Lon sK is defined by

δ(W,L)n(sK) := ⊕p,qWn+qKp,q =

∑r

s(Wn+rK) ∩ Lr(sK).

where Lr(sK) = s(Kp,q)q≥r.For a bifiltered complex (K, W,F ) with a decreasing F , the sum over F is natural.

52 FOUAD EL ZEIN

8.2. Properties. We have

Grδ(W,L)n (sK) ' ⊕pGrW

n+pK.,p[−p]

In the case of a DG+−complex defined as the hypercohomology of a complex(K,W ) on a simplicial space X., we have

Grδ(W,L)n RΓK ' ⊕pRΓ(Xp, GrW

n+pK.,p)[−p].

and for a bifiltered complex with a decreasing F

Grδ(W,L)n (RΓK, F ) ' ⊕pRΓ(Xp, (GrW

n+pK, F ))[−p].

Next we need the following

Lemma 8.8. If H = (HA, W, F ) is an A−MHS, a filtration L of HA is a filtrationof in the category of MHS, if and only if for all n, (Grn

LHA, GrnL(W ), Grn

L(F ) is anA−MHS.

Theorem 8.9. Let K be a graded differential A−MHC (for example defined by acosimplicial A−MHC).i) Then (sK, δ(W,L), F ) is an A−MHC.The first terms of the weight spectral sequence

δ(W,L)Epq1 (sK ⊗Q) =

∑p=m−n,q=r+n

Ha(GrWn K .,m)

form the simple complex (δ(W,L)Epq1 , d1) of A ⊗ Q−Hodge structures of weight q

associated to the double complex

Hq−(n+1)(GrWn+1K

.,m+1) δ−→ Hq−n(GrWn K .,m+1) δ−→ Hq−(n−1)(GrW

n−1K.,m+1)

d′′ ↑ d′′ ↑Hq−n(GrW

n K .,m) δ−→ Hq−(n−1)(GrWn−1K

.,m)

ii) The terms LEr for r > 0, of the spectral sequence defined by (sKA⊗Q, L) are en-dowed with a natural A−MHS, with differentials dr compatible with such structures.iii) The filtration L on H∗(sK) is a filtration in the category of MHS and

GrpL(Hp+q((sK), δ(W,L)[p + q], F ) = (Epq

∞ ,W, F ).

8.2.1. In the case of a logarithmic simplicial variety, the cohomology groups Hn(U.,Z)are endowed with MHS defined by the following MHCK := RΓ(U.,Z), (K ⊗Q := RΓ(U.,Q), δ(W,L)),(K ⊗ C := RΓ(U.,Ω∗X.(LogY.)), δ(W,L)), F )satisfying

Grδ(W,L)n RΓ(U.,Q) ' ⊕m(GrW

n+mRΓ(Um, ,Q))[−p])[−m] ' ⊕mRΓ(Y n+mm ,Q)[−n−2m].

withδ(W,L)E

−a,b1 = ⊕p+2r=b,q−r=−aHp(Y r

q , Q) ⇒ H−a+b(U.,Q).

the filtration F induces on δ(W,L)E−a,b1 a HS of weight b and the differentials d1

are compatible with the HS. The term E1 is the simple complex associated to thedouble complex of HS of weight b where G is an alternating Gysin map

Hb−(2r+2)(Y r+1q+1 ,Q)(−r − 1) G−→ Hb−2r(Y r

q+1,Q)(−r) G−→ Hb−2(r−2)(Y r−1q+1 ,Q)(−r + 1)∑

i(−1)iδi ↑∑

i(−1)iδi ↑Hb−2r(Y r

q ,Q)(−r) G−→ Hb−2(r−2)(Y r−1q ,Q)(−r + 1)


where the indices of Eab1 satisfy −a = q − r.

Proposition 8.10. i) The MHS on Hn(U.,Z) is defined by the graded differentialMHC associated to the simplicial MHC defined by the logarithmic complex on eachterm of X. and it is functorial in the couple (U.,X.),ii) The rational weight spectral sequence degenerates at rank 2 and the HS on E2

induced by E1 is isomorphic to the HS on GrW Hn(U.,Q).iii)The Hodge numbers hpq of Hn(U.,Q) vanish for p /∈ [0, n] or q /∈ [0, n].iv) For Y. = ∅, the Hodge numbers hpq of Hn(X.,Q) vanish for p /∈ [0, n] or q /∈ [0, n]or p + q > n.

Definition 8.11. The MHS on the cohomology of a complex algebraic varietyHn(X,Z) is defined by any logarithmic simplicial resolution of X via the isomor-phism with Hn(U.,Z) defined by the augmentation a : U. → X. It doesn’t dependson the resolution and it is functorial in X.

8.3. MHS on the cohomology of a complete embedded algebraic variety.For embedded varieties into smooth varieties , the MHS on cohomology can bededuced by a simple method, using exact sequences from MHS already constructedfor NCD, which should easily convince of the natural aspect of this theory. Thetechnical ingredients consist of Poincare duality and the trace (or Gysin ) mor-phism.Let p : X ′ → X be a proper morphism of complex smooth varieties of same dimen-sion, Y a closed subvariety of X and Y ′ = p−1(Y ). We suppose that Y ′ is a NCDin X ′ and the restriction of p induces an isomorphism p/X′−Y ′ : X ′−Y ′ ∼−→ X−Y .

Y ′ i′−→ X ′ j′← X ′ − Y ′

↓ pY ↓ p ↓ pX−Y

Yi−→ X

j← X − Y

The trace morphism Trp is defined as Poincare dual to the inverse image p∗ oncohomology. It can be defined at the level of sheaf resolutions of ZX′ and ZX

constructed by Verdier, that is in derived category Tr p : Rp∗ZX′ → ZX hence wededuce by restriction morphisms depending on the embeddings of Y and Y ′ into X ′.(Tr p)/Y : Rp∗ZY ′ → ZY , (Tr p)/Y : Hi(Y ′,Z) → Hi(Y,Z), Hi

c(Y′,Z) → Hi

c(Y,Z).

Remark 8.12. Let U be a neighbourhood of Y in X, retract by deformation ontoY s.t. U ′ = p−1(U) is a retract by deformation onto Y ′; this is the case if Y isa sub-variety of X. Then the morphism (Tr p)/Y is deduced from Tr(p/U ) in thediagram

Hi(Y ′,Z) ∼←− Hi(U ′,Z)↓ (Tr p)/Y ↓ Tr(p/U )Hi(Y,Z) ∼←− Hi(U,Z)

Consider now the diagram

RΓc(X ′ − Y ′,Z)j′∗−→ RΓ(X ′,Z) i′∗−→ RΓ(Y ′,Z)

Trp ↓ ↓ Trp ↓ (Trp)|YRΓc(X − Y,Z)

j∗−→ RΓ(X,Z) i∗−→ RΓ(Y,Z)

Proposition 8.13. i) The morphism (pY )∗ : Hi(Y,Z) → Hi(Y ′,Z) is injective withretraction (Tr p)/Y .

54 FOUAD EL ZEIN

ii) We have a quasi-isomorphism of i∗ZY with the cone C(i′∗−Tr p) of the morphismi′∗ − Tr p. The long exact sequence associated to the cone splits into short exactsequences

0 → Hi(X ′,Z)i′∗−Tr p−−−−−→ Hi(Y ′,Z)⊕Hi(X,Z)

(Tr p)/Y +i∗−−−−−−−−→ Hi(Y,Z) → 0Moreover i′∗ − Tr p is a morphism of MHS.

Definition 8.14. The MHS of Y is defined as cokernel of i′∗ − Tr p via its iso-morphism with Hi(Y,Z), induced by (Tr p)/Y + i∗. It coincides with Deligne’sMHS.

This result shows the uniqueness of the theory of MHS, once the MHS of theNCD Y ′ has been constructed.The above technique consists in the realization of the MHS on the cohomology ofY as relative cohomology with MHS structures.

Books

Carlson J., Muller-Stach S., Peters C.: Period mappings and Period Domains.Griffiths P., Harris J.: Principles of algebraic geometry.Voisin C.: Hodge theory and complex algebraic geometryTu Loring: An Introduction to Manifolds, Universitext, Springer.

References

[1] BEILINSON A.A., BERNSTEIN J., DELIGNE P.: Faisceaux Pervers, in Analyse et Topologiesur les espaces singuliers Vol.I, Asterisque 100(1982).

[2] CARTAN H., EILENBERG S.: Homological Algebra, Princeton University Press (1956).[3] DIEUDONNE J.: A history of algebraic and differential topology, 1900-1960. Birkhauser

Boston 1989.[4] DELIGNE P.: Theoreme de Lefschetz et criteres de degenerescence de suites spec-

trales.Publ.Math.I.H.E.S., 35 (1968), 107-126.[5] DELIGNE P.: Theorie de Hodge I. Actes du congres international. Nice (1970) Gauthier-

Villars (1971) I, 425-430.[6] DELIGNE P.: Theorie de Hodge II Publ. Math. IHES 40 (1972) 5-57.[7] DELIGNE P.: Theorie de Hodge III Publ. Math. IHES 44 (1975) 6-77.[8] GODEMENT R.: Topologie algebrique et theorie des faisceaux.Publ.Inst. Math. Univ. Stras-

bourg XIII, Paris Hermann (1958).[9] GRIFFITHS P., SCHMID W.: Recent developments in Hodge theory, in Discrete subgroups

of Lie groups Oxford university press, (1973).[10] GROTHENDIECK A.: Surquelques points d’algebre homologique, Tohoku Math. J. 9 (

(1957) 119-221.[11] GROTHENDIECK A.: On the De Rham cohomology of Algebraic Varieties, Publ. Math.

IHES 29 (1966).[12] SPANIER E.: Algebraic topology. New York, Mac Graw Hill (1966).[13] VERDIER J.L.:Des categories derives des categories abeliennes, Astrisque 239 (1996).

indico.ictp.itindico.ictp.it/event/a09153/material/2/0.pdf · mixed hodge structures fouad el zein...

Documents