Hodge Structures and the Topology of Algebraic Varieties

Hodge structures and the topology of algebraic varieties

Claire Voisin

CNRS, Institut de math´ematiquesde Jussieu

CMSA Harvard, 30 September 2020 Plan of the talk

• Analysis and diﬀerential geometry Hodge and Lefschetz decompositions.

• + Algebra Consequences on topology. • The importance of polarizations (signs and Hodge-Riemann relations). • Missing. Variations of Hodge structures. Kählerand projective complex manifolds Complex manifold= manifold equipped with an atlas ∼ n Ui = Vi ⊂ C , with holomorphic change of coordinates maps. • The tangent space at each point is endowed with a structure de 2 C-vector space, hence an operator I, I = −Id, of almost complex structure acting on TX,R. Newlander-Nirenberg integrability condition. • Notion of Hermitian metric on X. In local holomorphic coordinates, P 1 P h = ij hijdzi ⊗ dzj, with imaginary part ω = ι ij ωijdzi ∧ dzj, ωij = Im hij. This is a 2-form “of type (1, 1)”. Definition. The Hermitian metric is Kählerif dω = 0. [ω]= Kählerclass. N ∗ • On CP : Fubini-Study Kählermetric. The Kählerclass equals c1(H ), N where H is the Hopf line bundle, hence is integral. Idem for X ⊂ CP complex submanifold. • Kodaira embedding theorem. A compact Kählermanifold is projective iff it admits a Kählerform with integral cohomology class. Kählerand projective complex manifolds Complex manifold= manifold equipped with an atlas ∼ n Ui = Vi ⊂ C , with holomorphic change of coordinates maps. • The tangent space at each point is endowed with a structure de 2 C-vector space, hence an operator I, I = −Id, of almost complex structure acting on TX,R. Newlander-Nirenberg integrability condition. • Notion of Hermitian metric on X. In local holomorphic coordinates, P 1 P h = ij hijdzi ⊗ dzj, with imaginary part ω = ι ij ωijdzi ∧ dzj, ωij = Im hij. This is a 2-form “of type (1, 1)”. Definition. The Hermitian metric is Kählerif dω = 0. [ω]= Kählerclass. N ∗ • On CP : Fubini-Study Kählermetric. The Kählerclass equals c1(H ), N where H is the Hopf line bundle, hence is integral. Idem for X ⊂ CP complex submanifold. • Kodaira embedding theorem. A compact Kählermanifold is projective iff it admits a Kählerform with integral cohomology class. The Frölicherspectral sequence

• X complex manifold, z1, . . . zn = local holomorphic coordinates. Holomorphic vector bundle ΩX generated over OX by dzi. Transition matrices given by holomorphic Jacobian matrices. k Vk • Holomorphic de Rham complex ΩX := ΩX , with exterior diﬀerential d. Thm. (Holomorphic Poincar´elemma). The complex

d d d n OX → ΩX → ... → ΩX → 0

is exact in degrees > 0. This is a resolution of the constant sheaf C. k k • Corollary.H (X, C) = H (X, ΩX ). p • •≥p • Filtration “bête” F ΩX := ΩX Frölicherspectral sequence. p,q p+q E1 ⇒ H (X, C). p,q q p • E1 = H (X, ΩX ), d1 = d. • On the abutment : “Hodge” filtration p k k •≥p k • p,q p k F H (X, C) := Im (H (X, ΩX ) → H (X, ΩX )), E∞ = GrF H (X, C). Quasiprojective manifolds and logarithmic de Rham complexes • j : U,→ X, U = X \ Y , with Y ⊂ X closed analytic. • (Hironaka) By successive blow-ups of X along smooth centers supported over Y , one can assume that Y is a normal crossing divisor: i.e. Y is locally defined by a single holomorphic equation of the form f = z1 . . . zk in adequate holomorphic coordinates.

• Define ΩX (log Y ) as the holomorphic vector bundle generated over OX dz1 dzk by ,..., , dzk+1, . . . , dzn. z1 zk k Vk k k+1 • ΩX (log Y ) = ΩX (log Y ), d :ΩX (log Y ) → ΩX (log Y ). • Their sections (= forms with logarithmic growth) are the forms with pole order 1 along Y , whose differential also has pole order 1 along Y . • • Thm. The inclusion of the subcomplex ΩX (log Y ) ⊂ j∗ΩU is a quasiisomorphism. k k • Corollary. H (U, C) = H (X, ΩX (log Y )) and Frölicherspectral sequence. k k • k • also H (U, C) = H (U, ΩU ) hence two Hodge filtrations on H (U, C). Quasiprojective manifolds and logarithmic de Rham complexes • j : U,→ X, U = X \ Y , with Y ⊂ X closed analytic. • (Hironaka) By successive blow-ups of X along smooth centers supported over Y , one can assume that Y is a normal crossing divisor: i.e. Y is locally defined by a single holomorphic equation of the form f = z1 . . . zk in adequate holomorphic coordinates.

• Define ΩX (log Y ) as the holomorphic vector bundle generated over OX dz1 dzk by ,..., , dzk+1, . . . , dzn. z1 zk k Vk k k+1 • ΩX (log Y ) = ΩX (log Y ), d :ΩX (log Y ) → ΩX (log Y ). • Their sections (= forms with logarithmic growth) are the forms with pole order 1 along Y , whose differential also has pole order 1 along Y . • • Thm. The inclusion of the subcomplex ΩX (log Y ) ⊂ j∗ΩU is a quasiisomorphism. k k • Corollary. H (U, C) = H (X, ΩX (log Y )) and Frölicherspectral sequence. k k • k • also H (U, C) = H (U, ΩU ) hence two Hodge filtrations on H (U, C). The Hodge decomposition theorem 2 • X= compact oriented Riemannian manifold. L -metric on forms. R (α, β)L2 = X α ∧ ∗β. ∗ ∗ ∗ • Formal adjoint d = ± ∗ d∗. Laplacian ∆d = d ◦ d + d ◦ d.

Harmonic forms. ∆dα = 0. X compact and α harmonic ⇒ α is closed. Thm. (Hodge) Each de Rham cohomology class contains a unique harmonic representative. P • Forms of type (p, q) on X =cplx mﬂd: α = |I|=p,|J|=q αIJ dzI ∧ dzJ . P p,q Any k-form writes uniquely as a sum p+q=k α . p,q Thm. (Hodge) X K¨ahler ⇒ ∆dα is of type (p, q). P p,q p,q Corollary. α harmonic, α = p,q α ⇒ each α is harmonic. Thm. (Hodge) Let Hp,q(X) := {classes of closed forms of type (p, q)}. p,q ∼ q p k p,q Then H (X) = H (X, ΩX ) and H (X, C) = ⊕p+q=kH (X). • Hodge symmetry. Hp,q(X) = Hq,p(X).

Cor. The Fr¨olicherspectral sequence of X degenerates at E1 (E1 = E∞). • Consequences in deformation theory. For example BTT theorem. The Hodge decomposition theorem 2 • X= compact oriented Riemannian manifold. L -metric on forms. R (α, β)L2 = X α ∧ ∗β. ∗ ∗ ∗ • Formal adjoint d = ± ∗ d∗. Laplacian ∆d = d ◦ d + d ◦ d.

Cor. The Frölicherspectral sequence of X degenerates at E1 (E1 = E∞). • Consequences in deformation theory. For example BTT theorem. The category of Hodge structures Definition. (Hodge structure) A Hodge structure of weight k= lattice L p,q p,q q,p + decomposition LC = ⊕p+q=kL , with L = L . p r,k−r • Hodge decomposition Hodge filtration: F LC := ⊕r≥pL . p,q p q Conversely L = F LC ∩ F LC, p + q = k. • p k−p+1 Condition on F : LC = F LC ⊕ F LC. • Variants. (a) Rational coefficients. (b) Effective Hodge structure : Lp,q = 0 if p < 0 or q < 0. Definition. (L, F pL ), (L0,F pL0 ) Hodge structures of weights k, k + 2r. C C A morphism of Hodge structures between them is φ : L → L0, s.t. p,q 0p+r,q+r φC(L ) ⊂ L . n n 1,0 Example T = C /Γ ΓC C with kernel Γ ⊂ ΓC is an equivalence of categories (Complex tori )! (effective weight 1 Hodge structures). • Complex tori up to isogeny ! Weight 1 rational Hodge structures. Fact. The category of rational Hodge structures is not semi-simple. There 0 are morphisms of complex tori T T which do not split up to isogeny. The category of Hodge structures Definition. (Hodge structure) A Hodge structure of weight k= lattice L p,q p,q q,p + decomposition LC = ⊕p+q=kL , with L = L . p r,k−r • Hodge decomposition Hodge filtration: F LC := ⊕r≥pL . p,q p q Conversely L = F LC ∩ F LC, p + q = k. • p k−p+1 Condition on F : LC = F LC ⊕ F LC. • Variants. (a) Rational coefficients. (b) Effective Hodge structure : Lp,q = 0 if p < 0 or q < 0. Definition. (L, F pL ), (L0,F pL0 ) Hodge structures of weights k, k + 2r. C C A morphism of Hodge structures between them is φ : L → L0, s.t. p,q 0p+r,q+r φC(L ) ⊂ L . n n 1,0 Example T = C /Γ ΓC C with kernel Γ ⊂ ΓC is an equivalence of categories (Complex tori )! (effective weight 1 Hodge structures). • Complex tori up to isogeny ! Weight 1 rational Hodge structures. Fact. The category of rational Hodge structures is not semi-simple. There 0 are morphisms of complex tori T T which do not split up to isogeny. Hodge structures from geometry; functoriality k Thm. X compact Kähler.The cohomology H (X, Z)/Tors carries an effective Hodge structure of weight k. • φ : X → Y holomorphic map, with X,Y compact Kähler. ∗ k k φ : H (Y, Z)tf → H (X, Z)tf is a morphism of Hodge structures. k k−2d Prop. The Gysin morphism φ∗ : H (X, Z)tf → H (Y, Z)tf , d = dim X − dim Y , is a morphism of Hodge structures. ∗ • Explanation. (a) Via Poincaréduality, φ∗ is the transpose of φ . ∗ (b) weight k Hodge structure on L ! weight −k Hodge structure on L : ∗−p,−q r,s L is defined as the orthogonal of ⊕(r,s)6=(p,q)L . 2n−k (c) the Hodge structure on H (X, Z)tf is dual to the Hodge structure k on H (X, Z)tf up to a shift of bidegree (for “type reasons”: R p,q p0,q0 0 0 X α ∧ β = 0 for (p , q ) 6= (n − p, n − q)). qed 0 Construction. Hodge structure on L, resp.M of weights k, resp. k Weight k + k0 Hodge structure on L ⊗ M : p,q r,s 0r0,s0 (LC ⊗ MC) = ⊕r+r0=p,s+s0=qL ⊗ L . Hodge structures from geometry; functoriality k Thm. X compact Kähler.The cohomology H (X, Z)/Tors carries an effective Hodge structure of weight k. • φ : X → Y holomorphic map, with X,Y compact Kähler. ∗ k k φ : H (Y, Z)tf → H (X, Z)tf is a morphism of Hodge structures. k k−2d Prop. The Gysin morphism φ∗ : H (X, Z)tf → H (Y, Z)tf , d = dim X − dim Y , is a morphism of Hodge structures. ∗ • Explanation. (a) Via Poincaréduality, φ∗ is the transpose of φ . ∗ (b) weight k Hodge structure on L ! weight −k Hodge structure on L : ∗−p,−q r,s L is defined as the orthogonal of ⊕(r,s)6=(p,q)L . 2n−k (c) the Hodge structure on H (X, Z)tf is dual to the Hodge structure k on H (X, Z)tf up to a shift of bidegree (for “type reasons”: R p,q p0,q0 0 0 X α ∧ β = 0 for (p , q ) 6= (n − p, n − q)). qed 0 Construction. Hodge structure on L, resp.M of weights k, resp. k Weight k + k0 Hodge structure on L ⊗ M : p,q r,s 0r0,s0 (LC ⊗ MC) = ⊕r+r0=p,s+s0=qL ⊗ L . Hodge structures from geometry; functoriality k Thm. X compact Kähler.The cohomology H (X, Z)/Tors carries an effective Hodge structure of weight k. • φ : X → Y holomorphic map, with X,Y compact Kähler. ∗ k k φ : H (Y, Z)tf → H (X, Z)tf is a morphism of Hodge structures. k k−2d Prop. The Gysin morphism φ∗ : H (X, Z)tf → H (Y, Z)tf , d = dim X − dim Y , is a morphism of Hodge structures. ∗ • Explanation. (a) Via Poincaréduality, φ∗ is the transpose of φ . ∗ (b) weight k Hodge structure on L ! weight −k Hodge structure on L : ∗−p,−q r,s L is defined as the orthogonal of ⊕(r,s)6=(p,q)L . 2n−k (c) the Hodge structure on H (X, Z)tf is dual to the Hodge structure k on H (X, Z)tf up to a shift of bidegree (for “type reasons”: R p,q p0,q0 0 0 X α ∧ β = 0 for (p , q ) 6= (n − p, n − q)). qed 0 Construction. Hodge structure on L, resp.M of weights k, resp. k Weight k + k0 Hodge structure on L ⊗ M : p,q r,s 0r0,s0 (LC ⊗ MC) = ⊕r+r0=p,s+s0=qL ⊗ L . Functoriality and Hodge classes Definition. (Hodge classes) A Hodge class on a weight 2k Hodge structure L is an element of L ∩ Lk,k. Example. Hodge classes on L∗ ⊗ M, L of weight k, M of weight k + 2r, are the morphisms of Hodge structures L → M. Corollary. Hodge classes on a product X × Y of compact Kähler manifolds identify with the morphisms of Hodge structures ∗ ∗+2r H (X, Z)tf → H (Y, Z)tf . Example. Z ⊂ X closed analytic subset of codimension k has a class 2k [Z] ∈ H (X, Z). If X is compact Kähler,this is a Hodge class. Conjecture. (Hodge conjecture) X smooth complex projective. Rational Hodge classes on X are algebraic, i.e. generated by cycles classes.

Example. Künnethcomponents of the diagonal. δk IdHk(X,Z). • Known in degree 2 (Lefschetz (1, 1)-thm) and 2n − 2 by hard Lefschetz. • Wrong in the compact Kählersetting, even in a weaker form replacing cycle classes by Chern classes of coherent sheaves (Voisin). Functoriality and Hodge classes Definition. (Hodge classes) A Hodge class on a weight 2k Hodge structure L is an element of L ∩ Lk,k. Example. Hodge classes on L∗ ⊗ M, L of weight k, M of weight k + 2r, are the morphisms of Hodge structures L → M. Corollary. Hodge classes on a product X × Y of compact Kähler manifolds identify with the morphisms of Hodge structures ∗ ∗+2r H (X, Z)tf → H (Y, Z)tf . Example. Z ⊂ X closed analytic subset of codimension k has a class 2k [Z] ∈ H (X, Z). If X is compact Kähler,this is a Hodge class. Conjecture. (Hodge conjecture) X smooth complex projective. Rational Hodge classes on X are algebraic, i.e. generated by cycles classes.

Example. Künnethcomponents of the diagonal. δk IdHk(X,Z). • Known in degree 2 (Lefschetz (1, 1)-thm) and 2n − 2 by hard Lefschetz. • Wrong in the compact Kählersetting, even in a weaker form replacing cycle classes by Chern classes of coherent sheaves (Voisin). Mixed Hodge structures Definition. A rational mixed Hodge structure = a Q-vector space L with an increasing (weight) filtration WiL and a decreasing (Hodge) filtration p W F LC, such that: the induced filtration on Gri L defines a Hodge structure of weight i. Thm. (Deligne) The cohomology of quasiprojective complex varieties, or analytic-Zariski open in compact Kählermanifolds, or relative (co)homology of such pairs, carries functorial mixed Hodge structures. j • Smooth case: U = X \ Y ,→ X, Y =normal crossing divisor. Use k k • • H (U, C) = H (X, ΩX (log Y )). Filtration F on ΩX (log Y ) is the usual • one (“bête”).Filtration W on ΩX (log Y ): up to a shift, this is given by • i •−i WiΩX (log Y ) = ΩX (log Y ) ∧ ΩX . • A posteriori, the induced W - filtration is defined on rational cohomology and related to the Leray filtration of j.

• The s.s. for F degenerates at E1, the s.s. for W degenerates at E2. k • In this case, the smallest weight part of H (U, Q) is ∗ k k Im (j : H (X, Q) → H (U, Q)) (weight k). Mixed Hodge structures Definition. A rational mixed Hodge structure = a Q-vector space L with an increasing (weight) filtration WiL and a decreasing (Hodge) filtration p W F LC, such that: the induced filtration on Gri L defines a Hodge structure of weight i. Thm. (Deligne) The cohomology of quasiprojective complex varieties, or analytic-Zariski open in compact Kählermanifolds, or relative (co)homology of such pairs, carries functorial mixed Hodge structures. j • Smooth case: U = X \ Y ,→ X, Y =normal crossing divisor. Use k k • • H (U, C) = H (X, ΩX (log Y )). Filtration F on ΩX (log Y ) is the usual • one (“bête”).Filtration W on ΩX (log Y ): up to a shift, this is given by • i •−i WiΩX (log Y ) = ΩX (log Y ) ∧ ΩX . • A posteriori, the induced W - filtration is defined on rational cohomology and related to the Leray filtration of j.

• The s.s. for F degenerates at E1, the s.s. for W degenerates at E2. k • In this case, the smallest weight part of H (U, Q) is ∗ k k Im (j : H (X, Q) → H (U, Q)) (weight k). Formal properties and application to the coniveau • Morphisms of MHS: φ : L → L0, φ(W L) ⊂ W L0, φ (F pL ) ⊂ F pL0 . i i C C C Thm. (Deligne) Morphisms of mixed Hodge structures are strict for both filtrations (i.e.: F pL0 ∩ Im φ = φ (F pL ), W L0 ∩ Im φ = φ(W L)). C C C C i i Sketch of proof. Follows from an algebra lemma: There exists a p,q functorial decomposition LC = ⊕p,qL for mixed Hodge structures p r,q p,q (L, W, F ), with F LC = ⊕r≥p,qL , WiLC = ⊕p+q≤iL . 0 P p,q p,q Let α ∈ WiL ∩ Im φ. Write α = φ(β), β = p,q β . Then φ(β ) = 0 0 0 P p,q for p + q > i so α = φ(β ) with β = p+q≤i β ∈ WiLC. qed k Definition. A class α ∈ H (X, Q) is of coniveau ≥ c if α|X\Y = 0 with Y closed analytic of codim ≥ c. If X is smooth compact, j : Y,→ X, equivalent condition: α = j∗β in H2n−k(X, Q) for some β ∈ H2n−k(Y, Q). Strictness ⇒ If X is smooth projective, j : Y,→ X with desingularization ˜ ˜ j : Ye → X, then Im j∗ = Im j∗ ⊂ H2n−k(X, Q). Corollary. (Deligne) The set of cohomology classes of coniveau ≥ c is a k Hodge substructure of H (X, Q), of Hodge coniveau ≥ c. Formal properties and application to the coniveau • Morphisms of MHS: φ : L → L0, φ(W L) ⊂ W L0, φ (F pL ) ⊂ F pL0 . i i C C C Thm. (Deligne) Morphisms of mixed Hodge structures are strict for both filtrations (i.e.: F pL0 ∩ Im φ = φ (F pL ), W L0 ∩ Im φ = φ(W L)). C C C C i i Sketch of proof. Follows from an algebra lemma: There exists a p,q functorial decomposition LC = ⊕p,qL for mixed Hodge structures p r,q p,q (L, W, F ), with F LC = ⊕r≥p,qL , WiLC = ⊕p+q≤iL . 0 P p,q p,q Let α ∈ WiL ∩ Im φ. Write α = φ(β), β = p,q β . Then φ(β ) = 0 0 0 P p,q for p + q > i so α = φ(β ) with β = p+q≤i β ∈ WiLC. qed k Definition. A class α ∈ H (X, Q) is of coniveau ≥ c if α|X\Y = 0 with Y closed analytic of codim ≥ c. If X is smooth compact, j : Y,→ X, equivalent condition: α = j∗β in H2n−k(X, Q) for some β ∈ H2n−k(Y, Q). Strictness ⇒ If X is smooth projective, j : Y,→ X with desingularization ˜ ˜ j : Ye → X, then Im j∗ = Im j∗ ⊂ H2n−k(X, Q). Corollary. (Deligne) The set of cohomology classes of coniveau ≥ c is a k Hodge substructure of H (X, Q), of Hodge coniveau ≥ c. Polarizations Thm. (Hard Lefschetz, proved by Hodge) Let X be compact Kähler of dimension n, ω a Kählerform on X. Then ∀k ≤ n, n−k n−k k 2n−k ∪[ω] := L : H (X, R) → H (X, R) is an isomorphism. • Projective case : One can take [ω] rational. Then the Lefschetz isomorphism is an isomorphism of Hodge structures. k r k−2r Coro. (Lefschetz decomp.) H (X, R) = ⊕k−2r≥0L H (X, R)prim, k−2r n−k+2r+1 k−2r where H (X, R)prim := Ker L ⊂ H (X, R). k R n−k • Lefschetz intersection pairing on H : (α, β)Lef = X L α ∪ β. k hLef (α, β) := i (α, β)Lef .

• easy: The Lefschetz decomposition is orthogonal for ( , )Lef , and the Hodge decomposition is orthogonal for hLef . (HR1). p+r Thm. 2nd H-R bilinear relations: (−1) h r p−r,q−r is Lef|L H (X,R)prim positive definite Hermitian (up to a global sign depending on k). (HR2). r k−2r Corollary. Let [ω] be rational. On L H (X, Q)prim, multiply ( , )Lef by r k (−1) : one gets a polarized Hodge structure on H (X, Q). Polarizations Thm. (Hard Lefschetz, proved by Hodge) Let X be compact Kähler of dimension n, ω a Kählerform on X. Then ∀k ≤ n, n−k n−k k 2n−k ∪[ω] := L : H (X, R) → H (X, R) is an isomorphism. • Projective case : One can take [ω] rational. Then the Lefschetz isomorphism is an isomorphism of Hodge structures. k r k−2r Coro. (Lefschetz decomp.) H (X, R) = ⊕k−2r≥0L H (X, R)prim, k−2r n−k+2r+1 k−2r where H (X, R)prim := Ker L ⊂ H (X, R). k R n−k • Lefschetz intersection pairing on H : (α, β)Lef = X L α ∪ β. k hLef (α, β) := i (α, β)Lef .

Proof. Choose a polarization ( , ) on H. First prove that ( , )|H0 is nondegenerate using HR2, then define H00 = H0⊥. H00 is a Hodge substructure by HR1. qed • Polarizations on the cohomology of smooth projective varieties are almost motivic, but one needs the Lefschetz decomposition and the change of signs. To make them motivic, one needs: Lefschetz standard conjecture. X projective. There exists a codimension k closed algebraic subset ZLef ⊂ X × X such that ∗ 2n−k k n−k −1 [ZLef ] : H (X, Q) → H (X, Q) is the inverse (L ) of the Lefschetz isomorphism. 2k • ([ZLef ] ∈ H (X × X, Q) = cohomology class of ZLef .) • Implied by the Hodge conjecture because (Ln−k)−1 is an iso of Hodge structures hence produces a Hodge class on X × X. Hodge structures on cohomology algebras and applications to topology • A cohomology algebra = graded, graded commutative, algebra of finite 2n dimension over Q, with A = Q and Poincaréduality. Definition. A Hodge structure on a cohomology algebra A∗, = Hodge structure of weight k on Ak, such that Ak ⊗ Al → Ak+l is a morphism of Hodge structures. ∗ Example. H (X, Q) for X compact Kähler. Thm. (Voisin) There exist compact Kählermanifolds (dim ≥ 4) whose ∗ cohomology algebra is not isomorphic to H (X, Q) for X complex projective. Idea of proof. (1) Construct an X such that the structure of its 1 2 cohomology algebra ⇒ the Hodge structure on H (X, Q) (or H (X, Q) for simply connected examples) has endomorphisms. (2) Certain endomorphisms on weight 1 (or weight 2) HS prevent the existence of a polarization. Case of dim 2 (Kodaira), dim 3 (Lin): Any compact Kähler X has small deformations which are projective. Hodge structures on cohomology algebras and applications to topology • A cohomology algebra = graded, graded commutative, algebra of finite 2n dimension over Q, with A = Q and Poincaréduality. Definition. A Hodge structure on a cohomology algebra A∗, = Hodge structure of weight k on Ak, such that Ak ⊗ Al → Ak+l is a morphism of Hodge structures. ∗ Example. H (X, Q) for X compact Kähler. Thm. (Voisin) There exist compact Kählermanifolds (dim ≥ 4) whose ∗ cohomology algebra is not isomorphic to H (X, Q) for X complex projective. Idea of proof. (1) Construct an X such that the structure of its 1 2 cohomology algebra ⇒ the Hodge structure on H (X, Q) (or H (X, Q) for simply connected examples) has endomorphisms. (2) Certain endomorphisms on weight 1 (or weight 2) HS prevent the existence of a polarization. Case of dim 2 (Kodaira), dim 3 (Lin): Any compact Kähler X has small deformations which are projective. Polarizations+MHS: Topology of families; global invariant cycles theorem Thm. (Blanchard, Deligne) If f : X → Y is smooth projective, the Leray spectral sequence of f with Q-coefficients degenerates at E2.

Proof. Relative Lefschetz operator L = c1(L)∪ acts on the whole spectral sequence, and induces Lefschetz decomposition k r k−2r R f∗Q = ⊕rL (R f∗Q)prim. Suffices to prove d2α = 0 for p q n−q+1 n−q+1 α ∈ H (Y,R f∗Qprim). But L α = 0 ⇒ L d2α = 0. But p+2 q−1 n−q+1 q−1 ∼ 2n−q+1 d2α ∈ H (Y,R f∗Q) and L : R f∗Q) = R f∗Q. qed k • Monodromy. Local system R f∗Q monodromy representation k k ρ 0 k ρ : π1(Y, 0) → Aut H (X0, Q). Thus H (X0, Q) = H (Y,R f∗Q) k k = Im (H (X, Q) → H (X0, Q)) by degeneracy at E2. Thm (Deligne) X ⊂ X smooth projective, f : X → Y as above with Y k ρ k k quasi-projective. Then H (X0, Q) = Im (H (X, Q) → H (X0, Q)). This k is a Hodge substructure of H (X0, Q). k k Proof. H (X, Q) → H (X0, Q) is a morphism of mixed Hodge structures. On the right, pure of weight k. On the left, the weight k part k k is Im (H (X, Q) → H (X, Q)). Then apply strictness. qed The Hodge bundles • Algebraic de Rham complex Ω• , relative version Ω• for f : X → Y X/C X/Y algebraic, smooth morphism. Thm. (Serre-Grothendieck) X smooth quasiprojective over C. Then k • ∼ k H (X, ΩX /C) = HB(X, C). • So, for X projective, the Hodge filtration and Frölichers.s. are algebraic. • Relative version ⇒ If f : X → Y is algebraic, smooth projective, then the Hodge bundles Hk,F pHk, Hp,q are algebraic on Y . • • Katz-Oda construction : relative holomorphic de Rham complex ΩX/Y . k • ∼ k k p k k •≥p R f∗ΩX/Y = H := H ⊗ OY . Hodge filtration F H = R f∗ΩX/Y with p k fiber F H (Xt). 2 • ∗ 2 •−2 • Let L ΩX := f ΩY ∧ ΩX . •−1 ∗ • 2 • • • Exact sequence. 0 → ΩX/Y ⊗ f ΩY → ΩX /L ΩX → ΩX/Y → 0 k k Thm. (Katz-Oda) The Gauss-Manin connection ∇ : H → H ⊗ ΩY is the connecting map. Corollary. The Gauss-Manin connection is algebraic. The Hodge bundles • Algebraic de Rham complex Ω• , relative version Ω• for f : X → Y X/C X/Y algebraic, smooth morphism. Thm. (Serre-Grothendieck) X smooth quasiprojective over C. Then k • ∼ k H (X, ΩX /C) = HB(X, C). • So, for X projective, the Hodge filtration and Frölichers.s. are algebraic. • Relative version ⇒ If f : X → Y is algebraic, smooth projective, then the Hodge bundles Hk,F pHk, Hp,q are algebraic on Y . • • Katz-Oda construction : relative holomorphic de Rham complex ΩX/Y . k • ∼ k k p k k •≥p R f∗ΩX/Y = H := H ⊗ OY . Hodge filtration F H = R f∗ΩX/Y with p k fiber F H (Xt). 2 • ∗ 2 •−2 • Let L ΩX := f ΩY ∧ ΩX . •−1 ∗ • 2 • • • Exact sequence. 0 → ΩX/Y ⊗ f ΩY → ΩX /L ΩX → ΩX/Y → 0 k k Thm. (Katz-Oda) The Gauss-Manin connection ∇ : H → H ⊗ ΩY is the connecting map. Corollary. The Gauss-Manin connection is algebraic.