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NOTES on MIXED HODGE STRUCTURES Srni – January 2005 Andrzej Weber Instytut Matematyki, Uniwersytet Warszawski

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NOTES on MIXED HODGE STRUCTURES Srni – January 2005 Andrzej Weber Instytut Matematyki, Uniwersytet Warszawski NOTES ON MIXED HODGE STRUCTURES Srni { January 2005 Andrzej Weber Instytut Matematyki, Uniwersytet Warszawski This is a short introduction to mixed Hodge structures. The notes contain basic definitions and statements. There are no proofs. The motivic nature of the theory is exposed. In the end some applications to equivariant cohomology are given Topology of algebraic varieties was studied from the very beginning of algebraic topology. Usual topological invariants often carry additional structures. For example: the cohomology of a complex smooth projective variety is equipped with a pure Hodge structure. Every cohomology class can be decomposed into (p; q){types. The decomposi- tion has well known symmetries which allow to form the famous Hodge diamond. If we deal with singular or open varieties the situation is much more complicated, but on the other hand we obtain an interesting and reach structure. The structure in question is derived from the fact that each variety can be built up from smooth projective ones by means of set-theoretic operations and blow-ups. The result is a filtration in cohomology which does not depend on the way the variety was constructed. This filtration is highly functorial. It is called the weight filtration. Each piece of the associated gradation looks like the cohomology of a projective variety, i.e. it carries a pure Hodge structure. This way we arrive to the notion of the mixed Hodge structure, in short MHS. This theory was constructed by Pierre Deligne, who managed to transpose his results arising from Weil conjectures into the category of complex varieties. Now the theory of mixed Hodge structure is an indispensable tool for studying singular varieties. After giving some simple but instructive examples I will describe the linear algebra which is hidden under the name of mixed Hodge structure. Then, I'll construct MHS for smooth open varieties using the logarithmic complex. To deal with singularities I'll construct MHS for simplicial varieties. I'll also give applications for equivariant cohomology when a linear algebraic group acts on an algebraic variety. Our approach is explained by the diagram: X 7−! M(X) 7−! H∗(X) smooth projective variety 7−! pure motive 7−! pure Hodge structure open=singular variety 7−! mixed motive 7−! mixed Hodge structure The reader can find a sufficient introduction to Chow motives e.g. in [M]. In fact the nature of motives is irrelevant. Only the interactions between them are important. Pure motives are organized into a category and the \interactions" are just the morphisms. We define the morphisms to be the correspondences. A correspondence from X to Y is an algebraic cycle in the product X × Y . To compose morphisms we have to be able to intersect algebraic cycle and therefore we have to divide them by a relation which allows to develop intersection theory. We will not explain further the subject of motives, but we keep in mind that the homological constructions we describe have a geometric background. 2 In this exposition we do not mention many important issues such as variation of Hodge structures or their degeneration (see e.g. [St], [Cl]). In general we refer to the book of C. Voisin [Vo]. Also the vast theory of mixed Hodge modules [Sa] is not presented, although it seems unavoidable for subtle analysis of singularities. Plan: 1. Introduction motivation, simple examples of cohomology of complex algebraic varieties. 2. Linear algebra pure Hodge structures, category of mixed Hodge structures, mixed Hodge complexes, degeneration of spectral sequences. 3. Open smooth varieties logarithmic complex, Poincar´eresidues, construction of MHS, cohomology with compact supports. 4. Singular varieties simplicial varieties, hypercovering of a singular variety, construction of MHS. virtual Poincar´epolynomials. 5. Actions of algebraic groups simplicial varieties arising from a group action, formality of equivariant cohomology, degeneration of Eilenberg-Moore spectral sequence. 1. Introduction 1.1 Motivation. Let X be an algebraic variety defined over Fq and φ be the induced Frobenius ∗ automorphism acting on the ´etalecohomology Het´ (X ⊗ Fq; Q`). (1.1.1) Corollaries from Weil conjectures: i { all eigenvalues of φ are algebraic numbers of the module q 2 (with respect to any embedding Q` ,−! C), n { if X is smooth and projective then all eigenvalues of φ on H (X ⊗ q; `) have the n et´ F Q module equal to q 2 , ∗ { let Vi be the subspace of Het´ (X ⊗Fq; Q`) corresponding to the eigenvalues of the module i 2 L equal to q . Then the weight filtration Wp = i≤p Vi can be computed in a geometric way. Moreover, it is strictly functorial. (1.1.2) Goal: Construct a weight filtration for complex algebraic varieties and prove its functoriality. The Hodge structure on the associated graded group should play the rigidifying role of the Frobenius automorphism. 3 1.2 Examples of cohomology of smooth open varieties. In this section we consider cohomology with arbitrary coefficients, but later the coefficients will be rational or complex numbers. (1.2.1) Let X be a smooth variety, X its smooth compactification. Suppose, that D = XnX is smooth, c = codim D. Then we have Gysin exact sequence j ∗ −−! Hn−2c(D) −−!! Hn(X) −−!i Hn(X) −−!res Hn−2c+1(D) −−! ; where res is the residue map. The weight filtration of Hn(X) looks as follows: ∗ n Wn−1 = 0;Wn = im(i ) = ker(res);Wn+1 = H (X) : (1.2.2) In general the bottom term of the weight filtration of a smooth variety X can be defined as follows n ∗ n n WnH (X) = im(i : H (X) ! H (X)) ; where i : X ! X is any smooth compactification of X. (1.2.3) Proposition: The subgroup im(i∗ : Hn(X) ! Hn(X)) ⊂ Hn(X) does not depend on the compactification. The proposition holds for arbitrary coefficients of cohomology. Proof: For any two compactifications X1 and X2 construct another one: let X3 be a resolution of the closure of the diagonal embedding X,−! X1 × X2. The projection n n pi : X3 ! Xi for i = 1; 2 induces a surjection (pi)! : H (X3) ! H (Xi) since it is a map n n of closed manifold of degree 1. The restriction map H (X3) ! H (X) factors through n n n n (pi)!. Therefore im(H (Xi) ! H (X)) = im(H (X3) ! H (X)). (1.2.4) We define another filtration in cohomology: Hodge filtration. Let Ω• denote the complex of holomorphic differential forms. It contains a subcomplex Ω•≥p. We identify Hn(X;Ω•) with Hn(X; C). Let p n •≥p n • n F = im(H (X;Ω ) ! H (X;Ω )) ⊂ H (X; C) : Hodge filtration is decreasing. If X is compact then M F p = Hr;n−r(X) : r≥p Moreover p p;n−p n p F \ F n−p = H (X) and H (X; C) = F ⊕ F n−p+1 : We say that the filtration F p is a pure Hodge structure of the weight n. (1.2.5) Proposition: In the example 1.2.1 Hodge filtration induces a pure Hodge structure W W of the weight n on Grn and of the weight n + 1 on Grn+1. 4 We have to choose a suitable subcomplex of Ω• and its resolution. Then the assertion follows from 2.1.8. We obtain a mixed Hodge structure in H∗(X). 1.3 Examples of cohomology of compact varieties. Let X be a compact variety with the smooth singular locus Z. Let µ : Xe ! X be a resolution of singularities. Suppose E = µ−1Z is smooth. We have Mayer-Vietoris exact sequence δ δ −−! Hn−1(E) −−! Hn(X) −−! Hn(Xe) ⊕ Hn(Z) −−! Hn(E) −−! : The weight filtration of Hn(X) looks as follows: n Wn−2 = 0;Wn−1 = im(δ);Wn = H (X) : ∗ n n Since µjE : H (Z) ! H (E) is injective we have n n Wn−1 = ker(H (X) ! H (Xe)) : Also in this case one constructs a Hodge filtration in Hn(X; C). (1.3.1) In general the term Wn−1 of the weight filtration of a compact variety X can be defined as follows n ∗ n n Wn−1H (X) = ker(µ : H (X) ! H (Xe)) ; where µ : Xe ! X is any resolution of singularities of X. (1.3.2) Proposition: The subgroup ker(µ∗ : Hn(X) ! Hn(Xe)) ⊂ Hn(X) does not depend on the resolution of singularities. The proposition holds for arbitrary coefficients of cohomology. Proof is analogous to 1.2.3. n (1.3.3) One can show that for a compact algebraic variety the group Wn−1H (X) ⊗ Q is a topological invariant of X. In fact it is the kernel of the map to intersection cohomology of Goresky-MacPherson [GoMP], see [We]. 2. Linear algebra 2.1 Pure Hodge structures. (2.1.1) Definition: A pure Hodge structure of the weight n is a finite dimensional Q-vector p space VQ and a decreasing filtration F of VC = VQ ⊗ C such that p n−p+1 VC = F ⊕ F : If p + q = n we define V p;q = F p \ F q : (2.1.2) An alternative but equivalent definition of the Hodge structure is a decomposition L p;q p;q q;p p L r;n−r VC = p+q=n V such that V = V . Then we define F = r≥p V : 5 (2.1.3) Example: The cohomology of a projective variety Hn(X) with the Hodge filtration constructed in 1.2.4 is a pure Hodge structure of the weight n. (2.1.4) Additionally one may consider an integral Hodge structure, i.e. a lattice VZ in VQ.
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