# Hodge Structures. October 8, A few examples of symmetric spaces

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1 Hodge Structures October 8, A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H g is the quotient of Sp 2g (R) by its maximal compact subgroup U(g). H g is a complex manifold, with an action of Sp 2g (R) by holomorphic automorphisms. Thus if Γ Sp 2g (R) is a discrete subgroup, then Γ\H g has a chance at being an algebraic variety. In fact, if Γ Sp 2g (Z) is a congruence subgroup, then this is in fact the case; the theorem of Baily-Borel shows that Γ\H g is an open subset of a projective variety, known as a Siegel modular variety. Just how general is this process? If we start with a group such as SL n (R), and take its quotient by (say) the compact subgroup SO(n), the result can be characterized as the set of positive definite quadratic forms of rank n and determinant 1. Its dimension is n(n+1)/2 1. For n = 3, this is an odd number, so there can t be a complex structure on SL 3 (R)/ SO(3). For arithmetic subgroups Γ SL 3 (Z), one can still form the quotient Γ\ SL 3 (R)/SO(3) to get an interesting manifold, but it won t be an algebraic variety. What about SL 2 (C)? That group has a complex structure at least. SL 2 (C) acts on the set of 2 2 hermitian matrices M, by the action g M = gmg, and preserves the determinant. The determinant, considered as a quadratic form on the space of hermitian matrices, has signature (1, 3). Thus we get a homomorphism SL 2 (C) O(1, 3), whose kernel is just {±I}. The Lorentz group O(1, 3) acts transitively on the set H 3 of 4-tuples (t, x, y, z) such that t 2 x 2 y 2 z 2 = 1, t > 0; the stabilizer of (1, 0, 0, 0) is O(1) O(3) which is the maximal compact subgroup. Hence the symmetric space for the complex Lie group SL 2 (C) is actually H 3, a hyperbolic 3-manifold! 1

2 In the following, we ll get to the bottom of which spaces G/K have complex structures, where G is a real Lie group and K G is a maximal compact subgroup. In fact, there s a special class of these, the hermitian symmetric domains, which are the most interesting and which are required to define Shimura varieties in general. 2 Classifaction of PPAVs Consider the problem of classifying principally polarized abelian varieties (PPAVs) over C. As we know, If A/C is an abelian variety of dimension g, then as topological spaces we have A = V/Λ, where Λ = H 1 (A, Z) and V = Λ Z R. To give a polarization λ: A A is to give a Riemann form E : Λ Λ Z; that is, an alternating bilinear form satisfying the properties 1. The extension E R : V V R satisfies E R (iv, iw) = E R (v, w), and 2. If H(v, w) = E R (iv, w) + ie R (v, w), then H : V V C is a positive definite hermitian form on V. The condition that λ be principal corresponds to the condition that E is a perfect pairing, so that it identifies Λ with its Z-linear dual. Thus if A is a principally polarized abelian variety of dimension g, then Λ = H 1 (A, Z) is a free Z-module of rank 2g equipped with a perfect symplectic form E. We can find a basis x 1,..., x 2g for Λ with respect to which E has some convenient form, such as J 0 = ( 0 Ig I g 0 ). Let Ψ be the corresponding symplectic form on Z 2g, so that Ψ(v, w) = v t J 0 w. Now we can ask a slightly different problem. Begin with Z 2g together with its symplectic form Ψ. We will classify pairs (A, λ, α) where (A, λ) is a PPAV and α: Z 2g H 1 (A, Z) is an isomorphism which is compatible with the symplectic forms on either space: Ψ(v, w) = E(α(v), α(w)), v, w, Z 2g. 2

3 Proposition Triples (A, λ, α) are in bijection with points of Siegel upper half space { } H g = X + iy M g g (C) Xt = X, Y > 0. Pairs (A, λ) are classified by the quotient Sp 2g (Z)\H g. (In the above definition of H g, X and Y are real matrices, and Y > 0 means that Y is positive definite. Thus H 1 is the usual upper half plane.) The second claim follows from the first: if (A, λ) is a PPAV, then there always exists an α of the required form, and any two such differ by an automorphism of Z 2g with preserves Ψ; that is, by an element of Sp 2g. To prove the first claim, we need to think of H g not as a subset of M g g (C) but rather as the quotient Sp 2g (R)/U(g). The symplectic group Sp 2g (R) acts transitively on H g via fractional linear transformations, and the stabilizer of ii g is U(g) (exercise). First, we ll start with a triple (A, λ, α) and construct the appropriate coset in Sp 2g (R)/U(g). Write A = V/Λ We can use α to identify the symplectic lattice (Z 2g, Ψ) with (Λ, E). Now when we tensor with R, we have an isomorphism between R 2g and Λ R = V, which is a complex vector space. The action of i on the V corresponds to a map J : R 2g R 2g with J 2 = I; that is, J is a complex structure on R 2g. Furthermore, we have the properties that E R (iv, iw) = E R (v, w) and E R (iv, v) > 0 for nonzero v V, and these correspond to the properties 1. J is symplectic: Ψ R (Jv, Jw) = Ψ R (v, w) 2. J is positive: Ψ R (v, Jv) > 0 for v R 2g, v 0. Going the other way, if J is a complex structure on R 2g which is symplectic and positive, then R 2g becomes a complex vector space V (where the action of i is J), and E becomes a principal Riemann form on Z 2g, so that A = V/Z 2g becomes a PPAV together with an isomorphism of its H 1 with Z 2g. How to classify symplectic and positive complex structures J on R 2g? Let J 0 be any such structure (such as the matrix J 0 noted above). Then any other such structure J is conjugate to J 0 by a symplectic matrix S: J = SJ 0 S 1 (exercise). Here S Sp 2g (R) is a matrix which is well-defined up to the group of matrices U which commute with J 0. But commuting with 3

4 J 0 means that U is C-linear on V = R 2g (considered as a complex vector space). Furthermore, our assumption on J 0 means that the form H(v, w) = Ψ(v, iw) + iψ(v, w) turns R 2g into a hermitian inner product space; the fact that S is symplectic means that S is unitary with respect to H. Thus S is well-defined as an element of Sp 2g (R)/U(g). 3 The formalism of Hodge structures In the foregoing discussion we considered the moduli space of complex structures J on a vector space V = R 2g, which preserved some extra structure (a symplectic pairing), and which were subject to some positivity condition. The moduli space ended up being the quotient of a real algebraic group Sp 2g (R) by a maximal compact subgroup. We re now going to introduce Hodge structures on a real vector space, which are a generalization of complex structures. If V is a real vector space of dimension d, then V C = V R C is a complex vector space of dimension d. Let τ be complex conjugation; then 1 τ is a C-semilinear endomorphism of V C, which we denote as v v. Let V be a real vector space. A Hodge structure on V is a bigrading V C = p,q V pq, such that V pq = V qp. A Hodge structure is of type S Z Z if V pq = 0 for all (p, q) S. The idea behind this definition comes from cohomology. Let X/C is a smooth projective variety of dimension d. Then X has the structure of a real manifold of dimension 2d. We have the singular cohomology groups H i (X, Z) = Hsing(X, i Z), which are zero outside the range i = 1, 2,..., 2d. For each i, the pairing of i-forms and i-cochains gives an isomorphism H i (X, R) = H i dr(x, R), where HdR i (X, R) is closed i-forms modulo exact i-forms. What s more, the complex space HdR i (X, C) breaks up according to the Hodge decomposition HdR(X, i C) = H p,q (X), p+q=i where H p,q (X) is the subspace of HdR i (X, C) consisting of classes having a representative which is of type (p, q). Complex conjugation maps (p, q)-forms 4

5 onto (q, p)-forms, so that we get a Hodge structure on each H i (X, R). (By the Dolbeault theorem, H p,q (X) is isomorphic to H q (X, Ω p ), where Ω p is the sheaf of holomorphic p-forms.) We let Fil p V be the direct summand Fil p V = p p V p q, so that Fil p V V C is a complex subspace, and Fil p V decreases with p. Say that V has weight n if p + q = n for every pair (p, q) with V pq 0. If V has weight n, note that the filtration {Fil p V } determines the Hodge structure completely, since V pq = Fil p V Fil q V. An integral Hodge structure is a Z-module Λ together with a (real) Hodge structure on Λ R. Thus in the above situation, H i (X, Z) is an integral Hodge structure. Let s see how complex structures are examples of Hodge structures. If V is a real vector space, and J GL(V ) is a complex structure, then V C breaks up into two subspaces V 1,0 and V 0,1, these being the subspaces where J = i and J = i, respectively. Thus a complex structure is the same as a Hodge structure of type {(1, 0), (0, 1)}. In the situation of abelian varieties, we have to be careful about duals. If A is an abelian variety, then the Hodge decomposition applied to A makes H 1 (A, Z) into an integral Hodge structure of type {(1, 0), (0, 1)}. But H 1 (A, Z), its Z-linear dual, is a Hodge structure of type {( 1, 0), (0, 1)}. If V is a Hodge structure, then define an action h: C GL(V ) as follows. First define an action h of C on V C by h(z)(v pq ) = z p z q v pq, all v pq V pq. Then since V pq = V qp, we have h(z)(v) = h(z)(v) for all v V C. Thus h(z) preserves the real subspace V V C, this being the space of vectors invariant under v v. We get an action h of C on V. It will be important to view this action in terms of algebraic groups. Let S be the restriction of scalars of G m from C to R. Thus, for an R-algebra A, S(A) is the group of pairs (a, b) A A which satisfy a 2 + b 2 0, under the multiplication law which tells you how to multiply complex numbers a + bi together. We have S(R) = C. The following is an exercise: 5

6 Proposition Morphisms of real algebraic groups S GL(V ) correspond to Hodge structures on V. This suggests defining Hodge structures relative to an algebraic group other than GL(n). So let G be a real algebraic group, and say that a Hodge structure on G is a homomorphism h: G of real algebraic groups. If V is a faithful representation of G, then h induces a Hodge structure on V as above. A polarization of a Hodge structure h on V is an alternating bilinear map of Hodge structures Ψ: V V R for which Ψ(v, Jw) is positive definite, where J = h(i). The statement that Ψ be a morphism of Hodge structures is the statement that Ψ(Jv, Jw) = Ψ(v, w). Example The set of polarized Hodge structures on R 2g {( 1, 0), (0, 1)} is the Siegel upper half-space H g. of type 6

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