Hodge Structure - Hodge Structures

Hodge Structures

A pure Hodge structure of weight n (n ∈ Z) consists of an abelian group H_Z and a decomposition of its complexification H into a direct sum of complex subspaces H p,q, where p + q = n, with the property that the complex conjugate of Hp,q is Hq,p:

An equivalent definition is obtained by replacing the direct sum decomposition of H by the Hodge filtration, a finite decreasing filtration of H by complex subspaces FpH (p ∈ Z), subject to the condition

for all p + q = n + 1.

The relation between these two descriptions is given as follows:

For example, if X is a compact Kähler manifold, H_Z = Hn(X,Z) is the nth cohomology group of X with integer coefficients, then H = Hn(X, C) is its nth cohomology group with complex coefficients and Hodge theory provides the decomposition of H into a direct sum as above, so that these data define a pure Hodge structure of weight n. On the other hand, the Hodge-de Rham spectral sequence supplies Hn with the decreasing filtration by FpH as in the second definition.

For applications in algebraic geometry, namely, classification of complex projective varieties by their periods, the set of all Hodge structures of weight n on H_Z is too big. Using the Riemann bilinear relations, it can be substantially cut down. A polarized Hodge structure of weight n consists of a Hodge structure (H_Z, H p,q) and a nondegenerate integer bilinear form Q on H_Z (polarization), which is extended to H by linearity, and satisfying the conditions:

In terms of the Hodge filtration, these conditions imply that

where C is the Weil operator on H, given by C=i p−q on H p,q.

Yet another definition of a Hodge structure is based on the equivalence between the Z-grading on a complex vector space and the action of the circle group U(1). In this definition, an action of the multiplicative group of complex numbers C*, viewed as a two-dimensional real algebraic torus, is given on H. This action must have the property that a real number a acts by an. The subspace Hp,q is the subspace on which z ∈ C* acts as multiplication by

In the theory of motives, it becomes important to allow more general coefficients for the cohomology. The definition of a Hodge structure is modified by fixing a Noetherian subring A of the field R of real numbers, for which A ⊗ _ZR is a field. Then a pure Hodge A-structure of weight n is defined as before, replacing Z with A. There are natural functors of base change and restriction relating Hodge A-structures and B-structures for A a subring of B.