Hodge Theory - Hodge Structures

Hodge Structures

An abstract definition of (real) Hodge structure is now given: for a real vector space W, a Hodge structure of integer weight k on W is a direct sum decomposition of WC = W ⊗ C, the complexification of W, into graded pieces Wp, q where k = p + q, and the complex conjugation of WC interchanges this subspace with Wq, p.

The basic statement in algebraic geometry is then that the singular cohomology groups with real coefficients of a non-singular complex projective variety V carry such a Hodge structure, with having the required decomposition into complex subspaces Hp, q. The consequence for the Betti numbers is that, taking dimensions

where the sum runs over all pairs p, q with p + q = k and where

The sequence of Betti numbers becomes a Hodge diamond of Hodge numbers spread out into two dimensions.

This grading comes initially from the theory of harmonic forms, that are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian (generalising harmonic functions, which must be locally constant on compact manifolds by their maximum principle). In later work (Dolbeault) it was shown that the Hodge decomposition above can also be found by means of the sheaf cohomology groups in which Ωp is the sheaf of holomorphic p-forms. This gives a more directly algebraic interpretation, without Laplacians, for this case.

In the case of singularities or noncompact varieties, the Hodge structure has to be modified to a mixed Hodge structure, where the double-graded direct sum decomposition is replaced by a pair of filtrations. This case is much used, for example in monodromy questions.