Variation of Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. A mixed Hodge structure is a generalization, defined by Pierre Deligne (1970), that applies to all complex varieties (even if they are singular and non-complete). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by P. A. Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by M. Saito (1989).
A pure Hodge structure of weight n (n ∈ Z) consists of an abelian group HZ and a decomposition of its complexification H into a direct sum of complex subspaces Hp,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
The relation between these two descriptions is given as follows:
For example, if X is a compact Kähler manifold, HZ = 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.
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