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Concept
Hodge structure
Summary
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. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989).
A pure Hodge structure of integer weight n consists of an abelian group and a decomposition of its complexification H into a direct sum of complex subspaces , where , with the property that the complex conjugate of is :
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 subject to the condition
The relation between these two descriptions is given as follows:
For example, if X is a compact Kähler manifold, is the n-th cohomology group of X with integer coefficients, then is its n-th 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 with the decreasing filtration by 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 is too big. Using the Riemann bilinear relations, in this case called Hodge Riemann bilinear relations, it can be substantially simplified. A polarized Hodge structure of weight n consists of a Hodge structure and a non-degenerate integer bilinear form Q on (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 on .
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