Editing Hodge theory (section)

==Generalizations==
'''Mixed Hodge theory''', developed by [[Pierre Deligne]], extends Hodge theory to all complex algebraic varieties, not necessarily smooth or compact. Namely, the cohomology of any complex algebraic variety has a more general type of decomposition, a [[mixed Hodge structure]].

A different generalization of Hodge theory to singular varieties is provided by '''[[intersection homology]]'''. Namely, [[Morihiko Saito]] showed that the intersection homology of any complex projective variety (not necessarily smooth) has a pure Hodge structure, just as in the smooth case. In fact, the whole Kähler package extends to intersection homology.

A fundamental aspect of complex geometry is that there are continuous families of non-isomorphic complex manifolds (which are all diffeomorphic as real manifolds). [[Phillip Griffiths]]'s notion of a '''[[variation of Hodge structure]]''' describes how the Hodge structure of a smooth complex projective variety <math>X</math> varies when <math>X</math> varies. In geometric terms, this amounts to studying the [[period mapping]] associated to a family of varieties. Saito's theory of [[Hodge module]]s is a generalization. Roughly speaking, a mixed Hodge module on a variety <math>X</math> is a sheaf of mixed Hodge structures over <math>X</math>, as would arise from a family of varieties which need not be smooth or compact.