Hodge theory for real manifolds[edit]
De Rham cohomology[edit]
The Hodge theory references the de Rham complex. Let M be a smooth manifold. For a non-negative integer k, let Ωk(M) be the real vector space of smooth differential forms of degree k on M. The de Rham complex is the sequence of differential operators
Generalizations[edit]
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 varies when varies. In geometric terms, this amounts to studying the period mapping associated to a family of varieties. Saito's theory of Hodge modules is a generalization. Roughly speaking, a mixed Hodge module on a variety is a sheaf of mixed Hodge structures over , as would arise from a family of varieties which need not be smooth or compact.
