Hodge Structure - Hodge Modules

Hodge Modules

Hodge modules are a generalization of variation of Hodge structures on a complex manifold. They can be thought of informally as something like sheaves of Hodge structures on a manifold; the precise definition (Saito 1989) is rather technical and complicated. There are generalizations to mixed Hodge modules, and to manifolds with singularities.

For each smooth complex variety, there is an abelian category of mixed Hodge modules associated with it. These behave formally like the categories of sheaves over the manifolds; for example, morphisms f between manifolds induce functors f_*, f*, f_!, f! between (derived categories of) mixed Hodge modules similar to the ones for sheaves.