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).