Hodge Structure - Variation of Hodge Structure

A variation of Hodge structure (Griffiths 1968, 1968, 1970) is a family of Hodge structures parameterized by a complex manifold X. More precisely a variation of Hodge structure of weight n on a complex manifold X consists of a locally constant sheaf S of finitely generated abelian groups on X, together with a decreasing Hodge filtration F on S⊗O_X, subject to the following two conditions:

• The filtration induces a Hodge structure of weight n on each stalk of the sheaf S
• (Griffiths transversality) The natural connection on S⊗O_X maps Fn into Fn−1⊗Ω1_X.

Here the natural (flat) connection on S⊗O_X induced by the flat connection on S and the flat connection d on O_X, and O_X is the sheaf of holomorphic functions on X, and Ω1_X is the sheaf of 1-forms on X.

A variation of mixed Hodge structure can be defined in a similar way, by adding a grading or filtration W to S.