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⊗OX, 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⊗OX maps Fn into Fn−1⊗Ω1X.
Here the natural (flat) connection on S⊗OX induced by the flat connection on S and the flat connection d on OX, and OX is the sheaf of holomorphic functions on X, and Ω1X 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.
Read more about this topic: Hodge Structure
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