Variation of Hodge structure
A variation of Hodge structure of weight on a complex manifold is a couple where is a locally constant sheaf of finitely-generated Abelian groups on , and is a finite decreasing filtration of by holomorphic subbundles, subject to the following conditions: i) the flat connection on defined by , for , local sections of and , respectively, satisfies (Griffiths' transversality); ii) for each , the pair is a Hodge structure of weight .
A polarization of a variation of Hodge structure is a flat bilinear form which induces a polarization of the Hodge structure for each . Similar notions exist for replaced by or , [a2]. If is a proper smooth morphism of algebraic varieties over , then is the underlying local system of a polarizable variation of Hodge structure on . By a result of A. Borel, for a polarized variation of Hodge structure on a complex manifold of the form , where is compact and is a divisor with normal crossings, the monodromy around each local component of is quasi-unipotent [a3] (monodromy theorem). A polarized variation of Hodge structure over gives rise to a holomorphic period mapping from to a classifying space of Hodge structures (see Period mapping).
If with a compact Kähler manifold and a divisor with normal crossings on , then for a polarized variation of Hodge structure on , the sheaf has a minimal extension to a perverse sheaf on and carries a pure Hodge structure [a4]–[a6]. In fact, is part of a polarized Hodge module [a7]. Generalizations are the notions of variation of mixed Hodge structure [a8], [a9] and mixed Hodge module [a10].
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Variation of Hodge structure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Variation_of_Hodge_structure&oldid=24589