Lowest Weights in Cohomology of Variations of Hodge Structure (II)

Details Lowest Weights in Cohomology of Variations of Hodge Structure (II) Lowest Weights in Cohomology of Variations of Hodge Structure (II)

2008-12-11

arxiv.org/abs/0812.2132v1

Mathematics

Algebraic Geometry

Extends results of preprint (arXiv:0708.0130v2) by the first author with the same title in the analytic context. Accepted for

Scientific paper

Let $X$ be an irreducible complex analytic space with $j:U\into X$ an immersion of a smooth Zariski open subset, and let $\bV$ be a variation of Hodge structure of weight $n$ over $U$. Assume $X$ is compact K\"ahler. Then provided the local monodromy operators at infinity are quasi-unipotent, $IH^k(X, \bV)$ is known to carry a pure Hodge structure of weight $k+n$, while $H^k(U,\bV)$ carries a mixed Hodge structure of weight $\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,\bV) \to H^k(U,\bV)$ is the lowest weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement $X-U$ is not a hypersurface.

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