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Lowest weights in cohomology of variations of Hodge structure

Published online by Cambridge University Press:  11 January 2016

Chris Peters  and
Morihiko Saito
Chris Peters
Affiliation:
Institut Fourier – UMR CNRS 5582 Université Grenoble 1, 38402-Saint-Martin d’Hères, France, chris.peters@ujf-grenoble.fr
Morihiko Saito
Affiliation:
Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502, Japan, msaito@kurims.kyoto-u.ac.jp
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Abstract

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Let X be an irreducible complex analytic space with j: U ↪ X an immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weight n over U. Assume that X is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, IHk (X, 𝕍) is known to carry a pure Hodge structure of weight k+n, while Hk (U, 𝕍) carries a mixed Hodge structure of weight at least k +n. In this note it is shown that the image of the natural map IHk (X, 𝕍) → Hk (U, 𝕍) 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.

Keywords

14C3032S35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 206 , June 2012 , pp. 1 - 24
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

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