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Non-vanishing of the twisted cohomology on the complement of hypersurfaces

Author: Yukihito Kawahara
Journal: Proc. Amer. Math. Soc. 136 (2008), 1967-1975
MSC (2000): Primary 14F40; Secondary 14C20, 32S22
Published electronically: February 15, 2008
MathSciNet review: 2383503
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Abstract: Generically, the cohomology with coefficients in a local system of rank one on the complement in $ \mathbb{P}^n$ of the union of a finite number of hypersurfaces vanishes except in the highest dimension. We study the non-generic case, in which the cohomology in other dimensions does not vanish. When the hypersurfaces are hyperplanes, many examples of this kind are known. In this paper, we consider the case in which the hypersurfaces need not be hyperplanes. We prove that the hypersurfaces given by some particular linear systems have non-vanishing local system cohomologies.

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Additional Information

Yukihito Kawahara
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan

Keywords: Local system, twisted cohomology, linear system, hypersurface complement, hyperplane arrangement.
Received by editor(s): February 26, 2006
Received by editor(s) in revised form: May 7, 2007
Published electronically: February 15, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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