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Characteristic varieties and logarithmic differential 1-forms

Part of: Cycles and subschemes Singularities Curves (Co)homology theory

Published online by Cambridge University Press:  22 January 2010

Alexandru Dimca
Alexandru Dimca*
Affiliation:
Laboratoire J.A. Dieudonné, UMR du CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France (email: [email protected])
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Abstract

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We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

Keywords

twisted cohomologylogarithmic de Rham complexzeroes of 1-formsHodge theory

MSC classification

Secondary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14F40: de Rham cohomology 14H52: Elliptic curves 32S22: Relations with arrangements of hyperplanes
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 129 - 144
Copyright
Copyright © Foundation Compositio Mathematica 2009

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