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Rank one local systems and forms of degree one. (English) Zbl 1404.14013

Summary: Cohomology support loci of rank one local systems of a smooth quasi-projective complex algebraic variety are finite unions of torsion-translated complex subtori of the character variety of the fundamental group. Tangent spaces of the character variety are (partially) represented by logarithmic 1-forms. In this paper, we give a relation between cohomology support loci and the natural strata of 1-forms given by the dimension of the vanishing locus. This relation generalizes the one for the projective case due to Green and Lazarsfeld and also generalizes the partial relation due to Dimca in the quasi-projective case.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F40 de Rham cohomology and algebraic geometry
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