Cohomology jump loci of quasi-projective varieties

Abstract

We prove that the cohomology jump loci in the space of rank one local systems over a smooth quasi-projective variety are finite unions of torsion translates of subtori. The main ingredients are a recent result of Dimca-Papadima, some techniques introduced by Simpson, together with properties of the moduli space of logarithmic connections constructed by Nitsure and Simpson. MB(X) = Hom(π1(X),C ∗ ) to be the variety of C ∗ representations of π1(X). Then MB(X) is a direct product of (C ∗ ) b1(X) and a finite abelian group. For each point ρ ∈ MB(X), there exists a unique rank one local system Lρ, whose monodromy representation is ρ. The cohomology jump loci of X are the natural strata � i (X) = {ρ ∈ MB(X) | dimC H i (X,Lρ) ≥ k}. � i (X) is a Zariski closed subset of MB(X). A celebrated result of Simpson says that if X is a smooth projective variety defined over C, theni (X) is a union of torsion