Geometry of cohomology support loci for local systems I

Abstract

Let X be a Zariski open subset of a compact Kaehler manifold. In this paper, we study the set $\Sigma^k(X)$ of one dimensional local systems on X with nonvanishing kth cohomology. We show that under certain conditions (X compact, X has a smooth compactification with trivial first Betti number, or k=1) $\Sigma^k(X)$ is a union of translates of sets of the form $f^*H^1(T,C^*)$, where $f:X \to T$ is a holomorphic map to a complex Lie group which is an extension of a compact complex torus by a product of C^*'s (these correspond to semiabelian varieties in the algebraic category). This generalizes earlier work of Beauville, Green, Lazarsfeld, Simpson and the author in the compact case. The main novelty lies in the proofs which involve consideration of Higgs fields with logarithmic poles. While a completely satifactory theory of such objects is still lacking, we are able to work out what we need in the rank one case by borrowing ideas from mixed Hodge theory. This will appear in the Journal of Algebraic Geometry.