Geometry of cohomology support loci for local systems I

TL;DR

This paper investigates the structure of cohomology support loci for local systems on certain complex manifolds, revealing they are unions of translated subvarieties related to holomorphic maps to complex Lie groups, extending previous results.

Contribution

It generalizes earlier results by describing the geometry of cohomology support loci for open subsets of compact Kähler manifolds, especially under specific conditions, using novel Higgs field techniques.

Findings

$ extstyleigcup$ of translates of $f^*H^1(T,C^*)$ describes $oldsymbol{ extstyleigcup ext{ of translates of }f^*H^1(T,C^*)}$

Supports are unions of translates of subvarieties related to holomorphic maps to complex Lie groups

Main novelty involves Higgs fields with logarithmic poles and mixed Hodge theory insights.

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.