Geometry of cohomology support loci II: integrability of Hitchin's map

TL;DR

This paper studies the geometry of cohomology support loci for Higgs bundles, showing they form degenerations of abelian varieties and that their fibers are lagrangian, extending classical theorems to a nonabelian setting.

Contribution

It proves that certain cohomology loci in the moduli space of Higgs bundles are unions of abelian varieties and that fibers of the Hitchin map are lagrangian, generalizing Green-Lazarsfeld's theorem.

Findings

Cohomology support loci are Zariski closed and degenerate to unions of abelian varieties.

Fibers of the Hitchin map are lagrangian subvarieties.

Establishment of a generic vanishing theorem for these loci.

Abstract

In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of abelian varieties. More precisely, we consider the subset of the moduli space of Higgs bundles satisfying the analogous cohomological condition. We show that this set is Zariski closed and that if Sigma is the normalization of an irreducible component containing a stable point, then a connected component of a general fiber of the restriction of the Hitchin map to Sigma is an abelian variety. This should be interpreted as a nonabelian version of a theorem of Green and Lazarsfeld. The Hitchin map in this setting is in fact Simpson's generalization of it. The key point is to show that the general fibers of this map are lagrangian (where the target of the map is taken to be the image). This hinges on the fact that this property is essentially hereditary for hyperkaehler submanifolds. We establish various other properties of these sets including a codimension estimate which may be viewed as a generic vanishing theorem.