Cohomology jump loci in the moduli spaces of vector bundles

TL;DR

This paper generalizes the understanding of cohomology jump loci from line bundles to higher-dimensional vector bundles on compact Kahler manifolds, showing they are defined by linear equations in the moduli space.

Contribution

It extends the classical results on cohomology jump loci to vector bundles, demonstrating their linearity in the local moduli space context.

Findings

Cohomology jump loci are linear in the moduli space of vector bundles.

Generalization of Green-Lazarsfeld's results to higher rank vector bundles.

Moduli space locally modeled by a quadratic cone with linear cohomology conditions.

Abstract

Two decades ago, as part of their work of generic vanishing theorems, Green-Lazarsfeld showed that over a compact Kahler manifold $X$, the cohomology jump loci in the $Pic^\tau(X)$ are all translates of subtori. In this paper, we generalize this result to higher dimensional vector bundles. It is showed by Nadel that locally the moduli space of vector bundles with vanishing chern classes is canonically isomorphic to a quadratic cone in the Zariski tangent space of a point. We prove that under the isomorphism, the cohomology jump loci are defined by linear equations.