Cohomology jump loci of differential graded Lie algebras

TL;DR

This paper introduces cohomology jump functors for differential graded Lie algebra pairs to analyze infinitesimal deformation problems with cohomology constraints, extending classical results in the field.

Contribution

It develops a new framework for studying cohomology jump loci in DGLA pairs and applies it to various geometric moduli spaces, broadening the understanding of their local structure.

Findings

Describes the analytic germs of cohomology jump loci in moduli spaces.

Extends previous results of Goldman-Millson, Green-Lazarsfeld, and others.

Provides a unified approach to cohomology constraints in deformation theory.

Abstract

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman-Millson, Green-Lazarsfeld, Nadel, Simpson, Dimca-Papadima, and of the second author.