L-infinity pairs and applications to singularities

TL;DR

This paper introduces finite-dimensional L-infinity pairs to control deformation problems with cohomology constraints, leading to new insights into the structure of cohomology jump loci and restrictions on fundamental groups in algebraic geometry.

Contribution

It demonstrates that deformation problems controlled by infinite-dimensional pairs can be effectively managed by finite-dimensional L-infinity pairs, with applications to algebraic varieties and topological spaces.

Findings

Cohomology jump loci components are tori under certain conditions.

Restrictions on fundamental groups of algebraic varieties, links, and Milnor fibers.

Finite-dimensional L-infinity pairs effectively control deformation problems.

Abstract

Over a field of characteristic zero, every deformation problem with cohomology constraints is controlled by a pair consisting of a differential graded Lie algebra together with a module. Unfortunately, these pairs are usually infinite-dimensional. We show that every deformation problem with cohomology constraints is controlled by a typically finite-dimensional L-infinity pair. As a first application, we show that for complex algebraic varieties with no weight-zero 1-cohomology classes, the components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori. This imposes restrictions on the fundamental groups. The same holds for links and Milnor fibers.