Cohomology jump loci of quasi-compact K\"ahler manifolds

TL;DR

This paper applies the exponential Ax-Lindemann Theorem to study the structure of cohomology jump loci, showing linearity properties for certain topological spaces and quasi-compact K"ahler manifolds, advancing understanding in algebraic geometry and topology.

Contribution

It demonstrates the linearity of cohomology jump loci in specific geometric contexts using the exponential Ax-Lindemann Theorem, linking topological and algebraic properties.

Findings

Linearity of cohomology jump loci around the trivial local system for spaces with finite models.

Linearity of cohomology jump loci for rank one local systems on quasi-compact K"ahler manifolds.

Application of exponential Ax-Lindemann Theorem to local systems in algebraic geometry.

Abstract

We give two applications of the exponential Ax-Lindemann Theorem to local systems. One application is to show that for a connected topological space, the existence of a finite model of the real homotopy type implies linearity of the cohomology jump loci around the trivial local system. Another application is the linearity of the cohomology jump loci of rank one local systems on quasi-compact K\"ahler manifolds.