Mixed Hodge structures and representations of fundamental groups of algebraic varieties

TL;DR

This paper develops a functorial mixed Hodge structure on the local ring of representation varieties of fundamental groups of complex algebraic varieties, using advanced algebraic and geometric tools, with applications to Kähler and quasi-projective cases.

Contribution

It introduces a new framework for constructing mixed Hodge structures on representation varieties using mixed Hodgediagrams and $L_$ algebras, applicable to various geometric contexts.

Findings

Constructed a functorial mixed Hodge structure on local rings of representation varieties.

Applied the framework to compact Kähler varieties with monodromy of Hodge structures.

Extended the approach to smooth quasi-projective varieties with finite image representations.

Abstract

Given a complex variety $X$, a linear algebraic group $G$ and a representation $\rho$ of the fundamental group $\pi\_1(X,x)$ into $G$, we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of the representation variety of $\pi\_1(X,x)$ into $G$ at $\rho$ using mixed Hodgediagrams and methods of $L\_\infty$ algebras. We apply it in two geometric situations: either when $X$ is compact K{\"a}hler and $\rho$ is the monodromy of a variation of Hodge structure, or when $X$ is smooth quasi-projective and $\rho$ has finite image.