Deformations of representations of fundamental groups of complex varieties

TL;DR

This paper studies the local structure of representation varieties of fundamental groups of complex varieties at special monodromy representations, using mixed Hodge structures and deformation theory to generalize existing theorems.

Contribution

It constructs a mixed Hodge structure on the local deformation ring of representations and relates it to the weight filtration, extending previous results for compact and finite cases.

Findings

Constructed a mixed Hodge structure on the local deformation ring.

Linked the weight filtration to a weighted-homogeneous presentation.

Generalized theorems of Eyssidieux-Simpson and Kapovich-Millson.

Abstract

We describe locally the representation varieties of fundamental groups for smooth complex varieties at representations coming from the monodromy of a variation of mixed Hodge structure. Given such a manifold $X$ and such a linear representation $\rho$ of its fundamental group $\pi_1(X,x)$, we use the theory of Goldman-Millson and pursue our previous work that combines mixed Hodge theory with derived deformation theory to construct a mixed Hodge structure on the formal local ring $\widehat{\mathcal{O}}_\rho$ to the representation variety of $\pi_1(X,x)$ at $\rho$. Then we show how a weighted-homogeneous presentation of $\widehat{\mathcal{O}}_\rho$ is induced directly from a splitting of the weight filtration of its mixed Hodge structure. In this way we recover and generalize theorems of Eyssidieux-Simpson ($X$ compact) and of Kapovich-Millson ($\rho$ finite).