Mixed Hodge structures and representations of fundamental groups of algebraic varieties - Archive ouverte HAL Access content directly
Journal Articles Advances in Mathematics Year : 2019

## Mixed Hodge structures and representations of fundamental groups of algebraic varieties

Louis-Clément Lefèvre

#### 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 Hodge diagrams and methods of $L_\infty$ algebras. We apply it in two geometric situations: either when $X$ is compact Kähler and $\rho$ is the monodromy of a variation of Hodge structure, or when $X$ is smooth quasi-projective and $\rho$ has finite image.

#### Domains

Mathematics [math] Algebraic Geometry [math.AG]

### Dates and versions

hal-01809625 , version 1 (06-06-2018)

### Identifiers

• HAL Id : hal-01809625 , version 1
• ARXIV :
• DOI :

### Cite

Louis-Clément Lefèvre. Mixed Hodge structures and representations of fundamental groups of algebraic varieties. Advances in Mathematics, 2019, 349, pp.869-910. ⟨10.1016/j.aim.2019.04.028⟩. ⟨hal-01809625⟩

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