arXiv subjects

Louis-Clément Lefèvre

Publications and source records attributed to Louis-Clément Lefèvre.

4 recordsLinked to original sources

arXiv2026

Deformations of representations of fundamental groups of complex varieties

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 $ρ$ of its fundamental group $π_1(X,x)$, 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 $\widehat{\mathcal{O}}_ρ$ to the representation variety of $π_1(X,x)$ at $ρ$. Then we show how a weighted-homogeneous presentation of $\widehat{\mathcal{O}}_ρ$ 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 ($ρ$ finite).

math.AG↗

arXiv2021

Mixed Hodge structures on cohomology jump ideals

In previous work, we constructed for a smooth complex variety $X$ and for a linear algebraic group $G$ a mixed Hodge structure on the complete local ring $\widehat{\mathcal{O}}_ρ$ to the moduli space of representations of the fundamental group $π_1(X,x)$ into $G$ at a representation $ρ$ underlying a variation of mixed Hodge structure. We now show that the jump ideals $J_k^i \subset \widehat{\mathcal{O}}_ρ$, defining the locus of representations such the the dimension of the cohomology of $X$ in degree $i$ of the associated local system is greater than $k$, are sub-mixed Hodge structures; this is in accordance with various known motivicity results for these loci. In rank one we also recover, and find new cases, where these loci are translated sub-tori of the moduli of representations. Our methods are first transcendental, relying on Hodge theory, and then combined with tools of homotopy and algebra.

math.AG↗

arXiv2018

Mixed Hodge structures and representations of fundamental groups of algebraic varieties

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

math.AG↗

arXiv2015

A criterion for quadraticity of a representation of the fundamental group of an algebraic variety

Let $Γ$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $ρ:Γ\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(Γ, G)$. It is known from the work of Goldman and Millson that if $Γ$ is the fundamental group of a compact K{ä}hler manifold and $ρ$ has image contained in a compact subgroup then $ρ$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(Γ, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion holds.

math.AG↗