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

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.
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https://hal.archives-ouvertes.fr/hal-01809625
Contributor : Louis-Clément Lefèvre <>
Submitted on : Wednesday, June 6, 2018 - 10:30:05 PM
Last modification on : Tuesday, May 11, 2021 - 11:36:03 AM
Long-term archiving on: : Friday, September 7, 2018 - 3:10:13 PM

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

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