# Deformations of representations of fundamental groups of complex varieties

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).
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-02399676
Contributor : Louis-Clément Lefèvre <>
Submitted on : Sunday, April 11, 2021 - 3:59:25 PM
Last modification on : Tuesday, April 13, 2021 - 3:32:42 AM

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### Identifiers

• HAL Id : hal-02399676, version 2
• ARXIV : 1912.04787

### Citation

Louis-Clément Lefèvre. Deformations of representations of fundamental groups of complex varieties. 2021. ⟨hal-02399676v2⟩

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