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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1912.04787 |
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| _version_ | 1866910029313998848 |
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| author | Lefèvre, Louis-Clément |
| author_facet | Lefèvre, Louis-Clément |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_04787 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Deformations of representations of fundamental groups of complex varieties Lefèvre, Louis-Clément Algebraic Geometry Algebraic Topology 14D07, 14C30, 18D50 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). |
| title | Deformations of representations of fundamental groups of complex varieties |
| topic | Algebraic Geometry Algebraic Topology 14D07, 14C30, 18D50 |
| url | https://arxiv.org/abs/1912.04787 |