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Deformations of representations of fundamental groups of complex varieties

Louis-Clément Lefèvre

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 $ρ$ 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).

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 and such a linear representation of its fundamental group , 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 to the representation variety of at . Then we show how a weighted-homogeneous presentation of 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 ( compact) and of Kapovich-Millson ( finite).

Paper Structure

This paper contains 32 sections, 24 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Topology of algebraic varieties
  3. Deformations of representations of fundamental groups
  4. Deformations via DG Lie algebras
  5. Hodge theory
  6. Results
  7. Acknowledgments
  8. Some updates
  9. Reminder on mixed Hodge theory
  10. Mixed Hodge structures
  11. Variations of mixed Hodge structure
  12. Mixed Hodge complexes
  13. Thom-Whitney functors
  14. Lax monoidal symmetric functors
  15. Thom-Whitney resolutions
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\rho$ be the monodromy representation of an admissible variation of mixed Hodge structure over a smooth complex algebraic variety or a quasi-Kähler manifold $X$ with unipotent monodromy at infinity. Then there is a mixed Hodge structure on $\widehat{\mathcal{O}}_\rho$, functorial in $X$ and the

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2: SteenbrinkZucker
  • Definition 2.3
  • Definition 2.4: Lefevre2
  • Definition 2.5
  • Remark 2.6
  • ...and 41 more