# Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures

### Philippe Eyssidieux

Université Joseph Fourier, Grenoble, Saint-Martin-d'Hères, France### Carlos Simpson

Université de Nice Sophia Antipolis, France

## Abstract

Let $X$ be a compact Kähler manifold, $x\in X$ be a base point and $\rho: \pi_1(X,x) \to GL_N(\mathbb C)$ be the monodromy representation of a $\mathbb C$-VHS. Building on Goldman–Millson’s classical work, we construct a mixed Hodge structure on the complete local ring of the representation variety at $\rho$ and a variation of mixed Hodge structures whose monodromy is the universal deformation of $\rho$.

## Cite this article

Philippe Eyssidieux, Carlos Simpson, Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures. J. Eur. Math. Soc. 13 (2011), no. 6, pp. 1769–1798

DOI 10.4171/JEMS/293