The Hodge theory of character varieties

TL;DR

This paper explores the deep geometric structure of character varieties by linking the weight filtration in cohomology with the perverse Leray filtration via non-Abelian Hodge theory, revealing new insights into their topology.

Contribution

It proves that for certain groups, the non-Abelian Hodge theorem aligns the weight and perverse Leray filtrations on cohomology, connecting Hodge theory and perverse sheaves.

Findings

Identification of filtrations for specific groups

Application of decomposition and support theorems

Clarification of the geometric structure of character varieties

Abstract

This is a report on joint work with T. Hausel and L. Migliorini, where we prove, for each of the groups GL(2,C), PGL(2,C), SL(2,C), that the non-Abelian Hodge theorem identifies the weight filtration on the cohomology of the character variety with the perverse Leray filtration on the cohomology of the domain of the Hitchin map. We review the decomposition theorem, N\^go's support theorem, the geometric description of the perverse filtration and the sub-additivity of the Leray filtration with respect to the cup product.