Character varieties of virtually nilpotent K\"ahler groups and G-Higgs bundles

TL;DR

This paper proves that for virtually nilpotent K"ahler groups, the character variety deformation retracts onto the compact subgroup, and extends the C^* action on G-Higgs bundles to a C action, revealing new geometric structures.

Contribution

It establishes a natural strong deformation retraction of character varieties for virtually nilpotent K"ahler groups and extends the C^* action to a C action on G-Higgs bundles.

Findings

Deformation retraction from Hom(Gamma, G)/G to Hom(Gamma, K)/K

Extension of C^* action to a C action on G-Higgs bundles

Structural insight into character varieties of virtually nilpotent K"ahler groups

Abstract

Let G be a connected complex reductive affine algebraic group, and let K be a maximal compact subgroup. Let X be a compact connected K\"ahler manifold whose fundamental group Gamma is virtually nilpotent. We prove that the character variety Hom(Gamma, G)/G admits a natural strong deformation retraction to the subset Hom(Gamma, K)/K. The natural action of C^* on the moduli space of G-Higgs bundles over X extends to an action of C. This produces the deformation retraction.