Uniformization of compact complex manifolds by Anosov representations

TL;DR

This paper investigates how certain compact complex manifolds, formed as quotients of flag varieties by Anosov representations, can be uniformly described and deformed, extending classical uniformization results to higher-rank settings.

Contribution

It establishes conditions under which deformations of complex structures correspond to deformations of Anosov homomorphisms, generalizing uniformization theorems to higher-rank Lie groups.

Findings

Deformations of complex structures are realized by deformations of Anosov homomorphisms.

Character variety maps are locally homeomorphic to Teichmüller spaces.

Provides a higher-rank analogue of the Bers uniformization theorem.

Abstract

We study uniformization problems for compact manifolds that arise as quotients of domains in complex flag varieties by images of Anosov homomorphisms. We focus on Anosov homomorphisms with "small" limit sets, as measured by the Riemannian Hausdorff codimension in the flag variety. Under such a codimension hypothesis, we show that all first-order deformations of complex structure on the associated compact complex manifolds are realized by deformations of the Anosov homomorphism. With some mild additional hypotheses we show that the character variety maps locally homeomorphically to the (generalized) Teichm\"uller space of the manifold. In particular this provides a local analogue of the Bers Simultaneous Uniformization Theorem in the setting of Anosov homomorphisms to higher-rank complex semisimple Lie groups.