On the Teichm\"uller stack of homogeneous spaces of SL(2,C)

TL;DR

This paper demonstrates that the admissible character stack forms an open substack of the Teichmüller stack for homogeneous spaces of SL(2,C), generalizing previous work on deformations of complex structures.

Contribution

It establishes the relationship between the character stack and the Teichmüller stack, showing the tautological family over the representation variety is always complete.

Findings

The admissible character stack is an open substack of the Teichmüller stack.

The tautological family over the representation variety is complete.

Generalizes Ghys's work on deformations of complex structures.

Abstract

In this paper, we show that the (admissible) character stack, which is a stack version of the character variety, is an open substack of the Teichm\"uller stack of homogeneous spaces of SL(2,C). We show that the tautological family over the representation variety, given by deforming the holonomy, is always a complete family. This is a generalisation of the work of E. Ghys about deformations of complex structures these homogeneous spaces.