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

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.