  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.
