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

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