The Moduli Space of Marked Branched Projective Structures

Holomorphic projective structures are structures on Riemann surfaces that play an important role in the theory of differential equations on Riemann surfaces, as well as in in the uniformization theorem. The notion of a branched holomorphic projective structure is a much more flexible concept. In particular, any representation of a surface group can be realized as the monodromy of a branched projective structure. One of the nicest feature of projective structures is the structure of the space of such structures on a given differential surface : it is a complex manifold with nice algebraic properties. We will show that most of these features extend to the moduli space of branched projective structures.