New uniformizations of some moduli spaces of curves with marked points

I will report on some work in collaboration with Julien
Marché, in which we constructed some compactification of the moduli
spaces of curves of genus g with n marked points carrying a complex
hyperbolic structure, in the cases (g,n)= (0,4),(0,5),(1,2),(1,3) and
(2,1). This extends classical works of Hirzebruch (case (g,n)=(0,5)),
Livne (case (g,n)=(1,2)), and this gives a negative answer to a
question asked by Siu in the eighties: there exists a holomorphic
submersion X-->Y between compact complex hyperbolic manifolds X,Y of
dimensions dim X>dim Y >1. These constructions are based on the
analysis of some topological properties of quantum projective
representations of the mapping class group.