Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

Stefano Francaviglia, Lorenzo Ruffoni

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus at least 2, which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor, and we establish a partial converse for hyperelliptic structures.