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

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus g≥ 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