Local deformations of branched projective structures: Schiffer variations and the Teichm\"uller map

TL;DR

This paper investigates how branched complex projective structures on closed surfaces can be deformed without changing their holonomy, revealing that branch points must lie on a canonical divisor under certain conditions.

Contribution

It characterizes local deformations of branched projective structures that preserve holonomy and branch point order, linking branch point arrangements to canonical divisors and hyperelliptic structures.

Findings

Branch points lie on a canonical divisor when the structure's complex structure is infinitesimally preserved.

A partial converse is established for hyperelliptic structures.

Deformations preserve holonomy and branch point order in the studied class.

Abstract

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus $g\geq 2$, 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.