Branched Projective Structures on a Riemann Surface and Logarithmic Connections

Abstract

We study the set ${\mathcal{P}}_S$ consisting of all branched holomorphic projective structures on a compact Riemann surface $X$ of genus $g\ge 1$ and with a fixed branching divisor $S:={\sum }_{i=1}^d{n}_i\cdot x_i$, where $x_i\in X$. Under the hypothesis that $n_i,=1$, for all $i$, with $d$ a positive even integer such that $d\ne 2g-2$, we show that ${\mathcal{P}}_S$ coincides with a subset of the set of all logarithmic connections with singular locus $S$, satisfying certain geometric conditions, on the rank two holomorphic jet bundle $J^1(Q)$, where $Q$ is a fixed holomorphic line bundle on $X$ such that $Q^{\otimes 2}=TX\otimes {\mathcal{O}}_X(S)$. The space of all logarithmic connections of the above type is an affine space over the vector space $H^0(X,K_X^{\otimes 2}\otimes {\mathcal{O}}_X(S))$ of dimension $3g-3+d$. We conclude that ${\mathcal{P}}_S$ is a subset of this affine space that has codimenison $d$ at a generic point.