# Branched Projective Structures on a Riemann Surface and Logarithmic Connections

### Indranil Biswas

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India### Sorin Dumitrescu

Université Côte d'Azur, Nice, France### Subhojoy Gupta

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

## Abstract

We study the set $P_{S}$ consisting of all branched holomorphic projective structures on a compact Riemann surface $X$ of genus $g≥1$ and with a fixed branching divisor $S:=∑_{i=1}n_{i}⋅x_{i}$, where $x_{i}∈X$. Under the hypothesis that $n_{i},=1$, for all $i$, with $d$ a positive even integer such that $d=2g−2$, we show that $P_{S}$ 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 $Q_{⊗2}=TX⊗O_{X}(S)$. The space of all logarithmic connections of the above type is an affine space over the vector space $H_{0}(X,K_{X}⊗O_{X}(S))$ of dimension $3g−3+d$. We conclude that $P_{S}$ is a subset of this affine space that has codimenison $d$ at a generic point.

## Cite this article

Indranil Biswas, Sorin Dumitrescu, Subhojoy Gupta, Branched Projective Structures on a Riemann Surface and Logarithmic Connections. Doc. Math. 24 (2019), pp. 2299–2337

DOI 10.4171/DM/726