## Branched Projective Structures on a Riemann Surface and Logarithmic Connections

##### Doc. Math. 24, 2299-2337 (2019)
DOI: 10.25537/dm.2019v24.2299-2337

### Summary

We study the set $\mathcal{P}_S$ consisting of all branched holomorphic projective structures on a compact Riemann surface $X$ of genus $g \geq 1$ 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 $d neq 2g-2$, 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 $H^0(X, K^{\otimes 2}_X \otimes\mathcal{O}_X(S))$ 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.

### Mathematics Subject Classification

51N15, 30F30, 32G15

### Keywords/Phrases

Riemann surface, branched projective structure, logarithmic connection, meromorphic quadratic differential, residue, local monodromy, second fundamental form

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### Affiliation

Biswas, Indranil
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Dumitrescu, Sorin
Université Côte d'Azur, Nice, France
Gupta, Subhojoy
Department of Mathematics, Indian Institute of Science, Bangalore 560012, Indi