Branched Projective Structures on a Riemann Surface and Logarithmic Connections

Biswas, Indranil ; Dumitrescu, Sorin ; Gupta, Subhojoy

Documenta mathematica, Tome 24 (2019), pp. 2299-2337.

Résumé

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.

Détail

Classification : 30F10, 30F30
Keywords: Riemann surface, branched projective structure, logarithmic connection, quadratic differential