Branched projective structures, branched SO(3,C)-opers and logarithmic connections on jet bundle

TL;DR

This paper explores the relationship between branched holomorphic projective structures on Riemann surfaces, branched SO(3,C)-opers, and logarithmic connections, establishing bijections and geometric characterizations of these structures.

Contribution

It introduces branched SO(3,C)-opers and shows their correspondence with branched holomorphic projective structures and certain logarithmic connections on jet bundles.

Findings

Branched holomorphic projective structures are in bijection with branched SO(3,C)-opers.

These structures correspond to specific logarithmic connections on jet bundles.

The work provides a geometric framework linking projective structures, opers, and connections.

Abstract

We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S\, =\, \sum_{i=1}^d x_i$, where $x_i \,\in\, X$ are distinct points. After defining branched ${\rm SO}(3,{\mathbb C})$--opers, we show that the branched holomorphic projective structures on $X$ are in a natural bijection with the branched ${\rm SO}(3,{\mathbb C})$--opers singular at $S$. It is deduced that the branched holomorphic projective structures on $X$ are also identified with a subset of the space of all logarithmic connections on $J^2((TX)\otimes {\mathcal O}_X(S))$ singular over $S$, satisfying certain natural geometric conditions.