Branched SL(r,C)-opers

TL;DR

This paper introduces branched SL(r,C)-opers, exploring their properties and establishing a characterization in terms of logarithmic connections on a fixed holomorphic vector bundle, highlighting differences from classical SL(r,C)-opers.

Contribution

It defines branched SL(r,C)-opers and shows how their underlying vector bundle depends on the oper, unlike the classical case, and characterizes them via associated logarithmic connections.

Findings

The underlying bundle depends on the branched oper.

A fixed bundle with logarithmic connection characterizes branched SL(r,C)-opers.

The properties differ from classical SL(r,C)-opers.

Abstract

We define the branched analog of SL(r,C)-opers and investigate their properties. For the usual SL(r,C)-opers, the underlying holomorphic vector bundle is independent of the opers. For the branched SL(r,C)-opers, the underlying holomorphic vector bundle depends on the oper. Given a branched SL(r,C)-oper, we associate to it another holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle does not depend on the branched oper. We characterize the branched SL(r,C)-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.