Branched SL(r, ℂ)-Opers

Abstract

Abstract Branched projective structures were introduced by Mandelbaum [22, 23], and opers were introduced by Beilinson and Drinfeld [2, 3]. We define the branched analog of ${\textrm SL}(r, {\mathbb C})$-opers and investigate their properties. For the usual ${\textrm SL}(r, {\mathbb C})$-opers, the underlying holomorphic vector bundle is actually determined uniquely up to tensoring with a holomorphic line bundle of order $r$. For the branched ${\textrm SL}(r, {\mathbb C})$-opers, the underlying holomorphic vector bundle depends more intricately on the oper. While the holomorphic connection for a branched ${\textrm SL}(r, {\mathbb C})$-oper is nonsingular, given a branched ${\textrm SL}(r, {\mathbb C})$-oper, we associate to it a certain holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle in question supporting a logarithmic connection does not depend on the branched oper. We characterize the branched ${\textrm SL}(r, {\mathbb C})$-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.