Generalized B-Opers

TL;DR

This paper introduces generalized B-opers, extending the concept of opers by incorporating bilinear forms, and explores their structure and relationship with jet bundles and geometric structures on Riemann surfaces.

Contribution

It defines and studies generalized B-opers, a new class of geometric objects that extend traditional opers with bilinear form compatibility.

Findings

Characterization of B-opers in relation to jet bundles

Analysis of B-opers' structure on Riemann surfaces

Connections between B-opers and geometric structures

Abstract

Opers were introduced by Beilinson-Drinfeld [arXiv:math.AG/0501398]. In [J. Math. Pures Appl. 82 (2003), 1-42] a higher rank analog was considered, where the successive quotients of the oper filtration are allowed to have higher rank. We dedicate this paper to introducing and studying generalized $B$-opers (where "$B$" stands for "bilinear"), obtained by endowing the underlying vector bundle with a bilinear form which is compatible with both the filtration and the connection. In particular, we study the structure of these $B$-opers, by considering the relationship of these structures with jet bundles and with geometric structures on a Riemann surface.