A remark on subbundles of symplectic and orthogonal vector bundles over curves

TL;DR

This paper reviews symplectic and orthogonal vector bundles over curves, providing criteria for extensions and subbundles, and describing their properties using graph-based methods.

Contribution

It introduces new criteria for symplectic and orthogonal structures on vector bundle extensions and characterizes their subbundles via graph representations.

Findings

Criteria for symplectic and orthogonal extensions

Graph-based description of subbundles

Conditions for isotropy of subbundles

Abstract

We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles.