Lagrangian subbundles of symplectic bundles over a curve

TL;DR

This paper establishes a sharp upper bound for the invariant measuring Lagrangian subbundles of symplectic bundles over curves, generalizing classical invariants and analyzing stratifications in moduli spaces.

Contribution

It introduces a new bound for the invariant of symplectic bundles and explores the stratification of moduli spaces, especially for rank four bundles.

Findings

Derived a sharp upper bound for the invariant ng

Analyzed stratifications on moduli spaces of symplectic bundles

Provided a complete description for rank four cases

Abstract

A symplectic bundle over an algebraic curve has a natural invariant $\sLag$ determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound on $\sLag$ which is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced by $\sLag$ on moduli spaces of symplectic bundles, and get a full picture for the case of rank four.