Simplicity of tangent bundles on the moduli spaces of symplectic and orthogonal bundles over a curve

TL;DR

This paper proves the simplicity of tangent bundles on moduli spaces of symplectic and orthogonal bundles over a curve, extending previous results and analyzing minimal rational tangents and tangent maps.

Contribution

It introduces a new approach using tangent maps of Hecke curves to establish simplicity of tangent bundles for symplectic and orthogonal bundles.

Findings

Nondegeneracy of the variety of minimal rational tangents

Simplicity of tangent bundles on moduli spaces

Tangent map is an embedding for large genus

Abstract

The variety of minimal rational tangents associated to Hecke curves was used by J.-M.Hwang [8] to prove the simplicity of the tangent bundle on the moduli of vector bundles over a curve. In this paper, we use the tangent maps of the symplectic and orthogonal Hecke curves to prove an analogous result for symplectic and orthogonal bundles. In particular, we show the nondegeneracy of the associated variety of minimal rational tangents, which implies the simplicity of the tangent bundle on the moduli spaces of symplectic and orthogonal bundles over a curve. We also show that for large enough genus, the tangent map is an embedding for a general symplectic or orthogonal bundle.