Isomorphisms between moduli stacks of vector bundles with fixed determinant

TL;DR

This paper classifies all isomorphisms between moduli stacks of vector bundles with fixed determinant on complex curves of genus at least 4, revealing they are composed of curve isomorphisms, dualizations, and tensorings.

Contribution

It provides a complete classification of isomorphisms between these moduli stacks, detailing their composition from known geometric operations.

Findings

Isomorphisms are characterized by curve isomorphisms, dualizations, and tensorings.

The structure of the automorphism 2-group of the moduli stack is compared to that of the moduli space.

The classification applies to curves of genus at least 4, ensuring generality in the high-genus case.

Abstract

We classify all isomorphisms between moduli stacks of vector bundles of fixed determinant on a smooth complex projective of genus at least 4. It is shown that each isomorphism between two different moduli stacks can be described as a composition of a pullback using an isomorphism of curves, dualization of vector bundles and tensoring with the pullback of a line bundle on the curve. We finally compare the 2-group of automorphisms of the moduli stack of vector bundles with the group of automorphisms of the moduli space of semistable vector bundles.