Isomorphisms between moduli stacks of vector bundles with fixed determinant
David Alfaya, Indranil Biswas, Tomás L. Gómez
TL;DR
The paper classifies all isomorphisms between moduli stacks ${\mathcal{M}}(X,r,\xi)$ of vector bundles with fixed determinant on complex projective curves of genus at least $4$, showing every stack isomorphism is, up to isomorphism, a basic transformation ${\mathcal{T}}_{\sigma,L,s}$ arising from a curve isomorphism, a line bundle, and a sign, with determinant compatibility. It extends known results for moduli schemes to the stack setting by describing the automorphism 2-group via stack-level pullbacks, duality, and twists, and by recovering the semistable locus through beyond-GIT techniques and a 2-dimensional density extension that guarantees unique extension from the semistable locus to the whole stack. The work leverages a fibered-category viewpoint, descent data, and ind-scheme techniques (e.g., ${\rm Div}_X^{r,d}$) to control extensions and codimension of non-semistable loci, culminating in a complete classification of stack isomorphisms. The results clarify how stack automorphisms relate to automorphisms of the moduli space and reveal subtle phenomena in low genus (notably $X=\mathbb{P}^1$) where stack actions on morphisms can be nontrivial even when the moduli space is simple. Overall, the paper provides a comprehensive, stack-level Torelli-type description for vector bundles with fixed determinant.
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.