Geometry of Moduli Stacks of $(k,l)$-stable vector bundles over an algebraic curve

TL;DR

This paper investigates the geometric and cohomological properties of moduli stacks of $(k, l)$-stable vector bundles over algebraic curves, introducing a filtration of open substacks and analyzing their coarse moduli spaces.

Contribution

It provides a detailed stacky framework for $(k, l)$-stability, revealing new geometric structures and their relation to coarse moduli spaces.

Findings

Moduli stacks of $(k, l)$-stable bundles admit coarse moduli spaces for certain $(k, l)$.

The paper characterizes the geometric and cohomological properties of these stacks.

A filtration of the moduli stack by open substacks is constructed.

Abstract

We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from $(k, l)$-stable vector bundles. $(k, l)$-stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of $(k, l)$-stable vector bundles. For particular pairs $(k, l) $ of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss their interplay.