$\mathbb{A}^1$--connectedness of moduli stack of semi-stable and parabolic semi-stable vector bundles over a curve

TL;DR

This paper proves that various moduli stacks of semi-stable and parabolic semi-stable vector bundles over a curve are $ ext{A}^1$-connected, extending understanding of their geometric and topological properties.

Contribution

It establishes $ ext{A}^1$-connectedness for moduli stacks of semi-stable, quasi-parabolic, and parabolic vector bundles with fixed data over a curve.

Findings

Moduli stack of semi-stable vector bundles is $ ext{A}^1$-connected.

Moduli stack of quasi-parabolic vector bundles is $ ext{A}^1$-connected.

Open substack of $oldsymbol{eta}$-semistable parabolic bundles is $ ext{A}^1$-connected for generic weights.

Abstract

Let $C$ be an irreducible smooth projective curve of genus $g\geq 2$ over an algebraically closed field. We prove that the moduli stack of semi-stable vector bundles on $C$ of fixed rank and determinant is $\mathbb{A}^1$--connected. We also show that the moduli stack of quasi-parabolic vector bundles with a fixed determinant and a given quasi-parabolic data along a set of points in $C$ is $\mathbb{A}^1$-connected. Moreover, for small and generic weights $\boldsymbol{\alpha}$ with $\gcd(n, \deg L) = 1$, the open substack of $\boldsymbol{\alpha}$-semistable parabolic vector bundles is also $\mathbb{A}^1$-connected.