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

Abstract

Let $C$ be an irreducible smooth projective curve of genus $g\ge 2$ 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 $\mathbf{\alpha }$ with $\gcd (n,°L)=1$, the open substack of $\mathbf{\alpha }$-semistable parabolic vector bundles is also ${\mathbb{A}}^1$-connected.

arXiv commentSubstantial changes have been made in Section 2 to address a gap in the earlier version. The main result in Section 2 remains unchanged. In Section 3, some additional assumptions on weights have been added for the final theorem. Comments are welcome