$\mathbb{A}^1$-connectedness of moduli of semistable bundles and symplectic bundles on a curve

Abstract

In this note, we show that over a geometrically irreducible smooth projective curve over an infinite field $k$, the moduli stack of semistable vector bundles of fixed determinant is $\mathbb{A}^1$-connected if and only if the moduli stack admits a $k$-rational point. For this we use Langton's elementary modifications for vector bundles at the level of families. In addition, we prove the $\mathbb{A}^1$-connectedness of the moduli stack of symplectic bundles with forms valued in a fixed line bundle $L$ on a smooth projective curve of genus $g \ge 2$ over an infinite field $k$ with $C(k)\neq \emptyset$. As an application, we deduce the $\mathbb{A}^1$-connectedness of moduli stack of quasi-parabolic symplectic vector bundles with forms valued in a fixed line bundle.