Etale Fundamental group of moduli of torsors under Bruhat-Tits group scheme over a curve

TL;DR

This paper proves that the étale fundamental group of the moduli stack of torsors under certain Bruhat-Tits group schemes over a curve matches that of principal G-bundles, with implications for the simply-connectedness of the moduli space.

Contribution

It establishes an isomorphism of étale fundamental groups between moduli stacks of torsors under Bruhat-Tits group schemes and principal G-bundles, extending known results to more general group schemes.

Findings

Étale fundamental group of the moduli stack of torsors under Bruhat-Tits group schemes is isomorphic to that of principal G-bundles.

Open substack of regularly stable torsors has complement of codimension at least two for genus ≥ 3.

The moduli space of G-torsors is simply-connected.

Abstract

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a Bruhat-Tits group scheme on $X$ which is generically semi-simple and trivial. We show that the \'etale fundamental group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of torsors under $\mathcal{G}$ is isomorphic to that of the moduli stack $\mathcal{M}_X(G)$ of principal $G$-bundles. For any smooth, noetherian and irreducible stack $\mathcal{X}$, we show that an inclusion of an open substack $\mathcal{X}^\circ$, whose complement has codimension at least two, will induce an isomorphism of \'etale fundamental group. Over $\mathbb{C}$, we show that the open substack of regularly stable torsors in $\mathcal{M}_X(\mathcal{G})$ has complement of codimension at least two when $g_X \geq 3$. As an application, we show that the moduli space $M_X(\mathcal{G})$ of $\mathcal{G}$-torsors is simply-connected.