|
Let X be a smooth projective curve over an algebraically closed field k . Let G be an almost simple simply-connected group over k . Let \(\mathcal {G}\) be a Bruhat–Tits group scheme on X which is generically the trivial group scheme with fibers G . We show that the étale 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. Our main goal is to prove that for any smooth, noetherian and irreducible stack \(\mathcal {X}\) , the inclusion of any non-empty open substack \(\mathcal {X}^\circ \) , whose complement has codimension at least two induces an isomorphism of étale 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 \(g_X \ge 3\) . As an application, we show that over \({\mathbb {C}}\) the moduli space \(M_X(\mathcal {G})\) of \(\mathcal {G}\) -torsors is simply-connected.
|
