# Torsors in the analytic topology versus torsors in the etale topology

Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$.

Is every analytic $G$-torsor over $S$ algebraic?

In other words, is the set $H^1_{et}(S,G)$ in bijection with the set of holomorphic $G$-torsors over $\mathbb C$ (via the natural map)?

In yet other terms: is every holomorphic morphism of analytic stacks $\mathbb C\to BG^{an}$ induced by a unique morphism of algebraic stacks $S\to BG$?

If the answer is negative, are there any positive results in this direction?

Yes, but for trivial reasons. By the Grauert-Oka principle, every analytic torsor over $\mathbb{C}$ is trivial.
• But already if you replace the line with a punctured elliptic curve, the statement is no longer true: every analytic torsor is trivial, while a $\mathbb{G}_m$-torsor may be algebraically non-trivial. – Roman Fedorov Dec 27 '14 at 19:57