An Oka principle for Stein G-manifolds

Abstract

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X:X\to Q_X$ and $p_Y:Y\to Q_Y$ be the quotient mappings. Assume that we have a biholomorphism $Q:=Q_X\to Q_Y$ and an open cover $\{U_i\}$ of $Q$ and $G$-biholomorphisms ${\Phi }_i:p_X^{-1}(U_i)\to p_Y^{-1}(U_i)$ inducing the identity on $U_i$. There is a sheaf of groups $\mathcal{A}$ on $Q$ such that the isomorphism classes of all possible $Y$ is the cohomology set $H^1(Q,\mathcal{A})$. The main question we address is to what extent $H^1(Q,\mathcal{A})$ contains only topological information. For example, if $G$ acts freely on $X$ and $Y$, then $X$ and $Y$ are principal $G$-bundles over $Q$, and Grauert's Oka Principle says that the set of isomorphism classes of holomorphic principal $G$-bundles over $Q$ is canonically the same as the set of isomorphism classes of topological principal $G$-bundles over $Q$. We investigate to what extent we have an Oka principle for $H^1(Q,\mathcal{A})$.

12 pages, minor changes, to appear in Indiana University Math. J