An Oka principle for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on normal Stein spaces $X$ and $Y$, which are locally $G$-biholomorphic over a common categorical quotient $Q$. When is there a global $G$-biholomorphism $X\to Y$? If the actions of $G$ on $X$ and $Y$ are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that $X$ and $Y$ are $G$-biholomorphic if $X$ is $K$-contractible, where $K$ is a maximal compact subgroup of $G$, or if $X$ and $Y$ are smooth and there is a $G$-diffeomorphism $ψ:X\to Y$ over $Q$, which is holomorphic when restricted to each fibre of the quotient map $X\to Q$. We prove a similar theorem when $ψ$ is only a $G$-homeomorphism, but with an assumption about its action on $G$-finite functions. When $G$ is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of $G$-biholomorphisms from $X$ to $Y$ over $Q$. This sheaf can be badly singular, even for a low-dimensional representation of $\mathrm{SL}_2(\C)$. Our work is in part motivated by the linearisation problem for actions on $\C^n$. It follows from one of our main results that a holomorphic $G$-action on $\C^n$, which is locally $G$-biholomorphic over a common quotient to a generic linear action, is linearisable.

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