Homotopy principles for equivariant isomorphisms

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. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $\varphi$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $\varphi$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $\Phi :X\to Y$, inducing $\varphi$, which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then $\Phi$ is homotopic, through $G$-diffeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second result roughly says that if we have a $G$-homeomorphism $\Phi :X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X^{-1}(q)$ and $p_Y^{-1}(\varphi (q))$ for $q\in Q_X$ and if $X$ satisfies an auxiliary property (which holds for most $X$), then $\Phi$ is homotopic, through $G$-homeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon earlier work of the authors and use new ideas and techniques.

51 pages, minor corrections in section 3. Final version, to appear in Transactions of the AMS