Homotopy principles for equivariant isomorphisms

TL;DR

This paper establishes homotopy principles for when equivariant biholomorphisms exist between Stein manifolds with reductive complex Lie group actions, based on quotient and Luna stratification conditions.

Contribution

It introduces two homotopy principles that connect $G$-diffeomorphisms and $G$-homeomorphisms to $G$-equivariant biholomorphisms under certain conditions, improving previous results.

Findings

Homotopy from $G$-diffeomorphisms to $G$-biholomorphisms under fiber conditions.

Homotopy from $G$-homeomorphisms to $G$-biholomorphisms with auxiliary properties.

Enhanced techniques for establishing equivariant isomorphisms in complex geometry.

Abstract

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ 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 $\phi$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $\phi$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $\Phi\colon X\to Y$, inducing $\phi$, 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 $\Phi\colon X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X^{-1}(q)$ and $p_Y^{-1}(\phi(q))$ 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.