Gromov's Oka principle for equivariant maps

TL;DR

This paper develops an equivariant version of Gromov's Oka theory, establishing fundamental properties and an equivariant Oka principle for finite group actions on complex manifolds, with conjectures extending to reductive groups.

Contribution

It introduces equivariant versions of Oka properties, proves an equivariant Oka principle for finite groups, and explores potential generalizations to reductive complex Lie groups.

Findings

Defined equivariant Oka properties and verified basic properties.

Proved an equivariant Oka principle for finite group actions.

Presented examples and partial results for reductive group actions.

Abstract

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group $G$ acts on a Stein manifold $X$ and another manifold $Y$ in such a way that $Y$ is $G$-Oka, then every $G$-equivariant continuous map $X\to Y$ can be deformed, through such maps, to a $G$-equivariant holomorphic map. Approximation on a $G$-invariant holomorphically convex compact subset of $X$ and jet interpolation along a $G$-invariant subvariety of $X$ can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.