Equivariant Oka theory: Survey of recent progress

TL;DR

This survey reviews recent advances in equivariant Oka theory, highlighting homotopy principles, applications to linearisation, bundle sections, and a new equivariant Gromov Oka principle, advancing understanding of complex Lie group actions.

Contribution

It compiles recent progress in equivariant Oka theory, introducing new principles and applications, including a $G$-Oka notion and classification methods for principal bundles.

Findings

Homotopy principles for equivariant isomorphisms established.

A parametric Oka principle for bundle sections proven.

An equivariant Gromov Oka principle based on $G$-Oka manifolds introduced.

Abstract

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group $G$ acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle $E$ of homogeneous spaces for a group bundle $\mathscr G$, all over a reduced Stein space $X$ with compatible actions of a reductive complex group on $E$, $\mathscr G$, and $X$. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a new notion of a $G$-manifold being $G$-Oka.