Parametric equivariant Oka principle

TL;DR

This paper establishes a parametric equivariant Oka principle for reductive complex Lie groups acting on Stein spaces, showing a homotopy equivalence between holomorphic and continuous maps under certain conditions.

Contribution

It proves a new equivariant Oka principle incorporating parametric and interpolation aspects, extending classical results with homotopy-theoretic and analytic methods.

Findings

Holomorphic $G$-maps are weakly homotopy equivalent to continuous $K$-maps.

Develops equivariant versions of homotopy approximation and nonlinear splitting lemmas.

Strengthens the principle to include interpolation on $G$-invariant subvarieties.

Abstract

Let $G$ be a reductive complex Lie group and $K$ be a maximal compact subgroup of $G$. Let $X$ be a reduced Stein $G$-space and $Y$ be a $G$-elliptic manifold. We prove the following parametric equivariant Oka principle. The inclusion of the space of holomorphic $G$-maps $X\to Y$ into the space of continuous $K$-maps $X\to Y$ is a weak homotopy equivalence with respect to the compact-open topology. The proof is divided into a homotopy-theoretic part, which is handled by an abstract theorem of Studer, and an analytic part, for which we prove equivariant versions of the homotopy approximation theorem and the nonlinear splitting lemma that are key tools in Oka theory. The principle can be strengthened so as to allow interpolation on a $G$-invariant subvariety of $X$.