Parametric equivariant Oka principle

TL;DR

We address extending the parametric Oka principle to an equivariant setting with interpolation for a reductive group action. The approach combines Studer’s abstract homotopy framework with analytic tools tailored to equivariant holomorphic approximation, including $G$-ellipticity, Kempf–Ness reduction, and dominating $G$-sprays, together with parametric and nonlinear splitting techniques. The main result establishes that the inclusion of holomorphic $G$-maps $X\to Y$ into continuous $K$-maps $X\to Y$ is a weak homotopy equivalence, with a version that interpolates on a $G$-invariant subvariety, and it provides an equivariant analytic toolkit (EPHAP, equivariant Cartan–Oka–Weyl) that works in this generalized setting. This advances equivariant Oka theory beyond homogeneous targets and yields a framework applicable to Danielewski manifolds and related spaces, thereby enabling topological-homotopy control of equivariant holomorphic maps in broad complex-analytic contexts.

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 $X$.

Theorems & Definitions (74)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• Lemma 2.5