An equivariant parametric Oka principle for bundles of homogeneous spaces

TL;DR

This paper establishes a parametric Oka principle for equivariant sections of holomorphic fiber bundles with homogeneous space fibers on Stein spaces, linking holomorphic and continuous sections via weak homotopy equivalence.

Contribution

It introduces a new equivariant parametric Oka principle for bundles with homogeneous fibers, extending previous classification and isomorphism results in complex geometry.

Findings

Inclusion of equivariant holomorphic sections into continuous sections is a weak homotopy equivalence.

The result applies to a broad class of bundles with homogeneous space fibers.

Strengthens existing classification and isomorphism theorems for equivariant principal bundles.

Abstract

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of $\mathscr G$, with the complexification $K^\mathbb C$ of a compact real Lie group $K$ acting on $X$, $\mathscr G$, and $E$. Our main result is that the inclusion of the space of $K^\mathbb C$-equivariant holomorphic sections of $E$ over $X$ into the space of $K$-equivariant continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch's classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.