Deformations of Oka manifolds

TL;DR

This paper studies how the Oka property behaves under deformations of compact complex manifolds, providing conditions for when fibers are Oka and analyzing special cases like tori and noncompact fibers.

Contribution

It establishes a characterization of Oka fibers in families, introduces a new uniform Oka property, and explores the Oka nature of projections with specific fiber types.

Findings

Oka fibers form a G-delta set in the base of a family.

A new uniform Oka property characterizes limits of Oka fibers.

Projections with torus fibers are Oka maps.

Abstract

We investigate the behaviour of the Oka property with respect to deformations of compact complex manifolds. We show that in a family of compact complex manifolds, the set of Oka fibres corresponds to a G-delta subset of the base. We give a necessary and sufficient condition for the limit fibre of a sequence of Oka fibres to be Oka in terms of a new uniform Oka property. We show that if the fibres are tori, then the projection is an Oka map. Finally, we consider holomorphic submersions with noncompact fibres.