Holomorphic flexibility properties of compact complex surfaces

TL;DR

This paper introduces stratified Oka manifolds, proves their strong dominability, and explores their properties and examples among compact complex surfaces, including Kummer surfaces and class VII surfaces, with implications for holomorphic flexibility.

Contribution

It defines stratified Oka manifolds, proves their strong dominability, and analyzes the Oka property in various complex surface classes and under modifications.

Findings

Kummer surfaces are strongly dominable.

The Oka property is not closed in families of complex manifolds.

Behavior of Oka property under blowing up and down is characterized.

Abstract

We introduce the notion of a stratified Oka manifold and prove that such a manifold $X$ is strongly dominable in the sense that for every $x\in X$, there is a holomorphic map $f:\C^n\to X$, $n=\dim X$, such that $f(0)=x$ and $f$ is a local biholomorphism at 0. We deduce that every Kummer surface is strongly dominable. We determine which minimal compact complex surfaces of class VII are Oka, assuming the global spherical shell conjecture. We deduce that the Oka property and several weaker holomorphic flexibility properties are in general not closed in families of compact complex manifolds. Finally, we consider the behaviour of the Oka property under blowing up and blowing down.