Holomorphic flexibility properties of compact complex surfaces

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.Comment: Version 2: Theorem 11 reformulated and its proof corrected. Minor improvements to the exposition. Version 3: A few minor improvements. To appear in International Mathematics Research Notice