Surjective holomorphic maps onto Oka manifolds

TL;DR

The paper demonstrates that for Oka manifolds, any continuous map from a Stein manifold can be homotoped to a surjective, strongly dominating holomorphic map, introducing a new property called the basic Oka property with surjectivity.

Contribution

It establishes the surjectivity and strong domination of holomorphic maps onto Oka manifolds and proposes a new flexibility property characterizing Oka manifolds.

Findings

Every continuous map from a Stein manifold to an Oka manifold can be homotoped to a surjective strongly dominating holomorphic map.

Existence of strongly dominating algebraic morphisms from affine space onto algebraically subelliptic manifolds.

Introduction of the basic Oka property with surjectivity as a potential characterization of Oka manifolds.

Abstract

Let $X$ be a connected Oka manifold, and let $S$ be a Stein manifold with $\mathrm{dim} S \geq \mathrm{dim} X$. We show that every continuous map $S\to X$ is homotopic to a surjective strongly dominating holomorphic map $S\to X$. We also find strongly dominating algebraic morphisms from the affine $n$-space onto any compact $n$-dimensional algebraically subelliptic manifold. Motivated by these results, we propose a new holomorphic flexibility property of complex manifolds, the basic Oka property with surjectivity, which could potentially provide another characterization of the class of Oka manifolds.