Elliptic characterization and localization of Oka manifolds

TL;DR

This paper proves that Gromov's ellipticity condition characterizes Oka manifolds, providing new insights into their properties, implications for Gromov's conjectures, and establishing a localization principle for these complex manifolds.

Contribution

The paper demonstrates that Gromov's ellipticity condition $ ext{Ell}_1$ precisely characterizes Oka manifolds, offering new proofs and applications, including the localization principle and non-invariance under bimeromorphic maps.

Findings

Ellipticity condition characterizes Oka manifolds.

Subellipticity implies the Oka property.

Localization principle for Oka manifolds established.

Abstract

We prove that Gromov's ellipticity condition $\mathrm{Ell}_1$ characterizes Oka manifolds. This characterization gives another proof of the fact that subellipticity implies the Oka property, and affirmative answers to Gromov's conjectures. As another application, we establish the localization principle for Oka manifolds, which gives new examples of Oka manifolds. In the appendix, it is also shown that the Oka property is not a bimeromorphic invariant.