Every projective Oka manifold is elliptic

A complex manifold $Y$ is said to be an Oka manifold if the homotopy principle holds for holomorphic maps from Stein manifolds and Stein spaces to $Y$. One of the simplest known characterisations of this class of manifolds is the convex approximation property, asking that every holomorphic map from a convex domain in a complex Euclidean space to $Y$ is a limit of entire maps. In 1989, Mikhail Gromov introduced a geometric sufficient condition for a manifold to be Oka, called ellipticity, which asks for the existence of a dominating holomorphic spray on $Y$. It was a long-standing open problem whether, conversely, every Oka manifold is elliptic. The first counterexamples for non-compact manifolds were found only recently by Yuta Kusakabe. In this talk, I will show that every projective Oka manifold is elliptic.
This is joint work with Finnur Larusson.