Every projective Oka manifold is subelliptic

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 characterizations 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 $Y$. An ostensibly weaker condition, subellipticity, asks for the existence of finitely many holomorphic sprays on $Y$ which together dominate. It was a long-standing open problem whether these sufficient conditions for the Oka property are also necessary. The first counterexamples for open (non compact) manifolds were found only recently by Yuta Kusakabe. In this talk, I will show that every projective Oka manifold is subelliptic. Whether every such manifold is elliptic remains to be seen.