Oka complements of countable sets and nonelliptic Oka manifolds

Abstract

We study the Oka properties of complements of closed countable sets in C n ( n > 1 ) \mathbb {C}^{n}\ (n>1) which are not necessarily discrete. Our main result states that every tame closed countable set in C n ( n > 1 ) \mathbb {C}^{n}\ (n>1) with a discrete derived set has an Oka complement. As an application, we obtain nonelliptic Oka manifolds which negatively answer a long-standing question of Gromov. Moreover, we show that these examples are not even weakly subelliptic. It is also proved that every finite set in a Hopf manifold has an Oka complement and an Oka blowup.