# Oka complements of countable sets and non-elliptic Oka manifolds

## TL;DR

This paper investigates the Oka properties of complements of certain countable sets in complex Euclidean spaces and Hopf manifolds, revealing new non-elliptic Oka manifolds and advancing understanding of complex manifold structures.

## Contribution

It demonstrates that complements of tame countable sets with discrete derived sets are Oka, providing new examples of non-elliptic Oka manifolds and analyzing their properties.

## Key findings

- Complement of tame countable sets with discrete derived sets are Oka.
- Constructs non-elliptic Oka manifolds answering Gromov's question.
- Shows finite sets in Hopf manifolds have Oka complements and blowups.

## Abstract

We study the Oka properties of complements of closed countable sets in $\mathbb{C}^{n}\ (n>1)$ which are not necessarily discrete. Our main result states that every tame closed countable set in $\mathbb{C}^{n}\ (n>1)$ with a discrete derived set has an Oka complement. As an application, we obtain non-elliptic 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.