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# Mathematical Research Letters

## Volume 27 (2020)

### Number 6

### Holomorphic families of Fatou–Bieberbach domains and applications to Oka manifolds

Pages: 1697 – 1706

DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n6.a5

#### Authors

#### Abstract

We construct holomorphically varying families of Fatou–Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $\mathbb{C}^n$ for $n \gt 1$. This provides a simple proof of the recent result of $Y$. Kusakabe to the effect that the complement $\mathbb{C}^n \setminus K$ of any polynomially convex subset $K$ of $\mathbb{C}^n$ is an Oka manifold. The analogous result is obtained with $\mathbb{C}^n$ replaced by any Stein manifold with the density property.

The first-named author is supported by the research program P1-0291 and grant J1-9104 from ARRS, Republic of Slovenia, and by the Stefan Bergman Prize 2019 from the American Mathematical Society. The second-named author is supported by the RCN grant 240569 from Norway.

Received 25 May 2020

Accepted 30 September 2020

Published 17 February 2021