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.

中文翻译：

---

Fatou–Bieberbach域的全纯族及其在Oka流形中的应用

我们以给定的中心在$ \ mathbb {C} ^ n $的任何紧凑的多项式凸子集$ K $的补码中，以给定的中心构造Fatou–Bieberbach域的全纯变族，以$ n \ gt 1 $为准。这提供了$ Y $最近结果的简单证明。Kusakabe表示$ \ mathbb {C} ^ n $的任何多项式凸子集$ K $的补数$ \ mathbb {C} ^ n \ setminus K $是Oka流形。用$ \ mathbb {C} ^ n $替换为具有密度属性的任何Stein流形都可以得到类似的结果。