AI 中文总结
本文在奥卡理论中构造了若干例子与反例，证明了部分复欧几里得空间的补集是奥卡的，同时指出某些嵌入的爆破不是奥卡的，还验证了$\mathbb{C}^n\setminus\{0\}$不满足代数基本奥卡性质。
AI 中文摘要
本文给出奥卡理论中的若干例子与反例：两个关于从复欧几里得空间中删除闭集的例子，一个关于沿连通奥卡中心爆破的例子，还有一个关于将连续映射变形为正则映射的例子。两个正面结果表明，对每个闭集$S\subset\mathbb{R}^3$，$\mathbb{C}^3\setminus S$是奥卡的；闭哈特格斯三角形在$\mathbb{C}^2$中的补集也是奥卡的。相反，对每个$n\ge3$，存在一个真全纯嵌入$\mathbb{C}\hookrightarrow\mathbb{C}^n$，其像$A$是与$\mathbb{C}$双全纯的闭连通光滑曲线，但它的爆破$Bl_A\mathbb{C}^n$是布罗迪体积双曲的，因此不是奥卡的。最后，对每个$n\geq 2$，存在光滑连通仿射代数簇$X$和连续映射$X\to\mathbb{C}^n\setminus\{0\}$，它与任何正则映射$X\to\mathbb{C}^n\setminus\{0\}$都不同伦；等价地，$\mathbb{C}^n\setminus\{0\}$不满足代数基本奥卡性质（aBOP）。
英文摘要
This paper gives examples and counterexamples in Oka theory: two about deleting closed sets from complex Euclidean space, one about blowing up along a connected Oka center, and one about deforming continuous maps to regular maps. Two positive results show that $\mathbb{C}^3\setminus S$ is Oka for every closed set $S\subset\mathbb{R}^3$, and that the complement of the closed Hartogs triangle in $\mathbb{C}^2$ is Oka. In contrast, for every $n\ge3$ there is a proper holomorphic embedding $\mathbb{C}\hookrightarrow\mathbb{C}^n$ whose image $A$ is a closed connected smooth curve biholomorphic to $\mathbb{C}$, but whose blow-up $Bl_A\mathbb{C}^n$ is Brody volume hyperbolic and hence not Oka. Finally, for every $n\geq 2$, there exist a smooth connected affine algebraic variety $X$ and a continuous map $X\to\mathbb{C}^n\setminus\{0\}$ that is not homotopic to any regular map $X\to\mathbb{C}^n\setminus\{0\}$; equivalently, $\mathbb{C}^n\setminus\{0\}$ fails the algebraic basic Oka property (aBOP).
