Some Examples and Counterexamples in Oka Theory

Abstract: 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 $S\subset\mathbb{R}<sup>3$, and that the complement of the closed Hartogs triangle in $\mathbb{C}<sup>2$ is Oka. In contrast, for every $n\ge3$ there is a proper holomorphic embedding $\mathbb{C}\hookrightarrow\mathbb{C}<sup>n$ whose image $A$ is a closed connected smooth curve biholomorphic to $\mathbb{C}$, but whose blow-up $Bl_A\mathbb{C}<sup>n$ 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 $X\to\mathbb{C}<sup>n\setminus{0}$ that is not homotopic to any regular map $X\to\mathbb{C}<sup>n\setminus{0}$; equivalently, $\mathbb{C}<sup>n\setminus{0}$ fails the algebraic basic Oka property (aBOP).