论文标题
欧几里得空间中的Oka域
Oka domains in Euclidean spaces
论文作者
论文摘要
在本文中，我们在欧几里得空间中发现了令人惊讶的小oka域，$ \ mathbb c^n $ dimension $ n> 1 $在可能的限制下。在封闭的无界凸套装上的温和几何假设下，$ \ mathbb c^n $中的$ e $我们表明，$ \ mathbb c^n \ setMinus e $是OKA域。特别是，有OKA域仅比半空间大一点，后者既不是Oka也不是双曲线。 This gives families of smooth real hypersurfaces $Σ_t\subset \mathbb C^n$ $(t\in\mathbb R)$ dividing $\mathbb C^n$ in an unbounded hyperbolic domain and an Oka domain such that at the threshold value $t=0$ the hypersurface $Σ_0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in $\mathbb C^n$ for $n>1$ whose projective closure $\overline E\subset\mathbb C\mathbb P^n$ avoids a hyperplane $Λ\subset\mathbb C\mathbb P^n$ and is polynomially convex in $\mathbb C\mathbb p^n \setMinusλ\ cong \ mathbb c^n $，然后$ \ mathbb c^n \ setMinus e $是OKA域。
In this paper we find surprisingly small Oka domains in Euclidean spaces $\mathbb C^n$ of dimension $n>1$ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set $E$ in $\mathbb C^n$ we show that $\mathbb C^n\setminus E$ is an Oka domain. In particular, there are Oka domains which are only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives families of smooth real hypersurfaces $Σ_t\subset \mathbb C^n$ $(t\in\mathbb R)$ dividing $\mathbb C^n$ in an unbounded hyperbolic domain and an Oka domain such that at the threshold value $t=0$ the hypersurface $Σ_0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in $\mathbb C^n$ for $n>1$ whose projective closure $\overline E\subset\mathbb C\mathbb P^n$ avoids a hyperplane $Λ\subset\mathbb C\mathbb P^n$ and is polynomially convex in $\mathbb C\mathbb P^n\setminus Λ\cong\mathbb C^n$, then $\mathbb C^n\setminus E$ is an Oka domain.
