Year: 2026
Type: preprint
Abstract: Let $D\subset\mathbb{P}^n$, $n\geqslant2$, be an arbitrary cubic hypersurface, and let $D_{\mathrm{red}}$ denote its reduced support. We prove that $\mathbb{P}^n\setminus D$ is holomorphically elliptic, and hence Oka, unless $D_{\mathrm{red}}$ is the union of three distinct hyperplanes containing a common codimension-two linear subspace. In the exceptional case, $\mathbb{P}^n\setminus D\cong(\mathbb{C}\setminus\{0,1\})\times\mathbb{C}^{n-1}$, so the complement is not Oka. As applications, we prove that, for every elliptic curve $E$, the space of degree-three holomorphic maps $E\to\mathbb{P}^1$, and the space of degree-three holomorphic self-maps of $\mathbb{P}^1$, are both holomorphically elliptic, and hence Oka. The second application is connected with the classification through an irreducible cubic hypersurface in $\mathbb{P}^4$.
Source: arXiv (Cornell University)
Author: Xie, Song-Yan
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1603846912, G7519385794, G8533620077
OpenAlex ID: W7172378003