Elliptic complements of cubic hypersurfaces

Abstract

Let $D\subset {\mathbb{P}}^n$, $n⩾2$, 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 ${\mathbb{P}}^4$.

28 pages