Regular immersions directed by algebraically elliptic cones

Abstract

Let $M$ be an open Riemann surface and $A$ be the punctured cone in ${\mathbb{C}}^n\setminus \{0\}$ on a smooth projective variety $Y$ in ${\mathbb{P}}^{n-1}$. Recently, Runge approximation theorems with interpolation for holomorphic immersions $M\to {\mathbb{C}}^n$, directed by $A$, have been proved under the assumption that $A$ is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by $A$ from a smooth affine curve $M$ into ${\mathbb{C}}^n$. The Oka property is naturally replaced by the stronger assumption that $A$ is algebraically elliptic, which it is if $Y$ is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.

To appear in J. Reine Angew. Math. (Crelle's J.)