Regular immersions directed by algebraically elliptic cones

TL;DR

This paper extends approximation and interpolation results for holomorphic immersions directed by algebraically elliptic cones from the complex to the algebraic setting, using homotopy-theoretic conditions.

Contribution

It establishes algebraic approximation and interpolation theorems for regular immersions directed by algebraically elliptic cones, replacing the Oka property with algebraic ellipticity.

Findings

Homotopy-theoretic conditions characterize approximation and interpolation.

Many cases of algebraically elliptic cones satisfy these conditions.

Results apply to smooth affine curves into complex affine space.

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.