Regular immersions directed by algebraically elliptic cones

TL;DR

This work develops algebraic Runge approximation with interpolation for regular immersions directed by punctured algebraic cones, replacing the Oka condition with algebraic ellipticity (guaranteed when Y is uniformly rational). It establishes a homotopy-theoretic criterion that is necessary and sufficient for simultaneous approximation and interpolation of A-immersions from smooth affine curves into $\mathbb{C}^n$, and proves this criterion in broad classes of cones, including the large cone and the punctured null quadric. Central to the approach is the construction of regular sections of the A-bundle with prescribed periods and poles, enabled by an algebraic analogue of Forstnerič’s section-theoretic framework and an ample supply of holomorphic sprays. The results also extend to directed harmonic maps, yielding approximations by regular harmonic A-maps with prescribed flux, and recover known minimal-surface phenomena in the null-quadric setting. Overall, the paper provides a robust algebraic-ellipticity toolkit for directed regular immersions and associated harmonic maps with interpolation and period control.

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.

Theorems & Definitions (37)

• Theorem 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Proposition 1.5
• Example 1.6
• Corollary 1.7
• Corollary 1.8
• Theorem 1.10
• proof : Proof of Proposition \ref{['pro:sufficient']}