Regular immersions directed by algebraically elliptic cones

2023/12/05 by Alarcon, Antonio, Larusson, Finnur · 1 citation
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.02795

Abstract

Let M be an open Riemann surface and A be the punctured cone in ℂⁿ∖\0\ on a smooth projective variety Y in ℙ^n-1. Recently, Runge approximation theorems with interpolation for holomorphic immersions M→ℂⁿ, 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 ℂⁿ. 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.

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• From Stein manifolds to Oka manifolds: the h-principle in complex analysis

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