Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

TL;DR

This paper explores the algebraic analogue of Oka theory for regular maps from affine varieties to algebraic manifolds, focusing on approximation, interpolation, and the algebraic Oka property, with new examples and limitations.

Contribution

It introduces the algebraic Oka property, compares it with the analytic version, and provides new examples such as smooth nondegenerate toric varieties, while also showing certain properties fail for compact and projective manifolds.

Findings

Algebraic Oka property is stronger than the analytic one.

Smooth nondegenerate toric varieties satisfy the algebraic Oka property.

Certain algebraic properties fail for all compact and projective manifolds.

Abstract

We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.