An implicit function theorem for sprays and applications to Oka theory

TL;DR

This paper introduces an implicit function theorem for sprays to address core problems in Oka theory, providing elementary proofs of approximation-interpolation equivalence and establishing Oka properties of blowups.

Contribution

It develops a novel implicit function theorem for sprays and applies it to prove key results in Oka theory, including approximation-interpolation equivalence and Oka properties of blowups.

Findings

Elementary proof that approximation yields interpolation in Oka theory

Blowup of an algebraically Oka manifold along a smooth algebraic center remains Oka

Characterization of equivariantly Oka manifolds via Gromov's condition Ell_1

Abstract

We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and L\'{a}russon's elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov's condition $\mathrm{Ell}_{1}$, and the equivariant localization principle is also given.