On formal maps between generic submanifolds in complex space

TL;DR

This paper proves the convergence of formal maps between real-analytic generic submanifolds in complex space under certain conditions, extending known results and providing new approximation theorems.

Contribution

It establishes convergence criteria for formal maps between generic submanifolds, including hypersurfaces, and introduces a new Artin type approximation theorem.

Findings

Convergence of formal maps between minimal and holomorphically nondegenerate submanifolds.

New convergence results for formal maps between hypersurfaces without holomorphic curves.

A novel Artin type approximation theorem for formal maps of full rank.

Abstract

Let H:(M,p)->(M',p') be a formal mapping between two germs of real-analytic generic submanifolds in C^N with nonvanishing Jacobian. Assuming M to be minimal at p and M' holomorphically nondegenerate at p', we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M' are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank.