Equivalences of real submanifolds in complex space

TL;DR

This paper proves that, outside a certain subvariety, formal equivalence of real-analytic submanifolds in complex space coincides with biholomorphic equivalence, with extensions to k-equivalences.

Contribution

It establishes a precise condition under which formal and biholomorphic equivalences are equivalent for real-analytic submanifolds, including more general k-equivalence results.

Findings

Formal and biholomorphic equivalences coincide outside a proper subvariety.

Results apply to germs of submanifolds at generic points.

Extensions to k-equivalences are provided.

Abstract

We show that for any real-analytic submanifold M in C^N there is a proper real-analytic subvariety V contained in M such that for any point p in M\V, any real-analytic submanifold M' in C^N, and any point p' in M', the germs of the submanifolds M and M' at p and p' respectively are formally equivalent if and only if they are biholomorphically equivalent. More general results for k-equivalences are also stated and proved.