Real submanifolds of maximum complex tangent space at a CR singular point, II

TL;DR

This paper classifies and normalizes quadratic CR singularities of real submanifolds in complex space with maximal complex tangent space, revealing divergence in normal forms and establishing a unique formal normal form under certain conditions.

Contribution

It provides a classification of quadratic CR singularities and constructs a unique formal normal form, highlighting divergence phenomena in the normalization process.

Findings

Normal forms can be divergent.

A unique formal normal form exists under non-degeneracy.

Classification depends on deck transformations and complexification.

Abstract

We study a germ of real analytic n-dimensional submanifold of $C^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck transformations, we first classify holomorphically its quadratic CR singularity. We then study its transformation to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We first conjugate formally its associated reversible map $\sigma$ to suitable normal forms and show that all these normal forms can be divergent. We then construct a unique formal normal form under a non degeneracy condition.