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

TL;DR

This paper investigates the local structure of real submanifolds with maximum complex tangent space at CR singularities, establishing conditions for holomorphic equivalence to normal forms and analyzing divergence issues.

Contribution

It introduces new normal form results for real submanifolds at CR singularities, including conditions for holomorphic equivalence and divergence of formal normal forms.

Findings

Normal forms can be divergent in general.

Holomorphic equivalence to normal forms is achieved under abelian and small divisors conditions.

Existence of complex submanifolds intersecting real submanifolds transversally at CR singularities.

Abstract

We study a germ of real analytic $n$-dimensional submanifold of ${\mathbf C}^n$ 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 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. If the singularity is {\it abelian}, we show, under some assumptions on the linear part of $\sigma$ at the singularity, that the real submanifold is holomorphically equivalent to an analytic normal form. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we prove that, in general, there exists a complex submanifold of positive dimension in ${\mathbf C}^n$ that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a CR singularity of the {\it complex type}.