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

TL;DR

This paper investigates the local structure of real analytic submanifolds in complex space with maximal complex tangent space at CR singularities, establishing normal forms, equivalence conditions, and intersection properties under certain assumptions.

Contribution

It provides a normal form classification for such submanifolds, proves holomorphic equivalence under small divisors conditions, and explores intersection phenomena at CR singularities.

Findings

Normal form classification for real submanifolds with maximal complex tangent space.

Holomorphic equivalence to quadrics under small divisors condition.

Existence of complex submanifolds intersecting real submanifolds transversally.

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 some assumptions , we show its equivalence to a normal form under a local biholomorphism at the singularity. 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 investigate the existence of a complex submanifold of positive dimension in C n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non-isolated CR singularity.