Deformations of CR maps and applications

TL;DR

This paper investigates the deformation theory of CR maps between real submanifolds in complex spaces, analyzing the structure and singularities of the mapping locus where such maps exist.

Contribution

It characterizes the structure of the mapping locus for CR maps, showing it is semi-analytic and providing examples with specific singularities.

Findings

The mapping locus is semi-analytic.

Examples of the locus with prescribed singularities.

Structural properties of deformations of CR maps.

Abstract

We study the deformation theory of CR maps in the positive codimensional case. In particular, we study structural properties of the {\em mapping locus} $E$ of (germs of nondegenerate) holomorphic maps $H \colon (M,p) \to M'$ between generic real submanifolds $M \subset \mathbb C^N$ and $M' \subset \mathbb C^{N'}$, defined to be the set of points $p' \in M'$ which admit such a map with $H(p) = p'$. We show that this set $E$ is semi-analytic and provide examples for which $E$ posseses (prescribed) singularities.