Vari\'et\'es CR polaris\'ees et G-polaris\'ees, partie I

TL;DR

This paper explores the complex geometry of polarized and G-polarized CR manifolds, focusing on their structural properties and deformations, revealing the rich interplay between foliation and CR structures.

Contribution

It introduces a detailed study of polarized and G-polarized CR manifolds, emphasizing their deformation theory and structural properties.

Findings

Analysis of the interplay between foliation and CR structures.

Characterization of deformation spaces of polarized CR manifolds.

Insights into the geometric richness of G-polarized CR structures.

Abstract

Polarized and $G$-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\Cal F$ (given by the action of a Lie group $G$ in the $G$-polarized case) and a transverse CR distribution $(E,J)$. Polarized means that $(E,J)$ is roughly speaking invariant by $\Cal F$. Both structures are therefore linked up. The interplay between them gives to polarized CR-manifolds a very rich geometry. In this paper, we study the properties of polarized and $G$-polarized manifolds, putting special emphasis on their deformations.