Levi-Kahler reduction of CR structures, products of spheres, and toric geometry

TL;DR

This paper introduces the Levi-Kahler quotient process to construct Kähler metrics from CR structures, focusing on toric CR manifolds and products of spheres, and identifies extremal quotients in this setting.

Contribution

It develops the Levi-Kahler quotient method for CR structures with torus actions, providing explicit descriptions and extremal examples for products of spheres.

Findings

Explicit descriptions of Levi-Kahler quotients of toric CR manifolds

Characterization of extremal Levi-Kahler quotients of products of 3-spheres

Introduction of a weighted extremality concept for these quotients

Abstract

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.