Mathematical Research Letters

Volume 27 (2020)

Number 6

Levi–Kähler reduction of CR structures, products of spheres, and toric geometry

Pages: 1565 – 1629



Vestislav Apostolov (Département de Mathématiques, Université du Québec à Montréal, Québec, Canada)

David M. J. Calderbank (Department of Mathematical Sciences, University of Bath, United Kingdom)

Paul Gauduchon (Centre de Mathématiques, École Polytechnique, Palaiseau, France)

Eveline Legendre (Université Paul Sabatier, Institut de Mathématiques de Toulouse, France)


We introduce a process, which we call Levi–Kähler reduction, for constructing Kähler manifolds and orbifolds from CR manifolds (of arbitrary codimension) with a transverse torus action. Most of the paper is devoted to the study of Levi–Kähler reductions of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of the orbifolds obtained by such reductions, and find that the Levi–Kähler reductions of products of $3$-spheres are extremal in a weighted sense introduced by G. Maschler and the first author [11], and further studied by A. Futaki and H. Ono [34].

V.A. was supported in part by an NSERC discovery grant. E.L. was partially supported by France ANR project EMARKS No ANR-14-CE25-0010. The authors are grateful to the London Mathematical Society, the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, and the Labex CIMI (Toulouse) for financial support. We are very grateful to the referee for the careful reading and the constructive suggestions which improved the manuscript.

Received 11 March 2020

Accepted 5 August 2020

Published 17 February 2021