Journal of Symplectic Geometry

Volume 17 (2019)

Number 5

Presymplectic convexity and (ir)rational polytopes

Pages: 1479 – 1511



Tudor Ratiu (School of Math. Sci. and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China; Section de Mathématiques, Université de Genève, Switzerland; and Ecole Polytechnique Fédérale de Lausanne, Switzerland)

Nguyen Tien Zung (School of Mathematics, Shanghai Jiao Tong University, Shanghai, China; and Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France)


In this paper, we extend the Atiyah–Guillemin–Sternberg convexity theorem and Delzant’s classification of symplectic toric manifolds to presymplectic manifolds. We also define and study the Morita equivalence of presymplectic toric manifolds and of their corresponding framed momentum polytopes, which may be rational or non-rational. Toric orbifolds [16], quasifolds [3] and noncommutative toric varieties [14] may be viewed as the quotient of our presymplectic toric manifolds by the kernel isotropy foliation of the presymplectic form.

The first author was partially supported by the National Natural Science Foundation of China grant number 11871334 and by NCCR SwissMAP grant of the Swiss National Science Foundation.

Received 4 June 2017

Accepted 5 September 2018

Published 20 November 2019