Mathematical Research Letters

Volume 24 (2017)

Number 2

Convexity for Hamiltonian torus actions on $b$-symplectic manifolds

Pages: 363 – 377



Victor Guillemin (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Eva Miranda (Department of Mathematics, Universitat Politècnica de Catalunya, Spain; and Barcelona Graduate School of Mathematics, Barcelona, Spain)

Ana Rita Pires (Department of Mathematics, Fordham University, New York, N.Y., U.S.A.)

Geoffrey Scott (Department of Mathematics, University of Toronto, Ontario, Canada)


In [GMPS] we proved that the moment map image of a b-symplectic toric manifold is a convex $b$-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on $b$-symplectic manifolds. The modular weights of the action on the connected components of the exceptional hypersurface play a fundamental role: either they are all zero and the moment map behaves as in classic symplectic case, or they are all nonzero and the moment map behaves as in the toric $b$-symplectic case studied in [GMPS].

Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize, by the Ministerio de Economía y Competitividad project with reference MTM2015-69135-P/FEDER and by the Generalitat de Catalunya project with reference 2014SGR634.

Ana Rita Pires was partially supported by a Short Visit Grant from the European Science Foundation’s “Contact and Symplectic Topology” network, and by the National Science Foundation under agreement number DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Received 9 December 2014

Accepted 10 May 2016

Published 24 July 2017