Displacing Lagrangian toric fibers by extended probes

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff’s original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space $ℙ (1, 3, 5)$ after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

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Keywords

symplectic manifolds, Lagrangian tori, nondisplaceable, toric manifolds

Mathematical Subject Classification 2010

Primary: 53D12

Secondary: 14M25, 53D40

References

Publication

Received: 29 May 2012

Accepted: 28 August 2013

Published: 30 January 2014

Authors

Miguel Abreu
Centro de Análise Mathemática, Geometria e Sistemas Dinâmicos Departamento de Mathemática Instituto Superior Técnico 1049-001 Lisboa Portugal

Matthew Strom Borman
Department of Mathematics University of Chicago Chicago, IL 60637 USA

Dusa McDuff
Mathematics Department Barnard College and Columbia University New York, NY 10027 USA

Volume 14, issue 2 (2014)

Miguel Abreu, Matthew Strom Borman and Dusa McDuff

Abstract