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Displacing Lagrangian toric fibers by extended probes

Miguel Abreu, Matthew Strom Borman and Dusa McDuff

Algebraic & Geometric Topology 14 (2014) 687–752

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 (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.

symplectic manifolds, Lagrangian tori, nondisplaceable, toric manifolds
Mathematical Subject Classification 2010
Primary: 53D12
Secondary: 14M25, 53D40
Received: 29 May 2012
Accepted: 28 August 2013
Published: 30 January 2014
Miguel Abreu
Centro de Análise Mathemática, Geometria e Sistemas Dinâmicos
Departamento de Mathemática
Instituto Superior Técnico
1049-001 Lisboa
Matthew Strom Borman
Department of Mathematics
University of Chicago
Chicago, IL 60637
Dusa McDuff
Mathematics Department
Barnard College and Columbia University
New York, NY 10027