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DISPLACING LAGRANGIAN TORIC FIBERS BY EXTENDED MIGUEL ABREU, MATTHEW STROM BORMAN, AND DUSA MCDUFF
 

Summary: DISPLACING LAGRANGIAN TORIC FIBERS BY EXTENDED
PROBES
MIGUEL ABREU, MATTHEW STROM BORMAN, AND DUSA MCDUFF
Abstract. 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. Ex-
tended 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.
1. Introduction and main results 2
1.1. Introduction 2
1.2. Main results 3
2. Symmetric extended probes 6
2.1. Moment polytopes and integral affine geometry 6
2.2. Probes and extended probes 8
2.3. Hirzebruch surfaces 10
3. Extended probes with flags: parallel case 12
3.1. The definition and the displaceability method 13

  

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

 

Collections: Mathematics