Displacing Lagrangian toric fibers by extended probes

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. 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 $\mathbb{P}\left(1,3,5\right)$ 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.