Displacing Lagrangian toric fibers by extended probes

TL;DR

This paper introduces extended probes, a new method to displace Lagrangian fibers in toric symplectic manifolds, expanding the ability to analyze fiber displaceability beyond previous techniques.

Contribution

It generalizes McDuff's probes by using deflections with auxiliary probes, enabling displacement of more fibers in Hirzebruch surfaces and weighted projective spaces.

Findings

Displaced all fibers in Hirzebruch surfaces except known nondisplaceable ones.

Displaced an open dense set of fibers in P(1,3,5) after resolving singularities.

Identified many cases with unknown displaceability status.

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 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.