Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

TL;DR

This paper demonstrates that non-displaceability of certain Lagrangian submanifolds can be established using Floer cohomology with non-unitary line bundles, extending previous results to all toric Fano manifolds.

Contribution

It introduces a method using non-unitary line bundles in Floer cohomology to prove non-displaceability, broadening the class of toric Fano manifolds studied.

Findings

Non-displaceability proven for all monotone Lagrangian torus fibers in toric Fano manifolds.

Extended Floer cohomology computations to all toric Fano manifolds, removing convexity restrictions.

Established non-displaceability for some non-monotone Lagrangian torus fibers.

Abstract

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states), some non-monotone Lagrangian torus fibers. We also extend the results by Oh and the author about the computations of Floer cohomology of Lagrangian torus fibers to the case of all toric Fano manifolds, removing the convexity assumption in the previous work.