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

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 a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and 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.