Floer cohomology of torus fibers and real lagrangians in Fano toric manifolds

TL;DR

This paper studies Floer cohomology between torus fibers and real Lagrangians in Fano toric manifolds, providing conditions for its definition and applications to non-displaceability.

Contribution

It offers a combinatorial description of Floer complexes and links non-zero Floer cohomology of fibers to that of pairs, advancing understanding of Lagrangian intersections.

Findings

Floer cohomology is non-zero under certain conditions.

Non-displaceability of Lagrangians is established.

Applications to intersection number bounds are demonstrated.

Abstract

In this article, we consider the Floer cohomology (with $\Z_2$ coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. We show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. We use this result to develop some applications to non-displaceability and the minimum number of intersection points under Hamiltonian isotopy.