Detecting non-displaceable toric fibers on compact toric manifolds via tropicalizations

Abstract

We provide a combinatorial way to locate non-displaceable Lagrangian toric fibers on any compact toric manifold. By taking the intersection of certain tropicalizations coming from its moment polytope, one can detect all Lagrangian toric fibers having non-vanishing Floer cohomology ([K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151(1) (2010) 23–174; K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Selecta Math. (N.S.) 17(3) (2011) 609–711.]). The intersection completely characterizes all non-displaceable toric fibers, in some cases including pseudo symmetric smooth Fano varieties ([G. Ewald, On the classification of toric Fano varieties, Discrete Comput. Geom. 3(1) (1988) 49–54.]).