Lagrangian Floer theory in symplectic fibrations

Abstract

Consider a fibration of symplectic manifolds, with an induced fibration of Lagrangians. We develop a new version of Lagrangian Floer theory that is well defined when the fiber Lagrangian is monotone and the base is rational and unobstructed. Then, we write down a Leray-Serre type spectral sequence that computes the Floer cohomology of the total Lagrangian from the Floer complexes of the base and fiber. We use this theory to discover fibered Floer-non-trivial tori in complex flag manifolds and ruled projective surfaces.