# A Leray-Serre spectral sequence for Lagrangian Floer theory

## TL;DR

This paper develops a spectral sequence framework for computing Floer cohomology of fibered Lagrangians in symplectic fibrations, revealing vanishing results and applications to Gelfand-Cetlin fibers.

## Contribution

It introduces a Leray-Serre spectral sequence for Lagrangian Floer theory in symplectic fibrations, linking fiber and total Floer cohomology.

## Key findings

- Vanishing Floer cohomology of fiber Lagrangians implies vanishing for total Lagrangians.
- Application to Gelfand-Cetlin fibers shows their Floer cohomology vanishes.
- Provides computational tools for Floer cohomology in fibered symplectic manifolds.

## Abstract

We consider symplectic fibrations as in Guillemin-Lerman-Sternberg, and derive a spectral sequence to compute the Floer cohomology of certain fibered Lagrangians sitting inside a compact symplectic fibration with small monotone fibers and a rational base. We show if the Floer cohomology with field coefficients of the fiber Lagrangian vanishes, then the Floer cohomology with field coefficients of the total Lagrangian also vanishes. We give an application to certain non-torus fibers of the Gelfand-Cetlin system in Flag manifolds, and show that their Floer cohomology vanishes.