Floer cohomology of the Chiang Lagrangian

TL;DR

This paper investigates holomorphic discs bounded by a specific Lagrangian in complex projective space, establishing conditions under which its Floer cohomology is non-zero, revealing deep algebraic and geometric properties.

Contribution

It provides new transversality and classification results for holomorphic discs on Lagrangians with group symmetries, applied to a Chiang Lagrangian in CP^3, and determines Floer cohomology conditions.

Findings

Floer cohomology of the Chiang Lagrangian is non-zero only in characteristic 5.

The Chiang Lagrangian generates the split-closed derived Fukaya category.

Classification results for holomorphic discs with boundary on symmetric Lagrangians.

Abstract

We study holomorphic discs with boundary on a Lagrangian submanifold $L$ in a Kaehler manifold admitting a Hamiltonian action of a group $K$ which has $L$ as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in $\mathbf{CP}^3$ first noticed by Chiang. We prove that this Lagrangian has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5, in which case it generates the split-closed derived Fukaya category as a triangulated category.