Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds

TL;DR

This paper proves that certain 6-dimensional symplectic manifolds with Hamiltonian torus actions and specific GKM-graph properties satisfy the hard Lefschetz property, linking symplectic topology and combinatorial graph conditions.

Contribution

It establishes the hard Lefschetz property for 6D GKM manifolds with index-increasing moment map images under Hamiltonian torus actions.

Findings

Hard Lefschetz property holds for specified GKM manifolds.

GKM-graph index-increasing condition is crucial.

Results connect symplectic geometry with combinatorial graph structures.

Abstract

In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,\omega)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will show that if the moment map image of $M$ is a GKM-graph and if the graph is index-increasing, then $(M,\omega)$ satisfies the hard Lefschetz property.