# GKM theory and Hamiltonian non-K\"ahler actions in dimension $6$

## TL;DR

This paper classifies 6-dimensional GKM manifolds using their GKM graphs and shows that certain Hamiltonian non-K"ahler actions are diffeomorphic to a known twisted flag manifold, revealing their geometric structure.

## Contribution

It demonstrates that the diffeomorphism type of simply-connected 6D GKM T^2-manifolds is determined by their GKM graph and identifies specific Hamiltonian non-K"ahler actions with a twisted flag manifold.

## Key findings

- GKM graph encodes the diffeomorphism type of 6D GKM T^2-manifolds
- Hamiltonian non-K"ahler actions by Tolman and Woodward are diffeomorphic to Eschenburg's twisted flag manifold
- These manifolds admit a noninvariant K"ahler structure

## Abstract

Using the classification of $6$-dimensional manifolds by Wall, Jupp and \v{Z}ubr, we observe that the diffeomorphism type of simply-connected, compact $6$-dimensional integer GKM $T^2$-manifolds is encoded in their GKM graph. As an application, we show that the $6$-dimensional manifolds on which Tolman and Woodward constructed Hamiltonian, non-K\"ahler $T^2$-actions with finite fixed point set are both diffeomorphic to Eschenburg's twisted flag manifold $SU(3)//T^2$. In particular, they admit a noninvariant K\"ahler structure.