GKM theory and Hamiltonian non-Kähler actions in dimension 6

Abstract

Using the classification of 6-dimensional manifolds by Wall, Jupp and Žubr, we observe that the diffeomorphism type of simply-connected, compact 6-dimensional integer GKM T2-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-Kähler T2-actions with finite fixed point set are diffeomorphic to Eschenburg's twisted flag manifold SU(3)//T2. In particular, they admit a noninvariant Kähler structure.