GKM theory and Hamiltonian non-Kähler actions in dimension 6

Abstract

Using the classification of 6-dimensional manifolds by Wall, Jupp andŽubr, 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-Kähler 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 Kähler structure.

Symplectic and Kähler structures on $$\mathbb CP^1$$-bundles over $$\mathbb CP^2$$

“…The following theorem was proven in [16]. Namely, it was proven in [16] that Tolman's manifold is diffeomorphic to Eschenburg's manifold.…”

“…The following theorem was proven in [16]. Namely, it was proven in [16] that Tolman's manifold is diffeomorphic to Eschenburg's manifold. On the other hand, Eschenburg's manifold is diffeomorphic to the projectivisation of a rank 2 bundle over CP 2 [12,Theorem 2].…”

“…This permits one to apply a theorem of Jupp to get a diffeomorphism between M T and the projectivization of a rank two bundle E over CP 2 . Our proof is similar to [16], but the approach to calculating the invariants is different. As for the cohomology classes of the forms ω λ 1 ,λ 2 , they are calculated by evaluating them at the collection of 9 spheres in M T which are T 2 -invariant.…”

“…In [31] Tolman constructed a remarkable family of symplectic forms ω λ 1 ,λ 2 on a compact 6-manifold M T , that have a Hamiltonian T 2 -action, and yet don't admit a T 2invariant compatible Kähler metric. 1 Recently, in [16,17] Goertsches, Konstantis, and Zoller proved that Tolman's manifold M T is diffeomorphic to the projectivisation P(E) of a complex rank two bundle E over CP 2 , and hence it admits some Kähler structure.…”

Remark on the Betti numbers for Hamiltonian circle actions

“…Tolman [17] and Woodward [19] constructed a six-dimensional closed Hamiltonian T 2 -manifold with only isolated fixed points and with no T 2 -invariant Kähler metric. Surprisingly Goertsches-Kostantis-Zoller [10] have recently shown that the examples of Tolman and Woodward indeed admit Kähler metrics that are not T 2 -invariant. Thus their result provides a positive evidence for the conjecture of the existence of Kähler metrics.…”