Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions

Abstract

We classify fibrations of abstract $3$ -regular GKM graphs over $2$ -regular ones, and show that all fibrations satisfying the known necessary conditions for realizability are, in fact, realized as the projectivization of equivariant complex rank- $2$ vector bundles over quasitoric $4$ -manifolds or $S^4$ . We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures on the total space. In this way, we obtain infinitely many Kähler manifolds with Hamiltonian non-Kähler actions in dimension $6$ with prescribed one-skeleton, in particular with a prescribed number of isolated fixed points.