Realization of GKM fibrations and new examples of Hamiltonian non-K\"ahler actions

TL;DR

This paper classifies certain GKM graph fibrations and constructs new examples of 6-dimensional Hamiltonian manifolds that are K"ahler but admit non-K"ahler actions, expanding the understanding of symplectic and complex structures.

Contribution

It classifies GKM fibrations and constructs infinitely many 6D K"ahler manifolds with Hamiltonian non-K"ahler actions, providing new examples with prescribed fixed points.

Findings

All fiberwise signed fibrations are realized as projectivizations over quasitoric 4-folds or S^4.

Existence of invariant almost complex, symplectic, and K"ahler structures on total spaces.

Construction of infinitely many 6D K"ahler manifolds with Hamiltonian non-K"ahler actions.

Abstract

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