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Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 10 / October 2023
- Published online by Cambridge University Press:
- 24 August 2023, pp. 2149-2190
- Print publication:
- October 2023
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- Article
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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.
