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

Part of: Homology and cohomology theories Symplectic geometry, contact geometry Differential topology Complex manifolds

Published online by Cambridge University Press:  24 August 2023

Oliver Goertsches ,
Panagiotis Konstantis  and
Leopold Zoller
Oliver Goertsches
Affiliation:
Philipps-Universität Marburg, Hans-Meerwein-Straße 6, D-35043 Marburg, Germany goertsch@mathematik.uni-marburg.de
Panagiotis Konstantis
Affiliation:
Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße 6, D-35043 Marburg, Germany pako@mathematik.uni-marburg.de
Leopold Zoller
Affiliation:
Ludwig-Maximilians-Universität München, Theresienstr. 39, D-80333 München, Germany leopold.zoller@mathematik.uni-muenchen.de
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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.

Keywords

equivariant topologyGKM fibrationsrealization of GKM graphscohomogeneity one manifoldsHamiltonian non-Kähler actionsequivariant geometric structuresequivariant obstruction theory

MSC classification

Primary: 57R91: Equivariant algebraic topology of manifolds 55N91: Equivariant homology and cohomology 53D05: Symplectic manifolds, general 32Q15: Kähler manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 10 , October 2023 , pp. 2149 - 2190
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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