A compact symmetric symplectic non-Kaehler manifold
Eugene Lerman (University of Illinois at Urban-Champaign)

TL;DR
This paper constructs a specific compact symplectic manifold with a Hamiltonian circle action that is non-Kaehler, challenging previous classification results that all such manifolds are Kaehler.
Contribution
It provides an explicit example of a compact symplectic manifold with a Hamiltonian circle action that is non-Kaehler, expanding the understanding of symplectic geometry beyond known classifications.
Findings
Constructed a non-Kaehler symplectic manifold with Hamiltonian circle action.
Showed not all symplectic manifolds with Hamiltonian torus actions are Kaehler.
Demonstrated existence of manifolds with no invariant complex structures or polarizations.
Abstract
In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of classification of Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon classification of Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of such manifolds are Kaehler. Delzant's classification of -dimensional symplectic manifolds with Hamiltonian action of -dimensional tori showed that all such manifolds are projective toric varieties, hence Kaehler. An example in this paper show that not all compact symplectic manifolds that admit Hamiltonian torus actions are Kaehler. Similar technique allows us to construct a compact symplectic…
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometry and complex manifolds · Advanced Algebra and Geometry
