A compact symmetric symplectic non-Kaehler manifold

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 $2n$-dimensional symplectic manifolds with Hamiltonian action of $n$-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 manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.