A compact symmetric symplectic non-kaehler manifold

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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 shows that not all compact symplectic manifolds that admit non-trivial Hamiltonian torus actions are Kaehler. Similar techniques allow us to construct a compact symplectic manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.

Original languageEnglish (US)
Pages (from-to)587-590
Number of pages4
JournalMathematical Research Letters
Issue number5
StatePublished - Jan 1 1996

ASJC Scopus subject areas

  • Mathematics(all)


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