Post Made Community Wiki by Scott Morrison
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I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian S^1 action with isolated fixed points, that does not admit a compatible S^1 invariant Kahler strucutre? One would say, of course there should be such an example. But I have not seen any...

Added. Apperently this is an open problem.

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I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian S^1 action with isolated fixed points, that is not a Kahler manifold (I.e. it does not admit a compatible S^1 invariant Kahler strucutre)strucutre? One would say, of course there should be such an example. But I have not seen any...

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