# Hamiltonian S^1 actions with isolated fixed points

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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wiki'd, per policy for known open problems. –  Scott Morrison Nov 16 '09 at 21:56

Sue Tolman has shown that there are non-Kahler 6-dimensional manifolds admitting Hamiltonian 2-torus actions all of whose fixed points are isolated ( http://arxiv.org/abs/dg-ga/9511007 ) . This might be a good first place to look to see if one of the components of the action satisfy the criteria you want.

Interestingly, Karshon has shown ( in dg-ga/9510004 [sorry, I can only post one link as a new user]) that if a 4-manifold admits a Hamiltonian circle action, it must be Kahler.

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I now this article. But surprisingly, as far as I understand, you can not just take a subgroup S^1 in this example to get something that does not admit a compatible Kahler metric. At least this is what I understood from Dusa McDuff. Apperently you do need a T^2 in this example... –  Dmitri Oct 24 '09 at 18:49
Surprising and interesting! Perhaps the next thing to do is do a literature search for papers that cite Tolman's paper. –  C. Lee Oct 24 '09 at 18:56
Isn't it toric-action? –  Ilya Nikokoshev Oct 24 '09 at 18:17
Actually, may be circle-action or localization? –  Ilya Nikokoshev Oct 24 '09 at 18:18