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

Lerman has constructed such an example see , well it has an isolated fixed point but also some fixed connected submanifolds, so maybe not exactly what you want.

The construction has been later revisited by Kim

(note to editors: I guess the creation of 'torus-action' tag migh be useful.)

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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
It's definitely 'torus-action', see e.g. this famous book by Audin That said we might also need 'toric-variety' or 'toric-geometry' at some point, but that's something else. – Thomas Sauvaget Oct 24 '09 at 18:22
Thanks a lot for both refferences! But still I want an action such that all fixed points are isolated. Though indeed it is nice to know that there are non-Kahler examples with at least one isolated point. – Dmitri Oct 24 '09 at 18:28

Sue Tolman has shown that there are non-Kahler 6-dimensional manifolds admitting Hamiltonian 2-torus actions all of whose fixed points are isolated ( ) . 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

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