# 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

Lerman has constructed such an example see http://arxiv.org/abs/dg-ga/9601012v1 , 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 http://www.mathjournals.org/mrl/2002-009-004/2002-009-004-002.pdf

(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 books.google.fr/books?id=uKXO9BDsk_kC 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