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,** 2009. Apparently this is an open problem.


**PS,** 2019. Not anymore! 



Hamiltonian $S^1$ actions with isolated fixed points
Nick Lindsay and have just proved that such a manifold indeed exists. And surprise, surprise, this is Tolman's manifold. See Theorem 1.3 and Corollary 1.4 of our paper: <https://arxiv.org/abs/1912.02785>


