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!