Hamiltonian S^1 8-dim manifold with minimal number of fixed points Let M be a compact symplectic manifold of dimension 8, acted on by S^1, with isolated fixed points, and such that the Betti numbers are the same as the Betti numbers of CP^4.
Let "c1" be the first Chern class of the tangent bundle, "x" the generator of H^2(M;Z) and "k" an integer such that c1=kx.
Is there an explicit example of M when k=1?
(E.g.: for k=5 one has CP^4).
 A: I am pretty sure that the answer to this question is unknown. And I would guess it should be hard (via impossible) to construct such a manifold.
Here is some argumentation:
In dimension $6$ the classification of symplectic manifolds with same 
homology as $\mathbb CP^3$ admitting a Hamiltonian $S^1$ action with isolated fixed points was  done by McDuff http://arxiv.org/abs/0808.3549 .
After the classification is done McDuff notices that all the (four) examples are in fact three-dimensional algebraic Fano varieties (i.e., the symplectic structure comes from a Kahler form). 
On the other hand, as far as I understand for the moment (as ridiculous as it sounds) there is no single example of a  symplectic manifold admitting a Hamiltonian $S^1$ action with isolated fixed points, that is known to be non-algebraic (see for example: Hamiltonian S^1 actions with isolated fixed points)
If we would now indeed try to look for an 8 dimensional example different from $\mathbb CP^4$ that is additionally algebraic, we would need to look for such a Fano four fold. BUT, here the situation is as follows: in all even dimensions $\mathbb CP^{2n}$ are the only known (as for today) Fano varieties with $H^{2k}=\mathbb Z$, $H^{2k-1}=0$. It is true though, that $4$-dimensional Fano varieties are not yet classified, contrary to $3$-dimensional ones.  
If on the other hand by any miracle we will find some non-algebraic example, this will answer to the following question, which I am sure is still open at the moment: Compact Symplectic Fano (strongly monotone) manfiolds
I guess the surest way to answer this question will be to try to see if McDuff's classification can be generalised to this dimension... 
