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).

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Susan Tolman's paper arxiv.org/abs/0903.4918 also considers the 6-dimensional case. From the argument (Remark 2.11) with the $\chi_y$-characteristic one should be able to deduce the other Chern classes of your putative example. –  Martin O Dec 15 '10 at 21:49

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...

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