|
3
1
|
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). |
|||
|
|
3
|
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: http://mathoverflow.net/questions/2323/hamiltonian-s1-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: http://mathoverflow.net/questions/12263/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... |
||
|
|
