Hamiltonian S^1 8-dim manifold with minimal number of fixed points - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:53:30Z http://mathoverflow.net/feeds/question/49539 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49539/hamiltonian-s1-8-dim-manifold-with-minimal-number-of-fixed-points Hamiltonian S^1 8-dim manifold with minimal number of fixed points Sil 2010-12-15T16:06:10Z 2010-12-15T21:02:16Z <p>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.</p> <p>Is there an explicit example of M when k=1?</p> <p>(E.g.: for k=5 one has CP^4).</p> http://mathoverflow.net/questions/49539/hamiltonian-s1-8-dim-manifold-with-minimal-number-of-fixed-points/49567#49567 Answer by Dmitri for Hamiltonian S^1 8-dim manifold with minimal number of fixed points Dmitri 2010-12-15T21:02:16Z 2010-12-15T21:02:16Z <p>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:</p> <p>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 <a href="http://arxiv.org/abs/0808.3549" rel="nofollow">http://arxiv.org/abs/0808.3549</a> .</p> <p>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). </p> <p>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: <a href="http://mathoverflow.net/questions/2323/hamiltonian-s1-actions-with-isolated-fixed-points" rel="nofollow">http://mathoverflow.net/questions/2323/hamiltonian-s1-actions-with-isolated-fixed-points</a>)</p> <p>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. </p> <p>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: <a href="http://mathoverflow.net/questions/12263/compact-symplectic-fano-strongly-monotone-manfiolds" rel="nofollow">http://mathoverflow.net/questions/12263/compact-symplectic-fano-strongly-monotone-manfiolds</a></p> <p>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... </p>