Same betti numbers as \$\Bbb{CP}^n\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:18:16Z http://mathoverflow.net/feeds/question/101042 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101042/same-betti-numbers-as-bbbcpn Same betti numbers as \$\Bbb{CP}^n\$ The Common Crane 2012-07-01T06:01:27Z 2012-11-02T14:13:58Z <p>I am sure that there is an answer out there for the following question. If one is given an n dimensional Kahler manifold \$X\$ with Betti numbers that are the same as in the case of \$\Bbb{CP}^n\$, then is \$X\$ holomorphically diffeomorphic to \$\Bbb{CP}^n\$? This is of course true in the one dimensional case, but other then that I am clueless. If this is textbook stuff that I might have missed, references will be appreciated :).</p> http://mathoverflow.net/questions/101042/same-betti-numbers-as-bbbcpn/104122#104122 Answer by Dmitri for Same betti numbers as \$\Bbb{CP}^n\$ Dmitri 2012-08-06T17:15:47Z 2012-11-02T14:13:58Z <p>Dear Common Crane,</p> <p>let me list several results and question going in the direction of your intuition (all the information below I learned from Sergey Galkin)</p> <p>1) Theorem. Hirzebruch-Kodaira, Yau. A Kahler manifold homeomorphic to \$\mathbb CP^n\$ is biholomorphic to \$\mathbb CP^n\$. There is a nice exposition of this result by Valentino Tossati: <a href="http://www.math.northwestern.edu/~tosatti/cpn.pdf" rel="nofollow">http://www.math.northwestern.edu/~tosatti/cpn.pdf</a></p> <p>2) There is a question of Wilson: </p> <p><em>If \$V\$ is a complex projective manifold of even dimension \$n>k(V)\$ and which has the same rational cohomology as \$P^n\$, is \$V\$ isomorphic to \$P^n\$?</em> </p> <p>3) In dimension 3 there are "four" complex projective manifolds with same Betti numbers as \$\mathbb CP^3\$ (these manifolds are \$P^3\$, the quadric, the manifold called \$V_5\$ and finally \$V_{22}\$. The first three manifolds are rigid, but the last one has six-dimensional moduli space of deformations).</p> <p>4) Finally you can find some interesting recent development related to the question in the preprint <a href="http://member.ipmu.jp/sergey.galkin/papers/ipmu-11-0100.pdf" rel="nofollow">http://member.ipmu.jp/sergey.galkin/papers/ipmu-11-0100.pdf</a> where you will as well find references for articles in 1), 2).</p>