Highly connected, compact complex manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:24:42Z http://mathoverflow.net/feeds/question/28063 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28063/highly-connected-compact-complex-manifolds Highly connected, compact complex manifolds Greg Kuperberg 2010-06-13T20:33:49Z 2010-06-14T00:06:31Z <p>Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:</p> <ol> <li><p>If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a <a href="http://sbseminar.wordpress.com/2008/02/14/complex-manifolds-which-are-not-algebraic/" rel="nofollow">blog posting by David Speyer</a>, you still have $H^2(M,\mathbb{R}) \ne 0$ even if $M$ is non-projective but algebraic.)</p></li> <li><p>An interesting first example of a non-Kähler manifold is a Hopf manifold, by definition $(\mathbb{C}^n\setminus 0)/\Gamma_r$, where $\Gamma_r$ is a rescaling by $r$ with $|r| \ne 0,1$. This example has $H^1(M,\mathbb{R}) \ne 0$.</p></li> <li><p>On the other hand, even-dimensional, compact Lie groups have left-invariant complex structures. If $M$ is such a manifold and is simply connected, then it is also 2-connected. $H^1(M,\mathbb{Z}) = H^2(M,\mathbb{Z}) = 0$ and $M$ is manifestly not Kähler. On the other hand, no such example is 3-connected and you always have $H^3(M,\mathbb{R}) \ne 0$.</p></li> <li><p>There is (or was) a long-standing conjecture that no even-dimensional sphere other than $S^2$ has a complex structure.</p></li> </ol> <p>So, question: Is there for each $n$, a compact, complex manifold $M$ which is $n$-connected?</p> http://mathoverflow.net/questions/28063/highly-connected-compact-complex-manifolds/28067#28067 Answer by Tim Perutz for Highly connected, compact complex manifolds Tim Perutz 2010-06-13T21:27:46Z 2010-06-13T21:27:46Z <p>E. Calabi, B. Eckmann, <i>A class of compact, complex manifolds which are not algebraic.</i> Ann. of Math. (2) 58, (1953). 494–500. </p> <p>From Chern's MR review (MR0057539): </p> <blockquote> This paper defines on the topological product $S^{2p+1} \times S^{2q+1}$ of two spheres of dimensions $2p+1$ and $2q+1$ respectively, $p$ > 0, a complex analytic structure. The complex manifold so obtained ... admits a complex analytic fibering, with two-dimensional tori as fibers and having as base space the product $\mathbb{P}^p \times \mathbb{P}^q$ of complex projective spaces of (complex) dimensions $p$ and $q$ respectively. </blockquote> http://mathoverflow.net/questions/28063/highly-connected-compact-complex-manifolds/28068#28068 Answer by algori for Highly connected, compact complex manifolds algori 2010-06-13T21:30:43Z 2010-06-13T21:30:43Z <p>As shown by Calabi and Eckmann, products of odd-dimensional spheres admit complex structures. See Anns of Maths 58, 1953, 494-500.</p>