Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:

  1. If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a blog posting by David Speyer, you still have $H^2(M,\mathbb{R}) \ne 0$ even if $M$ is non-projective but algebraic.)

  2. 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 $H^1(M,\mathbb{R}) \ne 0$.

  3. 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 $H^3(M,\mathbb{R}) \ne 0$.

  4. There is (or was) a long-standing conjecture that no even-dimensional sphere other than $S^2$ has a complex structure.

So, question: Is there for each $n$, a compact, complex manifold $M$ which is $n$-connected?

share|improve this question

2 Answers 2

up vote 18 down vote accepted

E. Calabi, B. Eckmann, A class of compact, complex manifolds which are not algebraic. Ann. of Math. (2) 58, (1953). 494–500.

From Chern's MR review (MR0057539):

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.
share|improve this answer
    
A slam dunk for the question! I did do a few Google searches (including Google Books and Google Scholar) and I just didn't see it. –  Greg Kuperberg Jun 13 '10 at 21:36
1  
There's also a Wikipedia page, with a construction that's very similar to that of a Hopf manifold. en.wikipedia.org/wiki/Calabi%2dEckmann_manifold –  Greg Kuperberg Jun 13 '10 at 21:53
    
My googling was directed by your Lie group examples, among which there are certain products of spheres! (BTW, when you described the Hopf manifolds, I expect you meant to delete zero.) –  Tim Perutz Jun 13 '10 at 23:07
    
Yeah, thanks, I fixed the typo. –  Greg Kuperberg Jun 14 '10 at 0:06

As shown by Calabi and Eckmann, products of odd-dimensional spheres admit complex structures. See Anns of Maths 58, 1953, 494-500.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.