MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Are there known examples of compact complex n-dimensional manifolds with betti numbers $b_1=b_2=b_n=0$ for $n >3$? (The case of $n=3$ is the question of integrable complex structures on homology 6-spheres.)

share|cite|improve this question
Compact? Obviously there are non-compact examples. – Jason Starr Dec 23 '12 at 18:14
Yes, I should have put in compact. – Andrew McHugh Dec 23 '12 at 20:12
@Andrew McHugh: Am I missing something? Isn't the $n=3$ case the question of whether a homology $6$-sphere admits a complex structure? Or is a homology $6$-sphere admitting a complex structure automatically $S^6$? – Daniel Litt Dec 23 '12 at 21:00
I think you are right. I should have said homology 6-sphere instead of topological $S^6$. I believe it is still a well known open question as to whether $S^6$ has an integrable complex structure. – Andrew McHugh Dec 23 '12 at 21:55
I edited the question to specify compact. – Andrew McHugh Dec 23 '12 at 23:20
up vote 6 down vote accepted

The product of two spheres of odd dimensions admits a complex structure (Calabi-Eckmann)

share|cite|improve this answer
yes, but it won't satisfy the OP's condition on the Betti numbers! – YangMills Dec 25 '12 at 1:14
oops, I guess I read the OP's condition as $b_1=b_2=b_3=...=b_n=0$, which is apparently not what was asked! – YangMills Dec 25 '12 at 1:15
Right $-$ it's enough for the odd dimensions to be distinct (and neither equal $1$). – Noam D. Elkies Dec 25 '12 at 1:39
Yes. provided I am answering indeed the question Andrew had in mind. – Dmitri Dec 25 '12 at 2:33
@Dmitri, Yes, this answers my question. Thank you! – Andrew McHugh Dec 25 '12 at 23:55

Your Answer


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.