# Compact Complex n-folds with Betti numbers $b_1=b_2=b_n=0$ for $n >3$

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.)

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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

## 1 Answer

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

http://en.wikipedia.org/wiki/Calabi%E2%80%93Eckmann_manifold

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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