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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 1

up vote 6 down vote accepted

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
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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

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