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.)
$\begingroup$
$\endgroup$
6
-
1$\begingroup$ Compact? Obviously there are non-compact examples. $\endgroup$– Jason StarrCommented Dec 23, 2012 at 18:14
-
$\begingroup$ Yes, I should have put in compact. $\endgroup$– Andrew McHughCommented Dec 23, 2012 at 20:12
-
1$\begingroup$ @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 $S^6$? $\endgroup$– Daniel LittCommented Dec 23, 2012 at 21:00
-
$\begingroup$ 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. $\endgroup$– Andrew McHughCommented Dec 23, 2012 at 21:55
-
$\begingroup$ I edited the question to specify compact. $\endgroup$– Andrew McHughCommented Dec 23, 2012 at 23:20
|
Show 1 more comment
1 Answer
$\begingroup$
$\endgroup$
5
The product of two spheres of odd dimensions admits a complex structure (Calabi-Eckmann)
http://en.wikipedia.org/wiki/Calabi%E2%80%93Eckmann_manifold
answered Dec 25, 2012 at 1:09
-
$\begingroup$ yes, but it won't satisfy the OP's condition on the Betti numbers! $\endgroup$ Commented Dec 25, 2012 at 1:14
-
1$\begingroup$ 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! $\endgroup$ Commented Dec 25, 2012 at 1:15
-
2$\begingroup$ Right $-$ it's enough for the odd dimensions to be distinct (and neither equal $1$). $\endgroup$ Commented Dec 25, 2012 at 1:39
-
$\begingroup$ Yes. provided I am answering indeed the question Andrew had in mind. $\endgroup$ Commented Dec 25, 2012 at 2:33
-
$\begingroup$ @Dmitri, Yes, this answers my question. Thank you! $\endgroup$ Commented Dec 25, 2012 at 23:55