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.)



Compact Complex n-folds with Betti numbers $b_1=b_2=b_n=0$ for $n >3$
The product of two spheres of odd dimensions admits a complex structure (Calabi-Eckmann) 


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


