Are there known examples of compact complex ndimensional 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 6spheres.)
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The product of two spheres of odd dimensions admits a complex structure (CalabiEckmann) http://en.wikipedia.org/wiki/Calabi%E2%80%93Eckmann_manifold 

