Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?

share|cite|improve this question
up vote 7 down vote accepted

It is known that the only spheres which can possibly be complex manifolds are $S^2$ and $S^6$. Of course $S^2$ is known to admit an integrable complex structure as the Riemann sphere, but it is still an open problem whether $S^6$ admits an integrable complex structure. By the uniformization theorem, $S^2$ admits a unique complex manifold structure.

See the theorem on page 212 of May's A Concise Course in Algebraic Topology for a proof using characteristic classes that $S^2$ and $S^6$ are the only spheres that even admit an almost complex structure. The notes are freely available on his website here.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.