Which Spheres are Complex Manifolds? - MathOverflow [closed]

Which Spheres are Complex Manifolds?

2010-06-29T20:27:00Z

Possible Duplicate:
complex structure on S^n

The two sphere $S^2$ is a real manifold of dimension $2$, while the three sphere $S^3$ is a real manifold of dimension $3$. Now $S^2$ is a complex manifold, while $S^3$ being odd dimensional is not. Is it true that all spheres of the form $S^{2N}$ are complex manifolds?

Answer by Charles Siegel for Which Spheres are Complex Manifolds?

2010-06-29T20:31:31Z

That's VERY much false. This question is looking for a reference to the fact that $S^2$ and $S^6$ are the only ones with even almost complex structures, and it's open if $S^6$ admits a complex structure (the almost complex structure known is not one)

Edit: Other questions on complex structures on spheres are here and here.

Answer by Justin Curry for Which Spheres are Complex Manifolds?

2010-06-29T22:52:47Z

It's a nice exercise in characteristic classes to show that S^4k for all k are NOT complex manifolds.

EDIT: I will answer Charlie's comment here and provide a sketch of the proof.

Let $\omega=TS^{4k}$ be the tangent space to the $4k$-sphere. If $S^{4k}$ was actually a complex manifold then $\omega$ would be a complex vector bundle. In this case the complexification of the underlying real vector bundle $\omega_{\mathbb{R}}$ would be canonically isomorphic to the Whitney sum $\omega\oplus \bar{\omega}$ (Milnor&Stasheff page 176). Now by corollary 15.5 in Milnor&Stasheff $$p_k(\omega_{\mathbb{R}})=c_k^2(\omega)-2c_{k-1}c_{k+1}(\omega)+\cdots\mp 2c_{2k}(\omega)$$

This then shows that the top Pontrjagin number $$< p_k,[S^{4k}]>=<\mp 2c_{2k},[S^{4k}]>=\mp 4$$ but we also know that spheres are boundries of an oriented manifold and thus have higher Pontrjagin number 0. Contradiction.

On another note, according to C.C. Hsiung's book Almost Complex and Complex Structures on page 233 he says "In fact, the absence of an almost complex structure on $S^{4k}$ for $k\geq 1$ and $S^{2n}$ for $n\geq 4$ was proved by Wu and jointly Borel and Serre respectively."