Which Spheres are Complex Manifolds? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-22T06:47:47Z http://mathoverflow.net/feeds/question/29964 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29964/which-spheres-are-complex-manifolds Which Spheres are Complex Manifolds? Mihail Matrix 2010-06-29T20:27:00Z 2010-07-01T14:20:38Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/11664/complex-structure-on-sn" rel="nofollow">complex structure on S^n</a> </p> </blockquote> <p>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?</p> http://mathoverflow.net/questions/29964/which-spheres-are-complex-manifolds/29965#29965 Answer by Charles Siegel for Which Spheres are Complex Manifolds? Charles Siegel 2010-06-29T20:31:31Z 2010-06-29T20:31:31Z <p>That's VERY much false. This <a href="http://mathoverflow.net/questions/25557/references-on-almost-complex-structures-on-spheres" rel="nofollow">question</a> 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)</p> <p>Edit: Other questions on complex structures on spheres are <a href="http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere" rel="nofollow">here</a> and <a href="http://mathoverflow.net/questions/11664/complex-structure-on-sn/11667#11667" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/29964/which-spheres-are-complex-manifolds/29977#29977 Answer by Justin Curry for Which Spheres are Complex Manifolds? Justin Curry 2010-06-29T22:52:47Z 2010-07-01T14:20:38Z <p>It's a nice exercise in characteristic classes to show that S^4k for all k are NOT complex manifolds.</p> <p><strong>EDIT:</strong> I will answer Charlie's comment here and provide a sketch of the proof.</p> <p>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&amp;Stasheff page 176). Now by corollary 15.5 in Milnor&amp;Stasheff $$p_k(\omega_{\mathbb{R}})=c_k^2(\omega)-2c_{k-1}c_{k+1}(\omega)+\cdots\mp 2c_{2k}(\omega)$$</p> <p>This then shows that the top Pontrjagin number $$&lt; p_k,[S^{4k}]>=&lt;\mp 2c_{2k},[S^{4k}]>=\mp 4$$ but we also know that spheres are boundries of an oriented manifold and thus have higher Pontrjagin number 0. Contradiction.</p> <p>On another note, according to C.C. Hsiung's book <a href="http://books.google.com/books?id=-yz03cskUOgC&amp;printsec=frontcover#v=onepage&amp;q=complex%2520structures%2520on%2520spheres&amp;f=false" rel="nofollow"><em>Almost Complex and Complex Structures</em></a> 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."</p>