As observed by [Calabi](http://dx.doi.org/10.2307/1993108) a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is not integrable.


Is it known whether $S^2\times S^4$ admits an integrable complex structure?


A few remarks. 


* This is stated as an open problem in Calabi's paper, but perhaps it has been solved in the meantime?
* This is similar to the case of $S^6$, which is still open (see  [this question](https://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere)).
* One can also ask the same question for $\Sigma\times S^4$ for $\Sigma$ any compact Riemann surface
* It seems that some people believe that every almost-complex manifold of real dimension $6$ or more admits an integrable complex structure (see [this other question](https://mathoverflow.net/questions/52960/which-almost-complex-manifolds-admit-a-complex-structure)).
* Even more generally (and this is obviously still open), one can ask about an arbitrary finite product of even dimensional spheres (excluding $S^0$). [It is known](http://dx.doi.org/10.1112/jlms/s1-40.1.705) that this is almost complex iff the only factors that appear are $S^2, S^6$ and $S^2\times S^4$.
* If one allows connected sums, then for example $(S^2\times S^4)\#2(S^3\times S^3)$ is a complex manifold, and in fact it has complex structures with trivial canonical bundles (see for example [here](http://projecteuclid.org/euclid.jdg/1197320602) and [here](http://dx.doi.org/10.2140/gt.2010.14.1723)).



Is S^2 x S^4 a complex manifold?
This is still an open problem. See [this paper](http://arxiv.org/abs/1301.6835) for some progress, which was prompted by this MO question.


