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I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau's list was published, so what is the status of this problem today?

Obviously it isn't hasn't been shown to be true, because we're still looking for complex structures on the six-sphere, but I have a vague feeling of having read that this doesn't hold. So do we know any counterexamples to this question? If not, then is anyone working on this problem?

Also, Yau only stated the problem for manifolds of dimension $n \geq 3$. We know this is true in dimension one, because there we have isothermal coordinates which give complex structures, but why didn't Yau mention almost complex surfaces? Do we know this holds there, or are there counterexamples in dimension 2?

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Added as a comment (because you've already got two great answers). In dim 4 you can get every finitely presented group as the fundamental group of an almost complex 4-mfd (symplectic even, by a thm of Gompf, but this is harder). On the other hand, the classification of complex surfaces tells you that the possible topology of complex surfaces is much more constrained. What changes drastically in dim 6 is that any finitely presented group arises as pi_1 of a closed complex threefold. This was first proved by Taubes as a corollary of an existence thm for self-dual metrics. –  Joel Fine Feb 11 '11 at 2:25
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I should add I think that this is a fascinating problem, but have not the slightest idea how to start trying to attack it. It seems to me like a question waiting for a "big idea". I would guess the total failure to understand the six-sphere is what puts most people off, but perhaps that is overly negative. Sometimes the less topology you have the harder things are. E.g, we know a lot about differential topology of various 4-manifolds, but still nothing about simply connected ones with b_2=0... –  Joel Fine Feb 11 '11 at 2:31

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In complex dimension 3 or more it is still an open conjecture (which was re-stated Yau a couple of years ago in his UCLA lectures). There is not a single known example of an almost complex manifold of dimension $\geq$ 3 not admitting a complex structure.

In dimension 2 it is easy, of course, because the non-Kähler complex surfaces are understood much better than Kähler ones: every non-Kähler surface with $b_1 >1$ is diffeomorphic to a blow-up of a locally trivial elliptic fibration over a curve. Hence any 4-dimensional compact almost complex manifold with odd $b_1 >1$ and a fundamental group not virtually isomorphic (*) to a cross-product of a fundamental group of a curve and $\mathbb{Z}$, cannot be a complex surface.

(*) Here "virtually isomorphic" means "isomorphic up to a finite index subgroup".

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There are actually counterexamples in real dimension $4$.

The first examples of compact almost complex $4$-manifolds admitting no complex structure were produced by Van de Ven in his paper "On the Chern numbers of some complex and almost-complex manifolds".

In fact, he obtained restrictions on the Chern numbers of an algebraic surface and constructed some almost complex $4$-manifolds violating them, hence showing that no almost complex structure in these examples could be integrable.

Later on, Brotherton constructed some counterexamples with trivial tangent bundle, see the article "Some parallelizable 4-manifolds not admitting a complex structure".

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Excellent, thank you Francesco. But we have no similar results for higher dimensions, i.e. all the known restrictions are the same for complex and almost complex manifolds? –  Gunnar Magnusson Jan 24 '11 at 16:23
    
I'm not aware of any results in higher dimensions. But I must say that I'm not a specialist in the field –  Francesco Polizzi Jan 24 '11 at 16:56

About the 6-dimensional sphere, there exists a paper in arxiv, which states that $S^{6}$ is a complex manifold. (See also the link of Mathoverflow)

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a guess it is an big open problem? –  Siqi He Sep 12 '12 at 4:31

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