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?
Best Answer
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".