Let $M$, $N$ be $n$-dimensional real manifolds. Does $M\times N$ admits a complex structure? If not, are there known condidtions ensuring that $M\times N$ admits a complex structure?



Complex structure on product of two $n$-dimensional real manifolds
I'm going to take $M$ and $N$ to be connected. Note that they cannot have boundary. Furthermore, as complex manifolds are orientable, $M$ and $N$ must also be orientable.


If $n = 1$, then $M, N \in \{\mathbb{R}, S^1\}$. Note that $\mathbb{R}\times\mathbb{R}$ has a complex structure, in fact, precisely two: $\mathbb{D}$ and $\mathbb{C}$. In addition, $\mathbb{R}\times S^1 = S^1\times \mathbb{R}$ and $S^1\times S^1$ have uncountably many complex structures (annuli and tori respectively).


If $n = 2$, then $M$ and $N$ are orientable surfaces. These always admit complex structures, and therefore, so do their products.


Already for $n = 3$, we do not have a complete answer. In this case, the manifold $M\times N$ always admits an almost complex structure, but it is not clear when it admits an integrable one (i.e. a complex structure). This is a major open problem in complex geometry:



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The answer is known to be false in real dimension four, but completely open in higher dimensions.


For $n = 4$, the manifold $M\times N$ may not even admit an almost complex structure. For example, $M = N = S^4$. Theorem 1 of [this paper](https://www.sciencedirect.com/science/article/pii/0040938370900315) of Heaps completely characterises which closed eight-manifolds admit an almost complex structure. In principle, this could be used to determine what $M$ and $N$ could be, at least in the compact case.


