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?
['https://mathoverflow.net/questions/346122', 'https://mathoverflow.net', 'https://mathoverflow.net/users/147073/']
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.
I assume you are asking if $M \times N$ admits a complex structure.
I believe that there is no result known on this question, although we might have something to say about the existence of an almost complex structure for which both factors are totally real.
Once again we can quote [Gromov](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/SpacesandQuestions.pdf): Page 30. "How much do we gain in global understanding of a compact $(V, J)$ by assuming that the structure $J$ is integrable (i.e. complex)? It seems nothing at all: there is no single result concerning all compact complex manifolds".