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$\newcommand{\C}{\mathbb{C}}\newcommand{\S}{\mathbb{S}}\newcommand{\Z}{\mathbb{Z}}$Complex tori, $\C^d/\Lambda$, where $\Lambda$ is a lattice in $\C^d$, generalise $\S^1\times\cdots\times\S^1$, by considering "slanted" versions of the latter. E.g. for $d=1$, all complex tori are isomorphic to a torus of the form $\C/(\Z+\tau\,\Z)$, with $\tau$ in the upper-half-plane, which is literally a slanted $\S^1\times\S^1$.

Is there a similar construction generalising $\S^n\times\cdots\times\S^n$ and $\S^{n_1}\times\cdots\times\S^{n_k}$?

ɪdɪət strəʊlə
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    Saying complex tori $\mathbb{C}^d/\Lambda$ generalise tori $T^{2d}$ is misleading. Instead, it is more accurate to say that complex tori are complex manifolds which are diffeomorphic to $T^{2d}$. Alternatively, you could say that complex tori $\mathbb{C}^d/\Lambda$ are a generalisation of the standard complex torus $\mathbb{C}^d/\mathbb{Z}^{2n}$. With this in mind, I believe you're asking if there are complex manifolds diffeomorphic to products of spheres. Is that correct? – Michael Albanese Jun 21 '22 at 19:44
  • @MichaelAlbanese Fair enough. With this correction, your rephrasing of my question makes sense. – ɪdɪət strəʊlə Jun 21 '22 at 19:55
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    Products of spheres of dimension greater than $1$ are simply-connected, so they admit no non-trivial coverings at all. Can you say more about the sense in which you seek "generalizing"? (With Michael's interpretation, you might be interested in Calabi-Eckmann manifolds.) – Andrew D. Hwang Jun 21 '22 at 22:40
  • @ɪdɪətstrəʊlə: Does my answer below address the question you intended to ask, or did you want to ask something different? – Michael Albanese Jun 22 '22 at 11:00
  • @MichaelAlbanese thanks. It partially addresses it. I'll try to digest it and if I get confused I'll ask for clarifications/extensions, otherwise I'll accept the answer :) – ɪdɪət strəʊlə Jun 22 '22 at 15:43
  • I edited the title back to better reflect the question I asked. – ɪdɪət strəʊlə Jul 01 '22 at 14:27

1 Answers1

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No, there is no similar construction for arbitrary products of spheres, or for products of spheres of the same dimension. In general, it is not known which products of spheres admit complex structures, or even almost complex structures. However, there are several partial results. Below, I only consider spheres of positive dimension.

The only spheres which admit almost complex structures are $S^2$ and $S^6$. Of course $S^2$ admits complex structures, and they are all biholomorphic to $\mathbb{CP}^1$. It is a famous open problem to determine the existence or non-existence of a complex structure on $S^6$. See this answer for more details.

The product of two odd-dimensional spheres always admits a complex structure. For example, a one-dimensional complex torus is diffeomorphic to $S^1\times S^1$, primary Hopf manifolds are diffeomorphic to $S^1\times S^{2n-1}$, and Calabi-Eckmann manifolds are diffeomorphic to $S^{2m-1}\times S^{2n-1}$ where $m, n > 1$. It follows that a product of an even number of odd-dimensional spheres admits a complex structure.

It is unknown when a product of two even-dimensional spheres admits a complex structure, but it is known which ones admit an almost complex structure: $S^{2m}\times S^{2n}$ admits an almost complex structure if and only $(m, n) \in \{(1, 1), (1, 2), (1, 3), (2, 1), (3, 1), (3, 3)\}$, see this paper. Among these manifolds, only $S^2\times S^2$ is known to admit complex structures (they're all biholomorphic to even Hirzebruch surfaces). I am not aware of any results concerning the product of three or more even-dimensional spheres. Having said that, an elementary characteristic class argument (as is used in this answer) shows that $S^{4k_1}\times\dots\times S^{4k_d}$ does not admit an almost complex structure for any choice $d$ and $k_1, \dots, k_d$.

Finally, we can consider the product of spheres where some of the spheres have odd dimension, and some have even dimension. These all admit almost complex structures (provided they have even dimension) because they are parallelisable: the product of two or more spheres is parallelisable if and only if at least one of the factors is odd-dimensional (see this answer for example). Again, it is unknown which of these manifolds admit complex structures. It is worth noting that there are examples which do not arise as the products of the examples I have mentioned above. For example, the product $S^1\times S^7\times S^6$ admits a complex structure, see this paper.

Michael Albanese
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