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In 1986 C. C. Hsiung published a paper "Nonexistence of a Complex Structure on the Six-Sphere" and in 1995 he even wrote a monograph "Almost Complex and Complex Structures" to further elaborate on his proof. Yet answers to the 2009 question on this site all agree that the existence of complex structures on $S^6$ is still an open problem. Some recent preprints answering the question with opposite answers are also cited there. I would like to know if there are any known mistakes in Hsiung's approach and if so I would appreciate some reference to a paper that points them out.

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    $\begingroup$ You might ask the same question about the paper of A.Adler "The second fundamental forms of $S^{6}$ and $P^{n}({\bf C})$", Amer. J. Math. 91 (1969) 657–670, which claims to prove the same result. In this case Steve Krantz says here books.google.com/… that he saw a paper of Y.T.Siu which explains precisely where Adler's error is. However, I couldn't find any paper of Siu where this is discussed. $\endgroup$ – YangMills Jul 10 '12 at 22:07
  • $\begingroup$ Having said this, I don't know exactly where Hsiung's mistake is, I will search a bit and see what I can find. $\endgroup$ – YangMills Jul 10 '12 at 22:08
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    $\begingroup$ I vaguely recall knowing about 3 separate published proofs that were not taken seriously by other experts but I don't remember the third one anymore. I've never found anyone who could explain to me what the error in either Adler's or Hsiung's proof is. But I never thought of asking Siu. $\endgroup$ – Deane Yang Jul 11 '12 at 2:45
  • $\begingroup$ @Deane: maybe you were thinking about S.S.Chern's last preprint (2004)? It was never published, but you might have seen it. $\endgroup$ – YangMills Jul 11 '12 at 2:57
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    $\begingroup$ @YangMills: As for the error(s) in Adler's 1969 paper, the main breakdown occurs because his whole argument relies on Proposition 2 from his earlier paper The Second Fundamental Form of a Kähler Metric, Amer. J. Math. 89 (1967), 260–274, and this Proposition is simply wrong. His error is that he assumes that the first Chern form of the LC-connection of an Hermitian metric on a complex manifold is necessarily of type $(1,1)$. (He just writes that it's "a known property", with no argument.) This false claim is crucial for his construction of a Kähler metric when his inequality is satisfied. $\endgroup$ – Robert Bryant Aug 12 '17 at 12:34
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While it's good to have a source, such as Datta's paper that points out the error, I find that his explanation of why the key equation is wrong is not as clear as it could be. In fact, with a little thought (requiring essentially no computation), it's clear why this equation must be wrong and what is wrong with the approach. Since it's relatively short, I thought I'd put it in:

On page 263 of Hsuing's monograph "Almost Complex and Complex Structures", he claims the following result, from which, if it were correct, the non-existence of a complex structure on the $6$-sphere would follow immediately (and, in fact, Hsiung 'applies' this result to get exactly this 'conclusion'):

Theorem 6.1. Let $J$ be an almost complex structure on a Riemannian $2n$-manifold $M^{2n}$ ($n\ge2$) with a Riemannian metric $g_{ij}$ but without a flat metric or a nonzero constant sectional curvature or both, and let $J_i^j$ and $R_{hijk}$ be respectively the components of the tensor of $J$ and the Riemann curvature tensor of $M^{2n}$ with respect to $g_{ij}$, where all indices take the values $1,2,\ldots,2n$. If $J$ is complex structure on $M^{2n}$, then $$ J_{i_1}^iJ_{i_2}^jR_{iji_3k}+J_{i_2}^iJ_{i_3}^jR_{iji_1k}+J_{i_3}^iJ_{i_1}^jR_{iji_2k}=0 $$ for all $i_1,i_2,i_3,k$.

Now, this result cannot possibly be correct, as you can see from the following observations.

First, note that no relation between $g$ and $J$ is supposed. If it weren't for the peculiar assumptions about $M$ not admitting a flat or constant curvature metric (which might have nothing to do with $g$), this would be a purely local statement, but, no matter, let's let $M$ be $\mathbb{CP}^n$ and note that, since $n\ge2$, $M$ cannot carry either kind of metric. Let $J$ be the standard complex structure on $M$. Then the above 'Theorem' would imply that, for any metric $g$ on $M$, its Riemann curvature tensor $R$ would satisfy the above equation. Since any metric in dimension $2n$ can be locally transplanted onto $\mathbb{CP}^n$ and since all complex structures are locally equivalent, it follows easily that the above 'Theorem' implies that the above relation (which is a purely pointwise statement) must hold identically as an algebraic relation for any local pair $J$ and $g$. (Moreover, since this doesn't involve any derivatives of $J$, the hypothesis that $J$ be integrable is irrelevant.)

Second, it's easy to check that this 'identity' does not hold: Just choose a metric $g$ of nonzero constant sectional curvature and any local $J$ that is $g$-orthogonal, and you'll see that this says that the $2$-form $\Omega$ associated to $J$ by $g$ must satisfy $\Omega^2 = 0$, contradicting the fact that $\Omega^n$ cannot vanish because $\Omega$ must be nondegenerate. (This is, in fact, Hsuing's argument as to why $S^6$ can't carry an integrable complex structure, because it has a metric of constant sectional curvature.)

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  • $\begingroup$ Great answer! In the "erratum" paper (p.673 of the same book) Hsiung still states the same identical "theorem"... $\endgroup$ – YangMills Jul 12 '12 at 2:21
  • $\begingroup$ @YangMills: Thanks. You must have a different edition of the book than I do, since the copy in our library of Hsuing's "Almost Complex and Complex Structures" does not have more than 300 pages. $\endgroup$ – Robert Bryant Jul 13 '12 at 5:26
  • $\begingroup$ Sorry, I misread your answer. I was referring to the book of Hsiung's collected papers here books.google.com/books?id=s85vcMYZDJgC&pg=PA673 $\endgroup$ – YangMills Jul 16 '12 at 18:13
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I just found this paper by B. Datta (later published in J. Indian Math. Soc. 60 (1994), no. 1-4, 171–190) that explains in details why one key equation in Hsiung's paper is wrong. See the whole discussion in section 4.

Also, in a 2001 book containing Hsiung's selected papers he added a short paper titled "Nonexistence of a Complex Structure on the Six-Sphere II" which looks like an erratum to his other one. He still claims the same result about $S^6$. This paper was never published.

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There was a paper published in China (1987?) by Dong (and Guan). It gave a reason why Hsiung was wrong. That was basically the note of Guan when he sat in a talk of Hsiung. Dong came late and missed the talk. Guan gave him his note ant told him that the proof was wrong. After Guan left to America, Dong published the note and added his own proof that there is no complex structure on S^6. After Guan received Dong's note, he warned him about his mistake. But it was too late and the note was published under the joint authors Dong and Guan. Later on Guan met Hsiung in Lehigh University. Hsiung showed Guan his book and said that Yau said that his proof is correct. Guan again told Hsiung about his mistake without any conclusion. It is more like politics to Guan.

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    $\begingroup$ Are you Guan the coauthor? $\endgroup$ – S. Carnahan Aug 7 '13 at 23:45
  • $\begingroup$ Here is the author:webvpn.ucr.edu/+CSCO+0h756767633A2F2F6A6A6A2E6E7A662E626574++/… $\endgroup$ – Guan Aug 8 '13 at 0:24
  • $\begingroup$ The link is broken (it requires ucr.edu vpn), but I assume the answer to Scott's question is "Yes". $\endgroup$ – Igor Rivin Aug 8 '13 at 2:08
  • $\begingroup$ It is MR1307945 (96b:43008). He is from Henan University, China. Ask Kefeng Liu if you really want get to that person to get a copy of that paper. $\endgroup$ – Guan Aug 8 '13 at 3:11

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