Question

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.

Answer 1

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.)

Answer 2

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.