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A property of a hypothetical complex structure on the six sphere. (English) Zbl 0891.53018
A long standing problem in differential geometry is the existence of a complex structure on the six-dimensional sphere $$S^6$$. Several proofs have been given, but none has been accepted by the mathematical community. In this paper, the author supposes that there exists a complex structure on the six-sphere. For such a hypothetical complex structure, Dolbeault cohomology groups $$H^{p,q} (S^6)$$ would be defined. It is proved that then $$\dim H^{0,1} (S^6)\geq 1$$.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C55 Global differential geometry of Hermitian and Kählerian manifolds
##### Keywords:
6-sphere; complex structure; Dolbeault cohomology groups