A property of a hypothetical complex structure on the six sphere. (English) Zbl 0891.53018

Gray, Alfred

Boll. Unione Mat. Ital., VII. Ser., B 11, No. 2, Suppl., 251-255 (1997).

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 \(\dim H^{0,1} (S^6)\geq 1\).

Reviewer: M.Fernandez (Bilbao)

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