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Group actions on \(S^6\) and complex structures on \(\mathbb{P}_3\). (English) Zbl 0971.53024

Does there exist a complex manifold structure on \(S^6\) ? This is a classical question whose answer is not yet known. However, many papers deal with the possible consequences of an affirmative answer to this question. In this very interesting paper, the authors show that such a complex structure is not almost homogeneous, i.e., the group of holomorphic automorphisms does not have an open orbit. As a corollary, one may infer that if \(S^6\) admits a complex structure, then there is a \(1\)-dimensional family of complex structures on \(\mathbb{P}_3\). Also a complex structure on \(S^6\) could carry at most two linearly independent holomorphic vector fields.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
32M12 Almost homogeneous manifolds and spaces
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