Group Actions on S⁶ and complex structures on P₃

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It is proved that if S⁶ possesses an integrable complex structure, then there exists a 1-dimensional family of pairwise different exotic complex structures on P₃(C). This follows immediately from the main result of the paper: S⁶ is not the underlying differentiable manifold of an almost homogeneous complex manifold X. Via elementary Lie theoretic techniques this is reduced to ruling out the possibility of a C*-action on a certain non-normal surface E in X. A contradiction is reached by analyzing combinatorial aspects of the non-normal locus N of E and its preimage in the normalization of E.