Group Actions on S^6 and complex structures on P_3

Abstract

It is proved that if S^6 possesses an integrable complex structure, then there exists a 1-dimensional family of pairwise different exotic complex structures on P_3(C). This follows immediately from the main result of the paper: S^6 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.