Group actions on S6 and complex structures on ℙ3

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 P3(C). This follows immediately from the main result of the paper: S 6 is not the underlying differentiable manifold of an almost homoge- neous 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 EX. A contradiction is reached by analyzing combinatorial aspects of the non-normal locus N of E and its preimage ˆ N in the normalization ˆ E.