Orthogonal complex structures on certain Riemannian 6-manifolds∗

It is shown that the Hermitian-symmetric space CP1 × CP1 × CP1 and the flag manifold F1,2 endowed with any left invariant metric admit no compatible integrable almost complex structures (even locally) different from the invariant ones. As an application it is proved that any stable harmonic immersion from F1,2 equipped with an invariant metric into an irreducible Hermitian symmetric space of compact type is equivariant. It is also shown thatCP1×CP1×CP1 and F1,2 with its invariant Kähler–Einstein structures are the only compact Kähler–Einstein spin 6-manifolds of non-negative, non-identically vanishing holomorphic sectional curvature that admit another orthogonal complex structure of Kähler type. A necessary and sufficient condition on a compact oriented 6-manifold to admit three mutually commuting almost complex structures is given; it is used to characterize CP1 × CP1 × CP1 and F1,2 as Fano 3-folds admitting three mutually commuting complex structures which satisfy certain compatibility conditions.