Orthogonal complex structures on certain Riemannian 6-manifolds
Let \(M\) be the Hermitian-symmetric space \({\mathbb P}^1{\mathbb C}\times {\mathbb P}^1{\mathbb C}\times{\mathbb P}^1{\mathbb C}\) or the flag manifold \[ F_{1 2}= U(3)/(U(1)\times U(1)\times U(1)) \] equipped with any left invariant metric. It is shown that any integrable almost complex structure on \(M\) which is compatible with both, the Riemann metric and the orientation of \(M\), must be invariant, even if the integrable almost complex structure is only locally defined. Moreover, a compact Kähler-Einstein spin 6-manifold whose holomorphic sectional curvature is non-negative and not identically zero, must be one of the two examples above, if it admits another compatible Kähler structure.
scientific article; zbMATH DE number 1454703