Chern numbers and the geometry of partial flag manifolds

Abstract

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds $F_n=SU(n+2)/S(U(n)\times U(1)\times U(1))$. For all $n>1$ there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of $C{P}^{n+1}$. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler structure, because $F_n$ is a 3-symmetric space. We explain the relations between the different structures and their Chern classes, and we prove that $F_n$ is not geometrically formal.