JournalscmhVol. 84 , No. 3DOI 10.4171/cmh/174

Chern numbers and the geometry of partial flag manifolds

  • Dieter Kotschick

    Universität München, Germany
  • S. Terzić

    University of Montenegro, Podgorica, Montenegro
Chern numbers and the geometry of partial flag manifolds cover

Abstract

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds Fn = SU(n + 2)/S(U(n) × U(1) × 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 ℂPn + 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 Fn is a 3-symmetric space. We explain the relations between the different structures and their Chern classes, and we prove that Fn is not geometrically formal.