# Chern numbers and the geometry of partial flag manifolds

### D. Kotschick

Universität München, Germany### S. Terzić

University of Montenegro, Podgorica, Montenegro

## Abstract

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds $F_{n}=SU(n+2)/S(U(n)×U(1)×U(1))$. For all $n>1$ there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of $CP_{n+1}$. 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.

## Cite this article

D. Kotschick, S. Terzić, Chern numbers and the geometry of partial flag manifolds. Comment. Math. Helv. 84 (2009), no. 3, pp. 587–616

DOI 10.4171/CMH/174