Chern numbers and the geometry of partial flag manifolds

TL;DR

This paper computes the Chern classes and numbers for natural structures on partial flag manifolds, revealing their geometric properties and relationships among various invariant structures.

Contribution

It provides explicit calculations of Chern classes and explores the relations between different invariant structures on partial flag manifolds.

Findings

Chern classes and numbers are computed for all n>1.

The projectivization of the cotangent bundle is identified as the twistor space of a Grassmannian.

F_n is shown not to be geometrically formal.

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 complex projective spaces. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic K\"ahler manifold. There is also an invariant nearly K\"ahler 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.