Chern numbers and the geometry of partial flag manifolds

Details Chern numbers and the geometry of partial flag manifolds Chern numbers and the geometry of partial flag manifolds

2007-09-19

arxiv.org/abs/0709.3026v2

Comment. Math. Helv. 84 (2009), 587--616

Mathematics

Differential Geometry

final version to appear in Comment. Math. Helv

Scientific paper

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.

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