Equivariant complex structures on homogeneous spaces and their cobordism classes

Abstract

We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the complex, almost complex and stable complex structures which are invariant under the canonical left action of the maximal torus T^k on G/H. As it is known, on G/H we may have many such structures and the computations of their toric genus in terms of fixed points for the same torus action give the constraints on possible collections of weights for the corresponding representations of T^k in the tangent spaces at the fixed points, as well as on the signs at these points. In that context, the effectiveness is also approached due to an explicit description of the relations between the weights and signs for an arbitrary couple of such structures. Special attention is devoted to the structures which are invariant under the canonical action of the group G. Using classical results, we obtain an explicit description of the weights and signs in this case. We consequently obtain an expression for the cobordism classes of such structures in terms of coefficients of the formal group law in cobordisms, as well as in terms of Chern numbers in cohomology. These computations require no information on the cohomology ring of the manifold G/H, but, on their own, give important relations in this ring. As an application we provide an explicit formula for the cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular interesting examples.