Классы кобордизмов однородных пространств

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 talk is to present 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 $G/H$. The effectiveness is due to the explicit description of the universal toric genus in terms of local data at the fixed points. Special attention is devoted to the structures which are invariant under the canonical action of the group $G$. Using classical results, we obtain in this case an explicit description of the local data (weights and signs) at the fixed points. We consequently obtain an expression for the cobordism classes of such structures in terms of coeffcients of the formal group law in cobordisms, as well as in terms of Chern numbers in cohomology appealing to the Chern-Dold character theory. As an application we provide an explicit formulas 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 other interesting cases.
The talk is based on the results of the joint work with Victor M. Buchstaber.