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) × U (n − k)) and some particular interesting examples.Date: November 8, 2018; MSC 2000: primary 57R77, 22F30; secondary 57R20, 14M15 . 1 G/H and multiplicative rules related to that basis. We hope that the relations in the cohomology rings H * (G/H, Z) obtained from the calculation of the universal toric genus of the manifold G/H may lead to new results in that direction.In the paper [28], which opened a new stage in the development of the cobordism theory, S. P. Novikov proposed a method for the description of the fixed points for actions of groups on manifolds, based on the formal group law for geometric cobordisms. That paper rapidly stimulated active research work which brought significant results. These results in the case of S 1 -actions are mainly contained in the papers [15], [16], [21], [22] and also in [23]. Our approach to this problem uses the results on the universal toric genus, which was introduced in [5] and described in details in [7]. Let us note that the formula for the universal toric genus in terms of fixed points is a generalization of Krichever's formula [21] to the case of stable complex manifolds. For the description of the cobordism classes of manifolds in terms of their Chern numbers we appeal to the Chern-Dold character theory, which is developed in [4].The universal toric genus can be constructed for any even dimensional manifold M 2n with a given torus action and stable complex structure which is equ...

Toric Genera of Homogeneous Spaces and their Fibrations

“…Section 3 is devoted to the universal toric genus on compact homogeneous spaces of positive Euler characteristic with an invariant almost complex structure and the canonical action of a maximal torus. We generalize our result from [15] to the flag manifolds endowed with an arbitrary invariant almost complex structure as well as to the generalized Grassmann manifolds. We compute the top Chern numbers s n for these manifolds and note that most of them vanish.…”

“…We consider a homogeneous space G/H with the canonical action of the maximal torus T k and an invariant almost complex structure J. It is proved in [15] that the weights for the canonical action of the maximal torus T k on G/H at the fixed point w ∈ W G /W H for this action related to the structure J are given with w(ǫ i α i ), 1 ≤ i ≤ n. Consequently it is deduced in [15] that the universal toric genus for (G/H, J) is given by the formula…”

“…., where a i ∈ Ω −2i U (Z). It follows [15] that the complex cobordism class for (G/H, J) can be obtained as the coefficient in t n in the polynomial…”

“…The other class of well known classical manifolds with such action are compact homogeneous spaces G/H of positive Euler characteristic with the action of the maximal torus for G. In this case due to representation and Lie group theories it is possible to effectively describe invariant almost complex structures as well as the corresponding weights at the fixed points for the canonical action of a maximal torus [15]. It allows us to, using toric genera, solve an outstanding problem of description of complex cobordism classes of homogeneous spaces [15]. In particular, we effectively computed complex cobordism classes of homogeneous spaces such as flag and Grassmann manifolds related to the standard invariant complex structure.…”

Upper bounds for the dimension of tori acting on GKM manifolds

“…We also note that G 2 /SU (3) ≃ S 6 (diffeomorphic). Therefore, by using Corollary 1.3 and Example 2.13, the following well-known fact can be proved (see also [3]):…”