We prove that all generalised symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.

The family of the complex Grassmann manifolds G n,k with the canonical action of the torus T n = T n and the analogue of the moment map µ : G n,k → ∆ n,k for the hypersimplex ∆ n,k , is well known. In this paper we study the structure of the orbit space G n,k /T n by developing the methods of toric geometry and toric topology. We use a subdivision of G n,k into the strata W σ . Relying on this subdivision we determine all regular and singular points of the moment map µ, introduce the notion of the admissible polytopes P σ such that µ(W σ ) = • P σ and the notion of the spaces of parameters F σ , which together describe W σ /T n as the product • P σ ×F σ . To find the appropriate topology for the set ∪ σ • P σ ×F σ we introduce also the notions of the universal space of parametersF and the virtual spaces of parametersF σ ⊂F such that there exist the projections F σ → F σ . Having this in mind, we propose a method for the description of the orbit space G n,k /T n . The existence of the action of the symmetric group S n on G n,k simplifies the application of this method. In our previous paper we proved that the orbit space G 4,2 /T 4 , which is defined by the canonical T 4 -action of complexity 1, is homeomorphic to ∂∆ 4,2 * CP 1 . We prove in this paper that the orbit space G 5,2 /T 5 , which is defined by the canonical T 5 -action of complexity 2, is homotopy equivalent to the space which is obtained by attaching the disc D 8 to the space Σ 4 RP 2 by the generator of the group π 7 (Σ 4 RP 2 ) = Z 4 . In particular, (G 5,2 /G 4,2 )/T 5 is homotopy equivalent to ∂∆ 5,2 * CP 2 .The methods and the results of this paper are very important for the construction of the theory of (2l, q)-manifolds we have been recently developing, and which is concerned with manifolds M 2l with an effective action of the torus T q , q ≤ l and an analogue of the moment map µ : M 2l → P q , where P q is a q-dimensional convex polytope. 14

In the focus of our paper is a system of axioms that serves as a basis for introducing structural data for (2n, k)-manifolds M 2n , where M 2n is a smooth, compact 2n-dimensional manifold with a smooth effective action of the k-dimensional torus T k . In terms of these data a construction of the model space E with an action of the torus T k is given, such that there exists a T k -equivariant homeomorphism E → M 2n . This homeomorphism induces a homeomorphism E/T k → M 2n /T k . The number d = n − k is called the complexity of an (2n, k)-manifold. Our theory comprises toric geometry and toric topology, where d = 0. It is shown that the class of homogeneous spaces G/H of compact Lie groups, where rk G = rk H, contains (2n, k)-manifolds that have non zero complexity. The results are demonstrated on the complex Grassmann manifolds G k+1,q with an effective action of the torus T k . 1

Abstract. We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F n D SU.n C 2/=S.U.n/ U.1/ 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 CP nC1 . The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler 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. Mathematics Subject Classification (2000). Primary 53C30, 57R20; Secondary 14M15, 53C26, 53C55.

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...