Uma V

Let G be a split connected reductive algebraic group. Let (Formula presented.) be a (Formula presented.) -torsor over a smooth base scheme (Formula presented.) and X be a regular compactification of G. We describe the Grothendieck ring of the associated fibre bundle (Formula presented.), as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle over (Formula presented.). These are relative versions of the corresponding results on the Grothendieck ring of X in the case when (Formula presented.) is a point, and generalize the classical results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.

Let X be a 2n-dimensional torus manifold with a locally standard T ≅ (S1)n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p : E → B and let π: E(X) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H∗(B)-algebra and the topological K-ring of E(X) as a K∗(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.

We describe the G-equivariant Grothendieck ring of a regular compactification X of an adjoint symmetric space G/H of minimal rank. This extends the results of Brion and Joshua for the equivariant Chow ring of wonderful symmetric varieties of minimal rank in (Brion, M., Joshua, R. 13, 471–493 (2008)) and generalizes the results on the regular compactification of an adjoint semisimple group in (Uma, V. 12(2), 371-406 (2007)).

Let X be a toric hyperKähler manifold. The purpose of this note is to describe the topological K-ring K*(X) of X. We give a presentation for the topological K-ring in terms of generators and relations similar to the known description of the cohomology ring of these manifolds.

Generalized Bott manifolds (over C and R) have been defined by Choi, Masuda and Suh in [4]. In this article we extend the results of [7] on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with Z2-coefficients, we derive a combinatorial characterization for orientablity and spin structure.

We describe the equivariant algebraic cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties. © 2012 The Author(s).

We describe the G × G-equivariant K-ring of X, where G is a factorial covering of a connected complex reductive algebraic group G, and X is a regular compactification of G. Furthermore, using the description of KG×G(X), we describe the ordinary K-ring K(X) as a free module (whose rank is equal to the cardinality of the Weyl group) over the K-ring of a toric bundle over G/B whose fibre is equal to the toric variety T+ associated with a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [1]). We also give an explicit presentation of KG×G(X) and K(X) as algebras over KG××G(Gad) and K(Gad) respectively, where Gad is the wonderful compactification of the adjoint semisimple group Gad. In the case when X is a regular compactification of Gad, we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over Gad.

In this paper we describe the G × G-equivariant K-ring of X, where X is a regular compactification of a connected complex reductive algebraic group G. Furthermore, in the case when G is a semisimple group of adjoint type, and X its wonderful compactification, we describe its ordinary K-ring K(X). More precisely, we prove that K(X) is a free module over K(G/B) of rank the cardinality of the Weyl group. We further give an explicit basis of K(X) over K(G/B), and also determine the structure constants with respect to this basis. © Birkhauser Boston 2007.

We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring KT(G/B) of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in KT(G/B) to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of K(X)Q where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure constants of Schubert varieties in the Grothendieck ring of G/B. © Instytut Matematyczny PAN, 2013.

In this note we shall give a description of the K-ring of a quasi-toric manifolds in terms of generators and relations. We apply our results to describe the K-ring of Bott-Samelson varieties.