In this article 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 Kring 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.
Abstract. Let p : E−→B be a principal bundle with fibre and structure group the torus T ∼ = (C * ) n over a topological space B. Let X be a nonsingular projective T -toric variety. One has the X-bundle π :. This is a Zariski locally trivial fibre bundle in case p : E−→B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H * (B; Z)-algebra, (ii) the topological K-ring of K * (E(X)) as a K * (B)-algebra when B is compact. When p : E−→B is algebraic over an irreducible, nonsingular, noetherian scheme over C, we describe (iii) the Chow ring of A * (E(X)) as an A * (B)-algebra, and (iv) the Grothendieck ring K 0 (E(X)) of algebraic vector bundles on E(X) as a K 0 (B)-algebra.
In this article we describe the G × G-equivariant Kring of X, where G is a factorial cover of a connected complex reductive algebraic group G, and X is a regular compactification of G. Furthermore, using the description of K G× G (X), we describe the ordinary K-ring K(X) as a free module of rank the cardinality of the Weyl group, over the K-ring of a toric bundle over G/B, with fibre the toric variety T + , associated to a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [19]). Further, we give an explicit presentation of K G× G (X) as well as K(X) as an algebra over the K G× G (G ad ) and K(G ad ) respectively, where G ad is the wonderful compactification of the adjoint semisimple group G ad . Finally, we identify the equivariant and ordinary Grothendieck ring of X respectively with the corresponding rings of a canonical toric bundle over G ad with fiber the toric variety T + .
We describe the equivariant algebraic cobordism rings of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.
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