The Higher K-Theory of Complex Varieties

Higher K-theory of Toric Varieties

1.1. Statements. In [Gu1] we showed that vector bundles on affine toric varieties are trivial. An easy generalization of this result to the stable situation yields the equality K0(R) = K0(R[M ]) for every regular ring R and every seminormal monoid M (see [Gu2], also [Sw, Corollary 1.4]). The key case is the class of finitely generated normal monoids which, up to isomorphism, can be described as...

Equivariant K-theory and Higher Chow Groups of Smooth Varieties

For any variety X with an action of a split torus over a field, we establish a spectral sequence connecting the equivariant and the non-equivariant higher Chow groups of X . For X smooth and projective, we give an explicit formula relating the equivariant and the non-equivariant higher Chow groups. This in particular implies that the spectral sequence degenerates. These results also hold with r...

The Higher K-Theory of a Complex Surface

LetX be a smooth complex variety of dimension at most two, and let F be its function ¢eld. We prove that the K -groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-¢nite.We also describe the torsion in the K -groups of F and X. Mathematics Subject Classi¢cations (2000). Primary: 19E08; secondary: 19E20, 14C35, 14F20.

Varieties of higher-order theory

K-theory of Minuscule Varieties

Based on Thomas and Yong’s K-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the K-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without ref...