Higher G-theory of simplicial toric varieties and vanishing of Chow groups

TL;DR

This paper computes G-theory groups for various classes of simplicial toric varieties, establishes the dimension of G_0, and proves the vanishing of the Chow group A^2 for affine smooth toric varieties, advancing understanding of their algebraic K-theory and Chow groups.

Contribution

It provides explicit G-theory computations for several classes of simplicial toric varieties and proves the vanishing of A^2 for affine smooth toric varieties, supporting a conjecture on Chow groups.

Findings

G-theory groups G_0, G_1, G_2 computed for specific toric varieties.

Dimension of G_0 tensor Q equals sum of even Betti numbers.

Chow group A^2 vanishes for affine smooth toric varieties.

Abstract

This paper gives computations of all the $G$-theory groups of several classes of simplicial toric varieties, including all affine toric surfaces when the base field is algebraically closed and has characteristic zero, all weighted projective spaces over any field and resolution of singularities of all affine toric surfaces $\operatorname{Spec}(k[x,xy,xy^2,...,xy^d])$ over any field $k$. The $G$-theory groups $G_0,G_1,G_2$ are computed for the product of any two weighted projective spaces over any field. The dimension of the rational vector space $G_0(X)\otimes\mathbb{Q}$ for any complete, simplicial toric variety $X$ over an algebraically closed field of characteristic zero is shown to be equal to the sum of the Betti numbers of even degrees. We also prove that the Chow group $A^2(X)$ of codimension 2 cycles vanishes for any affine, smooth toric variety, thereby proving a special case of my conjecture that the order of this Chow group $A^2(X)$ divides the determinant of the matrix whose columns are the minimal generators of the fan for any affine, simplicial toric variety $X$.