标题： 高阶G理论与单纯性环面簇及 Chow群的消失 显示英文标题

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

摘要： 本文给出了几个类别的单纯环面簇的$G$理论群的计算，包括当基域是代数闭域且特征为零时的所有仿射环面曲面，任何域上的加权射影空间以及任何域上的仿射环面曲面$\operatorname{Spec}(k[x,xy,xy^2,...,xy^d])$的奇点解消$k$。对于任何域上的任意两个加权射影空间的乘积，$G$理论群$G_0,G_1,G_2$被计算出来。对于代数闭域且特征为零的任何完备、单纯环面簇$X$，有理向量空间$G_0(X)\otimes\mathbb{Q}$的维数被证明等于偶数次贝蒂数之和。 我们还证明了对于任何仿射、光滑的扇形簇，2维循环的Chow群$A^2(X)$为零，从而证明了我提出的猜想的一个特例，即对于任何仿射、纯的扇形簇$X$，这个Chow群$A^2(X)$的阶整除由扇形的最小生成元作为列组成的矩阵的行列式。 显示英文摘要

摘要： 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 $X$.