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arXiv 1005.1124math.AGmath.AT
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动机双变特征类

Motivic bivariant characteristic classes

Shoji Yokura

备注：
This paper has been withdrawn since its much improved version with Joerg Schuermann joining as a co-author has been posted as arXiv:1110.2166v1 [math.AG] with the same title
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摘要：

相对Grothendieck群$K_0(\m V/X)$是由$X$上的复代数簇的同构类模“关系”生成的自由交换群。基元Hirzebruch类${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$是一个唯一的自然变换，满足对于非奇异簇$X$，恒等式$X \xrightarrow {id_X} X$的同构类的值${T_y}_*([X \xrightarrow {\op {id}_X} X])$是切丛$TX$的Hirzebruch上同调类的Poincaré对偶。它“统一”了著名的三个奇异变种的特征类：MacPherson的Chern类，Baum-Fulton-MacPherson的Todd类（或Riemann-Roch）和Goresky-MacPherson的L-类或Cappell-Shaneson的L-类。本文构造了一个双变相对Grothendieck群$\bK_0(\m V/X \to Y)$，使得当$Y$为点时，它等于原相对Grothendieck群$K_0(\m V/X)$.我们还构造了一个唯一的Grothendieck变换$T_y: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ[y]$满足一定的规范化条件的光滑态射，使它等于motivic Hirzebruch类${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$时，$Y$是一个点。当$y =0$，$T_0: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ$是富尔顿-麦克弗森的双变量黎曼-罗奇$\ga_{td}^{\op {FM}}:\bK_{alg}(X \to Y) \to \bH(X \to Y) \otimes \bQ$的“动机”提升。

英文摘要：

The relative Grothendieck group $K_0(\m V/X)$ is the free abelian group generated by the isomorphism classes of complex algebraic varieties over $X$ modulo the "scissor relation". The motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$ is a unique natural transformation satisfying that for a nonsingular variety $X$ the value ${T_y}_*([X \xrightarrow {\op {id}_X} X])$ of the isomorphism class of the identity $X \xrightarrow {id_X} X$ is the Poincaré dual of the Hirzebruch cohomology class of the tangent bundle $TX$. It "unifies" the well-known three characteristic classes of singular varieties: MacPherson's Chern class, Baum-Fulton-MacPherson's Todd class (or Riemann-Roch) and Goresky-MacPherson's L-class or Cappell-Shaneson's L-class. In this paper we construct a bivariant relative Grothendieck group $\bK_0(\m V/X \to Y)$ so that it equals the original relative Grothendieck group $K_0(\m V/X)$ when $Y$ is a point. We also construct a unique Grothendieck transformation $T_y: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ[y]$ satisfying a certain normalization condition for a smooth morphism so that it equals the motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$ when $Y$ is a point. When $y =0$, $T_0: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ$ is a "motivic" lift of Fulton-MacPherson's bivariant Riemann-Roch $\ga_{td}^{\op {FM}}:\bK_{alg}(X \to Y) \to \bH(X \to Y) \otimes \bQ$.

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