Product formula for twisted MacPherson classes
Product formula for twisted MacPherson classes
复制标题
扭曲麦弗逊类的乘积公式
DOI：
10.3792/pjaa.68.167
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发表时间：
1992
期刊：
影响因子：
--
通讯作者：
Shoji Yokura
中科院分区：
文献类型：
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作者：
M. Kwieciński;Shoji Yokura
导论.拓扑欧拉特征Z是乘法的，即，z（X x Y）z（X）z（Y）。对于流形X，z（X）到高维上同调类的推广是陈上同调类c*（X），它满足叉积公式c*（X x Y）= c*（X）x c*（Y）。对于（可能是奇异的）紧致复代数簇X，z（X）到更高维同调类的推广是Schwartz-MacPherson同调类c，（X），在光滑情况下，它只是通常的Chern上同调类c* 的Poincar对偶（X），其第0个分量等于Z（X）[1，4，5]。最近，关于提升Schwartz-MacPherson类到交同调[2]，第一作者[3]证明了Schwartz-MacPherson类的乘积公式，即，c，（X x Y）c，（X）x c，（Y）.第二位作者[6，71]定义了“扭曲的”MacPherson类ct，（X），其中包括SchwartzMacPherson类c，（X）作为特例，即，cl，（X）c，（X）. ct（X）的第0个分量是“分层加权”欧拉特征Z。t（X）是t的次dimX多项式，也涉及奇异欧拉特征，并且当t 1时等于z（X）。在[7]中，第二作者证明了Z t的乘法性，即，xt（X X Y）= zt（X）zt（Y）。本文通过加强和修改文[3]的证明，给出了乘积公式ct，（X × Y）ct，（X）× ct，（Y），从而乘积公式c，（X × Y）c，（X）× c，（Y）和zt（X × Y）·t（X）·t（Y）是它们的特例.更一般地，我们显示了一个产品公式的变换ct，作用于可构造的功能与多项式系数（定理4）。1.你的助手。我们考虑的簇都是（可能是奇异的）紧致复代数簇。设F是可构造函数协变函子，F（X）是X的特征函数1 w对W自由生成的交换群.对于态射f：X --* Y，前推f，F（X）-F（Y）由（f，lw）（Y）z（f-（y）f W）定义。设H，（;Z）是通常的Z-同调共变函子. Deligne，Grothendieck和MacPherson [3]证明了存在唯一的自然变换c，F ~* H，（;Z）满足对任意光滑X的附加条件c，（lx）= c ~*（X）fX].为了构造变换c，MacPherson首先观察到F（X）也是自由的，
Introduction. The topological Euler characteristic Z is multiplicative, i.e., z(X x Y) z(X)z(Y). For a manifold X, a generalization of z(X) to higher dimensional cohomology classes is the Chern cohomology class c*(X), which satisfies the cross-product formula c*(X x Y)= c*(X) x c*(Y). For a (possibly singular) compact complex algebraic variety X, a generalization of z(X) to higher dimensional homology classes is the Schwartz-MacPherson homology class c,(X), which in the smooth case is just the Poincar dual of the usual Chern cohomology class c*(X) and the 0-th component of which is equal to Z (X)[1, 4, 5]. Very recently, in connection with lifting Schwartz-MacPherson classes to intersection homology [2], the first author [3] proved the product formula for Schwartz-MacPherson classes, i.e., c,(X x Y) c,(X) x c,(Y). The second author [6, 71 defined the "twisted" MacPherson class ct,(X), which includes SchwartzMacPherson class c,(X) as a special case, i.e., cl,(X) c,(X). The 0-th component of ct,(X) is the "stratified weighted" Euler characteristic Z.t(X), which is a degree-dimX polynomial of t, involves Euler characteristic of sigularities also and equals to z(X) when t 1. In [7] the second author showed the multiplicativity of Z t, i.e., xt(X x Y)= zt(X)zt(Y). In this note, by strengthening and modifying the proof of [3] we show the product formula ct,(X x Y) ct,(X) x ct,(Y), thus the product formulae c,(X x Y) c,(X) x c,(Y) and zt(X x Y) .t(X).t(Y) follow as special cases. More generally we show a product formula for the transformation ct, acting on constructible functions with polynomial coefficients (Theorem 4). 1. Preliminaries. The varieties we consider are all (possibly singular) compact complex algebraic varieties. Let F be the constructible function covariant functor, where F(X) is the abelian group freely generated by characteristic functions 1w for subvarieties W of X. For a morphism f:X --* Y the pushforward f, F (X) -F (Y) is defined by (f,lw) (Y) z(f-(y) f W). Let H,(;Z) be the usual Z-homology covariant functor. Deligne and Grothendieck conjectured and MacPherson [3] proved that there exists a unique natural transformation c, F--* H,(;Z) satisfying the extra condition that c,(lx)= c*(X) f X] for any smooth X. To construct the transformation c,, MacPherson first observes that F(X) is also freely