A remark on the Chern classes of local complete intersections
A remark on the Chern classes of local complete intersections
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关于局部完全交集的陈省级的评论
DOI：
10.3792/pjaa.73.93
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
Shoji Yokura
中科院分区：
文献类型：
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作者：
T. Ohmoto;T. Suwa;Shoji Yokura
0. 简介。对于可能奇异的复代数或解析簇 X，有（至少）三种可用的 Chern 类。一类是 Chern-Schwartz-MacPherson 类 [3 和 17]，表示为 cSt(X)。它首先由 M.-H. 建造。 Schwartz 使用径向矢量场，然后 P. Deligne 和 A. Grothendieck 猜想了它作为函子的自然变换的存在性，并由 R. MacPherson 证明了。另一个是 Clern-Mather 类，表示为 Ct(X)。这是通过纳什爆炸定义的，粗略地说，是 X 的平滑部分的极限切丛的 Chern 类。这两个类之间的关系是麦克弗森理论的另一个方面，它用 C(M) 和奇异轨迹上支持的额外项来表达 cS(x)。该定理通过引入局部欧拉阻碍来证明，该阻碍也出现在 Dubson-Kashiwara 指数中 [4 和 12]。第三个是规范类或 Fulton-Johnson 的 Chern 类 [7 和 8]，表示为 CFI(X)。这是根据 X 的 Segre 类定义的，当 X 是局部完全交集时相对容易理解。当品种没有奇点时，这三个类别是相同的，因此它们之间的差异预计可以用奇点的某些不变量来表达。对于具有孤立奇点的（强）局部完全交集 X，在[19]中证明了用 CFI(X) 和奇点的 Milnor 数表示 cSt(X) 的公式。本说明的目的是报告一个观察结果，即该公式与其他已知公式一起意味着一个有趣且可能有前途的与此类 X 品种的 Ct(X) 和 CFY(X) 相关的公式（参见下面的定理 3.3）。
0. Introduction. For a possibly singular complex algebraic or analytic variety X there are (at least) three kinds of Chern classes available. One is the Chern-Schwartz-MacPherson class [3 and 17], denoted cSt(X). This was first constructed by M.-H. Schwartz using radial vector fields, then its existence as a natural transformation of functors was conjectured by P. Deligne and A. Grothendieck and was proved by R. MacPherson. Another is the Clern-Mather class, denoted Ct(X). This is defined via the Nash blow-up and is, roughly speaking, the Chern class of the limiting tangent bundle of the smooth part of X. The relation between these two classes is another aspect of MacPherson’s theory, which expresses cS(x) in terms of C(M) and the extra terms supported on the singular locus. This theorem is proved by introducing the local Euler obstruction, which also appears in the Dubson-Kashiwara index [4 and 12]. The third is the canonical class or Fulton-Johnson’s Chern class [7 and 8], denoted CFI(X). This is defined in terms of the Segre class of X and is relatively easy to understand when X is a local complete intersection. These three classes are identical when the variety has no singularities, thus the differences among them are expected to be expressible in terms of certain invariants of singularities. For a (strong) local complete intersection X with isolated singularities, in [19] is proved a formula expressing cSt(X) in terms of CFI(X) and the Milnor numbers of the singularities. The purpose of this note is to report an observation that this formula together with other already known formulas implies an interesting and possibly promising formula relating Ct(X) and CFY(X) for such varieties X (see Theorem 3.3 below).