Vector fields and Chern numbers
Vector fields and Chern numbers
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向量场和陈数
DOI：
10.1007/bf01425242
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发表时间：
1977
影响因子：
1.4
通讯作者：
D. Lieberman
中科院分区：
文献类型：
--
作者：
J. Carrell;D. Lieberman
本文的目的是揭示Bott [-3，41]和Baum-Bott [-5]工作的基础的基本Kozsul复形论证。他们的研究结果将X的全局Chern不变量与局部残基计算联系起来。他们的结果给出的治疗目前的方法是有效的分析和代数类别，我们认为，它澄清的作用Grothendieck剩余符号的公式。本文方法的一个更重要的结果是，对于具有非平凡零点的向量场的K~ ihler流形，可以证明X的Chern类（而不仅仅是Chern数）可以在零点集上计算。进一步，我们系统地研究了关于~值向量场V的等变丛的概念，即O的一个截面，其中O是全纯切丛，~是向量丛.我们很容易证明等变丛的陈数是在V的零点上确定的（XK~ ihler，~ F”平凡的和零（V）~~的陈类也是如此）。等方差的概念是非常富有成效的，因为它大大增加了定理的适用性。若X是Kihler，平凡，零（V）非空，则所有线丛都是等变的.进一步地，如果~/~是一个充分充裕的线丛，那么任何给定的丛8对所有的q值向量场都是等变的，而且这些向量场将大量存在（见§ 1）。定义了一个从全纯形式到~* 的映射i（V）：O 1 ~/-。因此，我们得到了V（f）= i（V）（dr）的一个导子V：Cx-~~。一个丛~称为V-等变的，如果可以将V提升到I~：~-~~®使得：
The purpose of this note is to expose an elementary Kozsul complex argument which underlies the work of Bott [-3, 41 and Baum-Bott [-5]. Their results relate global Chern invariants of X to local residue calculations. The treatment of their results given by the present approach is valid for both the analytic and algebraic categories, and we feel that it clarifies the role of the Grothendieck Residue symbol in the formulae. A more important consequence of the present approach is that for K~ ihler manifolds having a vector field with non-trivial zeroes one can prove that the Chern classes (and not merely the Chern numbers) of X may be computed on the zero set. Furthermore, we study systematically the notion of bundles equivariant with respect to a~-valued vector field V, ie a section of O®~,, where O is the holomorphic tangent bundle and~ a vector bundle. One proves easily that the Chern numbers of equivariant bundles are determined on the zeroes of V (as are the Chern classes for XK~ ihler,~ F" trivial, and zero (V)~~). The notion of equivariance is extremely fruitful since it vastly increases the applicability of the theorem. All line bundles are equivariant if X is K~ ihler, is trivial and zero (V) is nonempty. Further if~/~ is taken to be a sufficiently ample line bundle then any given bundle 8 will be equivariant for all q-valued vector fields, which will exist, moreover, in abundance (see § 1). A section V of O®~=(O1)*®~ is viewed as defining a map i (V): Ol~/-from the holomorphic forms to~.. We obtain therefore a derivation V: Cx-~~ by V (f)= i (V)(dr). A bundle~ is called V-equivariant if one may lift V to I~:~-~~® such that: