Chern classes of graph hypersurfaces and deletion-contraction
Chern classes of graph hypersurfaces and deletion-contraction
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图超曲面和删除收缩的 Chern 类
DOI：
10.17323/1609-4514-2012-12-4-671-700
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
P. Aluffi
中科院分区：
文献类型：
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作者：
P. Aluffi
研究了图的Chern超曲面类在边的删除-收缩操作下的行为。当边满足两个技术条件时，我们得到了一个显式公式，并证明了当边在图中是重边时，这两个条件都成立。这将导致递归的陈类图超曲面的图形，通过添加平行的边缘到一个给定的（定期）边缘。
类似的结果的情况下，Grothendieck类的图超曲面在以前的工作。Grothendieck类和Chern类都被用来定义“代数几何”费曼规则。本文的结果进一步证明了由图超曲面的Chern-Schwartz-MacPherson类定义的多项式Feynman规则密切地反映了相应图的组合学，证明主要结果的关键是一个关于横截交的Chern-Schwartz-MacPherson类的更一般的公式，它可能是独立的.我们还描述了一个更几何的方法，使用设备的“Verdier专业化”。
We study the behavior of the Chern classes of graph hypersurfaces under the operation of deletion-contraction of an edge of the corresponding graph. We obtain an explicit formula when the edge satisfies two technical conditions, and prove that both these conditions hold when the edge is multiple in the graph. This leads to recursions for the Chern classes of graph hypersurfaces for graphs obtained by adding parallel edges to a given (regular) edge.
Analogous results for the case of Grothendieck classes of graph hypersurfaces were obtained in previous work. Both Grothendieck classes and Chern classes were used to define `algebro-geometric' Feynman rules. The results in this paper provide further evidence that the polynomial Feynman rule defined in terms of the Chern-Schwartz-MacPherson class of a graph hypersurface reflects closely the combinatorics of the corresponding graph.The key to the proof of the main result is a more general formula for the Chern-Schwartz-MacPherson class of a transversal intersection, which may be of independent interest. We also describe a more geometric approach, using the apparatus of `Verdier specialization'.