Intersection theory and the Alesker product
Intersection theory and the Alesker product
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交集理论和 Alesker 产品
DOI：
10.1512/iumj.2016.65.5846
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Joseph H. G. Fu
中科院分区：
文献类型：
--
作者：
Joseph H. G. Fu
Alesker在光滑流形$M上引入了光滑赋值的空间$数学V^inty(M)，并证明了它允许自然交换乘法。虽然阿莱斯克最初的建造技术含量很高，但从道德角度来看，这款产品简直就是两套相交操作的神器。随后，Alesker和Bernig用微分形式给出了乘积的一个表达式。我们展示了Alesker-Bernig公式是如何自然地从交集解释中产生的，并应用这一见解给出了一个新的公式，用于计算一般赋值与以足够丰富的光滑多面体族的交集表示的赋值的乘积。
Alesker has introduced the space $\mathcal V^\infty(M)$ of {\it smooth valuations} on a smooth manifold $M$, and shown that it admits a natural commutative multiplication. Although Alesker's original construction is highly technical, from a moral perspective this product is simply an artifact of the operation of intersection of two sets. Subsequently Alesker and Bernig gave an expression for the product in terms of differential forms. We show how the Alesker-Bernig formula arises naturally from the intersection interpretation, and apply this insight to give a new formula for the product of a general valuation with a valuation that is expressed in terms of intersections with a sufficiently rich family of smooth polyhedra.