Intersection theory and the Alesker product

TL;DR

This paper explores the intersection-theoretic foundation of Alesker's product on smooth valuations, providing new formulas and insights into its geometric interpretation and applications.

Contribution

It demonstrates how Alesker's product naturally arises from intersection theory and derives new formulas for valuations involving intersections with smooth polyhedra.

Findings

Alesker's product can be derived from intersection operations.

New formula for valuation products involving smooth polyhedra.

Intersection interpretation simplifies understanding of valuation algebra.

Abstract

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.