Relative Mather discrepancy on arc spaces

TL;DR

This paper introduces a relative Mather discrepancy function on arc spaces for generically étale morphisms, linking it to differential kernels and motivic integration, and explores a new equivalence notion affecting motivic classes.

Contribution

It defines the relative Mather discrepancy on arc spaces, relates it to differential kernels, and introduces e9quivalence, impacting motivic class computations.

Findings

The relative Mather discrepancy computes the kernel dimension of the differential map.

The discrepancy relates to the change-of-variable formula in motivic integration.

c7fe9quivalence implies the same motivic class for varieties.

Abstract

Given any generically \'etale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.