Motivic vanishing cycles as a motivic measure

TL;DR

This paper demonstrates that motivic vanishing cycles define a motivic measure on the Grothendieck ring of varieties over the affine line, linking it to previous constructions using matrix factorizations.

Contribution

It establishes a new motivic measure derived from motivic vanishing cycles and relates it to earlier measures based on matrix factorizations.

Findings

Motivic vanishing cycles induce a motivic measure on the Grothendieck ring.

The measure relates to previous constructions using categories of matrix factorizations.

Provides a conceptual link between different motivic measures.

Abstract

We show that the motivic vanishing cycles introduced by J. Denef and F. Loeser give rise to a motivic measure on the Grothendieck ring of varieties over the affine line. We discuss the relation of this motivic measure to the motivic measure we constructed earlier using categories of matrix factorizations.