Motivic Volumes of Fibers of Tropicalization

TL;DR

This paper introduces a motivic volume approach to fibers of tropicalization for subvarieties of algebraic tori, linking geometric tropicalization with motivic integration and providing a formula for motivic zeta functions.

Contribution

It defines a tropicalization map on arc schemes, proves the constructibility and motivic volume of its fibers, and expresses the generating function as a rational function related to geometric tropicalization.

Findings

Fibers of the tropicalization map are constructible and have motivic volumes.

The generating function of motivic volumes is rational under certain conditions.

Provides a formula for Denef and Loeser's motivic zeta function.

Abstract

Let $T$ be an algebraic torus over an algebraically closed field, let $X$ be a smooth closed subvariety of a $T$-toric variety such that $U = X \cap T$ is not empty, and let $\mathscr{L}(X)$ be the arc scheme of $X$. We define a tropicalization map on $\mathscr{L}(X) \setminus \mathscr{L}(X \setminus U)$, the set of arcs of $X$ that do not factor through $X \setminus U$. We show that each fiber of this tropicalization map is a constructible subset of $\mathscr{L}(X)$ and therefore has a motivic volume. We prove that if $U$ has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev's theory of geometric tropicalization. We explain how this result, in particular, gives a formula for Denef and Loeser's motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization.