Singularities of symmetric powers and irrationality of motivic zeta functions

TL;DR

This paper investigates the singularities of symmetric powers of varieties and their impact on the rationality of motivic zeta functions, revealing that certain singularity conditions imply irrationality in higher dimensions.

Contribution

It proves that $ ext{L}$-rational singularities are preserved under symmetric powers and extends the irrationality criterion of motivic zeta functions to all dimensions.

Findings

Symmetric powers of varieties with $ ext{L}$-rational singularities also have $ ext{L}$-rational singularities.

Rationality of the motivic zeta function implies the variety has negative Kodaira dimension.

Varieties with rational motivic zeta functions lack nonzero even-degree differential forms.

Abstract

Let $K_0(\mathcal{V}_{K})$ be the Grothendieck ring of varieties over a field $K$ of characteristic zero, and let $\mathbb{L} = [\mathbb{A}^1_{K}]$ denote the Lefschetz class. We prove that if a $K$-variety has $\mathbb{L}$-rational singularities, then all its symmetric powers also have $\mathbb{L}$-rational singularities. We then use this result to show that, for a smooth complex projective variety $X$ of dimension greater than one, the rationality of its Kapranov motivic zeta function $Z(X, t)$ (viewed as a formal power series over $K_0(\mathcal{V}_{\mathbb{C}})$) implies that the Kodaira dimension of $X$ is negative and that $X$ does not admit global nonzero differential forms of even degree. This extends the irrationality part of the Larsen-Lunts rationality criterion from the surface case to arbitrary dimension. We also discuss some applications of these results.