Singularities of symmetric powers and irrationality of motivic zeta functions
Vladimir Shein
TL;DR
The paper develops a framework around the Grothendieck ring of varieties with the Lefschetz class to study how singularities behave under symmetric powers and how this impacts motivic zeta functions. It proves that varieties with $\\mathbb{L}$-rational singularities have symmetric powers that inherit the same property, and, using this, extends Larsen–Lunts irrationality results to higher dimensions by analyzing the Kapranov motivic zeta function $Z(X,t)$ with motivic measures $\\mu_k$. The main results show that if $X$ is smooth, projective, and $\\dim X>1$, then $Z(X,t)$ is not pointwise rational whenever $\\kappa(X)\\ge 0$ or $H^0(X,\\Omega_X^{2i})\\neq 0$ for some $i>0$, implying strong geometric constraints on rationality and the existence of even-degree differential forms. The paper also provides applications to Severi–Brauer varieties and to the study of stable birationality among symmetric powers, highlighting the utility of motivic measures in detecting non-rational behavior.
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.