Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities

TL;DR

This paper introduces an involution on a graded variant of the Grothendieck ring of varieties, explores its properties, and applies it to study singular varieties, compactifications, and irrationality of certain zeta functions.

Contribution

It constructs a natural involution on a graded Grothendieck ring extension and analyzes its interaction with symmetric powers and singular varieties.

Findings

Involution $ extbf{D}$ commutes with symmetric powers up to zero divisors.

Defined and studied $ extbf{D}$-singular varieties with applications.

Provided insights into compactifications and irrationality of Kapranov zeta functions.

Abstract

We realize a graded variant $K_0(Var_k^{dim})$ of the Grothendieck ring of varieties as a quadratic extension of the subring $K_0(Var_k^{sp})$ spanned by classes of smooth and proper varieties. As such, there exists a natural involution $\mathbb{D}$ on $K_0(Var_k^{dim})$. We show that $\mathbb{D}$ commutes with the symmetric power operations $Sym^m$ up to zero divisors. Moreover, we study varieties which are smooth up to cut-and-paste relations, which we call $\mathbb{D}$-singular varieties, and we give applications to compactifications of varieties and the irrationality of Kapranov zeta functions.