Motivic Serre invariants modulo the square of $\mathbb{L}-1$

TL;DR

This paper extends the motivic Serre invariants by lifting them to a finer level modulo the square of 1, using rational coefficients and specific assumptions, enhancing their algebraic structure.

Contribution

It introduces a method to lift motivic Serre invariants to modulo (1)^2 after tensoring with , providing a more detailed algebraic framework.

Findings

Lifting of motivic Serre invariants to (1)^2

Use of tensoring with  over  ring

Conditional assumptions for the lifting process

Abstract

Motivic Serre invariants defined by Loeser and Sebag are elements of the Grothendieck ring of varities modulo $\mathbb{L}-1$. In this paper, we show that we can lift these invariants to modulo the square of $\mathbb{L}-1$ after tensoring the Grothendieck ring with $\mathbb{Q}$, under certain assumptions.