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

Takehiko Yasuda

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.

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

Abstract

Motivic Serre invariants defined by Loeser and Sebag are elements of the Grothendieck ring of varities modulo . In this paper, we show that we can lift these invariants to modulo the square of after tensoring the Grothendieck ring with , under certain assumptions.

Paper Structure

This paper contains 6 sections, 4 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preparatory reductions
  3. The case $d'=1$.
  4. The case $d'=2$.
  5. The case $d'\ge3$.
  6. Closing comments

Key Result

Theorem 1.4

Let $X$ be a smooth projective $K$-variety. Under Assumption assu:assumption-desing-weak-fac, the invariant $\tilde{S}(\mathcal{X})$ is independent of the chosen regular snc model $\mathcal{X}$ and depends only on $X$.

Theorems & Definitions (8)

  • Remark 1.1
  • Definition 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof