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Logarithmic purity and logarithmic Nori fundamental group

Sara Mehidi

Abstract

We generalize the logarithmic purity theorem of Fujiwara-Kato to torsors which arise in the Kummer log flat topology under finite flat linearly reductive group schemes. As an application, we construct the logarithmic Nori fundamental group of a log regular log scheme classifying those torsors, and compare it to classical Nori fundamental group and tame fundamental group.

Logarithmic purity and logarithmic Nori fundamental group

Abstract

We generalize the logarithmic purity theorem of Fujiwara-Kato to torsors which arise in the Kummer log flat topology under finite flat linearly reductive group schemes. As an application, we construct the logarithmic Nori fundamental group of a log regular log scheme classifying those torsors, and compare it to classical Nori fundamental group and tame fundamental group.
Paper Structure (11 sections, 15 theorems, 51 equations)

This paper contains 11 sections, 15 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Logarithmic purity for linearly reductive torsors
  3. Logarithmic Nori fundamental group
  4. Future Directions
  5. Log torsors
  6. Log Purity Theorem
  7. Log regular base
  8. Log torsors over nodal curves
  9. Log Nori fundamental group
  10. Log étale fundamental group
  11. Log Nori Fundamental Group

Key Result

Theorem 1.1

(Theorem lin-red) Let $S$ be a log regular log scheme with integral underlying scheme, and let $U \subset S$ be the dense open locus of trivial log structure. Let $G$ be a finite flat linearly reductive group scheme over $S$. Then the restriction functor is an equivalence of categories.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2: Logarithmic torsor
  • Example 2.3
  • Theorem 3.1: GillibertTame
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 21 more