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The log homotopy exact sequence

Mattia Talpo

Abstract

We show exactness of the homotopy sequence for the logarithmic fundamental group in the case of log smooth, finitely presented, proper and saturated morphisms of fs log schemes over a field. This generalizes earlier results of Hoshi in the log regular case. In passing, we also construct a "log Stein factorization" in some particular cases.

The log homotopy exact sequence

Abstract

We show exactness of the homotopy sequence for the logarithmic fundamental group in the case of log smooth, finitely presented, proper and saturated morphisms of fs log schemes over a field. This generalizes earlier results of Hoshi in the log regular case. In passing, we also construct a "log Stein factorization" in some particular cases.
Paper Structure (10 sections, 16 theorems, 9 equations)

This paper contains 10 sections, 16 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Log schemes
  4. Root stacks
  5. The logarithmic fundamental group
  6. Log geometric fibers
  7. Fkét morphisms via root stacks
  8. Full faithfulness of pullback
  9. Log Stein factorization
  10. The homotopy exact sequence

Key Result

Theorem 1

Let $f\colon X\to S$ be a proper, finitely presented, log smooth and saturated morphism of fs log schemes over $k$, with connected log geometric fibers. Let $x\to X$ be a log geometric point of $X$, and $s\to S$ be its image in $S$. Then there is an induced exact sequence of log fundamental groups \

Theorems & Definitions (49)

  • Theorem : Theorems \ref{['theorem:main']} and \ref{['thm:left.exact']}
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 39 more