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The motivic fundamental groupoid at tangential basepoints

Sofian Tur-Dorvault

Abstract

We give a general construction of the motivic fundamental groupoid at tangential basepoints, extending previous works of P. Deligne, A. B. Goncharov, and M. Levine, which were limited to ordinary basepoints or to specific varieties. Given a smooth variety over a field endowed with a simple normal crossings divisor, we encode its tangential basepoints using the language of logarithmic geometry. Building on the recent construction by F. Binda, D. Park, and P. A. Østvær of a stable $\infty$-category of $\mathbb{A}^1$-invariant logarithmic motives and its comparison with the usual $\infty$-category of motives, we define in a functorial manner the associated motivic pointed path spaces. In the presence of a motivic $t$-structure, truncating yields the motivic fundamental groupoid. In general, we construct Betti and de Rham realization functors for logarithmic motives (linearizing the construction of F. Binda, D. Park and P. A. Østvær for the Betti case) and we show that the periods of the motivic fundamental groupoid are given by regularized iterated integration of logarithmic differential $1$-forms, thus yielding a general version of Chen's theorem with tangential basepoints.

The motivic fundamental groupoid at tangential basepoints

Abstract

We give a general construction of the motivic fundamental groupoid at tangential basepoints, extending previous works of P. Deligne, A. B. Goncharov, and M. Levine, which were limited to ordinary basepoints or to specific varieties. Given a smooth variety over a field endowed with a simple normal crossings divisor, we encode its tangential basepoints using the language of logarithmic geometry. Building on the recent construction by F. Binda, D. Park, and P. A. Østvær of a stable -category of -invariant logarithmic motives and its comparison with the usual -category of motives, we define in a functorial manner the associated motivic pointed path spaces. In the presence of a motivic -structure, truncating yields the motivic fundamental groupoid. In general, we construct Betti and de Rham realization functors for logarithmic motives (linearizing the construction of F. Binda, D. Park and P. A. Østvær for the Betti case) and we show that the periods of the motivic fundamental groupoid are given by regularized iterated integration of logarithmic differential -forms, thus yielding a general version of Chen's theorem with tangential basepoints.
Paper Structure (36 sections, 40 theorems, 225 equations, 1 figure)

This paper contains 36 sections, 40 theorems, 225 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The case of ordinary basepoints
  3. The case of tangential basepoints
  4. Content of the paper
  5. Remarks and relations with other works
  6. Tangential basepoints in log geometry
  7. The log normal space and pointed diagrams
  8. The log normal space at a point
  9. The functoriality of tangential basepoints
  10. The category of pointed diagrams
  11. Virtual points
  12. Log motives and motivic tangential basepoints
  13. Log motivic categories
  14. log-cdh-descent for log motives
  15. Motivic tangential basepoints
  16. ...and 21 more sections

Key Result

Theorem 1.0.1

Let $X$ be a smooth scheme of finite type over an arbitrary field $k$, endowed with a simple normal crossings divisor $D$ and $\mathbf{x},\mathbf{y}$ be tangential basepoints for the pair $(X,D)$ (definition tangent).

Figures (1)

  • Figure 1: The topological space $\,X'$

Theorems & Definitions (145)

  • Theorem 1.0.1: Theorem \ref{['thm Betti real pi']}, Theorem \ref{['deRhamreal']}, Theorem \ref{['end']} Definition \ref{['pathspace']}
  • Remark 1.0.2
  • Remark 1.0.3
  • Definition 1.3.1
  • Theorem 1.3.2: Theorem \ref{['thm Betti real pi']}
  • Theorem 1.3.3: Theorem \ref{['end']}
  • Definition 2.1.1
  • Remark 2.1.2
  • Lemma 2.1.3
  • proof
  • ...and 135 more