Groupes de Galois motivques et p\'eriodes

TL;DR

This paper discusses the development of motivic Galois theory, connecting it to periods of algebraic varieties and showing that relations between periods originate from Stokes' formula, advancing understanding of algebraic and motivic structures.

Contribution

It introduces a new tannakian approach based on Voevodsky's mixed motives, providing deeper insights into periods and their relations in algebraic geometry.

Findings

Relations between periods arise from Stokes' formula

Voevodsky's theory underpins the new tannakian approach

Advances understanding of motivic Galois groups

Abstract

In the mid sixties, A. Grothendieck envisioned a vast generalization of Galois theory to systems of polynomials in several variables, motivic Galois theory, and introduced tannakian categories on this occasion. In characteristic zero, various unconditional approaches were later proposed. The most precise one, due to J. Ayoub, relies on Voevodsky theory of mixed motives and on a new tannakian theory. It sheds new light on periods of algebraic varieties, and shows in particular that polynomial relations between periods of a pencil of algebraic varieties always arise from Stokes formula.