Une introduction aux p\'eriodes

TL;DR

This survey introduces periods as complex numbers defined by integrals of rational functions over polynomial domains, discusses their properties, and explores their interpretation via algebraic de Rham cohomology and singular homology, highlighting recent advances.

Contribution

It provides an accessible overview of periods, their fundamental properties, and their interpretation through algebraic topology, connecting classical analysis with modern algebraic geometry.

Findings

Periods can be expressed as integrals of rational functions over polynomial domains.

The interpretation of periods via algebraic de Rham cohomology and singular homology is central to recent research.

The Kontsevich-Zagier conjecture suggests all algebraic relations among periods derive from basic calculus rules.

Abstract

This survey article is the outgrowth of two talks given at the Journ\'ees X-UPS "P\'eriodes et transcendance" at \'Ecole polytechnique. Periods are complex numbers whose real and imaginary parts can be written as integrals of rational functions over domains defined by polynomial inequalities, everything with rational coefficients. According to a conjecture by Kontsevich-Zagier, every algebraic relation among these numbers should follow from the obvious rules of calculus: additivity, change of variables, Stokes's formula. I first explain the definition of periods and some ensuing elementary properties, by illustrating them with a host of examples. Then I gently move to the interpretation of these numbers as entries of the integration pairing between algebraic de Rham cohomology and singular homology of algebraic varieties defined over $\mathbf{Q}$, the point of view which is at the origin of all recent breakthroughs in their study.