Une introduction aux périodes

Abstract

This survey article is the outgrowth of two talks given at the Journées X-UPS "Périodes et transcendance" at École 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.