Vari\'et\'es rationnellement connexes sur un corps alg\'ebriquement clos

TL;DR

This paper provides an accessible overview of rationally connected varieties, emphasizing geometric aspects, and includes a detailed proof of Shokurov's rational connectedness conjecture following Hacon and McKernan.

Contribution

It offers a comprehensive, reader-friendly exposition on rationally connected varieties and presents the proof of a significant conjecture in the field.

Findings

Clarifies geometric properties of rationally connected varieties

Includes a detailed proof of Shokurov's conjecture

Aims to be accessible to a broad mathematical audience

Abstract

These are lectures notes on rationally connected varieties, written for the "Etats de la Recherche" of the French Mathematical Society held in Strasbourg (May 2008). We focus on geometric aspects. These notes have been written in order that a wide audience can easily read them, except maybe the last section, a bit more technical, where we give the proof of Shokurov rational connectedness conjecture following Hacon and McKernan. ----- Ce sont les notes d'un mini-cours sur les vari\'et\'es rationnellement connexes, \'ecrit pour les Etats de la Recherche de la Soci\'et\'e Math\'ematique de France (Strasbourg, 2008). On met l'accent sur les aspects g\'eom\'etriques. Ce cours est r\'edig\'e dans l'espoir de s'adresser \`a un public large, \`a l'exception peut-\^etre du \S 7, o\`u nous donnons les grandes lignes de la preuve de la conjecture de connexit\'e rationnelle de Shokurov par Hacon et McKernan, plus technique et o\`u les pr\'erequis sont un peu plus importants.