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Variétés Rationnellement Connexes: Aspects Géométriques et Arithmétiques
L. Bonavero, Université de Grenoble 1, Saint-Martin d'Hères, France, B. Hassett, Rice University, Houston, TX, J. M. Starr, Stony Brook University, NY, and O. Wittenberg, Écolé Normale Supérieure, Paris, France
A publication of the Société Mathématique de France.
 Panoramas et Synthèses 2010; 221 pp; softcover Number: 31 ISBN-10: 2-85629-339-5 ISBN-13: 978-2-85629-339-3 List Price: US$60 Member Price: US$48 Order Code: PASY/31 Over the last twenty years, rationally connected varieties have played an important role in the classification program of higher dimensional varieties. Over the last ten years, a number of their arithmetic properties have been discovered. It is the goal of this volume to report on many of these advances, as well as on a number of open questions. This volume gathers the contributions of the four speakers at the CNRS/SMF workshop "Etats de la Recherche", which was organized by J.-L. Colliot-Thélène, O. Debarre, and A. Höring in Strasbourg in May 2008. L. Bonavero discusses the fundamental geometric properties of rationally connected varieties and also offers an opening on modern birational classification techniques. O. Wittenberg surveys the arithmetic properties of rationally connected varieties, mostly over local fields and over finite fields (deformation techniques and cohomological techniques). B. Hassett reports on the weak approximation property for families of rationally connected varieties over a complex curve. The emerging notion of simply rationally connected variety is at the heart of J. Starr's contribution. Starr's paper starts with a study of sections of families of such varieties over a complex surface and culminates with a partly simplified proof of the theorem by de A. J. Jong, J. Starr, and X. He: Serre's Conjecture II for principal homogeneous spaces holds over function fields in two variables over the complex field. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians. Table of Contents J.-L. Colliot-Thélène -- Introduction O. Wittenberg -- La connexité rationnelle en arithmétique L. Bonavero -- Variétés rationnellement connexes sur un corps algébriquement clos J. M. Starr -- Rational points of rationally simply connected varieties B. Hassett -- Weak approximation and rationally connected varieties over function fields of curves