Mémoires de la Société Mathématique de France 2010; 120 pp; softcover Number: 120 ISBN10: 2856292968 ISBN13: 9782856292969 List Price: US$42 Member Price: US$33.60 Order Code: SMFMEM/120
 The author interprets the theory of HarderNarasimhan polygons by the language of \(\mathbb R\)filtrations. By using a variant version of Fekete's lemma and a combinatoric argument on monomials, he establishes the uniform convergence of polygons associated to a graded algebra equipped with filtrations. This leads to the existence of several arithmetic invariants, a very particular case of which is the sectional capacity. Two applications in Arakelov geometry are developed: the arithmetic HilbertSamuel theorem and the existence and the geometric interpretation of the asymptotic maximal slope. 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 interested in algebra and algebraic geometry. Table of Contents  Introduction
 Rappels et préliminaires
 Filtrations de HarderNarasimhan
 Convergence des polygones
 Applications
 Bibliographie
