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 Astérisque 2006; 117 pp; softcover Number: 309 ISBN-10: 2-85629-226-7 ISBN-13: 978-2-85629-226-6 List Price: US$39 Individual Members: US$35.10 Order Code: AST/309 The author establishes the correspondence between tame harmonic bundles and $$\mu _L$$-polystable parabolic Higgs bundles with trivial characteristic numbers. He also shows the Bogomolov-Gieseker type inequality for $$\mu _L$$-stable parabolic Higgs bundles. The author shows that any local system on a smooth quasiprojective variety can be deformed to a variation of polarized Hodge structure. He then concludes that some kind of discrete groups cannot be a split quotient of the fundamental group of a smooth quasiprojective variety. 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 vector bundles on surfaces and higher-dimensional varieties, and their moduli. Table of Contents Introduction Preliminary Parabolic Higgs bundle and regular filtered Higgs bundle An ordinary metric for a parabolic Higgs bundle Parabolic Higgs bundle associated to tame harmonic bundle Preliminary correspondence and Bogomolov-Gieseker inequality Construction of a frame Some convergence results Existence of adapted pluri-harmonic metric Torus action and the deformation of representations $$G$$-harmonic bundle Bibliography