The author establishes the correspondence between tame harmonic bundles and \(\mu _L\)polystable parabolic Higgs bundles with trivial characteristic numbers. He also shows the BogomolovGieseker 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 higherdimensional 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 BogomolovGieseker inequality
 Construction of a frame
 Some convergence results
 Existence of adapted pluriharmonic metric
 Torus action and the deformation of representations
 \(G\)harmonic bundle
 Bibliography
