Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

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.