Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

We establish the correspondence between tame harmonic bundles and µ_L-polystable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for µ_L-stable parabolic Higgs bundles.

Then we show that any local system on a smooth quasiprojective variety can be deformed to a variation of polarized Hodge structure. As a consequence, we can conclude that some kind of discrete groups cannot be a split quotient of the fundamental group of a smooth quasiprojective variety.