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.