  We establish the correspondence between tame harmonic bundles and
$\mu_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We
also show the Bogomolov-Gieseker type inequality for $\mu_L$-stable parabolic
Higgs bundles.
  Then we show that any local system on a smooth quasi projective 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 quasi projective variety.
