  In this paper, we consider the weight $i$ de Rham--Gauss--Manin bundles on a
smooth variety arising from a smooth projective morphism $f:X\_U\lrar U$ for
$i\geq 0$. We associate to each weight $i$ de Rham bundle, a certain parabolic
bundle on $S$ and consider their parabolic Chern characters in the rational
Chow groups, for a good compactification $S$ of $U$. We show the triviality of
the alternating sum of these parabolic bundles in the (positive degree)
rational Chow groups. This removes the hypothesis of semistable reduction in
the original result of this kind due to Esnault and Viehweg.
