  Given an open subset U of a projective curve Y and a smooth family f:V-->U of
curves, with semi-stable reduction over Y, we show that for a sub variation of
Hodge structures of rank >2 the Arakelov inequality must be strict. For
families of n-folds we prove a similar result under the assumption that the
(n,0) component of the Higgs bundle defines fibrewise a birational map.
