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Given an open subset U of a projective curve Y and a smooth family f:V-->U ofcurves, with semi-stable reduction over Y, we show that for a sub variation ofHodge structures of rank >2 the Arakelov inequality must be strict. Forfamilies 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.

Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds