Families of curves with Higgs field of arbitrarily large kernel

In this note we consider the flat bundle U and the kernel K of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion of U in K can be in the geometric case. More precisely, for any smooth projective curve C of genus g (at least 2) and any k (between 0 and g-1), we construct non-isotrivial deformations of C over a quasi-projecive base such that K has rank k and U has rank at most (g+1)/2.