Families of curves with Higgs field of arbitrarily large kernel

In this article, 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 U⊆K can be in the geometric case. More precisely, for any smooth projective curve C of genus g⩾2 and any r=0,…,g−1, we construct non‐isotrivial deformations of C over a quasi‐projective base such that rkK=r and rkU⩽g+12.