Families of curves with prescribed rank of Higgs field.
The Hodge bundle of a semistable family of curves has two nested subbundles: the flat unitary subbundle (spanned by flat sections with respect to the Gauss-Manin connection), and the kernel of the Higgs field, which has seminegative definite curvature and contains the flat subbundle. Though there is no reason to expect equality, we also don't know of examples explicitly showing a strict inclusion. In this talk I will present a construction of such examples, with kernels of any possible rank, some of them even over a projective base curve. This is work in progress with Sara Torelli.