Chern classes of Gau{\ss}-Manin bundles of weight 1 vanish
  We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at $\infty$ dies up to torsion in the Chow ring, except in codimension 0. 
H\'el\`ene Esnault, Eckart Viehweg