Fibred surfaces and their unitary rank

Let f S→ B a fibred surface with fibres of genus g≥ 2 and base curve of genus b. Let uf be its unitary rank. We prove many new slope inequalities involving uf and some other invariants of the fibration. As applications: (1) we prove a new Xiao-type bound on uf with respect to g for non-isotrivial fibrations: \[ u_f< g5g-2/6g-3. \] In particular this imples that if f is not locally trivial and uf=g-1 is maximal, then g≤ 6; (2) we prove a result in the direction of the Coleman-Oort conjecture: a new constrain on the rank of the (-1,0) part of the maximal unitary Higgs subbundle of a curve generically contained in the Torelli locus; (3) we study in the detail the extremal case (for non-isotrivial f) where uf=g-1, giving many constrains for the case $g=6$, uf=5.