Fibred surfaces and their unitary rank

Let $f\colon S\to B$ a complex fibred surface with fibres of genus $g\geq 2$. Let $u_f$ be its unitary rank, i.e., the rank of the maximal unitary summand of the Hodge bundle $f_*\omega_f$. We prove many new slope inequalities involving $u_f$ and some other invariants of the fibration. As applications: (1) we prove a new Xiao-type bound on $u_f$ with respect to $g$ for non-isotrivial fibrations: $$u_f< g\frac{5g-2}{6g-3}.$$ In particular this implies that if $f$ is not locally trivial and $u_f=g-1$ is maximal, then $g\leq 6$; (2) we prove a result in the direction of the Coleman-Oort conjecture: a new constraint on the rank of the $(-1,0)$ part of the maximal unitary Higgs subbundle of a curve generically contained in the Torelli locus.