Fibred surfaces and their unitary rank

TL;DR

This paper establishes new inequalities involving the unitary rank of fibred surfaces, providing bounds and constraints that relate to the geometry of the fibration and the Coleman-Oort conjecture.

Contribution

It introduces novel slope inequalities linking the unitary rank to other invariants of fibred surfaces and derives bounds that impact the understanding of the Torelli locus.

Findings

Proves a new bound on the unitary rank for non-isotrivial fibrations: u_f < g(5g-2)/(6g-3).

Shows that if u_f = g-1 is maximal, then the genus g is at most 6.

Provides a new constraint related to the Coleman-Oort conjecture on the Higgs bundle.

Abstract

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.