Fibred surfaces and their unitary rank

Abstract

Let $f:S\to B$ a complex fibred surface with fibres of genus $g\ge 2$. Let $u_f$ be its unitary rank, i.e., the rank of the maximal unitary summand of the Hodge bundle $f_{\ast }{\omega }_f$. 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 $g\le 6$; (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.

Shortened revised version, 22 pages; accepted for publication in SCIENCE CHINA Mathematics. To appear in a special issue dedicated to the memory of Gang Xiao

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Cited by in corpus (1)

• Locally non-trivial fibred surfaces with maximal unitary rank