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<p>Let <i>f</i>: <i>S</i> → <i>B</i> be a complex fibred surface with fibres of genus <i>g</i> ⩾ 2. Let <i>u</i><sub><i>f</i></sub> be its unitary rank, i.e., the rank of the maximal unitary summand of the Hodge bundle <i>f</i><sub>*</sub><i>ω</i><sub><i>f</i></sub>. We prove many new slope inequalities involving <i>u</i><sub><i>f</i></sub> and some other invariants of the fibration. As applications,<OrderedList> <ListItem> <ItemNumber>(1)</ItemNumber> <ItemContent> <p>we prove a new Xiao-type bound on <i>u</i><sub><i>f</i></sub> with respect to <i>g</i> for non-isotrivial fibrations: <Equation ID="Equ1"> <EquationSource Format="TEX">\(u_f&lt;g\frac{5g-2}{6g-3};\)</EquationSource> </Equation> in particular, this implies that if <i>f</i> is not locally trivial and <i>u</i><sub><i>f</i></sub> = <i>g</i> − 1 is maximal, then <i>g</i> ⩽ 6;</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(2)</ItemNumber> <ItemContent> <p>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 sub-bundle of a curve generically contained in the Torelli locus.</p> </ItemContent> </ListItem> </OrderedList></p>

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Fibred surfaces and their unitary rank

  • 期刊论文

  • 发表日期：

    2025年7月24日

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  • Lidia Stoppino

摘要

Let f: S → B be a complex fibred surface with fibres of genus g ⩾ 2. Let uf be its unitary rank, i.e., the rank of the maximal unitary summand of the Hodge bundle f*ωf. 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: \(u_f<g\frac{5g-2}{6g-3};\) in particular, this implies 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 constraint on the rank of the (−1, 0) part of the maximal unitary Higgs sub-bundle of a curve generically contained in the Torelli locus.