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, we prove a new Xiao-type bound on uf with respect to g for non-isotrivial fibrations: (Formula presented.) in particular, this implies that if f is not locally trivial and uf = g − 1 is maximal, then g ⩽ 6; 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.
Fibred surfaces and their unitary rank
Stoppino, Lidia
2025-01-01
Abstract
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, we prove a new Xiao-type bound on uf with respect to g for non-isotrivial fibrations: (Formula presented.) in particular, this implies that if f is not locally trivial and uf = g − 1 is maximal, then g ⩽ 6; 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
