Fibred surfaces and their unitary rank
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912413395189760 |
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| author | Stoppino, Lidia |
| author_facet | Stoppino, Lidia |
| contents | 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 $f_*ω_f$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fibred surfaces and their unitary rank Stoppino, Lidia Algebraic Geometry 14J10, 14D06, 14D07, 14G35, 14H40 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 $f_*ω_f$. 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. |
| title | Fibred surfaces and their unitary rank |
| topic | Algebraic Geometry 14J10, 14D06, 14D07, 14G35, 14H40 |
| url | https://arxiv.org/abs/2406.15909 |