Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Lifting problem for minimally wild covers of Berkovich curves


Authors: Uri Brezner and Michael Temkin
Journal: J. Algebraic Geom. 29 (2020), 123-166
DOI: https://doi.org/10.1090/jag/728
Published electronically: September 13, 2019
MathSciNet review: 4028068
Full-text PDF

Abstract | References | Additional Information

Abstract: This work continues the study of residually wild morphisms $f\colon Y\to X$ of Berkovich curves initiated in [Adv. Math. 303 (2016), pp. 800-858]. The different function $\delta _f$ introduced in that work is the primary discrete invariant of such covers. When $f$ is not residually tame, it provides a non-trivial enhancement of the classical invariant of $f$ consisting of morphisms of reductions $\widetilde f\colon \widetilde Y\to \widetilde X$ and metric skeletons $\Gamma _f\colon \Gamma _Y\to \Gamma _X$. In this paper we interpret $\delta _f$ as the norm of the canonical trace section $\tau _f$ of the dualizing sheaf $\omega _f$ and introduce a finer reduction invariant $\widetilde \tau _f$, which is (loosely speaking) a section of $\omega _{\widetilde f}^{\operatorname {log}}$. Our main result generalizes a lifting theorem of Amini-Baker-Brugallé-Rabinoff from the case of residually tame morphism to the case of minimally residually wild morphisms. For such morphisms we describe all restrictions the datum $(\widetilde f,\Gamma _f,\delta |_{\Gamma _Y},\widetilde \tau _f)$ satisfies and prove that, conversely, any quadruple satisfying these restrictions can be lifted to a morphism of Berkovich curves.


References [Enhancements On Off] (What's this?)

References


Additional Information

Uri Brezner
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv’at Ram, Jerusalem, 91904, Israel
Email: uri.brezner@mail.huji.ac.il

Michael Temkin
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv’at Ram, Jerusalem, 91904, Israel
MR Author ID: 332870
Email: michael.temkin@mail.huji.ac.il

Received by editor(s): February 14, 2018
Received by editor(s) in revised form: June 30, 2018, and July 19, 2018
Published electronically: September 13, 2019
Additional Notes: This work was supported by the Israel Science Foundation (grant No. 1159/15)
Article copyright: © Copyright 2019 University Press, Inc.