Lifting problem for minimally wild covers of Berkovich curves

Abstract

This work continues the study of residually wild morphisms f : Y → X f\colon Y\to X of Berkovich curves initiated in [Adv. Math. 303 (2016), pp. 800-858]. The different function δ f \delta _f introduced in that work is the primary discrete invariant of such covers. When f f is not residually tame, it provides a non-trivial enhancement of the classical invariant of f f consisting of morphisms of reductions f ~ : Y ~ → X ~ \widetilde f\colon \widetilde Y\to \widetilde X and metric skeletons Γ f : Γ Y → Γ X \Gamma _f\colon \Gamma _Y\to \Gamma _X . In this paper we interpret δ f \delta _f as the norm of the canonical trace section τ f \tau _f of the dualizing sheaf ω f \omega _f and introduce a finer reduction invariant τ ~ f \widetilde \tau _f , which is (loosely speaking) a section of ω f ~ log \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 ( f ~ , Γ f , δ | Γ Y , τ ~ f ) (\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.