Lifting problem for minimally wild covers of Berkovich curves

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.