Lifting problem for minimally wild covers of berkovich curves

This work continues the study of residually wild morphisms f : Y → X of Berkovich curves initiated in [Adv. Math. 303 (2016), pp. 800-858]. The different function δf 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 f : Y → X and metric skeletons τf : τY → τX. In this paper we interpret δf as the norm of the canonical trace section τf of the dualizing sheaf ωf and introduce a finer reduction invariant τf , which is (loosely speaking) a section of ωlogf . Our main result generalizes a lifting theorem of Amini-Baker-Brugall Le-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, τf , δ|τY , Τf) satisfies and prove that, conversely, any quadruple satisfying these restrictions can be lifted to a morphism of Berkovich curves.