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 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 mor- phisms of reductions ( f) over tilde: (Y) over tilde -> (X) over tilde and metric skeletons Gamma(f) : Gamma(Y) -> 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 (tau) over tilde (f) which is (loosely speaking) a section of omega(log)((f) over tilde). Our main result generalizes a lifting theorem of Amini-Baker-Brugalle-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) over tilde, Gamma(f), delta vertical bar(Gamma Y), (tau) over tilde (f)) satisfies and prove that, conversely, any quadruple satisfying these restrictions can be lifted to a morphism of Berkovich curves.