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.
Bibliographical noteFunding Information:
Received February 14, 2018, and, in revised form, June 30, 2018 and July 19, 2018. This work was supported by the Israel Science Foundation (grant No. 1159/15).
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