Appears in collection : Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems

One can often assign to a geometric object over a non-archimedean field geometric objects over the group of values - a skeleton or a tropicalization, and over the residue field - a reduction. Sometimes one can combine them in a single object, that we call tropical reduction. In simple cases it is something well known, like a log variety, but often the definition is still rather ad hoc. In my talk I will discuss two cases which reveal a surprising similarity : a wildly ramified cover of curves with minimal wild ramification (joint work with U. Brezner), and a curve with a differential form when the residue characteristic is zero (joint work with I. Tyomkin). In both cases there is a lifting theorem indicating that our definition is the correct one and we do not lose any residual/tropical information : any compatible tropical and residual data can be lifted to an object over the non-archimedean field.