A Riemann–Hurwitz formula for skeleta in non-Archimedean geometry

Abstract

Let k be an algebraically closed non-Archimedean non-trivially real valued field which is complete with respect to its valuation. Let \(\phi :C' \rightarrow C\) be a finite morphism between smooth projective irreducible k-curves. The morphism \(\phi \) induces a morphism \(\phi ^{\mathrm {an}} :C'^{\mathrm {an}} \rightarrow C^{\mathrm {an}}\) between the Berkovich analytifications of the curves. We construct a pair of deformation retractions of \(C'^{\mathrm {an}}\) and \(C^{\mathrm {an}}\) which are compatible with the morphism \(\phi ^{\mathrm {an}}\) and whose images \(\Sigma _{C'^{\mathrm {\mathrm {an}}}},\Sigma _{C^{\mathrm {\mathrm {an}}}}\) are closed subspaces of that are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces \(\Sigma _{C'^{\mathrm {\mathrm {an}}}}\) and \(\Sigma _{C^{\mathrm {\mathrm {an}}}}\) are such that their complements in their respective analytifications decompose into the disjoint union of isomorphic copies of Berkovich open disks. The pair of compatible deformation retractions forces the morphism \(\phi ^{\mathrm {an}}\) to restrict to a map \(\Sigma _{C'^{\mathrm {\mathrm {an}}}} \rightarrow \Sigma _{C^{\mathrm {\mathrm {an}}}}\). The first Betti number of the skeleton C is well defined and an invariant of the curve which we call \(g^{\mathrm {an}}(C)\). We study how the genus of \(\Sigma _{C'^{\mathrm {\mathrm {an}}}}\) can be calculated using the morphism \(\phi ^{\mathrm {an}}_{|\Sigma _{C'^{\mathrm {\mathrm {an}}}}}\) and invariants defined on \(\Sigma _{C^{\mathrm {an}}}\) via the deformation retractions.