TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Summer Semester 2023)

Antoine Ducros (Sorbonne Université, Paris)

Abstract: Skeletons are subsets of non-archimedean spaces (in the sense of Berkovich) that inherit from the ambiant space a natural PL (piecewise-linear) structure, and if S is such a skeleton, for every invertible holomorphic function f defined in a neighborhood of S, the restriction of log |f| to S is PL.
In this talk, I will present a joint work with E.Hrushovski F. Loeser and J. Ye in which we consider an irreducible algebraic variety X over an algebraically closed, non-trivially valued and complete non-archimedean field k, and a skeleton S of the analytification of X defined using only algebraic functions, and consisting of Zariski-generic points. If f is a non-zero rational function on X then log |f| induces a PL function on S, and if we denote by E the group of all PL functions on S that are of this form, we  prove the following finiteness result on the group E: it is stable under min and max, and there exist finitely many non-zero rational functions f_1,…f_m on X such that E is generated, as a group equipped with min and max operators, by the log |f_i| and the constants |a| for a in k^*. Our proof makes a crucial use of Hrushovski-Loeser’s model-theoretic approach of Berkovich spaces.