May 5, 2023 - First meeting in the Summer Semester 2023
The talks will be given in a hybrid format. If you are close-by, please join us in Frankfurt in room 711 (groß), Robert-Mayer-Str. 10, for three in-person talks. Otherwise, we're hoping to see you on Zoom. The Zoom info will be sent out to the mailing list as usual.
14:00-15:00 Léonard Pille-Schneider (ENS, Paris): The SYZ conjecture for families of hypersurfaces
15:30-16:30 Loujean Cobigo (Universität Tübingen): Tropical spin Hurwitz numbers
16:45-17:45 Antoine Ducros (Sorbonne Université, Paris): Tropical functions on skeletons: a finiteness result
Léonard Pille-Schneider: The SYZ conjecture for families of hypersurfaces
Abstract: Let X -> D* be a polarized family of complex Calabi-Yau manifolds, whose complex structure degenerates in the worst possible way. The SYZ conjecture predicts that the fibers X_t, as t ->0, degenerate to a tropical object; and in particular the program of Kontsevich and Soibelman relates it to the Berkovich analytification of X, viewed as a variety over the non-archimedean field of complex Laurent series.
I will explain the ideas of this program and some recent progress in the case of hypersurfaces.
Loujean Cobigo: Tropical spin Hurwitz numbers
Abstract: Classical Hurwitz numbers count the number of branched covers of a fixed target curve that exhibit a certain ramification behaviour. It is an enumerative problem deeply rooted in mathematical history. A modern twist: Spin Hurwitz numbers were introduced by Eskin-Okounkov-Pandharipande for certain computations in the moduli space of differentials on a Riemann surface. Similarly to Hurwitz numbers they are defined as a weighted count of branched coverings of a smooth algebraic curve with fixed degree and branching profile. In addition, they include information about the lift of a theta characteristic of fixed parity on the base curve.
In this talk we investigate them from a tropical point of view. We start by defining tropical spin Hurwitz numbers as result of an algebraic degeneration procedure, but soon notice that these have a natural place in the tropical world as tropical covers with tropical theta characteristics on source and target curve.
Our main results are two correspondence theorems stating the equality of the tropical spin Hurwitz number with the classical one.
Antoine Ducros: Tropical functions on skeletons: a finiteness result
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.