Forschungsseminar: Komplexe Geometrie

23.05.2014 Tony Yue Yu (Université Paris-Sud, Orsay)

Tropicalization of the moduli space of stable maps

Abstract: We study tropical geometry using Berkovich's deformation retraction to the skeleton. A basic fact of this "global tropicalization" procedure is that analytic curves in the analytic space map to piecewise linear graphs in the skeleton. This fact can be proved either using the functorial property of tropicalization by formal models or using quantifier eliminations from model theory. I will explain the latter approach, which will then be generalized to study the tropicalization of the moduli space of stable maps. I will prove that the tropicalization of the moduli space of stable maps of bounded genus and degree is a compact finite polyhedral complex. The proof has four major ingredients:

• (i) the k-analytic/tropical Kähler structures,
• (ii) the analytic-tropical continuity theorem via formal models and balancing conditions, (iii) the non-Archimedean analytic Gromov compactness theorem, (iv) the model theory of rigid subanalytic sets develop by Lipshitz. Due to limit of time, I will not go into technical details, so this talk can be followed independently from the previous two talks. The reference is the Section 7 of arXiv:1401.6452.