Albanese varieties of non-Archimedean uniformized varieties /

• Autores: Iago Giné Vázquez
• Directores de la Tesis: Francesc Xavier Xarles Ribas (dir. tes.)
• Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2017
• Idioma: español
• Tribunal Calificador de la Tesis: Martín Sombra (presid.) , Joaquim Roé Vellvé (secret.) , Xavier Guitart (voc.)
• Enlaces
  • Tesis en acceso abierto en:  DDD  TDX 
• Resumen

  • In this PhD thesis, we give a conjectural construction of the Albanese variety of a non-Archimedean uniformized variety by means of looking at a metrical structure contained in the last one and called its skeleton (which is, essentially, a tropical curve). In order to do that we make a parallel (real, tropical) construction on the skeleton, which becomes the skeleton of the conjectural Albanese we were seeking for, we rise this to an analytic construction over the given variety and we use that its skeleton is the quotient of a certain locally finite subbuilding of a Bruhat-Tits building, which appears also as the skeleton of the uniformizing variety. Later, we relate the harmonic cochains on the buildings with the harmonic measures on the ends as a key step in the proof that one gets the Albanese variety.

    The thesis has two main parts. One is devoted to describe with complete generality this construction in dimension 1, while in the second we study the structure of the Bruhat-Tits building and we make a construction that we expect it is the Albanese variety, under the assumption that the ground field has a discrete valuation.

    We start by study the Jacobian of a graph, in the chapter 1 without more structure, in the chapter 2 a metric graph. Our work, together with others, shows that our description of the Jacobian of a metric graph in terms of integration on the ends of the universal covering metric tree extends in some way the (discrete) Jacobian of a graph without metric structure. Here we introduce harmonic cochains on the trees and harmonic measures on the ends, and we prove that they are isomorphic, as an important step to the main result.

    In the chapter 3 we develope the theory of Mumford curves and their Jacobians in the setting of Berkovich geometry, we relate them with their skeletons by means of the retraction map and we introduce the multiplicative integrals. Then, we extend to our general hypotheses several known results about them in particular cases (like for a local ground field) with those new tools, and we use the results on the Jacobian of a metric graph applied to the corresponding skeleton to get that the construction we do with multiplicative integrals and harmonic measures is an abelian variety. After that, we prove that it is the Jacobian by means of the theory of theta functions, developed from the new perspective of Berkovich geometry using tropical functions.

    In the chapter 4 we adapt this construction to higher dimension, giving a natural candidate to be the Albanese variety of a non-Archimedean uniformized variety as it was built by Mustafin as a generalization of Mumford curves. In order to do it, we extend the notion of Schottky group to any dimension following the work of Mustafin, and we study deeply the structure of the Bruhat-Tits buildings over a complete field with a discrete valuation. Then, we restrict to dimension 2 to define the harmonic cochains over certain chamber subcomplexes, and we prove that they are isomorphic to the harmonic measures over a certain compact set of the rational points of the corresponding projective space when the associated subcomplex is a building. Finally we use it to start to prove that the construction we do is an analytic torus.