Cluster Poisson varieties at infinity

• V. V. Fock ^[1] ; A. B. Goncharov ^[2]
  1. [1] University of Strasbourg

     University of Strasbourg

     Arrondissement de Strasbourg-Ville, Francia

     
  2. [2] Yale University

     Yale University

     Town of New Haven, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 22, Nº. 4 (Special Issue: The Mathematics of Joseph Bernstein), 2016, págs. 2569-2589
• Idioma: inglés
• DOI: 10.1007/s00029-016-0282-6
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• Resumen

  • A positive space is a space with a positive atlas, i.e., a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes Thurston’s compactification of a Teichmüller space. A cluster Poisson variety, originally called cluster X -variety [2], is covered by a collection of coordinate tori (C∗)n, which form a positive atlas of a specific kind.

    We define a special completion X of X . It has a stratification whose strata are cluster Poisson varieties. The coordinate tori of X extend to coordinate affine spaces An in X . We define completions of Teichmüller spaces for decorated surfaces S with marked points at the boundary. The set of positive points of the special completion of the cluster Poisson variety XPGL2,S related to the Teichmüller theory on S [1] ] is a part of the completion of the Teichmüller space (see Fig. 1 on the next page).