Resumen de Cluster Poisson varieties at infinity

V. V. Fock, A.B. Goncharov

• 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).