Cluster Poisson varieties at infinity

Abstract

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 ${\mathcal X}$-variety [2], is covered by a collection of coordinate tori $({{\mathbb C}}^*)^n$, which form a positive atlas of a specific kind. We define a special completion $\widehat{\mathcal X}$ of ${\mathcal X}$. It has a stratification whose strata are cluster Poisson varieties. The coordinate tori of ${\mathcal X}$ extend to coordinate affine spaces ${\mathbb A}^n$ in $\widehat{\mathcal X}$. We define completions of Teichmüller spaces for decorated surfaces ${\mathbb S}$ with marked points at the boundary. The set of positive points of the special completion of the cluster Poisson variety ${\mathcal X}_{ PGL _2, {\mathbb S}}$ related to the Teichmüller theory on ${\mathbb S}$ [1] is a part of the completion of the Teichmüller space (see Fig. 1 on the next page).