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Article Dans Une Revue Selecta Mathematica (New Series) Année : 2016

Cluster Poisson varieties at infinity

Résumé

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 X is covered by a collection of coordinate tori , 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 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 $\mathbb{S}$ [FG1] is a part of the completion of the Teichmüller space.
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Dates et versions

hal-01897341 , version 1 (17-10-2018)

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Vladimir V. Fock, Alexander Goncharov. Cluster Poisson varieties at infinity. Selecta Mathematica (New Series), 2016, 22 (4), pp.2569 - 2589. ⟨10.1007/s00029-016-0282-6⟩. ⟨hal-01897341⟩
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