[Paper Review] Cluster X-varieties at infinity

V. V. Fock, A. B. Goncharov|arXiv (Cornell University)|Apr 3, 2011

Algebraic structures and combinatorial models9 references18 citations

TL;DR

This paper introduces a special completion of cluster Poisson varieties—called the tropical compactification—that generalizes Thurston’s compactification of Teichmüller space. It defines a stratified completion where coordinate tori extend to affine spaces, and the boundary strata correspond to simple X-laminations, with codimensions varying from one to even numbers, unifying positive geometry and Teichmüller theory via tropical methods.

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 the Thurston compactification of a Teichmuller space. The tropical boundary of a positive space is a sphere with a piecewise linear structure. Cluster X-varieties are positive spaces of rather special type. We define special completions of cluster X-varieties. They have a stratification whose strata are (affine closures of) cluster X-varieties. The original coordinate tori extend to coordinate affine spaces in the completion. We define completions of Teichmuller spaces for surfaces with marked points at the boundary. The set of positive points of the special completion of the corresponding cluster X-variety is a part of the completion of the Teichmuller space.

Motivation & Objective

• To define a canonical completion of cluster Poisson varieties that extends their positive atlas to a stratified structure.
• To generalize Thurston’s compactification of Teichmüller space using the framework of positive spaces and tropical geometry.
• To establish a correspondence between boundary strata of the completion and simple X-laminations on decorated surfaces.
• To develop a theory of convex subsets in tropical spaces using extremal positive functions and Minkowski sums.
• To unify the geometric compactification of Teichmüller spaces with algebraic structures in cluster varieties.

Proposed method

• Define a tropical compactification of a positive space using the closure of the positive real points in a toric variety associated to the tropicalization of the atlas.
• Construct the special completion $\widehat{\cal X}$ of a cluster Poisson variety $\cal X$, where coordinate tori $({\mathbb{C}}^*)^n$ extend to affine spaces $\mathbb{A}^n$.
• Use the semiring $\mathbb{L}_+({\cal X})$ of positive regular functions and its extremal elements $\mathbb{E}({\cal X})$ to define convex subsets in the tropical space $\cal X(\mathbb{A}^t)$.
• Define spherical convex subsets via inequalities $F^t(x) \leq 0$ for $F \in \mathbb{L}_+({\cal X})$, with Minkowski sum structure $S_{F_1} * S_{F_2} = S_{F_1F_2}$.
• Establish that convex subsets in the tropical space are intersections of basic convex sets defined by extremal functions and rational constants.
• Apply the framework to decorated surfaces $\mathbb{S}$, showing that the completion of the enhanced Teichmüller space arises as a special case of the completion of $\cal X_{PGL_2,\mathbb{S}}$.

Experimental results

Research questions

• RQ1 How can the Teichmüller space of a decorated surface be compactified using cluster Poisson geometry?
• RQ2 What is the structure of the boundary strata in the special completion of a cluster Poisson variety?
• RQ3 How do X-laminations on decorated surfaces parametrize the strata of the completion?
• RQ4 What is the role of positive regular functions and their tropicalizations in defining convexity in tropical spaces?
• RQ5 How does the Minkowski sum structure on spherical convex subsets relate to the algebraic structure of the positive semiring?

Key findings

• The special completion $\widehat{\cal X}$ of a cluster Poisson variety $\cal X$ is a stratified space where each stratum is itself a cluster Poisson variety.
• The coordinate tori $({\mathbb{C}}^*)^n$ of $\cal X$ extend to affine spaces $\mathbb{A}^n$ in $\widehat{\cal X}$, generalizing the notion of compactification.
• The boundary strata of the completion correspond bijectively to simple X-laminations on the decorated surface $\mathbb{S}$, with codimension one strata arising from non-boundary paths and codimension two from loops.
• The completion of the enhanced Teichmüller space for a decorated surface $\mathbb{S}$ is realized as the positive real points of $\widehat{\cal X}_{PGL_2,\mathbb{S}}$.
• Convex subsets in the tropical space $\cal X(\mathbb{A}^t)$ are defined via inequalities $E^t(x) \leq a_E$, with Minkowski sum structure $A_1 * \cdots * A_n$ given by summing the constants $a_E^{(i)}$.
• The spherical convex subsets form a semiring under intersection (addition) and Minkowski sum (multiplication), with the map $F \mapsto S_F$ a semiring morphism from $\mathbb{L}_+({\cal X})$ to the semiring of convex subsets.