Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

TL;DR

This paper investigates algebraic and symplectic methods to understand compactifications of two-dimensional finite type cluster varieties, linking algebraic polytopes with symplectic fibrations and del Pezzo surfaces.

Contribution

It introduces a dual algebraic and symplectic framework for classifying compactifications of cluster varieties of finite type in complex dimension two.

Findings

Identification of cluster varieties as complements of divisors in del Pezzo surfaces

Description of these varieties via almost toric fibrations

Comparison of algebraic and symplectic compactifications and their polytopes

Abstract

In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17]. In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over $\R^2$ (after completion). Once identifying them as almost toric manifolds, one can symplectically view them inside other del Pezzo surfaces. So we can identify other symplectic compactifications of the same cluster variety, which we expect should also correspond to different algebraic compactifications. Both viewpoints are presented here and several compactifications have their corresponding polytopes compared. The finiteness of the cluster mutations are explored to provide cycles in the graph describing monotone Lagrangian tori in del Pezzo surfaces connected via almost toric mutation [34].