Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

Abstract: Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry, and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Hülya Argüz.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

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