Toric degenerations of cluster varieties and cluster duality

Lara Bossinger, Bosco Frías-Medina, Timothy Magee, and Alfredo Nájera Chávez

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Submission date: 28. Sep. 2018
Bibtex
MSC-Numbers: 14J33, 13F60, 14M25
Keywords and phrases: Toric degeneration, cluster varieties
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
We introduce the notion of a Y -pattern with coefficients and its geometric counterpart: a cluster 𝒳-variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed cluster 𝒳-variety to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with each stratum encoded by Star(τ) for some cone τ of the g-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to 𝒜_prin of Gross-Hacking-Keel-Kontsevich, and the fibers cluster dual to 𝒜_t. Finally, we give two applications. First, we use our construction to identify the Rietsch-Williams toric degeneration of Grassmannians with the Gross-Hacking-Keel-Kontsevich degeneration in the case of Gr₂(ℂ⁵). Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.