Toric degenerations of cluster varieties and cluster duality

We introduce the notion of a Y-pattern with coefficients and its geometric counterpart: an X-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed X-cluster variety (X) over bar 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 strata the specially completed X-varieties encoded by Star(tau) for each cone tau 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 and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497-608], and the fibers cluster dual to A(t). Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437-3527] with the Gross-Hacking-Keel-Kontsevich degeneration in the case of Gr(2)(C-5). Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.