Toric degenerations of cluster varieties and cluster duality. (7th October 2020) - Bossinger, Lara; Frías-Medina, Bosco; Magee, Timothy; Nájera Chávez, Alfredo
Abstract : 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 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(τ ) for each 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 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.