Toric degenerations of cluster varieties and cluster duality. (7th October 2020)

• Record Type: Journal Article
• Authors: Bossinger, Lara; Frías-Medina, Bosco; Magee, Timothy; Nájera Chávez, Alfredo
• Is Part Of: Compositio mathematica. Volume 156:Number 10(2020)
• Page Start: 2149
• Page End: 2206
• Publication Date: 2020-10-07
• Subjects: cluster algebras, -- cluster varieties, -- toric degenerations; 13F60, -- 14M25, -- 14D06; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X2000740X
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : We introduce the notion of a $Y$ -pattern with coefficients and its geometric counterpart: an $\mathcal {X}$ -cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$ -cluster variety $\widehat {\mathcal {X} }$ 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 $\mathcal {X}$ -varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$ -fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ 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 $\mathcal {A} _t$ . 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 $\operatorname {Gr}_2(\mathbb {C} ^{5})$ . Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.