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Toric degenerations of cluster varieties and cluster duality. (English) Zbl 1454.13030

Summary: 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 \(\mathbf{g} \)-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 M. Gross et al. [J. Am. Math. Soc. 31, No. 2, 497–608 (2018; Zbl 1446.13015)], 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 K. Rietsch and L. Williams [Duke Math. J. 168, No. 18, 3437–3527 (2019; Zbl 1439.14142)] 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.

MSC:

13F60 Cluster algebras
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14D06 Fibrations, degenerations in algebraic geometry
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