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MiS Preprint
84/2018
Toric degenerations of cluster varieties and cluster duality
Lara Bossinger, Bosco Frías-Medina, Timothy Magee and Alfredo Nájera Chávez
We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: a cluster $\mathcal X$-variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed cluster $\mathcal X$-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 each stratum encoded by $\text{Star}(\tau)$ for some 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 $\mathcal A_{\text{prin}}$ of Gross-Hacking-Keel-Kontsevich, and the fibers cluster dual to $\mathcal A_t$. 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 $\text{Gr}_2(\mathbb C^5)$. Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.