Toric degenerations of cluster varieties and cluster duality

Abstract

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 $\hat{\mathcal{X}}$ 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 $\mathrm{Star}(\tau )$ 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-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 ${\mathrm{Gr}}_2({\mathbb{C}}^5)$. Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

53 pages. Published in Compositio Mathematica. Fixed typo in Figure 1 -- thanks to Antoine Bourget for spotting this and pointing it out