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 $\widehat{\mathcal{X}}$ 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 $\mathrm{Star}(τ)$ for each cone $τ$ 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.

Figures (9)

• Figure 1: Visualization of $\widehat{\mathscr{X} }_s$ defined by a $Y$-pattern with principal coefficients in type $A_2$. For more details see Table \ref{['tab:A2_principal coeff']}.
• Figure 2: Quivers associated to a type $A_2$ cluster pattern with principal coefficients at $v_0$. The boxes refer to frozen vertices.
• Figure 3: Comparison of $\mathbf{g}$-fan $\Delta^+_F$ and cluster complex ${\Delta^+}$ for $A_3$ quiver with one frozen direction. Here $Q = \textcolor{blue}{1}\leftarrow 2 \leftarrow 3$ and $F= \left\{1\right\}$. A schematic of the $\mathbf{g}$-fan appears on the left and corresponds to the case in which $A_1$ is non-invertible. The cluster complex is on the right, corresponding to an invertible $A_1$ variable.
• Figure 4: Two adjacent cones $\sigma,\sigma'\subset N_{\mathbb R}$ with $\sigma\cap\sigma'=\tau$ on the left and their dual cones $\sigma^\vee,\sigma'^\vee\subset\tau^\vee\subset M_{\mathbb R}$ on the right.
• Figure 5: $G^{\mathrm{rec}}_{3,5}$ with perfect orientation, as in RW17.

Theorems & Definitions (129)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Definition 2.1
• Example 2.2: Tropical semifields
• Example 2.3: Universal semifields
• Lemma 2.4: Universal property of universal semifields
• Example 2.5