Toric degenerations of cluster varieties and cluster duality
Toric degenerations of cluster varieties and cluster duality
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DOI：
10.1112/s0010437x2000740x
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发表时间：
2018-09
影响因子：
1.8
通讯作者：
L. Bossinger;Bosco Fr'ias-Medina;Timothy Magee;Alfredo Nájera Chávez
中科院分区：
文献类型：
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作者：
L. Bossinger;Bosco Fr'ias-Medina;Timothy Magee;Alfredo Nájera Chávez
我们引入了具有系数的$Y$-模式及其几何对应的概念：具有系数的$\数学{X}$-簇簇。利用这些构造，我们建立了每一个特殊完成的斜对称化的$\mathcal{X}$-簇簇$\widehat{\mathcal{X}}$到它的g-扇的环簇的平坦退化。此外，我们还证明了这个家族的纤维是以一种自然的方式分层的，其中的层是特别完成的数学{X}-簇，由$\操作符名称{Star}(\tau)$对$\mathbf{g}$-扇的每个锥体$\tau$编码。这些层退化为中央纤维的相关环状层。我们进一步证明了该族是Gross，Hating，Keel和Kontsevich[簇代数的典范基]的$\mathcal{A}_{\mathm{prin}}$的簇对偶.数学课。SoC。31(2018)，497-608]，并且纤维簇偶为$\mathcal{A}_t$。最后，我们给出了两个应用。首先，我们利用我们的构造从Rietsch和Williams[牛顿-奥孔科夫体，簇对偶，以及Grassmannians的镜像对称性，Duke Math]中识别Grassmannians的Toric简并。J.168(2019)，3437-3527]在$\OPERATOR NAME{Gr}_2(\mathbb{C}^{5})$的情况下，具有Gross-hking-Keel-Kontsevich退化。接下来，在镜像对称的背景下，我们用它将簇对偶与Gorenstein环面Fanos的Batyrev-Borisv对偶联系起来。
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.