论文标题
Fock-Goncharov双重群集品种和毛siebert镜子
Fock-Goncharov dual cluster varieties and Gross-Siebert mirrors
论文作者
论文摘要
群集品种成对出现：对于任何$ \ MATHCAL {x} $ cluster品种，都有关联的Fock-Goncharov dual $ \ Mathcal {a} $ cluster variety。另一方面，在镜像对称的背景下，与任何log calabi-yau品种相关的是镜偶，可以在Gross-Siebert程序框架中使用理性曲线的枚举几何形状构建。在本文中，我们将群集品种的理论桥接了大镜子对称性的代数几何框架。特别是，我们证明了$ \ Mathcal {x} $ cluster品种的镜子是Fock-Goncharov dual $ \ Mathcal {a} $ cluster virets的变性，反之亦然。为此，我们研究了毛keel-keel-kontsevich的群集散射图如何与Gross-Siebert定义的规范散射图进行比较，以在任意维度中构造镜子偶。因此，我们得出了群集散射图的枚举解释。一路上，我们证明了Frobenius结构的猜想，用于将作为折叠品种的爆炸获得的一类log calabi-yau品种。
Cluster varieties come in pairs: for any $\mathcal{X}$ cluster variety there is an associated Fock-Goncharov dual $\mathcal{A}$ cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi-Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross-Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross-Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{X}$ cluster variety is a degeneration of the Fock-Goncharov dual $\mathcal{A}$ cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich compares with the canonical scattering diagram defined by Gross-Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi-Yau varieties obtained as blow-ups of toric varieties.
