论文标题
平滑，散射和福卡亚的猜想
Smoothing, scattering, and a conjecture of Fukaya
论文作者
论文摘要
在2002年，福卡亚通过引入两个对应关系提出了一个镜子对称性的显着解释：在calabi-yau歧管$ \ check {x} $上的伪旋晶曲线理论之间\ check {x} \ to \ check check {b} $与镜子$ x $的变形理论与$ \ check {b} $的相同多价值摩尔斯理论之间的另一个。在本文中，我们证明了对福卡亚第二信函中的主要猜想的重新制定，其中基本$ \ check {b} $的多价值摩尔斯理论被legendre dual dual $ b $上的热带几何形状取代。在证明中，我们应用了以前的作品中开发的渐近分析技术，以热带DGBV前代数，该代数控制了我们最近的另一项作品中引入的最大变性calabi-yau对数的平滑。然后，此热带代数与与半流动部分的变形理论相关的DGBV代数之间的比较，$ x _ {\ text {sf}} \ subseteq x $允许我们从适当的Maurer-Cartan解决方案中提取一致的散射图。
In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.
