Derived equivalences of gentle algebras via Fukaya categories
Yankı Lekili and Alexander Polishchuk King’s College London University of Oregon, National Research University, Higher School of Economics Moscow, Russia, Korea Institute for Advanced Study, Seoul, South Korea
Abstract
Following the approach of Haiden-Katzarkov-Kontsevich [15], to any homologically smooth $\mathbb{Z}$ -graded gentle algebra $A$ we associate a triple $(\Sigma_{A},\Lambda_{A};\eta_{A})$ , where $\Sigma_{A}$ is an oriented smooth surface with non-empty boundary, $\Lambda_{A}$ is a set of stops on $\partial\Sigma_{A}$ and $\eta_{A}$ is a line field on $\Sigma_{A}$ , such that the derived category of perfect dg-modules of $A$ is equivalent to the partially wrapped Fukaya category of $(\Sigma_{A},\Lambda_{A};\eta_{A})$ . Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $\Sigma_{A}$ on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella-Alaminos-Geiss in [7], as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in [21].
中文速览
给定一个温和代数（gentle algebra），它可以对应一个带边界曲面上的Fukaya范畴，但如何判断两个这样的代数具有相同的导出范畴（即导出等价）一直缺乏完整的判据。本文以Haiden-Katzarkov-Kontsevich的框架为基础，先系统分类了带边界曲面上"线场（line field）"在映射类群作用下的轨道，给出由卷绕数（winding number）、一个关于线场是否来自向量场的0/1不变量以及Arf不变量构成的完整数值判据；再利用这套判据，对同调光滑的分次温和代数建立了导出等价的充分条件，推广并细化了Avella-Alaminos–Geiss（AAG）不变量的适用范围。作为应用，文章不仅找到了大量AAG不变量即可决定导出Morita类的新情形，还为带栈节点曲线（stacky nodal curves）建立了一批新的导出等价，为代数与几何两侧的分类问题提供了统一而可操作的工具。
原文 arXiv:1801.06370;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1801.06370v5