Derived equivalences of gentle algebras via Fukaya categories
Derived equivalences of gentle algebras via Fukaya categories
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DOI：
10.1007/s00208-019-01894-5
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发表时间：
2018-01
影响因子：
1.4
通讯作者：
Yankı Lekili;A. Polishchuk
中科院分区：
文献类型：
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作者：
Yankı Lekili;A. Polishchuk
遵循 Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017) 的方法，对于任何同调平滑分级的温和代数，我们将一个三元组关联起来，其中 是一个具有非空边界的定向光滑表面，是一组停止点 onand 是一个线场，使得 A 的完美 dg 模的派生范畴等于部分包裹的 Fukaya 类别。修改Johnson和Kawazumi的论点，我们对线场同伦类上的（辛）映射类群的作用的轨道分解进行分类。结果，我们获得了同调平滑分级温和代数导出等价的充分标准。我们的标准使用数值不变量，概括了 Avella–Alaminos–Geiss 在 Avella 等人中给出的数值不变量。 (J Pure Appl Algebra 212(1):228–243, 2008)，以及其他一些数值不变量。作为一个应用程序，当 AAG 不变量确定派生的 Morita 类时，我们发现了许多新情况。作为另一个应用，我们在 Lekili 和 Polishchuk 中考虑的堆叠节点曲线之间建立了一些导出的等价关系（J Topology 11:615–444, 2018）
Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), to any homologically smooth-graded gentle algebraAwe associate a triple, whereis an oriented smooth surface with non-empty boundary,is a set of stops onandis a line field on, such that the derived category of perfect dg-modules ofAis equivalent to the partially wrapped Fukaya category of. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group ofon 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 Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), 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 Lekili and Polishchuk (J Topology 11:615–444, 2018)