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Derived equivalences of gentle algebras via Fukaya categories

Research output: Contribution to journalArticle

Yanki Lekili, Alexander Polishchuk

Original languageEnglish
Pages (from-to)187-225
Number of pages39
JournalMathematische Annalen
Volume376
Issue number1-2
Early online date29 Aug 2019
DOIs
Accepted/In press20 Aug 2019
E-pub ahead of print29 Aug 2019
Published1 Feb 2020

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Abstract

Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), to any homologically smooth Z-graded gentle algebra A we associate a triple (ΣA,ΛA;ηA), where ΣA is an oriented smooth surface with non-empty boundary, ΛA is a set of stops on ∂ΣA and ηA is a line field on ΣA, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of (ΣA,ΛA;ηA). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of ΣA 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 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)

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