Derived equivalences of gentle algebras via Fukaya categories
| Autor: | Alexander Polishchuk, Yanki Lekili |
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| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Derived category Mathematics::Operator Algebras General Mathematics Homotopy 010102 general mathematics Sigma Boundary (topology) 01 natural sciences Mapping class group Mathematics - Algebraic Geometry Mathematics - Symplectic Geometry 0103 physical sciences FOS: Mathematics Symplectic Geometry (math.SG) 010307 mathematical physics 0101 mathematics Orbit (control theory) Representation Theory (math.RT) Fukaya category Algebraic Geometry (math.AG) Mathematics - Representation Theory Symplectic geometry Mathematics |
| Zdroj: | Lekili, Y & Polishchuk, A 2020, ' Derived equivalences of gentle algebras via Fukaya categories ', Mathematische Annalen, vol. 376, no. 1-2, pp. 187-225 . https://doi.org/10.1007/s00208-019-01894-5 |
| DOI: | 10.48550/arxiv.1801.06370 |
| Popis: | Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth 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 math/0607348, 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 arXiv:1705.06023. Comment: 35 pages. To appear in Mathematische Annalen |
| Databáze: | OpenAIRE |
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