Derived equivalences of gentle algebras via Fukaya categories

By Yankı Lekili, Alexander Polishchuk. Mathematische Annalen, 2018

101 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1e6bbc380b55c8ca0af3016b921df78378b05e86

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, methodology): “Indeed, the line field considered in [35] is tangent to the ◦-dissection.”
Auslander orders over nodal stacky curves and partially wrapped Fukaya categories (2017, background): “In a follow-up paper [18], we described explicitly invariants of line fields under this action.”