Auslander orders over nodal stacky curves and partially wrapped Fukaya categories

Abstract

It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order ${\mathcal{A}}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409.8611 that for punctured surfaces $X$ with stops $\Lambda$ at their boundary, the partially wrapped Fukaya category $\mathcal{W}(X,\Lambda )$ provides a categorical (smooth and proper) resolution of the compact Fukaya category $\mathcal{F}(X)$. Inspired by this analogy, we establish an equivalence between the derived category of modules over the Auslander orders over certain nodal stacky curves and partially wrapped Fukaya categories associated to punctured surfaces of arbitrary genus equipped with stops at their boundary. As an application, we deduce equivalences between derived categories of coherent sheaves (resp. perfect complexes) on such nodal stacky curves and the wrapped (resp. compact) Fukaya categories of punctured surfaces of arbitrary genus.

34 pages, 12 figures, accepted for publication by the Journal of Topology

References in corpus (3)

• Duality between Lagrangian and Legendrian invariants
• Arithmetic mirror symmetry for the 2-torus
• Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions

Cited by in corpus (16)

• A geometric model for the derived category of gentle algebras
• Homological mirror symmetry for higher dimensional pairs of pants
• The symplectic geometry of higher Auslander algebras: Symmetric products of disks
• q-Stability conditions on Calabi-Yau-X categories
• On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems
• On auto-equivalences and complete derived invariants of gentle algebras
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Homological mirror symmetry for invertible polynomials in two variables
• Non-commutative nodal curves and derived tame algebras
• Morita theory for non-commutative noetherian schemes
• Lagrangian Cobordisms in Liouville manifolds
• A mnemonic for the Lipshitz-Ozsváth-Thurston correspondence
• The Hall Algebras of Surfaces I
• Homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Fukaya categories of higher-genus surfaces and pants decompositions
• Line fields on punctured surfaces and twisted derived categories