Geometric models for the derived categories of Ginzburg algebras of n-angulated surfaces via local-to-global principles

Abstract

We consider a class of relative $n$-Calabi--Yau dg-algebras, referred to as relative Ginzburg algebras, associated with marked surfaces equipped with a decomposition into $n$-gons ($n$-angulation). We relate their derived categories to the geometry of the surface. Results include the description of a subset of the objects in the derived categories in terms of curves in the surfaces and their Homs in terms of intersections. The description of these derived categories as the global sections of perverse schobers greatly facilitates the construction of these geometric models, as the construction reduces to gluing local data. This approach may be considered as a generalized, algebraic analogue of matching sphere constructions appearing in the symplectic geometry of Lefschetz fibrations. Most results also hold for the perverse schobers defined over any commutative ring spectrum. As an application of the geometric model in the case $n=3$, we match certain Ext-groups in the derived categories of these relative Ginzburg algebras and the extended mutation matrices of a class of cluster algebras with coefficients, associated to multi-laminated marked surfaces by Fomin-Thurston.