Orlov spectra: bounds and gaps

TL;DR

This paper investigates the Orlov spectrum of triangulated categories, introducing gaps and bounds, with applications to mirror symmetry, singularities, and symplectic geometry, providing new examples, bounds, and geometric insights.

Contribution

It introduces the notion of gaps in the Orlov spectrum, provides the first examples of finite ultimate dimension in categories of singularities, and connects generation times to geometric and symplectic invariants.

Findings

Finite ultimate dimension for categories of singularities of isolated hypersurface singularities

Bounded generation time for objects in derived categories of Calabi-Yau hypersurfaces

Lower bounds on the ultimate dimension of derived Fukaya categories of higher genus surfaces

Abstract

The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator's generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one.