On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

Abstract

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs. We describe upper and lower bounds for this invariant in terms of various quantities of interest in symplectic topology. By combining these bounds, we get information about concrete geometric questions such as: (1) given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly-supported Hamiltonian diffeomorphism? ( 2) what is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers?Using homological mirror symmetry, we also obtain improved upper bounds for the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most 3. As a result, we resolve a well-known conjecture of Orlov for a large class of new examples, including all toric 3-folds and certain log Calabi-Yau surfaces. The proof of these upper bounds crucially relies on the arborealization theorem of Álvarez-Gavela-Eliashberg-Nadler and is therefore an application of symplectic flexibility to a problem in algebraic geometry.

Resolutions of toric subvarieties by line bundles and applications

“…holds for all smooth quasi-projective curves, Orlov conjectures that (1) holds for all smooth quasiprojective schemes. This conjecture has received considerable attention and been established in a variety of cases (see [BC21] for a thorough discussion of known cases). However, the general result remains elusive.…”