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

TL;DR

This paper investigates the Rouquier dimension of wrapped Fukaya categories in symplectic geometry and uses it to resolve a conjecture of Orlov in algebraic geometry, providing new bounds and insights into the structure of derived categories and symplectic invariants.

Contribution

It introduces a novel method combining symplectic flexibility and mirror symmetry to bound Rouquier dimensions and resolves Orlov's conjecture for a broad class of algebraic varieties, including toric 3-folds.

Findings

Bounds on Rouquier dimension are sharp in dimension up to 3.

Resolved Orlov's conjecture for toric 3-folds and certain log Calabi--Yau surfaces.

Established lower bounds for intersection points and critical points in symplectic geometry.

Abstract

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound 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. On the symplectic side, we study various quantitative 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? We give lower bounds for these quantities which are to our knowledge the first to go beyond the basic flexible/rigid dichotomy.