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 algebrogeometric 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 an application, we resolve a wellknown conjecture of Orlov for new classes of examples (e.g. toric $3$ folds, certain log Calabi–Yau surfaces). We also discuss applications to noncommutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (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 the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.
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The six functors for Zariskiconstructible sheaves in rigid geometry
We prove a generic smoothness result in rigid analytic geometry over a characteristic zero nonarchimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a sixfunctor formalism for Zariskiconstructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a wellbehaved theory of perverse sheaves.
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 NSFPAR ID:
 10339580
 Date Published:
 Journal Name:
 Compositio Mathematica
 Volume:
 158
 Issue:
 2
 ISSN:
 0010437X
 Page Range / eLocation ID:
 437 to 482
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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