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

   https://doi.org/10.1112/S0010437X22007886  

   Bai, Shaoyun; Côté, Laurent (March 2023, Compositio Mathematica)

   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 an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $$3$$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative 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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2. Homological mirror symmetry for higher-dimensional pairs of pants

   https://doi.org/10.1112/S0010437X20007150  

   Lekili, Yankı; Polishchuk, Alexander (July 2020, Compositio Mathematica)

   Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $$\mathbb{CP}^{n}$$ , for $$k\geqslant n$$ , with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $$\mathbb{C}P^{n}$$ ( $$n$$ -dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $$x_{1}x_{2}\ldots x_{n+1}=0$$ . By localizing, we deduce that the (fully) wrapped Fukaya category of the $$n$$ -dimensional pair of pants is equivalent to the derived category of $$x_{1}x_{2}\ldots x_{n+1}=0$$ . We also prove similar equivalences for finite abelian covers of the $$n$$ -dimensional pair of pants. 

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3. Covariantly functorial wrapped Floer theory on Liouville sectors

   https://doi.org/10.1007/s10240-019-00112-x  

   Ganatra, Sheel; Pardon, John; Shende, Vivek (August 2019, Publications mathématiques de l'IHÉS)

   We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid’s generation criterion follows. 

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4. Fukaya categories of surfaces, spherical objects and mapping class groups

   https://doi.org/10.1017/fms.2021.21  

   Auroux, Denis; Smith, Ivan (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $$2$$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $$1$$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included. 

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5. Arboreal singularities from Lefschetz fibrations

   Shende, Vivek (January 2021, Proceedings of 26th Gökova Geometry-Topology Conference)

   Nadler introduced certain Lagrangian singularities indexed by trees, and determined their microlocal sheaves to be the category of modules over the corresponding tree quiver. Another family of spaces indexed by trees: the tree plumbings of spheres. The Fukaya-Seidel category of the Lefschetz fibration with this plumbing as fiber and all spheres as vanishing cycles is well known to also be modules over the tree quiver. Here we upgrade this matching of categories to a matching of geometry. 

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