On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov
On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov
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DOI：
10.1112/s0010437x22007886
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发表时间：
2021-10
影响因子：
1.8
通讯作者：
Shaoyun Bai;Laurent Cot'e
中科院分区：
文献类型：
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作者：
Shaoyun Bai;Laurent Cot'e
我们研究了Liouville流形和流形对的包裹福谷范畴的Rouquier维数，并将此不变量应用于代数和辛几何中的各种问题。在代数几何方面，我们引入了一种新的方法，基于辛的灵活性和镜像对称绑定的Rouquier维数的衍生类别的相干层在某些复杂的代数簇和堆栈。这些边界在维度上是尖锐的，最多为3美元。作为一个应用程序，我们解决了一个著名的猜想奥尔洛夫的新类的例子（如复曲面3 $-倍，某些日志卡-丘表面）。我们还讨论了部分包裹福谷范畴上的非交换动机的应用。在辛方面，我们研究各种定量问题，包括以下内容。(1)给定一个Weinstein流形，在一个一般的紧支撑的Hamilton同构下，骨架和它的像之间的最小交点数是多少？(2)在具有Weinstein纤维的Liouville流形上，Lefschetz纤维化的临界点的最小数目是多少？我们给出了这些数量的下界，据作者所知，它们是第一个超越基本的柔性/刚性二分法的。
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.