分类作用的局部滤波和辛不变量的增长

IF 1.2 1区 数学 Q1 MATHEMATICS Journal fur die Reine und Angewandte Mathematik Pub Date : 2021-08-12 DOI:10.1515/crelle-2023-0037
Lauren Cote, Yusuf Barış Kartal
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引用次数: 13

摘要

摘要本文建立了作用过滤及其相关增长不变量的纯范畴理论。当专门用于几何兴趣范畴时,例如Weinstein流形的包裹Fukaya范畴,以及光滑代数变化上相干束的有界派生范畴,我们的范畴作用滤波本质上恢复了先前研究的几何起源滤波。我们的方法是围绕光滑分类紧化的概念建立的。我们证明了光滑范畴紧化诱导了定义良好的生长不变量,并且在这种紧化的锯齿形下保持了生长不变量。本文的技术核心是用(bi)模的某些极限的增长来计算这些增长不变量的方法。在实践中,这样的边界出现在两种感兴趣的几何设置中。主要应用有:(1)对同调镜像对称的“定量”改进,它将辛几何侧的reeb长度滤过的增长与无穷远处由极点阶定义的代数几何侧的滤过的增长联系起来(通常可以用支撑轴的尺寸来表示)。(2)证明了Weinstein流形上辛上同调和包花上同调的Reeb-length增长最多是指数增长。(3)作用于Fukaya类的某些自然内函子的熵和多项式熵的下界。
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Categorical action filtrations via localization and the growth as a symplectic invariant
Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.
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来源期刊
CiteScore
2.50
自引率
6.70%
发文量
97
审稿时长
6-12 weeks
期刊介绍: The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle"s Journal. In the almost 180 years of its existence, Crelle"s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals. It belongs to the very top mathematics periodicals, as listed in ISI"s Journal Citation Report.
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