Categorical action filtrations via localization and the growth as a symplectic invariant

TL;DR

This paper develops a categorical framework for action filtrations and growth invariants, connecting geometric and algebraic categories, and applies it to mirror symmetry and symplectic invariants.

Contribution

It introduces the notion of smooth categorical compactification to define and compute growth invariants, linking geometric filtrations with categorical growth measures.

Findings

Growth invariants are preserved under zig-zags of compactifications.

Reeb-length growth in symplectic cohomology is at most exponential.

Provides lower bounds for entropy of endofunctors on Fukaya categories.

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.