Abstract categorical residues and Calabi-Yau structures

TL;DR

This paper explores the relationship between categorical residues and Calabi-Yau structures, showing that certain categorical constructions inherit Calabi-Yau properties from their original categories, with applications in symplectic and algebraic geometry.

Contribution

It establishes that the categorical formal punctured neighborhood of infinity inherits a weak proper Calabi-Yau structure from a given weak smooth Calabi-Yau category, extending duality principles.

Findings

The categorical neighborhood of infinity has a weak proper Calabi-Yau structure of dimension n-1.

Applications include Calabi-Yau structures on Rabinowitz Fukaya categories and Orlov's singularity categories.

The results connect geometric dualities with categorical Calabi-Yau structures.

Abstract

Inspired by the simple fact that a compact n-dimensional manifold-with-boundary which satisfies Poincar\'e-Lefschetz duality of dimension n has a boundary which itself satisfies Poincar\'e duality of dimension n, we show that the categorical formal punctured neighborhood of infinity, a canonical categorical construction associated to every $A_{\infty}$ category, has a weak proper Calabi-Yau structure of dimension n-1 whenever the original $A_{\infty}$ category admits a weak smooth Calabi-Yau structure of dimension n. Applications include proper Calabi-Yau structures on Rabinowitz Fukaya category of a Liouville manifold and Orlov's singularity category of a proper singular Gorenstein scheme of finite type.