Residues and duality for singularity categories of isolated Gorenstein singularities

TL;DR

This work develops explicit Serre duality in the singularity category for isolated Gorenstein singularities by constructing complete injective resolutions and expressing the duality pairing through local cohomology and generalised fractions. The main result yields a natural, nondegenerate pairing on morphism spaces, which, when evaluated via a residue map, recovers the Kapustin–Li residue formula in the hypersurface setting. The approach unifies Auslander duality with constructive homological tools, enabling explicit residue computations for complete intersections and matrix factorisations. The framework clarifies how boundary observables in physics-inspired categories correspond to closed states via the boundary–bulk map and residue pairings, with Calabi–Yau-type symmetry emerging from the $(d-1)$-Calabi–Yau structure of the singularity category.

Abstract

We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules.

Theorems & Definitions (67)

• Theorem
• Definition 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Theorem 3.5
• proof
• Definition 4.1