Residues and duality for singularity categories of isolated Gorenstein singularities

TL;DR

This paper explores Serre duality in the singularity category of isolated Gorenstein singularities, providing explicit formulas involving residues and generalised fractions, and recovering known results for hypersurfaces.

Contribution

It introduces an explicit construction of complete injective resolutions for maximal Cohen-Macaulay modules, enabling concrete formulas for duality pairings.

Findings

Derived explicit residue formulas for duality pairings

Connected singularity categories with string theory residue formulas

Constructed complete injective resolutions for Cohen-Macaulay modules

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.