Jacobian discrepancies and rational singularities

TL;DR

This paper introduces Jacobian discrepancy as a new invariant for singular varieties, extending classical discrepancy concepts, and uses it to characterize rational and Du Bois singularities with broad implications for algebraic geometry.

Contribution

It develops the Jacobian discrepancy invariant, establishing a comprehensive framework for adjunction and inversion of adjunction applicable to all singular varieties.

Findings

Provides a formula for the gap between dualizing and Grauert-Riemenschneider sheaves.

Characterizes rational and Du Bois singularities via Jacobian discrepancies.

Extends theorems on singularities to cases with Q-Cartier canonical class.

Abstract

Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Riemenschneider canonical sheaf of a normal variety. As an application, we give characterizations for rational and Du Bois singularities on normal Cohen-Macaulay varieties in terms of Jacobian discrepancies. In the case when the canonical class of the variety is Q-Cartier, our result provides the necessary corrections for the converses to hold in theorems of Elkik, of Kova\'cs, Schwede and Smith, and of Kolla\'r and Kova\'cs on rational and Du Bois singularities.