Multiplier ideals and klt singularities via (derived) splittings

TL;DR

This paper provides a new characterization of multiplier ideals and klt singularities using derived splittings and regular alterations, extending known concepts to a broader algebraic geometric context.

Contribution

It introduces an alternative definition of multiplier ideals via regular alterations and offers a derived splinter perspective on klt singularities, connecting to rational singularities.

Findings

New characterization of multiplier ideals using maps from alterations

Derived splinter characterization of klt singularities

Description of test ideals in characteristic p>2

Abstract

Let $X$ be a normal, excellent, noetherian scheme over $\operatorname{Spec}\mathbb{Q}$ with a dualizing complex. In this note, we find an alternate characterization of the multiplier ideal of $X$, as defined by de Fernex-Hacon, by considering maps $\pi_*\omega_Y\to\mathcal{O}_X$ where $\pi:Y\to X$ ranges over all regular alterations. As a corollary to this result, we give a derived splinter characterization of klt singularities, akin to the characterization of rational singularities given by Kov\'acs and Bhatt. We also give an analogous description of the test ideal in characteristic $p>2$ as a corollary to a result of Epstein-Schwede.