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Multiplier ideals and klt singularities via (derived) splittings

Peter M. McDonald

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 $π_*ω_Y\to\mathcal{O}_X$ where $π:Y\to X$ 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 Kovács 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.

Multiplier ideals and klt singularities via (derived) splittings

Abstract

Let be a normal, excellent, noetherian scheme over with a dualizing complex. In this note, we find an alternate characterization of the multiplier ideal of , as defined by de Fernex-Hacon, by considering maps where 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 Kovács and Bhatt. We also give an analogous description of the test ideal in characteristic as a corollary to a result of Epstein-Schwede.
Paper Structure (3 sections, 22 theorems, 42 equations)

This paper contains 3 sections, 22 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Characteristic zero
  3. Positive characteristic

Key Result

Theorem 1.1

Let $X$ be a normal, excellent, noetherian scheme over $\operatorname{Spec}\mathbb{Q}$ with a dualizing complex and $I=\prod\mathscr{J}_k^{a_k}$ be a formal effective $\mathbb{Q}$-linear combination of ideal sheaves on $X$. The multiplier ideal $\mathscr{J}(X,I)$ can be realized as where $\pi:Y\to X$ ranges over all log regular alterations of $(X,I)$ with $E_Y=\sum a_kE_k$ for $\mathscr{J}_k\math

Theorems & Definitions (48)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3: Bha12 Theorem 2.12, Kov00 Theorem 3
  • Corollary 1.4
  • Proposition 1.5
  • Remark 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 38 more