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Higher Du Bois and Higher Rational Pairs

Haoming Ning, Brian Nugent

TL;DR

This work develops a unified framework for higher Du Bois and higher rational singularities in the setting of pairs $(X,\Sigma)$ within the minimal model program. It introduces and analyzes the log Du Bois complex for pairs, along with intrinsic duality relations that connect it to higher rational notions, and proves a generalized Kovács–Schwede-type injectivity theorem for pairs. The paper proves that higher rational singularities imply higher Du Bois singularities for pairs, and establishes stability results, Bertini-type theorems, and behavior under finite maps, hyperplane sections, and cyclic covers. Together, these results extend and unify previous single-variety theories (SVV23, Kov25, Kov16a/16b) to the pair setting, enabling inductive and descent techniques in birational geometry and the MMP. The framework provides a robust toolkit for handling higher singularities in paired contexts, with implications for cohomological behavior and resolution of singularities in the MMP.

Abstract

We extend the notions of higher Du Bois and higher rational singularities to pairs in the sense of the minimal model program. We extend numerous results to these higher pairs, including Bertini type theorems, stability under finite maps and that m-rational pairs are m-Du Bois. We prove these using a generalized Kovács-Schwede-type injectivity theorem for pairs, the main technical result of this paper.

Higher Du Bois and Higher Rational Pairs

TL;DR

This work develops a unified framework for higher Du Bois and higher rational singularities in the setting of pairs within the minimal model program. It introduces and analyzes the log Du Bois complex for pairs, along with intrinsic duality relations that connect it to higher rational notions, and proves a generalized Kovács–Schwede-type injectivity theorem for pairs. The paper proves that higher rational singularities imply higher Du Bois singularities for pairs, and establishes stability results, Bertini-type theorems, and behavior under finite maps, hyperplane sections, and cyclic covers. Together, these results extend and unify previous single-variety theories (SVV23, Kov25, Kov16a/16b) to the pair setting, enabling inductive and descent techniques in birational geometry and the MMP. The framework provides a robust toolkit for handling higher singularities in paired contexts, with implications for cohomological behavior and resolution of singularities in the MMP.

Abstract

We extend the notions of higher Du Bois and higher rational singularities to pairs in the sense of the minimal model program. We extend numerous results to these higher pairs, including Bertini type theorems, stability under finite maps and that m-rational pairs are m-Du Bois. We prove these using a generalized Kovács-Schwede-type injectivity theorem for pairs, the main technical result of this paper.
Paper Structure (9 sections, 38 theorems, 82 equations)

This paper contains 9 sections, 38 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logarithmic Du Bois Complex of a Pair
  4. Singularities
  5. Two Out of Three
  6. Hyperplane Sections and Cyclic Covers
  7. Bertini Theorems
  8. Injectivity Theorem and Applications
  9. Proof of the Injectivity Theorem

Key Result

Lemma 2.3

Let $X$ be a noetherian scheme and let $\Sigma$ be a reduced divisor on $X$. Let be an exact sequence of coherent sheaves on $X$. If $\mathscr{F}_0$ is $S_2$ and torsion free on $X$ and $\mathscr{F}_1$ is torsion free on $\Sigma$ then $\mathscr{F}$ is $S_2$ and torsion free on $X$.

Theorems & Definitions (85)

  • Definition 2.1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4
  • ...and 75 more