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Inversion of adjunction for higher rational singularities

Tatsuro Kawakami, Jakub Witaszek

TL;DR

This paper proves inversion of adjunction for higher rational singularities ($m$-rational). It develops a framework based on Deligne–Du Bois complexes and their higher variants, employing the Gr DR functor from mixed Hodge modules to translate singularity questions into perverse-constructible data and local-cohomology criteria. The main result shows that if $D$ is a normal Cartier prime divisor on a normal variety $X$ and $D$ is $m$-rational, then $X$ is $m$-rational along $D$; a related fibrewise statement holds for morphisms to curves. The work also establishes several equivalent characterizations and interrelations among higher Du Bois, IC-rational, IO, and DDB complexes, thereby extending classical inversion of adjunction to higher singularity classes and providing tools for applications in deformation theory and degenerations of complex varieties.

Abstract

We prove inversion of adjunction for higher rational singularities.

Inversion of adjunction for higher rational singularities

TL;DR

This paper proves inversion of adjunction for higher rational singularities (-rational). It develops a framework based on Deligne–Du Bois complexes and their higher variants, employing the Gr DR functor from mixed Hodge modules to translate singularity questions into perverse-constructible data and local-cohomology criteria. The main result shows that if is a normal Cartier prime divisor on a normal variety and is -rational, then is -rational along ; a related fibrewise statement holds for morphisms to curves. The work also establishes several equivalent characterizations and interrelations among higher Du Bois, IC-rational, IO, and DDB complexes, thereby extending classical inversion of adjunction to higher singularity classes and providing tools for applications in deformation theory and degenerations of complex varieties.

Abstract

We prove inversion of adjunction for higher rational singularities.

Paper Structure

This paper contains 19 sections, 38 theorems, 221 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equivalent definitions of higher singularities
  3. Outline
  4. Acknowlegements
  5. Preliminaries
  6. Review of $t$-structure
  7. Review of perverse coherent $t$-structure
  8. Review of perverse constructible $t$-structure
  9. Hodge modules
  10. $\mathop{\mathrm{Gr\,DR}}\nolimits$ functor
  11. Intersection and dual Deligne--Du Bois complexes
  12. Vanishing results
  13. Restriction to general hyperplane
  14. Miscellaneous
  15. Higher Du Bois singularities
  16. ...and 4 more sections

Key Result

Theorem 1

Let $X$ be a normal connected $d$-dimensional variety defined over $\mathbb{C}$ and let $D$ be a normal Cartier prime divisor. Assume that $D$ is $m$-rational. Then $X$ is $m$-rational along $D$.

Figures (1)

  • Figure 1: Spectral sequence

Theorems & Definitions (103)

  • Remark 1.1
  • Theorem 1: Theorem \ref{['thm:inversion-of-adjunction']}
  • Corollary 2
  • Remark 1.2
  • Definition 1.3
  • Proposition 3: Proposition \ref{['prop:lc-def-ofkDuBois']}
  • Proposition 4: Corollary \ref{['cor:lc-def-ofkRational']}
  • Lemma 5
  • Lemma 2.1
  • proof
  • ...and 93 more