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Inversion of Adjunction for the minimal exponent

Qianyu Chen

TL;DR

This work establishes an inversion-of-adjunction principle for the minimal exponent $\widetilde{\alpha}(X,Z)$ of a local complete intersection $Z$ inside a smooth irreducible variety $X$, connecting the behavior under cutting by a hypersurface to the global invariant. The authors develop a Hodge-theoretic approach using mixed Hodge modules, $V$-filtrations, and log-de Rham complexes to prove the hypersurface case and then lift it to general LCI settings via a Thom–Sebastiani-type construction. Consequently, they deduce inversion of adjunction for higher Du Bois and higher rational singularities in the LCI context, and establish a precise correspondence between $\widetilde{\alpha}(X,Z)$ and the singularity types $k$-Du Bois and $k$-rational. The results unify minimal exponent theory with Hodge-theoretic invariants, providing tools for analyzing singularities beyond hypersurfaces and enabling refined control over singularity types via the reduced Bernstein-Sato polynomial.

Abstract

We prove that the minimal exponent for local complete intersections satisfies an Inversion-of-Adjunction property. As a result, we also obtain the Inversion of Adjunction for higher Du Bois and higher rational singularities for local complete intersections.

Inversion of Adjunction for the minimal exponent

TL;DR

This work establishes an inversion-of-adjunction principle for the minimal exponent of a local complete intersection inside a smooth irreducible variety , connecting the behavior under cutting by a hypersurface to the global invariant. The authors develop a Hodge-theoretic approach using mixed Hodge modules, -filtrations, and log-de Rham complexes to prove the hypersurface case and then lift it to general LCI settings via a Thom–Sebastiani-type construction. Consequently, they deduce inversion of adjunction for higher Du Bois and higher rational singularities in the LCI context, and establish a precise correspondence between and the singularity types -Du Bois and -rational. The results unify minimal exponent theory with Hodge-theoretic invariants, providing tools for analyzing singularities beyond hypersurfaces and enabling refined control over singularity types via the reduced Bernstein-Sato polynomial.

Abstract

We prove that the minimal exponent for local complete intersections satisfies an Inversion-of-Adjunction property. As a result, we also obtain the Inversion of Adjunction for higher Du Bois and higher rational singularities for local complete intersections.
Paper Structure (14 sections, 13 theorems, 108 equations)

This paper contains 14 sections, 13 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline
  4. Acknowledgement
  5. Background
  6. Mixed Hodge modules
  7. The minimal exponent
  8. Higher Du Bois and higher rational singularities
  9. A preparation lemma
  10. Proof of Main Results
  11. Some reductions
  12. Proof of Theorem \ref{['thm:strinratbd']}
  13. Log comparison
  14. Proof of Theorem \ref{['thm:inratbd']}

Key Result

Theorem 1.1

Let $X$ be an irreducible and smooth complex algebraic variety and $Z$ be a local complete intersection closed subscheme in $X$. If there is a hypersurface $H$ in $X$ containing no irreducible component of $Z$ and a positive rational number $c$ such that, then we have $\widetilde{\alpha}(X,Z)>c$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Remark 2.7
  • Lemma 2.8: CDM*Lemma 3.5
  • ...and 11 more