Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Du Bois complex and extension of forms beyond rational singularities

Sung Gi Park

Abstract

We establish a characterization of the Du Bois complex of a reduced pair $(X,Z)$ when $X\smallsetminus Z$ has rational singularities. As an application, when $X$ has normal Du Bois singularities and $Z$ is the locus of non-rational singularities of $X$, holomorphic $p$-forms on the smooth locus of $X$ extend regularly to forms on a resolution of singularities for $p\le\mathrm{codim}_X Z-1$, and to forms with log poles over $Z$ for $p\ge\mathrm{codim}_X Z$. If $X$ is not necessarily Du Bois, then $p$-forms extend regularly for $p\le\mathrm{codim}_X Z-2$. This is a generalization of the theorems of Flenner, Greb-Kebekus-Kovács-Peternell, and Kebekus-Schnell on extending holomorphic (log) forms. A by-product of our methods is a new proof of the theorem of Kollár-Kovács that log canonical singularities are Du Bois. We also show that the Proj of the log canonical ring of a log canonical pair is Du Bois if this ring is finitely generated. The proofs are based on Saito's theory of mixed Hodge modules.

Du Bois complex and extension of forms beyond rational singularities

Abstract

We establish a characterization of the Du Bois complex of a reduced pair when has rational singularities. As an application, when has normal Du Bois singularities and is the locus of non-rational singularities of , holomorphic -forms on the smooth locus of extend regularly to forms on a resolution of singularities for , and to forms with log poles over for . If is not necessarily Du Bois, then -forms extend regularly for . This is a generalization of the theorems of Flenner, Greb-Kebekus-Kovács-Peternell, and Kebekus-Schnell on extending holomorphic (log) forms. A by-product of our methods is a new proof of the theorem of Kollár-Kovács that log canonical singularities are Du Bois. We also show that the Proj of the log canonical ring of a log canonical pair is Du Bois if this ring is finitely generated. The proofs are based on Saito's theory of mixed Hodge modules.
Paper Structure (19 sections, 40 theorems, 186 equations)

This paper contains 19 sections, 40 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Overview of Saito's theory of mixed Hodge modules
  3. Graded de Rham complexes of mixed Hodge modules
  4. A description of rational singularities via Hodge modules
  5. A characterization of Du Bois complexes of pairs
  6. Proof of the characterization Theorem \ref{['thm: Du Bois complex, rational singularities']}
  7. Log rational pairs and inversion of adjunction
  8. Proof of Corollary \ref{['cor: criterion for log rational pair']}
  9. Proof of Corollary \ref{['cor: inversion of adjunction']}
  10. Results on extensions of forms
  11. Extension criteria using Hodge modules
  12. Proofs of the main theorems on extending forms
  13. Proof of Theorem \ref{['thm: p-forms extend']}
  14. Proof of Theorem \ref{['thm: p-log form extends']}
  15. Proof of Theorem \ref{['thm: extending forms']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X$ be an irreducible normal variety and $Z\subset \mathrm{Sing}(X)$ be a closed subvariety such that $X\smallsetminus Z$ has rational singularities. Let $\mu:(\widetilde{X},E)\to (X,Z)$ be a log resolution of singularities with $E=\mu^{-1}(Z)$. Suppose $X$ has Du Bois singularities away from a

Theorems & Definitions (77)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Definition 1.9
  • Corollary 1.10
  • ...and 67 more