Du Bois complex and extension of forms beyond rational singularities
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911773317136384 |
|---|---|
| author | Park, Sung Gi |
| author_facet | Park, Sung Gi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15159 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Du Bois complex and extension of forms beyond rational singularities Park, Sung Gi Algebraic Geometry 14B05, 14F10, 14E30, 32S35 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. |
| title | Du Bois complex and extension of forms beyond rational singularities |
| topic | Algebraic Geometry 14B05, 14F10, 14E30, 32S35 |
| url | https://arxiv.org/abs/2311.15159 |