Higher Du Bois and higher rational singularities
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915484548464640 |
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| author | Friedman, Robert Laza, Radu |
| author_facet | Friedman, Robert Laza, Radu |
| contents | We prove that the higher direct images $R^qf_*Ω^p_{\mathcal Y/S}$ of the sheaves of relative Kähler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have $k$-Du Bois local complete intersection singularities, for $p\leq k$ and all $q\geq 0$, generalizing a result of Du Bois (the case $k=0$). We then propose a definition of $k$-rational singularities extending the definition of rational singularities, and show that, if $X$ is a $k$-rational variety with either isolated or local complete intersection singularities, then $X$ is $k$-Du Bois. As applications, we discuss the behavior of Hodge numbers in families and the unobstructedness of deformations of singular Calabi-Yau varieties.
In an appendix, Morihiko Saito proves that, in the case of hypersurface singularities, the $k$-rationality definition proposed here is equivalent to a previously given numerical definition for $k$-rational singularities. As an immediate consequence, it follows that for hypersurface singularities, $k$-Du Bois singularities are $(k-1)$-rational. This statement has recently been proved for all local complete intersection singularities by Chen-Dirks-Mustaţă. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04729 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher Du Bois and higher rational singularities Friedman, Robert Laza, Radu Algebraic Geometry We prove that the higher direct images $R^qf_*Ω^p_{\mathcal Y/S}$ of the sheaves of relative Kähler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have $k$-Du Bois local complete intersection singularities, for $p\leq k$ and all $q\geq 0$, generalizing a result of Du Bois (the case $k=0$). We then propose a definition of $k$-rational singularities extending the definition of rational singularities, and show that, if $X$ is a $k$-rational variety with either isolated or local complete intersection singularities, then $X$ is $k$-Du Bois. As applications, we discuss the behavior of Hodge numbers in families and the unobstructedness of deformations of singular Calabi-Yau varieties. In an appendix, Morihiko Saito proves that, in the case of hypersurface singularities, the $k$-rationality definition proposed here is equivalent to a previously given numerical definition for $k$-rational singularities. As an immediate consequence, it follows that for hypersurface singularities, $k$-Du Bois singularities are $(k-1)$-rational. This statement has recently been proved for all local complete intersection singularities by Chen-Dirks-Mustaţă. |
| title | Higher Du Bois and higher rational singularities |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2205.04729 |