V-filtrations and minimal exponents for locally complete intersection singularities
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866911791094693888 |
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| author | Chen, Qianyu Dirks, Bradley Mustaţă, Mircea Olano, Sebastián |
| author_facet | Chen, Qianyu Dirks, Bradley Mustaţă, Mircea Olano, Sebastián |
| contents | We define and study a notion of minimal exponent for a locally complete intersection subscheme $Z$ of a smooth complex algebraic variety $X$, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara-Malgrange $V$-filtration associated to $Z$. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology $H^r_Z({\mathcal O}_X)$, where $r$ is the codimension of $Z$ in $X$. We also study its relation to the Bernstein-Sato polynomial of $Z$. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension $1$ case. A key ingredient for our main result is a description of the Kashiwara-Malgrange $V$-filtration associated to any ideal $(f_1,\ldots,f_r)$ in terms of the microlocal $V$-filtration associated to the hypersurface defined by $\sum_{i=1}^rf_iy_i$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_03277 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | V-filtrations and minimal exponents for locally complete intersection singularities Chen, Qianyu Dirks, Bradley Mustaţă, Mircea Olano, Sebastián Algebraic Geometry 14F10, 14B05, 14J17 We define and study a notion of minimal exponent for a locally complete intersection subscheme $Z$ of a smooth complex algebraic variety $X$, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara-Malgrange $V$-filtration associated to $Z$. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology $H^r_Z({\mathcal O}_X)$, where $r$ is the codimension of $Z$ in $X$. We also study its relation to the Bernstein-Sato polynomial of $Z$. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension $1$ case. A key ingredient for our main result is a description of the Kashiwara-Malgrange $V$-filtration associated to any ideal $(f_1,\ldots,f_r)$ in terms of the microlocal $V$-filtration associated to the hypersurface defined by $\sum_{i=1}^rf_iy_i$. |
| title | V-filtrations and minimal exponents for locally complete intersection singularities |
| topic | Algebraic Geometry 14F10, 14B05, 14J17 |
| url | https://arxiv.org/abs/2208.03277 |