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V-filtrations and minimal exponents for locally complete intersection singularities
- Publication Year :
- 2022
-
Abstract
- 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 $\sum_{i=1}^rf_iy_i$.<br />Comment: 34 pages; v.2: new, simpler argument for Theorem 1.4. V.3: final version, to appear in Crelle's Journal
- Subjects :
- Mathematics - Algebraic Geometry
14F10, 14B05, 14J17
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2208.03277
- Document Type :
- Working Paper