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- Title
푉-filtrations and minimal exponents for local complete intersections.
- Authors
Chen, Qianyu; Dirks, Bradley; Mustaţă, Mircea; Olano, Sebastián
- Abstract
We define and study a notion of minimal exponent for a local complete intersection subscheme 푍 of a smooth complex algebraic variety 푋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 푉-filtration associated to 푍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r (O X) , where 푟 is the codimension of 푍 in 푋. We also study its relation to the Bernstein–Sato polynomial of 푍. 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 푉-filtration associated to any ideal (f 1 , ... , f r) in terms of the microlocal 푉-filtration associated to the hypersurface defined by ∑ i = 1 r f i y i .
- Subjects
EXPONENTS; ALGEBRAIC varieties; HYPERSURFACES; COHOMOLOGY theory
- Publication
Journal für die Reine und Angewandte Mathematik, 2024, Vol 2024, Issue 811, p219
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2024-0023