DOI: 10.1515/crelle-2024-0023 ISSN: 0075-4102

𝑉-filtrations and minimal exponents for local complete intersections

Qianyu Chen, Bradley Dirks, Mircea Mustaţă, Sebastián Olano
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Abstract

We define and study a notion of minimal exponent for a local 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 Z r ⁢ ( O X ) \mathcal{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 , … , f r ) (f_{1},\ldots,f_{r})
in terms of the microlocal 𝑉-filtration associated to the hypersurface defined by
∑ i = 1 r f i ⁢ y i \sum_{i=1}^{r}f_{i}y_{i}
.

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