V-filtrations and minimal exponents for locally complete intersection singularities

2022/08/05 by Qianyu Chen, Bradley Dirks, Chen, Qianyu +5
Mathematics · #14B05 #14F10 #14J17 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2208.03277

openalex publication_date 2022/08/05 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

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 (f₁,…,f_r) in terms of the microlocal V-filtration associated to the hypersurface defined by ∑_i=1^rf_iy_i.

Related

• The minimal exponent and k-rationality for local complete intersections
• Some applications of microlocalization for local complete intersection subvarieties
• Hodge ideals for Q-divisors, V-filtration, and minimal exponent
• Multiplier ideals, V-filtration, and spectrum
• Hodge ideals and microlocal V-filtration