V-filtrations and minimal exponents for locally complete intersection singularities

TL;DR

This paper extends the concept of minimal exponents from hypersurfaces to locally complete intersection singularities using V-filtrations, linking it to Hodge theory and Bernstein-Sato polynomials.

Contribution

It introduces a new definition of minimal exponent for locally complete intersections via V-filtrations, generalizing Saito's invariant and relating it to local cohomology and Bernstein-Sato polynomials.

Findings

Defines minimal exponent for locally complete intersections.

Shows the minimal exponent measures the agreement of Hodge and order filtrations.

Relates the invariant to hypersurface cases through reduction techniques.

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 $\sum_{i=1}^rf_iy_i$.