V-filtrations and minimal exponents for locally complete intersection singularities

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