On Fitting ideals of logarithmic vector fields and Saito's criterion

TL;DR

This paper investigates the algebraic and geometric properties of logarithmic vector fields associated with analytic sets, providing new criteria and interpretations for their Fitting ideals and extending Saito's criterion for free divisors.

Contribution

It introduces new sufficient conditions and geometric interpretations for Fitting ideals of logarithmic vector fields, extending Saito's criterion to more general settings.

Findings

Fitting ideals alone are insufficient to determine the full module of logarithmic vector fields.

Provided new algebraic and geometric conditions for the equality of modules in hypersurface cases.

Extended Saito's criterion for free divisors to broader classes of analytic sets.

Abstract

The germ of an analytic set $(X,p)$ in $\mathbb{C}^n$ has an associated $\mathscr{O}_{\mathbb{C}^n,p}$-module $\mathrm{Der}(-\log X)$ of `logarithmic vector fields', the ambient germs of holomorphic vector fields tangent to the smooth locus of $X$. For a module $L\subseteq \mathrm{Der}(-\log X)$ let $I_k(L)$ be the ideal generated by the $k\times k$ minors of a matrix of generators for $L$; these are the Fitting ideals of $\mathrm{Der}_{\mathbb{C}^n,p}/L$. We aim to: (i) find sufficient conditions on $\{I_k(L)\}$ to prove $L=\mathrm{Der}(-\log X)$; (ii) identify $\{I_k(\mathrm{Der}(-\log X))\}$, to provide a necessary condition for equality; and (iii) provide a geometric interpretation of these ideals. Even for $(X,p)$ smooth, an example shows that Fitting ideals alone are insufficient to prove equality, although we give a different criterion. Using (ii) and (iii) in the smooth case, we give partial answers to (ii) and (iii) for arbitrary $(X,p)$. When $(X,p)$ is a hypersurface, we give sufficient algebraic or geometric conditions for the reflexive hull of $L$ to equal $\mathrm{Der}(-\log X)$; for $L$ reflexive, this answers (i) and generalizes criteria of Saito for free divisors and Brion for linear free divisors.