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

Original · EN

The germ of an analytic set (X,p) in Cⁿ has an associated OCₙ,ₚ-module Der(- X) of `logarithmic vector fields', the ambient germs of holomorphic vector fields tangent to the smooth locus of X. For a module L Der(- X) let Iₖ(L) be the ideal generated by the k× k minors of a matrix of generators for L; these are the Fitting ideals of DerCₙ,ₚ/L. We aim to: (i) find sufficient conditions on {Iₖ(L)} to prove L=Der(- X); (ii) identify {Iₖ(Der(- 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 Der(- X); for L reflexive, this answers (i) and generalizes criteria of Saito for free divisors and Brion for linear free divisors.