A Criterion for the Logarithmic Differential Operators to Be Generated by Vector Fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give • a sufficient criterion for the property, • a simple proof of F.J. Calderón–Moreno’s theorem that free divisors have the property, • a proof that divisors in dimension 3 with only isolated quasi–homogeneous singularities have the property, • an example of a non–free divisor with non–isolated singularity having the property, • an example of a divisor not having the property, and • an algorithm to compute the V–filtration along a divisor up to a given order.