A vanishing theorem for a class of logarithmic D-modules

Let OX (resp. DX) be the sheaf of holomorphic functions (resp. the sheaf of linear differential operators with holomorphic coefficients) on X = Cn. Let D X be a locally weakly quasi-homogeneous free divisor defined by a polynomial f. In this paper we prove that, locally, the annihilating ideal of 1/fk over DX is generated by linear differential operators of order 1 (for k big enough). For this purpose we prove a vanishing theorem for the extension groups of a certain logarithmic DX–module with OX. The logarithmic DX–module is naturally associated with D (see Notation 1.1). This result is related to the so called Logarithmic Comparison Theorem