A vanishing theorem for a class of logarithmic D-modules

Let $O_X$ (resp. $D_X$) be the sheaf of holomorphic functions (resp. the sheaf of linear differential operators with holomorphic coefficients) on $X$ (=the complex affine n-space). Let $Y$ be a locally weakly quasi-homogeneous free divisor defined by a polynomial $f$. In this paper we prove that, locally, the annihilating ideal of $1\mathrm{/}{f}^k$ over $D_X$ 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 $D_X$–module with $O_X$. The logarithmic $D_X$–module is naturally associated with $Y$. This result is related to the so called Logarithmic Comparison Theorem.