On the logarithmic comparison theorem for integrable logarithmic connections

LetX be a complex analytic manifold, D ⊂ X a Koszul free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), j : U = X −D →֒ X the corresponding open inclusion, E an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of E on U . In this paper we prove that the canonical morphisms ΩX(logD)(E(kD)) −→ Rj∗L, j!L −→ Ω • X(logD)(E(−kD)) are locally isomorphisms in the derived category of sheaves of complex vector spaces for k ≫ 0.