On the logarithmic comparison theorem for integrable logarithmic connections

Abstract

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