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 (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 Ω • X(log D)(E(kD)) −→ Rj∗L, j!L −→ Ω • X(log D)(E(−kD)) are isomorphisms in the derived category of sheaves of complex vector spaces for k ≫ 0 (locally on X).Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona