Dualite et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres

Let X be a complex analytic manifold and D subset X a free divisor. Integrable logarithmic connections along D can be seen as locally free {cal O}_X-modules endowed with a (left) module structure over the ring of logarithmic differential operators {cal D}_X(log D). In this paper we study two related results: the relationship between the duals of any integrable logarithmic connection over the base rings {cal D}_X and {cal D}_X(log D), and a differential criterion for the logarithmic comparison theorem. We also generalize a formula of Esnault-Viehweg in the normal crossing case for the Verdier dual of a logarithmic de Rham complex.