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We show that the cohomology of a rank 1 local system on the complement of aprojective hyperplane arrangement can be calculated by the Aomoto complex incertain cases even if the condition on the sum of the residues of connectiondue to Esnault et al is not satisfied. For this we have to study thelocalization of Hodge-logarithmic differential forms which are defined by usingan embedded resolution of singularities. As an application we can computecertain monodromy eigenspaces of the first Milnor cohomology group of thedefining polynomial of the reflection hyperplane arrangement of type $G_{31}$without using a computer.

Rank one local systems on complements of hyperplanes and Aomoto complexes