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