Efficiency of hyperplane arrangements.

For a projective hyperplane arrangement, we study sufficient conditions in terms of combinatorial data for ESV-calculability of the monodromy eigenspaces of the first Milnor fiber cohomology for eigenvalues of order $m>1$. This can be reduced to the line arrangement case by Artin's theorem. When $m>4$, we usually get their vanishing by the ESV-calculation using the corresponding Aomoto complex in case one of the sufficient conditions holds. This vanishing could be conjectured for $m>4$ in view of non-existence of $k$-multinets for $k>4$ and a close relation between resonance varieties and multinets due to Falk, Yuzvinsky and others. The above sufficient conditions are often unsatisfied if the arrangement is close to a reflection arrangement or efficiency of the combinatorics of arrangement is high. To measure the latter, we introduce the notion of $m$-efficiency. This is a positive rational number, and is easily calculable, although a verification of the sufficient conditions is rather complicated in general. It may be expected that the ESV-calculability holds if this number is at most 2, except for some special cases.