  We give a vanishing theorem for the monodromy eigenspaces of the Milnor
fibers of complex line arrangements. By applying the modular bound of the local
system cohomology groups given by Papadima-Suciu, the result is deduced from
the vanishing of the cohomology of certain Aomoto complex over finite fields.
In order to prove this, we introduce degeneration homomorphisms of
Orlik-Solomon algebras.
