Homology graph of real arrangements and monodromy of Milnor fiber

Abstract

We study the first homology group H1(F,C) of the Milnor fiber F of sharp arrangements A‾ in PR2. Our work relies on the minimal complex C⁎(S(A)) of the deconing arrangement A and its boundary map. We describe an algorithm which computes possible eigenvalues of the monodromy operator h1 of H1(F,C). We prove that, if a condition on some intersection points of lines in A is satisfied, then the only possible nontrivial eigenvalues of h1 are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.