Homology graph of real arrangements and monodromy of Milnor Fiber

Abstract: We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the deconing arrangement and its boundary map. We describe an algorithm which computes possible eigenvalues of the first monodromy operator. We prove that, if a condition on some intersection points of lines is satisfied, then the only possible non trivial eigenvalues 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.