TWISTED COHOMOLOGY OF ARRANGEMENTS OF LINES AND MILNOR FIBERS

Abstract. Let A be an arrangement of ane lines in C2; with complementM(A): The (co)homology of M(A) with twisted coecients is strictly related to the cohomology of the Milnor bre associated to the conied arrangement, endowed with the geometric monodromy. Although several partial results are known, even the rst Betti number of the Milnor ber is not understood. We give here a vanishing conjecture for the rst homology, which is of a dierent nature with respect to the known results. Let be the graph of double points of A : we conjecture that if is connected then the geometric monodromy acts trivially on the rst homology of the Milnor ber (so the rst Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of A: We prove it in some cases with stronger hypotheses. In the nal parts, we introduce a new description in terms of the group given by the quotient ot the commutator subgroup of 1(M(A)) by the commutator of its length zero subgroup. We use that to deduce some new interesting cases of a-monodromicity, including a proof of the conjecture under some extra conditions. 1.