On the Twisted Cohomology of Affine Line Arrangements

Let A be an ane line arrangement in C2; with complement M(A): The twisted (co)homology of M(A) is an interesting object which has been considered by many people. In this paper we give a vanishing conjecture of a dierent nature with respect to the known results: namely, we conjecture that if the graph of double points of the arrangement is connected then there is no-nontrivial monodromy. This conjecture is obviously combinatorial (meaning that it depends only on the lattice of the intersections). We prove it in some cases with stronger hypoteses. We also consider the integral case, relating the property of having trivial monodromy over Z with a certain property of "commutativity" of the fundamental group up to some subgroup. At the end, we give several examples and computations.