Nonresonance conditions for arrangements. (Q1417739)

Let \(\mathcal A\) be an arrangement of hyperplanes in complex projective space \(\mathbb {CP}^n\) and \(N= \bigcup_{H \in \mathcal A}H\) the union of hyperplanes of \(\mathcal A\). Let \(M(\mathcal A)= \mathbb {CP}^n \setminus N\) be the complement of \(N\). An edge is a nonempty intersection of hyperplanes. An edge is dense if the subarrangement of hyperplanes containing it is irreducible: the hyperplanes cannot be partitioned into nonempty sets so that after a change of coordinates hyperplanes in different sets are in different coordinates. Let \(D(\mathcal A)\) denote the set of dense edges of \(\mathcal A\). There is a canonical way to obtain an embedded resolution of \(N\) in \(\mathbb {CP}^n\). \(D=p^{-1}(N)\) is a normal crossing divisor in an appropriate variety \(Z\) with smooth irreducible components \(D_X\) corresponding to the edges \(X \in D(\mathcal A)\). We denote by \(\mathcal L\) a complex local system of rank r on the complement \(M(\mathcal A)\) associated to a representation \[ \rho: \pi^1(M(\mathcal A),a) \to GL_r(\mathbb C). \] To each irreducible component \(D_X\) of the normal crossing divisor \(D\) corresponds a conjugacy class \(T_X\) in \(GL_r(\mathbb C)\) obtained as the monodromy of the local system \(\mathcal L\) along a small loop turning once in the positive direction about the hypersurfacee \(D_X\). The objective of the paper under review is the proof of the following theorem. Theorem. Assume that there is a hyperplane \(H \in \mathcal A\) such that for any dense edge \(X \in D(\mathcal A)\) with \(X \subseteq H\) the corresponding monodromy operator \(T_X\) does not admit 1 as eigenvalue. Then \(H^k(M(\mathcal A),\mathcal L)=0\) for any \(k \neq q\). The following problems are also discussed: the local systems which arise from flat connections on trivial bundles, the implications of the theorem in that special case and comparison with other resonance cases; a brief application to Milnor fibers associated to line arrangements in \(\mathbb {P^2}\) is given, which strengthens a result of Massey; a strategy to handle arrangements of more general hypersurfaces is suggested.