Characteristic Varieties and Logarithmic Differential 1-forms

We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M , see Theorem (3.1) and Corollaries (3.2) and (4.2). A logarithmic resonance variety is also considered in Proposition (4.5). As an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve, see Proposition (5.1). Finally, for a logarithmic one form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M , see Corollary (1.1). This question was inspired by the recent work by D. Cohen, G. Denham, M. Falk and A. Varchenko.