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 $\alpha$ on $M$ we investigate the relation between the resonance degree of $\alpha$ and the codimension of the zero set of $\alpha$ 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.Comment: 18 pages, in this new version Remark 6.4 is extended, a reference to a result by Green and Lazarsfeld is added and some minor corrections are don
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