Characteristic Varieties and Resonance Varieties

Abstract

Characteristic varieties (resp. resonance varieties) are jumping loci for some cohomology groups which are topologically (resp. algebraically) defined. We explain the relation between the characteristic varieties and the homology of finite abelian covers. The polynomial periodicity properties of the first Betti numbers of such covers, and the smooth surfaces obtained as coverings of $\mathbb {P}^2$ ramified over a line arrangement, are also discussed. The main results in this chapter are the Tangent Cone Theorem explaining the close relation between the two types of jumping loci and the relation with the multinet structures introduced by M. Falk and S. Yuzvinsky. After a brief discussion of the translated components of the characteristic varieties, we treat in great detail the deleted $B_3$-line arrangement.