Characteristic varieties and Betti numbers of free abelian covers
Abstract
The regular r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H1(X,). Moving about this variety, and recording when the Betti numbers b1,..., bi of the corresponding covers are finite carves out certain subsets ir(X) of the Grassmannian. We present here a method, essentially going back to Dwyer and Fried, for computing these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. Using the exponential tangent cones to these jump loci, we show that each -invariant is contained in the complement of a union of Schubert varieties associated to an arrangement of linear subspaces in H1(X,). The theory can be made very explicit in the case when the characteristic varieties of X are unions of translated tori. But even in this setting, the -invariants are not necessarily open, not even when X is a smooth complex projective variety. As an application, we discuss the geometric finiteness properties of some classes of groups.