Characteristic varieties and Betti numbers of free abelian covers
Alexander I. Suciu · 2013
The regular Zr-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H1 (X,Q). Moving about this variety, and recording when the Betti numbers b1,...,bi of the corresponding covers are finite carves out certain subsets Ωi r (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 Ω-invariant is contained in the complement of a union of Schubert varieties associated to an arrangement of linear subspaces in H1 (X,Q). 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 Ω-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.