Characteristic Varieties and Betti Numbers of Free Abelian Covers

The regular $\mathbb Z^r$-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in $H^1(X,\mathbb Q)$. Moving about this variety and recording when the Betti numbers b1,…,bi of the corresponding covers are finite carves out certain subsets $\Omega ^i_r(X)$ 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 $H^1(X,\mathbb Q)$. 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.