Characteristic varieties and Betti numbers of free abelian covers
  The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i 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 \Omega-invariant is contained in the complement of a union of Schubert varieties associated to an arrangement of linear subspaces in H^1(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 \Omega-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. 
Alexander I. Suciu
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