Characteristic varieties and Betti numbers of free abelian covers

TL;DR

This paper develops a method to analyze the Betti numbers of free abelian covers of finite complexes using jump loci and tangent cones, revealing geometric properties and finiteness conditions of associated groups.

Contribution

It introduces a new approach to compute \\Omega-invariants via exponential tangent cones and Schubert varieties, extending previous work to broader classes of spaces.

Findings

\\Omega-invariants are contained in complements of unions of Schubert varieties.

The theory applies explicitly when characteristic varieties are unions of translated tori.

\\Omega-invariants are not necessarily open, even for smooth complex projective varieties.

Abstract

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.