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Autori: A.I. Suciu
Editorial: Oxford University Press, International Mathematics Research Notices, 2014, p.1063-1124, 2014.
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.
Cuvinte cheie: Free abelian cover, characteristic variety, exponential tangent cone, Dwyer-Fried set, special Schubert variety, translated subtorus, Kaehler manifold, quasi-Kaehler manifold, hyperplane arrangement, property FP_n