Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

Details Bieri-Neumann-Strebel-Renz invariants and homology jumping loci Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

2008-12-14

arxiv.org/abs/0812.2660v2

Proceedings of the London Mathematical Society 100 (2010), no. 3, 795-834

Mathematics

Group Theory

40 pages; accepted for publication in the Proceedings of the London Mathematical Society

Scientific paper

10.1112/plms/pdp045

We investigate the relationship between the geometric Bieri-Neumann-Strebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as K\"ahler and quasi-K\"ahler manifolds.

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