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Autori: A. I. Suciu
Editorial: Topology of Algebraic Varieties and Singularities, J.I. Cogolludo-Agustin, E. Hironaka, American Math. Soc., Contemporary Mathematics, 538, p.179-223, 2011.
We survey the cohomology jumping loci and the Alexander-type
invariants associated to a space, or to its fundamental group. Though most
of the material is expository, we provide new examples and applications,
which in turn raise several questions and conjectures.
The jump loci of a space X come in two basic flavors: the characteristic
varieties, or, the support loci for homology with coefficients in rank 1 local
systems, and the resonance varieties, or, the support loci for the homology of
the cochain complexes arising from multiplication by degree 1 classes in the
cohomology ring of X. The geometry of these varieties is intimately related
to the formality, (quasi-) projectivity, and homological finiteness properties
We illustrate this approach with various applications to the study of hyperplane
arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin groups.
Cuvinte cheie: Fundamental group, Alexander polynomial, characteristic variety, resonance variety, abelian cover, formality, Bieri-Neumann-Strebel-Renz invariant, right-angled Artin group, Kaehler manifold, quasi-Kaehler manifold, hyperplane arrangement, Milnor fibration, boundary manifold