Minicourse: Cohomology jump loci in geometry and topology. Part III: Finiteness properties (original) (raw)

The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.