Cohomology Jumping Loci

Northeastern University, Boston MA

Investigators

Abstract

This project explores from various angles the deep and intricate connections between the topology and geometry of a space, the algebraic structure of its fundamental group, and the algebraic geometry of the associated cohomology jumping loci. The study is done in a context which abstracts two of the crucial features that make the dictionary between topology and algebra work so well for complements of hyperplane arrangements: formality (which leads to the Tangent Cone Theorem, relating the characteristic and resonance varieties), and quasi-projectivity (which insures that the characteristic varieties consist of possibly translated subtori). Together, these two properties put strong constraints on the geometry of the resonance varieties, leading to powerful obstructions to finitely presented groups being realizable as fundamental groups of smooth, (quasi-) projective complex varieties. Recently, new connections have emerged, relating the cohomology jumping loci of a space to the homological finiteness properties of its free abelian covers, and thereby to the structure of the Dwyer-Fried and Bieri-Neumann-Strebel-Renz invariants. Generalizations of the classical jump loci---from rank one local systems to the non-abelian setting, and from cohomology rings to differential graded algebras---extend the scope of the investigation, and broaden the range of its applicability. The topological and group-theoretic methods utilized in this project shed new light on the combinatorial and geometric structure of objects occurring in a variety of contexts, allowing for cross-pollination between different fields, with ideas originating in a given area being fruitfully applied in new settings. The theory of cohomology jumping loci impacts the study of a wide array of spaces and groups, including toric complexes and moment-angle complexes; right-angled Artin and Coxeter groups; real and complex quasi-toric manifolds; Kaehler and quasi-Kaehler manifolds; Milnor fibrations of hyperplane arrangements; as well as configuration spaces and compactifications of moduli spaces. The study of these objects, with their multiple connections to the theory of singularities, graph theory, and low-dimensional topology, provides a rich interplay between algebra, topology, and combinatorics, yielding applications to areas ranging from topological robotics to theoretical computer science. The investigation is being conducted together with several collaborators, as well as graduate and undergraduate students. Participation in intensive research periods and workshops is meant to introduce a new generation of students and young researchers to a very active, interdisciplinary area of study.