Cohomology Jumping Loci

点击翻译按钮获取中文摘要

This project explores from various angles the deep andintricate connections between the topology and geometryof a space, the algebraic structure of its fundamentalgroup, and the algebraic geometry of the associatedcohomology jumping loci. The study is done in a contextwhich abstracts two of the crucial features that makethe dictionary between topology and algebra work sowell for complements of hyperplane arrangements:formality (which leads to the Tangent Cone Theorem,relating the characteristic and resonance varieties), andquasi-projectivity (which insures that the characteristicvarieties consist of possibly translated subtori). Together,these two properties put strong constraints on the geometryof the resonance varieties, leading to powerful obstructionsto finitely presented groups being realizable as fundamentalgroups of smooth, (quasi-) projective complex varieties.Recently, new connections have emerged, relating the cohomologyjumping loci of a space to the homological finiteness propertiesof its free abelian covers, and thereby to the structure of theDwyer-Fried and Bieri-Neumann-Strebel-Renz invariants.Generalizations of the classical jump loci---from rank onelocal systems to the non-abelian setting, and from cohomologyrings to differential graded algebras---extend the scope ofthe investigation, and broaden the range of its applicability.The topological and group-theoretic methods utilizedin this project shed new light on the combinatorial andgeometric structure of objects occurring in a varietyof contexts, allowing for cross-pollination betweendifferent fields, with ideas originating in a given areabeing fruitfully applied in new settings. The theory ofcohomology jumping loci impacts the study of a widearray of spaces and groups, including toric complexesand moment-angle complexes; right-angled Artin andCoxeter groups; real and complex quasi-toric manifolds;Kaehler and quasi-Kaehler manifolds; Milnor fibrationsof hyperplane arrangements; as well as configurationspaces and compactifications of moduli spaces.The study of these objects, with their multipleconnections to the theory of singularities, graphtheory, and low-dimensional topology, provides arich interplay between algebra, topology, andcombinatorics, yielding applications to areasranging from topological robotics to theoreticalcomputer science. The investigation is being conductedtogether with several collaborators, as well as graduateand undergraduate students. Participation in intensiveresearch periods and workshops is meant to introduce anew generation of students and young researchers to avery active, interdisciplinary area of study.