Reduced resonance schemes and Chen ranks
arXiv:2303.07855 · doi:10.1515/crelle-2024-0051
Abstract
The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.
arXiv comment36 pages. Final version, to appear in the Journal fuer die reine und angewandte Mathematik (Crelle)
References in corpus (17)
- Topology and geometry of cohomology jump loci
- Algebraic invariants for right-angled Artin groups
- Fundamental groups of line arrangements: Enumerative aspects
- Chen Lie algebras
- Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
- Algebraic invariants for Bestvina-Brady groups
- Formality properties of finitely generated groups and Lie algebras
- Koszul modules and Green's conjecture
- Very stable Higgs bundles, equivariant multiplicity and mirror symmetry
- Vanishing resonance and representations of Lie algebras
- Chen ranks and resonance
- Characteristic varieties and Betti numbers of free abelian covers
- Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces
- On the geometric nature of characteristic classes of surface bundles
- The number of fiberings of a surface bundle over a surface
- Homological finiteness of abelian covers
- Higher resonance schemes and Koszul modules of simplicial complexes