Topology and geometry of cohomology jump loci

Abstract

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.

41 pages; this is a major revision of arXiv:math/0512480; accepted for publication in Duke Mathematical Journal

References in corpus (7)

• Algebraic invariants for right-angled Artin groups
• Toric complexes and Artin kernels
• Non-finiteness properties of fundamental groups of smooth projective varieties
• Which 3-manifold groups are Kähler groups?
• Characteristic varieties of nilpotent groups and applications
• Quasi-Kähler Bestvina-Brady groups
• Cohomology Jumping Loci and Relative Malcev Completion

Cited by in corpus (60)

• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• Toric complexes and Artin kernels
• Multinets, parallel connections, and Milnor fibrations of arrangements
• Hyperplane arrangements and Milnor fibrations
• Geometric and algebraic aspects of 1-formality
• Formality properties of finitely generated groups and Lie algebras
• Koszul modules and Green's conjecture
• Fundamental groups, Alexander invariants, and cohomology jumping loci
• Cohomology jump loci of differential graded Lie algebras
• Characteristic varieties of quasi-projective manifolds and orbifolds
• Vanishing resonance and representations of Lie algebras
• Quasi-Kähler groups, 3-manifold groups, and formality
• The local fundamental group of a Kawamata log terminal singularity is finite
• Chen ranks and resonance
• On the fundamental groups of normal varieties
• Algebraic monodromy and obstructions to formality
• Characteristic varieties and Betti numbers of free abelian covers
• Combinatorial covers and vanishing of cohomology
• The pure braid groups and their relatives
• Around the tangent cone theorem
• Non-formality of Milnor fibers of line arrangements
• Pure virtual braids, resonance, and formality
• Resonance varieties and Dwyer-Fried invariants
• Coxeter groups and Kähler groups
• Non-abelian resonance: product and coproduct formulas
• Cup products, lower central series, and holonomy Lie algebras
• Pure braid groups are not residually free
• Chen ranks and resonance varieties of the upper McCool groups
• Geometric and homological finiteness in free abelian covers
• Characteristic varieties and logarithmic differential 1-forms
• Recent results on cohomology jump loci
• On the geometry and topology of partial configuration spaces of Riemann surfaces
• Quasi-projectivity of even Artin groups
• Reduced resonance schemes and Chen ranks
• Finite Galois covers, cohomology jump loci, formality properties, and multinets
• Naturality properties and comparison results for topological and infinitesimal embedded jump loci
• Algebraic models, Alexander-type invariants, and Green-Lazarsfeld sets
• Infinitesimal finiteness obstructions
• Cohomology jump loci of compact Kähler manifolds
• Cohomology jump loci of 3-manifolds
• Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds
• Cohomology jump loci in the moduli spaces of vector bundles
• Poincaré duality and resonance varieties
• Taylor expansions of groups and filtered-formality
• Fundamental groups of links of isolated singularities
• Milnor fibrations of arrangements with trivial algebraic monodromy
• Flat bundles and Hyper-Hodge decomposition on solvmanifolds
• Sigma-invariants and tropical varieties
• Mixed Hodge structures on cohomology jump ideals
• Twisted cohomology of arrangements of lines and Milnor fibers
• First Milnor cohomology of hyperplane arrangements
• Admissibility of local systems for some classes of line arrangements
• Solving the isomorphism problems for two families of parafree groups
• Universal KZB Equations for arbitrary root systems
• Vanishing cycles and analysis of singularities of Feynman diagrams
• On webs, polylogarithms and cluster algebras
• $L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties
• Highly connected non-formal Milnor fibers via polyhedral products
• Some computations on the characteristic variety of a line arrangement
• Generalized Broughton polynomials and characteristic varieties