Around the tangent cone theorem

Abstract

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.

39 pages; to appear in the proceedings of the Configurations Spaces Conference (Cortona 2014), Springer INdAM series

References in corpus (7)

• Geometric and algebraic aspects of 1-formality
• Formality properties of finitely generated groups and Lie algebras
• Combinatorial covers and vanishing of cohomology
• On combinatorial invariance of the cohomology of Milnor fiber of arrangements and Catalan equation over function field
• Characteristic Varieties of Hypersurface Complements
• Algebraic models, Alexander-type invariants, and Green-Lazarsfeld sets
• Examples of topological spaces with arbitrary cohomology jump loci

Cited by in corpus (7)

• Cohomology jump loci of 3-manifolds
• Formality and finiteness in rational homotopy theory
• Cohomology, Bocksteins, and resonance varieties in characteristic 2
• Twisted cohomology of arrangements of lines and Milnor fibers
• Sigma-invariants and tropical varieties
• Higher resonance schemes and Koszul modules of simplicial complexes
• The homotopy type of elliptic arrangements