Fundamental groups of line arrangements: Enumerative aspects

Abstract

This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratification of the character variety of the fundamental group, G, by the jumping loci for cohomology with coefficients in rank 1 local systems. Counting certain torsion points on these "characteristic" varieties yields information about the homology of branched and unbranched covers of the complement, as well as on the number of low-index subgroups of its fundamental group. We conclude with two conjectures, expressing the lower central series quotients of G/G'' (and, in some cases, G itself) in terms of the closely related "resonance" varieties. We illustrate the discussion with a number of detailed examples, some of which reveal new phenomena.

Revised and expanded; 37 pages, 13 figures; to appear in Contemporary Mathematics (Proceedings of the AMS Special Session ``Enumerative Geometry in Physics," held in Lowell, MA, April 2000)

Cited by in corpus (36)

• Lower central series and free resolutions of hyperplane arrangements
• Chen Lie algebras
• Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
• Hyperplane arrangements and Milnor fibrations
• On the monodromy action on Milnor fibers of graphic arrangements
• When does the associated graded Lie algebra of an arrangement group decompose?
• Fundamental groups, Alexander invariants, and cohomology jumping loci
• Torsion in Milnor fiber homology
• Chen ranks and resonance
• Characteristic varieties and Betti numbers of free abelian covers
• The pure braid groups and their relatives
• Resonance varieties and Dwyer-Fried invariants
• Pure virtual braids, resonance, and formality
• The line geometry of resonance varieties
• Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology
• Chen ranks and resonance varieties of the upper McCool groups
• Fundamental groups of real arrangements and torsion in the lower central series quotients
• On the Milnor monodromy of the irreducible complex reflection arrangements
• Multinets, resonance varieties, and pencils of plane curves
• Reduced resonance schemes and Chen ranks
• Homology, lower central series, and hyperplane arrangements
• Cohomology Jumping Loci and Relative Malcev Completion
• Milnor fibrations of arrangements with trivial algebraic monodromy
• Cohomology, Bocksteins, and resonance varieties in characteristic 2
• Combinatorics of line arrangements and dynamics of polynomial vector fields
• Alexander invariants and cohomology jump loci in group extensions
• On the second nilpotent quotient of higher homotopy groups, for hypersolvable arrangements
• The homology groups of finite cyclic covering of line arrangement complement
• On algebraic surfaces associated to line arrangements
• The Orlik-Terao algebra and 2-formality
• Fundamental groups of some special quadric arrangements
• Computing untwisted Dijkgraaf-Witten invariants for arborescent links
• $L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties
• Resonance varieties over fields of positive characteristic
• Configurations of points and topology of real line arrangements
• Fundamental groups of some quadric-line arrangements