Results 1  10
of
117
Fibred surfaces, varieties isogenous to a product and related moduli spaces.
 Amer. J. Math.
, 2000
"... ..."
(Show Context)
Yuzvinski: Cohomology of the OrlikSolomon algebras and local systems
 Compositio Math. 121 (2000),337–361. Laboratoire J.A. Dieudonné, UMR du CNRS 6621, Université de Nice SophiaAntipolis, Parc Valrose, 06108 Nice Cedex 02
"... The paper provides a combinatorial method to decide when the space of local systems with non vanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements ..."
Abstract

Cited by 77 (9 self)
 Add to MetaCart
(Show Context)
The paper provides a combinatorial method to decide when the space of local systems with non vanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of OrlikSolomon algebra and cohomology of local systems are given. The methods are based on VinbergKac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components. 1
Characteristic varieties of arrangements
 MATH. PROC. CAMBRIDGE PHIL. SOC. 127 (1999), 33–53. MR 2000M:32036
, 1999
"... Abstract. The kth Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, Vk(A), of the algebraic torus (C∗) n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a co ..."
Abstract

Cited by 66 (20 self)
 Add to MetaCart
Abstract. The kth Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, Vk(A), of the algebraic torus (C∗) n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of Vk(A). For any arrangement A, we show that the tangent cone at the identity of this variety coincides with R1 k (A), one of the cohomology support loci of the OrlikSolomon algebra. Using work of Arapura [1] and Libgober [18], we conclude that all positivedimensional components of Vk(A) are combinatorially determined, and that R1 k (A) is the union of a subspace arrangement in Cn, thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.
ARRANGEMENTS AND LOCAL SYSTEMS
 MATHEMATICAL RESEARCH LETTERS 7, 299–316 (2000)
, 2000
"... We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish ..."
Abstract

Cited by 57 (8 self)
 Add to MetaCart
(Show Context)
We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish the relationship between the cohomology support loci of the complement and the resonance varieties of the OrlikSolomon algebra for any arrangement, and show that the latter are unions of subspace arrangements in general, resolving a conjecture of Falk. We also obtain lower bounds for the local system Betti numbers in terms of those of the OrlikSolomon algebra, recovering a result of Libgober and Yuzvinsky. For certain local systems, our results provide new combinatorial upper bounds on the local system Betti numbers. These upper bounds enable us to prove that in nonresonant systems the cohomology is concentrated in the top dimension, without using resolution of singularities.
Fundamental groups of line arrangements: Enumerative aspects
 Contemporary Mathematics
, 2001
"... 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 stratificati ..."
Abstract

Cited by 56 (14 self)
 Add to MetaCart
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 lowindex 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.
Multinets, resonance varieties, and pencils of plane curves
, 2006
"... We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a “multinet,” a multiarrangement with a partition into three or more equinumerous classes which have equal multiplicities at each interclass ..."
Abstract

Cited by 50 (8 self)
 Add to MetaCart
We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a “multinet,” a multiarrangement with a partition into three or more equinumerous classes which have equal multiplicities at each interclass intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement. We derive numerical conditions which impose restrictions on the number of classes, and the line and point multiplicities that can appear in multinets, and allow us to detect whether the associated pencils yield nonlinear fiberings of the complement.
Topology and geometry of cohomology jump loci
 DUKE MATHEMATICAL JOURNAL
, 2009
"... Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk a ..."
Abstract

Cited by 49 (27 self)
 Add to MetaCart
Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk and Rk are analytically isomorphic, if the group is 1formal; in particular, the tangent cone to Vk at 1 equals Rk. These new obstructions to 1formality 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 quasiprojective 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.
Translated tori in the characteristic varieties of complex hyperplane arrangements
 TOPOLOGY AND ITS APPLICATIONS
, 2002
"... We give examples of complex hyperplane arrangements for which the top characteristic variety contains positivedimensional irreducible components that do not pass through the origin of the character torus. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhib ..."
Abstract

Cited by 40 (12 self)
 Add to MetaCart
(Show Context)
We give examples of complex hyperplane arrangements for which the top characteristic variety contains positivedimensional irreducible components that do not pass through the origin of the character torus. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the OrlikSolomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.
Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers
 Adv. Math
, 2009
"... Abstract. The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
(Show Context)
Abstract. The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasiprojective case results of GreenLazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers h q,0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to SarnakAdams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers. 1.