Results 1  10
of
60
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 63 (6 self)
 Add to MetaCart
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
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 48 (7 self)
 Add to MetaCart
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.
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 41 (12 self)
 Add to MetaCart
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.
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 30 (18 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.
Yuzvinsky: Multinets, resonance varieties, and pencils of plane curves
"... Abstract. 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 int ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
Abstract. 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. 1.
Hyperplane arrangement cohomology and monomials in the exterior algebra
, 2000
"... Abstract. We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
Abstract. We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result. Let X be the complement of a complex hyperplane arrangement A. Inthispaper we study the singular homology H∗(X) as a module over the exterior algebra E on the first singular cohomology V: = H1 (X) always with coefficients in a fixed field K. Our first main result (Section 1) asserts that H∗(X) is generated in a single degree and has a linear free resolution; this amounts to an infinite sequence of statements about the multiplication in the OrlikSolomon algebra H ∗ (X). We also analyze other topological examples from the point of view of resolutions over the exterior algebra. In Section 2 we study an invariant of an Emodule N called the singular variety,
Characteristic varieties and constructible sheaves
 Rend. Lincei Mat. Appl
"... Abstract. We explore the relation between the positive dimensional irreducible components of the characteristic varieties of rank one local systems on a smooth surface and the associated (rational or irrational) pencils. Our study, which may viewed as a continuation of D. Arapura’s paper [1], yields ..."
Abstract

Cited by 19 (7 self)
 Add to MetaCart
Abstract. We explore the relation between the positive dimensional irreducible components of the characteristic varieties of rank one local systems on a smooth surface and the associated (rational or irrational) pencils. Our study, which may viewed as a continuation of D. Arapura’s paper [1], yields new geometric insight into the translated components relating them to the multiplicities of curves in the associated pencil, in a close analogy to the compact situation treated by A. Beauville [4]. The new point of view is the key role played by the constructible sheaves naturally arising from local systems. Contents
Formality, Alexander invariants, and a question of Serre
, 2005
"... We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the Iadic completion of the Alexander invariant of a 1formal group G is determined solely by the cupproduct map in low degrees. It follows that the germs at the origin of the chara ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the Iadic completion of the Alexander invariant of a 1formal group G is determined solely by the cupproduct map in low degrees. It follows that the germs at the origin of the characteristic and resonance varieties of G are analytically isomorphic; in particular, the tangent cone to Vd(G) at 1 equals Rd(G). This provides new and powerful obstructions to 1formality. A detailed analysis of the irreducible components of the first resonance variety yields even stronger obstructions to realizing a 1formal 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.
Bieri–Neumann–StrebelRenz invariants and homology jumping loci
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2010
"... We investigate the relationship between the geometric Bieri–Neumann– Strebel–Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential ta ..."
Abstract

Cited by 16 (15 self)
 Add to MetaCart
We investigate the relationship between the geometric Bieri–Neumann– Strebel–Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as Kähler and quasiKähler manifolds. Contents