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51
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 ..."
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Cited by 59 (9 self)
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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.
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 ..."
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Cited by 53 (8 self)
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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 ..."
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Cited by 51 (26 self)
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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 ..."
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Cited by 42 (13 self)
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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 ..."
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Cited by 32 (4 self)
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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.
CHARACTERISTIC VARIETIES AND CONSTRUCTIBLE SHEAVES
, 2007
"... 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 geom ..."
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Cited by 22 (10 self)
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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 [3]. The new point of view is the key role played by the constructible sheaves naturally arising from local systems.
MULTIVARIABLE ALEXANDER INVARIANTS OF HYPERSURFACE COMPLEMENTS
, 2007
"... We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin’s vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomolog ..."
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Cited by 20 (13 self)
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We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin’s vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to de Rham spectral sequence with the purity of some cohomology groups.
Characteristic varieties of quasiprojective manifolds and orbifolds, preprint arXiv:1005.4761v4
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PENCILS OF PLANE CURVES AND CHARACTERISTIC VARIETIES
, 2006
"... We give a geometric approach to the relation between the irreducible components of the characteristic varieties of local systems on a plane curve arrangement complement and the associated pencils of plane curves. In the case of line arrangements, this relation was recently discovered by M. Falk and ..."
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Cited by 12 (3 self)
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We give a geometric approach to the relation between the irreducible components of the characteristic varieties of local systems on a plane curve arrangement complement and the associated pencils of plane curves. In the case of line arrangements, this relation was recently discovered by M. Falk and S. Yuzvinsky [10] and the geometric point of view was already hinted at by A. Libgober and S. Yuzvinsky, see [15], Section 7. Our study yields new geometric insight on the translated components of the characteristic varieties, relating them to the multiplicities of curves in the associated pencil. We show that the irreducible components W with dimW ≥ 2 are combinatorically determined up to finite ambiguity in the case of a line arrangement.
First Milnor cohomology of hyperplane arrangements
, 2009
"... We show a combinatorial formula for a lower bound of the dimension of the nonunipotent monodromy part of the first Milnor cohomology of a hyperplane arrangement satisfying some combinatorial conditions. This gives exactly its dimension if a stronger combinatorial condition is satisfied. We also p ..."
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Cited by 12 (5 self)
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We show a combinatorial formula for a lower bound of the dimension of the nonunipotent monodromy part of the first Milnor cohomology of a hyperplane arrangement satisfying some combinatorial conditions. This gives exactly its dimension if a stronger combinatorial condition is satisfied. We also prove a noncombinatorial formula for the dimension of the nonunipotent part of the first Milnor cohomology, which apparently depends on the position of the singular points. The latter generalizes a formula previously obtained by the second named author.