Cohomology rings and nilpotent quotients of real and complex arrangements

Matei, Daniel; Suciu, Alexander I.

doi:10.2969/aspm/02710185

Cited by 38 publications

(91 citation statements)

References 23 publications

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“…These varieties are homotopy-type invariants of M (A), which inform on the homology of certain abelian covers [9,16], and also on the structure of the second nilpotent quotient of the fundamental group [13]. It follows from work of Arapura [2] that the characteristic varieties, or cohomology support loci, are unions of torsion-translated subtori of (C * ) n in more general circumstances.…”

Section: Introductionmentioning

confidence: 99%

Triples of arrangements and local systems

Cohen

2002

Proc. Amer. Math. Soc.

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Abstract. For a triple of complex hyperplane arrangements, there is a wellknown long exact sequence relating the cohomology of the complements. We observe that this result extends to certain local coefficient systems, and use this extension to study the characteristic varieties of arrangements. We show that the first characteristic variety may contain components that are translated by characters of any order, thereby answering a question of A. Suciu.

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Section: Introductionmentioning

confidence: 99%

Triples of arrangements and local systems

Cohen

2002

Proc. Amer. Math. Soc.

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“…The partitions need not correspond to multinets, and the corresponding varieties need not be linear or disjoint, as our Examples 4.4 and 4.3 show. Example 4.2, due to D. Matei and A. Suciu [26], shows that the stratification of R(A, k) need not coincide with stratification by dimension.…”

Section: Resonance and Characteristic Varietiesmentioning

confidence: 99%

Resonance Varieties over Fields of Positive Characteristic

Falk

2007

International Mathematics Research Notices

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Let A be a hyperplane arrangement, and k a field of arbitrary characteristic. We show that the projective degree-one resonance variety R 1 (A, k) of A over k is ruled by lines, and identify the underlying algebraic line complex L(A, k) in the Grassmannian G(2, k n ), n = |A|. L(A, k) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of A. Each linear line complex is the intersection of a family of special Schubert varieties corresponding to a subspace arrangement determined by the partition.In case k has characteristic zero, the resulting ruled varieties are linear and pairwise disjoint, by results of A. Libgober and S. Yuzvinsky. We give examples to show that each of these properties fails in positive characteristic. The (4,3)-net structure on the Hessian arrangement gives rise to a nonlinear component in R 1 (A, Z3), a cubic hypersurface in P 4 with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted B3 arrangement has linear resonance components over Z2 that intersect nontrivially. Resonance and characteristic varietiesArising out of the study of local system cohomology and fundamental groups, characteristic and resonance varieties of complex hyperplane arrangements have become the object of much of the current research in the field. The study of resonance varieties in particular has led to surprising connections with other areas of mathematics: generalized Cartan matrices [23], Latin squares and loops, nets [40], special pencils of plane curves [23,13], graded resolutions and the BGG correspondence [10,33,34], and the Bethe Ansatz [7]. In this paper we establish a connection between resonance varieties and projective line complexes, which becomes evident only when one works over a field of positive characteristic.Keywords and phrases. arrangement, resonance variety, line complex, Orlik-Solomon algebra.2000 Mathematics Subject Classification. Primary 52C35 Secondary 16S99, 14J26, 05B35. 1Let A be a (central) complex hyperplane arrangement of rank ℓ, k a field, and A = A k (A) the Orlik-Solomon algebra of A over k. Since A is a quotient of an exterior algebra, left-multiplication by a fixed element a ∈ A 1 defines a chain complex (A, a):The d th resonance variety of A over k isAn element a ∈ A 1 lies in R(A, k) if and only if there exists b ∈ A 1 not proportional to a with ab = 0. Identifying A 1 with k n using the canonical basis corresponding to the hyperplanes of A, R(A, k) is seen to be a homogeneous affine algebraic variety. Hence R(A, k) determines a projective variety R(A, k) in P n−1 , which we call the (degree-one) projective resonance variety of A over k.If k has characteristic zero, R(A, k) has remarkable algebraic and combinatorial features. It consists of disjoint linear subspaces, each of which corresponds to a combinatorial structure, called a multinet, on a subarrangement of A. If k = C, the multinet structure arises from a pencil of plane projective curves whose singular elements include at l...

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“…We note that the characteristic zero assumption is necessary; the aforementioned results depend on this. A thorough treatment of resonance varieties over arbitrary fields (and even commutative rings) can be found in a recent paper of Falk [17] (see also [24]). …”

Section: Resonance Varietiesmentioning

confidence: 99%

“…The matrix ∆ lin appears in the statement of Theorem 4.6 and in Remark 4.7 from [8]; see also [24] and [28] for more general contexts. The reason for the terminology is as follows: Viewing Λ ⊗ C as the coordinate ring of the algebraic torus (C * ) n and S ⊗ C as the coordinate ring of C n , the entries of ∆ lin are the derivatives at 1 ∈ (C * ) n of the corresponding entries of ∆.…”

Section: Linearized Alexander Invariantmentioning

confidence: 99%

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

Schenck

Suciu

2005

Trans. Amer. Math. Soc.

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Abstract. If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π 1 (X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H * (X, k), viewed as a module over the exterior algebra E on A:where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for, where h r is the number of r-dimensional components of the projective resonance variety R 1 (A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R 1 (A) and a localization argument, we establish the inequalityfor arbitrary A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R 1 (A), such that θ k (G) = P(k), for all k 0.

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Cohomology rings and nilpotent quotients of real and complex arrangements

Cited by 38 publications

References 23 publications

Triples of arrangements and local systems

Triples of arrangements and local systems

Resonance Varieties over Fields of Positive Characteristic

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

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