Characteristic Varieties of Arrangements
نویسنده
چکیده
The k Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, Vk(A), of the complex algebraic torus (C). 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 the characteristic variety V1(A) coincides with R1(A), the first-cohomology support locus of the Orlik-Solomon algebra. Using work of Arapura [1] and Libgober [18], we conclude that the variety V1(A) is combinatorially determined, and that R1(A) is the union of a subspace arrangement in C, thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.
منابع مشابه
Translated Tori in the Characteristic Varieties of Complex Hyperplane Arrangements
Abstract. We give examples of complex hyperplane arrangements A for which the top characteristic variety, V1(A), contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus (C∗)|A|. 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 Or...
متن کاملMultinets, Parallel Connections, and Milnor Fibrations of Arrangements
The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces. One tool we use is the interpretation ...
متن کاملTriples of Arrangements and Local Systems
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 o...
متن کاملHomotopy Types of Complements of 2-arrangements in R
We study the homotopy types of complements of arrangements of n transverse planes in R4, obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the...
متن کاملCombinatorics of Line Arrangements and Characteristic Varieties
The complement M of a complex line arrangementA in C gives rise to combinatorial data, namely, the intersection lattice L(A). We prove that if the arrangement A is the complexification of a real arrangement, the characteristic varieties of complex rank one local systems on M are determined by the combinatorics of the arrangement A.
متن کاملAround the tangent cone theorem
A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several cl...
متن کامل