Complements and Higher Resonance Varieties of Hyperplane Arrangements
نویسندگان
چکیده
Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some non-linear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using homological algebra results and vector bundles techniques, of the fact that all resonance varieties are determinantal.
منابع مشابه
Triples 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 کاملGeometric and Homological Finiteness in Free Abelian Covers
We describe some of the connections between the Bieri–Neumann– Strebel–Renz invariants, the Dwyer–Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, tran...
متن کاملTranslated 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 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 کاملHigher Homotopy Groups of Complements of Complex Hyperplane Arrangements
We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Zπ1-module presentation of πp, the first non-vanishing higher homotopy g...
متن کامل