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 Orlik-Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg-Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components. 1

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 Orlik-Solomon 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 Orlik-Solomon 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 non-resonant systems the cohomology is concentrated in the top dimension, without using resolution of singularities.

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We give examples of complex hyperplane arrangements for which the top characteristic variety contains positive-dimensional 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 Orlik-Solomon 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.

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 multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class 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.

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 Orlik-Solomon 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 E-module N called the singular variety,

The Orlik-Solomon algebra A(G) of a matroid G is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing G. On the other hand, some features of the matroid G are reflected in the algebraic structure of A(G). In this mostly expository article, we describe recent developments in the construction of algebraic invariants of A(G). We develop a categorical framework for the statement and proof of recently discovered isomorphism theorems which suggests a possible setting for classification theorems. Several specific open problems are formulated. The Orlik-Solomon algebra of a matroid Let G be a simple matroid with ground set [n]: = {1,..., n}. The Orlik-Solomon

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the I-adic completion of the Alexander invariant of a 1-formal group G is determined solely by the cup-product 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 1-formality. A detailed analysis of the irreducible components of the first resonance variety yields even stronger obstructions to realizing a 1-formal group as the fundamental group of a smooth, complex quasi-projective 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.

Abstract. 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. 1.

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 1-formal; in particular, the tangent cone to Vk at 1 equals Rk. These new obstructions to 1-formality 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 quasi-projective 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.