25 pages. In this version a new Section was added, containing a presentation for the cohomology ring of toric arrangements. Furthermore the other sections were expanded to improve readability
Given an arrangement of subtori of arbitrary codimension in a torus, we compute the cohomology groups of the complement. Then, using the Leray spectral sequence, we describe the multiplicative structure on the graded cohomology. We also provide a differential model for the cohomology ring by considering a toric wonderful model and its Morgan algebra. Finally we focus on the divisorial case, proving a new presentation for the cohomology of toric arrangements.
We consider the arrangements of subtori in a flat d - dimensional torus T. Let us consider an arrangement on n subtori of codimension one, let f be the number of connected components of the complement in T to the union of subtori. We found the set of all possible numbers f for given n and d and arbitrary arrangements of subtori.
We compute the equivariant cohomology $H^*_{T_I}(\mathcal Z_K)$ of moment-angle complexes $\mathcal Z_K$ with respect to the action of coordinate subtori $T_I \subset T^m$. We give a criterion for the equivariant formality of $\mathcal Z_K$ and obtain specifications for the cases of flag complexes and graphs.
A toric arrangement is a finite collection of codimension-$1$ subtori in a torus. These subtori stratify the ambient torus into faces of various dimensions. Let $f_i$ denote the number of $i$-dimensional faces; these so-called face numbers satisfy the Euler relation $\sum_i (-1)^i f_i = 0$. However not all tuples of natural numbers satisfying this relation arise as face numbers of some toric arrangement. In this paper we focus on toric arrangements in a $2$-dimensional torus and obtain a characterization of face numbers. In particular we show that the convex hull of these face numbers is a con…
A subspace arrangement is a finite collection of affine subspaces in $\mathbb{R}^n$. One of the main problems associated to arrangements asks up to what extent the topological invariants of the union of these spaces, and of their complement are determined by the combinatorics of their intersection. The most important result in this direction is due to Goresky and MacPherson. As an application of their stratified Morse theory they showed that the additive structure of the cohomology of the complement is determined by the underlying combinatorics. In this paper we consider toric arrangements; a …
We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain: -a "combinatorial" version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements, -a uniqueness result for realizations of a…
In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type A. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group.
We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over $\mathbb Z$. The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the ar…
A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$.
We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determined by the arithmetic matroid (however it is determined by the poset of layers).
Some projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T were constructed in [3]. In this paper we describe their integer cohomology rings by generators and relations.
In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.
We prove that the cohomology algebra of elliptic arrangements depends only on the poset of layers. In the particular case of braid elliptic arrangements, we study the cohomology as representation and we compute some Hodge numbers. Finally, we discuss 1-formality for graphic elliptic arrangements.
From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI$_W$-module theory which Wilson…
A bi-arrangement of hyperplanes in a complex affine space is the data of two sets of hyperplanes along with a coloring information on the strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call its motive. The motivation for studying such relative cohomology groups comes from the notion of motivic period. More generally, we suggest the systematic study of the motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute the structure of these motives. Our main object is the Orlik-Solom…
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.
The notions of deleted and restricted arrangements have been useful in the study of arrangements of hyperplanes. If A is an arrangement of hyperplanes, x in A and A', A'' the deleted and restricted arrangements, there is a formula connecting the Poincare polynomials of the complement spaces M(A), M(A') and M(A''). In this paper, we consider the extension of this formula to arbitrary subspaces arrangements. The main result is the existence of a long exact sequence connecting the rational cohomology of M(A), M(A') and M(A''). Using this sequence, we obtain new results connecting the Betti number…
We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset $\mathfrak{L}$, and then define the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \mathcal{C})$ over a commutative ring with unit, which is built by the classical Orlik--Solomon algebra and a monoidal cosheaf $\mathcal{C}$ as coefficients. Furthermore, we construct a monoidal cosheaf $\hat{\mathcal{C}}(\mathcal{A})$ associated with $\mathcal{A}$, so that the…
An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian arrangements, we provide a combinatorial presentation for a differential graded algebra whose cohomology is isomorphic to the rational cohomology of the complement. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers.
A new relation between a class of complex polynomials with a good behavior at infinity studied by A. Némethi and A. Zaharia and the cohomology groups of affine complex hyperplane arrangement complements with rank one local system coefficients is introduced and explored. This approach gives in particular new upper-bounds for the dimension of the twisted cohomology groups of line arrangement complements in the complex affine plane.
A referee found an error in the proof of the Theorem 2 that we could not fix. More precisely, the proof of Lemma 2.1 is incorrect. Hence the fact that integer cohomology of complement of toric Weyl arrangements is torsion free is still a conjecture. ----- A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we prove that if $\Cal T_{\wdt W}$ is the toric arrangement defined by the \textit{cocharacters} lattice of a Weyl group $\wdt W$, then the integer cohomology of its complement is torsion free.
We prove that the complement of a toric arrangement has the homotopy type of a minimal CW complex. As a corollary we obtain that the integer cohomology of these spaces is torsion free. We use Discrete Morse Theory, providing a sequence of cellular collapses that leads to a minimal complex.
We give a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement of type A_{n-1} as a graded representation of S_n, and we give a conjectural description of this representation in terms of the cohomology of the configuration space of n points in SU(2) modulo translation. We also give a version of this conjecture for more general graphical arrangements.
We prove vanishing results of the cohomology groups of Aomoto complex over arbitrary coefficient ring for real hyperplane arrangements. The proof is using minimality of arrangements and descriptions of Aomoto complex in terms of chambers. Our methods also provide a new proof for the vanishing theorem of local system cohomology groups which was first proved by Cohen, Dimca and Orlik.
Associated to any subspace arrangement is a "De Concini-Procesi model", a certain smooth compactification of its complement, which in the case of the braid arrangement produces the Deligne-Mumford compactification of the moduli space of genus 0 curves with marked points. In the present work, we calculate the integral homology of real De Concini-Procesi models, extending earlier work of Etingof, Henriques, Kamnitzer and the author on the (2-adic) integral cohomology of the real locus of the moduli space. To be precise, we show that the integral homology of a real De Concini-Procesi model is iso…
We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the co…
Let $\A$ be an arrangement of affine lines in $\C^2,$ with complement $\M(\A).$ The (co)homo-logy of $\M(\A)$ with twisted coefficients is strictly related to the cohomology of the Milnor fibre associated to the conified arrangement, endowed with the geometric monodromy. Although several partial results are known, even the first Betti number of the Milnor fiber is not understood. We give here a vanishing conjecture for the first homology, which is of a different nature with respect to the known results. Let $Γ$ be the graph of \emph{double points} of $\A:$ we conjecture that if $Γ$ is connecte…
In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of the coordinate ring of a complement of a hyperplane arrangement. We give a presentation of these rings by generators and defining relations. This presentation was used by Holler and I. Kriz to calculate the $\mathbb{Z}$-graded coefficients of localizations of ordinary $RO((\mathbb{Z}/p)^n)$-graded equivariant cohomology at a given set of representation spheres, and also more recently by the author in a generalization to the case of an arbitrary finite group. We also give…
We study a topological method to calculate the first Milnor fiber cohomology of a defining polynomial of a reduced projective hyperplane arrangement $X$ of degree $d$. We can show the vanishing of a monodromy eigenspace of the first Milnor fiber cohomology with eigenvalue of order $m\ge 2$ if $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle})$ or more generally $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle}\cup X_d)$ is connected. Here $X^{[(m)]}$ is the set of points of $X$ with multiplicity divisible by $m$, and $X^{\langle 3\rangle}:=\bigcup_{i,j,k}X_i\cap X_j\cap X_k$ with $X_i$ the irreducib…
We compute the mixed Hodge structure on the cohomology ring of complements of complex coordinate subspace arrangements. The mixed Hodge structure can be described in terms of the special bigrading on the cohomology ring of complements of complex coordinate subspace arrangements. Originally this bigrading was introduced in the setting of toric topology by V.M. Buchstaber and T.E. Panov.
The integral cohomology ring of the complement of an arrangement of linear subspaces of a finite dimensional complex projective space is determined by combinatorial data, i.e. the intersection poset and the dimension function.
Deletion-restriction is a fundamental tool in the theory of hyperplane arrangements. Various important results in this field have been proved using deletion-restriction. In this paper we use deletion-restriction to identify a class of toric arrangements for which the cohomology algebra of the complement is generated in degree $1$. We also show that for these arrangements the complement is formal in the sense of Sullivan.
We show that the monodromy operator action on the first cohomology group of the Milnor fiber is combinatorially determined for line arrangements with at most triple points and containing at most 18 lines, with one possible exception.
An n-dimensional polytope P^n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P^n with m codimension-one faces defines an arrangement of even-dimensional planes in R^{2m}. We construct a free action of the group R^{m-n} on the complement of this arrangement. The corresponding quotient is a smooth manifold Z_P invested with a canonical action of the compact torus T^m with the orbit space P^n. For each smooth projective toric variety M^{2n} defined by a simple polytope P^n with the given lattice of faces there exists a subgroup T…
In this survey article we present several new developments of `toric topology' concerning the cohomology of face rings (also known as Stanley-Reisner algebras). We prove that the integral cohomology algebra of the moment-angle complex Z_K (equivalently, of the complement U(K) of the coordinate subspace arrangement) determined by a simplicial complex K is isomorphic to the Tor-algebra of the face ring of K. Then we analyse Massey products and formality of this algebra by using a generalisation of Hochster's theorem. We also review several related combinatorial results and problems.
We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and th…
Church-Ellenberg-Farb used the language of FI-modules to prove that the cohomology of certain sequences of hyperplane arrangements with S_n-actions satisfies representation stability. Here we lift their results to the level of the arrangements themselves, and define when a collection of arrangements is "finitely generated". Using this notion we greatly widen the stability results to: 1) General linear subspace arrangements, not necessarily of hyperplanes. 2) A wide class of group actions, replacing FI by a general category C. We show that the cohomology of such collections of arrangements sati…
We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology rin…
A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arrangement is defined by a Weyl group we also provide an algebraic description, very handy for cohomology computations. In the last part we give a description in terms of tableaux for a toric arrangement appearing in robotics.
We show that the center of a flat graded deformation of a standard Koszul algebra behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming the Koszul dual algebra. This is an algebraic version of a duality discovered by Goresky and MacPherson b…
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 classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.
