M. W. Davis, T. Januszkiewicz, I. J. Leary
We compute the l^2-Betti numbers of the complement of a finite collection of affine hyperplanes in complex space. At most one of the l^2-Betti numbers is non-zero.
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We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.
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.
Over the complex numbers, the complement of a collection of hyperplanes is a widely-studied object; the cohomology ring, in particular, is known to have a structure depending only on the combinatorial properties of the intersection of hyperplanes. The fundamental group, on the other hand, requires specific knowledge about the particular embedding in complex space, though the tower of nilpotent quotients can still be determined (see arXiv:math/9805056). Over the quaternions, however, since hyperplane complements are simply connected, the topological properties of hyperplane complements are simu…
Cohomology of $\mathbb{Z}$-local systems on complex hyperplane arrangement complements
本文证明了复超平面排列补集上秩一 $\mathbb{Z}$-局部系的 Cohen-Dimca-Orlik 型定理，从而解决了 S. Sugawara 最近提出的猜想。
Suppose $\Cal R$ is the complement of an essential arrangement of toric hyperlanes in the complex torus $(\C^*)^n$ and $π=π_1(\Cal R)$. We show that $H^*(\Cal R;A)$ vanishes except in the top degree $n$ when $A$ is one of the following systems of local coefficients: (a) a system of nonresonant coefficients in a complex line bundle, (b) the von Neumann algebra $\cnπ$, or (c) the group ring $\zz π$. In case (a) the dimension of $H^n$ is $|e(\Cal R)|$ where $e(\Cal R)$ denotes the Euler characteristic, and in case (b) the $n^{\mathrm{th}}$ $\eltwo$ Betti number is also $|e(\Cal R)|$.
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.
Mod 2 magnitude cohomology ring of real hyperplane arrangements
本文针对有限中心实超平面排列的顶图，改进Koizumi的模长同调结果，利用典范上同调基确定其交叉分次模长上同调环，明确了直和项的秩性质。
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 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.
Maximal twisted Betti numbers of complex hyperplane arrangement complements
本文证明本质复超平面排列补空间上局部系统的贝蒂数仅在局部系统为常值时最大，正面回答了吉永与第一作者近期提出的问题。
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…
Affine hyperplane arrangements at finite distance
本文研究了仿射超平面排列的有限距离上同调，提出了一种Orlik-Solomon型描述，并将其与正几何中的典范形式联系起来。
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…
Two standard invariants used to study the fundamental group G of the complement X of a hyperplane arrangement are the Malcev completion of G and the cohomology groups of X with coefficients in rank one local systems. In this paper, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. This is a prosolvable group that is tightly controlled by the cohomology groups of X with coefficients in rank one local systems. The prounipotent radical U_p of the relative completion S_p corresponds to a pronilpotent L…
The complement of an arrangement A of a finite number of affine hyperplanes in complex n-space has the structure of a poset of spaces indexed by the intersection poset, L(A). The space corresponding to G in L(A) is homotopy equivalent to the complement of the hyperplanes in the central arrangement A_G normal to G. This poset of spaces structure can be used to repair a spectral sequence argument in two earlier papers of Davis, Januszkiewicz, Leary and Okun for computing certain cohomology groups of arrangement complements. Similarly, toric hyperplane arrangements have the structure of a diagram…
The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real numbers. The cells and attaching maps are described in combinatorial terms of chambers. We also discuss the cellular chain complex with coefficients in a local system and a presentation for the fundamental group associated to the minimal CW-decomposition for the complement.
Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of a complex hyperplane arrangement has the homotopy type of a CW complex in which the number of p-cells equals the p-th betti number.
The cohomology on the complement of hyperplanes with the coefficients in the rank one local system associated to a generic weight vanishes except in the highest dimension. In this paper, we construct matroids or arrangements and its weights with non-vanishing cohomology of Orlik-Solomon algebras, using decomposable relations arising from Latin hypercubes.
We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n>1 the fundamental group of M-S satisfies Mostow-type Rigidity, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hy…
We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the complement to a line arrangement of a given combinatorial type with respect to isomorphisms inducing the canonical isomorphism of the first homology groups.
This note is mostly an expository survey, centered on the topology of complements of hyperplane arrangements, their Milnor fibrations, and their boundary structures. An important tool in this study is provided by the degree 1 resonance and characteristic varieties of the complement, and their tight relationship with orbifold fibrations and multinets on the underlying matroid. In favorable situations, this approach leads to a combinatorial formula for the first Betti number of the Milnor fiber and the algebraic monodromy. We also produce a pair of arrangements for which the respective Milnor fi…
In this paper we explore a relationship between the topology of the complex hyperplane complements $\mathcal{M}_{BC_n} (\mathbb{C})$ in type B/C and the combinatorics of certain spaces of degree-$n$ polynomials over a finite field $\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras $H^*(\mathcal{M}_{BC_n} (\mathbb{C});\mathbb{C})$, and an asymptotic stability result for certain polynomial statistics on monic squarefree…
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 Zπ_1-module presentation of π_p, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the π_1-coinvariants of π_p. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of π_2, and study some of the pro…
In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we provide combinatorial formulas for these lengths. As by-products, we also obtain in this case combinatorial formulas for the intersection cohomology Betti numbers of rank one local systems on the complement with same monodromy around the planes.
Under the generic situation, the cohomology with the coefficients in the local system on complements of hypersurfaces vanishes except in the highest dimension. Our problem is of when the local system cohomology does not vanish. In the case of arrangements of hyperplanes, many examples were founded. In this paper, we shall generalize their examples to hypersurfaces. We obtain that hypersurfaces given by some linear system have non-vanishing local system cohomologies.
We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us t…
Uniform approximation of Betti numbers
本文将Lefschetz关于仿射概形上同调逼近的原理实现为均匀形式，利用Jouanolou技巧、Deligne一般基变换公式的动形版本及反常层理论，提出了一种独立于上同调界定域上拟射影概形Betti数的新方法。
It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, i.e., they depend uniquely on the underlying matroid. To do this, we associate to every such matroid a topological space, that we call the reduced realization …
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 prove that the complement of any affine 2-arrangement in R^d is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its i-th rational Betti number. For the proof, we provide a Lefschetz-type hyperplane theorem for complements of 2-arrangements, and introduce Alexander duality for combinatorial Morse functions. Our results greatly generalize previous work by Falk, Dimca--Papadima, Hattori, Randell, and Salvetti--Settepanella and others, and they demonstrate that in contrast to previous investigations, a purely combinatorial approach suffices to show minimali…
We compute the complete Fadell-Husseini index of the 8 element dihedral group D_8 acting on S^d \times S^d, both for F_2 and for integer coefficients. This establishes the complete goup cohomology lower bounds for the two hyperplane case of Gr"unbaum's 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably-chosen hyperplanes in R^d? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of D_8.
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.
Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\mathcal{A}$ of locally flat, codimension-1 submanifolds that intersect like hyperplanes. To such a collection we associate two combinatorial objects: the face category and the intersection poset. We also associate a topological space to the arrangement called the tangent bundle complement. It is the complement of union of tangent bundles of these submanifolds inside the tangent bundle of the ambient manifo…
We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. The main tools are polar curves and the affine Lefschetz theory developped by H. Hamm and A. Némethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain an independant proof for the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result).
We use the Taylor resolution of a monomial ideal to compute the Tor algebra of the Stanley-Reisner ring of a simplicial complement of a simplicial complex.
Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive…
We use covering space theory and the fundamental group of complements of complexified-real line arrangements to explore the associated Milnor fiber. This work yields a combinatorially determined upper bound on the rank of the first homology group of the Milnor fiber. Under certain combinatorial conditions, we then show that one may determine the exact rank of the group and show that it is torsion free.
A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,σ}(Λ^{*,*}[P], d)$ over a principle ideal domain k and proved that $H_{*,*}(Λ[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)$, $link_{K}σ$, $star_{K}σ$ and the cohomology of the generalized moment-angle complexes.
In this paper we show that in a stable range the cohomology of the space of regular algebraic sections of a line bundle $\mathscr{L}$on a curve $X$ is isomorphic to the cohomology of the space of regular $C^{\infty}$sections of the same line bundle.
We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main tools are the polar curves and the affine Lefschetz theory developped by H. Hamm, D.T. Lê and A. Némethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result). This is then…
We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster than for a hypersurface as above. Our conjecture is supported by the results due to L. Narv\' ez Macarro and M. Saito on the roots of Bernstein-Sato polynomials of such hypersurfaces, by all the examples computed so far, and by one p…
For the complement of a hyperplane arrangement we construct a dual homology basis to the no broken circuit basis of cohomology. This is based on the theory of wonderful embeddings and nested sets.
The cohomology groups of finite cyclic covers of complexified real arrangement complements
本文研究复化实构形补空间有限循环覆盖的上同调群，利用Yoshinaga的室上链复形得到贝蒂数上界，引入类CDO组合条件证明此类覆盖整上同调无挠，部分推广了作者与Liu的相关结果。
Even torsions in the homology group of the Milnor fiber boundary of hyperplane arrangements in $\mathbb{C}^3$
本文研究三维复空间中超平面排列的Milnor纤维边界同调群，证明在某些条件下，第一同调群的偶阶torsion分解数大于等于项目化补集的欧拉示性。
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