Hyperplane arrangements and Lefschetz's hyperplane section theorem
Masahiko Yoshinaga
备注：
47 pages, 13 figures; v.2: typos corrected, references added, v.3: minor change, v.4: minor change, v.5: final form to appear in Kodai Math. Journal (2007)
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.
We prove a certain 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties, similar to a result of Goresky and MacPherson (over complex numbers). This statement easily yields certain (vast) generalizations of the 'ordinary' Weak Lefschetz and Barth's theorems in arbitrary characteristic (that do not require any stratified Morse theory for their proof).
In this article, we give a counterexample to the Lefschetz hyperplane theorem for non-singular quasi-projective varieties. A classical result of Hamm-Lê shows that Lefschetz hyperplane theorem can hold for hyperplanes in general position. We observe that the condition of ``hyperplane'' is strict in the sense that it is not possible to replace it by higher degree hypersurfaces. The counterexample is very simple: projective space minus finitely many points. Moreover, as an intermediate step we prove that the Grothendieck-Lefschetz theorem also fails in the quasi-projective case.
In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements.
We show that the Quantum Lefschetz Hyperplane Principle can fail for certain orbifold hypersurfaces and complete intersections. It can fail even for orbifold hypersurfaces defined by a section of an ample line bundle.
The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.
The perverse filtration in cohomology and in cohomology with compact supports is interpreted in terms of kernels of restrictions maps to suitable subvarieties by using the Lefschetz hyperplane theorem and spectral objects. Various mixed-Hodge-theoretic consequences for intersection cohomology and for the decomposition theorem are derived.
We describe the standard and Leray filtrations on the cohomology groups with compact supports of a quasi projective variety with coefficients in a constructible complex using flags of hyperplane sections on a partial compactification of a related variety. One of the key ingredients of the proof is the Lefschetz hyperplane theorem for perverse sheaves and, in an appendix, we discuss the base change maps for constructible sheaves on algebraic varieties and their role in a proof, due to Beilinson, of the Lefschetz hyperplane theorem.
In this article, we study the $k$-Lefschetz properties for non-Artinian algebras, proving that several known results in the Artinian case can be generalized in this setting. Moreover, we describe how to characterize the graded algebras having the $k$-Lefschetz properties using sectional matrices. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements, with particular attention to the class of free arrangements.
The main theorem of the paper provides a way to produce examples such that the movable cone of an ample divisor does not coincide with the movable cone of its ambient variety.
We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For $Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence of deformation retracts and attachments of high-dimensional finite disc quotients. We use this to derive more familiar statements about the relative homotopy, homology, and cohomology groups of the pair $(X,Z)$. We also prove some preliminary results suggesting that the Lefschetz Hyperplane Theorem holds for Artin stacks as well. One technical innovation is to rei…
An effective algorithm of determining Gromov--Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov--Witten invariants of the ambient space is proposed. The Appendix is joint with E. Schulte-Geers.
We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let $J$ be the Jacobian of a curve $X$, and let $W_d \subset J$ be the locus of effective divisor classes of degree $d$. We show that the pair $(J^{an},W_d^{an})$ is $d$-connected, and thus in particular the inclusion of the analytification of the theta divisor $Θ^{an}$ into $J^{an}$ satisfies a Lefschetz hyperplane theorem for $\mathbb{Z}$-cohomology groups and homotopy groups. A key ingredient in our proof is a generalization, over arbitrar…
We study the topology of the space of affine hyperplanes $L \subset \CC^n$ which are in general position with respect to a given generic quadratic hypersurface $A$, and calculate the monodromy action of the fundamental group of this space on the relative homology groups $H_*(\CC^n, A \cup L)$ associated with such hyperplanes.
We prove a type of the Lefschetz hyperplane section theorem on Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities by using some degeneration method. As a byproduct, we obtain a new example of a Calabi-Yau 3-fold $X$ with Picard number one whose invariants are $$(H_X^3, c_2 (X) \cdot H_X, {e} (X)) = (8, 44, -88),$$ where $H_X$, $e(X)$ and $c_2(X)$ are an ample generator of $\Pic(X)$, the topological Euler characteristic number and the second Chern class of $X$ respectively.
The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We determine precisely when this phenomenon happens for two-dimensional arrangements.
We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.
We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropica…
Let X be a smooth projective variety over the complex numbers, and let D be an ample divisor in X. For which spaces Y is the restriction map r: Hom(X, Y) -> Hom(D, Y) an isomorphism? Using positive characteristic methods, we give a fairly exhaustive answer to this question. An example application of our techniques is: if dim(X) > 2, Y is smooth, the cotangent bundle of Y is nef, and dim(Y) < dim(D), the restriction map r is an isomorphism. Taking Y to be the classifying space of a finite group BG, the moduli space of pointed curves M_{g,n}, the moduli space of principally polarized Abelian var…
This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and …
We introduce the package \textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.
There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.
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.
One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on étale fundamental groups $π_1(H \cap U) \rightarrow π_1(U)$ is surjective. We prove a generalization, replacing the hyperplane by a general $\operatorname{PGL}_{n+1}$-translate of an arbitrary projective variety: If $U \subset \mathbb P^n$ is a normal quasi-projective variety, $X$ is a geometrically irreducible projective variety of dimension at least $n + 1 - \dim U$, and…
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.
Let X be a smooth Mori dream space of dimension at least 4. We show that, if X satisfies a suitable GIT condition which we call "small unstable locus", then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Neron-Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking "Mori dream hypersurfaces" of an ambient Mori dream space…
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $σ$-Gröbner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.
In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.
In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $\mathbb{Q}$.
This is the expanded notes of the lecture by the author in "Arrangements in Pyrenees", June 2012. We are discussing relations of freeness and splitting problems of vector bundles, several techniques proving freeness of hyperplane arrangements, K. Saito's theory of primitive derivations for Coxeter arrangements, their application to combinatorial problems and related conjectures.
We prove the existence of Gysin morphisms for hyperplane sections that may not satisfy the usual hypotheses of the Lefschetz hyperplane theorem. As an application, we show the triviality of the Alexander polynomial of a particular class of non-symmetric line arrangements, thus providing positive evidence for a conjecture of Papadima and Suciu.
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.
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of hyperplane arrangements attached to certain gain graphs.
Each irreducible component of the first resonance variety of a hyperplane arrangement naturally determines a codimension one foliation on the ambient space. The superposition of these foliations define what we call the resonance web of the arrangement. In this paper we initiate the study of these objects with emphasis on their spaces of abelian relations.
Faculty of Mathematics and Computer Science, University of Bucharest（布加勒斯特大学数学与计算机科学学院）；Simion Stoilow Institute of Mathematics（西蒙·斯托伊洛瓦数学研究所）；Institut für Mathematik, Humboldt-Universität zu Berlin（柏林洪堡大学数学研究所）；Department of Mathematics, University of Notre Dame（圣母大学数学系）；Department of Mathematics, Northeastern University（东北大学数学系）
We will consider some characterizations of freeness of a hyperplane arrangement, in terms of the following properties: locally freeness, factorization of characteristic polynomial and freeness of restricted multiarrangement. In the case of 3-arrangement, freeness is characterized by factorization of characteristic polynomial and coincidence of its roots with exponents of restricted multiarrangement. In the case of higher dimension, it is characterized by a kind of locally freeness and freeness of restricted multiarrangement. As an application, we prove the freeness of certain arrangements whic…
Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally …
If F is a master function corresponding to a hyperplane arrangement A and a collection of weights y, we investigate the relationship between the critical set of F, the variety defined by the vanishing of the one-form w = d log F, and the resonance of y. For arrangements satisfying certain conditions, we show that if y is resonant in dimension p, then the critical set of F has codimension at most p. These include all free arrangements and all rank 3 arrangements.
The minimal model program suggests a compactification of the moduli space of hyperplane arrangements which is a moduli space of stable pairs. Here, a stable pair consists of a scheme X which is a degeneration of projective space and a divisor D=D_1+..+D_n on X which is a limit of hyperplane arrangements. For example, in the 1-dimensional case, the stable pairs are stable curves of genus 0 with n marked points. Kapranov has defined an alternative compactification using his Chow quotient construction, which may be described fairly explicitly. We prove that these two compactifications coincide. W…
This is a slightly revised version (with references added in) of a survey article which appeared in the Spring 2005 edition of the MSRI newsletter, the Emissary. The article describes some of the themes from the Fall 2004 MSRI program on Hyperplane Arrangements and Applications.
The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given. It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over $\Q$, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial. We construct a hyperplane arrangement defined over $\Q$, whose Milnor fiber has a nontrivial monodromy o…
We give a presentation of the motivic cohomology ring of the complement of a hyperplane arrangement considered as algebra over the motivic cohomology of the ground field.
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.
We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.
We introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, i.e., plus-one generated arrangements have their logarithmic derivation modules generated by dimension plus one elements, with relations containing one linear form coefficient. We show that strictly plus-one generated arrangements can be obtained if we delete a hyperplane from free arrangements. We show a relative freeness criterion in terms of plus-one generatedness. In particular, for plane arrangements, we show…
We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.
One of the most important invariants in singularity theory is the Hodge spectrum. Calculating the Hodge spectrum is a difficult task and formulas exist for only a few cases. In this article the main result is the formula for reduced hyperplane arrangements in four variables.
The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from its Jacobian ideal.
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated discriminantal arrangement on the other hand. The type of hyperplane arrangements considered and the isomorphism classes have been defined precisely. As a consequence we enumerate such isomorphism classes by computing the characteristic polynomial of the discriminantal arrangement. With a certain restriction, the enumerated value is shown to be independent of the discri…
Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri and Marcolli posed questions about the $\mathbb{F}_1$-structure (in the sense of torification) on the complement of a hyperplane arrangement, especially for an arrangement defined in the space of cycles of a graph. In this paper, we prove that an arrangement has an $\mathbb{F}_1$-structure if and only if it is Boolean. We also prove that the arrangement in the cycle space of a graph is Boolean if and only if the cycle space has a basis consisting of cycles such that any two of them do not…
