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 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.
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.
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.
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 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.
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…
In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic conce…
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 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.
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.
Let $S$ be a complex projective surface. Lefschetz originally proved Lefschetz $(1, 1)$--Theorem by studying a Lefschetz pencil of hyperplane sections of $S$ and the Abel--Jacobi mapping. In this paper, we attack Lefschetz $(1, 1)$--Theorem by constructing certain two-parameter families of twice hyperplane sections of $S$ and then applying the topological Abel--Jacobi mapping. Our geometric constructions would give an inductive approach and some insight for higher dimensional cases. We prove a strong tube theorem which generalizes Schnell's tube theorem to integral homology groups for complex …
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…
Let $ϕ:X\rightarrow \mathbb{P}^n$ be a morphism of varieties. Given a hyperplane $H$ in $\mathbb{P}^n$, there is a Gysin map from the compactly supported cohomology of $ϕ^{-1}(H)$ to that of $X$. We give conditions on the degree of the cohomology under which this map is an isomorphism for all but a low-dimensional set of hyperplanes, generalizing results due to Skorobogatov, Benoist, and Poonen-Slavov. Our argument is based on Beilinson's theory of singular supports for étale sheaves.
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 propose a strengthening of the Grothendieck--Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities. Version 2: Main conjecture changed, following recent results of Bhatt and de Jong. Small changes to the proofs.
By the Lefschetz hyperplane theorem, if X is a smooth quasi-projective variety and C a general curve section of X then the fundamental group of C surjects onto the fundamental group of X. Here we consider when this conclusion holds for a general member of an arbitrary family of curves that covers X. The most interesting case is families of rational curves.
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.
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 prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $\mathbb{R}$ which is "close" to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and Bézout's theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation the…
According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we…
We show the existence of good hyperplane sections for schemes over discrete valuation rings with good or (quasi) semistable reduction, and the existence of good Lefschetz pencils for schemes with good reduction or ordinary quadratic reduction. As an application we prove that the reciprocity map introduced for smooth projective varieties over local fields by Bloch, Kato and Saito is an isomorphism after profinite completion, if the variety has good reduction or 'almost good' reduction.
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…
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…
We prove a strengthening of the Grothendieck-Lefschetz hyperplane theorem for local Picard groups conjectured by Kollár. Our approach, which relies on acyclicity results for absolute integral closures, also leads to a restriction theorem for higher rank bundles on projective varieties in positive characteristic.
We describe the Hilbert scheme components parametrizing lines and conics on the space of determinantal nets of conics, N. As an application, we use the quantum Lefschetz hyperplane principle to compute the instanton numbers of rational curves on a complete intersection Calabi-Yau threefold in N. We also compute the number of lines and conics on some Calabi-Yau sections of non-decomposable vector bundles on N. The paper contains a brief summary of the A-model theory leading up to Givental-Kim's quantum Lefschetz hyperplane principle.
Givental has defined a Lagrangian cone in a symplectic vector space which encodes all genus-zero Gromov-Witten invariants of a smooth projective variety X. Let Y be the subvariety in X given by the zero locus of a regular section of a convex vector bundle. We review arguments of Iritani, Kim-Kresch-Pantev, and Graber, which give a very simple relationship between the Givental cone for Y and the Givental cone for Euler-twisted Gromov-Witten invariants of X. When the convex vector bundle is the direct sum of nef line bundles, this gives a sharper version of the Quantum Lefschetz Hyperplane Princ…
Jacobian algebras and variation of hyperplane sections
Università degli Studi Di Napoli “Federico II”（那不勒斯费德里科二世大学）；Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高等研究所）；Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学）
In the present paper Mori extremal rays of a smooth projective manifold X are divided into two classes: L-supported and L-negligible (where ``L'' stands for ``Lefschetz'' since the division comes from Hard Lefschetz Theorem). Roughly speaking: L-supported rays are strongly distinguishable in topology while L-negligible rays have very mild geometry. Each L-supported ray R defines hyperplane in H^2(X,R) on which Lefschetz duality degenerates so it is a cohomology ring invariant. The hyperplane carries a multiplicity (cohomology ring invariant) which is related to the geometry of the ray R. The n…
We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $Θ$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $Θ$.
As is well known, the Lefschetz theorems for the étale fundamental group of SGA1 do not hold. We fill a small gap in the literature showing they do for tame coverings. Let $X$ be a regular projective variety over a field $k$, and let $D\hookrightarrow X$ be a strict normal crossings divisor. Then, if $Y$ is an ample regular hyperplane intersecting $D$ transversally, the restriction functor from tame étale coverings of $X\setminus D$ to those of $Y\setminus D\cap Y$ is an equivalence if dimension $X \ge 3$, and fully faithful if dimension $X=2$. The method is dictated by Grothendieck-Murre ("Th…
Given a very ample line bundle on a smooth projective variety, the variation of Hodge structure associated to the universal family of hyperplane sections can be thought of as a $D$-module with action generated by the Gauss-Manin connection. The decomposition theorem for a projective morphism and results similar to the Lefschetz hyperplane theorem tell us that the only nontrivial part of this VHS occurs in the middle degree corresponding to the vanishing cohomology of the hyperplane sections. We use Nori's connectivity theorem to give an explicit description of the cohomology sheaves of the de …
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we make some remarks on connections between the mentioned property of a projective variety and its adjunction properties.
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…
Given a vector bundle $F$ on a smooth Deligne-Mumford stack $\X$ and an invertible multiplicative characteristic class $\bc$, we define the orbifold Gromov-Witten invariants of $\X$ twisted by $F$ and $\bc$. We prove a "quantum Riemann-Roch theorem" which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A Quantum Lefschetz Hyperplane Theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of $\X$ and that of a complet…
We describe a new relation between the topology of hyperplane arrangements, Milnor fibers and global polar curves, via the affine Lefschetz theory developped by A. Némethi. In particular, we improve some results due to Orlik and Terao (see Math. Ann. 301(1995)) and complete/clarify a proof by Randell concerning the minimality of hyperplane arrangements (see math.AT/0011101).
Let $M$ be a complex projective manifold of dimension $n+1$ and $f$ a meromorphic function on $M$ obtained by a generic pencil of hyperplane sections of $M$. The $n$-th cohomology vector bundle of $f_0=f|_{M-\RR}$, where $\RR$ is the set of indeterminacy points of $f$, is defined on the set of regular values of $f_0$ and we have the usual Gauss-Manin connection on it. Following Brieskorn's methods in [bri], we extend the $n$-th cohomology vector bundle of $f_0$ and the associated Gauss-Manin connection to $\pl$ by means of differential forms. The new connection turns out to be meromorphic on t…
Given a holomorphic vector bundle $E:EX X$ over a compact Kähler manifold, one introduces twisted GW-invariants of $X$ replacing virtual fundamental cycles of moduli spaces of stable maps $f: Σ\to X$ by their cap-product with a chosen multiplicative characteristic class of $H^0(Σ, f^* E) - H^1(Σ, f^*E)$. Using the formalism of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for $X$. The result (Theorem 1) is a consequence of Mumford's Riemann -- Roch -- Grothendieck formula applied to the universal stable map. When $E$ is concave, a…
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).
Let $Y$ be a smooth complex projective variety. We study the cohomology of smooth families of hypersurfaces $X\to B$ for $B\subset{\bf P}H^0(Y,O(d))$ a codimension $c$ subvariety. We give an asymptotically optimal bound on $c$ and $k$ for $d\to\infty$ for the space $H^k(X,\C)$ not to be spanned by the image of $H^k(Y\times B,\C)$, thus extending the validity of Lefschetz Hyperplane section Theorem and Nori's Connectivity Theorem. Next, we construct in the limit case explicit families of higher Chow groups which span the non trivial cohomology classes in $X$. We give examples of indecomposable …
We discuss a paper of M. Green from a new algebraic perspective, and provide applications of its results to level and Gorenstein algebras, concerning their Hilbert functions and the weak Lefschetz property. In particular, we will determine a new infinite class of symmetric $h$-vectors that cannot be Gorenstein $h$-vectors, which was left open in a recent work of Migliore-Nagel-Zanello. This includes the smallest example previously unknown, $h=(1,10,9,10,1)$. As M. Green's results depend heavily on the characteristic of the base field, so will ours. The appendix will contain a new argument, kin…
We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve.
The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the exis…
We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V. Srinivas in the surface case. We then show that the Abel-Jacobi map for codimension 2 cycles and the Albanese map are bijective if a general hyperplane section is a surface for which Bloch's conjecture is proved. In certain cases we verify Nori's conjecture on the Griffiths group. We also prove a weak Lefschetz-type theorem for (higher) Chow groups, generalize a…
We describe the package "IncidenceCorrespondenceCohomology" for the computer algebra system Macaulay2. The main feature concerns the computation of characters and dimensions for the cohomology groups of line bundles on the incidence correspondence (the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it). Additionally, the package provides tools for (1) computing the multiplication in the graded Han-Monsky representation ring, (2) determining the splitting type of vector bundles of principal parts on the projective line, and (3) tes…
Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $\mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $\mathcal{A}_I$ is isomorphic to $H^*(\mbox{Hess}(N,I))$ and $H^*(\mbox{Hess}(S,I))^W$, the invariants in $H^*(\mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, an…
Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An example of such a birational isomorphism is the Abel-Jacobi map relating the Hilbert scheme of g points on a K3 of degree 2g-2 to an integrable system: the union of Jacobians of hyperplane sections (curves) of genus g. The main results are: 1) We construct a stratified version of a Mukai elementary transformation modeled after dual pairs of Springer resolutions…
Cohomology on the incidence correspondence and related questions
Busitema University（布西泰马大学）；Università degli Studi di Genova（热那亚大学）；CIMAT - Centro de Investigación en Matemáticas（墨西哥数学研究中心）；University of Notre Dame（圣母大学）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛瓦数学研究所）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
