As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. We show an arithmetic analog. Our primary technique is $p$-adic nonabelian Hodge theory. Conditional on certain foundational results in \emph{logarithmic} $p$-adic Hodge theory, we also show a logarithmic analog.
In this article we prove that the log Hodge de Rham spectral sequences of certain proper log smooth schemes of Cartier type in characteristic $p>0$ degenerate at $E_1$. We also prove that the log Kodaira vanishings for them hold when they are projective. We formulate the log weak Lefschetz conjecture for log crystalline cohomologies and prove that it is true in certain cases.
We give a short proof of a Grothendieck-Lefschetz Theorem for equivariant Picard groups of nonsingular varieties with the action of an affine algebraic group.
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 propose some variants of Lefschetz fixed point theorem for Fourier-Mukai functors on a smooth projective algebraic variety. Independently we also suggest a similar theorem for endo-functors on the category of perfect modules over a smooth and proper DG algebra.
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 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.
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.
Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational (k,k)-cohomology class is algebraic, i.e., the Hodge conjecture holds on them.
The theorem of Barth-Lefschetz is a statement about the cohomology of a submanifold X of some projective space, in a range depending on the codimension of the embedding. Here this is generalized to the case of a submanifold X of a smooth projective toric variety Y, under some natural conditions on X. In particular, in the Barth-Lefschetz range a formula for the Hodge numbers of X in terms of the toric, i.e. combinatorial data of Y is given.
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.
We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty). Our method is an adaptation of Griffiths and Harris' degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups, which generalize several known results.
We introduce the notion of lef line bundles on a complex projective manifold. We prove that lef line bundles satisfy the Hard Lefschetz Theorem, the Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations. We study proper holomorphic semismall maps from complex manifolds and prove that, for constant coefficients, the Decomposition Theorem is equivalent to the non-degeneracy of certain intersection forms. We give a proof of the Decomposition Theorem for the complex direct image of the constant sheaf when the domain and the target are projective by proving that the forms in question are…
We prove that for a normal projective variety $X$ in characteristic 0, and a base-point free ample line bundle $L$ on it, the restriction map of divisor class groups $\Cl(X)\to \Cl(Y)$ is an isomorphism for a general member $Y\in |L|$ provided that $\dim{X}\geq 4$. This is a generalization of the Grothendieck-Lefschetz Theorem, for divisor class groups of singular varieties.
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.
Theorems of Barth-Lefschetz type describe restrictions on the topology of varieties of small codimension. R. Schoen and J. Wolfson, using Morse theory on a path space, have described a technique to prove theorems of this kind for complex submanifolds of Kähler manifolds of non-negative holomorphic bisectional curvature. In this paper this program is carried out for the compact Hermitian symmetric spaces. The key technical point is to define and compute an invariant, called the {\it complex positivity}, that measures the ``amount'' of positive curvature, in a suitable sense.
In this paper we prove a concentration theorem for arithmetic $K_0$-theory, this theorem can be viewed as an analog of R. Thomason's result in the arithmetic case. We will use this arithmetic concentration theorem to prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. Such a formula was conjectured of a slightly stronger form by K. Köhler and D. Roessler and they also gave a correct route of its proof. Nevertheless our new proof is much simpler since it looks more natural and it doesn't involve too many complicated computations.
The goal of this paper is to generalize a theorem of Fujiwara (formerly Deligne's conjecture) to the situation appearing in a joint work [KV] with David Kazhdan on the global Langlands correspondence over function fields. Moreover, our proof is more elementary than the original one and stays in the realm of ordinary algebraic geometry, that is, does not use rigid geometry. We also include proof of the Lefschetz-Verdier trace formula and of the additivity of filtered trace maps, thus making the paper essentially self-contained.
We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over the complex numbers of dimension $\ge 4$ are completions of UFDs of finite type over the complex numbers.
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.
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…
In this note, we obtain a number of results related to the hard Lefschetz theorem for pseudoeffective line bundles, due to Demailly, Peternell and Schneider. Our first result states that the holomorphic sections produced by the theorem are in fact parallel, when viewed as currents with respect to the singular Chern connection associated with the metric. Our proof is based on a control of the covariant derivative in the approximation process used in the construction of the section. Then we show that we have an isomorphsim between such parallel sections and higher degree cohomology. As an applic…
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.
We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field $k$ of odd characteristics or characteristic 0. We follow the strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an application, we calculate the generic fibers of Hitchin systems over a smooth quintic surface of Picard number 1.
We prove the Relative Hard Lefschetz theorem and the Relative Hodge-Riemann bilinear relations for combinatorial intersection cohomology sheaves on fans.
We reprove the Lefschetz trace formula for Lubin-Tate spaces, based on the locally finite cell decompositions of these spaces obtained by Fargues, and Mieda's theorem of Lefschetz trace formula for certain open adic spaces (\cite{Mi1} theorem 3.13). This proof is rather different from those of Strauch in \cite{St} (theorem 3.3.1) and of Mieda in \cite{Mi1} (example 4.21), and is quite hopeful to generalized to some other Rapoport-Zink spaces as soon as there exist suitable cell decompositions. For example, we proved a Lefschetz trace formula for some unitary Rapoport-Zink spaces in \cite{Sh} b…
We show a Lefschetz theorem for irreducible overconvergent $F$-isocrystals on smooth varieties defined over a finite field. We derive several consequences from it.
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…
We study an analogue of the Noether-Lefschetz theorem in the context of Beilinson's Hodge conjecture. We estimate the dimension of the Noether-Lefschetz locus by the infinitesimal methods in Hodge theory.
Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact Kähler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible $f$-vectors of convex polytopes. While the statements of these theorems depend on the choice of a Kähler class, or its analog, there is usually a cone of possible Kähler classes. It is then natur…
The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a general polytope.
Let X be a very general complete intersection in complex projective space and we denote by $F_r(X)$ the variety of r-planes in X, for $r\geq 1$. We show that the Picard number of $F_r(X)$ is 1, as soon as $\dim F_r(X)\geq 2$, except when X is a quadric of dimension 2r or 2r+2, or X is a complete intersection of two quadrics of dimension 2r+2. We also apply this result to determine the cohomology class of the variety of planes of a cubic fivefold contained (by the Abel-Jacobi map) in the intermediate Jacobian.
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).
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…
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…
New updated edition by Yves Laszlo of the book ``Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2)'', Advanced Studies in Pure Mathematics 2, North-Holland Publishing Company - Amsterdam, 1968. Published by the Societe Mathematique de France http://smf.emath.fr/Publications/DocumentsMathematiques/ Original text also available in the LaTeX file. Dans cet ouvrage, on montre des théorèmes d'algébrisation et de pureté qui permettent d'obtenir des théorèmes de type Lefschetz pour le groupe fondamental ou de Picard. In this monograph algebraization and pu…
We prove the Weak Lefschetz theorem for simplicial PL-spheres. This result is weaker than the Hard Lefschetz theorem for more general spheres proved by Adiprasito (arXiv:1812.10454), but the proof here involves simple algebra and avoids the more complicated combinatorics.
Using the Grothendieck-Lefschetz theory (see \cite{[SGA2]}) we prove a criterion to deduce that certain subvarieties of $\mathbb P^n$ of dimension $\geq 2$ are not set-theoretic complete intersections (see Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the last part of the paper we prove that the arithmetic rank of a rational normal $d$-dimensional scroll $S_{n_1,...,n_d}$ in $\mathbb P^N$ is $N-2$, by producing an explicit set of $N-2$ homogeneous equations which define these scrolls set-theoretically (see Theorem 2 of the Introduction).
We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne's program in Weil II. The notes are based on some aspects of the material for the Santaló lectures at the Universidad Complutense de Madrid (October 2015) and the Rademacher lectures at the University of Pennsylvania (February 2016). They appear in Revista Matemática Complutense.
We continue our study of the Noether-Lefschetz loci in toric varieties and investigate deformation of pairs (V,X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a odd dimensional simplicial projective toric variety, with V\subset X. Under some assumptions, we prove that the cohomological class in H^{k,k}(X) associated to V remains of type (k,k) under an infinitesimal deformation if and only if V remains algebraic. Actually we prove that locally the Noether-Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 …
The Hard Lefschetz theorem for intersection cohomology of nonrational polytopes was recently proved by K. Karu [Ka]. This theorem implies the conjecture of R. Stanley on the unimodularity of the generalized $h$-vector. In this paper we strengthen Karu's theorem by introducing a canonical bilinear form $(\cdot ,\cdot)_Φ$ on the intersection cohomology $IH(Φ)$ of a complete fan $Φ$ and proving the Hodge-Riemann bilinear relations for $(\cdot ,\cdot)_Φ$.
