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 $Y$ is a general $\operatorname{PGL}_{n+1}$-translate of $X$, then the map $π_1(Y \cap U) \rightarrow π_1(U)$ is surjective.
The Lefschetz algebra $L^*(X)$ of a smooth complex projective variety $X$ is the subalgebra of the cohomology algebra of $X$ generated by divisor classes. We construct smooth complex projective varieties whose Lefschetz algebras do not satisfy analogues of the hard Lefschetz theorem and Poincaré duality.
We study the Lefschetz standard conjecture on a smooth complex projective variety X. In degree 2, we reduce it to a local statement concerning deformations of vector bundles on X. When X is hyperkähler, we show that the existence of nontrivial deformations of stable hyperholomorphic bundles implies the Lefschetz standard conjecture in codimension 2.
The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen-Vial and Fu-Tian-Vial on multiplicative Chow-Künneth decompositions.
Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue characteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction, (3) $K$ is $p$-adic and $X$ has "combinatorial reduction."' We also reprove a version of their result, Hodge symmetry for $H^1$, without the use of moduli spaces of semistable sheaves. All of this relies on cases of Kato's log hard Lefschetz conjecture, which we prove for $H^1$ and for lo…
It is known that all complete intersection Artinian standard graded algebras of codimension 3 have the Weak Lefschetz Property. Unfortunately, this property does not continue to be true when you increase the number of minimal generators for the ideal defining the algebra. For instance, it is not more valid for almost complete intersection Artinian standard graded algebras of codimension 3. On the other hand, the Hilbert functions of all Weak Lefschetz Artinian graded algebras are unimodal and with the positive part of their first difference forming an $O$-sequence (i.e. are Weak Lefschetz sequ…
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 investigate the Lefschetz standard conjecture for degree $2$ cohomology of hyper-Kähler manifolds admitting a covering by Lagrangian subvarieties. In the case of a Lagrangian fibration, we show that the Lefschetz standard conjecture is implied by the SYZ conjecture characterizing classes of divisors associated with Lagrangian fibration. In dimension $4$, we consider the more general case of a Lagrangian covered fourfold $X$, and prove the Lefschetz standard conjecture in degree $2$, assuming $ρ(X)=1$ and $X$ is general in moduli. Finally we discuss various links between Lefschetz cycles and…
We show a Lefschetz theorem for irreducible overconvergent $F$-isocrystals on smooth varieties defined over a finite field. We derive several consequences from it.
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport \cite{LR87} of deriving the Kottwitz's formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands-Rapoport conjecture), which replaces their Galois gerb theoretic arguments by more geometric …
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
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 …
Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of Mori theory, fully determines each other. When two varieties are related in this way, we call them birational twins. We explore this notion and its various flavors for other embeddings between Mori dream spaces.
This note is a short survey on the Lefschetz defect, an invariant of smooth Fano varieties that has been recently introduced; it is related to the Picard number rho(X) of X, and to the Picard number of prime divisors in X. We explain the definition of the Lefschetz defect delta(X) and its origin, the known results - mainly on the case delta(X)>1, and several examples, especially in dimensions 3 and 4. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with rho(X)>4, due to Mori and Mukai. We also review the known examples of Fa…
We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov-Witten techniques we also determine the generating series of Noether-Lefschetz numbers of a general pencil of Debarre-Voisin varieties. This reproves and extends a result of Debarre, Han, O…
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).
The goal of this article is to try understand where Hodge cycles on a singular complex projective variety X come from. As a first step we consider Hodge cycles on the maximal pure quotient $H^{2p}(X)/W_{2p-1}$, and introduce a class of algebraic cycles that we call homologically Cartier, that should conjecturally describe all such Hodge cycles. Secondly, given a singular complex projective variety $X$, we show that there is a cycle map from motivic cohomology group $H^{2p}_M(X,Q(p))$ to the space of weight 2p Hodge cycles in $H^{2p}(X,Q)$. We conjecture that this is surjective when X is define…
We study the positivity of complete intersections of nef classes. We first give a sufficient and necessary characterization on the complete intersection classes which have hard Lefschetz property on a compact complex torus, equivalently, in the linear case. In turn, this provides us new kinds of cohomology classes which have Hodge-Riemann property or hard Lefschetz property on an arbitrary compact Kähler manifold. We also give a complete characterization on when the complete intersection classes are non-vanishing on an arbitrary compact Kähler manifold. Both characterizations are given by the …
We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we discuss classification results for 4-dimensional complex smooth projective varieties admitting some particular fiber type contractions.
We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.
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.
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke correspondence acting on the compactly supported étale cohomology and verify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization of that formula. The main ingredients of our proof of the formula are a recent work of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory of Galois gerbs developed by Langlands and Rapoport \cite{LR87}. Es…
We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is …
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).
We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $\ell$ give maximal rank for all the multiplication maps $\times \ell: [A]_i \longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the \emph{non-Lefschetz locus}. For monomial complete …
We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperkähler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of the Beauville-Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a $K3$ surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of $X$ of Shen and Vial agrees with the eigenspace decomposition of a canonical lift …
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.
We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let $γ$ be a correspondence between two abelian varieties $A$ and $B$ that can be written as a linear combination of products of symmetric divisors. Assume that $A$ is isogenous to th…
We consider artinian algebras $A=\mathbb{C}[x_0,\ldots,x_m]/I$, with $I$ generated by a regular sequence of homogeneous forms of the same degree $d\geq 2$. We show that the multiplication by a general linear form from $A_{d-1}$ to $A_d$ is injective. We prove that the Weak Lefschetz Property holds for artinian complete intersection algebras as above, with $d=2$ and $m\leq 4$. Apparently, this was previously known only for $m\leq 3$. Although we are proposing only very limited progress towards the WLP conjecture for complete intersections, we hope that the methods of the present article can ill…
A Lefschetz (1,1) Theorem for normal projective complex varieties
我们证明了Lefschetz（1，1）定理的一个类似定理，它刻画了Cartier因子的上同调类（或等价的线丛的第一类Chern类）在第二积分上同调中的性质.设$X$是正规复射影簇.我们证明了$H^2(X,Z)$中的Cartier因子类恰好是$x$类，使得（i）$x$在$H^2(X,C)$中的像（复系数上同调）位于$F^1 H^2(X,C)$（Deligne混合Hodge结构的Hodge滤子的第一级），（ii）$x$是Zurki-local平凡的，即，存在$X$被Zerkki开集$U$覆盖使得$x$在$H^2(U,Z)$中具有零像。对于正常的拟射影簇，这肯定地回答了一个问题的Barbieri-Viale和Srinivas（J. Reine Ang. Math.450（1994）），其中给出的例子表明，除数类本身不具有上述条件（i）、（ii）中的任何一个的特征，这与非奇异变种的情况不同。本文件还包含一个例子，一个非正规的投射品种，其中（i）和（ii）不足以表征除数类。
Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of C. Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X,Y,Z]/(X^d,Y^d,Z^d) in terms of d and p. This answers a question of J. Migliore, R. M. Miro-Roig and U. Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X^d,Y^d,Z^d) on PP^2 satisfies the Grauert-Muelich theorem. As a corollary we obtain that for p=2 the algebra A has the weak Lefschetz property if and only if d=(2^t+1)/3 or d=(2^t-1)/3 for s…
We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Cech-de~Rham …
Given an ideal $I=(f_1,\ldots,f_r)$ in $\mathbb C[x_1,\ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $\mathbb C[x_1,\ldots,x_n]/I^k$ be? If $r\le n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S/I^k$, where $I$ is the ideal generated by the $d$'th powers of the v…
We prove the Relative Hard Lefschetz theorem and the Relative Hodge-Riemann bilinear relations for combinatorial intersection cohomology sheaves on fans.
