Nodes and the Hodge conjecture
R. P. Thomas
证明了Hodge猜想等价于一个关于具有普通二重点奇点的非常充足因子的同调问题。并讨论了结果的无穷小形式。
The Hodge conjecture is shown to be equivalent to a question about the homology of very ample divisors with ordinary double point singularities. The infinitesimal version of the result is also discussed.
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The paper provides a version of the rational Hodge conjecture for $\3\dg$ categories. The noncommutative Hodge conjecture is equivalent to the version proposed in \cite{perry2020integral} for admissible subcategories. We obtain examples of evidence of the Hodge conjecture by techniques of noncommutative geometry. Finally, we show that the noncommutative Hodge conjecture for smooth proper connective $\3\dg$ algebras is true.
We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.
As an application of our previous work on CM liftings of K3 surfaces and the Tate conjecture, we prove the Hodge standard conjecture for squares of K3 surfaces. We also deduce the Hodge standard conjecture for all the powers of certain K3 surfaces.
We describe in some details an idea of M. Kontsevich how one can try to find a counterexample to the Hodge conjecture using tropical geometry.
Despite the failure of the integral Hodge conjecture, we show that the rational Hodge conjecture implies an integral version (modulo torsion) of the absolute Hodge conjecture.
The Hodge conjecture is a major open problem in complex algebraic geometry. In this survey, we discuss the main cases where the conjecture is known, and also explain an approach by Griffiths-Green to solve the problem.
On the Hodge conjecture for products of certain surfaces
在本文中，我们证明了$S_1 \times ... S_n$形式的积的霍奇猜想，其中$S_i$是光滑的投影曲面，使得$p_g(S_i)=1, q(S_i)=2$。证明了K3曲面$X$的任意自积的Hodge猜想，使得$End_{hodge}(T(X))$是CM数域.
On Beilinson's Hodge and Tate conjectures for open complete intersections
内容发生了变化。
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).
The Hodge Conjecture for general Prym varieties
证明了一般Prym簇的Hodge圈空间是由它的Neron-Severi群生成的。
Variational Hodge conjecture for complete intersections on hypersurfaces in projective space
该研究针对射影空间中超曲面上完全交闭链的变分Hodge猜想，提出了一种全新的简化证明方法，为相关代数几何问题的研究提供了新的论证思路。
The Hodge conjecture for rationally connected fivefolds
霍奇猜想被证明是合理连接的fivefolds，或更一般的fivefolds的最大合理连接纤维化的基础是最多3维。
Grothendieck standard conjectures, morphic cohomology and Hodge index theorem
利用形态上同调，我们得到一个终止于Grothendieck标准猜想A的序列，称为形态上同调。给出了Hodge结构的一个加细，并在形态结构的假设下，证明了一个Hodge指标定理.通过假设Grothendieck标准猜想B，回答了Friedlander和Lawson的一个问题，并证明了在某些情况下，由形态上同调得到的拓扑滤子等价于Grothendieck算术滤子.
In this paper we discuss an obstruction to the integral Hodge conjecture, which arises from certain behavior of vanishing cycles. This allows us to construct new counter-examples to the integral Hodge conjecture. One typical such counter-example is the product of a very general hypersurface of odd dimension and an Enriques surface. Our approach generalizes the degeneration argument of Benoist-Ottem.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
The Hodge conjecture for powers of K3 surfaces of Picard number 16
本文针对一般Picard数为16的K3曲面族，证明在Kuga–Satake对应为代数的条件下，族内任意K3曲面的所有幂都满足Hodge猜想。
Hodge loci and absolute Hodge classes
我们结合德利涅的全球不变循环定理，代数定理的Cattani，德利涅和卡普兰，为连接组件的轨迹的霍奇类，得出结论，在简单的假设下，这些组件被定义在数域（假设初始家庭），预期从霍奇猜想。我们还表明，霍奇猜想（弱）绝对霍奇类减少到霍奇猜想（弱）绝对霍奇类的品种定义在数域。
The Beilinson-Hodge conjecture asserts the surjectivity of the cycle map $$H^n_M(X,\Q(n)) \to {\rm Hom}_{MHS}(\Q(-n),H^n(X,\Q))$$ for all positive integers $n$ and every smooth complex algebraic variety $X$. For $n=2$, we prove the conjecture if $X$ is rational.
Indecomposable K_1 and the Hodge-D-conjecture for K3 and Abelian Surfaces
证明了一般K3曲面和阿贝尔曲面的Hodge-D-猜想。这一结果的一些后果，例如，讨论了椭圆曲线乘积的高Chow群上的
The Hodge and Tate conjectures for hyper-Kähler sixfolds of generalized Kummer type
研究广义库默尔型六维超凯勒簇，证明其霍奇猜想和泰特猜想。
An inductive approach to the Hodge conjecture for abelian varieties
我们描述了一个归纳的方法，最适合的阿贝尔簇与一个虚二次域的行动。
A counterexample to the Hodge conjecture for Kaehler varieties
总结：霍奇猜想是问光滑射影流形上的有理霍奇类是否由代数子集类生成，或者等价地由相干层的陈类生成。在紧致Kaehler流形上，若代数子集被解析子集取代，则Hodge猜想是不成立的。在这里，我们表明，它甚至是假的，对于Kaehler流形，霍奇类生成的陈类的相干层。我们还表明，有限自由分解一般不存在的凝聚层紧Kaehler流形。
In this article we study the (cohomological) Hodge conjecture for singular varieties. We prove the conjecture for simple normal crossing varieties that can be embedded in a family where the Mumford-Tate group remains constant. We show how to produce such families. Furthermore, we show for varieties with worse singularities the conjecture can be expressed solely in terms of the algebraic classes.
Kuga-Satake varieties and the Hodge conjecture
本文介绍了Kuga-Satake簇，并讨论了与之相关的Hodge猜想的一些问题。Kuga-Satake簇是与一定权重的两个Hodge结构相关联的阿贝尔簇，例如K3曲面的第二上同调群。我们给一个详细的帐户建设的久贺佐竹品种和他们的分解简单的子品种。我们回顾了霍奇猜想，我们指出了霍奇猜想之间的连接交换四倍和Kuga-Satake品种。讨论了Hodge猜想对第二上同调群具有Kuga-Satake簇的曲面几何的影响.我们的结论与最近的一些结果的Kuga-Satake品种的霍奇结构上的一个虚二次域的作用。
We use a result of van Geemen to determine the endomorphism algebra of the Kuga--Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of $\PP^2$ which are ramified along six lines.
Let $X/C$ be a general product of elliptic curves. Our goal is to establish the Hodge-D-conjecture for $X$. We accomplish this when $\dim X \leq 5$. For $\dim X \geq 6$, we reduce the conjecture to a matrix rank condition that is amenable to computer calculation.
We explain intersection multiplicity defined by J. P. Serre, in terms of the Poincare product in Hodge theory by a modification of the chern character map. We also discuss a formulation of the Euler characteristic via the action of correspondences on the Chow groups of projective varieties, assuming the Grothendieck Standard conjectures over $Q_l$.
We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.
We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi-Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of int…
Congruences for the number of rational points, Hodge type and motivic conjectures for Fano varieties
这是一篇为《库博杂志》撰写的调查文章。讨论了有限域上有理点的存在性和个数，复数上的Hodge型，以及控制这些不变量的动机结构.我们提出了一个几何版本。
On the Log Hodge Theory of Toroidal Varieties and a Partial Proof of the Absolute Hodge Conjecture
本文构建加权环面结构与 Hodge 类障碍理论，借助 $E_1$-退化结果建立范畴 Hodge 对应，证明一类有理加权射影环面簇的绝对 Hodge 猜想。
The Generalized Hodge conjecture for 1-cycles and codimension two algebraic cycles
本文证明了“光滑射影簇$X$上的（广义）Hodge猜想对余维为2的圈成立“是一个双有理不变命题，即如果该命题对$X$成立，则对所有与$X$双有理等价的光滑簇$X'$也成立.我们还证明了1-圈的类似结果。作为直接推论，Hodge猜想对维数小于或等于5的光滑有理射影流形成立，广义Hodge猜想对维数小于或等于4的光滑有理射影流形成立.
On the integral Hodge conjecture for varieties with trivial Chow group
本文构造了全周群有限秩却违反整 Hodge 猜想的复光滑射影簇，并证明此类反例也可定义在数域上。
On the Hodge and Tate conjectures for moduli spaces of curves
本文综述了通过Hodge和Tate猜想视角研究稳定曲线模空间的上同调近期进展，特别是广义的锥形形式如何将Hodge结构和l-adic伽罗华表示与代数循环联系起来，并讨论了开放问题和未来研究方向。
We establish the real integral Hodge conjecture for 1-cycles on various classes of uniruled threefolds (conic bundles, Fano threefolds with no real point, some del Pezzo fibrations) and on conic bundles over higher-dimensional bases which themselves satisfy the real integral Hodge conjecture for 1-cycles. In addition, we show that rationally connected threefolds over non-archimedean real closed fields do not satisfy the real integral Hodge conjecture in general and that over such fields, Bröcker's EPT theorem remains true for simply connected surfaces of geometric genus zero but fails for some…
A Deformation Theoretic Reduction of the Hodge Conjecture via Derived Categories and Complete Intersections
本文提出几何与范畴方法，通过嵌入平坦族并利用导出范畴证明完全交情形的霍奇猜想，并将一般情形约化为极限代数性假设，统一了霍奇理论、形变与导出技术。
Hodge conjecture for projective hypersurface
本文通过证明霍奇类为解析对数De Rham类，并证明此类为代数闭链类，从而证明了光滑射影超曲面的霍奇猜想。
Hodge-theoretic variants of the Hopf and Singer Conjectures
本文提出了Hopf与Singer猜想的若干Hodge理论类比，并在特殊情形下给出了证明。
Hodge-type conjecture for higher Chow groups
设X是有理数域的代数闭包上的光滑拟射影簇。证明了高阶Chow群到Deligne上同调的圈映射是内射的，高阶Hodge圈是由Beilinson和Jannsen所证明的圈映射的象生成的，如果到Deligne上同调的圈映射是内射的，并且对于有理数域的代数闭包上的某些光滑投射簇，Hodge猜想成立.证明了光滑射影簇中一般超曲面并的补的满射性猜想。
In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincare polynomial of the associated cohomology ring.
We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a proper curve of compact type over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture …
Following ideas of Bloch, Esnault, and Kerz, the deformational part of Grothendieck's variational Hodge conjecture is established for proper, smooth schemes over $K[[t]]$, where $K$ is an algebraic extension of the rational numbers. The main tool is a pro Hochschild-Kostant-Rosenberg theorem for Hochschild homology.
We study the generalized Hodge conjecture for certain sub-Hodge structure defined as the kernel of the cup product map with a big cohomology class, which is of Hodge coniveau at least 1. As predicted by the generalized Hodge conjecture, we prove that the kernel is supported on a divisor, assuming the Lefschetz standard conjecture.
We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotica…
Non-abelian Hodge theory for non-proper varieties and the linear Shafarevich conjecture
本文综述了非交换Hodge理论在非 proper 代数变种中的最新进展，并利用这些工具构造代数Shafarevich映射，证明了对于任何代数变种的线性Shafarevich猜想版本。
Indecomposable cycles and the case of the twisted Hodge D conjecture
本文引入扭曲循环及其上同调调节子，证明一般光滑射影曲面上调节子满射的猜想，并在椭圆四重流形上构造不可分解扭曲循环以证明该映射非零。
Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".
Hodge integrals, partition matrices, and the lambda_g conjecture
证明了稳定点曲线的模空间上Hodge丛的顶Chern类对余切线类积分的一个闭公式。这些积分通过从Gromov-Witten理论中的虚局部化获得的关系来计算。需要分析由分区索引的几个自然矩阵。
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.
For a fixed integer $d$, we study here the locus of degree $d$ hypersurfaces $X$ in $\mathbb{P}^{2n+1}$ such that $H^{2n}(X,\mathbb{Q}) \cap H^{n,n}(X,\mathbb{C}) \not= \mathbb{Q}$. We call this locus \textit{the Noether-Lefschetz locus}. Any irreducible component of this locus is locally a Hodge locus. So, we see that the study of this locus is very closely related to the variational Hodge conjecture. In this article we show that the Hodge locus corresponding to the cohomology class of a complete intersection subscheme in $\mathbb{P}^{2n+1}$ of codimension $n+1$ and degree less than $d$ satis…
Counterexample to the Hodge Conjecture
我们构造了一个K3曲面，它的超越格具有一个自同构，它不是$\mathbb{Q}$上保持上积直到非零倍数的自同构的线性组合。它与自己的产品提供候选人的反例霍奇猜想这可能是感兴趣的。我们还研究了与这些例子有关的Kuga-Satake结构。
We use the universal generation of algebraic cycles to relate (stable) rationality to the integral Hodge conjecture. We show that the Chow group of 1-cycles on a cubic hypersurface is universally generated by lines. Applications are mainly in cubic hypersurfaces of low dimensions. For example, we show that if a generic cubic fourfold is stably rational then the Beauville--Bogomolov form on its variety of lines, viewed as an integral Hodge class on the self product of its variety of lines, is algebraic. In dimension $3$ and $5$, we relate stable rationality with the geometry of the associated i…
Hodge Conjecture via Singular Varieties
本文利用光滑射影簇与奇异超曲面的双有理关系，证明具有 $A_n$ 奇点的奇数维超曲面满足两种 Hodge 猜想，且其消解满足经典 Hodge 猜想。
The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties
通过奇异OG6型流形与Weil型判别式1的四维阿贝尔流形的几何关系，证明了Hodge猜想和Tate猜想。
Building upon the Bloch-Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of the second-named author implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Ojanguren and the first named author implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain…
We formulate the "real integral Hodge conjecture", a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi-Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel-Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the Chow group of 1-cycles of real threefolds. New results about these proble…
On algebraic 1-motives related to Hodge cycles
引入代数簇的Hodge 1-模，并给出相应的上同调Grothendieck-Hodge猜想，将经典的Hodge猜想推广到任意奇异的真概型上.
We know that semi-regular sub-varieties satisfy the variational Hodge conjecture i.e., given a family of smooth projective varieties $π:\mathcal{X} \to B$, a special fiber $\mathcal{X}_o$ and a semi-regular subvariety $Z \subset \mathcal{X}_o$, the cohomology class corresponding to $Z$ remains a Hodge class (as $\mathcal{X}_o$ deforms along $B$) if and only if $Z$ remains an algebraic cycle. In this article, we investigate examples of such sub-varieties. In particular, we prove that any smooth projective variety $Z$ of dimension $n$ is a semi-regular sub-variety of a smooth projective hypersur…
Degeneration of families of projective hypersurfaces and Hodge conjecture
该研究通过维数归纳法，结合光滑超曲面情形的既有结论、混合霍奇模的邻近循环函子等工具，证明了光滑复射影簇的霍奇猜想。
Hodge cycles on some moduli spaces
目标是在一定的模空间上验证Hodge猜想（以及一些相关的猜想）。证明了（广义）Hodge猜想对阿贝尔曲面或曲线上的向量丛的射影模空间成立，如果它对曲面或曲线的所有幂都成立.在后一种情况下，曲线可以由其雅可比矩阵代替。因此，当曲线非常一般时，广义Hodge猜想对这些模空间成立。对于任意曲面上点的Hilbert格式也证明了类似的结果。
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