K. H. Kim, F. W. Roush
We construct a K3 surface whose transcendental lattice has a self-isomorphism which is not a linear combination of self-isomorphisms over $\mathbb{Q}$ which preserve cup products up to nonzero multiples. Products of it with itself give candidates for counterexamples to the Hodge conjecture which may be of interest. We also study the Kuga-Satake construction in relation to these examples.
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Summary: The Hodge conjecture asks whether rational Hodge classes on a smooth projective manifolds are generated by the classes of algebraic subsets, or equivalently by Chern classes of coherent sheaves. On a compact Kaehler manifold, Hodge conjecture is known to be false if algebraic subsets are replaced with analytic subsets. Here we show that it is even false that for a Kaehler manifold, Hodge classes are generated by Chern classes of coherent sheaves. We also show that finite free resolution do not in general exist for coherent sheaves on compact Kaehler manifolds.
We study the integral Hodge conjecture in complex codimension $2$ and $3$ for approximations to the classifying space of groups of type A. In degree two, we prove a conjecture of Ben Antieau, extending his two counterexamples to a general family of groups. In particular, when reduced to a finite field, these spaces extend Antieau's counterexamples to the integral $\ell$-adic Tate conjecture from the cases $\ell = 2, 3$ to all primes $\ell$.
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.
We consider the conjectures from the paper by Katzarkov, Kontsevich, and Pantev about Landau-Ginzburg Hodge numbers associated to tamely compactifiable Landau-Ginzburg models. We test these conjectures in case of dimension two, verifying some and giving a counterexample to the other.
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.
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.
In this paper we prove the Hodge conjecture for products of the form $S_1 \times ... S_n$, where $S_i$ are smooth projective surfaces such that $p_g(S_i)=1, q(S_i)=2$. We also prove the Hodge conjecture for arbitrary self-products of a K3 surface $X$ such that $End_{hodge}(T(X))$ is a CM number field.
The contents is changed.
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 space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group.
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 is shown to hold for rationally connected fivefolds, or more generally for fivefolds for which the base of the maximal rationally connected fibration is at most 3 dimensional.
Using morphic cohomology, we produce a sequence of conjectures, called morphic conjectures, which terminates at the Grothendieck standard conjecture A. A refinement of Hodge structures is given, and with the assumption of morphic conjectures, we prove a Hodge index theorem. We answer a question of Friedlander and Lawson by assuming the Grothendieck standard conjecture B and prove that the topological filtration from morphic cohomology is equal to the Grothendieck arithmetic filtration for some cases.
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.
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.
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。
We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields.
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.
We prove the Hodge-D-conjecture for general K3 and Abelian surfaces. Some consequences of this result, e.g., on the levels of higher Chow groups of products of elliptic curves, are discussed.
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。
We describe an inductive approach most appropriate for abelian varieties with an action of an imaginary quadratic field.
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.
This paper gives an introduction to Kuga-Satake varieties and discusses some aspects of the Hodge conjecture related to them. Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We give a detailed account of the construction of Kuga-Satake varieties and of their decomposition in simple subvarieties. We recall the Hodge conjecture and we point out a connection between the Hodge conjecture for abelian fourfolds and Kuga-Satake varieties. We discuss the implications of the Hodge conjecture on the g…
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.
A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers
本文提出一个统一的三参数反例族，证明Batyrev非负猜想在至少五维失败，并给出完整分类，指出唯一例外情况。
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 $Q_l$.
On a conjecture on Hodge loci of linear combinations of linear subvarieties
本文针对 Movasati 关于包含相交 $k$-平面的三次和四次超曲面 Hodge 轨迹维数的猜想，在 $k \geq 5$ 时给出反例，并在 $k=3,4$ 时提供支持该猜想的新证据。
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…
This is meant to be a survey article for the Cubo Journal. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those invariants. We present a geometric version of it.
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。
In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is also true for all smooth varieties $X'$ which are birationally equivalent to $X$. We also prove the analogous result for 1-cycles. As direct corollaries, the Hodge Conjecture holds for smooth rational projective manifolds with dimension less than or equal to five, and, the Generalized Hodge Conjecture holds for smooth rational projective manifolds with dimensi…
Conditional on the Lefschetz standard conjecture in degree 2, we prove that the index of a Brauer class on a smooth projective variety divides a fixed power of its period, uniformly in smooth families. In the other direction, we reinterpret in more classical terms recent work of Hotchkiss which gives Hodge-theoretic lower bounds on the index of Brauer classes. We also prove versions of our results over arbitrary algebraically closed base fields, and as an application construct qualitatively new counterexamples to the integral Tate conjecture.
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。
Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injective and the Hodge conjecture is true for certain smooth projective varieties over the algebraic closure of the rational number field. We also verify the conjecture on the surjectivity in the case of the complement of a union of general hypersurfaces in…
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.
In this note we give a counterexample to a conjecture proposed by Ciliberto about special linear systems of P^n through multiple base points.
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。
A non-klt counterexample to Shokurov's index conjecture
该研究针对非klt情形的Shokurov指标猜想，构造了反例，为相关代数几何问题提供了否定性依据。
Counterexamples to the Kuznetsov--Shinder L-equivalence conjecture
通过构造K3表面sheaves模空间的反例，推翻了Kuznetsov-Shinder关于D等价变种与L等价性的猜想。
An elementary way to find a counterexample to the Jacobian Conjecture
本文提出一种初等数学方法，用于构造雅可比猜想的反例，所得反例与Levent Alpöge的已知例子在坐标变换下等价。
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".
We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. An analysis of several natural matrices indexed by partitions is required.
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.
A counterexample to the PIA conjecture for minimal log discrepancies
本文通过构造反例，推翻了关于最小 log 抛物线差异的 PIA 猜想，并对家庭中的 LSC 猜想也提供了反例。
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