Kenji Koike
We show the dominance of a certain Prym map. By the Scoens result, this implies the algebraicity of Weil Hodge classes of Abelian 6-folds for the Gauss number field with the discriminant 1.
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Please forget our result!! I would like to apologize that there are a fatal mistake in the proof of the algebraicity of Weil Hodge cycles. See contents for the reason. So please forget about the result of algebraicity.
The Hodge theory of complex algebraic varieties is at heart a transcendental comparison of two algebraic structures. We survey the recent advances bounding this transcendence, mainly due to the introduction of o- minimal geometry as a natural framework for Hodge theory.
Algebraicity of Hodge classes on some generalized Prym Varieties
本文通过重新审视Chad Schoen在特定Prym簇上构造的代数环，利用无分歧几何类域论证明某些广义Prym簇上Hodge类的代数性。
Constructive Degenerations and the Algebraicity of Limiting Hodge
本文提出一种基于构造性退化和极限混合霍奇结构的框架，给出(p, p)型有理类在退化极限下成为代数类的判别准则，并论证每个有理(p, p)类都可作为代数闭链的退化极限，从而为霍奇猜想提供新的有效化研究路径。
Algebraicity of Artin--Hasse--Weil L-series over global function fields
本文证明了整体函数域上阿贝尔簇的L-函数被Artin表示扭曲后的特殊值满足Deligne周期猜想的类比，并以椭圆曲线为例说明。
Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture
本文提出一个符号解析框架，通过精细不变量厄米特谱指纹的消失性来判定绝对霍奇类，从而为霍奇猜想提供新的统一判据与解决路径。
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 prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$ (a result originally due to Cattani-Deligne-Kaplan).
We give a Hodge-theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance on higher dimensional bases up to a set of codimension $\geq 2$. The main tool is variation of Hodge structures and variation of mixed Hodge structures. We also give a description on the codimension 2 locus for the moduli space of Calabi--Yau threefolds. We prove that the points lying on exactly one finite and one infinite divisor have infinite Weil--Petersson distance along angular slices. Finally, by giving a classification of the dominant term of the candidates of the Weil--Petersson potent…
We determine which codimension two Hodge classes on $J\times J$, where $J$ is a general abelian surface, deform to Hodge classes on a family of abelian fourfolds of Weil type. If a Hodge class deforms, there is in general a unique such family. We show how to determine the imaginary quadratic field acting on the fourfolds of Weil type in this family as well as their polarization. There are Hodge classes that may deform to more than one family. We relate these to Markman's Cayley classes.
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 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。
Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class
本文针对具有平凡判别式的一般韦伊型阿贝尔四维流形的霍奇猜想给出新证明。借助其到\(K3^{[3]}\)型超凯勒六维流形的映射，通过超凯勒流形双有理几何及形变理论成果，得出六维流形切丛第二陈类拉回相关代数类结论。
In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the…
Secant sheaves and Weil classes on abelian varieties
本文研究了 abelian 变分上的 secant sheaves 和 Weil 类，通过构造嵌入和极化策略，证明了 Weil 类的 algebraicity，并推导了 Hodge 猜想的结论。
We study the Hodge theory of twisted derived categories and its relation to the period-index problem. Our main contribution is the development of a theory of twisted Mukai structures for topologically trivial Brauer classes on arbitrary smooth proper varieties and in families. As applications, we construct Hodge classes whose algebraicity would imply period-index bounds; construct new counterexamples to the integral Hodge conjecture on Severi-Brauer varieties; and prove the integral Hodge conjecture for derived categories of Deligne-Mumford surfaces.
Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on algebraic cycles have led Green and Griffiths to conjecture the algebraicity of the zero locus of normal functions. This corresponds to a mixed version of the theorem of Cattani, Deligne and Kaplan. This result has been proven recently by Brosnan-Pearlstein, Kato-Nakayama-Usui, and Schnell building on work of M. Saito. We will present some of the ideas around…
Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds
本文利用阿贝尔 n 重上的割线层证明判别式 -1 的 Weil 型阿贝尔六重上 Weil 类的代数性，并经退化论证推出阿贝尔四重情形及 Hodge 猜想。
We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension are disjoint.
Recent papers by Markman and O'Grady give, besides their main results on the Hodge conjecture and on hyperkaehler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six. In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic repres…
We analyze the relationship between abelian fourfolds of Weil type and Hodge structures of type K3, and we extend some of these correspondences to the case of arbitrary dimension.
Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
本文受 Markman 与 Kontsevich 启发，阐明 Weil 型阿贝尔 2n 维族仅在判别式为 $(-1)^n$ 时具极大幂幺退化，并给出判别式定义及相关事实。
Complex vs etale Abel Jacobi map and algebraicity of the zero locus of etale normal functions
本文利用开代数簇的 $p$-进 Hodge 理论，证明了 étale Abel-Jacobi 映射消失蕴含复 Abel-Jacobi 映射消失，并由此推导出 étale 正规函数零轨迹包含于复正规函数零轨迹及其代数性结论。
The Hodge conjecture for Fermat fourfolds of odd degree at most 199
该研究通过计算机辅助方法，结合几何闭包准则与霍奇轨道普查，证明了次数不超过199的奇次数费马四重曲面的霍奇猜想，分离出13个特殊轨道并验证了相关代数性。
Using our recent results on the algebraicity of the Hodge locus for variations of Hodge structures of level at least $3$, we improve the results of Lawrence-Venkatesh in direction of the refined Bombieri-Lang conjecture.
The period-index conjecture for abelian threefolds and Donaldson-Thomas theory
本文通过发展三维Calabi-Yau范畴的约化Donaldson-Thomas不变量理论，证明了阿贝尔三维流形上非分歧Brauer类的周期-指数猜想，并由此推导出一般扭曲阿贝尔三维流形的积分Hodge猜想。
We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in $X\times X$ of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara's bound, which can be seen as a {\em second order} Weil upper bound, to establish a new {\em third order} Weil upper bound, and using {\tt magma} to produce numerical tables for {\em higher order} Weil upper bounds. We also give some interpretation for the defect…
In this paper we study Hodge classes on complex abelian varieties. We prove some general results that allow us, in certain cases, to compute the Hodge group of a product abelian variety $X = X_1 \times X_2$ once we know the Hodge groups of the two factors. Using these results we can compute the Hodge groups of all abelian varieties of dimension $\leq 5$. We prove that the Hodge ring of any such abelian variety $X$ is generated by divisor classes together with the so-called Weil classes on (quotients of) $X$.
Consider a complex abelian variety X on which a field F acts. Generalizing a construction of A. Weil, one associates to this a subspace W_F of the cohomology of X, which we call the space of Weil classes w.r.t. F. The purpose of this paper is to answer the following two questions: Q1: under what conditions on F does the space W_F contain, or even consist of, Hodge classes?, Q2: if W_F contains Hodge classes, under what conditions on F are these exceptional? In case X is defined over a number field, we also answer the analogous questions for Tate classes.
Secant sheaves on abelian n-folds with real multiplication and Weil classes on abelian 2n-folds with complex multiplication
本文利用实乘法阿贝尔簇上的极化与旋量表示构造割线束，并借 Orlov 等价得到 CM 阿贝尔簇上的对象，以研究其平坦形变与 Hodge–Weil 类的代数性。
Bounded core partitions and Borel-Weil-Bott
本文基于博雷-韦-博特定理，通过分析分解，提出两种计算相关Hodge数的公式，并引入新的整数钩积统计量和半标准 Young表，改进了Hodge数正性的已有界限。
In this paper, we represent the Hodge metric in terms of the Weil-Petersson metric and its Ricci curvature on the moduli spaces of polarized Calabi-Yau threefolds.
McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds
研究McMullen曲线与Weil位移在Abelian六折中的性质，证明其交点为有限且可能为空，揭示Hodge-Weil类的特殊性及Hodge猜想的新案例。
We prove, following Deligne and André, that the Hodge classes on abelian varieties of CM-type can be expressed in terms of divisor classes and split Weil classes, and we describe some consequences. In particular, we show that Grothendieck's standard conjecture of Lefschetz type implies the Hodge conjecture for abelian varieties (Abdulali, André, ...).
Let $A$ be an Abelian variety of dimension $n$. For $0<p<2n$ an odd integer, Sampson constructed a surjective homomorphism $π:J^p(A)\rightarrow A$, where $J^p(A)$ is the higher Weil Jacobian variety of $A$. Let $\widehatω$ be a fixed form in $H^{1,1}(J^p(A),\mathbb{Q})$, and $N=\dim (J^p(A))$. He observes that if the map $π_*(\widehat{ω}^{N-p-1}\wedge .): H^{1,1}(J^p(A),\mathbb{Q})\rightarrow H^{n-p,n-p}(A,\mathbb{Q})$ is injective, then the Hodge conjecture is true for $A$ in bidegree $(p,p)$. In this paper, we give some clarification of the approach and show that the map above is {not inject…
Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $tt^*$ Structures
本文严格证明了 $tt^*$ 结构的完整 CY/LG 对应，通过分析 LG 侧周期积分并修正实结构，保持了 Hodge 滤过、双线性型及实结构。
We establish an unexpected relation among the Weil-Petersson metric, the generalized Hodge metrics and the BCOV torsion. Using this relation, we prove that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not admit complete subvarieties. That is, there is no complete family for certain class of polarized Calabi-Yau manifolds. We also give an estimate of the complex Hessian of the BCOV torsion using the relation. After establishing a degenerate version of the Schwarz Lemma of Yau, we prove that the complex Hessian of the BCOV torsion is bounded by the Poincaré metric.
Analytically dense Mordell-Weil rank jumps on elliptic surfaces via transverse realization
本文研究椭圆面的高度模数，通过动机高度ζ函数的有理性与Kudla-Millson模性，探讨Mordell-Weil秩跃迁问题。
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