Mark Andrea A. de Cataldo, Luca Migliorini
We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new Hodge-theoretic results on the cohomology of projective varieties which extend naturally the classical theory and provide new applications.
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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.
The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.
We review Hodge structures, relating filtrations, Galois Theory and Jordan-Holder structures. The prototypical case of periods of Riemann surfaces is compared with the Galois-Artin framework of algebraic numbers.
We introduce the notion of mixed Hodge complex on an algebraic variety, improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge structure with the theory of mixed Hodge module. This was supposed to be true, but is quite nontrivial, because the theory of mixed Hodge module does not work on a simplicial scheme. Some application to the Du Bois singularity is given.
We get a new inequality on the Hodge number $h^{1,1}(S)$ of fibred algebraic complex surfaces $S$, which is a generalization of an inequality of Beauville. Our inequality implies the Arakelov type inequalities due to Arakelov, Faltings, Viehweg and Zuo, respectively.
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。
For a linear algebraic group $G$ over $\bf Q$, we consider the period domains $D$ classifying $G$-mixed Hodge structures, and construct the extended period domains $D_{\mathrm{BS}}$, $D_{\mathrm{SL}(2)}$, and $Γ\backslash D_Σ$. In particular, we give toroidal partial compactifications of mixed Mumford--Tate domains.
Hodge similarities, algebraic classes, and Kuga-Satake varieties
本文引入超凯勒流形超越格的霍奇相似性概念，证明特定 K3 曲面族平方的霍奇猜想，并利用 Kuga-Satake 构造的函子性推出广义 Kummer 形变型超凯勒流形的霍奇相似性均为代数的。
We prove a new inequality for the Hodge number h^1,1 of irregular complex smooth projective surfaces of general type without irregular pencils of genus >1. More specifically we show that if the irregularity q satisfies q=2^k+1 then h^1,1>4q-4.
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。
We provide a Hodge theoretical characterization of the set of algebraic numbers which arises from the complete list, due to A. Beauville, of semistable families of elliptic curves over $\mathbb{P}^1$ with four singular fibers. Our technical innovation is the analysis of the periodicity of the uniformizing Higgs bundle attached to $\mathbb{P}^1$ minus four points over the field of complex numbers.
Spectral Fingerprints of Algebraic Cycles: A Hodge-Theoretic Approach to the Hodge Conjecture and Special L-Values
本文提出检测复代数簇上代数闭链的“谱指纹理念”，通过霍奇理论与算术数据关联闭链代数性，证明Kummer K3曲面新除子的存在，为霍奇猜想研究提供新视角。
We prove that given any rational Hodge isometry $ψ:H^2(S_1,\mathbb{Q})\rightarrow H^2(S_2,\mathbb{Q})$ between any two Kähler $K3$ surfaces $S_1$ and $S_2$ the cohomology class of $ψ$ in $H^{2,2}(S_1\times S_2)$ is a polynomial in Chern classes of coherent analytic sheaves over $S_1 \times S_2$. Consequently, the cohomology class of $ψ$ is algebraic whenever $S_1$ and $S_2$ are algebraic.
Lecture notes of lectures delivered at the Hodge Theory Summer School in ICTP Trieste, June 2010.
Given a complex variety $X$, a linear algebraic group $G$ and a representation $ρ$ of the fundamental group $π\_1(X,x)$ into $G$, we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of the representation variety of $π\_1(X,x)$ into $G$ at $ρ$ using mixed Hodgediagrams and methods of $L\_\infty$ algebras. We apply it in two geometric situations: either when $X$ is compact K{ä}hler and $ρ$ is the monodromy of a variation of Hodge structure, or when $X$ is smooth quasi-projective and $ρ$ has finite image.
We prove that there exists a pencil of Enriques surfaces defined over $\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.
Algebraic cycles of some Fano varieties with Hodge structure of level one
本文研究了具有Hodge结构一级的Fano簇的代数循环，通过étale动机方法证明了光滑四次双五维流形的整数Hodge猜想。
Let $X$ be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products $Sym^{n}X$ when the cohomology of $X$ is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials $μ_{X^{n}}^{S_{n}}\left(t,u,v\right)$, codifying the permutation action of $S_{n}$ as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products $μ_{Sym^{n}X}\left(t,u,v\right)$. These formulas are then applied to the case of linear alg…
We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data of some Newton polyhedra. In particular, the limit mixed Hodge numbers and some part of the Jordan normal forms of the monodromies of such a family will be described very explicitly.
Relations for quadratic Hodge integrals via stable maps
该研究遵循Faber-Pandharipande方法，利用到$ \mathbb{P}^1 $的稳定映射模空间虚局部化公式计算霍奇积分关系，证明其某些生成函数为多项式，为二次霍奇积分研究提供新结果。
Spectral Analysis of Hodge Cycles: A Novel Approach to the Hodge Conjecture via Generalized Moments
本研究提出研究霍奇猜想的新方法，通过推广Zernike矩定义霍奇闭链的特征形式与谱展开系数，经玩具模型验证后构建适用于K3曲面的框架，为霍奇猜想研究提供新方向。
Let G be a reductive group over an algebraically closed field of positive characteristic. Let C be a smooth projective curve over k. We give a description of the moduli space of flat G-bundles in terms of the moduli space of G-Higgs bundles over the Frobenius twist C' of C. This description can be regarded as the non-abelian Hodge theory for curves in positive characteristic.
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…
Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves
本文证明了正特征代数曲线上驯顺 $G$-局部系统与对数 $G$-Higgs 丛间的非阿贝尔 Hodge 对应，并引入 parahoric 群概型语言完整刻画了非紧情形。
This paper has been superseded to a great extent by the following: Paper: math.AG/0511558 Title: The Neron-Severi group of a proper seminormal complex variety Authors: L. Barbieri-Viale, A. Rosenschon & V. Srinivas Paper: math.AG/0102150 Title: Deligne's Conjecture on 1-Motives Authors: Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito Journal-ref: Ann. of Math. (2), Vol. 158 (2003), no. 2, 593--633
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。
We give, for a complex algebraic variety $S$, a Hodge realization functor $\mathcal F_S^{Hdg}$ from the derived category of constructible motives $DA_c(S)$ to the derived category $D(MHM(S))$ of algebraic mixed Hodge modules over $S$. Moreover, for $f:T\to S$ a morphism of complex quasi-projective algebraic varieties, $\mathcal F_{-}^{Hdg}$ commutes with the four operation $f^*$,$f_*$,$f_!$,$f^!$ on $DA_c(-)$ and $D(MHM(-))$, making the Hodge realization functor a morphism of 2-functor wich for a given $S$ sends $DA_c(S)$ to $D(MHM(S)$, moreover $\mathcal F_S^{Hdg}$ commutes with tensor produc…
For an algebraic vector bundle $E$ over a smooth algebraic variety $X$, a monodromic $D$-module on $E$ is decomposed into a direct sum of some $O$-modules on $X$. We show that the Hodge filtration of a monodromic mixed Hodge module is decomposed with respect to the decomposition of the underlying $D$-module. By using this result, we endow the Fourier-Laplace transform $M^{\wedge}$ of the underlying $D$-module $M$ of a monodromic mixed Hodge module with a mixed Hodge module structure. Moreover, we describe the irregular Hodge filtration on $M^{\wedge}$ concretely and show that it coincides with…
We construct infinitely many non-isotrivial families of abelian varieties over given four punctured projective lines. These families lead to algebraic solutions of Painleve VI equation. Finally, based on a recent paper by Lin-Sheng-Wang, we prove a complete characterization for the locus of motivic Higgs bundles in the moduli space as fixed points of an ``additive'' self-map. This is a note based on the lecture given by the second named author on 04 Nov. 2022 at Tsinghua University.
In this paper, we show some applications to algebraic cycles by using higher Abel-Jacobi maps which were defined in [the author: Motives and algebraic de Rham cohomology]. In particular, we prove that the Beilinson conjecture on algebraic cycles over number fields implies the Bloch conjecture on zero-cycles on surfaces. Moreover, we construct a zero-cycle on a product of curves whose Mumford invariant vanishes, but not higher Abel-Jacobi invariant.
We study a property of cycle spaces in connection with degenerating Hodge structures of odd-weight, and construct maps from some partial compactifications of period domains to the Satake compatifications of Siegel spaces. These maps are a generalization of the maps from the toroidal compactifications of Siegel spaces to the Satake compactifications. We also show the continuity of these maps for the case of Calabi-Yau threefolds with h^{2,1}=1.
Let X be a nonsingular complex projective variety that is acted on by a reductive group $G$ and such that $X^{ss} \neq X_{(0)}^{s}\neq \emptyset$. We give formulae for the Hodge--Poincaré series of the quotient $X_{(0)}^s/G$. We use these computations to obtain the corresponding formulae for the Hodge--Poincaré polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.
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。
These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing sheaves of Kollár is extended to the sheaves of differential forms of arbitrary degrees in the case of a well prepared birational model. We will also prove the log versions of some of the results, i.e., the case where we allow horizontal boundary components.
Weak Hodge Theorem on Piecewise-Algebraic Spaces
在Kontsevich和Soibelman引入的分段代数空间上，证明了弱Hodge定理，将奇异上同调自然嵌入调和形式空间，而非建立同构。
We introduce the categories of geometric mixed Hodge modules on algebraic varieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, the categories of geometric $p$-adic mixed Hodge modules on algebraic varieties over a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We then give an Hodge realization functor on the de derived category of relative motives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodge realization functor on the derived category of relative motives over $k\subset K\subset\mathbb C_p$.
In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is shown that extension groups of arithmetic Hodge structure do not vanish even for degree $\geq2$. Moreover, we define higher Abel-Jacobi maps from Bloch's higher Chow groups of X to these extension groups. These maps essentially involve the classical Abel-Jacobi maps by Weil and Griffiths, and Mumford's infinitesimal invariants of 0-cycles on surfaces.
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.
Let X be a complete complex nonsingular algebraic variety and D a closed algebraic subset of X. We show that the cup product maps H^i(X \setminus D) \otimes H^j(X,D) \to H^{i+j}(X,D) are morphisms of mixed Hodge structures. As a corollary we obtain a new proof of a duality result of Fujiki. We also deal with the extraordinary cup product.
Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic
本文证明K3[n]型紧致超Kähler流形间第二有理上同调的Hodge等距可由解析对应诱导并提升为保持Mukai配对与分次的总上同调对应，且在射影情形下为代数对应。
With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's Mixed Hodge Structure on cohomology of complex algebraic varieties is described.
According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we…
Hodge theory, algebraic cycles of hyper-Kähler manifolds
本论文研究射影hyper-Kähler流形与严格Calabi-Yau流形的代数闭链，旨在推进Beauville与Voisin关于Chow环的猜想，并探讨相关双有理不变量。
Classification of the Mumford--Tate Groups of Rational Polarizable Hodge Structures
本文通过分解与极化有理Hodge结构相伴的pro-代数群G，将其商、导出群及伴随群分别刻画为Serre群、单连通覆盖及特定单代数群的乘积，从而分类了Mumford--Tate群。
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 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.
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…
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…
Sections of Hodge bundles I: Global theory and applications to period maps
本文通过变形论和矩阵表示两种方法，研究Hodge包的全局截面，证明提升后的周期映射像包含在复欧几里得子空间中，并构造Calabi-Yau类型流形的Teichmüller空间的全局复仿射结构。
On a smooth algebraic curve X with genus greater than 1 we consider a flat principal bundle with a reductive structure group S and a vector bundle associated with it. To this set of information we put in correspondence a pro-algebraic group on whose functional algebra we introduce a mixed Hodge structure. This construction, in fact, works for any smooth algebraic variety X which, considered as an analytic space, has a nonabelian first homotopy group, and the rest are trivial. The Hodge structure defined in this way can be expressed in terms of iterated integrals. Furthermore, considered in the…
We introduce an algebraic method for describing the Hodge filtration of degenerating hypersurfaces in projective toric varieties. For this purpose, we show some fundamental properties of logarithmic differential forms on proper equivariant morphisms of toric varieties.
Let $Y$ be a smooth complex projective variety of dimension $N+1$, $L$ an invertible sufficiently ample sheaf, $X\in |L|$ a smooth hypersurface and $λ\in F^kH^N(X,C)$ a vanishing cohomology class, where $F^{*}$ is the Hodge filtration and $k\in\{1,...,[N/2]\}$. Assume that $L$ is sufficiently ample and that the codimension in $|L|$ of the Hodge variety associated to $λ$ (locally defined as the locus where the image of $λ$ by flat transport over $|L|$ remains in $F^k$) is sufficiently small. I show that this forces $N$ to be even and $k=[N/2]$, and that the class $λ$ is a linear combination wit…
Hodge and Frobenius colevels of algebraic varieties
本文为代数簇的Hodge与Frobenius余级在所有上同调维度上提供了以维数和定义方程多重次数表示的新下界，从而增强了Esnault与第一作者所提问题的肯定回答。
Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. We give criterion for semistability and stability of system of Hodge bundles on $X$. We define notion of generalized opers on $X$, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle $E$ together with a connection $\nabla$ over $X$ implies semistability of the pair $(E, \nabla)$.
Multivariable Alexander invariants of algebraic links calculated in terms of algebro-geometric invariants (polytopes and ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity property of polytopes of quasiadjunction is discussed.
The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety $X$ to such invariants of singularities of proper algebraic maps defined on $X$. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in \cite{CS1, S}.
The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central algebraic object of their proof can be understood geometrically as a line bundle on a derived scheme. In this interpretation, the Deligne-Illusie result can be seen as a proof that this line bundle is trivial under certain assumptions. We give a criterion for the triviality of this line bundle in a more general context. The proof uses techniques from derived algebraic geometry, specifically arguments which show the formality of certain derive…
Let X be a smooth, projective variety over the field of complex numbers. On the space H of its rational cohomology of degree i we have the arithmetic filtration F^p. On the other hand, on the space of cohomology of degree i of X with complex coefficients we have the Hodge filtration G^p. It is well known that F^p is contained in the intersection of G^p with H, and that, in general, this inclusion is strict. In this paper we propose a natural substitute S^p for the Hodge filtration space G^p such that the intersection of S^p with H is the space F^p of the arithmetic filtration. In particular, S…
Let $k$ be a perfect field of odd characteristic and $X$ a smooth algebraic variety over $k$ which is $W_2$-liftable. We show that the exponent twisiting of the classical Cartier descent gives an equivalence of categories between the category of nilpotent Higgs sheaves of exponent $\leq p-1$ over $X/k$ and the category of nilpotent flat sheaves of exponent $\leq p-1$ over $X/k$, and it is equivalent up to sign to the inverse Cartier and Cartier transforms for these nilpotent objects constructed in the nonabelian Hodge theory in positive characteristic by Ogus-Vologodsky. In view of the crucial…
Algebraic cycles on hyper-Kähler varieties of generalized Kummer type
本文证明了任意四维广义Kummer型超Kähler簇的Hodge和Tate猜想，并证明任意广义Kummer型簇由$H^2$生成的有理上同调子代数中所有Hodge类均为代数类。
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