Local and global invariant cycle theorems for Hodge modules
Morihiko Saito
该研究论证了Hodge模的局部与整体不变循环定理可通过一般理论直接推导，并补充了相关研究的评述内容。
我们证明Hodge模的局部与整体不变循环定理可由一般理论轻松推导得出，同时对相关论文给出了若干评述。
We show that the local and global invariant cycle theorems for Hodge modules follow easily from the general theory. We also give some remarks about related papers.
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Hodge-Grothendieck classes and monodromy invariants of nearby cycles sheaves
研究邻近周期层的 Hodge-Grothendieck 类与单值化不变量，评述邻近周期函子并证明广义局部不变周期结果。
We give explicit formulas for the Hodge filtration on mixed Hodge modules associated with certain hypersurfaces.
We study an analogue of the Noether-Lefschetz theorem in the context of Beilinson's Hodge conjecture. We estimate the dimension of the Noether-Lefschetz locus by the infinitesimal methods in Hodge theory.
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…
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 semi-simplicity of the Hodge group is proved for a simple Abelian variety with a stable reduction of odd toric (reductive) rank. If, besides, the dimension of the Abelian variety is an odd integer, then the Hodge conjecture on algebraic cycles holds for it.
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…
This text is an expository survey on the interplay between polarized variation of Hodge structure (PVHS) and the formalism of Hodge modules. We specifically review the extensions of a PVMHS over their singularities and its relation to mixed Hodge modules.
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.
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 main subject of study in this paper is the Noether-Lefschetz locus for Beilinson-Hodge cycles on the complement of the union of hyperplane sections in a surface in $\Bbb P^3$. We give an analogue of the result by M.Green and C.Voisin on the maximal components in the Noether-Lefschetz locus.
For a smooth, projective complex variety, we introduce several mixed Hodge structures associated to higher algebraic cycles. Most notably, we introduce a mixed Hodge structure for a pair of higher cycles which are in the refined normalized complex and intersect properly. In a special case, this mixed Hodge structure is an oriented biextension, and its height agrees with the higher archimedean height pairing introduced in a previous paper by the first two authors. We also compute a non-trivial example of this height given by Bloch-Wigner dilogarithm function. Finally we study the variation of m…
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。
We prove that the lowest nonzero piece in the Hodge filtration of a mixed Hodge module is always weakly positive in the sense of Viehweg.
We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles.
In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization of curves in positive characteristics which are reduction of Shimura curve over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian fourfolds over k, char k=p. We show that some conditions on the crystalline Hodge cycles over such a curve imply that this curve can be lifted to a Shimura curve.
We give a rather informal introduction to the theory of mixed Hodge modules for young mathematicians.
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 …
For a smooth algebraic variety $X$, a monodromic $D$-module on $X\times \mathbb{C}$ is decomposed into a direct sum of some $D$-modules on $X$. We show that the Hodge filtration of a mixed Hodge module on $X\times \mathbb{C}$ whose underlying $D$-module is monodromic is also decomposed. Moreover, we show that there is an equivalence of categories between the category of monodromic mixed Hodge modules on $X\times \mathbb{C}$ and the category of ``gluing data''. As an application, we endow the Fourier-Laplace transformation of the underlying $D$-module of a monodromic mixed Hodge module with a m…
In this article we are interested in morphisms without slope for mixed Hodge modules. We first show the commutativity of iterated nearby cycles and vanishing cycles applied to a mixed Hodge module in the case of a morphism without slope. Then we define the notion "strictly without slope" for a mixed Hodge module and we show the preservation of this condition under the direct image by a proper morphism. As an application we prove the compatibility of the Hodge filtration and Kashiwara-Malgrange filtrations for some pure Hodge modules with support an hypersurface with quasi-ordinary singularitie…
We consider mixed Hodge module structures on GKZ-hypergeometric differential systems. We show that the Hodge filtration on these D-modules is given by the order filtration, up to suitable shift. As an application, we prove a conjecture on the existence of non-commutative Hodge structures on the reduced quantum D-module of a nef complete intersection inside a toric variety.
In this paper we give a geometrical interpretation of an extension of mixed Hodge structures (MHS) obtained from the canonical MHS on the group ring of the fundamental group of a hyperelliptic curve modulo the fourth power of its augmentation ideal. We show that the class of this extension coincides with the regulator image of a canonical higher cycle in a hyperelliptic jacobian. This higher cycle was introduced and studied by Collino.
We explain some fundamental differences between the theories of mixed Hodge modules and mixed twistor modules (including the difference in weight system on the nearby cycle functor) which do not seem to be clarified explicitly in the literature.
We prove the weak positivity of the kernels of Kodaira-Spencer- type maps for pure Hodge module extensions of generically defined variations of Hodge structure.
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。
Thom-Sebastiani Theorem for Hodge Modules
本文通过与Verdier特化的兼容性，证明了Hodge模块的Thom-Sebastiani定理，解决了混合Hodge模块的结构问题。
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.
We give a generalization to higher genera of the famous formula $12 λ=δ$ for genus 1. We also compute the classes of certain strata in the Satake compactification as elements of the push down of the tautological ring.
Mixed Hodge modules on stacks
本文借助混合 Hodge 模的 ∞-范畴增强，将其连同六运算与权推广到代数栈，并证明 Drew 的 motivic Hodge 模范畴全忠实嵌入且像为几何来源的混合 Hodge 模。
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 the injectivity and vanishing theorem for R-Hodge modules and R-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in \cite{Wu15}. In particular, the injectivity generalizes the fundamental injectivity of Esnault-Viehweg for normal crossing Q-divisors, while the vanishing generalizes Kawamata-Viehweg vanishing for Q-divisors. As a main application, we also deduce a Fujita-type freeness result for R-Hodge modules in the normal crossing case.
Let $X$ be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of $X$ is ample. Using the cylinder homomorphism associated with the family of complete intersections contained in $X$, we prove that the vanishing cycles in the middle homology group of $X$ are represented by topological cycles whose support is contained in a proper Zariski closed subset $T\subset X$ of certain codimension. In some cases, we can find such a Zariski closed subset $T$ with codimension equal to the upper bound obtained from the Hodge struc…
These notes are an extended version of the authors' lectures at the 2013 CMI Workshop "Mixed Hodge Modules and Their Applications". We give an overview, with an emphasis on applications, of recent developments on the interaction between characteristic class theories for singular spaces and Saito's theory of mixed Hodge modules in the complex algebraic context.
After explaining the definition of pure and mixed Hodge modules on complex manifolds, we describe some of Saito's most important results and their proofs, and then discuss two simple applications of the theory.
Height Pairing on Higher Cycles and Mixed Hodge Structures II
该研究针对恰当相交且余维互补的布洛赫高阶周环，通过赋予其带框混合霍奇结构定义两种阿基米德局部高度配对，推广了经典代数周环的双扩张阿基米德高度，应用于多对数变分并证明了相关高度性质。
We give some details of a simpler definition of mixed Hodge modules which has been announced in some papers. Compared with earlier arguments, this new definition is simplified by using Beilinson's maximal extension together with stability by subquotients systematically.
We establish a relationship between the graded quotients of a filtered holonomic D-module, their sheaf-theoretic duals, and the characteristic variety, in case the filtered D-module underlies a polarized Hodge module on a smooth algebraic variety. The proof is based on Saito's result that the associated graded module is Cohen-Macaulay, and on local duality on the cotangent bundle.
We exhibit a direct correspondence between the potential defining the H^{1,1} small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules.
The goal is to verify the Hodge conjecture (and some related conjectures) for certain moduli spaces. It is shown that the (generalized) Hodge conjecture holds for the projective moduli spaces of vector bundles over an abelian or K3 surface or a curve if it holds for all powers of the surface or curve. In the latter case, the curve can be replaced by its Jacobian. As a consequence the generalized Hodge conjecture holds for these moduli spaces when the curve is very general. A similar result is proved for the Hilbert schemes of points over an arbitrary surface.
Equivariant Hodge modules and rational singularities
本文定义了具有有理奇点的Hodge模，并推广Boutot定理，证明了在约化群作用于仿射簇且存在稳定点时，GIT商上的诱导等变Hodge模仍保持有理奇点。
We give a new proof of a theorem by Pareschi, Popa and Schnell that the direct image of the canonical bundle of a smooth projective variety along a morphism to an abelian variety admits a Chen-Jiang decomposition, without using the theory of Hodge modules.
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…
This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.
We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation of Hodge structures. In the case that $M$ is the minimal extension of a simple polarized variation of Hodge structures $V$, we show that the homotopy image of $\mathcal{P}_M$ is the minimal extension of the graph morphism of the usual period integral map for $V$.
In this article we describe three constructions of complex variations of Hodge structure, proving the existence of interesting opposite filtrations that generalize a construction of Deligne. We also analyze the relation between deformations of Frobenius modules and certain maximally degenerate variations of Hodge structures. Finally, under a certain generation hypothesis, we show how to construct a Frobenius manifold starting from a deformation of a Frobenius module.
On the Hodge and V-filtrations of mixed Hodge modules
本文通过Hodge滤波证明了V滤波的Beilinson型公式，并利用该公式研究了更高乘数理想与Hodge理想的关系，改进了相关结果并回答了若干开放问题。
The normal function associated to algebraic cycles in higher Chow groups defines a differential equation. This Picard-Fuchs equation defines an extension of $D$-modules as well as an extension of local systems. In this paper, we show that both extensions define the same extension of mixed Hodge modules determined by the normal function.
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.
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。
We construct a quasi-categorically enhanced Grothendieck six-functor formalism on schemes of finite type over the complex numbers. In addition to satisfying many of the same properties as M. Saito's derived categories of mixed Hodge modules, this new six-functor formalism receives canonical motivic realization functors compatible with Grothendieck's six functors on constructible objects.
Classification of can-Var diagrams of mixed Hodge structures and monodromic mixed Hodge modules on $\mathbb{C}$
本文对复平面上的幂单单值混合 Hodge 模进行分类，证明其等价于混合 Hodge 结构的 can-Var 图范畴，给出了其相伴分次图的分解形式及表示方法，并推广到单值无特征值 1 的情形。
De Rham-Higgs comparison for mixed Hodge modules in positive characteristic
本文基于正特征非阿贝尔霍奇理论，从混合霍奇模视角推广Deligne-Illusie分解定理，并在特殊情形下验证该推广，得到若干副产品。
As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge conjecture is inductively equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between the vanishing cycles of type (0,0) with unipotent monodromy around a singular hyperplane section defines a primitive Hodge class such that this singular hyperplane section is a generalized Tho…
Fourier transform and Radon transform for mixed Hodge modules
本文推广了微局部化与消失循环的关系至混合Hodge模，建立了局部化三角形同构，并用于比较Radon变换与Fourier-Laplace变换，进而应用于GKZ系统的Hodge模结构。
Given a family of Laurent polynomials, we will construct a morphism between its (proper) Gauss-Manin system and a direct sum of associated GKZ systems. The kernel and cokernel of this morphism are very simple and consist of free O-modules. The result above enables us to put a mixed Hodge module structure on certain classes of GKZ systems and shows that they have quasi-unipotent monodromy.
Generalizing a theorem of Macdonald, we show a formula for the mixed Hodge structure on the cohomology of the symmetric products of bounded complexes of mixed Hodge modules by showing the existence of the canonical action of the symmetric group on the multiple external self-products of complexes of mixed Hodge modules. We also generalize a theorem of Hirzebruch and Zagier on the signature of the symmetric products of manifolds to the case of the symmetric products of symmetric parings on bounded complexes with constructible cohomology sheaves where the pairing is not assumed to be non-degenera…
We use filtered log-$\mathscr{D}$-modules to represent the (dual) localization of Saito's Mixed Hodge Modules along a smooth hypersurface, and show that they also behave well under the direct image functor and the dual functor in the derived category of filtered log-$\mathscr{D}$-modules. The results of this paper can be used to generalize the result of M. Popa and C. Schnell about Kodaira dimension and zeros of holomorphic one-forms into the log setting.
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…
We extend the results of generic vanishing theory to polarizable real Hodge modules on compact complex tori, and from there to arbitrary compact Kähler manifolds. As applications, we obtain a bimeromorphic characterization of compact complex tori among compact Kähler manifolds, semi-positivity results, and a description of the Leray filtration for maps to tori.
Twisted Hodge groups and deformation theory of Hilbert schemes of points on surfaces via Hodge modules
本研究利用Hodge模理论计算紧复曲面上点的Hilbert概形的扭曲Hodge数，证明了相关猜想修正版，推广了经典公式，并确定了Douady空间形变理论的切空间与障碍空间。
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