Mikhail Kapranov
We identify the category of real mixed Hodge structures with the category of vector bundles with connections (not necessarily flat) on C, equivariant with respect to C^*. Here C is the complex plane considered as a 2-dimensional real manifold, and C^* is the multiplicative group of complex numbers considered as a real group.
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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…
Mixed Hodge Complexes on Algebraic Varieties
引入代数簇上的混合Hodge复形的概念，改进了Du Bois的滤子复形，并将Deligne的混合Hodge结构理论与混合Hodge模理论联系起来.这被认为是正确的，但这是非常不平凡的，因为混合霍奇模的理论不适用于单纯方案。杜波依斯奇异性的一些应用。
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.
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.
We give a rather informal introduction to the theory of mixed Hodge modules for young mathematicians.
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 promote Beilinson's triangulated equivalence between the bounded derived category of rational polarizable mixed Hodge structures and the derived category of rational polarizable mixed Hodge complexes to an equivalence of symmetric monoidal quasi-categories. We use this equivalence to construct a presheaf of commutative differential graded algebras in the ind-completion of the category of rational mixed Hodge structures which computes Deligne's mixed Hodge structure on the rational Betti cohomology of $\mathbf{C}$-schemes of finite type. This leads to a preshea--in the quasi-categorical sens…
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。
We construct toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We construct them as the spaces of nilpotent orbits.
Degenerations of Mixed Hodge structure
我们继续我们的工作的变化梯度极化混合霍奇结构的定义类似物的谐波度规方程过滤丛和证明一个精确的模拟Schmid的幂零轨道定理的1参数退化的梯度极化混合霍奇结构。
Mixed Hodge Structures of cluster varieties of dimension 3
本文计算光滑三维簇簇的混合霍奇数，证明其为混合泰特型，并研究奇异簇簇的上同调与交上同调的混合霍奇结构。
The Gysin map is compatible with mixed Hodge structures
我们证明了Gysin映射与混合Hodge结构相容。
We announce the construction of toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We include an application to the analyticity of zero loci of some functions.
We generalize the theorem of E. Cattani, P. Deligne, and A. Kaplan to admissible variations of mixed Hodge structure.
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…
In this paper, we show that the mixed Hodge structures of character varieties are of Hodge--Tate type and that the mixed Hodge polynomials are independent of the choice of generic eigenvalues, which is a conjecture due to Hausel, Letellier and Rodriguez-Villegas. Moreover, we investigate the mixed Hodge structures of the moduli space of semistable parabolic Higgs bundles and the moduli space of semistable regular singular parabolic connections. We show that the mixed Hodge structures of these moduli spaces are pure.
We construct an enlargement of the classifying space of mixed Hodge structures with polarized graded quotients, by adding mixed Hodge theoretic version of SL(2)-orbits. This space has a real analytic structure and a log structure with sign. The SL(2)-orbit theorem in several variables for mixed Hodge structures can be understood naturally with this space.
Variations of Mixed Hodge Structure attached to the deformation theory of a Complex Variation of Hodge Structures
我们在紧致Kähler流形上的VHS的完整性上构造了表示方案的局部完备环上的混合Hodge结构，并证明了相应的重言式表示是VMHS的完整性.为了进行这一建设，我们重温了著名的工作高盛和米尔森的变形理论的代表性凯勒集团。
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…
This note is inspired by the work of Deligne on the local behavior of Hodge structures at infinity. We study limit mixed Hodge structures of degenerating families of compact hyperkähler manifolds. We show that when the monodromy action on $H^2$ has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type.
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。
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.
Structures de Hodge mixtes et fibrés sur le plan projectif complexe
本文的目的是将混合Hodge结构的概念几何化。因此，我们将射影平面上的等变向量丛与三滤向量空间联系起来。用混合Hodge结构产生的滤子构造这个Rees结构，给出了混合Hodge结构范畴与${\bf P}^{2}_{\bf C}$上的等变向量丛范畴之间的范畴等价，这些范畴对于关于因子的半稳定条件是半稳定的。利用这个字典，我们可以从几何上恢复混合Hodge结构的某些性质，如混合Hodge结构范畴的阿贝尔性质或高阶扩张消失的事实。接着，我们在Rees丛的第二类Chern中定义了混合Hodge结构的${\bf R}$ -分裂水平，推广了${\bf R}$ -分裂混合Hodge结构的概念.这个不变量可以用Hodge数$h^{p,q}$和一些Hodge型整数$s^{p,q}$来表示，这些整数由Hodge和共轭滤子的相对位置给出。当Hodge数对于一族混合Hodge结构是常数时，这些数半连续地变化。这允许定义混合霍奇结构的变化的基础的分层。
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.
Mixed Hodge structures associated to geometric variations
本文证明了作者在数学AG/0301140 [Leray谱序列是motivic，Invent. 2005]对于Hodge结构的几何变化与M. Saito的混合Hodge模理论这篇笔记也包含了一个小的勘误表为早先的文件。
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.
The purpose of this article is to investigate the properties of the category of mixed plectic Hodge structures defined by Nekovář and Scholl. We give an equivalent description of mixed plectic Hodge structures in terms of the weight and partial Hodge filtrations. We also construct an explicit complex calculating the extension groups in this category.
This paper was motivated by the following question: Recall that for a smooth projective variety X whose polarized Hodge structure on H^n(X,Q)_{prim} leads to a period point ...
We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita--Kawamata semi-positivity theorem.
Nonabelian mixed Hodge structures
本文提出了“非阿贝尔混合Hodge结构”的定义，并给出了一个与光滑射影簇$X$和非阿贝尔混合Hodge结构$V$相关联的构造，即“$X$的非阿贝尔上同调系数在$V$中”，它是一个（前）非阿贝尔混合Hodge结构，记为$H=Hom(X_M, V)$ .我们描述了基本的定义，然后给出了一些说明应该发生什么。最后给出了一个例子：$V$的同伦类型是复化的2-球面，而它与$\pp ^1$的同伦类型是混合的Hodge结构。对于这个例子，我们表明，$Hom (X_M,V)$是一个namhs任何光滑的投射簇$X$。
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…
We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.
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.
Singularities of variations of mixed Hodge structure
我们证明了具有幂单值性的穿孔圆盘上的梯度极化混合Hodge结构的一个变体，在穿孔处具有一个极限混合Hodge结构（即，在[SZ]）的意义上是允许的，该[SZ]）在$\R$上分裂，当且仅当络合重量过滤的某些分级（平滑地取决于Hodge过滤）延伸穿过穿孔。特别地，该结果精确地补充了Schmid关于纯结构的分级变分定理，并给出了相对单值权滤子存在的Hodge理论条件.
We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric, the norm estimates and the asymptotics of the reduced limit Hodge filtration.
Building Mixed Hodge Structures
这主要是一篇简短的文章，介绍的第一句话几乎概括了这一点：这篇文章针对的是那些已经知道混合霍奇结构是什么以及它们有什么好处，但不知道如何构建它们的人。
In this article, we will give the Deligne 1-motives up to isogeny corresponding to the $\mathbb{Q}$-limiting mixed Hodge structures of semi-stable degenerations of curves, by using logarithmic structures and Steenbrink's cohomological mixed Hodge complexes associated to semi-stable degenerations of curves.
Periods of Limit Mixed Hodge Structures
本文是一个扩大版本的谈话在当前的发展数学会议去年11月（2002年）的工作Wilfred施密德期间的限制霍奇结构。本文首先阐述了Hodge结构的极限理论，重点是几何上的。接着，通过Deligne，Goncharov等人的工作，讨论了带负3点的射影直线的基本群（使之幂幺）上极限混合Hodge结构的周期及其与Spec Z上的混合Zeta数和混合Tate动机的关系。
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。
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.
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…
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.
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.
We give some remarks on limit mixed Hodge structure and spectrum. These are more or less well-known to the specialists, and do not seem to be stated explicitly in the literature. However, they do not seem to be completely trivial to the beginners, and may be worth writing down explicitly.
We construct a polarization on the relative log de Rham cohomology groups of a projective log deformation. To this end, we study the behavior of weight and Hodge filtrations under the cup product and construct a trace morphism for a log deformation.
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。
We study a period map for triple coverings of $\mathbf P^2$branching along special configurations of $6$ lines. Though the moduli space of special configurations isa two dimensional variety,the minimal models of the coverings form a oneparameter family of K3 surfaces.We extract extra one dimensionalinformation from the mixed Hodge structure on the second relative homology group.
Cup products and mixed Hodge structures
设X是完备复非奇异代数簇，D是X的闭代数子集。证明了上积映射H^i（X \setminus D）\otimes H^j（X，D）\to H^{i+j}（X，D）是混合Hodge结构的态射.作为推论，我们得到了Fujiki的一个对偶结果的新证明。我们也处理特殊的上积。
Variation of mixed Hodge structures(VMHS), introduced by P. Deligne, is a linear structure reflecting the geometry on cohomology of the fibers of an algebraic family, generalizing variation of Hodge structures for smooth proper families, introduced by P. Griffiths. Hence, it is a strong tool to study the variation of the geometric structure of fibers of a morphism. We describe here the degenerating properties of a VMHS of geometric origin and the existence of a relative monodromy filtration, as well the definition and properties of abstract admissible VMHS.
We give three new proofs of a theorem of C. Sabbah asserting that the weight filtration of the limit mixed Hodge structure at infinity of cohomologically tame polynomials coincides with the monodromy filtration up to a certain shift depending on the unipotent or non-unipotent monodromy part.
SL_2-orbits and degenerations of mixed Hodge structure
对于一类包含对数变形、1-模退化和阿基米德高度的混合Hodge结构的变分，我们证明了Schmid的$\text{\rm SL}_2$ -轨道定理的一个类似定理.特别地，作为这个定理的推论，我们得到了一个简单的公式的渐近行为的阿基米德高度的平坦族的代数圈，它只依赖于重量过滤和局部monodromy。
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.
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.
Niveaux de R-scindement de structures de Hodge mixtes
通过定义混合Hodge结构的${\bf R}$-分裂水平，我们提出了${\bf R}$ -分裂混合Hodge结构概念的推广.这是一个离散不变量，取值为正整数，并且对于${\bf R}$ -分裂混合霍奇结构等于0。我们描述了这些结构上的经典运算的不变量的一些性质，并给出了一些明确的计算。
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。
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。
We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural \textit{limiting} mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotnet orbit in several variables.
The mixed Hodge structure on the fundamental groups of the Collino surfaces
研究Collino表面开集根本群的混合Hodge结构，揭示第二延伸类与阿贝尔-雅各布不变量及标记点的关系。
The Mixed Hodge Structure on the Fundamental Group of a Punctured Riemann Surface
给定亏格为$g$的紧致黎曼曲面$\bar{X}$和$\bar{X}$上的点$q$，我们考虑具有基点$p\in X$的$X:=\bar{X}\setminus\{q\}$.研究了混合Hodge结构在基本群上（在Hain意义下）的扩张，其权为-1和-2.我们表明，它自然对应的元素$(2g q-2 p-K)$一方面在$\Pic^0(\bar{X})$，其中$K$代表典型的除数，另一方面，各自的扩展$\bar{X}$。最后，我们证明了一个点Torelli定理穿孔黎曼曲面。
We prove the structure theorem of the intersection complexes of toric varieties in the category of mixed Hodge modules. This theorem is due to Bernstein, Khovanskii and MacPherson for the underlying complexes with rational coefficients. As a corollary the Euler characteristic Hodge numbers of non-degenerate toric hypersurface can be determined by the Euler characteristic subtotal Hodge numbers together with combinatorial data of Newton polyhedra. This is used implicitly in an explicit formula by Batyrev--Borisov. Note that a formula for the Euler characteristic subtotal Hodge numbers in terms …
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