On the Hodge Filtration of Hodge Modules

非正则 Hodge 滤过的 Kodaira-Saito 消没

Kodaira-Saito vanishing for the irregular Hodge filtration

本文借助 Mochizuki 的最新成果修正并改进了非正则混合 Hodge 模范畴的定义，进而为该范畴的非正则 Hodge 滤过导出了 Kodaira-Saito 型消没定理。

Irregular Hodge filtration of some confluent hypergeometric systems

We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric $\mathcal{D}$-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations.

Hodge filtration, minimal exponent, and local vanishing

We bound the generation level of the Hodge filtration on the localization along a hypersurface in terms of its minimal exponent. As a consequence, we obtain a local vanishing theorem for sheaves of forms with log poles. These results are extended to Q-divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing cycles.

Hodge filtration and Hodge ideals for $\mathbb{Q}$-divisors with weighted homogeneous isolated singularities

We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-α}$ associated to the effective $\mathbb{Q}$-divisor $D=α\cdot Z$, where $0<α\le1$ and $Z=(f=0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic functio…

Opposite filtrations, variations of Hodge structure, and Frobenius modules

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-Newton filtration for p-divisible O-modules

The notions Hodge-Newton decomposition and Hodge-Newton filtration for F-crystals are due to Katz and generalize Messing's result on the existence of the local-étale filtration for p-divisible groups. Recently, some of Katz's classical results have been generalized by Kottwitz to the context of F-crystals with additional structures and by Moonen to $μ$-ordinary p-divisible groups. In this paper, we discuss further generalizations to the situation of crystals in characteristic p and of p-divisible groups with additional structure by endomorphisms.

关于混合Hodge模的Hodge和V滤波

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。

扭曲参数素除子的霍奇滤过

Hodge filtrations for twisted parametrically prime divisors

该研究将霍奇滤过相关工作推广到$\mathbb{Q}$-除子情形，针对特定参数素全纯函数$f$和$α\geq 0$，给出了$\mathscr{M}(f^{-α})$上霍奇滤过的显式表达式。

Hodge ideals for Q-divisors, V-filtration, and minimal exponent

We explicitly compute the Hodge ideals of Q-divisors in terms of the V-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein-Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

Hodge ideals and microlocal V-filtration

We show that the Hodge ideals in the sense of Mustata and Popa are quite closely related to the induced microlocal V-filtration on the structure sheaf, defined by using the microlocalization of the V-filtration of Kashiwara and Malgrange. More precisely the former coincide, module the ideal of the divisor, with the part of the latter indexed by positive integers, although they are different without modulo the ideal in general. This coincidence implies that the $j$-log-canonicity in their sense is determined by the microlocal log-canonical threshold of the divisor, which coincides with the maxi…

Hodge Filtrations on Tempered Hodge Modules

We show that the Hodge filtration of a tempered Hodge module is generated by the lowest piece of its Hodge filtration. As a consequence, we prove the main conjecture of [SV] in the special case of tempered representations of real reductive Lie groups.

E_1-degeneration of the irregular Hodge filtration (with an appendix by Morihiko Saito)

For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in arXiv:1203.2338 a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d+df, extending a definition of Deligne in the case of curves. In this article, we show the degeneration at E_1 of the spectral sequence attached to the irregular Hodge filtration, by using the method of arXiv:0804.4328. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on its proof of the corresponding E_1 degeneration, by reduction to characteris…

德利涅-芒福德栈上非正则霍奇滤子的$E_1$退化

$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks

针对粗空间拟射影的光滑分离德利涅-芒福德栈，利用Kontsevich复形的有限平展下降，将射影栈情形归约为Esnault–Sabbah–Yu的定理，证明其非正则霍奇滤子在所有有理指标处$E_1$退化，结果适用于各类栈紧化。

Singularities of variations of mixed Hodge structure

We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, extends across the puncture. In particular, the result exactly supplements Schmid's Theorem for pure structures, which holds for the graded variation, and gives a Hodge-theoretic condition for the relative monodromy weight filtration to exist.

Local vanishing and Hodge filtration for rational singularities

Given an n-dimensional variety Z with rational singularities, we conjecture that for a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, the (n-1)-th higher direct image of the sheaf of differential forms with log poles along E vanishes. We prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization of the structure sheaf along D is at most n-3.

Filtrations on the Chow groups and transcendence degree

For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending of the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of…

刚性上同调上的一个 Hodge 型滤过

A Hodge-Type Filtration on Rigid Cohomology

本文证明了 Gros 在刚性上同调上引入的 Hodge 型滤过与其构造中所选取浸入无关，填补了该结论长期缺乏正式证明的空白。

利用广义V-滤子对混合Hodge模的限制

Restrictions of mixed Hodge modules using generalized V-filtrations

本文利用Sabbah定义的广义V-滤子，通过循环覆盖与通常V-滤子的比较，计算了零截面包含映射的限制函子，并应用于研究完全交子簇的奇点，分类了加权齐次孤立完全交奇点的k-Du Bois和k-理性性质。

德利涅-芒福德栈上非正则霍奇滤子的紧化独立性

Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks

研究德利涅-芒福德栈上由Yu方法定义的非正则霍奇滤子，证明其不依赖紧化，该结果适用于Harder和Lee的orbifold非正则霍奇数研究，仅依赖栈Landau–Ginzburg模型。

Hypergeometric Hodge modules

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.

Spreading out the Hodge filtration in non-archimedean geometry

The goal of the current text is to study non-archimedean analytic derived de Rham cohomology by means of formal completions. Our approach is inspired by the deformation to the normal cone provided in \cite{Gaitsgory_Study_II}. More specifically, given a morphism $f \colon X \to Y$ of (derived) $k$-analytic spaces we construct the \emph{non-archimedean deformation to the normal cone} associated to $f$. The latter can be thought as an $\mathbf A^1_k$-parametrized deformation whose fiber at $1 \in \mathbf A^1_k$ coincides with the formal completion of $f$ and the fiber at $0 \in \mathbf A^1_k$ wi…

Simpson 滤过与 Hodge 模空间的闭包关系

Simpson filtrations and closure relations of Hodge moduli spaces

本文研究 Hodge 模空间中 Simpson 层的闭包关系，证明 Heinloth 权在特化改变固定分量时严格增加，并构造反例反驳 Simpson 嵌套猜想，同时从 $\Theta$ 分层角度讨论 Simpson 滤过。

Geometricity of the Hodge filtration on the $\infty$-stack of perfect complexes over $X_{DR}$

We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes with $λ$-connection structure on a smooth projective variety $X$. This maps to $A ^1 / G_m$, so it can be considered as the Hodge filtration of its fiber over 1 which is $M_{DR}(X,Perf)$, parametrizing complexes of $D_X$-modules which are $O_X$-perfect. We apply the result of Toen-Vaquie that $Perf(X)$ is locally geometric. The proof of geometricity of the map $M_{Hod}(X,Perf) \to Perf(X)$ uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a s…

The Hodge theory of character varieties

This is a report on joint work with T. Hausel and L. Migliorini, where we prove, for each of the groups GL(2,C), PGL(2,C), SL(2,C), that the non-Abelian Hodge theorem identifies the weight filtration on the cohomology of the character variety with the perverse Leray filtration on the cohomology of the domain of the Hitchin map. We review the decomposition theorem, Nĝo's support theorem, the geometric description of the perverse filtration and the sub-additivity of the Leray filtration with respect to the cup product.

The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements

The Hodge filtration of the module of derivations on the orbit space of a finite real reflection group acting on an $\ell$-dimensional Euclidean space was introduced and studied by K. Saito. The filtration is equivalent data to the flat structure or the Frobenius manifold structure. We will show that the Hodge filtration coincides with the filtration by the order of contacts to the reflecting hyperplanes. Moreover, a standard basis for the Hodge filtration is explicitly given.

Irregular Hodge theory

We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\expφ$ for any meromorphic function $φ$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also…

带分歧滤过的霍奇上同调，II

Hodge cohomology with a ramification filtration, II

本研究作为系列论文的第二部分，探讨由无穷远除子索引的霍奇上同调群滤过，证明其在带模motive范畴中可被表示，并得到霍奇上同调群的实现函子。

Hodge 滤过与参数素因子

The Hodge filtration and parametrically prime divisors

本文研究沿除子亚纯函数层的典范 Hodge 滤过，针对 Bernstein–Sato 根位于 (-2,0) 的函数芽给出第零部分公式及 f^{-1} 所在步的界，并在 Euler 齐次、参数素条件下计算全部滤过部分。

On the irregular Hodge filtration of exponentially twisted mixed Hodge modules

Given a mixed Hodge module and a meromorphic function f on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module, which extends the construction of arXiv:1302.4537. We show the strictness of the push-forward filtered D-module through any projective morphism, by using the theory of mixed twistor D-modules of T. Mochizuki. We consider the example of the rescaling of a regular function f, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss-Manin systems of f in ter…

约化除子的加权 Hodge 理想

Weighted Hodge ideals of reduced divisors

本文基于双有理几何与V-filtration研究超曲面定位上的Hodge与权滤子，定义了包含adjoint ideal和multiplier ideal的weighted Hodge ideals，并分析其性质以应用于光滑簇的超曲面奇点研究。

The Hodge filtration on complements of complex coordinate subspace arrangements and integral representations of holomorphic functions

We compute the Hodge filtration on cohomology groups of complements of complex coordinate subspace arrangements. By means of this result we construct integral representations of holomorphic functions such that kernels of these representations have singularities on complex coordinate subspace arrangements.

Mixed Hodge modules without slope

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…

On the Hodge-Newton filtration for p-divisible groups of Hodge type

A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystal…

计算局部化上的Hodge与权重滤过

Calculating Hodge and weight filtrations on localizations

本文介绍Macaulay2包MixedHodgeModules，用于计算多项式环局部化上的Hodge与权重滤过及相关不变量，并支持计算多种复形。

权重过滤与生成层级

Weight filtration and generating level

本文研究混合 Hodge 模的权重过滤，推广已知结果至加权情形，建立 Hodge 理论与双有理几何的新联系，推导过滤公式并得到最大权重与生成层级结果，还给出了若干除子类表达式。

Grothendieck standard conjectures, morphic cohomology and Hodge index theorem

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.

A remark on the Generalized Hodge Conjecture

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…

Hodge ideals for the determinant hypersurface

We determine explicitly the Hodge ideals for the determinant hypersurface as an intersection of symbolic powers of determinantal ideals. We prove our results by studying the Hodge and weight filtrations on the mixed Hodge module O_X(*Z) of regular functions on the space X of n x n matrices, with poles along the divisor Z of singular matrices. The composition factors for the weight filtration on O_X(*Z) are pure Hodge modules with underlying D-modules given by the simple GL-equivariant D-modules on X, where GL is the natural group of symmetries, acting by row and column operations on the matrix…

Irregular Hodge numbers for rigid $G_2$-connections

Certain rigid irregular $G_2$-connections constructed by the first-named author are related via pullbacks along a finite covering and Fourier transform to rigid local systems on a punctured projective line. This kind of property was first observed by Katz for hypergeometric connections and used by Sabbah and Yu to compute irregular Hodge filtrations for hypergeometric connections. This strategy can also be applied to the aforementioned $G_2$-connections and we compute jumping indices and dimensions for their irregular Hodge filtrations.

Hodge ideals of free divisors

We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein-Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples.

Weak positivity via mixed Hodge modules

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.

$F$-幂零性与孤立奇点之外的Hodge滤过

$F$-nilpotence and Hodge filtrations beyond isolated singularities

本文通过Hodge理论刻画特征零簇的开$F$-幂零型与开弱$F$-幂零型，将Srinivas-Takagi结果推广至非孤立奇点情形，并分别无条件及在弱普通性猜想下证明两个方向。

超几何微分方程的不规则 Hodge 滤过

Irregular Hodge filtration of hypergeometric differential equations

本文研究有理参数超几何联络的不规则Hodge滤过，借助几何解释重新证明Fedorov与Sabbah–Yu的Hodge数公式，并证明相关超几何和在乘法群上处处通常。

Hodge滤链与Nygaard滤链的栈化比较

A stacky comparison of the Hodge and Nygaard filtrations

本文通过Drinfeld与Bhatt–Lurie的栈化方法，将棱镜上同调的Nygaard滤链与de Rham上同调的Hodge滤链的比较结果推广到任意规范系数情形，并发展出能同时捕捉共轭滤链与Sen算子的衍射Hodge上同调栈化理论。

Asymptotic Hodge Theory of Vector Bundles

We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.

Hodge filtration on local cohomology, Du Bois complex, and local cohomological dimension

We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions, and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing, and reflexive differentials associated to Z.

Hodge-DeRham theory with degenerating coefficients

Let ${\cal L}$ be a local system on the complement $X^{\star}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{\star} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe a weight filtration $W$ on the direct image ${\bf j}_{\star}{\cal L} $ and in case a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)$, the weight filtration on the complex of nearby cocycles $Ψ_f ({\cal L})$ on $Y$. A comparison theorem shows that the filtration coincides with the weight defined by the logarithm of the monodromy and …

Deligne-Hodge-DeRham theory with coefficients

Let ${\cal L}$ be a variation of Hodge structures on the complement $X^{*}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{*} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe the weight filtration $W$ on a combinatorial logarithmic complex computing the (higher) direct image ${\bf j}_{*}{\cal L} $, underlying a mixed Hodge complex when $X$ is proper, proving in this way the results in the note [14] generalizing the constant coefficients case. When a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)…

Hodge cycles and the Leray filtration

This is loosely a continuation of the author's previous paper arXiv:1802.09496. In the first part, given a fibered variety, we pull back the Leray filtration to the Chow group, and use this to give some criteria for the Hodge and Tate conjectures to hold for such varieties. In the second part, we show that the Hodge conjecture holds for a good desingularization of a self fibre product of a non-isotrivial elliptic surface under appropriate conditions. We also show that the Hodge and Tate conjectures hold for natural families of abelian varieties parameterized by certain Shimura curves. This use…

Weighted multiplier ideals of reduced divisors

We use methods from birational geometry to study the Hodge and weight filtrations on the localization along a hypersurface. We focus on the lowest piece of the Hodge filtration of the submodules arising from the weight filtration. This leads to a sequence of ideal sheaves called weighted multiplier ideals. The last ideal of this sequence is a multiplier ideal (and a Hodge ideal), and we prove that the first is the adjoint ideal. We also study the local and global properties of weighted multiplier ideals and their applications to singularities of hypersurfaces of smooth varieties.

The Hodge filtration on nonabelian cohomology

This is partly a survey article on nonabelian Hodge theory, but we also give proofs of results that have only been announced elsewhere. In the introduction we discuss a wide range of recent work on this subject and give some references. In the body of the paper, we discuss Corlette's nonabelian Hodge theorem, Hitchin's quaternionic structure on the moduli space of representations of $π_1(X)$ (for a compact Kähler manifold $X$), and Deligne's construction of the resulting twistor space. We then mention the interpretation of Deligne's space of $λ$-connections as the Hodge filtration of the nonab…

On Hodge Structures and Periods -- Part I: Algebraic Aspects

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.

Integral p-adic Hodge filtrations in low dimension and ramification

Given an integral p-adic variety, we observe that if the integral Hodge--de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p-adic Hodge theory, we show that such a nice behavior is guaranteed if the p-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne--Illusie and Fontaine--Messing. Lastly, we discuss an example illustrating the necessity of t…

二次微分非变化层上Hodge丛的过滤与分裂

Filtration and splitting of the Hodge bundle on the non-varying strata of quadratic differentials

该研究刻画了二次微分非变化层内Teichmüller曲线的Hodge丛的Harder–Narasimhan过滤，证明远离非正则分支的Hodge丛可分裂为线丛直和，进而确定相关Teichmüller曲线的所有Lyapunov指数并得到其渐近行为的新结论。