A bound on the Hodge filtration of the de Rham cohomology of supervarieties
Alexander Polishchuk, Dmitry Vaintrob
本文考察真光滑超簇及其约化簇，建立二者 de Rham 上同调 Hodge 滤过之间的关系界。
我们研究真光滑超簇 $X$ 的 de Rham 上同调的 Hodge 滤过，与相应约化簇 $X_0$ 的通常 Hodge 滤过之间的关系。
We study the relation between the Hodge filtration of the de Rham cohomology of a proper smooth supervariety $X$ and the usual Hodge filtration of the corresponding reduced variety $X_0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
De Rham cohomology for supervarieties
研究超簇的 de Rham 上同调与 Hodge 到 de Rham 谱序列。
We give a definition and study the basic properties of the irregular Hodge filtration on the exponentially twisted de Rham cohomology of a smooth quasi-projective complex variety.
Kato's Ramification filtration via de Rham-Witt complex and applications
通过de Rham-Witt复形的分层，研究Kato分层的描述及其在对偶定理和Lefschetz定理中的应用。
We prove the Künneth formula for the irregular Hodge filtrations on the exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-projective complex varieties. The method involves a careful comparison of the underlying chain complexes under a certain elimination of indeterminacy.
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.
We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring $\mathbb{F}_q[[t]]$, where $\mathbb{F}_q$ is a finite field. As an application, we obtain a new filtration on the maximal abelian quotient $π^{\text{ab}}_1(U)$ of the étale fundamental groups $π_1(U)$ of an open subscheme $U \subseteq X$, which gives a measure of ramification along a divisor $D$ with normal crossing and $\text{Supp}(D) \subseteq X-U$. This filtration coincides with the Brylinski-Kato-Matsuda filtration in the relative dimension zero case.
Hochschild homology and the derived de Rham complex revisited
本文通过普遍性质重新审视Hochschild同调与导出de Rham复形的关系，证明了HKR滤层的关联graded与导出de Rham复形相等，并给出了新的构造方法。
Filtrations and cohomology I: crystallization
本文比较了多种滤过导出交换环的概念，探讨了HKR-滤过Hochschild同调与Hodge-滤过上同调，并证明de Rham上同调是无穷小上同调的结晶化。
Filtrations and cohomology II: the Gauss-Manin connection
该研究用导出方法分析多类同调/上同调中的高斯-马宁联络，给出零幂不变性等问题的新方法，推广Bhatt特征零结果至混合特征，揭示与哈特肖恩复形的关联并解释特征零棱柱上同调特征。
We define formal vector bundles with marked sections on Hilbert modular schemes and we show how to use them to construct modular sheaves with an integrable meromorphic connection and a filtration which, in degree 0, gives to us a $p$-adic interpolation of the usual Hodge filtration. We relate it with the sheaf of overconvergent Hilbert modular forms.
We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.
In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum.
Filtrations on Local Cohomology and Injectivity Theorems
本文证明局部上同调的Hodge滤过包含于符号Ext滤过，通过滤过对偶与Du Bois复形比较，并推广了高阶单射性猜想。
For a smooth projective scheme $X$ over a ring $R$ on which $p$ is nilpotent that meets some general assumptions we prove that the crystalline cohomology is equipped with the structure of a higher display which is a relative version of Fontaine's strongly divisible lattices. Frobenius-divisibilty is induced by the Nygaard filtration on the relative de Rham-Witt complex. For a nilpotent PD-thickening $S/R$ we also consider the associated relative display and can describe it explicitly by a relative version of the Nygaard filtration on the de Rham-Witt complex associated to a lifting of $X$ over…
$q$-de Rham cohomology and topological Hochschild homology over ku
本文证明：若环 $R$ 有球面 $\mathbb{E}_2$-提升，则 $\mathrm{TC}^-$ 的偶滤过分次分量给出 $q$-de Rham 上同调上的 $q$-形变 Hodge 滤过，并由此构造数域 Habiro 环。
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。
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…
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…
We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over $\mathcal O_K$ with $K$ being a $p$-adic field, we improve Breuil--Caruso's theory on comparison between torsion crystalline cohomology and torsion étale cohomology.
$q$-Hodge complexes over the Habiro ring
本文证明在 Habiro 环上存在 $q$-de Rham 复形的变体（$q$-Hodge 复形），当 $q$-de Rham 复形具有 $q$-Hodge 滤过时，该复形典范下降，推广了数域的 Habiro 环，并在光滑概形上给出存在条件。
A construction of tame sheaves and tame de Rham--Witt cohomology
本文提出了一种构造 tame sheaves 的一般方法，应用于大 de Rham--Witt sheaves 和 reciprocity sheaves，并比较了 tame syntomic cohomology 与 tame de Rham--Witt 复合物上的 Nygaard 过滤。
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。
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.
The aim of this article is to study degeneration of the variations of Hodge structure associated to a proper Kähler semistable morphism. We prove that the weight filtrations constructed in the author's previous paper coincide with the monodromy weight filtrations on the relative log de Rham cohomology groups of a proper Kähler semistable morphism. Moreover, we show that the limiting mixed Hodge structures form admissible variations of mixed Hodge structure.
On Kato's ramification filtration
本文利用滤过 de Rham-Witt 复形给出加藤分歧滤过的上同调描述，并由此证明有限域及局部域上的对偶定理和 Brauer 群的 Lefschetz 定理。
Milne's correcting factor, which appears in the Zeta-value at $s=n$ of a smooth projective variety $X$ over a finite field $\mathbb{F}_q$, is the Euler characteristic of the derived de Rham cohomology of $X/\mathbb{Z}$ modulo the Hodge filtration $F^n$. In this note, we extend this result to arbitrary separated schemes of finite type over $\mathbb{F}_q$ of dimension at most $d$, provided resolution of singularities for schemes of dimension at most $d$ holds. More precisely, we show that Geisser's generalization of Milne's factor, whenever it is well defined, is the Euler characteristic of the …
We prove that, for adic étale motives over $\mathbb{C}_p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G_{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sh…
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。
Divided Powers and Derived De Rham Cohomology
本研究构建导出除幂代数形式体系，在该框架下重新研究导出德拉姆与晶体上同调，给出其泛性质刻画，验证其与经典理论的一致性，并推导相关代数几何工具的应用。
TR and the $r$-Nygaard filtered prismatic cohomology
本文定义并研究绝对棱柱上同调上的 $r$-Nygaard 滤过，并将其用于分析 TR 的 motif 滤过及与拓扑循环同调、de Rham–Witt 复形的比较。
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…
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。
We show that the extended noncommutative de Rham complex of a cofibrant resolution, when completed at a certain Hodge filtration, is (reduced) quasi-isomorphic to the periodic cyclic complex, while each of its filtration piece is quasi-isomorphic to the negative cyclic complex. This extends a classical result of Feigin and Tsygan, which corresponds to the Hodge degree $0$ part of our quasi-isomorphism. This result is applied to the study of Calabi-Yau categories in \cite{Yeu1, Yeu2}.
Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice
本文定义了p进Hodge结构的Tannakian范畴，并展示了其与可接受对之间的等价关系，通过研究p进周期的超越性来推广一维形式群的结果。
Equivariant aspects of de-completing cyclic homology
本文通过等变同伦结构描述 Kaledin 的多项式周期循环同调，建立无需 Noetherian 假设的 Morita 不变性，并推广到拓扑情形，得到非交换晶体-de Rham 比较及 p=2 的扩展。
Over a subfield of the field of complex numbers, the Hodge realization of a geometrical motive is defined and represented as the cohomology of a mixed Hodge DG-complex in the sense of Deligne. Both filtrations are represented by truncation functors, on a Bondarko weight complex for the weight filtration and on the De Rham motivic complex for the Hodge one. The Deligne-Beilinson realization is also constructed. This preprint partially replaces the former "Réalisation des complexes motiviques de Voevodsky".
Periodic Cyclic Homology over Q
本文给出了特征0动画交换环上导出概形的周期循环同调关于Hodge完备化导出de Rham复形的完整描述，推广了Loday-Quillen的计算，并给出了使HKR滤过穷尽的显式条件。
Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic
本文阐述了正特征下的quasisyntomic下降与Nygaard滤过，提出了Illusie比较定理的新方法，重新审视了Ogus比较定理，并确定了乘以$n$在Abel簇fppf上同调上的作用。
We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered lo…
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…
TR with logarithmic poles and the de Rham-Witt complex
本文研究 Hesselholt 关于对数 TR 与对数 de Rham–Witt 复形关系的猜想，在 p-adic 复数域上的 p-完全光滑形式概形情形中分析对数 TR^r 的动机滤过及 S^1-同伦不动点。
For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we prove two structural results concerning the cohomology of Higgs moduli which are predicted by the P=W conjecture in non-abelian Hodge theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge-Tate decomposition for a character variety to each piece of th…
Given a smooth projective variety $X$ over a field $k$ of characteristic zero, we consider the composition of the de Rham cohomology cycle class map over $k$ from the Chow group $CH^q(X\times_kK)$, where $K$ is the field of fractions of henselization $A^h$ of the local ring of a smooth closed point of a variety over the field $k$ with an appropriate projection: $CH^q(X\times_kK)\longrightarrow\bigoplus_{p=1}^qgr_F^{q-p}N^{q-p} H^{2q-p}_{dR/k}(X)\otimes_kΩ^p_{A^h/k,{\rm closed}},$ where $F^{\bullet}$ and $N^{\bullet}$ are the Hodge and the coniveau filtrations on the de Rham cohomology, respect…
Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator
本文研究了特征 $p$ 下 de Rham 复形的分解障碍，证明了存在 Hodge-to-de Rham 谱序列不退化的可提升光滑射影簇，并基于交换代数与 Steenrod 运算揭示了 Sen 算子的非半单性及 $E_{\infty}$-代数形式性条件。
In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex $AΩ$ constructed in our previous work, and in equal characteristic $p$ to crystalline cohomology. Our construction of the filtration on $\mathrm{THH}$ is via flat descent to semiperfectoid rings. As one application, we refine the construction of the $AΩ$-complex by giving a cohomological construction o…
We look more closely at the higher nonabelian de Rham cohomology of a smooth projective variety or family of varieties that had been defined in some previous papers. We formalize using $n$-stacks the notion of shape underlying this nonabelian cohomology. A generalization of the de Rham construction to any appropriate formal category or family of formal categories, yields various algebraic aspects of Hodge theory for the de Rham shape, such as the Hodge filtration, the Gauss-Manin connection, Griffiths transversality, an extension of the Gauss-Manin connection with regular singularities across …
We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classi…
A note on weight filtrations at the characteristic
本文研究了在非可逆特征下，有理系数上射影Dedekind方案上的重量滤泡性质，并结合后续工作，证明了已知的对数上同调理论在项目ive sncd对上具有重量滤泡，同时展示了重量滤泡上同调是开部分的不变量。
A Hodge module construction of relative Du Bois complexes
本文通过混合 Hodge 模和 Grothendieck 对偶构造相对 Du Bois 复形，并建立有限滤过及纤维限制与符号 m-Du Bois 奇点的联系。
The intersection cohomology Hodge module of toric varieties
本文研究了托里安品种的交集上同调Hodge模的Hodge滤波，通过Ishida复形计算反射微分 sheaf 的更高直接像，并给出了与交集上同调留数的精确公式。
We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitr…
Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $TC^-$ of quasi-lci rings with bounded torsion, which recovers the BMS-filtration after $p$-adic completion. Then we compute the graded pieces of this filtration in terms of Hodge completed derived de Rham cohomology relative to the base ring $\mathbb{Z}$. We denote the cofiber of the canonical map from $\mathrm{gr}^{n}TC^-(-)$ to $\mathrm{gr}^{n}TP(-)$ by $LΩ^{<n}_{-/\mathbb{S}}[2n]$. Let $\mathcal{X}$ be a regular connected scheme of dimension $d$ proper over $\…
Deformation of rational singularities and Hodge structure
本文证明了在中心纤维具有有理奇点等条件下，一参数退化中前沿 Hodge 数的不变性，并通过计算诱导 $V$-滤除揭示了其与结构层上同调维数不变性的等价性及局部单值群幂零阶数的性质。
Delta theory of Anderson Modules II: Hodge-Pink structure
本文利用δ-几何理论，对任意阿贝尔安德森模E构造了带霍奇-平克结构的典范z-等晶体，证明其与de Rham上同调的相容关系及同构，还验证了Drinfeld模、Carlitz模的相关性质。
In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map is obtained. These results are applied in the second part to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。