Hélène Esnault, Daqing Wan
We improve the known Hodge type bound for the exotic cohomology of complete intersections. In the revised version, we included a simplification of our original argument due to Pierre Deligne. The note appears in the C. R. de l'Aca. des Sc. de Paris.
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We show that the description of Deligne--Beilinson cohomology is improved by using log Hodge theory. We consider the log relative version of it, and also present a fundamental conjecture in log Hodge theory.
Hodge decomposition of L_2-cohomology and intersection cohomology of a Shimura variety
本文证明了 Shimura 簇的平方可积上同调与 Baily-Borel 紧化的交上同调之间的 Hodge 分解在已有的 Zucker 猜想证明下是一致的。
We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$ satisfying a ``Hard Lefschetz Condition''. Furthermore, the total cohomology $H_{orb}^{*,*}(X)$ forms a mixed Hodge structure that is polarized by every element of the Kähler cone of $X$. Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of $X$. This construction should be considered as the analogue of the abstract polarized variation of Hodge s…
A bound on the Hodge filtration of the de Rham cohomology of supervarieties
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。
In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a variety to be a Hodge theoretic K(pi,1), which roughly means that the cohomology of variations of mixed Hodge structure can be determined from the group.
Hodge Classes in the Cohomology of Local Systems
本文研究球算术商和二阶西格尔域上代数簇族的霍奇猜想，并给出用自守形式表达霍奇数的公式。
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.
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.
We use Higgs cohomology to determine the Hodge numbers of the first intersection cohomology group of a local system V arising from the third direct image of a family of Calabi-Yau 3-folds over a smooth, quasi-projective curve. We give applications to Rhode's families of Calabi-Yau 3-folds without MUM.
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.
We constructed the analytic Milnor fiber is a non-archimedean model of the classical topological Milnor fibration. In the present paper, we describe the homotopy type of the analytic Milnor fiber in terms of a strictly semi-stable model, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. We give a similar expression for Denef and Loeser's motivic Milnor fiber in terms of a strictly semi-stable model.
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 recall the construction of the Hodge character and we show, using a result due to F. Bittner, that these can be constructed using classical pure Hodge theory only, sideskipping Deligne's construction of functorial mixed Hodge structures on the cohomology of algberaic varieties. We then give a proof, based on the weak factorization theorem, that the nearby fibre is well-defined in the Grothendieck group of varieties. Finally, various formulas for numerical invariants of Hodge-theoretic origin are presented which are associated to one-parameter degenerations of projective manifolds.
Étale Nori finite vector bundles are those bundles defined by representations of a finite étale group scheme in the usual way. In this note we show that in many cases the dimensions of the Hodge cohomology groups of such a vector bundle and of a twist of it by an automorphism of the ground field are the same. This generalizes to the higher rank case a result of Pink-Roessler.
We construct correspondences in logarithmic Hodge theory over a perfect field of arbitrary characteristic. These are represented by classes in the cohomology of sheaves of differential forms with log poles and, notably, log zeroes on cartesian products of varieties. From one perspective this generalizes work of Chatzistamatiou and Rülling, who developed (non-logarithmic) Hodge correspondences over perfect fields of arbitrary characteristic; from another we provide partial generalizations of more recent work of Binda, Park and Østvær on logarithmic Hodge correspondences by relaxing finiteness a…
We study the moduli space of Higgs bundles on a compact Riemann surface. It was shown by Thaddeus and Hausel (in rank 2) and Markman (in general rank) that the rational cohomology ring of this space is generated by universal classes. In this paper, we use their results to compute the Hodge structure on the cohomology of this moduli space. We show that the natural Hodge structure on H^k is pure of weight k.
We establish a blow-up formula for Hodge cohomology of locally free sheaves on smooth proper varieties over an algebraically closed field of positive characteristic. For this, we introduce a notion of relative Hodge sheaves and study their behavior under blow-ups along smooth centers. In particular, as an application, we study the blow-up invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence for smooth proper varieties.
It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not true, as already remarked by M. Saito. We explain how one has to modify the latter, show that the Picard group can be expressed by absolute Hodge cohomology, too, and introduce an intermediate object between Picard group and usual Deligne-Beilinson cohomology group. Similarly as in the case of Deligne cohomology one can relate line bundles with a regular connec…
Let $X$ be a compact Riemann surface, $Σ$ a finite set of points and $M = X\setminus Σ$. We study the $L^2$ cohomology of a polarized complex variation of Hodge structure on a Galois covering of the Riemann surface of finite type $M$. In this article we treat the case when the covering comes from a branched covering of $X$, and where $M$ is endowed with a metric asymptotic to a Poincaré metric. We prove that after tensorisation with the algebra of affiliated operators, the $L^2$ cohomology admits a pure Hodge structure.
We interpret syntomic cohomology of Nekovář-Nizioł as a $p$-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomology by Beilinson and generalizes the results of Bannai and Chiarellotto, Ciccioni, Mazzari in the good reduction case, and of Yamada in the semistable reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for proper and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated…
We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.
We study restrictions on cohomology algebras of Kaehler compact manifolds, not depending on the h^{p,q} numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the Lefschetz property but not having the cohomology algebra of a compact Kaehler manifold. We also prove the stability of these restrictions under products.
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 prove the vanishing modulo torsion of the higher direct images of the sheaf of Witt vectors (and the Witt canonical sheaf) for a purely inseparable projective alteration between normal finite quotients over a perfect field. For this, we show that the relative Hodge-Witt cohomology admits an action of correspondences. As an application we define Witt-rational singularities which form a broader class than rational singularities. In particular, finite quotients have Witt-rational singularities. In addition, we prove that the torsion part of the Witt vector cohomology of a smooth, proper scheme…
For a smooth and proper scheme over an artinian local ring with ordinary reduction over the perfect residue field we prove - under some general assumptions - that the relative de Rham-Witt spectral sequence degenerates and the relative crystalline cohomology, equipped with its display structure arising from the Nygaard complexes, has a Hodge-Witt decomposition into a direct sum of (suitably Tate-Twisted) multiplicative displays. As examples our main results include the cases of abelian schemes, complete intersections, varieties of K3 type and some Calabi-Yau $n$-folds.
Mixed Hodge structure in the cohomology of moment-angle complexes
本研究探讨矩角复形上同调的多重分次与坐标子空间排列补集上同调的混合霍奇结构多重分次的关系，证明二者一致，细化了相关定理，并将混合霍奇结构结果应用于有限域上的同类簇，得到有理点与全局函数的相关公式。
We prove two results regarding Hodge structures appearing in the cohomology of complex tori. First, we prove that if a polarizable Hodge structure appears in the cohomology of a complex torus $T$, it appears in the cohomology of an abelian variety isomorphic to a subquotient of $T$. Second, we prove a universality result for the Kuga-Satake construction applied to Hodge structures of $K3$ type that might not be polarized.
Insprired by the work of C. Simpson, it is shown that every variation of graded-polarized mixed Hodge structure defined over Q gives rise to a natural Higgs field on the underlying vector bundle. In the context of Mirror Symmetry it is then shown that these Higgs fields share an intimate relationship with Quantum Cohomology.
Hodge theory of degenerations, (II): vanishing cohomology and geometric applications
本文研究孤立超曲面奇点的加权谱和消失上同调，揭示其对光滑化极限混合Hodge结构的作用，并应用于紧化与镜像对称中的多类奇点。
Hodge theory of secant varieties
本文研究了secant varieties的Hodge理论，探讨了局部cohomology模块的缺陷及其与原始cohomology的关系，并计算了相关Hodge数和结构，同时改进了文献中的结果。
Products in absolute Hodge cohomology and mixed Hodge formality
本文证明混合Hodge形式性蕴含绝对Hodge上同调的平方零扩张分解，定义类似Massey积的阻碍积，并关联乘法Chow-Künneth分解。
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。
Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomol…
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。
v2: We improved a little bit according to the referee's wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the one of its $a$-th power $L^a$ if $a$ is prime to $n$, under the assumptions that $X$ lifts to $W_2(k)$ and $dim X\le p$, if $k$ has characteristic $p>0$. They show this if $k$ has characteristic 0 and if $n$ is prime to $p$ in characteristic $p>0$. We show the conjecture in characteristic $p>0$ if $n=p$ assuming in addition that $X$ is ordinary (in the sens…
Vanishing of Local Cohomology with Applications to Hodge Theory
本文利用混合霍奇模理论与局部上同调研究极化霍奇结构变分在紧化下的延拓，分析其 de Rham 复形的分次部分及低次数上同调的霍奇结构。
This paper studies the Hilbert scheme of a curve on a complete-intersection K-trivial threefold, in the case in which the curve is unobstructed in the ambient variety in which the threefold lives. The basic result is that the obstruction theory of the curve is completely determined by the scheme-theoretic Abel-Jacobi mapping. Several applications of this fact are given.
Let A be an abelian variety over C such that the semisimple part of the Hodge group of A is a product of copies of SU(p,1) for some p>1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology of A is isomorphic to a Hodge structure in the cohomology of some abelian variety.
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold.
We prove a Hodge-type decomposition for the de-Rham cohomology of $ p$-adically uniformized varieties by the product of Drinfeld's symmetric spaces. It is based on work of Schneider, Stuhler, Iovita and Spiess on the cohomology of Drinfeld's symmetric space.
We show that certain abelian varieties A have the property that for every Hodge structure V in the cohomology of A, every effective Tate twist of V occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.
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。
We use $L^2$-Higgs cohomology to determine the Hodge numbers of the parabolic cohomology $H^1(\bar S, j_*\V)$, where the local system $\V$ arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve $\bar S$. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.
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.
Let $X$ be an irreducible complex analytic space with $j:U\into X$ an immersion of a smooth Zariski open subset, and let $\bV$ be a variation of Hodge structure of weight $n$ over $U$. Assume $X$ is compact Kähler. Then provided the local monodromy operators at infinity are quasi-unipotent, $IH^k(X, \bV)$ is known to carry a pure Hodge structure of weight $k+n$, while $H^k(U,\bV)$ carries a mixed Hodge structure of weight $\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,\bV) \to H^k(U,\bV)$ is the lowest weight part of this mixed Hodge structure. In the algebraic c…
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 prove the Hodge symmetry type result on the Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Consequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles.
We give formulas for the Dolbeault numbers of \Mbar_{3,1}, using the first author's calculations of the weights of the cohomology of M_{2,2} and M_{2,3} and the second author's calculation of the weights of the cohomology of M_{3,1}.
This article constructs Von Neumann invariants for constructible complexes and coherent D-modules on compact complex manifolds, generalizing the work of the author on coherent L 2-cohomology. We formulate a conjectural generalization of Dingoyan's L 2-Mixed Hodge structures in terms of Saito's Mixed Hodge Modules and give partial results in this direction. 2020 AMS Classification: 32J27.
We present a detailed overview of the construction of the A_inf-cohomology theory from the preprint "Integral p-adic Hodge theory", joint with B. Bhatt and P. Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham--Witt complexes.
Weighted Cohomology, Hodge Theory and Intersection Cohomology of Shimura varieties
本文证明了复Shimura簇的Baily-Borel紧化交上同调与重合Hodge结构的顶权商相关，引入了与特殊循环相关的规范循环类，将非紧Shimura簇的分析几何体积与拓扑项联系起来。
We consider algebraic manifolds $Y$ of dimension 3 over $\Bbb{C}$ with $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the $D$-dimension of $X$ is not zero, then $Y$ is a fibre space over a smooth affine curve $C$ (i.e., we have a surjective morphism from $Y$ to $C$ such that general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of $X$ is $-\infty$ and the $D$-dimension …
We study the $L^2$--cohomology of certain local systems on non-compact arithmetic ball quotients $X=Γ\backslash \B_n$, in particular vanishing and non--vanishing results. We also give generalizations to higher dimensional ball quotients and study the mixed Hodge structure on the sheaf cohomology of a local system with the $L^2$-cohomology contributing to the lowest weight part.
A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k-1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondance for certain cubic 4-folds.
For complex parallelisable manifolds $Γ\backslash G$, with $G$ a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de-Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.
Mixed Hodge structures for vanishing cycles and orbifold cohomology
研究消失环的混合Hodge结构与orbifold上同调的极化关系，探讨Hodge-Tate情况下的上同调结构。
We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology for complex manifolds.
A-D-E diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology
本文研究了复Grassmannian中Hodge-Tate超平面截面的半单量子上同调性质，提出基于Fano指数和贝蒂数的判据，并验证了Benedetti和Perrin的猜想。
The purpose of this note is to give an account of a well-known folklore result: the Hodge structure on the second cohomology of a compact hyperkähler manifold uniquely determines Hodge structures on all higher cohomology groups. We discuss the precise statement and its proof, which are somewhat difficult to locate in the literature.
We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.
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