Morihiko Saito
We introduce the notion of mixed Hodge complex on an algebraic variety, improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge structure with the theory of mixed Hodge module. This was supposed to be true, but is quite nontrivial, because the theory of mixed Hodge module does not work on a simplicial scheme. Some application to the Du Bois singularity is given.
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We generalize some results in Hodge theory to generalized normal crossing varieties.
We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.
In this paper we investigate the exactness of the Grassmannian BGG complexes introduced in a previous work (arXiv:1211.2486), and obtain some inequalities between some Hodge numbers of some irregular varieties. In particular, we obtain sharp lower bounds for the Hodge numbers of smooth subvarieties of Abelian varieties, as well as some improvements of results of Lazarsfeld and Popa and Lombardi concerning threefolds and fourfolds.
We introduce an algebraic method for describing the Hodge filtration of degenerating hypersurfaces in projective toric varieties. For this purpose, we show some fundamental properties of logarithmic differential forms on proper equivariant morphisms of toric varieties.
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。
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 …
The space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group.
We describe an inductive approach most appropriate for abelian varieties with an action of an imaginary quadratic field.
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
If X is a complex projective variety with klt singularities, then the mixed Hodge structures on the first two singular cohomology groups are pure. We describe the pieces of the Hodge decomposition in terms of reflexive differential forms. Applications include a Lefschetz (1,1) Theorem and a weak analogue of the Hodge-Riemann bilinear relations for klt varieties.
This paper gives an introduction to Kuga-Satake varieties and discusses some aspects of the Hodge conjecture related to them. Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We give a detailed account of the construction of Kuga-Satake varieties and of their decomposition in simple subvarieties. We recall the Hodge conjecture and we point out a connection between the Hodge conjecture for abelian fourfolds and Kuga-Satake varieties. We discuss the implications of the Hodge conjecture on the g…
We describe in some details an idea of M. Kontsevich how one can try to find a counterexample to the Hodge conjecture using tropical geometry.
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-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces
本文为刚性空间上的 v-完备复形构造了从 Hodge-Tate 叠到 v-站点的全忠实函子，并用 Higgs 模导出范畴刻画完美复形，从而建立导出的 $p$-进 Simpson 函子。
Differential complexes and Hodge theory on log-symplectic manifolds
本文研究对数辛流形上的微分形式复形（含反向de Rham复形），将其与Poisson双矢量的退化除子及秩轨迹关联，并在Kähler情形下建立重数轨迹与Hodge数的关系，同时揭示Hodge数消失与无阻碍形变的联系。
Perverse-Hodge complexes for Lagrangian fibrations
本文研究 Lagrangian 纤维化的反常-Hodge 复形，提出其对称性猜想，该猜想分类了“反常=Hodge”恒等式并推广了松下的结构层高次直像定理，通过 Hodge 结构变化、Hilbert 概形和 Looijenga-Lunts-Verbitsky Lie 代数验证了若干情形。
The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate--Shafarevich group and the Tate conjecture of surfaces over finite fields.
In this paper we study Hodge classes on complex abelian varieties. We prove some general results that allow us, in certain cases, to compute the Hodge group of a product abelian variety $X = X_1 \times X_2$ once we know the Hodge groups of the two factors. Using these results we can compute the Hodge groups of all abelian varieties of dimension $\leq 5$. We prove that the Hodge ring of any such abelian variety $X$ is generated by divisor classes together with the so-called Weil classes on (quotients of) $X$.
In this article we construct a Koszul-type resolution of the p-th exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel-type vanishing theorem for Hodge ideals associated to effective \mathbb{Q}-divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.
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…
Perverse-Hodge complexes for Lagrangian fibrations and symplectic resolutions
该研究针对全纯辛簇的拉格朗日纤维化展开，证明了辛解消情形下反常霍奇复形的辛Hard Lefschetz型定理与对称性，推广了光滑情形的结果并验证了相关猜想，还给出了“反常=霍奇”数值结论的新证明，并应用于奇异Higgs模空间研究。
On the Log Hodge Theory of Toroidal Varieties and a Partial Proof of the Absolute Hodge Conjecture
本文构建加权环面结构与 Hodge 类障碍理论，借助 $E_1$-退化结果建立范畴 Hodge 对应，证明一类有理加权射影环面簇的绝对 Hodge 猜想。
Algebraicity of Hodge classes on some generalized Prym Varieties
本文通过重新审视Chad Schoen在特定Prym簇上构造的代数环，利用无分歧几何类域论证明某些广义Prym簇上Hodge类的代数性。
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 give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties. We also give a generalization to higher genera of the famous formula $12 λ_1=δ$ for genus 1.
On the integral Hodge conjecture for varieties with trivial Chow group
本文构造了全周群有限秩却违反整 Hodge 猜想的复光滑射影簇，并证明此类反例也可定义在数域上。
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.
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 Beilinson-Hodge conjecture asserts the surjectivity of the cycle map $$H^n_M(X,\Q(n)) \to {\rm Hom}_{MHS}(\Q(-n),H^n(X,\Q))$$ for all positive integers $n$ and every smooth complex algebraic variety $X$. For $n=2$, we prove the conjecture if $X$ is rational.
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.
Level $m$-stratifications on PEL Shimura varieties are defined and studied by Wedhorn using BT-$m$s with PEL structure, and then by Vasiu for general Hodge type Shimura varieties using Shimura $F$-crystals. The theory of foliations is established by Oort for Siegel modular varieties, and by Mantovan for PEL Shimura varieties. It plays an important role in Hamacher's work to compute the dimension of Newton strata of PEL Shimura varieties. We study level $m$ stratifications on good reductions at $p>2$ of Shimura varieties of Hodge type by constructing certain torsors together with equivariant mo…
This is meant to be a survey article for the Cubo Journal. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those invariants. We present a geometric version of it.
We know that semi-regular sub-varieties satisfy the variational Hodge conjecture i.e., given a family of smooth projective varieties $π:\mathcal{X} \to B$, a special fiber $\mathcal{X}_o$ and a semi-regular subvariety $Z \subset \mathcal{X}_o$, the cohomology class corresponding to $Z$ remains a Hodge class (as $\mathcal{X}_o$ deforms along $B$) if and only if $Z$ remains an algebraic cycle. In this article, we investigate examples of such sub-varieties. In particular, we prove that any smooth projective variety $Z$ of dimension $n$ is a semi-regular sub-variety of a smooth projective hypersur…
Non-abelian Hodge theory for non-proper varieties and the linear Shafarevich conjecture
本文综述了非交换Hodge理论在非 proper 代数变种中的最新进展，并利用这些工具构造代数Shafarevich映射，证明了对于任何代数变种的线性Shafarevich猜想版本。
Around a class version of the Hodge index theorem for singular varieties
本文概述了奇异复代数簇霍奇指标定理特征类版本的最新进展，澄清了文献中L类概念的关系，并给出了该猜想成立的多种新情况，包括紧 toric 簇、Schubert 簇等。
These are notes of lectures given at the school `Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.
The purpose of this work is to geometrize the notion of mixed Hodge structure. Therefore, we associate equivariant vector bundles on the projective plane to trifiltered vector spaces. Making this Rees construction with filtrations arising from mixed Hodge structures gives an equivalence of categories between the category of mixed Hodge structures and a category of equivariant vector bundles on ${\bf P}^{2}_{\bf C}$ that are semistable for a semistability condition with respect to a divisor. Using this dictionary we can recover geometrically some properties of mixed Hodge structures like the ab…
For any subfield K of the complex numbers which is not contained in an imaginary quadratic number field, we construct conjugate varieties whose algebras of K-rational (p,p)-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when K is contained in an imaginary quadratic number field; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut(\C)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on their second real cohomolog…
We construct canonical non-vanishing global sections of powers of the Hodge bundle on each Ekedahl-Oort stratum of a Hodge type Shimura variety. In particular we recover the quasi-affineness of the Ekedahl-Oort strata. In the projective case, this gives a very short proof of non-emptiness of Ekedahl-Oort strata. It follows that the Newton strata are also nonempty, by a result of S.Nie. We also deduce the fact that the $μ$-ordinary locus is determined by the Ekedahl-Oort strata of its image by any embedding.
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。
Summary: The Hodge conjecture asks whether rational Hodge classes on a smooth projective manifolds are generated by the classes of algebraic subsets, or equivalently by Chern classes of coherent sheaves. On a compact Kaehler manifold, Hodge conjecture is known to be false if algebraic subsets are replaced with analytic subsets. Here we show that it is even false that for a Kaehler manifold, Hodge classes are generated by Chern classes of coherent sheaves. We also show that finite free resolution do not in general exist for coherent sheaves on compact Kaehler manifolds.
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.
We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotica…
We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg structure, a notion introduced in [GNPR10] leading to a good calculation of the homotopy category in terms of (co)fibrant objects. This result provides a conceptual framework from which Beilinson's [Bei86] and Carlson's [Car80] results on mixed Hodge complexes and extensions of mixed Hodge structures follow easily.
Hodge symmetry and Lefschetz theorems for singular varieties
该研究证明奇点代数簇拓扑与霍奇理论新结果，揭示奇点复杂度条件与霍奇-杜波依斯菱形对称性的关联，关联有理同调流形理论并在低维有理奇点代数簇中刻画了它。
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.
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 study the structure of string theory flux compactification for a general family of elliptic CY 3-folds. We investigate the locus of the attractor points of the flux compactification in type IIB string theory on the boundary components of period domains. Specifically we give equations describing this locus through the asymptotic of nilpotent orbits on period domains. our approach is a mixture of techniques of asymptotic Hodge theory and the numerical period vectors used in physics.
Differential Forms and Hodge Structures on Singular Varieties
研究奇异复代数变种上的微分形式概念及其与Hodge滤波器的关联，引入并研究称为准有理的奇异性，探讨孤立奇异性条件。
Hodge Conjecture via Singular Varieties
本文利用光滑射影簇与奇异超曲面的双有理关系，证明具有 $A_n$ 奇点的奇数维超曲面满足两种 Hodge 猜想，且其消解满足经典 Hodge 猜想。
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 …
The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties
通过奇异OG6型流形与Weil型判别式1的四维阿贝尔流形的几何关系，证明了Hodge猜想和Tate猜想。
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。
Lengths of Hodge characters in Fermat varieties
本文通过引入长度缩减算法和提升操作，将Fermat簇的Hodge猜想简化为限制例外元组长度，并证明了度数小于65（除44、51、52）的所有Fermat簇的猜想。
In this article we study the (cohomological) Hodge conjecture for singular varieties. We prove the conjecture for simple normal crossing varieties that can be embedded in a family where the Mumford-Tate group remains constant. We show how to produce such families. Furthermore, we show for varieties with worse singularities the conjecture can be expressed solely in terms of the algebraic classes.
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。
Hodge integrals and $λ_{g}$ conjecture with target varieties
本文提出带靶簇Hodge积分的$\lambda_{g}$猜想，建立其与Virasoro猜想的关系，并在多类情形下证明该猜想，导出后代Gromov-Witten不变量的新型普适约束。
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