Christian Schnell, Ruijie Yang
The purpose of this paper is to study certain notions of metric positivity for the lowest nonzero piece in the Hodge filtration of a Hodge module. We show that the Hodge metric satisfies the minimal extension property. In particular, this singular Hermitian metric has semi-positive curvature.
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We show that the $\mathbb{Q}$-descents of the canonical $\mathbb{R}$-variation of Hodge structure of Calabi-Yau type over a tube domain of type $A$ can be realized as sub-variations of Hodge structure of certain $\mathbb{Q}$-variations of Hodge structure which are naturally associated to abelian varieties of (generalized) Weil type.
We prove that, for every totally real number field E_0, there exists a weight three variation of Hodge structure of Calabi-Yau type defined over the rational numbers with associated endomorphism algebra E_0 such that the unique irreducible factor of Calabi-Yau type of the corresponding real variation of Hodge structure is the canonical real VHS of CY type over the Hermitian symmetric domain II_6, associated to the real group SO^*(12). The main point is a rationality result for the half spin representations of a form of the group SO^*(4m) defined over a number field.
Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
本文受 Markman 与 Kontsevich 启发，阐明 Weil 型阿贝尔 2n 维族仅在判别式为 $(-1)^n$ 时具极大幂幺退化，并给出判别式定义及相关事实。
In this paper we study the $\mathbb{C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. A main result of this paper is a bound on this invariant which generalizes both the Milnor-Wood inequality of the Hermitian case and the Arakelov inequalities of classical variations of Hodge str…
Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.
Lefschetz theorems, Hodge-Riemann relations and Ample vector bundles
本文提出了一种新的赫米特度量，扩展了经典的霍奇理论，并应用于证明混合凯勒类的希格纳-里曼关系及 ample 向量丛的陈类关系。
We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.
We study the sheaf of locally square integrable holomorphic section of vector bundle with semi-positive curved singular Hermitian metric. We confirm the coherence when its induced determinant metric has analytic singularities.
An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with $p$-adic applications
本文给出 Hermitian 簇上 Hodge 型标准猜想的显式初等证明，利用对偶极图控制相交形式，并证明 $p$-adic 初等因子整除 $q^m$，从而得到代数环的数值等价性及猜想对有限覆盖簇的推广。
Equivariant Hodge modules and rational singularities
本文定义了具有有理奇点的Hodge模，并推广Boutot定理，证明了在约化群作用于仿射簇且存在稳定点时，GIT商上的诱导等变Hodge模仍保持有理奇点。
The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail.
We investigate the Hodge structure of the singular O'Grady's six and ten dimensional examples of irreducible symplectic varieties. In particular, we compute some of their Betti numbers and their Euler characteristic. As consequence, we deduce that these varieties do not have finite quotient singularities answering a question of Bakker and Lehn.
We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.
In this paper, we study the coherence of a higher rank analogue of a multiplier ideal sheaf. Key tools of the study are Hörmander's $L^2$-estimate and a singular version of a Demailly--Skoda type result.
We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic cu…
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.
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…
In this article, we propose a definition of Nakano semi-positivity of singular Hermitian metrics on holomorphic vector bundles. By using this positivity notion, we establish $L^2$-estimates for holomorphic vector bundles with Nakano positive singular Hermitian metrics. We also show vanishing theorems, which generalize both Nakano type and Demailly-Nadel type vanishing theorems. As applications, we specifically construct globally Nakano semi-positive singular Hermitian metrics for several bundles, and prove vanishing theorems associated with them.
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。
This supersedes 0704.0566. We prove the invariance of logarithmic plurigenera for a projective family of KLT pairs and the adjoint line bundle of KLT line bundles. The proof uses the canonical singular hermitian metrics on relative logcanonical bundles.
In this paper, we give complex geometric descriptions of the notions of algebraic geometric positivity of vector bundles and torsion-free coherent sheaves, such as nef, big, pseudo-effective and weakly positive, by using singular Hermitian metrics. As an applications, we obtain a generalization of Mori's result. We also give a characterization of the augmented base locus by using singular Hermitian metrics on vector bundles and the Lelong numbers.
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。
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 investigate singular Hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ esitimates and vanishing theorems usually require an assumption that vector bundles are Nakano positive. However there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theromes by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.
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.
Maximal Infinitesimal Variation of Hodge Structure for Singular Curves
研究代数曲线族的微分Hodge结构变分，将经典理论扩展到奇异和非平面情形，定义奇异极小变分，证明等奇异平面曲线的极小秩等于算术亏点，分解秩为几何和奇异贡献。
Hodge theory of degenerations, (III): a vanishing-cycle calculus for non-isolated singularities
本文研究退化 Hodge 理论第三部分，针对奇点轨迹为一维的非孤立奇点情形进行具体计算，涵盖 $k$-log-canonical 奇点、slc 与非 slc 曲面奇点及 Feynman integrals 中的五维奇点。
The purpose of this paper is to give two supplements for vanishing theorems: One is a relative version of the Kawamata-Viehweg-Nadel type vanishing theorem, which is obtained from an observation for the variation of the numerical dimension of singular hermitian line bundles. The other is an analytic injectivity theorem for log canonical pairs on surfaces, which can be seen as a partial answer for Fujino's conjecture.
Non-abelian Hodge correspondence over singular Kähler spaces
本文在奇异凯勒空间上建立了非阿贝尔霍奇对应，并通过调和丛和降维定理扩展了射影klt品种的结果，应用于具有大canonical除数的品种的准统一化定理。
We introduce a new class of canonical AZD's (called the supercanonical AZD's) on the canonical bundles of smooth projective varieties with pseudoeffective canonical classes. We study the variation of the supercanonical AZD $\hat{h}_{can}$ under projective deformations and give a new proof of the invariance of plurigenera.
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。
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。
We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type vanishing theorem of a vector bundle with a singular Hermitian metric. We do not use an approximation technique of a singular Hermitian metric. We apply these theorems to a singular Hermitian metric induced by holomorphic sections and a big vector bundle, and we obtain a generaliz…
We study one parameter degenerations of complex projective manifolds by introducing certain type of Hodge metrics coming from the pluricanonical forms. We show that degenerations with at most canonical singularities are all in the finite distance boundary of moduli spaces. We also propose the converse to be true in the sense that finite distance degenerations admit birational models which have at most canonical singularities in the degenerate fiber. We verify this for curves and show that the Calabi-Yau case follows from the minimal model conjecture.
This text is an expository survey on the interplay between polarized variation of Hodge structure (PVHS) and the formalism of Hodge modules. We specifically review the extensions of a PVMHS over their singularities and its relation to mixed Hodge modules.
Infinitesimal Variations of Hodge Structure for Singular Curves I
本文研究了奇异曲线的微分Hodge结构，揭示了节点奇点与其它奇点在微分变化中的不同贡献，并在特定条件下给出了最大变化的判定标准。
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.
Logarithmic geometry and Infinitesimal Hodge Theory
本文通过对数几何和残差理论，系统研究了奇异和等奇异族的微分Hodge结构变化，澄清了Jacobian环的出现及Severi簇的变形空间结构。
Let $\mathcal{X}_ΓG:=\mathrm{Hom}(Γ,G)/\!/G$ be the $G$-character variety of $Γ$, where $G$ is a complex reductive group and $Γ$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of $\mathcal{X}_ΓG$, and present some applications, focusing on the cases when $Γ$ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere.
We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.
Let $E$ be a holomorphic vector bundle endowed with a singular Hermitian metric $H$. In this paper, we develop the harmonic theory on $(E,H)$. Then we extend several canonical results of J. Kollár and K. Takegoshi to this situation. In the end, we generalise Nakano's vanishing theorem.
Weak Hodge Theorem on Piecewise-Algebraic Spaces
在Kontsevich和Soibelman引入的分段代数空间上，证明了弱Hodge定理，将奇异上同调自然嵌入调和形式空间，而非建立同构。
We give a new variant of $L^2$-extension theorem for the jets of holomorphic sections and discuss the relation between the extension problem of singular Hermitian metrics with semipositive curvature.
Hodge-to-singular correspondence for reduced curves
本文研究了约化谱曲线上 $\mathrm{GL}_n$ 的 Hitchin 系统分解定理的直和项，建立了其与超环面箭图簇拓扑的新对应关系，并描述了正则 Higgs 丛模空间交叉上同调群对度数的依赖性。
Singular Hodge theory for combinatorial geometries
本文引入了matroid的交集上同调模，并证明其满足Poincaré对偶、硬Lefschetz定理和Hodge-Riemann关系，进而证明Dowling和Wilson的Top-Heavy猜想及所有matroid的Kazhdan-Lusztig多项式系数非负性。
Local cohomology and singular cohomology of toric varieties via mixed Hodge modules
本文利用混合 Hodge 模块技术研究了环面簇的局部上同调与奇异上同调，证明了真环面簇奇异上同调的混合 Hodge-Tate 型性质，并详细计算了特定子类的局部上同调与 Betti 数。
Slope zero tensors, uniformizing variations of Hodge structure and quotients of tube domains
该研究建立了刻画带klt奇点与丰沛典范除子的复射影簇的两种方法的等价性，构造了斜率零张量与单值化Hodge结构形变的互逆对应，推广了经典单值化结果，并证明了奇异Kähler–Einstein度量和乐群的紧性。
The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We ap…
For a projective algebraic variety $V$ with isolated singularities, endowed with a metric induced from an embedding, we consider the analysis of the natural partial differential operators on the regular part of $V$. We show that, in the complex case, the Laplacians of the de Rham and Dolbeault complexes are discrete operators except possibly in degrees $n,n\pm 1$, where $n$ is the complex dimension of $V$. We also prove a Hodge theorem on the operator level and the $L^2$--Stokes theorem outside the degrees $n-1,n$. We show that the $L^2$-Stokes theorem may fail to hold in the case of real alge…
We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to b…
Over an arbitrary compact complex space or an arbitrary germ of complex space $X$, we provide fine resolutions of pure Hodge modules with strict supports $IC_X(\mathbb{V})$ via differential forms with locally $L^2$ boundary conditions. When $\mathbb{V}=\mathbb{C}_{X_{\rm reg}}$ is the trivial variation of Hodge structure, we give a solution to a Cheeger-Goresky-MacPherson type conjecture: For any compact complex space $X$, there is a complete hermitian metric $ds^2$ on $X_{\rm reg}$ such that there is a canonical isomorphism $$H^i_{(2)}(X_{\rm reg},ds^2)\simeq IH^i(X),\quad \forall i.$$ Such m…
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.
This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part.
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.
The goal of this article is to try understand where Hodge cycles on a singular complex projective variety X come from. As a first step we consider Hodge cycles on the maximal pure quotient $H^{2p}(X)/W_{2p-1}$, and introduce a class of algebraic cycles that we call homologically Cartier, that should conjecturally describe all such Hodge cycles. Secondly, given a singular complex projective variety $X$, we show that there is a cycle map from motivic cohomology group $H^{2p}_M(X,Q(p))$ to the space of weight 2p Hodge cycles in $H^{2p}(X,Q)$. We conjecture that this is surjective when X is define…
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 conjectural Hodge index theorem for finite quotient spaces of singular varieties
本文证明奇异簇有限商空间满足Brasselet-Schürmann-Yokura提出的Hodge指标定理特征类猜想，通过等变解析K-同调与代数G-理论识别局部化类，并给出无需等变K-理论版本时的充分条件。
Hodge theory and K-stability of some very symmetric hypersurfaces
研究某些自然出现在超曲面模空间周期映射中的超曲面，通过Sung Gi Park最近的工作，计算这些超曲面的Hodge结构和交叠上同调，并展示某些轻微奇异退化超曲面的K-多稳定性。
We give an example of a homogeneous reflexive sheaf over $\mathbb{C}^3$ which admits a non-conical Hermitian Yang-Mills connection. This is expected to model bubbling phenomenon along complex codimension 2 submanifolds when the Fueter section takes zero value.
Let (M,I, ω, Ω) be a nearly Kaehler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ωand the Hermitian form ωwhich satisfies $dω=3λ\ReΩ, d\ImΩ=-2λω^2$, for a non-zero real constant λ. We develop an analogue of Kaehler relations on M, proving several useful identities for various intrinsic Laplacians on M. When M is compact, these identities bring powerful results about cohomology of M. We show that harmonic forms on M admit the Hodge decomposition, and prove that H^{p,q}(M)=0 unless p=q or (p=1, q=2) or (p=2, q=1).
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