On the Lefschetz and Hodge-Riemann theorems

Noether-Lefschetz locus and a special case of the variational Hodge conjecture

For a fixed integer $d$, we study here the locus of degree $d$ hypersurfaces $X$ in $\mathbb{P}^{2n+1}$ such that $H^{2n}(X,\mathbb{Q}) \cap H^{n,n}(X,\mathbb{C}) \not= \mathbb{Q}$. We call this locus \textit{the Noether-Lefschetz locus}. Any irreducible component of this locus is locally a Hodge locus. So, we see that the study of this locus is very closely related to the variational Hodge conjecture. In this article we show that the Hodge locus corresponding to the cohomology class of a complete intersection subscheme in $\mathbb{P}^{2n+1}$ of codimension $n+1$ and degree less than $d$ satis…

Hodge Diamonds and Adjoint Orbits

We present a Macaulay2 package that produces compactifications of adjoint orbits and of the fibres of symplectic Lefschetz fibrations on them. We use Macaulay2 functions to calculate the corresponding Hodge diamonds, which then reveal topological information about the Lefschetz fibrations.

奇点代数簇的霍奇对称性与莱夫谢茨定理

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。

混合Lefschetz定理与Hodge-Riemann双线性关系

Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

类似于硬莱夫谢茨定理（HLT）和霍奇-黎曼双线性关系（HRR）的陈述在各种情况下都成立：它们对光滑紧致凯勒流形的上同调代数或投射复曲面簇的交上同调施加限制;它们限制霍奇结构的极化变体的局部单值性;它们对凸多胞体的可能$f$ -向量施加条件。虽然这些定理的陈述依赖于一个凯勒类的选择，或其类似物，通常有一个可能的凯勒类锥。然后很自然地会问HLT和HRR在混合背景下是否仍然正确。在这份说明中，我们提出了一个统一的方法来证明混合HLT和HRR，推广以前已知的结果，并证明它在新的情况下，如非有理多面体的相交上同调。

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for abelian varieties

We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let $γ$ be a correspondence between two abelian varieties $A$ and $B$ that can be written as a linear combination of products of symmetric divisors. Assume that $A$ is isogenous to th…

非有理多面体交上同调的硬Lefschetz定理和Hodge-Riemann关系

Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes

非有理多面体的交上同调的Hard Lefschetz定理最近由K.卡鲁[卡]。这个定理蕴涵了R. Stanley关于广义$h$ -向量的幺模性.本文通过在完备扇形$Φ$的交上同调$IH(Φ)$上引入一个标准双线性型$(\cdot ,\cdot)_Φ$，并证明了$(\cdot ,\cdot)_Φ$的Hodge-Riemann双线性关系，加强了Karu定理.

Hodge symmetry for rigid varieties via log hard Lefschetz

Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue characteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction, (3) $K$ is $p$-adic and $X$ has "combinatorial reduction."' We also reprove a version of their result, Hodge symmetry for $H^1$, without the use of moduli spaces of semistable sheaves. All of this relies on cases of Kato's log hard Lefschetz conjecture, which we prove for $H^1$ and for lo…

Notes on variation of Lefschetz star operator and $T$-Hodge theory

These notes were written to serve as an easy reference for \cite{Wang-AF}. All the results in this presentation are well-known (or quasi-well-known) theorems in Hodge theory. Our main purpose was to give a unified approach based on a variation formula of the Lefschetz star operator, following \cite{Wang-k}. It fits quite well with Timorin's $T$-Hodge theory, i.e. the Hodge theory on the space of differential forms divided by $T$ (i.e. forms like $T\wedge u$), where $T$ is a finite wedge product of Kähler forms.

紧致Kahler流形上的混合Hodge-Riemann双线性关系

The mixed Hodge-Riemann bilinear relations for compact Kahler manifolds

证明了在混合情形下紧致Kahler流形的Hodge-Riemann双线性关系、硬Lefschetz定理和Lefschetz分解。

Brilliant families of K3 surfaces: Twistor spaces, Brauer groups, and Noether-Lefschetz loci

We describe the Hodge theory of brilliant families of K3 surfaces. Their characteristic feature is a close link between the Hodge structures of any two fibres over points in the Noether-Lefschetz locus. Twistor deformations, the analytic Tate-Safarevic group, and one-dimensional Shimura special cycles are covered by the theory. In this setting, the Brauer group is viewed as the Noether-Lefschetz locus of the Brauer family or as the specialization of the Noether-Lefschetz loci in a family of approaching twistor spaces. Passing from one algebraic twistor fibre to another, which by construction i…

对于奇异方案的Lefschetz-Riemann-Roch定理

A Lefschetz-Riemann-Roch theorem for singular schemes

本文提出一个适用于奇异方案的Lefschetz-Riemann-Roch定理，扩展了之前针对奇异流形的结果。

XI所罗门莱夫谢茨纪念系列讲座：霍奇结构在非交换几何。（作者：Ernesto Lupercio）

XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in non-commutative geometry. (Notes by Ernesto Lupercio)

传统上，霍奇结构与复杂的投射簇相关联。在我的临时讲座中，我讨论了变形量子化和导出代数几何中霍奇结构的非对易推广。

A Weak (k,k)-Lefschetz Theorem for Projective Toric Orbifolds

Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational (k,k)-cohomology class is algebraic, i.e., the Hodge conjecture holds on them.

Grassmannian G（2，N）和Racah多项式的标准公式

Standard conjectures for the arithmetic Grassmannian G(2,N) and Racah polynomials

证明了射影空间中Grassmannian $G=G(2,N)$参数化直线的算术Hodge指标和硬Lefschetz结构，以及射影空间中$G$通过Plücker嵌入定义的自然算术Lefschetz算子的算术Hodge指标和硬Lefschetz结构. Hodge指数不等式的分析涉及到某些Racah多项式的估计。

The perverse filtration and the Lefschetz hyperplane theorem, II

The perverse filtration in cohomology and in cohomology with compact supports is interpreted in terms of kernels of restrictions maps to suitable subvarieties by using the Lefschetz hyperplane theorem and spectral objects. Various mixed-Hodge-theoretic consequences for intersection cohomology and for the decomposition theorem are derived.

Hodge-theoretic splitting mechanisms for projective maps (in appendix, a letter from P. Deligne)

According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we…

Lefschetz number formula for Shimura varieties of Hodge type

For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport \cite{LR87} of deriving the Kottwitz's formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands-Rapoport conjecture), which replaces their Galois gerb theoretic arguments by more geometric …

Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci

We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensi…

Fano 与 Calabi–Yau 三维簇中曲面的 Noether–Lefschetz 轨迹的密度

Density of Noether-Lefschetz loci for surfaces in Fano and Calabi-Yau threefolds

该研究利用 Hodge 轨迹的一般定理，考察 Fano 与 Calabi–Yau 三维簇中充分丰富线性系统曲面族的 Noether–Lefschetz 轨迹，并比较一般分支与例外分支并集的不同密度行为。

Relative Hard Lefschetz Theorem for Fans

We prove the Relative Hard Lefschetz theorem and the Relative Hodge-Riemann bilinear relations for combinatorial intersection cohomology sheaves on fans.

莱夫谢茨模的一个分解定理

A decomposition theorem for Lefschetz modules

本文研究莱夫谢茨模的分解问题，建立其与复射影簇态射分解定理平行的结构结果，可应用于组合霍奇理论并阐明代数几何中分解定理的霍奇理论性质。

自由线丛的硬莱夫谢茨定理

Hard Lefschetz theorems for free line bundles

本文通过半小性亏格定义代数映射的部分正性，并借此为自由线丛的陈类完全交建立硬莱夫谢茨定理与霍奇-黎曼双线性关系。

On a Lefschetz-type phenomenon for elliptic Calabi--Yaus

We consider 18 families of elliptic Calabi--Yaus which arise in constructing $F$-theory compactifications of string vacua, and show in each case that the upper Hodge diamond of a crepant resolution of the associated Weierstrass model coincides with the upper Hodge diamond of the (blown up) projective bundle in which the crepant resolution is naturally embedded. Such results are unexpected, as each crepant resolution we consider does \emph{not} satisfy the hypotheses of the Lefschetz hyperplane theorem. In light of such findings, we suspect that all ellipitic Calabi--Yaus satisfy such a `Lefsch…

热带超曲面的 Lefschetz 截面定理

Lefschetz section theorems for tropical hypersurfaces

本文建立环面簇热带超曲面的 Lefschetz 超平面截面定理，证明整系数热带同调群在紧致或 $\mathbb{R}^n$ 情形下无挠，并建立其与复超曲面 Hodge 数及 Hodge-Deligne 数的联系。

复曲面簇的Barth-Lefschetz定理

A Barth-Lefschetz theorem for toric varieties

巴特-莱夫谢茨定理是关于某个射影空间的子流形X的上同调的一个陈述，在一个依赖于嵌入余维的范围内。本文将其推广到光滑投射复曲面簇Y的子流形X的情形，在X上的某些自然条件下。特别地，在Barth-Lefschetz范围内，给出了X的Hodge数关于复曲面的公式，即Y的组合数据。

An arithmetic Lefschetz-Riemann-Roch theorem. With an appendix by Xiaonan Ma

In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic conce…

Galois gerbs and Lefschetz number formula for Shimura varieties of Hodge type

For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke correspondence acting on the compactly supported étale cohomology and verify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization of that formula. The main ingredients of our proof of the formula are a recent work of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory of Galois gerbs developed by Langlands and Rapoport \cite{LR87}. Es…

A Lefschetz (1,1) theorem for singular varieties

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…

硬Lefschetz定理与半小映射的拓扑

The Hard Lefschetz Theorem and the topology of semismall maps

在复射影流形上引入左线丛的概念。证明了左线丛满足硬Lefschetz定理、Lefschetz分解和Hodge-Riemann双线性关系。研究了复流形上的真全纯半小映射，证明了对于常系数，分解定理等价于某些交形式的非退化性。通过证明所讨论的形式是非退化的，给出了定义域与目标是射影的情况下常数层的复直象的分解定理的一个证明。我们的证明揭示了一个新的特点，即形式是确定的。

多面体的Hodge-Riemann关系：一种几何方法

Hodge-Riemann Relations for Polytopes: A Geometric Approach

本文利用多面体上的几何运算证明了K.Karu提出的多面体的组合交上同调的Hodge Riemann关系。

正规射影复簇的Lefschetz（1，1）定理

A Lefschetz (1,1) Theorem for normal projective complex varieties

我们证明了Lefschetz（1，1）定理的一个类似定理，它刻画了Cartier因子的上同调类（或等价的线丛的第一类Chern类）在第二积分上同调中的性质.设$X$是正规复射影簇.我们证明了$H^2(X,Z)$中的Cartier因子类恰好是$x$类，使得（i）$x$在$H^2(X,C)$中的像（复系数上同调）位于$F^1 H^2(X,C)$（Deligne混合Hodge结构的Hodge滤子的第一级），（ii）$x$是Zurki-local平凡的，即，存在$X$被Zerkki开集$U$覆盖使得$x$在$H^2(U,Z)$中具有零像。对于正常的拟射影簇，这肯定地回答了一个问题的Barbieri-Viale和Srinivas（J. Reine Ang. Math.450（1994）），其中给出的例子表明，除数类本身不具有上述条件（i）、（ii）中的任何一个的特征，这与非奇异变种的情况不同。本文件还包含一个例子，一个非正规的投射品种，其中（i）和（ii）不足以表征除数类。

关于法向交叉簇的全局光滑化的霍奇结构

On the Hodge Structures of Global Smoothings of Normal Crossing Varieties

本文研究半稳定退化中一般纤维的霍奇结构，给出满足∂∂-引理的判定准则和用中心纤维拓扑不变量计算霍奇指标的公式，并应用于ODP光滑化及非凯勒卡拉比-丘流形等例子。

The period-index problem and Hodge theory

Conditional on the Lefschetz standard conjecture in degree 2, we prove that the index of a Brauer class on a smooth projective variety divides a fixed power of its period, uniformly in smooth families. In the other direction, we reinterpret in more classical terms recent work of Hotchkiss which gives Hodge-theoretic lower bounds on the index of Brauer classes. We also prove versions of our results over arbitrary algebraically closed base fields, and as an application construct qualitatively new counterexamples to the integral Tate conjecture.

Lefschetz decompositions for eigenforms on a Kähler manifold

We show that the eigenspaces of the Laplacian $Δ_k$ on $k$-forms on a compact Kähler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of $Δ_k$ lies in the spectrum of $Δ_{k+1}$.

On Hodge-Riemann Cohomology Classes

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.

关于具有半稳定退化的紧复流形的Hodge结构

On Hodge structures of compact complex manifolds with semistable degenerations

研究半稳定退化紧复流形上Hodge理论性质的保持条件，证明在特定单值性条件下，Hodge对称性和Hodge-Riemann关系成立。

拟阵多面体的 Hodge 理论

Hodge Theory for Polymatroids

本文为任意构建集上的离散拟阵多面体建立 Leray 模型，推广 Goresky–MacPherson 公式，并证明其 Chow 环的 Poincaré 对偶、Hard Lefschetz 与 Hodge–Riemann 性质及相对 Lefschetz 分解。

真正规复簇的Neron-Severi群

The Neron-Severi group of a proper seminormal complex variety

证明了复数上真正规簇的一个Lefschetz（1，1）-定理.这个证明是一个非平凡的几何论证，应用于与H^2上的混合霍奇结构相关联的莱夫谢茨1-动机的同构类。

有孔Riemann Surface基本群上的混合Hodge结构

The Mixed Hodge Structure on the Fundamental Group of a Punctured Riemann Surface

给定亏格为$g$的紧致黎曼曲面$\bar{X}$和$\bar{X}$上的点$q$，我们考虑具有基点$p\in X$的$X:=\bar{X}\setminus\{q\}$.研究了混合Hodge结构在基本群上（在Hain意义下）的扩张，其权为-1和-2.我们表明，它自然对应的元素$(2g q-2 p-K)$一方面在$\Pic^0(\bar{X})$，其中$K$代表典型的除数，另一方面，各自的扩展$\bar{X}$。最后，我们证明了一个点Torelli定理穿孔黎曼曲面。

Riemann surface上向量丛模空间的Hodge数

The Hodge Numbers of the Moduli Spaces of Vector Bundles over a Riemann surface

Harder，Narasimhan，Desale和Ramanan利用数论方法和Weil定理得到了亏格至少为2的固定Riemann曲面上秩和度互质的稳定全纯向量丛的模空间的贝蒂数的归纳公式，Atiyah和Bott利用规范理论重新导出了该公式.在本文中，我们观察到，有类似的归纳公式确定这些模空间的Hodge数。

扩展的操作Chow群与Lefschetz (1,1)-定理

Extended operational Chow group and Lefschetz (1,1)-theorem

本文通过引入扩展的操作Chow群，研究了从操作Chow群到Hodge (p,p)-类的Bloch-Gillet-Soulé（BGS）周类映射，证明了在有理奇点的正规曲面和孤立奇点情形下，该映射的满射性，从而推广了Lefschetz (1,1)-定理。

Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motives

V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.

Real Seifert forms, Hodge numbers and Blanchfield pairings

In this survey article we present connections between Picard--Lefschetz invariants of isolated hypersurface singularities and Blanchfield forms for links. We emphasize the unifying role of Hermitian Variation Structures introduced by Némethi.

An Introduction to Hodge Structures

We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with conn…

量子黎曼-罗克、莱夫谢茨和塞尔

Quantum Riemann - Roch, Lefschetz and Serre

给定紧致Kähler流形上的一个全纯向量丛$E:EX X$，引入$X$的扭曲GW-不变量，用$H^0(Σ, f^* E) - H^1(Σ, f^*E)$的一个选择的乘法特征类的帽积来代替稳定映射$f: Σ\to X$的模空间的虚基本环.利用量子化的二次哈密顿量的形式，我们用$X$的下降势来表示扭曲理论的下降势。结果（定理1）是Mumford的Riemann-Roch- Grothendieck公式应用于普适稳定映射的一个推论。当$E$是凹的，并且选择逆$\CC^{\times}$ -等变欧拉类时，扭曲理论产生$EX$的GW-不变量。"非线性塞尔对偶原理“通过超流形$ΠE^*X$的GW-不变量表达了$EX$的GW-不变量，其中欧拉类和$E^*$取代了逆欧拉类和$E$。我们从定理1导出非线性塞尔对偶的一个非常一般的形式（推论2）。当丛$E$是凸的，且子流形$Y\subset X$由整体截面定义时，$ΠE X$的亏格0 GW-不变量与$Y$的亏格0 GW-不变量一致.通过$X$的亏格0 GW-不变量，证明了表示完备交$Y$亏格0 GW-不变量的“量子Lefschetz超平面截面原理”（定理2）.这推广了Y. P. Lee和A. Gathmann和产生大多数已知的复曲面完全相交的镜像公式。

Orbifold Quantum Riemann-Roch, Lefschetz and Serre

Given a vector bundle $F$ on a smooth Deligne-Mumford stack $\X$ and an invertible multiplicative characteristic class $\bc$, we define the orbifold Gromov-Witten invariants of $\X$ twisted by $F$ and $\bc$. We prove a "quantum Riemann-Roch theorem" which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A Quantum Lefschetz Hyperplane Theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of $\X$ and that of a complet…

A Deligne-Riemann-Roch isomorphism I: Preliminaries on virtual categories

This is the first article in an upcoming series of papers. They have arisen through an attempt to answer open questions of Deligne proposed in "Le determinant de la cohomologie", Contemp. Mathematics 67 (1987). It amounts to functorial and metrized versions of the Grothendieck-Riemann-Roch theorem, as well as a Lefschetz-Riemann-Roch formula in the sense of Thomason, cf. "Lefschetz-Riemann-Roch theorem and coherent trace formula" Ann. Sci. Ec. Norm. Sup. 18 (1985), Theorem 3.5. In this article we treat various preliminary results on virtual categories which are the categories where the Deligne…

Hodge Theory for Combinatorial Geometries

We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.

霍奇模与凯勒态射

Hodge modules and Kähler morphisms

本文证明了沿真凯勒态射的带整结构霍奇模分解定理，推广了 Saito 的射影情形结果，方法基于周期映射紧化与 Cattani-Kaplan-Schmid 的 L²-上同调定理。

Relations de Hodge--Riemann et combinatoire des matroïdes (d'après K. Adiprasito, J. Huh et E. Katz)

Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded "Chow ring" $A(M)$ for which they prove analogs of the Poincaré duality, the Hard Lefschet…

Hodge-Riemann bilinear relations for Schur classes of ample vector bundles

Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le λ_N\le λ_{N-1} \le \cdots \le λ_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class $s_λ(E)\in H^{d-2,d-2}(X)$ has the Hard Lefschetz property and satisfies the Hodge-Riemann bilinear relations. As a consequence we obtain various new inequalities between characteristic classes of ample vector bundles, including a higher-rank version of the Khovanskii-Teissier inequalities.

Mixed Hodge-Riemann bilinear relations and $m$-positivity

Motivated by our previous work on Hodge-index type inequalities, we give a form of mixed Hodge-Riemann bilinear relation by using the notion of $m$-positivity, whose proof is an adaptation of the works of Timorin and Dinh-Nguyên. This mixed Hodge-Riemann bilinear relation holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate along some directions. In particular, it applies to fibrations of compact Kähler manifolds.

Hodge-Riemann Relations for Schur Classes in the Linear and Kähler Cases

We prove a version of the Hodge-Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space.

非阿贝尔霍奇理论的渐近几何与黎曼–希尔伯特对应，秩三 $\widetilde{E}_6$ 情形

Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence, rank three $\widetilde{E}_6$ case

本文在三孔球面上证明秩三的几何 P=W 猜想，通过渐近阿贝尔化与等变方法给出相关映射的 WKB 展开，并分析其 Stokes 现象，从而刻画特征簇在无穷远处的拓扑。