Tien-Cuong Dinh, Viet-Anh Nguyen
We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.
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
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.
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.
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。
We give methods to compute l^2-cohomology groups of a covering manifolds obtained by removing pullback of a (normal crossing) divisor to a covering of a compact Kähler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.
A Sharp Lower Bound for the Spectrum of the Hodge Laplacian on Kähler Hyperbolic Manifolds and its Applications
本文研究了Kähler超曲面Hodge拉普拉斯算子谱的精确下界，并应用于有界对称域的谱分析。
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。
This is a technical introduction to the paper "Extension of twisted Hodge metrics for Kahler morphisms" by the authors.
The nonabelian Hodge correspondence (Corlette-Simpson correspondence), between the polystable Higgs bundles with vanishing Chern classes on a compact Kähler manifold $X$ and the completely reducible flat connections on $X$, is extended to the Fujiki class $\mathcal C$ manifolds.
Estimation and Vanishing Results for Hodge numbers
本文研究紧Kähler流形的拓扑刚性，通过Hodge数的消没与估计定理，在曲率特征值加权和为正的条件下推出其有理上同调环同于复射影空间，并证明Tachibana定理的Kähler类比。
We review the recent development of Hodge theory for almost complex manifolds. This includes the determination of whether the Hodge numbers defined by $\bar\partial$-Laplacian are almost complex, almost Kähler, or birational invariants in dimension four.
We give a way of constructing real variations of mixed Hodge structures over compact Kähler manifolds by using mixed Hodge structures on Sullivan's $1$-minimal models of certain differential graded algebras associated with real variations of Hodge structures.
On the existence of balanced metrics of Hodge-Riemann type
本文研究了非凯勒复流形上Hodge-Riemann类型平衡度量的存在性，证明了某些流形不存在此类结构，并构造了首个非凯勒的Hodge-Riemann平衡结构例子。
We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical Kähler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of $X$. These are obtained by removing from the Kuranishi family the two "superfluous" dimensions of…
We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kähler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kählerian analogue of the Weil conjectures carries over as well. Generalizing a construction of Looijenga and Lunts one obtains possibly infinite dimensional Lie algebras attached to Kähler-Riemann foliations. Finally using $(\mathfrak{g},K)$-cohomology we discuss a class of examples obtained by dividing a product of symmetric spaces by a cocompact lattice and considering t…
Kähler thresholds
本文提出一种退化Kähler流形（acK），证明在特定阈值下，其奇数Betti数为偶数、偶数Betti数为正，并利用紧性定理和Sobolev正则Kähler结构的Hodge理论给出证明。
We investigate properties of the Hodge metric of a mixed period domain. In particular, we calculate its curvature and the curvature of the Hodge bundles. We also consider when the pull back metric via a period map is Kähler. Several applications in cases of geometric interest are given, such as for normal functions and biextension bundles.
Positivity in the shadow of Hodge index theorem
本文在紧致Kähler流形上利用Hodge指标定理研究Lorentzian类，刻画Lefschetz型算子的核面，为Teissier比例性问题提供新方法，并建立Brunn-Minkowski不等式极值与定量稳定性估计。
Vanishing theorems for Hodge numbers and the Calabi curvature operator
本文证明具有$\ rac{n}{2}$-正Calabi曲率算子的紧致Kähler流形具有复射影空间的有理上同调，并分类了$\ rac{n}{2}$-非负情形，同时改进了Kähler–Einstein度量下的Kähler曲率算子结果。
Let $X$ be a Calabi-Yau threefold. A conifold transition first contracts $X$ along disjoint rational curves with normal bundles of type $(-1,-1)$, and then smooth the resulting singular complex space $\bar{X}$ to a new compact complex manifold $Y$. Such $Y$ is called a Clemens manifold and can be non-Kähler. We prove that any small smoothing $Y$ of $\bar{X}$ satisfies $\partial\bar{\partial}$-lemma. We also show that the resulting pure Hodge structure of weight three on $H^3(Y)$ is polarized by the cup product. These results answer some questions of R. Friedman.
Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of $f$ can be extended over the degenerate point $0$ of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of $f$ denoted $Ω_f$. A mixed Hodge structure can be defined on $Ω_…
On the dual positive cones and the algebraicity of a compact Kähler manifold
研究具有正有理Hodge类的紧凯勒流形的代数性，证明若对偶凯勒锥包含有理类为内点则阿尔巴尼斯簇是射影的，并回答了Oguiso-Peternell问题。
Hodge Laplacian and geometry of Kuranishi family of Fano manifolds
本文利用 Bochner-Kodaira 公式估计 Fano 流形上 Hodge Laplacian 的特征值，并据此分析其形变 Kuranishi 族，证明 Kähler 形式的保持性、给出 Ricci 势公式且重新解释 Donaldson-Fujiki 图景。
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.
We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.
Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries
本文研究了具有(1,1)-形式元素的Griffiths正定矩阵行列式的Hodge-Riemann性质，在特定维度下肯定回答了Dinh-Nguyên的问题，并给出两个相关应用。
Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of $ω$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge index type inequality for positive currents.
We prove the convergence of the Bergman kernels and the $L^2$-Hodge numbers on a tower of Galois coverings $\{X_j\}$ of a compact Kähler manifold $X$ converging to an infinite Galois (not necessarily universal) covering $\widetilde{X}$. We also show that, as an application, sections of canonical line bundle $K_{X_j}$ for sufficiently large $j$ give rise to an immersion into some projective space, if so do sections of $K_{\widetilde{X}}$.
Bottom of the spectrum of complete noncompact Kähler manifolds
综述了完备非紧凯勒流形上Hodge Laplacian谱的下界，重点讨论凯勒双曲流形和有界对称域，以及Ricci和双截曲率假设下的上界定理与刚性结果。
An $L^2$-$\partial\overline\partial$-Lemma on a class of complete Kähler manifolds
本文研究了完备凯勒流形上的 $L^2$-$\partial\overline\partial$-引理，通过分析霍奇拉普拉斯算子的谱间隙性，推广了经典凯勒流形上的 $\partial\overline\partial$-引理。
On Kähler manifolds with non-negative mixed curvature
本文研究了具有非负或拟正混合曲率的紧Kähler流形，利用共形扰动方法证明了包含直线的完备Kähler流形的分裂定理及紧Kähler流形的结构定理，并得出拟正混合曲率下Hodge数必然消失的结论。
This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's $L^2$-index theory. More precisely, given a compact Kähler manifold $(M,h)$ of even complex dimension $2m$, we prove that $$σ(M)=\sum_{p,q=0}^{2m}(-1)^ph_{(2),Γ}^{p,q}(M)$$ where $σ(M)$ is the signature of $M$ and $h_{(2),Γ}^{p,q}(M)$ are the $L^2$-Hodge numbers of $M$ with respect to a Galois covering having $Γ$ as group of Deck transformations. Likewise we also prove an $L^2$-version of the Frölicher index theorem. Afterwards we give some applications of these two theorems and finally we conclude t…
We introduce a "qualitative property" for Bott-Chern cohomology of complex non-Kähler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the $\partial\overline\partial$-Lemma. This follows from a quantitative study of Bott-Chern cohomology. In this context, we also prove a new bound on the dimension of the Bott-Chern cohomology in terms of the Hodge numbers. We also give a generalization of this upper bound, with applications to symplectic cohomologies.
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。
For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. Such foliations are generalizations of holomorphic principal torus bundles. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bigraded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan's theory of m…
In this paper we prove that the cohomology of smooth projective tropical varieties verify the tropical analogs of three fundamental theorems which govern the cohomology of complex projective varieties: Hard Lefschetz theorem, Hodge-Riemann relations and monodromy-weight conjecture. On the way to establish these results, we introduce and prove other results of independent interest. This includes a generalization of the results of Adiprasito-Huh-Katz, Hodge theory for combinatorial geometries, to any unimodular quasi-projective fan having the same support as the Bergman fan of a matroid, a tropi…
We study the $\bar \partial $-equation first in Stein manifold then in complete Kähler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any $(p,q)$-form $ω$ in $L^{r}$ with compact support and $\bar \partial $-closed there is a $(p,q-1)$-form $u$ in $W^{1,r}$ with compact support and such that $\bar \partial u=ω.$ In the case of Kähler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar \partial $ equation is done by a classical theorem stating that the …
We construct a simply-connected compact complex non-Kähler manifold satisfying the $\partial\bar\partial$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $\partial\bar\partial$-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in \cite{rao-yang-yang, yang-yang, stelzig-blowup, stelzig-doublecomplex} with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology…
We characterize the boundaries of positive holomorphic chains (with both compact and non-compact support) in an arbitrary complex manifold. We then consider a compact oriented real submanifold of dimension 2p-1 in a compact Kahler manifold X and address the question of which relative homology classes in H_{2p}(X,M;Z) are represented by positive holomorphic chains. Specifically, we define what it means for a class u in H_{2p}(X,M;Z) to be of type (p,p) and positive. It is then shown that u has these properties if and only if u = [T+S] where T is a positive holomorphic chain with dT = du and S i…
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。