S. A. Merkulov
This is an expanded comment on Barannikov's paper math.AG/0006193. A symplectic version of his construction is discussed. It is shown that the duality transformation for mirror torus fibrations over the same Monge-Ampere manifold exchanges semi-infinite A-variations of Hodge structure introduced in this paper with Barannikov's semi-infinite B-variations of Hodge structure.
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Sections of Hodge bundles II: Deformation of $(p,p)$-classes and applications to Kähler geometry
本文通过Hodge包的截面研究Kähler流形的(p,p)-类变形，提出Kähler锥的上半连续性质及Hodge领域的内在描述。
We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.
In this note, we show that the cohomology groups of (virtually) nilpotent Kähler groups are naturally endowed with a mixed Hodge structure. These structures make the Hopf morphisms into mixed Hodge structures morphisms. We illustrate this fact with the study of known examples of non-abelian nilpotent Kähler groups.
We construct a Mixed Hodge Structure on the local complete ring of the representation scheme at the holonomy of a VHS on a compact Kähler manifold and prove that the corresponding tautological representation is the holonomy of a VMHS. In order to carry out this construction we revisit the well-known work of Goldman and Millson on the deformation theory of representations of Kähler groups.
Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic
本文证明K3[n]型紧致超Kähler流形间第二有理上同调的Hodge等距可由解析对应诱导并提升为保持Mukai配对与分次的总上同调对应，且在射影情形下为代数对应。
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。
A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact Kähler manifolds. Several isomorphisms between different cohomologies given.
The aim of this article is to study degeneration of the variations of Hodge structure associated to a proper Kähler semistable morphism. We prove that the weight filtrations constructed in the author's previous paper coincide with the monodromy weight filtrations on the relative log de Rham cohomology groups of a proper Kähler semistable morphism. Moreover, we show that the limiting mixed Hodge structures form admissible variations of mixed Hodge structure.
Hodge structures of the surface of planes in a cubic 5-fold
本文研究一般三次五重簇中平面模空间的几何与形变，证明其为典范丛丰沛的光滑射影曲面，且特定族的1次Hodge结构变分极大，核心技术为hyper-Kähler几何与Hodge数精细计算。
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…
On the Kähler cone of irreducible symplectic orbifolds
本文在不可约辛轨形上推广了全局Torelli定理及Kähler锥相关结果，并新定义了镜像对称作为模空间上的对合。
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。
We investigate the relation between the Hodge theory of a smooth subcanonical $n$-dimensional projective variety $X$ and the deformation theory of the affine cone $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a distinguished graded component of the module of first order deformations of $A_X$, and later on we show how to identify the whole primitive cohomology of $X$ as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over $X$. In the particular case of a projective smooth hypersurface $X$ we recover Griffiths' isomorp…
Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over…
The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness 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。
Comparing Kahler cone and symplectic cone of one-point blowup of Enriques surface
本文研究Enriques曲面单点爆破的Kähler锥与辛锥的差异，证明其存在非Kähler辛形式，通过代数几何不变量定量比较两者区别，推进了$p_g=0$的Kähler曲面上辛通用复结构相关问题的研究。
We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that can be used to decode the variational properties of the polarized Hodge-Lefschetz module structure on the fibres of our Higgs bundle. Thus we can use a generalized version of Lu's Hodge metric to study the curvature property of the complexified Kähler cone. In particular, it implie…
We treat two quite different problems related to changes of complex structures on Kähler manifolds by using global geometric method. First, by using operators from Hodge theory on compact Kähler manifold, we present a closed explicit extension formula for holomorphic canonical forms in different complex structures. As applications, we give a closed explicit formula for certain canonical sections of Hodge bundles on marked and polarized moduli spaces of projective manifolds, and provide a closed explicit extension formula for holomorphic pluricanonical forms under certain natural conditions. Se…
Assuming suitable convergence properties for the Gromov-Witten potential of a Calabi-Yau manifold $X$ one may construct a polarized variation of Hodge structure over the complexified Kähler cone of $X$. In this paper we show that, in the case of fourfolds, there is a correspondence between ``quantum potentials'' and polarized variations of Hodge structures that degenerate to a maximally unipotent boundary point. Under this correspondence, the WDVV equations are seen to be equivalent to the Griffiths' trasversality property of a variation of Hodge structure.
This is a technical introduction to the paper "Extension of twisted Hodge metrics for Kahler morphisms" by the authors.
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。
For several Hodge-type Shimura varieties of good reduction in characteristic $p$, we show that the cone of weights of automorphic forms is encoded by the stack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instances of a general conjecture formulated in previous papers by the authors. Furthermore, we prove in these cases that any mod $p$ automorphic form whose weight lies in a specific region of the weight space is divisible by a partial Hasse invariant. This generalizes to other Shimura varieties previous results of Diamond--Kassaei on Hilbert modular forms.
Mixed Hodge formality
研究引入混合霍奇形式性概念，通过发展障碍理论，来理解复代数簇上同调的混合霍奇结构，进而从混合霍奇意义上剖析某些紧致凯勒流形的非形式性。
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.
Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds
本文构造 Sasakian 流形对应的李超代数，并据此无坐标证明其与 Vaisman 流形上调和形式及上同调的 Hodge 分解结果，还显式计算了超对称代数。
This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.
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…
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。
Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformations of Kähler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian-Einstein metric on this bun…
This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2π. We also put all our technical results together to complete the proof of the main theorem that if a K-stable Fano manifold admits a Kahler-Einstein metric.
Given a hyperplane arrangement of some type in a projective space, the Dunkl system, developed by Couwenberg, Heckman and Looijenga, is used to study the geometric structures on its complement, and as a consequence it leads to the discovery of new ball quotients when the Schwarz conditions are imposed. In this paper, we study how the space, investigated in this system, looks like when there is no Schwarz conditions imposed. As a result, we show that the space in question is still of a particular type of structure, namely, the structure of a cone-manifold. With this point of view, Thurston's wo…
This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2π. We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.
We describe two approaches to classifying the possible monodromy cones C arising from nilpotent orbits in Hodge theory. The first is based upon the observation that C is contained in the open orbit of any interior point N in C under an associated Levi subgroup determined by the limit mixed Hodge structure. The possible relations between the interior of C its faces are described in terms of signed Young diagrams. The second approach is to understand the Tannakian category of nilpotent orbits via a category D introduced by Deligne in a letter to Cattani and Kaplan. In analogy with Hodge theory, …
Apollonian carpets and the boundary of the Kahler cone of a hyperkahler manifold
该研究探讨超Kähler流形丰沛锥的迷向边界与全纯自同构群动力学的关系，证明迷向边界上无理点均属极限集，实解析曲线含于测地球面，还研究了Apollonian地毯的分形几何及其与自同构群的关联。
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.
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