Existence of approximate Hermitian-Einstein structures on semistable principal bundles
Indranil Biswas, Adam Jacob, Matthias Stemmler
设E_G是紧致连通Kähler流形上的主G-丛，其中G是连通约化复线性代数群.我们证明了E_G是半稳定的当且仅当它包含近似的Hermitian-Einstein结构。
Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.
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We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.
On semistable principal bundles over a complex projective manifold
设G是定义在复数上的单线性代数群。设G的一个真抛物子群P和P的一个非平凡反占优特征标χ，证明了连通复射影流形M上的全纯主G-丛E是半稳定的，其伴随丛的第二类Chern类在有理上同调中为零当且仅当由χ定义的E/P上的线丛是数值有效的.类似的结果对于以约化线性代数群为结构群的主丛仍然有效。这些推广了Y. Miyaoka在那里给出了光滑射影曲线上半稳定向量丛的特征。利用这些特征也可以产生类似的标准抛物主丛的半稳定性在一个紧凑的黎曼曲面。
Approximate Hermitian-Yang-Mills structures on semistable Higgs bundles
本文综述了Higgs包的（弱）Hermitian-Yang-Mills结构和近似Hermitian-Yang-Mills结构的概念。然后，我们构造了紧致Kähler流形上Higgs丛的Donaldson泛函，并给出了它的一些基本性质，特别地，我们证明了它的梯度流可以用Hitchin-Simpson联络的中曲率表示.我们还研究了与该泛函相关的发展方程的一些性质。其次，我们研究了近似Hermitian-Yang-Mills结构的存在性问题及其与Mumford-Takemoto半稳定性的代数几何概念的关系，并证明了对于紧致Riemann曲面上的Higgs丛，近似Hermitian-Yang- Mills结构的概念实际上是半稳定性概念的微分几何对应物。最后，使用唐纳森热流，我们表明，在一个紧凑的Kähler流形（每一个维度）的希格斯丛的半稳定性意味着存在一个近似的厄米-杨-米尔斯结构。作为一个结果，我们推断，许多结果写在近似的厄米-杨-米尔斯结构的希格斯包可以翻译的半稳定性。
Principal bundles on projective varieties and the Donaldson-Uhlenbeck compactification
设$H$是半单代数群.证明了定义在域$\bc$上的{ \it光滑射影簇$X$ }上的$μ$--半稳定主$H$ --丛的半稳定约化定理.当$X$是{ \it光滑射影曲面}，$H$是单曲面时，我们构造了$μ$ -半稳定主$H$-丛的模空间的代数几何Donaldson-Uhlenbeck紧化，并刻画了它的点.对于大的特征类，我们证明了$μ$ --稳定主$H$ --丛的模空间是非空的.
A recent paper (arxiv.org:1810.00025) studied properties of a compactification of the moduli space of irreducible Hermitian-Yang-Mills connections on a hermitian bundle over a projective algebraic manifold. In this follow-up note, we show that the Yang-Mills flow at infinity on the space of semistable integrable connections defines a continuous map to the set of ideal connections used to define this compactification. Part of the proof involves a comparison between the topologies of the Grothendieck Quot scheme and the space of smooth connections.
Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures
我们研究了紧致Kähler流形上Higgs层的基本性质，并建立了一些关于半稳定性概念的结果;特别地，我们证明了具有相等斜率的半稳定Higgs层的任何扩张都是半稳定的。然后，我们使用压扁定理来构造一个正则化的任何扭转免费的希格斯层，我们表明，它实际上是一个希格斯丛。利用这一点，我们证明了任何厄米度量的正则化的扭转免费希格斯层诱导一个容许结构的希格斯层。最后，利用容许结构证明了半稳定Higgs层的一些性质。
In this note, by using the Yang-Mills-Higgs flow, we show that semistable Higgs bundles with vanishing the first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles.
In this paper, we consider the existence of approximate Hermitian-Einstein structure and the semi-stability on Higgs bundles over compact Gauduchon manifolds. By using the continuity method, we show that they are equivalent.
In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves…
Hermitian-Einstein connections on polystable parabolic principal Higgs bundles
给定光滑复射影簇X和X上的光滑因子D，证明了在D上具有抛物结构的多稳抛物主Higgs丛上，关于X - D上的Poincaré型度量的Hermitian-Einstein联络的存在性，并满足对D的限制条件.
Pseudo-real principal Higgs bundles on compact Kaehler manifolds
设$X$是一个紧连通Kähler流形，它具有一个与Kähler结构相容的反全纯对合.设$G$是一个具有实形式$σ_G$的连通复约化仿射代数群.我们定义伪真正的主要$G$-束上$X$，这些是真正的代数主要$G$ -束上的一个真正的代数簇的推广。接下来，我们定义稳定，半稳定和多稳定的伪实主$G$ -丛。研究了它们与通常的稳定、半稳定和多稳定主$G$ -丛的关系。然后证明了如下的Donaldson-Uhlenbeck-Yau型对应成立：一个伪实主$G$ -丛允许相容的Einstein-Hermitian联络当且仅当它是多稳定的.建立了以下两个集合之间的一个双射：1）多稳定伪实主$G$ -丛的同构类使得其基础拓扑主$G$ -丛的所有有理特征类为零. 2)在$G$的$σ_G$-不变极大紧子群中$X$的扩张基本群的扭表示的等价类. (The在伪实主$G$ -丛的定义中使用中心元素来定义扭曲表示。这些结果也推广到了赝实Higgs $G$ -丛。
Hermitian-Einstein connections on principal bundles over flat affine manifolds
设$M$是一个具有Gauduchon度量$g$和协变常数体积形式的无边界紧致连通特殊平坦仿射流形.设$G$是一个连通的还原复线性代数群或复还原群的分裂实型的实轨迹。证明了$M$上的平坦主$G$丛$E_G$具有Hermitian-Einstein结构当且仅当$E_G$是多稳定的. $M$上的多稳平坦主$G$-丛存在唯一的Hermitian-Einstein联络.我们还证明了$M$上平坦向量丛的Harder-Narasimhan滤子的存在唯一性。
Admissible Hermitian-Yang-Mills connections over normal varieties
本文在正规簇上证明完整版 Donaldson-Uhlenbeck-Yau 定理，建立 Hitchin-Kobayashi 对应，并由此得到 Bogomolov-Gieseker 不等式及其等号刻画，还给出平凡第一 Chern 类正规 Kaehler 簇为环面有限商的新判据。
Hermitian Yang-Mills connections on pullback bundles
研究全纯浸没全空间上拉回丛的埃尔米特-杨-米尔斯联络，给出严格半稳定向量丛拉回承载该联络的充要判据，结合Donaldson-Uhlenbeck-Yau对应证明稳定/不稳定丛拉回保持原性质并解决半稳定情形。
In this paper, we provide a systematic and constructive description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds. From this description we explicitly classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds using elements of representation theory of complex simple Lie algebras. Moreover, we also describe using Lie theory all homogeneous solutions of the Hermitian-Einstein-Weyl equation on compact homogeneous Hermitian-Weyl manifolds. As an application, we provide a huge class of explicit (nontrivial) examples of such structure…
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is described in a systematic way.
Principal bundles over a real algebraic curve
设X是一个紧连通Riemann曲面，它具有一个反全纯对合σ.设G是连通复约化仿射代数群，σ G是G的实形式.考虑X上的全纯主G-丛满足关于σ和σ_G的相容性条件。在{ \rm亏格}（X）> 2的假设下，证明了X上主G-丛模空间的光滑轨迹的$\mathbb R$上定义的点正是这些对象。稳定、半稳定和多稳定丛都是在这种情况下定义的。探讨了这些性质与相应的全纯主G-丛的性质之间的关系。建立了酉表示与多稳定对象之间的双射对应。
Algebraic limits of Hermitian-Yang-Mills connections on Kähler surfaces
本文提出几何方法统一执行 Hörmander $L^2$ 估计，证明 Kähler 曲面上 HYM 联络的 Uhlenbeck 收敛可实现为代数收敛，并应用于连续性定理与半稳定层有界性。
In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal $S^{1}$-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order to describe and classify these structures. We use these normal almost contact structures to explicitly describe a huge class of compact Hermitian non-Kähler manifolds obtained from products of principal $S^{1}$-bundles over complex flag manifolds. Moreover, we obtain from our de…
We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.
Let $G$ be a connected reductive complex affine algebraic group and $K\subset G$ a maximal compact subgroup. Let $M$ be a compact complex torus equipped with a flat Kähler structure and $(E_G ,θ)$ a polystable Higgs $G$-bundle on $M$. Take any $C^\infty$ reduction of structure group $E_K \subset E_G$ to the subgroup $K$ that solves the Yang--Mills equation for $(E_G ,θ)$. We prove that the principal $G$-bundle $E_G$ is polystable and the above reduction $E_K$ solves the Einstein--Hermitian equation for $E_G$. We also prove that for a semistable (respectively, polystable) Higgs $G$-bundle $(E_G…
In this paper, we study semistable Higgs sheaves over compact Kähler manifolds, we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a semi-stable reflexive Higgs sheaf.
Let $X$ be a compact connected Kähler--Einstein manifold with $c_1(TX)\, \geq\, 0$. If there is a semistable Higgs vector bundle $(E\,,θ)$ on $X$ with $θ\,\not=\,0$, then we show that $c_1(TX)=0$, any $X$ satisfying this condition is called a Calabi--Yau manifold, and it admits a Ricci--flat Kähler form \cite{Ya}. Let $(E\,,θ)$ be a polystable Higgs vector bundle on a compact Ricci--flat Kähler manifold $X$. Let $h$ be an Hermitian structure on $E$ satisfying the Yang--Mills--Higgs equation for $(E\,,θ)$. We prove that $h$ also satisfies the Yang--Mills--Higgs equation for $(E\,,0)$. A similar…
Principal Distribution Isomorphisms and Almost Hermitian geometry on Isoparametric Hypersurfaces
本文研究了OT--FKM类型等参超曲面的主分布同构关系，构造了主分布的显式同构，并利用这些同构诱导近正交结构，证明了相关超曲面的$*$-瑞奇曲率为零。
I consider principal Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced by Bruzzo and Graña Otero. I prove that a principal Higgs bundle $\mathfrak{E}=(E,φ)$ is H-nflat is either stable or there exists a Higgs reduction of $\mathfrak{E}$ to a parabolic subgroup $P$ of $G$ such that the principal $L$-bundle $\mathfrak{E}_L$ obtained by extending the reduced Higgs bundle $\mathfrak{E}_P$ to the Levi factor $L$ is H-nflat and stable; and as consequence, $H^{*}(\mathfrak{E},\mathbb{R})$ is isomorphic to the cohomology ring of the associated graded object $\ma…
Generalized Killing spinors as Lagrangian sections of principal spin bundles
本文建立了三维伪黎曼旋量流形上ε-单位广义Killing旋量与其主旋量丛上Lagrangian截面的一一对应关系。
We examine various examples of horosymmetric manifolds which exhibit interesting properties with respect to canonical metrics. In particular, we determine when the blow-up of a quadric along a linear subquadric admits Kähler-Einstein metrics, providing infinitely many examples of manifolds with no Kähler-Ricci solitons that are not K-semistable. Using a different construction, we provide an infinite family of Fano manifolds with no Kähler-Einstein metrics but which admit coupled Kähler-Einstein metrics. Finally, we elaborate on the relationship between Kähler-Ricci solitons and the more genera…
Stable principal bundles and reduction of structure group
设$E_G$是紧致连通Kaehler流形上的稳定主$G$ -丛，其中$G$是定义在复数上的连通约化线性代数群.设$H\subset G$是一个不一定连通的复约化子群，$E_H\subset E_G$是结构群的全纯约化.证明了$E_G$上的Einstein-Hermitian联络保持$E_H$.利用这一点，我们表明，如果$E_H$是一个最小的还原还原在这个意义上，没有复杂的还原真子群的$H$ $E_H$允许全纯约化的结构群，那么$E_H$是唯一的在以下意义上：对于任何其他最小的还原还原$(H', E_{H'})$的$E_G$，有一些元素$g$的$G$，使$H'= g^{-1}Hg$和$E_{H'}= E_Hg$。作为应用，我们对Balaji和Kollár的一个问题给出了肯定的回答。
Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group
研究紧致Kähler流形上的主$G$ -丛，其中$G$是复代数群，使得它的包含单位元的连通分支是约化的.定义（半）稳定性，这类丛，它表明，一个主$G$ -丛$E_G$承认一个Einstein-Hermitian联系当且仅当$E_G$是多稳定的。我们给出了（半）稳定性条件的等价形式。一个问题是将这个定义与数学的定义进行比较。
$t$-Gauduchon Ricci-flat metrics on non-Kähler Calabi-Yau manifolds
该研究在Picard数大于1的有理齐次簇上的紧非Kähler Calabi-Yau主环面丛中，对所有t<1构造了新的t-Gauduchon Ricci平坦度量，并给出了特定Fano三维簇上主T²-丛的多种Ricci平坦及平衡Hermitian度量新实例。
Hermitian integral geometry
我们给出了仿射酉群在$\C^n$上作用的主要运动学公式的显式形式，以及计算酉运动学公式的全部数组的一个简单的代数方法，表示为某些凸赋值，本质上，由H。田崎我们还介绍了其他几个典型的基础代数的酉不变的估值，探讨它们之间的相互关系，并在这些方面的特点锥的积极和单调的元素。
We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb C\mathbb P^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman, Gauduchon. Furthermore, we construct new examples …
We introduce a class of $G$-invariant connections on a homogeneous principal bundle $Q$ over a hermitian symmetric space $M=G/K$. The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution. The fixed points of the anti-holomorphic involution are precisely the integrable invariant complex structures on $Q.$ This normal variety is closely related to quiver varieties and, more generally, to varieties of commuting matrix tuples modulo simultaneous conjugation.
Let $(M^n, h)$ be a compact Hermitian manifold. Suppose $λ$ is the lowest eigenvalue of the complex Laplacian on $M$. We prove that $λ\geq C$ where $C$ depends only on the dimension $n$, the diameter $d$, the Ricci curvature of the Levi-Civita connection on $M$, and a norm, expressed in curvature, that determines how much $M$ fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng. We combine these results to obtain the main estimate.
Yang-Mills Connections On Orientable and Nonorientable Surfaces
在数学.SG/0605587中，我们研究了闭、连通、不可定向曲面上主G_R-丛的联络空间上的Yang-Mills泛函，其中G_R是任意紧连通李群.本文推广了Atiyah和Bott在“Riemann曲面上的Yang-Mills方程”中的讨论，并对数学进行了讨论。对于非酉典型群SO（n）和Sp（n），得到了Yang-Mills泛函在可定向曲面和不可定向曲面上Morse层的表示簇.事实证明，它与单一情况有很大不同。利用Laumon和Rapoport在“The Langlands lemma and the贝蒂数of stacks of G-bundles on a curve”中的方法，反演Atiyah-Bott递推关系，给出了全纯结构空间的半稳定层在主$SO(n,\bC)$ -丛或主$Sp(n,\bC)$ -丛上的有理等变Poincaré级数的显式表达式.
Chern correspondence for higher principal bundles
本文研究高阶 gauge 理论中的 Chern 对应关系，通过定义乘法上同调丛的连接结构，提出 2-群的复化概念，建立新的高阶连接概念，统一了文献中两种高阶连接定义。
Invariant operators on manifolds with almost Hermitian symmetric structures, II. Normal Cartan connections
在本系列论文的第一部分中，我们发展了关于Cartan连接的不变微分，我们用基本的主连接描述了这一过程，并讨论了这一理论在自然算子构造中的应用。在这一部分中，我们将\cite{Ochiai, 70}关于几乎Hermite对称结构流形上所谓正规Cartan联络的存在性和唯一性的结果推广到不允许挠自由线性联络的一阶结构。此外，对于这些结构中的每一个，我们得到明确的（普遍的）公式，这些典型的连接在底层的线性连接的曲率。
Invariant Complex Structures and Kähler Metrics on Principal Bundles
研究主丛上殆复结构和凯勒度量的存在性与分类，用王的不变联络理论给出不变可积复结构分类的几何证明，用列维 - 奇维塔联络证明凯勒准则，应用结果表明上半平面和复射影空间中不变可积复结构唯一。
Let $M$ be a compact connected complex manifold and $G$ a connected reductive complex affine algebraic group. Let $E_G$ be a holomorphic principal $G$--bundle over $M$ and $T\, \subset\, G$ a torus containing the connected component of the center of $G$. Let $N$ (respectively, $C$) be the normalizer (respectively, centralizer) of $T$ in $G$, and let $W$ be the Weyl group $N/C$ for $T$. We prove that there is a natural bijective correspondence between the following two: Torus subbundles $\mathbb T$ of ${\rm Ad}(E_G)$ such that for some (hence every) $x\, \in\, M$, the fiber ${\mathbb T}_x$ lies…
Let $M$ be a compact complex manifold of dimension at least three and $Π: M\rightarrow X$ a positive principal elliptic fibration, where $X$ is a compact Kähler orbifold. Fix a preferred Hermitian metric on $M$. In \cite{V}, the third author proved that every stable vector bundle on $M$ is of the form $L\otimes Π^*B_0$, where $B_0$ is a stable vector bundle on $X$, and $L$ is a holomorphic line bundle on $M$. Here we prove that every stable Higgs bundle on $M$ is of the form $(L\otimes Π^*B_0,Π^*Φ_X)$, where $(B_0, Φ_X)$ is a stable Higgs bundle on $X$ and $L$ is a holomorphic line bundle on $…
We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal b…
Geometry and symmetries of Hermitian-Einstein and instanton connection moduli spaces
本文研究Kähler带挠率流形上Hermitian-Einstein联络模空间及瞬子联络模空间的几何结构，证明特殊向量场诱导保持度量和复结构的作用，并将模空间几何建模为全环面主丛，同时探讨相关二维sigma模型的对称代数。
In this paper, we introduce the associated geodesic-Einstein flow for a relatively ample line bundle $L$ over the total space $\mathcal{X}$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $(\mathcal{X}, L)$ is nonlinear semistable if the {associated} Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time {existence property}. We also define the associated $S$-classes and $C$-classes for $(\mathcal{X}, L)$ and obtain two inequalities between them when $L$ admits a geodesic-Einstein metric. Finally, …
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