Changpeng Pan, Zhenghan Shen, Xi Zhang
In this paper, we study the non-Hermitian Yang-Mills (NHYM for short) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in \cite{KV}.
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In this paper, we study the hypercritical deformed Hermitian-Yang-Mills equation on compact Kähler manifolds and resolve two conjectures of Collins-Yau.
In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kähler manifold $(X,g)$ with the Hermitian metric $g$ satisfying the Gauduchon and Astheno-Kähler condition.
We construct local examples of singular Hermitian Yang-Mills connections over $B_1\subset \mathbb{C}^3$ with uniformly bounded $L^2$-energy, but the number of essential singular points can be arbitrarily large.
In this work, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold. We introduce the notions of coerciveness and properness of the $\mathcal{J}$-functional on the space of almost calibrated $(1,1)$-forms and show that they are both equivalent to the existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.
In this paper, we investigate the Chern number inequalities on $4$-dimensional Kähler manifolds admitting the deformed Hermitian-Yang-Mills metrics under the assumption $\hatθ\in (π,2π)$.
A note on curvature estimate of the Hermitian-Yang-Mills flow
本文研究了全纯向量丛上Hermitian-Yang-Mills流的曲率估计。在一个简单的情况下，我们证明了演化厄米特度量的曲率是一致有界远离由Harder-Narasimhan-Seshadri过滤的全纯向量丛的解析子簇。
The purpose of this paper is to investigate the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase in a smooth domain $Ω\subset \mathbb{C}^{n}$. By using $J$-functional, we are able to prove the convergence of solutions. As an application, we give an alternative proof of the Dirichlet problem for deformed Hermitian-Yang-Mills equation.
Geometry of non-Hermitian Yang--Mills moduli spaces
研究紧致Kähler流形上非厄米杨-米尔斯联络的模空间，利用归一化调和度量构造光滑轨迹上的自然Hermitian度量，并证明在Hermitian轨迹附近光滑轨迹具有与相关Riemann度量相容的几乎超复结构。
The limit of the Hermitian-Yang-Mills flow on reflexive sheaves
本文研究了自反层上Hermitian-Yang-Mills流的渐近行为。证明了极限自反层同构于Harder-Narasimhan-Seshadri滤子分次层的双重对偶，从而回答了Bando和Siu的一个问题.
Tangent cones of Hermitian Yang-Mills connections with isolated singularities
我们给出了一个简单的直接证明的唯一性的奇异射影厄米杨-米尔斯联系的自反层在孤立奇点上的$μ$ -多稳定全纯丛上建模$\mathbf{P}^{n-1}$。
Existence of Hermitian-Yang-Mills metrics under conifold transitions
我们首先研究了Hermitian-Yang-Mills度量序列关于Calabi-Yau三重$\hat{X}$上的平衡度量序列的退化，该平衡度量序列退化为Fu，Li和Yau在$\hat{X}$上的多条（-1，-1）-曲线的补上构造的平衡度量.然后在一定的假设下，证明了三重$X_t$上平凡正则丛的Hermitian-Yang-Mills度量的存在性，其中平凡正则丛是通过对$\hat{X}$进行锥变换得到的.
In this paper, we extend a result of Gao Chen regarding the solvability of the twisted deformed Hermitian Yang-Mills equations on compact Kähler manifolds to allow for the twisting function to be non-constant and slightly negative in all dimensions. Using this result, we prove that the twisted supercritical dHYM equation on compact, projective manifolds can be solved provided certain numerical conditions are satisfied.
In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.
Long-time behavior of the Hermitian-Yang-Mills flow on non-Kähler manifolds
本文研究了非Kähler流形上Hermitian-Yang-Mills流的长期行为，证明了特征值收敛到几何不变量，并将Atiyah-Bott-Bando-Siu问题推广到非Kähler情况。
The moduli space of Hermitian-Yang-Mills connections
本文研究紧致 Kähler 流形上 Hermitian-Yang-Mills 联络的模空间，计算其虚拟维数，并解析证明 Higgs 丛稳定性的开性条件。
In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant $C$ such that $-\frac{1}{C}ω<\sqrt{-1}F<Cω$. We also study the self-shrinker over $\mathbb{C}^n$ to the corresponding parabolic flow. We prove that the self-shrinker over $\mathbb{C}^n$ is a quadratic polynomial function. We also show the similar rigid theorem for the J-equations and the self-shr…
We give an algebraic criterion for the existence of projectively Hermitian-Yang-Mills metrics on a holomorphic vector bundle $E$ over some complete non-compact Kähler manifolds $(X,ω)$, where $X$ is the complement of a divisor in a compact Kähler manifold and we impose some conditions on the cohomology class and the asymptotic behaviour of the Kähler form $ω$. We introduce the notion of stability with respect to a pair of $(1,1)$-classes which generalizes the standard slope stability. We prove that this new stability condition is both sufficient and necessary for the existence of projectively …
Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors I
本文解决了预定的Hermitian-Yang-Mills张量问题，证明了在紧致Kähler流形上的全纯向量丛中，给定正定Hermitian张量时存在唯一的Hermitian度量使其Hermitian-Yang-Mills张量等于该张量，并导出了定量Chern数不等式。
Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors II
本文解决了紧复流形上Higgs丛的预定Hermitian-Yang-Mills张量问题，证明了在初始度量正定条件下，对任意正定张量存在唯一Hermitian度量满足方程，并建立了定量Chern数不等式。
Weak solutions of the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation: boundary cases
研究在可解区域边界上的上同调类中，广义Monge-Ampère方程和超临界变形Hermitian-Yang-Mills方程的弱解存在唯一性，并证明相关几何流在流意义下收敛到弱解，方法结合了粘性理论和多重位势理论。
The Hermitian-Yang-Mills Iteration on Stable Bundles
基于Fan-Wang-Yang-Yau的最新结果，本文提供了稳定全纯向量丛上Hermitian-Einstein度量的动力学构造，并推广到Higgs丛，同时用热流方法给出了扭曲预定HYM张量方程解的存在唯一性新证明。
Weak limits of the $J$-flow and the deformed Hermitian-Yang-Mills flow on Kähler surfaces: boundary cases
研究证明了在Kähler类$J$- nef时，$J$-流在Kähler表面收敛于Monge-Ampère方程的弱解，同时建立了变形辛-Yang-Mills流的相同收敛行为，方法基于粘性子解极限的性质。
Deformed Hermitian-Yang-Mills equation on the manifold of full flags
研究全旗流形上变形厄米-杨-米尔斯方程，构建小半径高阶不可约及大半径无穷多不同丛的dHYM联络示例，研究超临界外解，证明任意角度解存在并排除部分稳定性条件。
In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution.
From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mariño, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibrat…
Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds
我们证明了渐近圆柱Kähler流形的Donaldson-Uhlenbeck-Yau定理的一个类似：如果$\mathscr{E}$是ACyl Kähler流形上的自反层，它渐近于$μ$ -稳定的全纯向量丛，那么它允许一个渐近不变的保护性Hermitian Yang-Mills度量（在奇异集上具有$L^2_{\mathrm{loc}}$中的曲率）。我们的证明结合了Uhlenbeck和Yau [UY 86]的解析连续性方法与Bando和Siu [BS 94]引入的几何正则化方案。
The prescribed Hermitian-Yang-Mills flow I
本文引入一类广泛的流，包括规定的Hermitian-Yang-Mills流，证明其长时间收敛到极限度量，并用于求解规定的Hermitian-Yang-Mills张量方程。
Let $(X, ω)$ be a compact connected Hermitian manifold of dimension $n$. We consider the Bott-Chern cohomology and let $[χ] \in H^{1,1}_{\text{BC}}(X; \mathbb{R})$. We study the deformed Hermitian-Yang-Mills equation, which is the following nonlinear elliptic equation $\sum_{i} \arctan (λ_i) = h(x)$, where $λ_i$ are the eigenvalues of $χ$ with respect to $ω$.
The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact Kähler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen proved that the existence of the solution is equivalent to a uniform stability condition in terms of holomorphic intersection numbers along test families. In this paper, we establish an analogous stability result not involving a uniform constant in accordance with a recent work on the $J$-equation by Song, which makes further progress toward Collins-Jacob-Yau's origina…
Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation
本文是相关研究续篇，针对两个一维复环面乘积上稳定秩二全纯向量丛的近似埃尔米特-杨-米尔斯度量，证明大凯勒极限下该近似度量对精确度量具有任意阶指数精度的渐近描述，克服了全局C^0估计等解析难点。
In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.
This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kähler metrics $ω_ε$ when $ε\to 0$. Here $ω_ε$ are flat and have areas $ε$ and $ε^{-1}$ on the two elliptic curves respectively. A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get t…
Non-Hermitian Yang--Mills Connections with Vanishing Chern Classes
本文在紧致Kähler曲面的光滑平凡二秩丛上构造了非平坦非厄米杨--米尔斯联络，其陈类为零且两侧稳定，表明消失陈类不强制平坦性，并给出模空间收缩与能量估计。
On a Riemannian manifold of dimension $n$ we extend the known analytic results on Yang-Mills connections to the class of connections called $Ω$-Yang-Mills connections, where $Ω$ is a smooth, not necessarily closed, $(n-4)$-form. Special cases include $Ω$-anti-self-dual connections and Hermitian-Yang-Mills connections over general complex manifolds. By a key observation, a weak compactness result is obtained for moduli space of smooth $Ω$-Yang-Mills connections with uniformly $L^2$ bounded curvature, and it can be improved in the case of Hermitian-Yang-Mills connections over general complex man…
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 $X\rightarrow {\mathbb P}^1$ be an elliptically fibered $K3$ surface, admitting a sequence $ω_{i}$ of Ricci-flat metrics collapsing the fibers. Let $V$ be a holomorphic $SU(n)$ bundle over $X$, stable with respect to $ω_i$. Given the corresponding sequence $Ξ_i$ of Hermitian-Yang-Mills connections on $V$, we prove that, if $E$ is a generic fiber, the restricted sequence $Ξ_i|_{E}$ converges to a flat connection $A_0$. Furthermore, if the restriction $V|_E$ is of the form $\oplus_{j=1}^n\mathcal O_E(q_j-0)$ for $n$ distinct points $q_j\in E$, then these points uniquely determine $A_0$.
A note on the supercritical deformed Hermitian-Yang-Mills equation
该研究针对超临界形变Hermitian-Yang-Mills方程，证明紧Kähler流形上有解的实(1,1)类是满足Collins-Jacob-Yau数值条件的类的既开又闭真子集，推翻了Collins-Jacob-Yau的相关猜想。
This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of …
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。
Hermitian Yang-Mills instantons on Calabi-Yau cones
研究并构造了Calabi-Yau锥上的非交换厄米Yang-Mills（HYM）瞬子。通过一种特殊的保等距算子，HYM方程被简化为Einstein-Kahler基上的一种新的Higgs-Yang-Mills流。对于任何二维的卡-丘锥，我们发现明确的解决方案，对应于非平凡的SU（d）HYM瞬子的流动方程。这些可以看作是卡-丘锥自旋联系的变形。
We survey some recent progress on the deformed Hermitian-Yang-Mills (dHYM) equation. We discuss the role of geometric invariant theory (GIT) in approaching the solvability of the dHYM equation, following work of the first author and S.-T. Yau. We compare the GIT picture with the conjectural picture for dHYM involving Bridgeland stability. In particular, following Arcara-Miles, we show that on the blow-up of $\mathbb{P}^2$ any line bundle admitting a solution of the deformed Hermitian-Yang-Mills equation is Bridgeland stable, but not conversely. Finally, we survey some recent progress on heat f…
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct continuous solutions of the associated Dirichlet problem.
We introduce a new system of equations coupling Kähler-Einstein and Hermitian-Yang-Mills equations. We provide a moment map interpretation of these equations. We identify a Futaki type invariant as an obstruction to the existence of solutions to these equations. We also prove a Matsushima-Lichnerowicz type theorem. We prove a deformation result that produces nontrivial solutions of these equations under some conditions. We produce examples on some projective bundles using Calabi ansatz.
The prescribed Hermitian-Yang-Mills flow II
通过规定Hermitian-Yang-Mills流，证明了经典Donaldson-Uhlenbeck-Yau定理的类比，并应用于Fano流形上存在唯一Hermitian度量张量满足给定方程。
Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our proof uses gluing techniques, and is hence asymptotically explict. This recovers, through the Hitchin-Kobayashi correspondence, algebro-geometric results due to Buchdahl and Sibley.
Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles
我们推广的Hitchin-Kobayashi对应半稳定性和存在的近似Hermitian-Yang-Mills结构的情况下，主Higgs丛。证明了紧致Kaehler流形上的主Higgs丛是半稳定的当且仅当它具有近似Hermitian-Yang-Mills结构，其中结构群是连通的线性代数约化群.
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.
Decay estimates for Yang-Mills connections and Yang-Mills-Higgs pairs in higher dimensions
本文通过抽象衰减定理和Kato不等式，证明了高维流形上Yang-Mills联络与Yang-Mills-Higgs对在二次衰减假设下的改进衰减速率。
Stability of the Ricci Yang-Mills flow at Einstein Yang-Mills metrics
设$P$是闭流形$M$上的主U（1）-丛.在$P$上，人们可以定义一个修改版本的里奇流，称为里奇杨-米尔斯流，因为这些方程是里奇流和杨-米尔斯热流的耦合。利用极大正则性理论和Simonett关于抽象拟线性抛物型偏微分方程渐近行为的思想，研究了二维Einstein Yang-Mills度规下体积归一化Ricci Yang-Mills流的稳定性.在某些情况下，我们表明存在一个中心流形的不动点，而在其他情况下，我们表明存在一个渐近稳定的不动点。
$F$-Yang-Mills connections are critical points of $F$-Yang Mills functional on the space of connections of a principal fiber bundle, which is a generalization of Yang-Mills connections, $p$-Yang-Mills connections and exponential Yang-Mills connections and so on. Here, $F$ is a strictly increasing $C^{2}$-function. In this paper, we extend Simons theorem for an instability of Yang-Mills connections to $F$-Yang-Mills connections. We derive a sufficient condition that any non-flat, $F$-Yang-Mills connection over convex hypersurfaces in a Euclidean space is instable. In the sphere case, this condi…
Normal-Yang-Mills and Tangent-Yang-Mills submanifolds
本文研究子流形浸入单位球体时正法和切法曲率张量的L²范数变分问题，定义了正法-杨-米尔斯和切法-杨-米尔斯子流形，并推导了相应的欧拉-拉格朗日方程，构造了非平凡例子。
$J$-equations and deformed Hermitian-Yang-Mills equations on holomorphic submersions
本文研究了全纯投影上$J$-方程和变形的辛-杨-米尔斯方程的解的存在性及其逆向关系。
Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions
本文基于Arcara和Miles的猜想，证明了复平面上的线束在大尺度极限下稳定性与变形Hermite-Yang-Mills度量的存在等价，扩展了Stoppa对紧致托里表面的结果。
Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths-and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled.
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we prove a new Chern number inequality and discuss the relationship with algebraic stability conditions.
Morse Homology for the Yang-Mills Gradient Flow
利用闭Riemann曲面上联络空间上的Yang-Mills梯度流构造了Morse-Bott链复形.链群由Yang-Mills连接生成。边界算子是通过计算渐近收敛于杨-米尔斯联络的杨-米尔斯梯度流线的适当定义的模空间的元素来定义的。
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。
Blowing-up Hermitian Yang--Mills connections
本文研究Kähler流形爆破后拉回向量丛的埃尔米特杨-米尔斯联络，在温和假设下给出了该丛存在相应联络序列的充要数值判据，并证明联络收敛于分次对象的拉回联络。
We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kahler manifolds.
A deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. The dDT connection is an analogue of a deformed Hermitian Yang-Mills (dHYM) connection which is extensively studied recently. In this paper, we study the moduli spaces of dDT and dHYM connections. In the former half, we prove that the deformation of dDT connections is controlled by a subcomplex of the canonical complex, an elliptic complex …
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