Proof of Chern conjecture for flat affine manifolds
Mihail Cocos
本文通过变形论证证明了闭合平坦仿射流形的欧拉特征数为零，验证了Chern猜想。
我们证明了Chern猜想，该猜想指出闭合的平坦仿射流形的欧拉特征数为零。我们的关键创新是一个关于欧拉形式的变形论证。
We prove Chern conjecture, which states that the Euler characteristic vanishes for closed flat affine manifolds. Our key innovation is a deformation argument for the Euler form.
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Corrigendum to "Chern connection of a pseudo-Finsler metric as a family of affine connections"
本文给出了[2，命题3.3和定理3.4]的正确表述，并给出了用仿射联络$\nabla^V$的曲率张量$R^V$和陈氏张量$P$表示的陈氏曲率的一个公式.
We prove an old conjecture of S. S. Chern that the Euler characteristic of a closed affine manifold equals to zero.
Chern connection of a pseudo-Finsler metric as a family of affine connections
我们考虑（圆锥）伪Finsler流形$(M,L)$的Chern联络为任意开子集$Ω\subset M$上的线性联络$\nabla^V$与$Ω$上任意处处非零的向量场$V$相关联.这种联系是无挠的，几乎与基本张量$g$度量相容。然后，我们给出了与$\nabla^V$相关的曲率张量$R^V$的一些性质，特别是证明了$R^V$沿测地线的Jacobi算子与由陈曲率给出的Jacobi算子一致.
Quasi-Kähler manifolds with trivial Chern Holonomy
本文研究了具有拟Kähler Chern平坦度量的殆复流形（Chern平坦度量是指Chern联络的完整性是平凡的）.我们证明了在紧的情况下，这样的流形是所有的nilmanifold。一些部分分类结果的建立和我们证明了一个拟Kähler陈平坦结构可以驯服的辛形式当且仅当周围空间是同构的平坦环面。
Projectively flat and affinely flat parabolic subgroups of special linear groups
实数域和四元数域上的一种特殊线性李群存在一个射影平坦的仿射联络。证明了抛物子群是自平行子流形，并给出了诱导联络与平坦仿射联络射影等价的一个判据。
Notes on Chern's Affine Bernstein Conjecture
完备仿射极大曲面有两个著名的定理，一个是E.卡拉比，另一个给党卫军。陈大约十年前，通过研究相关的欧拉-拉格朗日方程，用不同的方法解决了这两个问题。本文综述了陈省身猜想的两个证明，特别是补充了文[L-X-S-J]中遗漏的辅助材料的证明细节。我们在引言中描述了相关背景。我们的调查是合适的，作为一个报告，最近的发展和技术，在某些蒙日-安培方程的研究。
This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. These groups could play an essential role in the study of homogeneous spaces $M=G/H$ admitting flat affine or flat projective structures invariant under the natural action of $G$ on $M$. A. Medina asked several years ago if the group of affine transformations of a flat affine Lie g…
A Kümmer construction for Chern-Ricci flat balanced manifolds
本文针对带有孤立奇点和Chern-Ricci平坦平衡度量的非Kähler Calabi-Yau轨形，通过粘合构造证明其所有crepant分解均存在Chern-Ricci平坦平衡度量，并探讨了在寻找Hull-Strominger系统解及奇异三维簇中的应用。
Holomorphic functions on complete Hermitian manifolds with flat Chern connection, II
本文研究完备Chern-flat流形上的全纯函数，证明其椭圆性、在挠率多项式增长下双全纯等价于复Lie群，并给出多项式增长全纯函数的尖锐维数估计及与复欧氏空间的双全纯等价条件，推广了先前结果。
Quasi-Kähler Chern-flat manifolds and complex 2-step nilpotent Lie algebras
拟Kähler Chern平坦几乎Hermitian流形的研究与反双不变几乎复李代数的研究密切相关。本文证明了紧致流形上的拟Kähler Chern平坦几乎Hermite结构对应于满足第二Gray恒等式的复可平行Hermite结构。从代数的角度来看，这种对应读起来像是李代数上的反双不变殆复结构与双不变复结构之间的自然对应。一些自然的代数问题进行了探讨，并仔细描述了一些异国情调的例子。
A characterization of hyperbolic affine flat, affine minimal surfaces in $\mathbb{A}^3$
研究等仿射空间$\mathbb{A}^3$中双曲仿射平坦、仿射极小曲面的几何性质。我们使用Cartan的方法的活动标架计算一套完整的局部不变量，这样的表面。利用这些不变量，我们给出了一个完整的局部分类，这样的表面和构造新的例子。
In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.
In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transfor…
Blowing up Chern-Ricci flat balanced metrics
本文将 Arezzo-Pacard 构造推广到平衡情形，证明陈-里奇平坦平衡轨形在光滑点爆破上存在常陈数量曲率平衡度量，并证明具有 crepant 消解的孤立奇点轨形总带有陈-里奇平坦平衡度量。
Solitons and affine projectively flat surfaces
本文的目的是给出仿射曲面的局部描述，其诱导Blaschke结构是射影平坦的。证明了具有常高斯仿射曲率和不定诱导Blaschke度量的仿射曲面可用孤子方程描述。
Chern flat manifolds that are torsion-critical
本文研究紧陈平坦流形上的挠率临界埃尔米特度量，证明陈平坦度量为挠率临界当且仅当对应的复李群是半单的，完善了挠率临界度量的结构刻画。
This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connection on a Lie group. In the last case, we show the existence of a Lie group endowed with a flat affine bi-invariant connection whose Lie algebra contains the Lie algebra of complete infinitesimal affine transformations of the given Lie group. We also prove some results about flat a…
Conformally flat affine hypersurfaces with semi-parallel cubic form
本文研究了具有消失 Weyl 曲率张量和半平行三次形式的局部强凸仿射超曲面，分类了仿射度量非平坦的情形，并完全确定了 2、3 维情形。
Chern-Simons field theory on the general affine group, $3d$-gravity and the extension of Cartan connections
该研究从几何力学角度探究结构群为一般仿射群时3d引力与Chern-Simons场论的对应，通过推广Cartan联络下的联络延拓约化概念，建立了主纤维丛上Chern-Simons型变分问题与常规理论的关联。
Scalar Flat Kähler Metrics on Affine Bundles over $\mathbb{CP}^1$
证明了$\mathbb{CP}^1$上任意负度仿射$\mathbb{C}$ -丛的全空间都存在ALE标量平坦Kähler度量.这里仿射丛的度是指丛的自然紧化中无穷远处截面的自相交数的负值，因此对于线丛，它与通常的度概念一致。
We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of aff…
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。
The deformations of flat affine structures on the two-torus
定义双环面上平坦仿射结构的变形空间的模问题的群作用是仿射群$\Aff(2)$在$\bbR^2$上的作用。由于这个作用有非紧稳定子$\GL(2,\bbR)$，所以其基础的局部齐次几何是高度非黎曼的。本文描述了双环面上所有平坦仿射结构的形变空间。在这种情况下，有趣的现象出现在变形空间的拓扑中，例如，它是\emph{not}一个T2空间。这与常曲率度量或曲面上的共形结构的情况形成对比，这在经典的泰希米勒理论中遇到。作为我们关于环面上平坦仿射结构的变形空间的主要结果，我们证明了从变形空间到环面的基本群到仿射群的同态的共轭类簇的完整映射是局部同胚。
The Gauss-Bonnet-Chern mass of conformally flat manifolds
本文证明了渐近平坦CF流形及其刚性的正质量定理和最近在\cite{GWW}中引入的Gauss-Bonnet-Chern质量的Penrose型不等式.
We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine Bäcklund theorem. We describe some geometrical conditions which imply the local symmetry of both induced connections. We give also some necessary and sufficient conditions under which the affine fundamental forms are proportional.
The vortex equation on affine manifolds
设M是一个紧连通的特殊仿射流形，它具有仿射Gauduchon度量。证明了由M上的平坦向量丛E和E的平坦非零截面E构成的向量对（E，E）存在涡方程的解当且仅当它是多稳定的.为了证明这一点，我们将Kähler流形上全纯对的降维技术应用于仿射流形上平坦对的情况。
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invariant flat affine connection. We characterize flat pseudo-Riemannian Lie groups. For a flat left-invariant pseudo-metric on a Lie group, we show the equivalence between the completeness of the Levi-Civita connection and unimodularity of the group. We emphasize the case of flat left i…
Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds
平坦伪黎曼流形和平坦仿射流形的理论与李群的准齐次仿射表示密切相关。在这篇文章中，我们展示了这种对应关系的几个方面。在我们介绍的核心是发展的理论的特征类和字符的预齐次仿射表示。我们给出了有关平坦仿射，以及伪黎曼和辛仿射平坦流形的应用。
In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold $(X, ω)$ with the Hermitian metric $ω$ satisfying the Gauduchon and Astheno-Kähler conditions. We prove that numerically flatness is equivalent to numerically effectiveness with vanishing first Chern number, semistablity with vanishing first and second Chern numbers, approximate Hermitian flatness and the existence of a filtration whose quotients are Hermitian flat. This gives an affirmative answer to the question proposed by Demailly, Peternell and Schneider.
In this paper we introduce a family of center of masses that complement the definition of the family of Gauss-Bonnet-Chern masses by Ge-Wang-Wu and Li-Nguyen. In order to prove the existence and the well-definedness of the center of mass, we use the formalism of double forms of Kulkarni and Labbi. This allows for transparent conceptual proofs, which apply to all known cases of asymptotic invariants of asymptotically flat manifolds.
In a previous article, we generalised the classical four-dimensional Chern-Gauss-Bonnet formula to a class of manifolds with finitely many conformally flat ends and singular points, in particular obtaining the first such formula in a dimension higher than two which allows the underlying manifold to have isolated conical singularities. In the present article, we extend this result to all even dimensions $n\geq 4$ in the case of a class of conformally flat manifolds.
We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a flat affine symplectic Lie group with bi-invariant symplectic connection contains a nontrivial one parameter subgroup formed by central translations. We give two methods for constructing flat affine symplectic Lie groups, thus obtaining all those having bi-invariant symplectic co…
Cohomology of flat bundles and a Chern-Simons functional
证明了具有紧连通结构群的平坦主丛及其李群的伴随丛与平凡丛具有相同的上同调，这是通过证明它们满足Leray-Hirsch定理的条件来实现的.这一信息已被用来构造一个平坦丛的伴随丛的上同调类，其对每个纤维的限制是Maurer-Cartan 3-形式的类。然后，我们使用这个结果来定义一个不变量的规范变换的平坦丛描述的效果的规范变换上的Chern-Simons功能。
Singular Semi-Flat Calabi-Yau Metrics on S^2
在具有坐标x^j和凸局部势函数f的仿射平坦流形上，如果仿射Kahler度量f_{ij} dx^i dx^j满足det f_{ij} = 1，则称其为半平坦的Calabi-Yau。最近Gross-Wilson在S^2 - 24奇点上构造了许多这样的度量，作为椭圆K3曲面上Calabi-Yau度量的退化极限。我们在S^2上构造更多这样的度量，在任意6个或更多点处是奇异的，并计算奇点附近的局部仿射结构。这些技巧涉及仿射微分几何和求解S^2负奇点上的半线性偏微分方程。我们还计算了镜像对称对结果曲面的作用。
New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry
本文从子流形理论与Calabi仿射几何视角研究仿射极大型曲面，刻画了已有显式欧氏完备反例的几何特征，并构造新的非二次欧氏完备仿射极大型超曲面，拓展了对应参数的取值范围。
Let $\mathcal{M}$ be a Type $\mathcal{A}$ affine surface. We show that $\mathcal{M}$ is linearly strongly projectively flat. We use the quasi-Einstein equation together with the condition that $\mathcal{M}$ is strongly projectively flat to examine to examine the geodesic completeness of $\mathcal{M}$.
Topological and affine structure of complete flat manifolds
本文的结果涉及具有循环和乐群的完备黎曼流形和低维可定向完备平坦流形的拓扑结构。我们还讨论了相关的结果，如可定向完备平坦4-流形的仿射分类，仿射等价的代数判据，完整同态与一些代数和几何不变量之间的关系。
Chern-Ricci flat balanced metrics on small resolutions of Calabi-Yau threefolds
本文通过粘合构造证明，射影节点 Kähler Calabi-Yau 三流形的所有小分解均容许 Chern-Ricci 平衡度量，从而解决了 Hull-Strominger 系统的伸缩子方程。
Low dimensional Lie groups admitting left invariant flat projective or affine structures
证明了\leq 5维的实李群存在一个左不变平坦射影结构.证明了\leq 5维的实李群L有左不变平坦仿射结构当且仅当L的李代数不完备.
Chern-Ricci flow and t-Gauduchon Ricci-flat condition
本文研究Chern-Ricci流下的t-Gauduchon Ricci平坦条件，提供了不保持该条件的例子、非平凡新解及几何流的Gromov-Hausdorff极限行为，明确描述了特定主丛上pluriclosed流的极限。
We give an explicit formula for the Gauss-Bonnet-Chern mass of an asymptotically flat graphical manifold of arbitrary codimension and use it to prove the positive mass theorem and the Penrose inequality for graphs with flat normal bundle.
Families of singular Chern-Ricci flat metrics
本文在一族 hermitian 簇上证明了退化复 Monge-Ampère 方程的一致先验估计，推广了已有定理，并应用于研究 conifold 转变中 Chern-Ricci 平坦势的一致有界性。
We define the notion of spectral network on manifolds of dimension $\le 3$. For a manifold $X$ equipped with a spectral network, we construct equivalences between Chern-Simons invariants of flat ${\mathrm {SL}}(2,{\mathbb C})$-bundles over $X$ and Chern-Simons invariants of flat ${\mathbb C}^\times$-bundles over ramified double covers $\widetilde X$. Applications include a new viewpoint on dilogarithmic formulas for Chern-Simons invariants of flat ${\mathrm {SL}}(2,{\mathbb C})$-bundles over triangulated 3-manifolds, and an explicit description of Chern-Simons lines of flat ${\mathrm {SL}}(2,{…
Hermitian-Einstein connections on principal bundles over flat affine manifolds
设$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。
Une caractérisation des variétés complexes compactes parallélisables admettant des structures affines
我们对复紧致可平行化流形进行了分类，它允许平坦无无挠的全纯仿射联络。我们展示了复杂的紧致流形承认全纯仿射联络，但没有平坦无挠的全纯仿射联络。
A Rigidity Theorem for Affine Kähler-Ricci Flat Graph
证明了$$\det(\frac{\partial^{2}u}{\partial ξ_{i}\partial ξ_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial ξ_{i}} - d_0\right\},$$的任何光滑严格凸整体解，其中$d_0$，$d_1$，.，$d_n$是常数，必须是二次多项式。这扩展了著名的Jörgens-Calabi-Pogorelov定理。
Flat extensions of principal connections and the Chern-Simons $3$-form
本文研究主连接的平坦扩展与陈-西蒙斯3-形式的关系，揭示了3流形浸入欧几里得空间的障碍条件。
The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field $K$ and a torsion-free subgroup $U$ in the group of units of the ring of integers of $K$, with rank of $U$ equal to the number of real embeddings of $K$. We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all non-trivial elements in $U$ are primitive in $K$, then $X$ contains no proper c…
These are lecture notes prepared for the summer school "Geometric, algebraic and topological methods in quantum field theory", held in Villa de Leyva in July 2017. Our goal is to provide an introduction to a conjecture of Chern that states that the Euler characteristic of a closed affine manifold vanishes. We present part of the history and motivation for the conjecture as well as some recent developments. All comments and corrections are most welcome!
Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. In this paper, we study the existence of a complete scalar-flat Kähler metric on $X \setminus D$ on the assumption that $D$ has a constant positive scalar curvature Kähler metric.
Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. In this paper, we show the existence of a complete Kähler metric on $X \setminus D$ whose scalar curvature is flat away from some divisor if there are positive integers $l(>n),m$ such that the line bundle $K_{X}^{-l} \otimes L_{X}^{m}$ is very ample and the ratio $m/l$ is sufficiently small.
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