Sebastien Gontard
We study the complete Kahler-Einstein metric in tube domains. We obtain estimates of this metric and its holomorphic bisectional curvatures near the weakly pseudoconvex boundary points.
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A formula of the renormalized volume of tubes over polalized Kähler-Einstein manifolds is given in terms of the Einstein constant and the volume of the polarization.
We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existence of coupled Kähler-Einstein metrics imply a certain algebraic stability condition, generalizing $K$-polystability.
We show the existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler-Einstein metric using the normalized Kähler-Ricci flow.
The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler-Einstein manifolds. In special cases, we give the explicit expressions of of these metrics. These examples show that there is a compact Kähler manifold $M$ and its subvariety $N$ whose codimension is greater than 1 such that there is a complete conformally Kä…
We prove a local boundary regularity result for the complete Kahler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary point. We also study the asymptotic behaviour of their holomorphic bisectional curvatures near such points.
In this note we shall prove that the complete Kähler-Einstein volume form on a bounded strongly pseudoconvex domain with $C^{\infty}$-boundary is the normalized limit of a sequence of Bergman kernels.
In this note, we prove that on a compact Kähler manifold $X$ carrying a smooth divisor $D$ such that $K_X+D$ is ample, the Kähler-Einstein cusp metric is the limit (in a strong sense) of the Kähler-Einstein conic metrics when the cone angle goes to $0$. We further investigate the boundary behavior of those and prove that the rescaled metrics converge to a cylindrical metric on $\mathbb C^*\times \mathbb C^{n-1}$.
Rigidity of complete Kähler--Einstein metrics under cscK perturbations
本文研究完备非紧Kähler-Einstein流形上cscK度量的刚性，证明Bergman度量在常标量曲率下为Kähler-Einstein，并结合Cheng猜想解决给出有界严格伪凸域的球形刻画。
For a holomorphic vector bundle over a compact Kähler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
An iterative construction of complete Kähler--Einstein metrics
本文提出了一种迭代方法，用于在非紧致凯勒流形上构造具有负标量曲率的凯勒-爱因斯坦度量，并探讨了其在相对canonical包上的半正曲率性质。
In this paper, we extend the existence and regularity theorems for Kähler-Einstein metrics having conic singularities along a simple normal crossing divisor to the case of normal crossing divisor, i.e. when components of the divisor are allowed to intersect themselves transversely.
In these notes we give an exposition of a result of G. Tian, which says that a Fano surfaces admits a Kahler-Einstein metric precisely when the Lie algebra of holomorphic vector fields is reductive.
We explain a characterization of Einstein-Fano manifolds in terms of the lower bound of the density of the volume of the Kähler-Ricci Flow. This is a direct consequence of Perelman's uniform estimate for the Kähler-Ricci Flow and a $C^0$ estimate of Tian and Zhu.
Kähler-Einstein metrics of negative curvature
本文构造了任意维数$n\geq 2$的紧致负截面曲率Kähler-Einstein流形，且该流形不被球覆盖。
Positively curved Kähler metrics on tube domains and their applications to optimal transport
本文研究管状域上具有 $O(n) \times \mathbb{R}^n$ 对称性的凯勒流形，证明非负双截曲率条件下的唯一性结果，刻画非负正交反双截曲率的充要条件，并探讨其在最优传输中合成曲率界的应用。
This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.
We extend the arguments of Tosatti-Zhang to reduce a well-known conjecture concerning the structure of the Gromov-Hausdorff limit in both the setting of degenerating Calabi-Yau manifolds and the Kähler-Ricci flow to a certain partial second-order estimate.
In this paper, we study the existence of a complete holomorphic vector fields on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler-Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler-Einstein metric whose differential has a constant length. Then we will construct a complete holomorphic vector field from the gradient vector field of the potential function.
We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler-Ricci flow provides weak convergence in…
This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of Kähler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations Kähler-Einstein metrics on bounded strongly pseudoconvex domains of dimension $2$. In addition, we discuss the previous result with general bounded pseudoconvex domain and local triviality of a family of bounded strongly pseudoconvex domains.
Let X be a Kähler manifold and D be a R-divisor with simple normal crossing support and coefficients between 1/2 and 1. Assuming that K_X+D is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on X\D having mixed Poincaré and cone singularities according to the coefficients of D. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair (X,D).
Limits of conical Kähler-Einstein metrics on rank one horosymmetric spaces
本文研究秩一horosymmetric Fano流形上锥Kähler-Einstein度量在锥角递减时的极限行为，证明其在补集上收敛到底空间的Kähler-Einstein度量，在对称空间纤维上重标定后收敛到Stenzel的Ricci平坦度量。
It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.
Let (X,D) be a klt pair. Assuming either K_X+D big or -(K_X+D) ample, and that the coefficients of D are greater than 1/2, we show that the Kähler-Einstein metric attached to (X,D) -whenever it exists- has cone singularities along D on the log-smooth locus of the pair intersected with the ample locus of K_X+D (in the negatively curved case).
This paper concerns obstruction flatness of hypersurfaces $Σ$ that arise as unit sphere bundles $S(E)$ of Griffiths negative Hermitian vector bundles $(E, h)$ over Kähler manifolds $(M, g).$ We prove that if the curvature of $(E, h)$ satisfies a splitting condition and $(M,g)$ has constant Ricci eigenvalues, then $S(E)$ is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and $(M,g)$ is complete, then we show that the corresponding ball bundle admits a complete Kähler-Einstein metric.
In this paper we characterize logarithmic surfaces which admit Kähler-Einstein metrics with negative scalar curvature and small edge singularities along a normal crossing divisor.
We present classical and recent results on Kähler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "Kähler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.
We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.
Strict positivity of Kähler-Einstein currents
本文证明了在全局光滑化或孤立可光滑化奇点条件下，Kähler-Einstein流在奇异轨迹附近控制一个Kähler形式，并应用于三维对数终端奇点空间上的非正曲率奇异Kähler-Einstein度量。
In joint work with Chen and Weber, the author has elsewhere shown that CP2#2(-CP2) admits an Einstein metric. The present paper gives a new and rather different proof of this fact. Our results include new existence theorems for extremal Kahler metrics, and these allow one to prove the above existence statement by deforming the Kahler-Einstein metric on CP2#3(-CP2) until bubbling-off occurs.
We apply Nadel's method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler-Einstein metric. In particular, all del Pezzo surfaces of degree $4,5$, or $6$ and certain special del Pezzo surfaces of lower degree are shown to have a Kähler-Einstein metric. This result is not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian-Yau.
Gauduchon metrics and Hermite-Einstein metrics on non-Kähler varieties
本文证明了任意紧Hermiteian簇上Gauduchon度量的存在性，借此在非Kähler紧正规簇上定义无挠凝聚层的斜率稳定性，并证明了斜率稳定反射层奇异Hermite-Einstein度量的存在性与唯一性。
It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and $L^2$-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Riemannian structure on canonical models. Our approach can be viewed as the Kodaira embedding theorem on singular metric spaces with canonical Kahler metrics.
We present a renormalized Gauss-Bonnet formula for approximate Kahler-Einstein metrics on compact complex manifolds with pseudo-Einstein CR boundaries. The boundary integral is given explicitly, and it is proved that it gives a pseudo-Einstein invariant, which generalizes the Burns-Epstein invariant.
Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2π(1-α)$ for $α\in (0, 1)$. In this paper we study how the existence of such Kähler-Einstein metrics depends on $α$. We show that in the negative scalar curvature case, if such Kähler-Einstein metrics exist for all small cone-angles then they exist for every $α\in(\frac{n+1}{n+2}, 1)$, where $n$ is the dimension. We also give a characterization of the pairs that admit negatively curved cone-edge Kähler-Einstein metrics with…
Smooth Kahler-Einstein metrics have been studied for the past 80 years. More recently, singular Kahler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a survey of some of these developments.
Psh variation of singular Kähler-Einstein metrics and Invariance of plurigenera for smooth Kähler families
本文证明光滑Kähler族中多重亏格维数不变性，解决Siu在2002年提出的猜想，方法涉及奇异Kähler-Einstein度量的Psh变化。
We show that on an open bounded smooth strongly pseudoconvex subset of $\CC^{n}$, there exists a Kähler-Einstein metric with positive Einstein constant, such that the metric restricted to the Levi distribution of the boundary is conformal to the Levi form. To achieve this, we solve an associated complex Monge-Ampère equation with Dirichlet boundary condition. We also prove uniqueness under some more assumptions on the open set.
The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain $Y_{II}(r,p;K)$ is obtained in this paper when the parameter $K$ equals $\frac p2+\frac 1{p+1}$. The estimate of holomorphic sectional curvature under this metric is also given which intervenes between $-2K$ and $-\frac{2K}p$ and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalent to the Bergman metric on $Y_{II}(r,p;K)$ when $K=\frac p2+\frac 1{p+1}$.
The curvature estimation of the complete Kähler-Einstein metrics on the disk bundles
本文计算了完备Kähler-Einstein流形上圆盘丛的完备Kähler-Einstein度量的截面曲率，并研究其负夹挤性；对伪凸Hartogs域，证明了Bergman度量为Kähler-Einstein当且仅当域双全纯等价于单位球。
We calculate Chow quotients of some families of symmetric \(T\)-varieties. In complexity two we obtain new examples of Kähler-Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional application we determine the homeomorpism class of the orbit space of the compact torus action.
In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubharmonic (psh) functions on any (reduced) complex space of pure dimension. As a by-product, we show that any weak Kähler-Einstein metric on \emph{singular} $\mathbb{Q}$-Fano/Calabi-Yau/general type varieties cannot be smooth, and that in general there exists no \emph{singular} normal Kähler complex space such that the Kä…
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