Some Properties of Balanced Hyperbolic Compact Complex Manifolds

紧致凯勒双曲流形的全纯形变

Holomorphic Deformations of Compact Kähler Hyperbolic Manifolds

本文研究紧致凯勒双曲流形的形变问题，通过提出凯勒双曲性的修正版本，为探讨 Gromov 经典凯勒双曲性的形变开性迈出第一步。

Holomorphic submersions onto Kähler or balanced manifolds

We study many properties concerning weak Kählerianity on compact complex manifolds which admits a holomorphic submersion onto a Kähler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the Kähler case), Michelson (the balanced case), Popovici (the sG case) and others.

极值Kähler度量与Fano流形的平衡度量

Balanced metrics for extremal Kähler metrics and Fano manifolds

本文先综述作者在代数几何平衡度量与稳定性概念方面的工作，再仅用Hopf--Rinow定理证明完备黎曼流形上测地凸函数临界点存在性的结论并推广至局部紧完备长度度量空间。

到平衡流形的映射的动量映射

A Moment Map for the Space of Maps to a Balanced Manifold

本文构建了保持体积形式的流形S的生物全变换群作用的动量映射，用于研究满足Bott-Chern上同调类条件的全纯映射，为未来构造平衡商奠定基础。

Anticanonically balanced metrics on Fano manifolds

We show that if a Fano manifold has discrete automorphism group and admits a polarized Kähler-Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the Kähler-Einstein metric. Our proof is based on a simplification of Donaldson's proof of the analogous result for balanced metrics, replacing a delicate geometric argument by the use of Berezin-Toeplitz quantization. We then apply this result to compute the asymptotics of the optimal rate of convergence to the fixed point of Donaldson's iterations in the anticanonical setting.

Radial balanced metrics on the unit ball of the Kepler manifold

We show that there is no radial balanced metric on the unit ball of the Kepler manifold with not too wild boundary behavior. Additionally, we identify explicitly the weights corresponding to radial metrics with such boundary behavior which satisfy the balanced condition as far as germs at the boundary are concerned. Related results for Poincare metrics are also established.

Kobayashi双曲黎曼流形的性质与例子

Properties and examples of Kobayashi hyperbolic Riemannian manifolds

本文在黎曼流形中推广了Brody引理，并构造了新的Kobayashi双曲流形例子。

Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds

We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li.

Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles

For a quasi-compact Kähler manifold $U$ endowed with a nilpotent harmonic bundle whose Higgs field is injective at one point, we prove that $U$ is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log general type. Moreover, we prove that there is a finite unramified cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ so that any projective compactification of $\tilde{U}$ is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic and is of general type. As a byproduct, we establish some criterion of pseudo-Picard hyperbolicity and pseudo-algebraic hyperbo…

SKT双曲与Gauduchon双曲紧复流形

SKT Hyperbolic and Gauduchon Hyperbolic Compact Complex Manifolds

本文为紧复流形引入SKT双曲性与Gauduchon双曲性，证明前者蕴含Kobayashi/Brody双曲性，后者蕴含除子双曲性，并给出SKT双曲流形万有覆盖上L^2调和空间的消失定理。

Hyperbolic $n$-dimensional manifolds with automorphism group of dimension $n^2$

We obtain a complete classification of complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$, for which the dimension of the group of holomorphic automorphisms is equal to $n^2$.

Covering stability of Bergman kernels on Kähler hyperbolic manifolds

This paper is a sequel to \cite{Xu}. In this paper, an estimation of the Bergman Kernel of Kähler hyperbolic manifold is given by the $L^2$ estimate and the Bochner formula. As an application, an effective criterion of the very ampleness of the canonical line bundle of Kähler hyperbolic manifold is given, which is a generalization of Yeung's result.

Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism group

In this paper we determine all Kobayashi-hyperbolic 2-dimensional complex manifolds for which the group of holomorphic automorphisms has dimension 3. This work concludes a recent series of papers by the author on the classification of hyperbolic $n$-dimensional manifolds, with automorphism group of dimension at least $n^2-1$, where $n\ge 2$.

平衡流形上的体积形式与Calabi-Yau方程

Volume forms on balanced manifolds and the Calabi-Yau equation

本文研究了平衡流形上混合体积形式空间中的测地方程及Calabi-Yau方程，证明了在正性假设下解的存在性。

Hyperbolic manifolds whose envelopes of holomorphy are not hyperbolic

We present a family of examples of two dimensional, hyperbolic complex manifolds whose envelopes of holomorphy are not hyperbolic.

由Carathéodory双曲流形一致化的拟射影流形及其子簇的双曲性

Quasi-projective manifolds uniformized by Carathéodory hyperbolic manifolds and hyperbolicity of their subvarieties

本研究探讨Carathéodory双曲复流形的分析性质，证明其上存在实解析有界严格多重次调和函数，完备Kähler情形下存在Bergman度量，强Carathéodory双曲的万有覆叠对应的拟射影簇的子簇为对数一般型或一般型，与Lang猜想一致。

Evolution Families and the Loewner Equation II: complex hyperbolic manifolds

We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.