Generic linear systems for projective CR manifolds

Projective manifolds whose tangent bundle contains a strictly nef subsheaf

Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group $π_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.

On projective Kähler manifolds of partially positive curvature and rational connectedness

In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of $k$-positive Ricci curvature and prove that for a projective $n$-dimensional Kähler manifold of $k$-positive Ricci curvature the MRC fibration has generic fibers of dimension at least $n-k+1$. We also establish an analogous result for projec…

On projective manifolds with pseudo-effective tangent bundle

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective manifold $Y$ such that its general fiber is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general non-minimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

A numerical criterion for generalised Monge-Ampere equations on projective manifolds

We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the $J$-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the $J$-equation, and prove a conjecture of Székelyhidi in the projective case on the solvability of certain inverse Hessian equations. The key new ingredient in improving Chen's result is a degenerate concent…

A decomposition theorem for projective manifolds with nef anticanonical bundle

Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle.

射影流形上阿贝尔覆盖上指数增长性的全纯函数

Holomorphic Functions of Exponential Growth on Abelian Coverings of a Projective Manifold

设M是射影流形，p：M_{G} --> M是M上具有自由交换变换群G的正则覆盖.我们描述了指数增长性的M_{G}上的全纯函数，它关于由从M拉回的度量定义的距离。作为推论，我们得到这样的功能Cartwright和刘维型定理。我们的方法结合了完备Kähler流形上全纯向量丛的L_{2}上同调技术和射影流形的几何性质。

Brody曲线和整曲线行为非常不同的投影流形

A projective manifold where Brody and entire curves behave very differently

我们给出了一个例子，射影流形的稠密整曲线，使每个布罗迪曲线是退化的。

Finiteness of meromorphic mappings from Kähler manifold into projective space

The purpose of this paper is to prove the finiteness theorems for meromorphic mappings of a complete connected Kähler manifold into projective space sharing few hyperplanes in subgeneral position without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Our results are extensions and generalizations of some recent ones.

Deformations of the tangent bundle of projective manifolds

We investigate when the tangent bundle of a projective manifold has a non-trivial first order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.

每个射影Oka流形都是椭圆的

Every projective Oka manifold is elliptic

本文证明了每个射影Oka流形在Gromov意义下都是椭圆的，从而肯定回答了一个长期悬而未决的问题。

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。

Albanese maps of projective manifolds with nef anticanonical bundles

Let $X$ be a projective manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the Albanese map $p: X \rightarrow Y$ is locally isotrivial. In particular, $p$ is a submersion.

负曲率Kähler-Einstein流形退化族的射影嵌入

Projective embedding of degenerating family of Kähler-Einstein manifolds of negative curvature

本文研究了负曲率Kähler-Einstein流形退化族的Bergman嵌入，并在特殊情况下证明通过构造正交基可使诱导嵌入收敛至极限空间的Bergman嵌入并带有气泡。

Uniqueness problem of meromorphic mappings from a complete Kähler manifold into a projective variety

The purpose of this paper has twofold. The first is to prove a unicity theorem for meromorphic mappings of a complete Kähler manifold M in P^n(C) sharing few hypersurfaces. The second is to give a unicity theorem for the case of differential nondegenerate meromorphic mappings sharing 2N-n+3+2ρ(2N-n+1) hyperplanes in N subgeneral position.

A characterization of complex quasi-projective manifolds uniformized by unit balls

In 1988 Simpson extended the Donaldson-Uhlenbeck-Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasi-projective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson. Several byproducts are also obtained in this paper.

A stability theorem for projective $CR$ manifolds

We consider smooth deformations of the $CR$ structure of a smooth $2$-pseudoconcave compact $CR$ submanifold $\textsf{M}$ of a reduced complex analytic variety $\textsf{X}$ outside the intersection $D\,{\cap}\,\textsf{M}$ with the support $D$ of a Cartier divisor of a positive line bundle $\texttt{F}_{\textsf{X}}.$ We show that nearby structures still admit projective $CR$ embeddings. Special results are obtained under the additional assumptions that $\textsf{X}$ is a projective space or a Fano variety.

The openness conjecture for projective manifolds

We give a proof of the openness conjecture of Demailly and Kollár for positively curved singular metrics on ample line bundles over projective varieties. As a corollary it follows that the openness conjecture for plurisubharmonic functions with isolated sigularities holds.

On h-principle and specialness for complex projective manifolds

We show that a complex projective manifold X which satisfies the Gromov's h-principle is `special', and raise some questions about the reverse implication, the extension to the quasi-K\" ahler case, and the relationships of these properties to the `Oka' property. The guiding principle is that the existence of many Stein manifolds which have degenerate Kobayashi pseudometric gives strong obstructions to the complex hyperbolicity of X satisfying the h-principle.

Quantization of Donaldson's heat flow over projective manifolds

Consider $E$ a holomorphic vector bundle over a projective manifold $X$ polarized by an ample line bundle $L$. Fix $k$ large enough, the holomorphic sections $H^0(E\otimes L^k)$ provide embeddings of $X$ in a Grassmanian space. We define the \textit{balancing flow for bundles} as a flow on the space of projectively equivalent embeddings of $X$. This flow can be seen as a flow of algebraic type hermitian metrics on $E$. At the quantum limit $k\to \infty$, we prove the convergence of the balancing flow towards the Donaldson heat flow, up to a conformal change. As a by-product, we obtain a numeri…

Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation

In this paper we prove that every quasi-projective base space $V$ of smooth family of minimal projective manifolds with maximal variation is pseudo Kobayashi hyperbolic, i.e. $V$ is Kobayashi hyperbolic modulo a proper subvariety $Z\subsetneq V$. In particular, $V$ is algebraically degenerate, that is, every nonconstant entire curve $f:\mathbb{C}\to V$ has image $f(\mathbb{C})$ which lies in that proper subvariety $Z\subsetneq V$. As a direct consequence, we prove the Brody hyperbolicity of moduli spaces of minimal projective manifolds, which answers a question by Viehweg-Zuo in 2003.

具有正常交叉极限的退化凯勒-爱因斯坦流形族的射影嵌入

Projective embedding of degenerating family of Kähler-Einstein manifolds with normal crossing limit

本文研究负曲率凯勒-爱因斯坦流形退化族的伯格曼嵌入，针对中央纤维仅含简单正常交叉奇点的情况，构造标准正交基实现伯格曼嵌入的收敛性。

复射影流形上的第二类有理积分及其本原上同调

Rational Integrals of the second kind on a complex projective manifold and its primitive cohomology

设X是一个n+1维复代数流形，它嵌入一个足够高维的复射影空间，Y是X的一般超平面截面。利用X上的有理积分刻画了H^p（X-Y，C）上的混合Hodge结构和Y的中本原上同调群H^n（Y，C）_0的Hodge滤子. (Key原上同调，第二类有理积分，广义Poincare剩余映射，Hodge滤子，混合Hodge结构）

Algebraicity of foliations on complex projective manifolds, applications

Contents 1. Algebraicity criterion: statement 2. Proof of the algebraicity criterion. 3. Pseudoeffectivity and movable classes. 4. Harder-Narasimhan filtrations and pseudo-effectivity. 5. Pseudo-effectivity of relative canonical bundles. 6. Rational curves and non-pseudoeffectivity of the canonical/cotangent bundles. 7. Birational stability of the cotangent bundle 8. Shafarevich-Viehweg conjecture. 9. Numerically trivial foliations. 10. The Beauville-Bogomolov-Yau decomposition 11. Some questions 11.1. Pseudoeffectivity of determinants. 11.2. Variations on Abundance. 11.3. Special manifolds. R…

On automorphisms of blowups of projective manifolds

In this paper we mainly study the following question: For what projective manifold $X$ of dimension $\geq 3$ that any $f\in Aut(X)$ has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where $X\rightarrow X_0$ is a finite blowup along smooth centers, here $X_0$ is a projective manifold of interest. Here we allow $X_0$ to be either one of the following manifolds: it has Picard number 1, or a Fano manifold, or it is a projective hyper-Kähler manifold. We also allow the centers of blowups to have large dimensions relative to that of $X_0$ …