The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties II

On holomorphic curves into complex projective varieties

In this paper, we study holomorphic curves satisfying the Fubini-Study derivative $\|f'(z)\|=O(|z|^σ)$ for some $σ>-1$ from the viewpoint of Nevanlinna theory

沿复解析簇的全纯1-形式的Bruce-Roberts数

The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties

本文定义全纯1-形式沿复解析簇的Bruce-Roberts数，证明其可用Ebeling--Gusein-Zade指标、Milnor数和Tjurina数表示，并应用于复二维全纯叶状结构。

沿复解析簇的全纯1-形式的Bruce-Roberts Tjurina数

The Bruce-Roberts Tjurina number of holomorphic 1-forms along complex analytic varieties

本文定义了全纯1-形式沿复解析簇对的Bruce-Roberts Tjurina数，证明其与多个已知不变量之间的关系，并应用于复二维全纯叶层芽的拟齐性问题。

从 ${\Bbb C}^{p}$ 到半阿贝尔簇和一般射影簇的亚纯映射

Meromorphic maps from ${\Bbb C}^{p}$ into semi-abelian varieties and general projective varieties

通过纤维积分降维法，将全纯曲线的Nevanlinna型结果推广到从${\Bbb C}^{p}$到射影簇的亚纯映射，包括Bloch定理、第二基本定理等。

Proper holomorphic disks in the complement of varieties in \C^2

For any closed analytic set X in C^2 there exists a proper holomorphic embedding of the unit disk into C^2 such that the image avoids X.

Nevanlinna theory for holomorphic curves from annuli into semi Abelian varieties

In this paper, we prove a lemma on logarithmic derivative for holomorphic curves from annuli into Kähler compact manifold and. As its application, a second main theorem for holomophic curves from annuli into semi abelian varieties intersecting with only one divisor is given.

Cyclic and Abelian coverings of real varieties

We describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties.

Truncated counting functions of holomorphic curves in Abelian varieties

A new proof of the Second Main Theorem with truncation level 1 for Zariski-dense holomorphic curves into Abelian varieties, which has just been proved by Yamanoi [Y2], is presented. Our proof is based on the idea of the "Radon transform" introduced in [K2] combined with consideration on certain singular perturbation of the probability measures on the parameter space which appears in the "Radon transform".

从复圆盘到射影簇和超曲面的全纯映射的亏量关系

Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces

本文针对复圆盘上有限增长指数的全纯映射，建立了与射影簇中任意超曲面族相交的第二主要定理，给出截断亏量之和的上界，并推广改进了先前相关结果。

Approximation of holomorphic maps from Runge domains to affine algebraic varieties

We present a geometric proof of the theorem saying that holomorphic maps from Runge domains to affine algebraic varieties admit approximation by Nash maps. Next we generalize this theorem.

Holomorphic curves in moduli spaces of polarized Abelian varieties

We study the value distribution of holomorphic curves from a general open Riemann surface into a smooth logarithmic pair $(X, D).$ By stochastic calculus, we first obtain a version of tautological inequality (proposed by McQuillan) and a logarithmic derivative lemma. Then, one uses them to establish a Second Main Theorem of Nevanlinna theory for pair $(X, D)$ under certain conditions. Finally, we apply the Second Main Theorem to study the holomorphic curves from a general open Riemann surface into certain special moduli spaces of polarized Abelian varieties intersecting boundary divisors.

Norm Preserving Extensions of Holomorphic Functions Defined on Varieties in ${\mathbb C}^n$

If $V$ is an analytic set in a pseudoconvex domain $Ω$, we show there is always a pseudoconvex domain $G \subseteq Ω$ that contains $V$ and has the property that every bounded holomorphic function on $V$ extends to a bounded holomorphic function on $G$ with the same norm. We find such a $G$ for some particular analytic sets. When $Ω$ is an operhedron we show there is a norm on holomorphic functions on $V$ that can always be preserved by extensions to $Ω$.

全纯映射到代数簇的非积分亏量关系

A non-integrated defect relation for holomorphic maps into algebraic varieties

本文推广Fujimoto关于全纯映射的非积分亏量理论，建立了全纯映射到射影簇的非积分亏量关系，将经典Nevanlinna-Cartan型结果从射影空间推广到一般射影簇。

Holomorphic curves into algebraic varieties intersecting moving hypersurface targets

In [Ann. of Math. 169 (2009)], Min Ru proved a second main theorem for algebraically nondegenerate holomorphic curves in complex projective varieties intersecting fixed hypersurface targets. In this paper, by introducing a new proof method for the case of projective varieties, we generalize this result to moving hypersurface targets.

Second main theorem for holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets

Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties intersecting moving hypersurfaces targets. The main purpose of this paper is to give some interesting improvements of Ru's second main theorem for moving hypersurfaces targets located in subgeneral position with index.

Holomorphic Curves into Algebraic Varieties Intersecting Divisors in Subgeneral Position

Recently, there are many developments on the second main theorem for holomorphic curves into algebraic varieties intersecting divisors in general position or subgeneral position. In this paper, we refine the concept of subgeneral position by introducing the notion of the index of subgeneral position. With this new notion we give some surprising improvement of the previous known second main theorem type results. Moreover, via the analogue between Nevanlinna theory and Diophantine approximation, the corresponding Schmidt's subspace type theorems are also established in the final section.

Projection on Segre varieties and determination of holomorphic mappings between real submanifolds

It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given.

单叶函数的Grunsky范数与阿贝尔全纯微分

The Grunsky norm of univalent functions and abelian holomorphic differentials

本研究证明圆盘上归一化单叶函数的Grunsky范数由全纯阿贝尔微分的平方决定，区别于关联所有可积全纯二次微分的Teichmuller范数，该结果可给出所有拟共形曲线Fredholm特征值的显式表示。

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

Let M be a projective manifold, p:M_{G} --> M a regular covering over M with a free abelian transformation group G. We describe holomorphic functions on M_{G} of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary we obtain for such functions Cartwright and Liouville type theorems. Our approach brings together L_{2} cohomology technique for holomorphic vector bundles on complete Kähler manifolds and geometric properties of projective manifolds.

Extending holomorphic maps from Stein manifolds into affine toric varieties

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstnerič, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular tar…