On Nevanlinna - Cartan theory for holomorphic curves with Tsuji characteristics

TL;DR

This paper extends Nevanlinna-Cartan theory to holomorphic curves on angular domains, establishing fundamental theorems and addressing the uniqueness problem via inverse images of hypersurfaces, with new results in this specific setting.

Contribution

It introduces the first known results on the uniqueness of holomorphic curves by inverse images of hypersurfaces on angular domains, expanding classical value distribution theory.

Findings

Proves fundamental theorems for holomorphic curves intersecting hypersurfaces on angular domains.

Establishes a new uniqueness theorem for holomorphic curves in an angle.

Improves existing results on uniqueness of holomorphic curves on the complex plane.

Abstract

In this paper, we prove some fundamental theorems for holomorphic curves on angular domain intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex projective variety with the level of truncation. As applications of the second main theorems for an angle, we will discuss the uniqueness problem of holomorphic curves in an angle instead of the whole complex plane. Detail, we establish a result for uniqueness problem of holomorphic curve by inverse image of a hypersurface. In my knowledge, this is the first result for uniqueness problem of holomorphic curve by inverse image of hypersurface on angular domain. On complex plane, we obtain a uniqueness result for holomorphic curves, it is improvement of some results before [5, 10] in this trend.

Equations

$$\Xi (\alpha ,\beta ;r)=\{z=t{e}^{i\theta }:\alpha <\theta <\beta ,1<t\le r(sin(k(\theta -\alpha )){)}^{1/k}\}.$$

$${\mathfrak{N}}_{\alpha \beta ,f}(r,D)$$

$$=\sum _{1<|a_n|<r(sin(k(\theta -\alpha )){)}^{1/k}}(\frac{sink({\alpha }_n-\alpha )}{|a_n{|}^k}-\frac{1}{r^k}),$$

$${\mathfrak{m}}_{\alpha \beta ,f}(r,D)=\frac{1}{2\pi }{\int }_{arcsin{r}^{-k}}^{\pi -arcsin{r}^{-k}}\log \frac{\Vert \text{f}(rsi{n}^{k^{-1}}\phi e^{i(\alpha +k^{-1}\phi )}){\Vert }^d}{|Q(\text{f})(rsi{n}^{k^{-1}}\phi e^{i(\alpha +k^{-1}\phi )})|}\frac{d\phi }{r^{k}si{n}^2\phi }.$$

$${\mathfrak{N}}_{\alpha \beta ,f}^{\delta }(r,D)$$

$$=\sum _{1<|a_n|<r(sin(k(\theta -\alpha )){)}^{1/k},\min \{{\text{ord}}_{Q(\text{f})}(a_n),\delta \}}(\frac{sink({\alpha }_n-\alpha )}{|a_n{|}^k}-\frac{1}{r^k}),$$

$$C_{\alpha \beta ,f}(r,D)=2\sum _{1\le {\rho }_n\le r,\alpha \le {\psi }_n\le \beta }(\frac{1}{{{\rho }_n}^k}-\frac{{{\rho }_n}^k}{r^{2k}})sink({\psi }_n-\alpha ),$$

$$C_{\alpha \beta ,f}^{\delta }(r,D)=2\sum _{1\le {\rho }_n\le r,\alpha \le {\psi }_n\le \beta ,\min \{{\text{ord}}_{Q(\text{f})}({\rho }_n{e}^{i{\psi }_n}),\delta \}}(\frac{1}{{{\rho }_n}^k}-\frac{{{\rho }_n}^k}{r^{2k}})sink({\psi }_n-\alpha ),$$

$$A_{\alpha \beta ,f}(r,D)=\frac{k}{\pi }{\int }_1^r(\frac{1}{t^k}-\frac{t^k}{r^{2k}})\log \frac{\Vert \text{f}(t{e}^{i\alpha }){\Vert }^d\Vert \text{f}(t{e}^{i\beta }){\Vert }^d}{|Q(\text{f})(t{e}^{i\alpha })Q(\text{f})(r{e}^{i\beta })|}\frac{dt}{t}$$

$$B_{\alpha \beta ,f}(r,D)=\frac{2k}{\pi r^k}{\int }_{\alpha }^{\beta }\log \frac{\Vert \text{f}(r{e}^{i\phi }){\Vert }^d}{|Q(\text{f})(r{e}^{i\phi })|}sin(k(\phi -\alpha ))d\phi ,$$

$$V\cap \text{Supp}{D}_{i_0}\cap \cdots \cap \text{Supp}{D}_{i_n}=\varnothing ,$$

$$d{S}_{\alpha \beta ,f}(r)=A_{\alpha \beta ,f}(r,D)+B_{\alpha \beta ,f}(r,D)+C_{\alpha \beta ,f}(r,D)+O(1).$$

$$d{\mathfrak{T}}_{\alpha \beta ,f}(r)={\mathfrak{m}}_{\alpha \beta ,f}(r,Q)+{\mathfrak{N}}_{\alpha \beta ,f}(r,Q)+O(1).$$

$$S_{\alpha \beta ,f}(r)=A_{\alpha \beta ,f}(r,H)+B_{\alpha \beta ,f}(r,H)+C_{\alpha \beta ,f}(r,H)+O(1).$$

$${\mathfrak{T}}_{\alpha \beta ,f}(r)={\mathfrak{m}}_{\alpha \beta ,f}(r,H)+{\mathfrak{N}}_{\alpha \beta ,f}(r,H)+O(1).$$

$$\Vert (q-n-1)S_{\alpha \beta ,f}(r)⩽\sum _{j=1}^q{C}_{\alpha \beta ,f}^n(r,H_j)+O(\log T_f(r)+\log r).$$

$$\Vert (q-n-1){\mathfrak{T}}_{\alpha \beta ,f}(r)⩽\sum _{j=1}^q{{\mathfrak{N}}^{\mathfrak{n}}}_{\alpha \beta ,f}(r,H_j)+O(\log {\mathfrak{T}}_{\alpha \beta ,f}(r)+\log r).$$

$$\Vert (n-(d+N+1)N){\mathfrak{T}}_{\alpha \beta ,f}(r)$$

$$\le {\mathfrak{N}}_{\alpha \beta ,f}^N(r,D)+o({\mathfrak{T}}_{\alpha \beta ,f}(r)).$$

$$D=\{z\in {\mathbb{P}}^N(\mathbb{C}),\sum _{i=0}^N(\sum _{t=0}^i{x}_t{)}^n{x}_i^d=0\}$$

$$M\ge \frac{n^n{d}^{n^2+n}(19nI({\epsilon }^{-1}){)}^n(\mathrm{deg}V{)}^{n+1}}{n!},$$

$$(q(1-\epsilon /3)-(n+1)-\epsilon /3){\mathfrak{T}}_{\alpha \beta ,f}(r)⩽\sum _{l=1}^q{d}_l^{-1}{{\mathfrak{N}}^M}_{\alpha \beta ,f}(r,Q_l)+O(\log {\mathfrak{T}}_{\alpha \beta ,f}(r)+\log r)$$

$$M\ge \frac{n^n{d}^{n^2+n}(19nI({\epsilon }^{-1}){)}^n(\mathrm{deg}V{)}^{n+1}}{n!},$$

$$(q(1-\epsilon /3)-(n+1)-\epsilon /3)S_{\alpha \beta ,f}(r)⩽\sum _{l=1}^q{d}_l^{-1}{C}_{\alpha \beta ,f}^M(r,Q_l)+O(\log T_f(r)+\log r)$$

$$\Omega (\alpha ,\beta ;r)=\Omega (\alpha ,\beta )\cap \{1<|z|<r\}.$$

$$A_{\alpha \beta }(r,f)$$

$$B_{\alpha \beta }(r,f)$$

$$C_{\alpha \beta }(r,f)$$

$$=2\sum _{1\le {\rho }_n\le r,\alpha \le {\psi }_n\le \beta }(\frac{1}{{{\rho }_n}^k}-\frac{{{\rho }_n}^k}{r^{2k}})sink({\psi }_n-\alpha ),$$

$$c_{\alpha \beta }(r,f)=\sum _{1<{\rho }_n\le r,\alpha \le {\psi }_n\le \beta }sin(k({\psi }_n-\alpha )),$$

$$S_{\alpha \beta }(r,f)=A_{\alpha \beta }(r,f)+B_{\alpha \beta }(r,f)+C_{\alpha \beta }(r,f).$$

$$C_{\alpha \beta }(r,\frac{1}{f})-C_{\alpha \beta }(r,f)$$

$$+\frac{2k}{\pi r^k}{\int }_{\alpha }^{\beta }\log |f(r{e}^{i\phi })|.sin(k(\phi -\alpha ))d\phi +O(1).$$

$$\Xi (\alpha ,\beta ;r)=\{z=t{e}^{i\theta }:\alpha <\theta <\beta ,1<t\le r(sin(k(\theta -\alpha )){)}^{1/k}\}.$$

Full text

2010 Mathematics Subject Classification. Primary 32H30.

Key words: Algebraic variety, General position, Hypersurface, Nevanlinna theory, Tsuji characteristics.

1. Introduction and main results

We denote $\Omega(\alpha,\beta)=\{z:\alpha<argz<\beta\}$ by the angle on complex plane, where $0<\beta-\alpha\leq 2\pi.$ Then, $\Omega(\alpha,\beta)$ is called an angular domain on complex plane. The Nevanlinna second main theorem for an angle was used in [17, 4, 6, 7, 22, 20], and [19] to investigate the growth of meromorphic functions with some radially distributed values. The usage of the second main theorem produces a basic and elementary method in the topic [20]. In [19], in view of the Tsuji second main theorem, we established a five-value uniqueness theorem and four-value uniqueness theorem for meromorphic functions in an angle. In 2015, J. Zheng [21] established the value distribution of holomorphic curves on an angular domain from the point of view of potential theory and established the first and second fundamental theorems corresponding to those theorems of Ahlfors-Shimizu, Nevanlinna, and Tsuji on meromorphic functions in an angular domain. We refer readers to [21] for comments on the results of the value distribution of holomorphic curves on an angular domain. These results motivate us to consider the case of holomorphic curves on $\overline{\Omega}(\alpha,\beta),$ $\Omega(\alpha,\beta)$ intersecting hypersurfaces. In this paper, we prove the fundamental theorems for holomorphic mappings from $\overline{\Omega}(\alpha,\beta)$, $\Omega(\alpha,\beta)$ to $\mathbb{P}^{n}(\mathbb{C})$ intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex projective variety with the level of truncation and the Nevanlinna functions have the form of Tsuji characteristics.

We denote by $k=\dfrac{\pi}{\beta-\alpha},$ and for any pair of real numbers $\alpha$ and $\beta$ in $[0,2\pi)$ with $0<\beta-\alpha\leq 2\pi,$

[TABLE]

Let $f:\Omega(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve. Let ${\large\text{f}}=(f_{0}:\dots:f_{n})$ be a reduced representation of $f,$ where $f_{0},\dots,f_{n}$ are holomorphic functions and without common zeros in $\Omega(\alpha,\beta).$ Set $||{\large\text{f}}(z)||=\max\{|f_{0}(z)|,\dots,|f_{n}(z)|\}.$ Let $D$ be a hypersurface in $\mathbb{P}^{n}(\mathbb{C})$ of degree $d$. Let $Q$ be the homogeneous polynomial of degree $d$ defining $D$. Under the assumption that $Q(\large\text{f})\not\equiv 0,$ the counting function $\mathfrak{N}_{\alpha\beta,f}(r,D)$ of $f$ with respect to $D$ is defined as

[TABLE]

where the $\mathfrak{n}_{\alpha\beta,f}(t,D)$ are the number zeros of $Q(\large\text{f})$ in the set $\Xi(\alpha,\beta;r)$ counting with multiplicity and $a_{n}=|a_{n}|e^{i\alpha_{n}}$ are zeros of $Q(\large\text{f})$ in the set $\Xi(\alpha,\beta;r).$

The proximity function of $f$ on $\Omega(\alpha,\beta)$ with respect to $D$ is defined as following:

[TABLE]

Now let $\delta$ be a positive integer, the truncated counting function of $f$ is defined by

[TABLE]

where the $\mathfrak{n}^{\delta}(t,D)$ are the number zeros of $Q(\large\text{f}),$ any zero of multiplicity greater than $\delta$ of $Q(\large\text{f})$ in $\Xi(\alpha,\beta;r)$ is “truncated” and counted as if it only had multiplicity $\delta.$

Let $f:\overline{\Omega}(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic map. Let ${\large\text{f}}=(f_{0}:\dots:f_{n})$ be a reduced representation of $f,$ where $f_{0},\dots,f_{n}$ are holomorphic functions and without common zeros in $\overline{\Omega}(\alpha,\beta)$. The counting function $C_{\alpha\beta,f}(r,D)$ of $f$ with respect to $D$ is defined as

[TABLE]

where the $\rho_{n}e^{i\psi_{n}}$ are the zeros of $Q(\large\text{f})$ in $\overline{\Omega}(\alpha,\beta)$ counting with multiplicity. For each zero $\rho_{n}e^{i\psi_{n}}$ of $Q(\large\text{f})$ in $\overline{\Omega}(\alpha,\beta)$ with multiple $m$, then term $2(\dfrac{1}{{\rho_{n}}^{k}}-\dfrac{{\rho_{n}}^{k}}{r^{2k}})sin\hskip 5.69046ptk(\psi_{n}-\alpha)$ is counted $m$ times in $C_{\alpha\beta,f}(r,D).$

Now let $\delta$ be a positive integer, the truncated counting function of $f$ is defined by

[TABLE]

where any zero of multiplicity greater than $\delta$ of $Q(\large\text{f})$ in $\overline{\Omega}(\alpha,\beta)$ is “truncated” and counted as if it only had multiplicity $\delta.$ This means that for each zero $\rho_{n}e^{i\psi_{n}}$ of $Q(f)$ in $\overline{\Omega}(\alpha,\beta)$ with multiple $m$, the terms $2(\dfrac{1}{{\rho_{n}}^{k}}-\dfrac{{\rho_{n}}^{k}}{r^{2k}})sin\hskip 5.69046ptk(\psi_{n}-\alpha)$ is counted $\mathop{\mathrm{min}}\{m,\delta\}$ times in $C_{\alpha\beta,f}(r,D).$

The angular proximity Nevanlinna of $f$ with respect to $D$ is defined as following:

[TABLE]

and

[TABLE]

where $\|f(z)\|=\max\{|f_{0}(z)|,\dots,|f_{n}(z)|\}.$

Let $V\subset\mathbb{P}^{N}(\mathbb{C})$ be a smooth complex projective variety of dimension $n\geq 1.$ Let $D_{1},\dots,D_{q}$ be hypersurfaces in $\mathbb{P}^{N}(\mathbb{C}),$ where $q>n.$ The hypersurfaces $D_{1},\dots,D_{q}$ are said to be in general position on V  if for every subset $\{i_{0},\dots,i_{n}\}\subset\{1,\dots,q\},$ we have

[TABLE]

where $\text{Supp}(D)$ means the support of the divisor $D.$ A map $f:\Omega(\alpha,\beta)\to V$ is said to be algebraically nondegenerate if the image of $f$ is not contained in any proper subvarieties of $V.$

In this paper, a notation $``\|"$ in the inequality is mean that the inequality holds for $r\in(1,\infty)$ outside a set with measure finite.

Our main results are

Theorem 1**.**

Let $D$ be a hypersurface in $\mathbb{P}^{n}(\mathbb{C})$ and $f:\overline{\Omega}(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained $D$. Then we have for any $1<r<\infty$,

[TABLE]

Theorem 2**.**

Let $D$ be a hypersurface in $\mathbb{P}^{n}(\mathbb{C})$ and $f:\Omega(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained $D$. Then we have for any $1<r<\infty$,

[TABLE]

Taking $d=1,$ we get the following results:

Corollary 1**.**

Let $H$ be a hyperplane in $\mathbb{P}^{n}(\mathbb{C})$ and $f:\overline{\Omega}(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained $H$. Then we have for any $1<r<\infty$,

[TABLE]

Corollary 2**.**

Let $H$ be a hyperplane in $\mathbb{P}^{n}(\mathbb{C})$ and $f:\Omega(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained $H$. Then we have for any $1<r<\infty$,

[TABLE]

Theorem 3**.**

Let $f:\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Then we have

[TABLE]

Theorem 4**.**

Let $f:\Omega(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Then we have

[TABLE]

Theorem 5**.**

Let $f:\Omega(\alpha,\beta)\to\mathbb{P}^{N}(\mathbb{C})$ be an algebraically nondegenerate holomorphic curve. Let $d$ and $n$ be two integers with $n>N(d+N+1).$ Let $\mathcal{H}_{i}=\{z\in\mathbb{P}^{N}(\mathbb{C}),\mathcal{H}_{i}(z)=0\},0\leq i\leq N,$ be hyperplanes in $\mathbb{P}^{N}(\mathbb{C}).$ Let $D_{i}=\{z\in\mathbb{P}^{N}(\mathbb{C}),Q_{i}(z)=0\},0\leq i\leq N,$ be hypersurfaces of degree $d$ such that the hypersurfaces $\{\mathcal{H}_{0}^{n}Q_{0}=0\},\dots,\{\mathcal{H}_{N}^{n}Q_{N}=0\}$ are in general position in $\mathbb{P}^{N}(\mathbb{C}).$ Let $D=\{z\in\mathbb{P}^{N}(\mathbb{C}),\sum_{i=0}^{N}\mathcal{H}_{i}^{n}Q_{i}=0\}.$ Then

[TABLE]

We give a hypersurfaces satisfying Theorem 5.

Example 6**.**

Let $D_{i}=\{z=(x_{0}:\dots:x_{N})\in\mathbb{P}^{N}(\mathbb{C}),x_{i}^{d}=0\},0\leq i\leq N,$ be hypersurfaces of degree $d.$ Let $\mathcal{H}_{i}=\{z=(x_{0}:\dots:x_{N})\in\mathbb{P}^{N}(\mathbb{C}),\sum_{t=0}^{i}x_{t}=0\}.$ We see that the hypersurfaces $\{(\sum_{t=0}^{i}x_{t})^{n}x_{i}^{d}=0\},0\leq i\leq N,$ are in general position in $\mathbb{P}^{N}(\mathbb{C}).$ Then

[TABLE]

satisfies the Theorem 5 with $n>N(d+N+1).$

As an application of Theorem 5, we prove the uniqueness theorem for holomorphic curves on angular domain by inverse image of a hypersurface.

Theorem 7**.**

Let $f,g:\Omega(\alpha,\beta)\to\mathbb{P}^{N}(\mathbb{C})$ be two algebraically nondegenerate holomorphic curves, and $n$ be a integer with $n>N(d+N+3).$ Let $D$ be a hypersurface as the same Theorem 5. Suppose that $f(z)=g(z)$ on $f^{-1}(D)\cup g^{-1}(D),$ then $f\equiv g.$

In my knowledge, up to now, Theorem 7 is a first result for uniqueness problem of holomorphic curve by inverse image of a hypersurface on angular domain.

When $\alpha=0,\beta=2\pi,$ this means $\Omega(\alpha,\beta)=\mathbb{C},$ we obtain some uniqueness results for holomorphic curves on complex plane as following:

Corollary 3**.**

Let $f,g:\mathbb{C}\to\mathbb{P}^{N}(\mathbb{C})$ be two algebraically nondegenerate holomorphic curves, and $n$ be a integer with $n>N(d+N+3).$ Let $D$ be a hypersurface as the same Theorem 5. Suppose that $f(z)=g(z)$ on $f^{-1}(D)\cup g^{-1}(D),$ then $f\equiv g.$

By using method of Ru [14] and Ru et. al. [2], we are easy to get some results as follows:

Theorem 8**.**

Let $V\subset\mathbb{P}^{N}(\mathbb{C})$ be a complex projective variety of dimension $n\geq 1.$ Let $D_{1},\dots,D_{q}$ be hypersurfaces in $\mathbb{P}^{N}(\mathbb{C})$ of degree $d_{j},$ located in general position on $V.$ Let $d$ be the least common multiple of the $d_{i},$ $i=1,\dots,q.$ Let $f:\Omega(\alpha,\beta)\rightarrow V$ be an algebraically non-degenerate holomorphic map. Let $\varepsilon>0$ and

[TABLE]

where $I(x):=\mathop{\mathrm{min}}\{k\in\mathbb{N}:k>x\}$ for a positive real number $x.$ Then

[TABLE]

holds for all $r\in(0,+\infty)$ outside a set of finite measure.

Theorem 9**.**

Let $V\subset\mathbb{P}^{N}(\mathbb{C})$ be a complex projective variety of dimension $n\geq 1.$ Let $D_{1},\dots,D_{q}$ be hypersurfaces in $\mathbb{P}^{N}(\mathbb{C})$ of degree $d_{j},$ located in general position on $V.$ Let $d$ be the least common multiple of the $d_{i},$ $i=1,\dots,q.$ Let $f:\mathbb{C}\rightarrow V$ be an algebraically non-degenerate holomorphic map. Let $\varepsilon>0$ and

[TABLE]

where $I(x):=\mathop{\mathrm{min}}\{k\in\mathbb{N}:k>x\}$ for a positive real number $x.$ Then

[TABLE]

holds for all $r\in(0,+\infty)$ outside a set of finite measure.

2. Some preliminaries in angular Nevanlinna theory for meromorphic functions

First, we remind some definitions which is contained the book of A. A. Goldberg and I. V. Ostrovskii. We consider the set

[TABLE]

Let $f$ be a meromorphic function on the angle $\overline{\Omega}(\alpha,\beta;r)$, $0<\beta-\alpha\leq 2\pi$, $1\leq r<\infty.$ We recall that

[TABLE]

where

[TABLE]

and $\rho_{n}e^{i\psi_{n}}$ are poles of $f(z)$ counted according with multiplicity. We denote $S_{\alpha\beta}(r,f)$ by the angular Nevanlinna characteristics on $\overline{\Omega}(\alpha,\beta;r)$ and defined as following:

[TABLE]

In order to prove theorems, we need the following lemmas.

Lemma 1**.**

[6](Carleman formula)* Let $f$ be a nonconstant meromorphic function in $\overline{\Omega}(\alpha,\beta;r).$ Then*

[TABLE]

For any pair of real numbers $\alpha$ and $\beta$ in $[0,2\pi)$ with $0<\beta-\alpha\leq 2\pi,$

[TABLE]

Let $f$ be a nonconstant meromorphic function on $\Omega(\alpha,\beta)$. We define

[TABLE]

where the $\mathfrak{n}_{\alpha\beta}(t,f)$ are the number poles of $f$ in the set $\Xi(\alpha,\beta;t)$ counting with multiplicity and $b_{n}=|b_{n}|e^{i\beta_{n}}$ are poles of $f$ in the set $\Xi(\alpha,\beta;t).$

The proximity function of $f$ on $\Omega(\alpha,\beta)$ is given by

[TABLE]

The characteristic function $\mathfrak{T}_{\alpha\beta}(r,f)$ of $f$ on $\Omega(\alpha,\beta)$ is defined by

[TABLE]

Lemma 2**.**

[19]** Let $f$ be a nonconstant meromorphic function in $\Omega(\alpha,\beta).$ Then

[TABLE]

Lemma 3**.**

[6]** Let $k$ be a natural number and $f$ be nonconstant meromorphic function on $\mathbb{C}.$ Then we have the estimate

[TABLE]

holds for $1<r<\infty$ outside a set of finite measure.

Lemma 4**.**

[19]** Let $k$ be a natural number and $f$ be nonconstant meromorphic function on $\Omega(\alpha,\beta).$ Then we have the estimate

[TABLE]

holds for $1<r<\infty$ outside a set of finite measure.

3. Proofs of Theorems

Proof of Theorem 1 and Theorem 2.

First, we prove the Theorem 1. Note that $C_{\alpha\beta}(r,Q(f))=0.$ By the definitions of $S_{\alpha\beta,f}(r)$, $A_{\alpha\beta,f}(r,D)$, $B_{\alpha\beta,f}(r,D)$ and apply to Lemma 1 for $Q(f)\not\equiv 0$, we have

[TABLE]

Hence, we get

[TABLE]

[TABLE]

This is conclusion of Theorem 1.

The end, we prove Theorem 2. We have $\mathfrak{N}_{\alpha\beta}(r,Q(f))=0.$ By the definitions of $\mathfrak{T}_{\alpha\beta,f}(r)$, $\mathfrak{m}_{\alpha\beta,f}(r,D)$, $\mathfrak{N}_{\alpha\beta,f}(r,D)$ and apply to Lemma 2 for $Q(f)\not\equiv 0$, we have

[TABLE]

Thus, we obtain

[TABLE]

We have completed the proof of Theorem 2.

∎

In order to prove the Theorem 3, and Theorem 4, we need some lemmas. First we recall the property of Wronskian.

Let $f_{0},f_{1},\dots,f_{n}$ be meromorphic functions on complex plane $\mathbb{C},$ then Wronskian of $f_{0},f_{1},\dots,f_{n}$ is defined by

[TABLE]

Lemma 5**.**

[9]** Let $f_{0},f_{1},\dots,f_{n}$ be meromorphic functions on $\mathbb{C},$ then

[TABLE]

Let $f=(f_{0}:\dots:f_{n}):\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be holomorophic curve, then Wronskian of $f$ is defined by

[TABLE]

We denote $C_{\alpha\beta,W}(r,0)$ by the counting function in zeros of $W(f_{0},\dots,f_{n})$ in $\overline{\Omega}(\alpha,\beta),$ this means

[TABLE]

We use the notation $\mathfrak{N}_{W}(r,0)$ talking the counting function in zeros of $W(f_{0},\dots,f_{n})$ in $\text{Imz}\geq 0,$ namely

[TABLE]

We call $\mathfrak{N}_{\alpha\beta,W}(r,0)$ the counting function in zeros of $W(f_{0},\dots,f_{n})$ in $\Omega(\alpha,\beta),$ namely

[TABLE]

Let $L_{0},\dots,L_{n}$ are linearly independent forms of $z_{0},\dots,z_{n}$. For $j=0,\dots,n$, set

[TABLE]

By the property of Wronskian there exists a constant $C\neq 0$ such that

[TABLE]

Lemma 6**.**

Let $f=(f_{0}:\dots:f_{n}):\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Then we have

[TABLE]

Here the maximum is taken over all subsets $K$ of $\{1,\dots,q\}$ such that $a_{j}$, $j\in K$, are linearly independent.

Proof.

First, we prove

[TABLE]

Let $K\subset\{1,\dots,q\}$ such that $a_{j},j\in K,$ are linearly independent. Without loss of generality, we may assume that $q\geqslant n+1$ and $\#K=n+1.$ Let $\mathcal{T}$ is the set of all injective maps $\mu:\{0,1,\dots,n\}\rightarrow\{1,\dots,q\}$. Then we have

[TABLE]

Thus, we obtain

[TABLE]

By the property of Wronskian, we see that

[TABLE]

where $C\neq 0$ is constant.

Thus, we get

[TABLE]

Take

[TABLE]

Apply to Lemma 5, we have

[TABLE]

For each $j\in\{1,\dots,n\}$ and $k\in\mathbb{N}^{*},$ using Lemma 3, we have the inequality as following

[TABLE]

Futhermore $T(r,g_{\mu(j)})\leq T_{f}(r)+O(1).$ Then from (3.8), we have

[TABLE]

Hence for any $\mu\in\mathcal{T}$ and from (3), we have

[TABLE]

This implies that

[TABLE]

Combining (3) and (3.9), we get the inequality (3). Similarly, we obtain

[TABLE]

and

[TABLE]

We may obtain the conclusion of Lemma 6 by adding (3), (3) and (3) and note that

[TABLE]

We have completed the proof of this lemma. ∎

Lemma 7**.**

Let $f=(f_{0}:\dots:f_{n}):\mathbb{C}\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Let $a_{j}$ be the vector associated with $H_{j}$ for $j=1,\dots,q$. Then

[TABLE]

Proof.

Let $a_{j}=(a^{j}_{0},\dots,a^{j}_{n})$ be the associated vector of $H_{j}$, $1\leqslant j\leqslant q,$ and let $\mathcal{T}$ be the set of all injective maps $\mu:\{0,1,\dots,n\}\longrightarrow\{1,\dots,q\}$. By hypothesis, $H_{1},\dots,H_{q}$ are in general position for any $\mu\in\mathcal{T}$, then the vectors $a_{\mu(0)},\dots,a_{\mu(n)}$ are linearly independent.

Let $\mu\in T$, we have

[TABLE]

Solve the system of linear equations (3.12), we get

[TABLE]

where\biggl{(}b_{j}^{\mu(t)}\biggl{)}_{t,j=0}^{n} is the inverse matrix of \biggl{(}a_{j}^{\mu(t)}\biggl{)}_{t,j=0}^{n}. So there is a constant $C_{\mu}$ satisfying

[TABLE]

Set $C=\max\limits_{\mu\in\mathcal{T}}C_{\mu}$. Then for any $\mu\in\mathcal{T}$, we have

[TABLE]

For any $z\in\overline{\Omega}(\alpha,\beta)$, there exists the mapping $\mu\in\mathcal{T}$ such that

[TABLE]

for $j\notin\{\mu(0),\dots,\mu(n)\}$. Hence

[TABLE]

We have

[TABLE]

[TABLE]

Thus

[TABLE]

Therefore, we conclude that

[TABLE]

This is conclusion of Lemma 7. ∎

By argument as Lemma 6 and Lemma 7, we are easy to get results as following:

Lemma 8**.**

Let $f=(f_{0}:\dots:f_{n}):\Omega(\alpha,\beta)\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Then we have

[TABLE]

Here the maximum is taken over all subsets $K$ of $\{1,\dots,q\}$ such that $a_{j}$, $j\in K$, are linearly independent.

Lemma 9**.**

Let $f=(f_{0}:\dots:f_{n}):\Omega(\alpha,\beta)\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Let $a_{j}$ be the vector associated with $H_{j}$ for $j=1,\dots,q$. Then

[TABLE]

Proof of Theorem 3 and Theorem 4.

First, we prove the Theorem 3. Using Lemma 6 and Lemma 7, we have

[TABLE]

By Corollary 1, we get that

[TABLE]

for any $j\in\{1,\dots,q\}$. So from (3), we have

[TABLE]

For $z_{0}\in\overline{\Omega}(\alpha,\beta)$, we may assume that $(a_{j},f)$ vanishes at $z_{0}$ for $1\leqslant j\leqslant q_{1}$, $(a_{j},f)$ does not vanish at $z_{0}$ for $j>q_{1}$. Hence, there exists a integer $k_{j}$ and nowhere vanishing holomorphic function $g_{j}$ in neighborhood $U$ of $z$ such that

[TABLE]

here $k_{j}=0$ for $q_{1}<j\leqslant q$. We may assume that $k_{j}\geqslant n$ for $1\leqslant j\leqslant q_{0}$, and $1\leqslant k_{j}<n$ for $q_{0}<j\leqslant q_{1}.$ By property of the Wronskian, we have

[TABLE]

where $h(z)$ is holomorphic function on $U$. Then $W(f)$ is vanishes at $z_{0}$ with order at least

[TABLE]

By the definition of $C_{\alpha\beta,f}(r,H),\ C_{\alpha\beta,W}(r,0)$ and $C_{\alpha\beta,f}^{n}(r,H)$, we have

[TABLE]

So from (3.14), we have

[TABLE]

The proof of Theorem 3 is completed.

Next, we prove the Theorem 4. By Lemma 8 and Lemma 9, we have

[TABLE]

Corollary 2 gives that

[TABLE]

for any $j\in\{1,\dots,q\}$. Hence from (3.15), we obtain

[TABLE]

For $z_{0}\in\Omega(\alpha,\beta;r)=\Omega(\alpha,\beta)\cap\{1<|z|<r\},$ we may suppose that $(a_{j},f)$ vanishes at $z_{0}$ for $1\leqslant j\leqslant q_{1}$, $(a_{j},f)$ does not vanish at $z_{0}$ for $j>q_{1}$. Hence, there exists a integer $k_{j}$ and nowhere vanishing holomorphic function $g_{j}$ in neighborhood $U$ of $z$ such that

[TABLE]

here $k_{j}=0$ for $q_{1}<j\leqslant q$. We may assume that $k_{j}\geqslant n$ for $1\leqslant j\leqslant q_{0}$, and $1\leqslant k_{j}<n$ for $q_{0}<j\leqslant q_{1}.$ By property of the Wronskian, we have

[TABLE]

where $h(z)$ is holomorphic function on $U$. Then $W(f)$ is vanishes at $z_{0}$ with order at least

[TABLE]

By the definition of $\mathfrak{N}_{f}(r,H),\mathfrak{N}_{W}(r,0)$ and $\mathfrak{N}_{f}^{n}(r,H)$, we have

[TABLE]

Thus from (3.17), we get the inequality

[TABLE]

This is statement of Theorem 4. ∎

Proof of Theorem 5.

Let ${\large\text{f}}=(f_{0}:\dots:f_{N})$ be a reduced representation of $f,$ where $f_{0},\dots,f_{N}$ are entire functions on $\Omega(\alpha,\beta)$ and have no common zeros. We consider the function $\phi_{i}=Q_{i}\circ{\large\text{f}}=Q_{i}(f_{0},\dots,f_{N}),0\leq i\leq N.$ Let $F=(\phi_{0}f_{0}^{n}:\dots:\phi_{N}f_{N}^{n}).$ Since the hypersurfaces $\{\mathcal{H}_{i}^{n}Q_{i}=0\},0\leq i\leq N,$ are located in general position in $\mathbb{P}^{N}(\mathbb{C}),$ then $F:\Omega(\alpha,\beta)\to\mathbb{P}^{N}(\mathbb{C})$ is a holomorphic curve. Let $\mathfrak{H}_{i},0\leq i\leq N,$ be the hypersurface defined by $\{\mathcal{H}_{i}^{n}Q_{i}=0\},0\leq i\leq N.$ From the hypothesis $\mathfrak{H}_{0},\dots,\mathfrak{H}_{N}$ are in general position, i.e.

[TABLE]

Thus by Hilbert’s Nullstellensatz [15], for any integer k, $0\leq k\leq N,$ there is an integer $m_{k}>n+d$ such that

[TABLE]

where $b_{0},\dots,b_{N}$ are homogeneous forms with coefficients in $\mathbb{C}$ of degree $m_{k}-(n+d).$ This implies

[TABLE]

where $c_{1}$ is a positive constant depending only on the coefficients of $b_{i},0\leq i\leq N,0\leq k\leq N,$ thus depending only on the coefficients of $Q_{i},0\leq i\leq N.$ Therefore,

[TABLE]

From (3.18) and the First Main Theorem, we have

[TABLE]

On the other hand, by applying Theorem 4 to $F$, and the hyperplanes

[TABLE]

and

[TABLE]

yields

[TABLE]

We have

[TABLE]

for all $i=0,\dots,N,$ where $\mathfrak{N}^{N}(r,\dfrac{1}{g})$ is counting function with level of truncation $N$ of $g.$ Hence

[TABLE]

for all $i=0,\dots,N.$ Also note $\mathfrak{N}_{\alpha\beta,F}^{N}(r,H_{N+1})=\mathfrak{N}_{\alpha\beta,f}^{N}(r,D).$ By combining (3) to (3), we obtain

[TABLE]

∎

Proof of Theorem 7.

We suppose that $f\not\equiv g,$ then there are two numbers $i,j\in\{0,\dots,N\},$ $i\neq j$ such that $f_{i}g_{j}\not\equiv f_{j}g_{i}.$ Assume that $z_{0}\in\Omega(\alpha,\beta)$ is a zero of $Q(f),$ where $Q$ is a homogeneous defining $D.$ From condition $f(z)=g(z)$ when $z\in f^{-1}(D)\cup g^{-1}(D),$ we get $f(z_{0})=g(z_{0}).$ This implies $z_{0}$ is a zero of $\dfrac{f_{i}}{f_{j}}-\dfrac{g_{i}}{g_{j}}.$ Therefore, we have

[TABLE]

Apply to Theorem 5, we obtain

[TABLE]

Similarly, we have

[TABLE]

Combining (3.22) and (3.23), we get

[TABLE]

This is a contradiction with $n>(d+N+3)N.$ Hence $f\equiv g.$ ∎

**Acknowledgements

**The author thanks to the Proffesor Jian-Hua Zheng for very helpful comments and useful suggestions in this paper.