# Values distribution of holomorphic curves on annuli into projective
varieties with hypersurface targets

Si Duc Quang  Department of Mathematics
Hanoi National University of Education
136-Xuan Thuy, Cau Giay, Hanoi, Vietnam<EMAIL_ADDRESS>and  Tran An Hai
Division of Mathematics
Banking Academy
12-Chua Boc, Dong Da, Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper has twofold. The first is to establish a Cartan-
Nochka theorem for holomorphic curves from annuli into projective varieties
intersecting hypersurfaces in subgeneral position with truncated counting
functions. The second is to give a second main theorem for such curves with
arbitrary families of hypersurfaces, where the truncation level of the
counting functions is explicitly estimated not depending on the number of
hypersurfaces. As an application, we give a uniqueness theorem for such curves
sharing a few hypersurfaces in a projective variety ignoring multiplicity. Our
results generalize and improve the recent previous results in this topic, such
as [10, 12, 18].

###### Key words and phrases:

Holomorphic curve, second main theorem, defect relation, Nochka weight,
distributive constant, annulus.

###### 2000 Mathematics Subject Classification:

Primary 32H30; Secondary 32H04, 32H25, 14J70.

##  1\. Introduction and Main results

The values distribution theory for non-constant meromorphic functions on the
complex plane was initiated by Nevanlinna [6] in 1926. And then, the theory of
Nevanlinna had been developed to the case of higher dimension by many authors.
Especially, in 1933, Cartan [1] established the most important second main
theorem for linearly nondegenerate meromorphic mappings from
${\mathbb{C}}^{m}$ into ${\mathbb{P}}^{n}({\mathbb{C}})$ and hyperplanes in
general position. By introducing the notion of Nochka weights, in 1983 Nochka
[7] extended the result of Cartan to the case of hyperplanes in subgeneral
position. He proved the following second main theorem.

Theorem A (cf. [7, 8]). Let
$f:{\mathbb{C}}^{m}\to{\mathbb{P}}^{n}({\mathbb{C}})$ be a linearly
nondegenerate meromorphic mapping and $\\{H_{i}\\}_{i=1}^{q}$ a family of
hyperplanes of ${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position. We
have

$\|\
(q-2N+n-1)T_{f}(r)\leq\sum_{i=1}^{q}N^{[n]}(r,\nu_{(H_{i},f)})+o(T_{f}(r)).$

Here, the notation “$\|\ P$” means the assertion $P$ holds for every
$r\in(0,+\infty)$ outside a finite linear measure set; $T_{f}(r)$ and
$N^{[n]}(r,\nu_{(H_{i},f)})$ denote the characteristic function of $f$ and the
truncated (to level $n$) counting function of the divisor $f^{*}H_{i}$,
respectively (see [9] for the definition); and the family
$\\{H_{i}\\}_{i=1}^{q}$ is said to be in $N$-subgeneral position if
$\bigcap_{j=0}^{N}H_{i_{j}}=\varnothing$ for any $1\leq
i_{0}<i_{1}<\cdots<i_{N}\leq q$.

Recently, Quang and An [11] have extended Theorem A to the case of
hypersurface targets. In order to state their result we need to recall the
following. Let $V$ be a complex projective subvariety of
${\mathbb{P}}^{n}({\mathbb{C}})$ of dimension $k\ (k\leq n)$ and
$Q_{1},\ldots,Q_{q}\ (q\geq k+1)$ hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$. The family $\\{Q_{i}\\}_{i=1}^{q}$ is said
to be in $N$-subgeneral position with respect to $V$ if
$V\cap(\bigcap_{j=0}^{N}Q_{i_{j}})=\varnothing$ for any $1\leq
i_{0}<\cdots<i_{N}\leq q$. We say that the family $\\{Q_{i}\\}_{i=1}^{q}$ is
in general position with respect to $V$ if it is in $k$-subgeneral position
with respect to $V$.

Denote by $I(V)$ the ideal of homogeneous polynomials in
${\mathbb{C}}[x_{0},\ldots,x_{n}]$ defining $V$. For a positive integer $d$,
denote by $H_{d}$ the ${\mathbb{C}}$-vector space of all homogeneous
polynomials in ${\mathbb{C}}[x_{0},\ldots,x_{n}]$ of degree $d$, including the
zero polynomial, and define

$I_{d}(V):=\dfrac{H_{d}}{I(V)\cap H_{d}}\text{ and
}H_{V}(d):={\mathrm{dim}}I_{d}(V).$

Then $H_{V}(d)$ is called the Hilbert function of $V$. Each element of
$I_{d}(V)$ which is an equivalent class of an element $D\in H_{d},$ is denoted
by $[D]$ and it defines a hypersurface in ${\mathbb{P}}^{n}({\mathbb{C}})$,
denoted by $D^{*}$. A meromorphic mapping $f$ from ${\mathbb{C}}^{m}$ into $V$
is said to be nondegenerate over $I_{d}(V)$ if there is no $D\in H_{d}$ with
$[D]\neq 0$ such that $f({\mathbb{C}}^{m})\subset D^{*}$. In [11], Quang and
An proved the following result.

Theorem B (cf. [11]). Let $V$ be a complex projective subvariety of
${\mathbb{P}}^{n}({\mathbb{C}})$ of dimension $k\ (k\leq n)$. Let
$\\{Q_{i}\\}_{i=1}^{q}$ be hypersurfaces of ${\mathbb{P}}^{n}({\mathbb{C}})$
in $N$-subgeneral position with respect to $V$ with $\deg Q_{i}=d_{i}\ (1\leq
i\leq q)$. Let $d$ be the least common multiple of $d_{i}^{\prime}$s, i.e.,
$d=lcm(d_{1},\ldots,d_{q})$. Let $f$ be a holomorphic curve of
${\mathbb{C}}^{m}$ into $V$ such that $f$ is nondegenerate over $I_{d}(V)$.
Then, we have

$\biggl{\|}\
\left(q-\dfrac{(2N-k+1)H_{V}(d)}{k+1}\right)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[H_{V}(d)-1]}(r,\nu_{(Q_{i},f)})+o(T_{f}(r)).$

In this theorem, the number $\frac{(2N-k+1)H_{V}(d)}{k+1}$ is called the above
bound of the total defect. We note that, in 1979, B. Shiffman [17] conjectured
that if $f$ is an algebraically nondegenerate holomorphic curve from
${\mathbb{C}}$ into ${\mathbb{P}}^{n}({\mathbb{C}})$ then the total defect of
$f$ with a family of hypersurfaces in general position is bounded above by
$n+1$. In 2004, M. Ru [15] solved this conjecture by establishing a second
main theorem for algebraically nondegenerate holomorphic curves and families
of hypersurfaces of ${\mathbb{P}}^{n}({\mathbb{C}})$ in general position,
where the total defect is bounded above exactly by $n+1$. Later on, in 2009,
M. Ru [16] generalizes his results to the case of hypersurfaces in general
position with respect to a projective variety. Recently, by introducing the
notion of distributive constant of an arbitrary family of hypersurfaces, Quang
[12] extended the results of Ru to the more general case. We recall the
following definition.

Theorem C (cf. [12]). Let $V$ be a projective subvariety of
${\mathbb{P}}^{n}({\mathbb{C}})$ of dimension $k\leq n$, and
$Q_{1},\ldots,Q_{q}$ hypersurfaces in ${\mathbb{P}}^{n}({\mathbb{C}})$. The
distributive constant of the family $\mathcal{Q}=\\{Q_{1},\ldots,Q_{q}\\}$
with respect to $V$ is defined by

$\Delta_{\mathcal{Q},V}:=\underset{\varnothing\neq\Gamma\subset\\{1,\ldots,q\\}}{\max}\frac{\sharp\Gamma}{k-{\mathrm{dim}}\left(\bigcap_{j\in\Gamma}Q_{j}\right)\cap
V}.$

In [12], Quang proved the following second main theorem.

Theorem D (cf. [12]). Let $V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be a smooth
complex projective variety of dimension $k\geq 1$. Let
$\mathcal{Q}=\\{Q_{1},\ldots,Q_{q}\\}$ be a family of hypersurfaces of
${\mathbb{P}}^{n}({\mathbb{C}})$ with the distributive constant $\Delta$ with
respect to $V$, $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$, and $d$ the least common
multiple of $d_{1},\ldots,d_{q}$. Let $f$ be an algebraically non-degenerate
meromorphic mapping from ${\mathbb{C}}^{m}$ into $V$. Then, for every
$\epsilon>0$,

$\bigl{\|}\
\left(q-\Delta(n+1)-\epsilon\right)T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d_{i}}N^{[M_{0}]}(r,\nu_{(Q_{i},f)}),$

where $M_{0}=\left\lfloor
d^{n^{2}+n}\deg(V)^{n+1}e^{n}\Delta^{n}(2n+4)^{n}(n+1)^{n}(q!)^{n}\epsilon^{-n}\right\rfloor$.
Here, by $\lfloor x\rfloor$ we denote the largest integer not exceeding the
real number $x$.

The purpose of this article is to generalize Theorem B and Theorem D to the
case of holomorphic curves from an annulus
${\mathbb{A}}(R_{0})=\\{z\in{\mathbb{C}}|\ \frac{1}{R_{0}}<|z|<R_{0}\\}\
(1<R_{0}\leq+\infty)$ into a projective subvariety of
${\mathbb{P}}^{n}({\mathbb{C}})$. Our results are stated as follows.

######  Theorem 1.1.

Let $V$ be a complex projective subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$
of dimension $k\ (k\leq n)$ and $\\{Q_{i}\\}_{i=1}^{q}$ hypersurfaces of
${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position with respect to
$V$ with $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$. Let $d$ be the least common
multiple of $d_{1},\ldots,d_{q}$. Let $f$ be a holomorphic curve from
${\mathbb{A}}(R_{0})\ (1<R_{0}\leq+\infty)$ into $V$ such that $f$ is
nondegenerate over $I_{d}(V)$. Assume that $q>\dfrac{(2N-k+1)H_{V}(d)}{k+1}.$
Then, we have

$\left(q-\dfrac{(2N-k+1)H_{V}(d)}{k+1}\right)T_{0}(r,f)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{i},f)})+S_{f}(r).$

Here, the characteristic function $T_{0}(r,f)$, the counting function
$N^{[m]}_{0}(r,\nu_{(Q_{i},f)})$ and the small term $S_{f}(r)$ are defined in
Section 2. In order to optimal the above bound of the total defect, we
generalize and improve Theorem D to the following.

######  Theorem 1.2.

Let $V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be a smooth complex projective
variety of dimension $k\geq 1$. Let $\mathcal{Q}=\\{Q_{1},\ldots,Q_{q}\\}$ be
a family of hypersurfaces of ${\mathbb{P}}^{n}({\mathbb{C}})$ with the
distributive constant $\Delta_{\mathcal{Q},V}$ with respect to $V$, $\deg
Q_{i}=d_{i}\ (1\leq i\leq q)$, and $d$ the least common multiple of
$d_{1},\ldots,d_{q}$. Let $f$ be an algebraically nondegenerate holomorphic
curve from ${\mathbb{A}}(R_{0})\ (1<R_{0}\leq+\infty)$ into $V$. Then, for
every $\epsilon>0$,

$(q-\Delta_{\mathcal{Q},V}(n+1)-\epsilon)T_{0}(r,f)\leq\sum_{i=1}^{q}\frac{1}{d_{i}}N^{[L_{0}-1]}_{0}(r,\nu_{(Q_{i},f)})+S_{f}(r),$

where $L_{0}=\left\lfloor d^{n^{2}+n}(\deg
V)^{n+1}e^{n}\Delta_{\mathcal{Q},V}^{n}(2n+5)^{n}((n+1)\Delta_{\mathcal{Q},V}\epsilon^{-1}+1)^{n}\right\rfloor$.

We would like to emphasize that, in the above result the truncation level
$L_{0}$ depends on $V,n,\mathcal{Q}$ and $\epsilon$, but does not depend on
$q$ as in Theorem D. Here, in order to estimate the truncation level $L_{0}$
not depending on $q$, we will use the new below bound of Chow weight in [12]
(see Lemma 4.2) and a new technique proposed in [14] to control the error term
occurring when estimating the Hilbert weights.

For the last purpose of this paper, we will give an application of Theorem
1.1. We prove the following uniqueness theorem for holomorphic curves on
annuli sharing hypersurfaces. Our last result is stated as follows.

######  Theorem 1.3.

Let $V$ be a complex projective subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$
of dimension $k\ (k\leq n)$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position with respect to
$V$ and $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$. Let $d$ be the least common
multiple of $d_{1},\ldots,d_{q}$. Let $f$ and $g$ be admissible holomorphic
curves from $\mathbb{A}(R_{0})$ into $V$ which are nondegenerate over
$I_{d}(V)$. Assume that

(i) ${\mathrm{dim}}f^{-1}(Q_{i}\cap Q_{j})=\varnothing$ for every $1\leq
i<j\leq q,$

(ii) $f=g$ on $\bigcup_{i=1}^{q}(f^{-1}(Q_{i})\cup g^{-1}(Q_{i})).$

Then the following assertions hold:

a) If $q>\dfrac{2(H_{V}(d)-1)}{d}+\dfrac{(2N-k+1)H_{V}(d)}{k+1},$ then $f=g;$

b) If $q>\dfrac{2(2N-k+1)H_{V}(d)}{k+1}$, then there exist $N+1$ hypersurfaces
$Q_{i_{0}},\ldots,Q_{i_{N}}$, $1\leq i_{0}<\cdots<i_{N}\leq q$, such that
$\dfrac{D^{d/d_{i_{0}}}_{i_{0}}({\bf f})}{D^{d/d_{i_{0}}}_{i_{0}}({\bf
g})}=\cdots=\dfrac{D^{d/d_{i_{N}}}_{i_{N}}({\bf
f})}{D^{d/d_{i_{N}}}_{i_{N}}({\bf g})},$ where $D_{i}\ (1\leq i\leq q)$ is a
homogeneous polynomial defining $Q_{i}$ with $\deg D_{i}=d_{i}$, ${\bf f}$ and
${\bf g}$ are reduced representations of $f$ and $g$, respectively.

In the case where $d=1$, Theorem 1.3b) gives us the following uniqueness
theorem for admissible holomorphic curves into
${\mathbb{P}}^{n}({\mathbb{C}})$, which may be linearly degenerate, sharing
few hyperplanes in general position.

######  Corollary 1.4.

Let $\\{H_{i}\\}_{i=1}^{q}$ be hyperplanes in ${\mathbb{P}}^{n}({\mathbb{C}})$
in general position. Let $f$ and $g$ be admissible holomorphic curves of
${\mathbb{A}}(R_{0})$ into ${\mathbb{P}}^{n}({\mathbb{C}})$. Assume that

(i) ${\mathrm{dim}}f^{-1}(H_{i}\cap H_{j})=\varnothing$ for every $1\leq
i<j\leq q,$

(ii) $f=g$ on $\bigcup_{i=1}^{q}(f^{-1}(H_{i})\cup g^{-1}(H_{i})).$

If $q>2(2n-k+1)$, where $k$ is the dimension of the smallest linear subspace
containing the image of $f$, then $f=g.$

Our results not only extend the previous result in the case of mappings on
${\mathbb{C}}^{m}$, but also generalize and improve the recent results for the
case of holomorphic curves on annuli and hypersurfaces targets of other
authors, such as Phuong-Padaphet [10] and Yang-Zhu [18]. The paper is
organized as follows. The basic notions in Nevanlinna theory is recalled in
Section 2. The Cartan-Nochka theorem on annuli for hypersurface targets
(Theorem 1.1) is proved in Section 3. Theorem 1.2 is proved in Section 4. In
the last section, we will prove the uniqueness theorem (Theorem 1.3) and the
Corollary 1.4.

##  2\. Basic notions and auxiliary results from Nevanlinna theory on annuli

(a) Nevanlinna theory for meromorphic functions on annuli.

For a divisor $\nu$ on ${\mathbb{A}}(R_{0})$, which we may regard as a
function on ${\mathbb{A}}(R_{0})$ with values in ${\mathbb{Z}}$ whose support
is a discrete subset of ${\mathbb{A}}(R_{0}),$ and for a positive integer $M$
(maybe $M=+\infty$), we define the counting function of $\nu$ as follows

$N_{0}^{[M]}(r,\nu)=\int\limits_{\frac{1}{r}}^{1}\dfrac{n_{0}^{[M]}(t)}{t}dt+\int\limits_{1}^{r}\dfrac{n_{0}^{[M]}(t)}{t}dt\quad(1<r<R_{0}),$
$\displaystyle\text{ where
}n_{0}^{[M]}(t)=\begin{cases}\sum\limits_{1\leq|z|\leq
t}\min\\{M,\nu(z)\\}&\text{ if }1\leq t<R_{0},\\\
\sum\limits_{t\leq|z|<1}\min\\{M,\nu(z)\\}&\text{ if
}\dfrac{1}{R_{0}}<t<1.\end{cases}$

For brevity we will omit the character [M] if $M=+\infty$.

For a meromorphic function $\varphi$ on ${\mathbb{A}}(R_{0})$, denote by
$\nu^{0}_{\varphi}$ (resp. $\nu^{\infty}_{\varphi}$) the divisor of zeros
(resp. divisor of poles) of $\varphi$. The divisor of $\varphi$ is defined by
$\nu_{\varphi}:=\nu^{0}_{\varphi}-\nu^{\infty}_{\varphi}.$ We define the
proximity function of $\varphi$ by

$m_{0}(r,\varphi)=\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log^{+}|\varphi(\dfrac{1}{r}e^{i\theta})|d\theta+\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log^{+}|\varphi(re^{i\theta})|d\theta-\dfrac{1}{\pi}\int\limits_{0}^{2\pi}\log^{+}|\varphi(e^{i\theta})|d\theta.$

The Nevanlinna characteristic function of $f$ is defined by

$T_{0}(r,\varphi)=m_{0}(r,\varphi)+N_{0}(r,\nu^{\infty}_{\varphi}).$

Throughout this paper, we denote by $S_{f}(r)$ quantities satisfying

  1. (i)

in the case $R_{0}=+\infty$,

$S_{\varphi}(r)=O(\log(rT_{0}(r,\varphi)))$

for $r\in(1,+\infty)$ except for a set $E$ such that
$\int_{E}r^{\lambda-1}dr<+\infty$ for some $\lambda\geq 0$,

  2. (ii)

in the case $R_{0}<+\infty$,

$S_{\varphi}(r)=O(\log(\dfrac{T_{0}(r,\varphi)}{R_{0}-r}))\text{ as
}r\longrightarrow R_{0}$

for $r\in(1,R_{0})$ except for a set $E^{\prime}$ such that
$\int_{E^{\prime}}\dfrac{dr}{(R_{0}-r)^{\lambda+1}}<+\infty$ for some
$\lambda\geq 0$.

The function $\varphi$ is said to be admissible if it satisfies

$\underset{r\longrightarrow+\infty}{\mathrm{limsup}}\dfrac{T_{0}(r,\varphi)}{\log
r}=+\infty\text{ in the case }R_{0}=+\infty$

or

$\underset{r\longrightarrow
R_{0}^{-}}{\mathrm{limsup}}\dfrac{T_{0}(r,\varphi)}{-\log(R_{0}-r)}=+\infty\text{
in the case }1<R_{0}<+\infty.$

Thus for an admissible meromorphic function $\varphi$ on the annulus
${\mathbb{A}}(R_{0})$, we have $S_{\varphi}(r)=o(T_{0}(r,\varphi))$ as
$r\longrightarrow R_{0}$ for all $1\leq r<R_{0}$ except for the set $E$ or the
set $E^{\prime}$ mentioned above, respectively (cf. [2]).

######  Lemma 2.1 (Lemma on logarithmic derivatives [2, 4, 5]).

Let $\varphi$ be a nonzero meromorphic function on ${\mathbb{A}}(R_{0})$. Then
for each positive integer $k$ we have

$m_{0}\left(r,\dfrac{\varphi^{(k)}}{\varphi}\right)=S_{\varphi}(r)\ (1\leq
r<R_{0}).$

Here by $\varphi^{(k)}$ we denote the derivative of order $k$ of the function
$f$.

######  Theorem 2.2 (First main theorem [2, 4, 5]).

Let $f$ be a meromorphic function on ${\mathbb{A}}(R_{0})$. Then for each
$a\in{\mathbb{C}}$ we have

$T_{0}(r,\varphi)=T_{0}\left(r,\dfrac{1}{\varphi-a}\right)+S_{\varphi}(r)\
(1\leq r<R_{0}).$

(b) Nevanlinna theory for holomorphic curves on annuli.

Let $f$ be a holomorphic mapping from an annulus ${\mathbb{A}}(R_{0})$ into
${\mathbb{P}}^{n}({\mathbb{C}})$ with a reduced representation ${\bf
f}=(f_{0},\ldots,f_{n})$. The characteristic function of $f$ is defined by

$T_{0}(r,f)=\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log\|{\bf
f}(re^{i\theta})\|d\theta+\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log\|{\bf
f}(\frac{1}{r}e^{i\theta})\|d\theta-\dfrac{1}{\pi}\int\limits_{0}^{2\pi}\log\|{\bf
f}(e^{i\theta})\|d\theta,$

where $\|{\bf f}\|=(|f_{0}|^{2}+\cdots+|f_{n}|^{2})^{\frac{1}{2}}$.

Let $Q$ be a hyperpsurface in ${\mathbb{P}}^{n}({\mathbb{C}})$ of degree $d$
and $D$ a homogeneous polynomial of degree $d$ defining $Q$, i.e.,
$Q=\\{(x_{0}:\cdots:x_{n})\in{\mathbb{P}}^{n}({\mathbb{C}})|D(x_{0},\ldots,x_{n})=0\\},$
where

$D(x_{0},\ldots,x_{n})=\sum_{I\in\mathcal{I}_{d}}a_{I}x^{I}$

with
$\mathcal{I}_{d}=\\{(i_{0},\ldots,i_{n})\in\mathbb{N}^{n+1}|i_{0}+\cdots+i_{n}=d\\},x^{I}=x_{0}^{i_{0}}\cdots
x_{n}^{i_{n}}$ for $I=(i_{0},\ldots,i_{n})\in\mathcal{I}_{d}$. Define

$D({\bf f})=\sum_{I\in\mathcal{I}_{d}}a_{I}f^{I},$

where $f^{I}=f_{0}^{i_{0}}\cdots f_{n}^{i_{n}}$ for $I=(i_{0},\ldots,i_{n})$.
Then the divisor of $D({\bf f})$ does not depend on the choice of the reduced
representation ${\bf f}$ and the defining homogeneous polynomial $D$ and we
denote it by $\nu_{(Q,f)}$. The proximity function of $f$ with respect to $Q$
is defined by

$\displaystyle m_{0}(r,f,Q)=$
$\displaystyle\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log\dfrac{\|{\bf
f}(re^{i\theta})\|^{d}}{|D({\bf
f})(re^{i\theta})|}d\theta+\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log\dfrac{\|{\bf
f}(\frac{1}{r}e^{i\theta})\|^{d}}{|D({\bf
f})(\frac{1}{r}e^{i\theta})|}d\theta-\dfrac{1}{\pi}\int\limits_{0}^{2\pi}\log\dfrac{\|{\bf
f}(e^{i\theta})\|}{|D({\bf f})(e^{i\theta})|}d\theta.$

By Jensen’s formula, we have the First Main Theorem as follows

$dT_{0}(r,f)=m_{0}(r,f,Q)+N_{0}(r,\nu_{(Q,f)}).$

##  3\. Cartan-Nochka theorem for hypersurfaces

The following lemma on Nochka weights for hypersurfaces is due to Quang-An
[11].

######  Lemma 3.1 (Cf. [11, Lemma 3]).

Let $V$ be a complex projective subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$
of dimension $k\ (k\leq n)$. Let $Q_{1},\ldots,Q_{q}$ be $q\ (q>2N-k+1)$
hypersurfaces in ${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position
with respect to $V$ of the common degree $d.$ Then there are positive rational
constants $\omega_{i}\ (1\leq i\leq q)$ satisfying the following:

i) $0<\omega_{i}\leq 1,\ \forall i\in\\{1,\ldots,q\\}$,

ii) Setting $\tilde{\omega}=\max_{j\in Q}\omega_{j}$, one gets

$\sum_{j=1}^{q}\omega_{j}=\tilde{\omega}(q-2N+k-1)+k+1.$

iii) $\dfrac{k+1}{2N-k+1}\leq\tilde{\omega}\leq\dfrac{k}{N}.$

iv) For $R\subset\\{1,\ldots,q\\}$ with $\sharp R=N+1$, then $\sum_{i\in
R}\omega_{i}\leq k+1$.

v) Let $E_{i}\geq 1\ (1\leq i\leq q)$ be arbitrarily given numbers. For
$R\subset\\{1,\ldots,q\\}$ with $\sharp R=N+1$, there is a subset
$R^{o}\subset R$ such that $\sharp R^{o}=\mathrm{rank}\\{Q_{i}\\}_{i\in
R^{o}}=k+1$ and

$\prod_{i\in R}E_{i}^{\omega_{i}}\leq\prod_{i\in R^{o}}E_{i}.$

The next lemma due to Quang-An [11] is stated for the case of the meromorphic
mappings on ${\mathbb{C}}^{m}$ but it automatically holds for the case of
holomorphic curves on annuli.

######  Lemma 3.2 (cf. [11, Lemma 4]).

Let $\\{Q_{i}\\}_{i\in R}$ be a set of hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$ of the common degree $d$ and let $f$ be a
holomorphic curve of $\mathbb{A}(R_{0})$ into
${\mathbb{P}}^{n}({\mathbb{C}})$. Assume that $\bigcap_{i\in R}Q_{i}\cap
V=\varnothing$. Then there exist positive constants $\alpha$ and $\beta$ such
that

$\alpha\|{\bf f}\|^{d}\leq\max_{i\in R}|D_{i}(f)|\leq\beta\|{\bf f}\|^{d},$

where ${\bf f}$ is a reduced representation of $f$ and $D_{i}\ (i\in R)$ is a
homogeneous polynomial defining $Q_{i}$ of degree $d$.

In order to proof Theorem 1.1, we first prove the general form of second main
theorem as follows.

######  Theorem 3.3.

Let $f$ be a linearly nondegenerate holomorphic curve from
${\mathbb{A}}(R_{0})\ (1<R_{0}\leq+\infty)$ into
${\mathbb{P}}^{n}({\mathbb{C}})$ with a reduced representation ${\bf
f}=(f_{0},\ldots,f_{n})$. Let $\\{L_{i}\\}_{i=1}^{q}$ be a set of linear form
in $n+1$ variables. Then, we have

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\max_{K_{1}}\sum_{i\in
K_{1}}\log\frac{\|{\bf f}(re^{i\theta})\|}{|L_{i}({\bf
f}(re^{i\theta}))|}d\theta$
$\displaystyle+\frac{1}{2\pi}\int_{0}^{2\pi}\max_{K_{2}}\sum_{i\in
K_{2}}\log\frac{\|{\bf f}(\frac{1}{r}e^{i\theta})\|}{|L_{i}({\bf
f}(\frac{1}{r}e^{i\theta}))|}d\theta$
$\displaystyle\leq(n+1)T_{0}(r,f)-N_{0}(r,\nu_{W})+S_{f}(r),$

where the maximums are taken over all subsets $K_{t}\ (t=1,2)$ of
$\\{1,\ldots,q\\}$ so that $\\{L_{i}|i\in K_{t}\\}$ is linear independent, and
$W$ is the wronskian of ${\bf f}$, i.e., $W=\det(f_{j}^{(i)};0\leq i,j\leq
n)$.

###### Proof.

By adding more linear forms if necessary, we suppose that rank$\\{L_{i};1\leq
i\leq q\\}=n+1$. Denote by $\mathcal{L}$ the set of all subsets
$K\subset\\{1,\ldots,q\\}$ such that $\sharp K=n+1$ and $\\{L_{i}|i\in K\\}$
is linear independent. Again, by adding more linear forms, we may assume that
the maximums in the desired inequality of the theorem are taken over all
subsets $K_{t}\in\mathcal{L}\ (t=1,2)$. Note that, for each $K\in\mathcal{L}$,
there is a positive constant $C_{K}$ such that

$|W|=C_{K}|\det(H_{j}({\bf f})^{(i)};j\in K,0\leq i\leq n)|.$

We set

$S_{K}=\log^{+}\left(\frac{|\det(H_{j}({\bf f})^{(i)};j\in K,0\leq i\leq
n)|}{\prod_{j\in K}|H_{j}({\bf f})|}\right).$

By Lemma 2.1, we have

(1)
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}S_{K}(re^{i\theta})d\theta+\frac{1}{2\pi}\int_{0}^{2\pi}S_{K}(\frac{1}{r}e^{i\theta})d\theta=S_{f}(r).$

On the other hand, one has

$\max_{K_{t}\in\mathcal{L}}\sum_{i\in K_{t}}\log\frac{\|{\bf
f}(z)\|\cdot|W(z)|}{|L_{i}({\bf
f}(z))|}\leq\sum_{K\in\mathcal{L}}S_{K}+(n+1)\log\|{\bf f}(z)\|+O(1)\
(t=1,2).$

Integrating both sides of this inequalities and using Jensen’s formula and
(1), we get

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\max_{K_{1}}\sum_{i\in
K_{1}}\log\frac{\|{\bf f}(re^{i\theta})\|}{|L_{i}({\bf
f}(re^{i\theta}))|}d\theta$
$\displaystyle+\frac{1}{2\pi}\int_{0}^{2\pi}\max_{K_{2}}\sum_{i\in
K_{2}}\log\frac{\|{\bf f}(\frac{1}{r}e^{i\theta})\|}{|L_{i}({\bf
f}(\frac{1}{r}e^{i\theta}))|}d\theta$
$\displaystyle+N_{0}(r,\nu_{W})\leq(n+1)T_{0}(r,f)+S_{f}(r).$

The theorem is proved. ∎

###### Proof of Theorem 1.1.

Let $D_{i}$ be a homogeneous polynomial defining $Q_{i}\ (1\leq i\leq q)$ with
$\deg D_{i}=d_{i}$ and ${\bf f}=(f_{0},\ldots,f_{n})$ a reduced representation
of $f$. Replaying $D_{i}$ by $D_{i}^{d/d_{i}}$ if necessary, we may suppose
that all $D_{i}\ (i=1,\ldots,q)$ do have the same degree $d$. It is easy to
see that there is a positive constant $\beta$ such that $\beta\|{\bf
f}\|^{d}\geq|D_{i}({\bf f})|$ for every $1\leq i\leq q.$ Set
$Q:=\\{1,\cdots,q\\}$. Let $\\{\omega_{i}\\}_{i=1}^{q}$ be as in Lemma 3.1 for
the family $\\{Q_{i}\\}_{i=1}^{q}$.

Take a ${\mathbb{C}}$-basis $\\{[A_{i}]\\}_{i=1}^{H_{V}(d)}$ of $I_{d}(V)$,
where $A_{i}\in H_{d}$. Let $F$ be the holomorphic curve from
${\mathbb{A}}(R_{0})$ into ${\mathbb{P}}^{H_{V}(d)-1}({\mathbb{C}})$ with the
reduced representation ${\bf F}=(A_{1}({\bf f}),\ldots,A_{1}({\bf f}))$. Since
$f$ is nondegenerate over $I_{d}(V)$, $F$ is linearly nondegenerate. Then

$W\equiv\det\bigl{(}(A_{j}({\bf f}))^{(i)};1\leq i\leq H_{V}(d),0\leq i\leq
H_{V}(d)-1\bigl{)}\not\equiv 0.$

We denote by $\mathcal{R}^{o}$ the family of all subsets $R^{o}$ of
$\\{1,\ldots,q\\}$ satisfying

$\mathrm{rank}\\{D_{i}\\}_{i\in R^{o}}=\sharp R^{o}=k+1.$

Let $z$ be a fixed point. For each $R\subset Q$ with $\sharp R=N+1,$ we choose
$R^{o}\subset R$ such that $R^{o}\in\mathcal{R}^{o}$ and $R^{o}$ satisfies
Lemma 3.1 v) with respect to numbers $\bigl{\\{}\dfrac{\beta\|{\bf
f}(z)\|^{d}}{|D_{i}({\bf f})(z)|}\bigl{\\}}_{i=1}^{q}$. On the other hand,
there exists $\bar{R}\subset Q$ with $\sharp\bar{R}=N+1$ such that
$|D_{i}({\bf f})(z)|\leq|D_{j}({\bf f})(z)|,\forall
i\in\bar{R},j\not\in\bar{R}$. Since $\bigcap_{i\in\bar{R}}Q_{i}=\varnothing$,
by Lemma 3.2, there exist positive constant $\alpha$ and $\alpha^{\prime}$
(chosen commonly for all $z$) such that

$\alpha\|{\bf f}(z)\|^{d}\leq\max_{i\in\bar{R}}|D_{i}({\bf
f}(z))|\leq\alpha^{\prime}\|{\bf F}(z)\|.$

Then, we get

$\displaystyle\dfrac{\|{\bf
f}(z)\|^{d(\sum_{i=1}^{q}\omega_{i})}}{|Q_{1}^{\omega_{1}}({\bf f}(z))\cdots
Q_{q}^{\omega_{q}}({\bf
f}(z))|}\leq\dfrac{1}{\alpha^{q-N-1}\beta^{N+1}}\prod_{i\in\bar{R}}\left(\dfrac{\beta\|{\bf
f}(z)\|^{d}}{|Q_{i}({\bf f}(z))|}\right)^{\omega_{i}}\leq
c\prod_{i\in\bar{R}^{o}}\left(\dfrac{\|{\bf F}(z)\|}{|L_{i}({\bf
F}(z))|}\right),$

where $c$ is a positive constant and $L_{i}\ (1\leq i\leq q)$ is linear form
in $H_{V}(d)$ variables such that
$L_{i}(A_{1}(x),\ldots,A_{H_{V}(d)}(x))=Q_{i}(x)$ for every
$x=(x_{0},\ldots,x_{n})$. Integrating both sides of the above inequality and
applying Theorem 3.3, we obtain

$d(\sum_{i=1}^{q}\omega_{i})T_{0}(r,f)-\sum_{i=1}^{q}\omega_{i}N_{0}(r,\nu_{(Q_{i},f)})\leq
H_{V}(d)T_{0}(r,F)-N_{0}(r,W)+S_{f}(r).$

Here, we note that $T_{0}(r,F)=dT_{0}(r,f)+S_{f}(r)$. Therefore, we get

(2)
$\displaystyle\begin{split}d&\left(q-2N+k-1-\dfrac{H_{V}(d)-k-1}{\tilde{\omega}}\right)T_{0}(r,f)\\\
&\leq\sum_{i=1}^{q}\dfrac{\omega_{i}}{\tilde{\omega}}N_{0}(r,\nu_{(Q_{i},f)})-\dfrac{1}{\tilde{\omega}}N_{0}(r,\nu_{W})+S_{f}(r).\end{split}$

######  Claim 3.4.

$\sum_{i=1}^{q}\omega_{i}N_{0}(r,\nu_{(Q_{i},f)})-N_{0}(r,\nu_{W})\leq\sum_{i=1}^{q}\omega_{i}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{i},f)})$.

Indeed, let $z$ be a zero of some $D_{i}({\bf f}(z))$. Since
$\\{Q_{i}\\}_{i=1}^{q}$ is in $N$-subgeneral position, $z$ is not zero of more
than $N$ functions $D_{i}({\bf f})$. Without loss of generality, we may assume
that $z$ is zero of $D_{i}({\bf f})$ for each $1\leq i\leq k\leq N$ and $z$ is
not zero of $D_{i}({\bf f})$ for each $i>N$. Put $R=\\{1,\ldots,N+1\\}.$
Choose $R^{1}\subset R$ such that $\sharp R^{1}=\mathrm{rank}\\{D_{i}\\}_{i\in
R^{1}}=k+1$ and $R^{1}$ satisfies Lemma 3.1 v) with respect to numbers
$\bigl{\\{}e^{\max\\{\nu_{(Q_{i},f)}(z)-H_{V}(d)+1,0\\}}\bigl{\\}}_{i=1}^{q}.$
Then we have

$\displaystyle\sum_{i\in
R}\omega_{i}\max\\{\nu_{(D_{i},f)}(z)-H_{V}(d)+1,0\\}\leq\sum_{i\in
R^{1}}\max\\{\nu_{(D_{i},f)}(z)-H_{V}(d)+1,0\\}.$

By usual argument in Nevanlinna theory, we have

$\displaystyle\nu_{W}(z)$ $\displaystyle\geq\sum_{i\in
R^{1}}\max\\{\nu_{L_{i}({\bf F})}(z)-H_{V}(d)+1,0\\}$
$\displaystyle=\sum_{i\in R^{1}}\max\\{\nu_{(Q_{i},f)}(z)-H_{V}(d)+1,0\\}$
$\displaystyle\geq\sum_{i\in
R}\omega_{i}\max\\{\nu_{(Q_{i},f)}(z)-H_{V}(d)+1,0\\}.$

Hence

$\displaystyle\sum_{i=1}^{q}$
$\displaystyle\omega_{i}\nu_{(Q_{i},f)}(z)-\nu_{W}(z)=\sum_{i\in
R}\omega_{i}\nu_{(Q_{i},f)}(z)-\nu_{W}(z)$ $\displaystyle=\sum_{i\in
R}\omega_{i}\min\\{\nu_{(Q_{i},f)}(z),H_{V}(d)-1\\}+\sum_{i\in
R}\omega_{i}\max\\{\nu_{(Q_{i},f)}(z)-H_{V}(d)+1,0\\}-\nu_{W}(z)$
$\displaystyle\leq\sum_{i\in
R}\omega_{i}\min\\{\nu_{(Q_{i},f)}(z),H_{V}(d)-1\\}=\sum_{i=1}^{q}\omega_{i}\min\\{\nu_{(Q_{i},f)}(z),H_{V}(d)-1\\}.$

This yields that

$\sum_{i=1}^{q}\omega_{i}N_{0}(r,\nu_{(Q_{i},f)})-N_{0}(r,\nu_{W})\leq\sum_{i=1}^{q}\omega_{i}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{i},f)}).$

This proves the claim.

Combining Claim 3.4 and (2), we obtain

$\displaystyle d(q-2N+k-1-\dfrac{H_{V}(d)-k-1}{\tilde{\omega}})T_{0}(r,f)$
$\displaystyle\leq\sum_{i=1}^{q}\dfrac{\omega_{i}}{\tilde{\omega}}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{i},f)})+S_{f}(r)$
$\displaystyle\leq\sum_{i=1}^{q}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{i},f)})+S_{f}(r).$

Since $\tilde{\omega}\geq\dfrac{k+1}{2N-k+1}$, the above inequality implies
that

$d\left(q-\dfrac{(2N-k+1)H_{V}(d)}{k+1}\right)T_{0}(r,f)\leq\sum_{i=1}^{q}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{i},f)})+S_{f}(r).$

The theorem is proved. ∎

##  4\. Second main theorem for arbitrary families of hypersurfaces

Let $X\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be a projective subvariety of
dimension $k$ and degree $\delta$. For
$\textbf{a}=(a_{0},\ldots,a_{n})\in\mathbb{Z}^{n+1}_{\geq 0}$ we write ${\bf
x}^{\bf a}$ for the monomial $x^{a_{0}}_{0}\cdots x^{a_{n}}_{n}$. Let $I_{X}$
be the prime ideal in ${\mathbb{C}}[x_{0},\ldots,x_{n}]$ defining $X$. Let
${\mathbb{C}}[x_{0},\ldots,x_{n}]_{u}$ be the vector space of homogeneous
polynomials of degree $u$ (including $0$) in
${\mathbb{C}}[x_{0},\ldots,x_{n}]$. For $u=1,2,\ldots,$ put
$(I_{X})_{u}:={\mathbb{C}}[x_{0},\ldots,x_{n}]_{u}\cap I_{X}$ and define the
Hilbert function $H_{X}$ of $X$ by

$\displaystyle
H_{X}(u):={\mathrm{dim}}{\mathbb{C}}[x_{0},\ldots,x_{n}]_{u}/(I_{X})_{u}.$

Let ${\bf c}=(c_{0},\ldots,c_{n})$ be a tuple in $\mathbb{R}^{n+1}_{\geq 0}$
and let $e_{X}({\bf c})$ be the Chow weight of $X$ with respect to ${\bf c}$.
The $u$-th Hilbert weight $S_{X}(u,{\bf c})$ of $X$ with respect to ${\bf c}$
is defined by

$\displaystyle S_{X}(u,{\bf c}):=\max\sum_{i=1}^{H_{X}(u)}{\bf a}_{i}\cdot{\bf
c},$

where the maximum is taken over all sets of monomials ${\bf x}^{{\bf
a}_{1}},\ldots,{\bf x}^{{\bf a}_{H_{X}(u)}}$ whose residue classes modulo
$I_{X}$ form a basis of ${\mathbb{C}}[x_{0},\ldots,x_{n}]_{u}/(I_{X})_{u}.$

The following theorem is due to Evertse and Ferretti [3].

######  Theorem 4.1 (see [3, Theorem 4.1]).

Let $X\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be an algebraic variety of
dimension $k$ and degree $\delta$. Let $u>\delta$ be an integer and let ${\bf
c}=(c_{0},\ldots,c_{n})\in\mathbb{R}^{n+1}_{\geqslant 0}$. Then

$\dfrac{1}{uH_{X}(u)}S_{X}(u,{\bf c})\geq\dfrac{1}{(k+1)\delta}e_{X}({\bf
c})-\dfrac{(2k+1)\delta}{u}\cdot\left(\max_{i=0,\ldots,n}c_{i}\right).$

The following lemma is due to the first author in [13].

######  Lemma 4.2 (see [13, Lemma 3.2]).

Let $Y$ be a projective subvariety of ${\mathbb{P}}^{R}({\mathbb{C}})$ of
dimension $k\geq 1$ and degree $\delta_{Y}$. Let $\ell\ (\ell\geq k+1)$ be an
integer and let ${\bf c}=(c_{0},\ldots,c_{R})$ be a tuple of non-negative
reals. Let $\mathcal{H}=\\{H_{0},\ldots,H_{R}\\}$ be a set of hyperplanes in
${\mathbb{P}}^{R}({\mathbb{C}})$ defined by $H_{i}=\\{y_{i}=0\\}\ (0\leq i\leq
R)$. Let $\\{i_{1},\ldots,i_{\ell}\\}$ be a subset of $\\{0,\ldots,R\\}$ such
that:

  * (1)

$c_{i_{\ell}}=\min\\{c_{i_{1}},\ldots,c_{i_{\ell}}\\}$,

  * (2)

$Y\cap\bigcap_{j=1}^{\ell-1}H_{i_{j}}\neq\varnothing$,

  * (3)

and $Y\not\subset H_{i_{j}}$ for all $j=1,\ldots,\ell$.

Let $\Delta_{\mathcal{H},Y}$ be the distributive constant of the family
$\mathcal{H}=\\{H_{i_{j}}\\}_{j=1}^{\ell}$ with respect to $Y$. Then

$e_{Y}({\bf
c})\geq\frac{\delta_{Y}}{\Delta_{\mathcal{H},Y}}(c_{i_{1}}+\cdots+c_{i_{\ell}}).$

###### Proof of Theorem 1.2.

Let $D_{1},\ldots,D_{q}$ be a homogeneous polynomial defining the
hypersurfaces $Q_{1},\ldots,Q_{q}$, respectively, where $\deg D_{i}=d_{i}\
(1\leq i\leq q)$. Replacing $D_{i}$ by $D^{d/d_{i}}$ if necessary, we may
assume that $D_{1},\ldots,D_{q}$ have the same degree $d$. It is suffice for
us to consider the case where $q>\Delta_{\mathcal{Q},V}(n+1)$.

Denote by $\sigma_{1},\ldots,\sigma_{n_{0}}$ all bijections from
$\\{0,\ldots,q-1\\}$ into $\\{1,\ldots,q\\}$, where $n_{0}=q!$. For each
$\sigma_{i}$, it is easy to see that $\bigcap_{j=0}^{q-2}Q_{\sigma_{i}(j)}\cap
V=\varnothing$. Then there exists a smallest index $\ell_{i}\leq q-2$ such
that $\bigcap_{j=0}^{\ell_{i}}Q_{\sigma_{i}(j)}\cap V=\varnothing$. Hence
$\bigcap_{j=0}^{\ell_{i}}Q_{\sigma_{i}(j)}\cap V=\varnothing$ for all
$i=1,\ldots,n_{0}$.

By Lemma 3.2, there is a positive constant $A$, chosen commonly for all
$\sigma_{i}$, such that

$\|{\bf f}(z)\|^{d}\leq A\max_{0\leq j\leq\ell_{i}}|D_{\sigma_{i}(j)}({\bf
f})(z)|\ \forall i=1,\ldots,n_{0}.$

Denote by $S(i)$ the set of all $z$ such that $D_{j}({\bf f})(z)\neq 0$ for
all $j=1,\ldots,q$ and

$|D_{\sigma_{i}(0)}({\bf f})(z)|\leq|D_{\sigma_{i}(1)}({\bf
f})(z)|\leq\cdots\leq|D_{\sigma_{i}(q-1)}({\bf f})(z)|.$

Therefore, for every $z\in S(i)$, we have

$\prod_{j=1}^{q}\dfrac{\|{\bf f}(z)\|^{d}}{|D_{j}({\bf f})(z)|}\leq
C\prod_{j=0}^{\ell_{j}}\dfrac{\|{\bf f}(z)\|^{d}}{|D_{\sigma_{i}(j)}({\bf
f})(z)|},$

where $C$ is a positive constant (which is commonly chosen for all
$\sigma_{i}$).

Consider the mapping $\Phi_{z}$ from $V$ into
${\mathbb{P}}^{q-1}({\mathbb{C}})$ which maps the point ${\bf
x}=(x_{0}:\cdots:x_{n})\in V$ into the point

$\Phi({\bf x})=(D_{1}(x):\cdots:D_{q}(x))\in{\mathbb{P}}^{q-1}({\mathbb{C}}),$

where $x=(x_{0},\ldots,x_{n})$. We set

$\tilde{\Phi}(x)=(D_{1}(x),\ldots,D_{q}(x)).$

Let $Y=\Phi(V)$. Since $V\cap\bigcap_{j=1}^{q}Q_{j}=\varnothing$, $\Phi$ is a
finite morphism on $V$ and $Y$ is a projective subvariety of
${\mathbb{P}}^{q-1}({\mathbb{C}})$ with ${\mathrm{dim}}Y=n$ and of degree

$\delta:=\deg Y\leq d^{n}.\deg V=\delta.$

For every ${\bf a}=(a_{1},\ldots,a_{q})\in\mathbb{Z}^{q}_{\geq 0}$ and ${\bf
y}=(y_{1},\ldots,y_{q})$ we denote ${\bf y}^{\bf a}=y_{1}^{a_{1}}\ldots
y_{q}^{a_{q}}$. Let $u$ be a positive integer. We set
$\xi_{u}:=\binom{q+u}{u}$ and define the ${\mathbb{C}}$-vector space

$Y_{u}:={\mathbb{C}}[y_{1},\ldots,y_{q}]_{u}/(I_{Y})_{u}\text{ and
}n_{u}:={\mathrm{dim}}Y_{u}-1.$

We fix a basis $\\{v_{0},\ldots,v_{n_{u}}\\}$ of $Y_{u}$ and consider the
holomorphic map $F:{\mathbb{C}}\rightarrow{\mathbb{P}}^{n_{u}}({\mathbb{C}})$
which has a reduced representation

${\bf F}=(v_{0}(\tilde{\Phi}\circ{\bf
f}),\ldots,v_{n_{u}}(\tilde{\Phi}\circ{\bf
f})):{\mathbb{A}}(R_{0})\rightarrow{\mathbb{C}}^{n_{u}+1}.$

Hence $F$ is linearly nondegenerate, since $f$ is algebraically nondegenerate.

Now, we fix a point $z\not\in\bigcup_{j=1}^{q}(D_{j}({\bf f}))^{-1}(0)$.
Suppose that $z\in S(i_{0})$. We define

${\bf c}_{z}=(c_{1,z},\ldots,c_{q,z})\in\mathbb{R}^{q},$

where

(3) $\displaystyle c_{j,z}:=\log\frac{\|{\bf f}(z)\|^{d}}{|D_{j}({\bf
f}(z))|}\text{ for }j=1,\ldots,q.$

We see that $c_{j,z}\geq 0$ for all $j$. By the definition of the Hilbert
weight, there are ${\bf a}_{0,z},\ldots,{\bf
a}_{n_{u},z}\in\mathbb{Z}^{q}_{\geq 0}$ with

${\bf a}_{i,z}=(a_{i1,z},\ldots,a_{iq,z})\text{ with
}a_{is,z}\in\\{0,\ldots,u\\},$

such that the residue classes modulo $I(Y)_{u}$ of ${\bf y}^{{\bf
a}_{0,z}},\ldots,{\bf y}^{{\bf a}_{n_{u},z}}$ form a basis of $I_{u}(Y)$ and

(4) $\displaystyle S_{Y}(u,{\bf c}_{z})=\sum_{i=0}^{n_{u}}{\bf
a}_{i,z}\cdot{\bf c}_{z}.$

We see that ${\bf y}^{{\bf a}_{i,z}}\in Y_{u}$ (modulo $I(Y)_{u}$). Then we
may rewrite

${\bf y}^{{\bf a}_{i,z}}=L_{i,z}(v_{0},\ldots,v_{n_{u}}),$

where $L_{i,z}\ (0\leq i\leq n_{u})$ are linearly independent linear forms
with coefficients in ${\mathbb{C}}$. We have

$\displaystyle\log\prod_{i=0}^{n_{u}}|L_{i,z}({\bf F}(z))|$
$\displaystyle=\log\prod_{i=0}^{n_{u}}\prod_{1\leq j\leq q}|D_{j}({\bf
f}(z))|^{a_{ij,z}}$ $\displaystyle=-S_{Y}(m,{\bf c}_{z})+\Delta
u(n_{u}+1)\log\|{\bf f}(z)\|+O(u(n_{u}+1)).$

This implies that

$\displaystyle\log\prod_{i=0}^{n_{u}}\dfrac{\|{\bf
F}(z)\|\cdot\|L_{i,z}\|}{|L_{i,z}({\bf F}(z))|}=$ $\displaystyle S_{Y}(u,{\bf
c}_{z})-du(n_{u}+1)\log\|{\bf f}(z)\|$ $\displaystyle+(n_{u}+1)\log\|{\bf
F}(z)\|+O(u(n_{u}+1)).$

Here we note that $L_{i,z}$ depends on $i$, $z$ and $u$, but the number of
these linear forms is finite. We denote by $\mathcal{L}$ the set of all
$L_{i,z}$ occurring in the above inequalities. Then,

(5) $\displaystyle\begin{split}S_{Y}(u,{\bf
c}_{z})\leq&\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}+\Delta u(n_{u}+1)\log\|{\bf f}(z)\|\\\
&-(n_{u}+1)\log\|{\bf F}(z)\|+O(u(n_{u}+1)),\end{split}$

where the maximum is taken over all subsets $\mathcal{J}\subset\mathcal{L}$
with $\sharp\mathcal{J}=n_{u}+1$ and $\\{L;L\in\mathcal{J}\\}$ is linearly
independent. From Theorem 4.1, we have

(6) $\displaystyle\dfrac{1}{u(n_{u}+1)}S_{Y}(u,{\bf c}_{z})\geq$
$\displaystyle\frac{1}{(n+1)\delta}e_{Y}({\bf
c}_{z})-\frac{(2n+1)\delta}{u}\max_{1\leq j\leq q}c_{j,z}.$

One remarks that

$\displaystyle\max_{1\leq j\leq q}c_{j,z}\leq\sum_{1\leq j\leq
q}\log\frac{\|{\bf f}(z)\|^{d}}{|D_{j}({\bf f}(z))|}.$

Combining (5), (6) and the above remark, we get

(7) $\displaystyle\begin{split}\frac{1}{(n+1)\delta}e_{Y}({\bf
c}_{z})\leq&\dfrac{1}{u(n_{u}+1)}\left(\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}-(n_{u}+1)\log\|{\bf F}(z)\|\right)\\\
&+d\log\|{\bf f}(z)\|+\frac{(2n+1)\delta}{u}\max_{1\leq j\leq
q}c_{j,z}+O(1)\\\
\leq&\dfrac{1}{u(n_{u}+1)}\left(\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}-(n_{u}+1)\log\|{\bf F}(z)\|\right)\\\
&+d\log\|{\bf f}(z)\|+\frac{(2n+1)\delta}{u}\sum_{1\leq j\leq
q}\log\frac{\|{\bf f}(z)\|^{d}}{|D_{j}({\bf f}(z))|}+O(1).\end{split}$

We have $V\cap\bigcap_{j=0}^{l_{i_{0}}}Q_{\sigma_{i_{0}}(j)}=\varnothing$.
Then by Theorem 4.2 and (7), we have

(8) $\displaystyle\begin{split}e_{Y}({\bf
c}_{z})&\geq\frac{\delta}{\Delta_{\mathcal{Q},V}}\cdot(c_{\sigma_{i_{0}}(0),z}+\cdots+c_{\sigma_{i_{0}}(l_{i_{0}}),z})=\frac{\delta}{\Delta_{\mathcal{Q},V}}\cdot\log\prod_{j=0}^{l_{i_{0}}}\frac{\|{\bf
f}(z)\|^{d}}{|Q_{\sigma_{i_{0}}(j)}({\bf f}(z))|}\\\
&\geq\frac{\delta}{\Delta_{\mathcal{Q},V}}\cdot\log\prod_{j=1}^{q}\frac{\|{\bf
f}(z)\|^{d}}{|D_{j}({\bf f}(z))|}+O(1).\end{split}$

Then, from (7) and (8) we have

(9)
$\displaystyle\begin{split}\frac{1}{\Delta_{\mathcal{Q},V}(n+1)}&\cdot\log\prod_{j=1}^{q}\frac{\|{\bf
f}(z)\|^{d}}{|D_{j}({\bf f}(z))|}\\\
&\leq\dfrac{1}{u(n_{u}+1)}\left(\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}-(n_{u}+1)\log\|{\bf F}(z)\|\right)\\\
&+d\log\|{\bf f}(z)\|+\frac{(2n+1)\delta}{u}\sum_{1\leq j\leq
q}\log\frac{\|{\bf f}(z)\|^{d}}{|D_{j}({\bf f}(z))|}+O(1),\end{split}$

where the term $O(1)$ does not depend on $z$.

By applying Theorem 3.3 to the linear nondegenerate holomorphic curve $F$ and
the system of linear forms $\mathcal{L}$, we get:

(10)
$\displaystyle\int_{0}^{2\pi}\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\left(\frac{\|{\bf
f}\|}{|L({\bf
f})|}(te^{i\theta})\right)\frac{d\theta}{2\pi}\leq(n_{u}+1)T_{0}(r,F)-N_{0}(r,\nu_{W({\bf
F})})+S_{f}(r),$

where the maximum is taken over all subsets $\mathcal{J}$ of $\mathcal{L}$,
such that $\sharp J=n_{u}+1$ and $\\{L|L\in\mathcal{J}\\}$ is linearly
independent.

Integrating both sides of (9), in the view of (10), we obtain

(11)
$\displaystyle\begin{split}\frac{1}{\Delta_{\mathcal{Q},V}}&\left(dqT_{0}(r,f)-\sum_{i=1}^{q}N_{Q_{i}(f)}(r)\right)\leq
d(n+1)T_{0}(r,f)-\frac{(n+1)}{u(n_{u}+1)}N_{0}(r,\nu_{W({\bf F})})\\\
&+\frac{(2n+1)(n+1)\delta}{u}\sum_{i=1}^{q}m_{0}(r,f,Q_{i})+S_{f}(r).\end{split}$

We now estimate $N_{0}(r,\nu_{W({\bf F})})$. Fix a point $z\in{\mathbb{C}}$.
We set $c_{i}=\max\\{0,\nu_{(Q_{i},f)}(z)-n_{u}\\}$ for $i=1,\ldots,q$, and

${\bf c}=(c_{1},\ldots,c_{q})\in\mathbb{Z}^{q}_{\geq 0}.$

Then, for $i=0,\ldots,n_{u}$, there are

${\bf a}_{i}=(a_{i1},\ldots,a_{iq}),a_{is}\in\\{1,\ldots,u\\}$

such that ${\bf y}^{{\bf a}_{0}},\ldots,{\bf y}^{{\bf a}_{n_{u}}}$ is a basic
of $I_{u}(Y)$ and

$S_{Y_{z}}(u,{\bf c})=\sum_{i=0}^{n_{u}}{\bf a}_{i}\cdot{\bf c}.$

Similarly as above, ${\bf y}^{{\bf a}_{i}}=L_{i}(v_{0},\ldots,v_{n_{u}})$,
where $L_{0},\ldots,L_{n_{u}}$ are linearly independent linear forms with
coefficients in ${\mathbb{C}}$. By the property of the Wronskian, we have

$W({\bf F})=c\det\bigl{(}L_{0}(\tilde{\Phi}({\bf
f}))^{(w)}),\ldots,L_{n_{u}}(\tilde{\Phi}({\bf f}))^{(w)}\bigl{)}_{0\leq w\leq
n_{u}},$

where $c$ is a non-zero constant. This yields that

$\nu_{W({\bf F})}(z)\geq\sum_{i=0}^{n_{u}}\min_{0\leq w\leq
n_{u}}\nu_{L_{i}({\bf f})^{(w)}}(z).$

We easily see that

$\nu_{L_{i}({\bf f})^{(w)}}(z)=\sum_{j=1}^{q}a_{ij}\nu_{D_{j}({\bf
f})^{(w)}}(z)\geq\sum_{j=1}^{q}a_{ij}\max\\{0,\nu_{D_{j}({\bf
f})}(z)-n_{u}\\}.$

Thus, we have

(12) $\displaystyle\nu_{W({\bf F})}(z)\geq\sum_{i=0}^{n_{u}}{{\bf
a}_{i}}\cdot{\bf c}=S_{Y}(u,{\bf c}).$

Take an index $i_{0}$ such that $c_{\sigma_{i_{0}}(0)}\geq
c_{\sigma_{i_{0}}(1)}\geq\cdots\geq c_{\sigma_{i_{0}}(q-1)}$. Hence,
$c_{\sigma_{i_{0}}(j)}=0$ for all $j\geq l_{i_{0}}$. Then by Lemma 4.2 we have

$\Delta_{\mathcal{Q},V}e_{Y}({\bf
c})\geq\delta(c_{\sigma_{i_{0}}(0)}+\cdots+c_{\sigma_{i_{0}}(l_{i_{0}})})=\delta\sum_{j=1}^{q}\max\\{0,\nu_{D_{j}({\bf
f})}(z)-n_{u}\\}.$

On the other hand, by Theorem 4.1 we have that

$\displaystyle\frac{1}{u(n_{u}+1)}S_{Y}(u,{\bf c})$
$\displaystyle\geq\frac{1}{(n+1)\delta}e_{Y}({\bf
c})-\frac{(2n+1)\delta}{u}\max_{1\leq i\leq q}c_{i}$
$\displaystyle\geq\left(\frac{1}{\Delta_{\mathcal{Q},V}(n+1)}-\frac{(2n+1)\delta}{u}\right)\sum_{j=1}^{q}\max\\{0,\nu_{D_{j}({\bf
f})}(z)-n_{u}\\}.$

Combining this inequality and (12), we have

$\displaystyle\dfrac{(n+1)}{u(n_{u}+1)}$ $\displaystyle\nu^{0}_{W({\bf
F})}(z)\geq\dfrac{(n+1)}{u(n_{u}+1)}S_{Y}(u,{\bf c})$
$\displaystyle\geq\left(\frac{1}{\Delta_{\mathcal{Q},V}}-\frac{(2n+1)(n+1)\delta}{u}\right)\sum_{j=1}^{q}\max\\{0,\nu_{D_{j}({\bf
f})}(z)-n_{u}\\}.$

Integrating both sides of this inequality, we obtain

$\dfrac{(n+1)}{u(n_{u}+1)}N_{0}(r,\nu_{W({\bf
F})})\geq\left(\frac{1}{\Delta_{\mathcal{Q},V}}-\frac{(2n+1)(n+1)\delta}{u}\right)\sum_{j=1}^{q}(N_{0}(r,\nu_{(Q_{j},f)})-N_{0}^{[n_{u}]}(r,\nu_{(Q_{j},f)})).$

Combining inequalities (11) and the above inequality and the first main
theorem, we get

$\displaystyle\frac{1}{\Delta_{\mathcal{Q},V}}$
$\displaystyle\left(dqT_{0}(r,f)-\sum_{i=1}^{q}N_{0}(r,\nu_{(Q_{i},f)})\right)\leq
d(n+1)T_{0}(r,f)$
$\displaystyle-\left(\frac{1}{\Delta_{\mathcal{Q},V}}-\frac{(2n+1)(n+1)\delta}{u}\right)\sum_{j=1}^{q}(N_{0}(r,\nu_{(Q_{j},f)})-N_{0}^{[n_{u}]}(r,\nu_{(Q_{j},f)}))$
$\displaystyle+\frac{(2n+1)(n+1)\delta}{u}\sum_{i=1}^{q}(dT_{0}(r,f)-N_{0}(r,\nu_{(Q_{i},f)}))+S_{f}(r).$

By setting
$m_{0}=\frac{1}{\Delta_{\mathcal{Q},V}}-\frac{(2n+1)(n+1)\delta}{u}$, the
above inequality implies that

$\displaystyle\left(q-\frac{n+1}{m_{0}}\right)T_{0}(r,f)\leq\sum_{j=1}^{q}\dfrac{1}{d}N^{[n_{u}]}_{0}(r,\nu_{(Q_{j},f)})+S_{f}(r).$

We choose $u=\lceil\Delta_{\mathcal{Q},V}(2n+1)(n+1)d^{n}\deg
V((n+1)\Delta_{\mathcal{Q},V}+\epsilon)\epsilon^{-1}\rceil$. Then we have

$u\geq\Delta_{\mathcal{Q},V}(2n+1)(n+1)\delta((n+1)\Delta_{\mathcal{Q},V}\epsilon^{-1}+1)$

and hence

$\displaystyle\frac{n+1}{m_{0}}$
$\displaystyle\leq(n+1)\Delta_{\mathcal{Q},V}+\epsilon,$ $\displaystyle n_{u}$
$\displaystyle\leq H_{Y}(u)-1\leq d^{n}\deg V\binom{u+n}{n}-1.$

Note that, we may suppose that $\epsilon<q-\Delta_{\mathcal{Q},V}(n+1)$.
Hence, if $n=1$ then

$\displaystyle n_{u}+1$ $\displaystyle<d^{n}\deg V(1+u)$
$\displaystyle<d^{n}\deg V(\Delta_{\mathcal{Q},V}(2n+1)(n+1)d^{n}\deg
V((n+1)\Delta_{\mathcal{Q},V}\epsilon^{-1}+1)+2)$ $\displaystyle\leq
d^{n^{2}+n}(\deg
V)^{n+1}e^{n}\Delta_{\mathcal{Q},V}^{n}(2n+5)^{n}((n+1)\Delta_{\mathcal{Q},V}\epsilon^{-1}+1)^{n}.$

Otherwise, if $n\geq 2$, we have

$\displaystyle n_{u}+1$ $\displaystyle<d^{n}\deg
Ve^{n}\left(1+\frac{u}{n}\right)^{n}$ $\displaystyle\leq d^{n}\deg
Ve^{n}\left(1+\frac{\Delta_{\mathcal{Q},V}(2n+1)(n+1)d^{n}\deg
V((n+1)\Delta_{\mathcal{Q},V}\epsilon^{-1}+1)+1}{n}\right)^{n}$
$\displaystyle\leq d^{n^{2}+n}(\deg
V)^{n+1}e^{n}\Delta_{\mathcal{Q},V}^{n}(2n+5)^{n}((n+1)\Delta_{\mathcal{Q},V}\epsilon^{-1}+1)^{n}.$

Therefore, we always have

$n_{u}\leq\left\lfloor d^{n^{2}+n}(\deg
V)^{n+1}e^{n}\Delta_{\mathcal{Q},V}^{n}(2n+5)^{n}((n+1)\Delta_{\mathcal{Q},V}\epsilon^{-1}+1)^{n}\right\rfloor-1=L_{0}-1.$

Then, we get

$\displaystyle(q-\Delta_{\mathcal{Q},V}(n+1)-\epsilon)T_{0}(r,f)\leq\sum_{j=1}^{q}\frac{1}{d}N^{[L_{0}-1]}_{0}(r,\nu_{(Q_{j},f)})+S_{f}(r).$

The proof of the theorem is completed. ∎

##  5\. Uniqueness theorem for holomorphic curves on annuli

###### Proof of Theorem 1.2.

Replacing $D_{i}$ by $D_{i}^{d/d_{i}}$ if necessary, without loss of
generality, we may assume that $d_{i}=d$ for all $1\leq i\leq q.$

a) Suppose that $f\neq g$. Then there exist two indexes $s,t$ with $0\leq
s<t\leq n$ such that $H:=f_{s}g_{t}-f_{t}g_{s}\not\equiv 0.$ By the assumption
(ii) of the theorem, we have $H=0$ on
$\bigcup_{i=1}^{q}(\mathrm{Zero}D_{i}({\bf f})\cup\mathrm{Zero}D_{i}({\bf
g}))$. Therefore, we have

$\nu^{0}_{H}\geq\sum_{i=1}^{q}\min\\{1,\nu^{0}_{(Q_{i},f)}\\}.$

This follows that

(13) $\displaystyle
N_{0}(r,\nu^{0}_{H})\geq\sum_{i=1}^{q}N^{[1]}_{0}(r,\nu_{(Q_{i},f)}).$

On the other hand, by the definition of the characteristic function and by the
Jensen formula, we have

$\displaystyle N_{0}(r,\nu^{0}_{H})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\log|(f_{s}g_{t}-f_{t}g_{s})(re^{i\theta})|d\theta+\frac{1}{2\pi}\int_{0}^{2\pi}\log|(f_{s}g_{t}-f_{t}g_{s})(\frac{1}{r}e^{i\theta})|d\theta+O(1)$
$\displaystyle\leq\sum_{h={\bf f},{\bf
g}}\left(\frac{1}{2\pi}\int_{0}^{2\pi}\log\|h(re^{i\theta})\|d\theta+\frac{1}{2\pi}\int_{0}^{2\pi}\log\|h(\frac{1}{r}e^{i\theta})\|d\theta\right)+O(1)$
$\displaystyle=T_{0}(r,f)+T_{0}(r,g)+S_{f}(r)+S_{g}(r).$

Combining this and (13), we obtain

$T_{0}(r,f)+T_{0}(r,g)\geq\sum_{i=1}^{q}N^{[1]}_{0}(r,\nu_{(Q_{i},f)})+S_{f}(r)+S_{g}(r).$

Similarly, we have

$T_{0}(r,f)+T_{0}(r,g)\geq\sum_{i=1}^{q}N^{[1]}_{0}(r,\nu_{(Q_{i},g)})+S_{f}(r)+S_{g}(r).$

Summing-up both sides of the above two inequalities and applying Theorem 1.1,
we have

$\displaystyle 2(T_{0}(r,f)+T_{0}(r,g))$
$\displaystyle\geq\sum_{i=1}^{q}(N^{[1]}_{0}(r,\nu_{(Q_{i},f)})+N^{[1]}_{0}(r,\nu_{(Q_{i},g)}))+S_{f}(r)+S_{g}(r)$
$\displaystyle\geq\sum_{h=f,g}\sum_{i=1}^{q}\dfrac{1}{H_{V}(d)-1}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{i},h)})+S_{f}(r)+S_{g}(r)$
$\displaystyle\geq\dfrac{d}{H_{V}(d)-1}\left(q-\dfrac{(2n-k+1)H_{V}(d)}{k+1}\right)(T_{0}(r,f)+T_{0}(r,g))$
$\displaystyle+S_{f}(r)+S_{g}(r).$

Letting $r\longrightarrow R_{0}$, we get

$2\geq\frac{d}{H_{V}(d)-1}\left(q-\frac{(2n-k+1)H_{V}(d)}{k+1}\right),$
$\text{i.e., }q\leq\frac{2(H_{V}(d)-1)}{d}+\frac{(2n-k+1)H_{V}(d)}{k+1}.$

This is a contradiction. Hence $f=g$. The assertion a) is proved.

b) Suppose that $f\neq g$. By changing indexes if necessary, we may assume
that

$\underbrace{\dfrac{D_{1}({\bf f})}{D_{1}({\bf
g})}\equiv\cdots\equiv\dfrac{D_{k_{1}}({\bf f})}{D_{k_{1}}({\bf g})}}_{\text{
group }1}\not\equiv\underbrace{\dfrac{D_{k_{1}+1}({\bf f})}{D_{k_{1}+1}({\bf
g})}\equiv\cdots\equiv\dfrac{D_{k_{2}}({\bf f})}{D_{k_{2}}({\bf g})}}_{\text{
group }2}$ $\not\equiv\underbrace{\dfrac{D_{k_{2}+1}({\bf
f})}{D_{k_{2}+1}({\bf g})}\equiv\cdots\equiv\dfrac{D_{k_{3}}({\bf
f})}{D_{k_{3}}({\bf g})}}_{\text{ group
}3}\not\equiv\cdot\cdot\cdot\not\equiv\underbrace{\dfrac{D_{k_{s-1}+1}({\bf
f})}{D_{k_{s-1}+1}({\bf g})}\equiv\cdots\equiv\dfrac{D_{k_{s}}({\bf
f})}{D_{k_{s}}({\bf g})}}_{\text{ group }s},$

where $k_{s}=q.$

By the supposition $f\neq g$, the number of elements of each group is at most
$n$. For each $1\leq i\leq q,$ we set

$\sigma(i)=\begin{cases}i+n&\text{ if $i+n\leq q$},\\\ i+n-q&\text{ if
$i+n>q$}\end{cases}$

and

$P_{i}=D_{i}({\bf f})D_{\sigma(i)}({\bf g})-D_{i}({\bf g})D_{\sigma(i)}({\bf
f}).$

Then $\dfrac{D_{i}({\bf f})}{D_{i}({\bf g})}$ and $\dfrac{D_{\sigma(i)}({\bf
f})}{D_{\sigma(i)}({\bf g})}$ belong to two distinct groups, and hence
$P_{i}\not\equiv 0$ for every $1\leq i\leq q.$ It is easy to see that

$\displaystyle\nu_{P_{i}}(z)\geq$
$\displaystyle\min\\{\nu_{(Q_{i},f)}(z),\nu_{Q_{i}(g)}(z)\\}+\min\\{\nu_{D_{\sigma(i)}({\bf
f})}(z),\nu_{D_{\sigma(i)}({\bf g})}(z)\\}$
$\displaystyle+\sum_{\overset{j=1}{j\neq
i,\sigma(i)}}^{q}\min\\{\nu_{(Q_{j},f)}(z),1\\}$ $\displaystyle\geq$
$\displaystyle\sum_{j=i,\sigma(i)}\biggl{(}\min\\{\nu_{(Q_{j},f)}(z),H_{V}(d)-1\\}+\min\\{\nu_{(Q_{j},g)}(z),H_{V}(d)-1\\}$
$\displaystyle-(H_{V}(d)-1)\min\\{\nu_{(Q_{j},f)}(z),1\\}\biggl{)}+\sum_{\overset{j=1}{j\neq
i,\sigma(i)}}^{q}\min\\{\nu_{(Q_{j},f)}(z),1\\}.$

for all $z$ in $\mathbb{A}(R_{0})$.

Integrating both sides of this inequality, we get

(14)
$\displaystyle\begin{split}N_{0}(r,\nu^{0}_{P_{i}})&\geq\sum_{j=i,\sigma(i)}\biggl{(}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{j},f)})+N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{j},g)})\\\
&-(H_{V}(d)-1)N^{[1]}_{0}(r,\nu_{(Q_{j},f)})\biggl{)}+\sum_{\overset{j=1}{j\neq
i,\sigma(i)}}^{q}N^{[1]}_{0}(r,\nu_{(Q_{j},f)}).\end{split}$

By Jensen’s formula and by the definition of the characteristic function, we
have

(15) $\displaystyle N_{P_{i}}(r)$ $\displaystyle\leq d(T_{0}(r,f)+T_{0}(r,g))$

From (14) and (15), we get

$\displaystyle d(T_{0}(r,f)+T_{0}(r,g))$
$\displaystyle\geq\sum_{j=i,\sigma(i)}\biggl{(}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{j},f)})+N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{j},g)})$
$\displaystyle-(H_{V}(d)-1)N^{[1]}_{0}(r,\nu_{(Q_{j},f)})\biggl{)}+\sum_{\overset{j=1}{j\neq
i,\sigma(i)}}^{q}N^{[1]}_{0}(r,\nu_{(Q_{j},f)}).$

Summing-up both sides of this inequality over all $1\leq i\leq q$, we obtain

$\displaystyle dq(T_{0}(r,f)+T_{0}(r,g))$ $\displaystyle\geq
2\sum_{h=f,g}\sum_{j=1}^{q}N^{[H_{V}(d)-1]}_{0}(r,\nu_{(Q_{j},h)})+(q-2H_{V}(d))\sum_{j=}^{q}N^{[1]}_{0}(r,\nu_{(Q_{i},f)})$
$\displaystyle\geq
2d\left(q-\dfrac{(2n-k+1)H_{V}(d)}{k+1}\right)\biggl{(}T_{0}(r,f)+T_{0}(r,g)\biggl{)}+S_{f}(r)+S_{g}(r).$

Letting $r\longrightarrow R_{0}$, we get

$\displaystyle dq$ $\displaystyle\geq
2d\left(q-\dfrac{(2n-k+1)H_{V}(d)}{k+1}\right),$ $\displaystyle\text{i.e., }q$
$\displaystyle\leq\dfrac{2(2n-k+1)H_{V}(d)}{k+1}.$

This is a contradiction.

Hence the assertion b) holds. The theorem is proved. ∎

###### Proof of Corollary 1.4.

Let ${\bf f}=(f_{0},\ldots,f_{n})$ and ${\bf g}=(g_{0},\ldots,g_{n})$ be
reduced representations of $f$ and $g$, respectively. Let $V(f)$ and $V(g)$ be
the smallest linear subspaces of ${\mathbb{P}}^{n}({\mathbb{C}})$ containing
the images of $f$ and $g$, respectively. It is easy to see that $V(f)$ (resp.
$V(g)$) is the intersection of all hyperplanes which contain
$f(\mathbb{A}(R_{0}))$ (resp. $g(\mathbb{A}(R_{0}))$). We may consider $f$
(resp. $g$) as a holomorphic curve into $V(f)$ (resp. $V(g)$) which is
nondegenerate over $I_{1}(V(f))$ (resp. $I_{1}(V(g))$). Of course,
$H_{1},\ldots,H_{q}$ are in $n$-subgeneral position with respect to both
$V(f)$ and $V(g)$.

Now, suppose that $H$ is a hyperplane in ${\mathbb{P}}^{n}({\mathbb{C}})$ such
that $f({\mathbb{A}}(R_{0}))\subset H$. We denoted again by $H$ the
homogeneous linear form defining the hyperplane $H$. If
$g({\mathbb{A}}(R_{0}))\not\subset H$, i.e., $H({\bf g})\not\equiv 0$, then
$H({\bf g})=H({\bf f})=0$ on $\bigcup_{i=1}^{q}g^{-1}(H_{i}),$ and hence

$\displaystyle T_{0}(r,g)$ $\displaystyle\geq
N_{0}(r,\nu_{(H,g)})\geq\sum_{i=1}^{q}N^{[1]}_{0}(r,\nu_{(H_{i},g)})+S_{g}(r)$
$\displaystyle\geq\dfrac{1}{H_{V(g)}(1)-1}\sum_{i=1}^{q}N^{[H_{V(g)}(1)-1]}_{0}(r,\nu_{(H_{i},g)})+S_{g}(r)$
$\displaystyle\geq\dfrac{1}{H_{V(g)}(1)-1}\left(q-2n+(H_{V(g)}-1)-1\right)T_{0}(r,g)+S_{g}(r)$
$\displaystyle\geq\dfrac{H_{V(g)}+1}{H_{V(g)}-1}T_{0}(r,g)+S_{g}(r),$

(here, note that $H_{V(g)}(1)-1={\mathrm{dim}}V(g)$ and $q\geq 2n+3$). This is
a contradiction. Therefore, $g(\mathbb{A}(R_{0}))\subset H$. This implies that
$g(\mathbb{A}(R_{0}))\subset V(f)$, and hence $V(g)\subset V(f)$. Similarly,
we have $V(f)\subset V(g)$. Then $V(f)=V(g)=V$.

We see that $q>\dfrac{2(2n-k+1)H_{V}(1)}{k+1}$, since $H_{V}(1)=k+1$.
Therefore, from Theorem 1.2 b), there exist $n+1$ hyperplanes
$H_{i_{0}},\ldots,H_{i_{n}}$, $1\leq i_{0}<\cdots<i_{n}\leq q$ such that

$\dfrac{H_{i_{0}}({\bf f})}{H_{i_{0}}({\bf g})}=\cdots=\dfrac{H_{i_{n}}({\bf
f})}{H_{i_{n}}({\bf g})}.$

This yields that $f=g$. ∎

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