# Non-integrated defect relation for meromorphic maps from a Kähler manifold
intersecting hypersurfaces in subgeneral of ${\mathbb{P}}^{n}({\mathbb{C}})$

Si Duc Quang  ,  Nguyen Thi Quynh Phuong  and  Nguyen Thi Nhung

###### Abstract.

In this article, we establish a truncated non-integrated defect relation for
meromorphic mappings from an $m$-dimensional complete Kähler manifold into
${\mathbb{P}}^{n}({\mathbb{C}})$ intersecting $q$ hypersurfaces
$Q_{1},...,Q_{q}$ in $k$-subgeneral position of degree $d_{i}$, i.e., the
intersection of any $k+1$ hypersurfaces is emptyset. We will prove that

$\sum_{i=1}^{q}\delta_{f}^{[u-1]}(Q_{i})\leq(k-n+1)(n+1)+\epsilon+\frac{\rho
u(u-1)}{d},$

where $u$ is explicitly estimated and $d$ is the least common multiple of
$d_{i}^{\prime}$s. Our result generalizes and improves previous results. In
the last part of this paper we will apply this result to study the
distribution of the Gauss map of minimal surfaces.

††footnotetext:  2010 Mathematics Subject Classification: Primary 32H30,
32A22; Secondary 30D35.
Key words and phrases: Nevanlinna, second main theorem, meromorphic mapping,
non-integrated defect relation.

##  1\. Introduction and Main result

Let $M$ be a complete Kähler manifold of dimension $m$. Let
$f:M\longrightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a meromorphic mapping
and $\Omega_{f}$ be the pull-back of the Fubini-Study form $\Omega$ on
${\mathbb{P}}^{n}({\mathbb{C}})$ by $f$. For a positive integer $\mu_{0}$ and
a hypersurface $D$ of degree $d$ in ${\mathbb{P}}^{n}({\mathbb{C}})$ with
$f(M)\not\subset D$, we denote by $\nu_{f}(D)(p)$ the intersection
multiplicity of the image of $f$ and $D$ at $f(p)$.

In 1985, H. Fujimoto [5] defined the notion of the non-integrated defect of
$f$ with respect to $D$ truncated to level $\mu_{0}$ by

$\delta_{f}^{[\mu_{0}]}:=1-\inf\\{\eta\geq 0:\eta\text{ satisfies condition
}(*)\\}.$

Here, the condition (*) means that there exists a bounded non-negative
continuous function $h$ on $M$ whose order of each zero is not less than
$\min\\{\nu_{f}(D),\mu_{0}\\}$ such that

$d\eta\Omega_{f}+\dfrac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log
h^{2}\geq[\min\\{\nu_{f}(D),\mu_{0}\\}].$

And then he gave a result analogous to the defect relation in Nevanlinna
theory as follows.

Theorem A (see [5, Theorem 1.1]). Let $M$ be an $m$-dimensional complete
Kähler manifold and $\omega$ be a Kähler form of $M.$ Assume that the
universal covering of $M$ is biholomorphic to a ball in $\mathbb{C}^{m}.$ Let
$f:M\to{\mathbb{P}}^{n}({\mathbb{C}})$ be a meromorphic map which is linearly
nondegenerate (i.e., its image is not contained in any hyperplane of
${\mathbb{P}}^{n}({\mathbb{C}})$). Let $H_{1},\cdots,H_{q}$ be hyperplanes of
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position. For some $\rho\geq 0$,
if there exists a bounded continuous function $h\geq 0$ on $M$ such that

$\rho\Omega_{f}+\mathrm{dd}^{c}\log h^{2}\geq\mathrm{Ric}\ \omega,$

then

$\sum_{i=1}^{q}\delta_{f}^{[n]}(H_{i})\leq n+1+\rho n(n+1).$

Recently, M. Ru-S. Sogome [10] generalized Theorem A to the case of
meromorphic mappings intersecting a family of hypersurfaces in general
position. After that, Q. Yan [12] extended Theorem A by consider the case
where the family of hypersurfaces in subgeneral position. He proved the
following.

Theorem B (see [12, Theorem 1.1]). Let $M$ be an $m$-dimensional complete
Kähler manifold and $\omega$ be a Kähler form of $M.$ Assume that the
universal covering of $M$ is biholomorphic to a ball in $\mathbb{C}^{m}.$ Let
$f$ be an algebraically nondegenerate meromorphic map of $M$ into
${\mathbb{P}}^{n}({\mathbb{C}})$. Let $Q_{1},...,Q_{q}$ be hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$ of degree $P_{Ij},$ in $k$-subgeneral
position in ${\mathbb{P}}^{n}({\mathbb{C}}).$ Let
$d=l.c.m.\\{Q_{1},...,Q_{q}\\}$ (the least common multiple of
$\\{Q_{1},...,Q_{q}\\}$). Denote by $\Omega_{f}$ the pull-back of the Fubini-
Study form of ${\mathbb{P}}^{n}({\mathbb{C}})$ by $f.$ Assume that for some
$\rho\geq 0,$ there exists a bounded continuous function $h\geq 0$ on $M$ such
that

$\rho\Omega_{f}+\mathrm{dd}^{c}\log h^{2}\geq\mathrm{Ric}\ \omega.$

Then, for each $\epsilon>0,$ we have

$\sum_{j=1}^{q}{\delta}_{f}^{[u-1]}(Q_{j})\leq k(n+1)+\epsilon+\dfrac{\rho
u(u-1)}{d},$

where $u=\bigl{(}^{N+n}_{\ \
n}\bigl{)}\leq(3ekdI(\epsilon^{-1}))^{n}(n+1)^{3n}$ and
$N=2kdn^{2}(n+1)^{2}I(\epsilon^{-1}).$

Here, for a real number $x$, we define $I(x):=\min\\{a\in\mathbf{Z}\ ;\
a>x\\}$.

However, the above result of Q. Yan does not yet completely extend the results
of H. Fujimoto and M. Ru-S. Sogome. Indeed, when the family of hypersurfaces
in general position, i.e., $k=n$, the first term in the right hand side of the
defect relation inequality is $n(n+1)$, which is bigger than $(n+1)$ as usual.
Recently, T. V. Tan and V. V. Truong in [11] also gave a non-integrated defect
relation for the family of hypersurfaces in subgeneral position, where this
term is equal to $n+1$. But their definition of “subgeneral position” is quite
special, which has an extra condition on the intersection of these $q$
hypersurfaces (see Definition 1.1(ii) in [11])

The first aim of this paper is to establish a non-integrated defect relation
for meromorphic mappings of complete Kähler manifolds into
${\mathbb{P}}^{n}({\mathbb{C}})$ sharing hypersurfaces located in subgeneral
position which generalizes the above mentioned results and improves the result
of Q. Yan. In usual principle, to treat with the case of family of
hypersurfaces in subgeneral position, we need to generalize the notion of
Nochka weights. However for the case of hypersurfaces, there is no Nochka
weights constructed. In order to over come this difficult, we will use a
technique “replacing hypersurfaces” proposed in [8, 9]. Before stating our
result, we recall the following.

Let $k\geq n$ and $q\geq k+1.$ Let $Q_{1},...,Q_{q}$ be hypersurfaces in
$\mathbb{P}^{n}(\mathbb{C}).$ The hypersurfaces $Q_{1},...,Q_{q}$ are said to
be in $k$-subgeneral position in $\mathbb{P}^{n}(\mathbb{C})$ if

$Q_{j_{1}}\cap\cdots\cap Q_{j_{k+1}}=\emptyset\text{ for every }1\leq
j_{1}<\cdots<j_{k+1}\leq q.$

If $\\{Q_{i}\\}_{i=1}^{q}$ is in $n$-subgeneral position then we say that it
is in general position.

Our main result is stated as follows.

######  Theorem 1.1.

Let $M$ be an $m$-dimensional complete Kähler manifold and $\omega$ be a
Kähler form of $M.$ Assume that the universal covering of $M$ is biholomorphic
to a ball in $\mathbb{C}^{m}.$ Let $f$ be an algebraically nondegenerate
meromorphic map of $M$ into ${\mathbb{P}}^{n}({\mathbb{C}})$. Let
$Q_{1},...,Q_{q}$ be hypersurfaces in ${\mathbb{P}}^{n}({\mathbb{C}})$ of
degree $d_{j},$ in $k$-subgeneral position in
${\mathbb{P}}^{n}({\mathbb{C}}).$ Let $d=l.c.m.\\{d_{1},...,d_{q}\\}$ (the
least common multiple of $\\{d_{1},...,d_{q}\\}$). Assume that for some
$\rho\geq 0,$ there exists a bounded continuous function $h\geq 0$ on $M$ such
that

$\rho\Omega_{f}+\mathrm{dd}^{c}\log h^{2}\geq\mathrm{Ric}\ \omega.$

Then, for each $\epsilon>0,$ we have

$\sum_{j=1}^{q}{\delta}^{[u-1]}_{f}(Q_{j})\leq p(n+1)+\epsilon+\dfrac{\rho
u(u-1)}{d},$

where $p=k-n+1,\ u=\bigl{(}^{N+n}_{\ \ n}\bigl{)}\leq
e^{n+2}(dp(n+1)^{2}I(\epsilon^{-1}))^{n}$ and
$N=(n+1)d+p(n+1)^{3}I(\epsilon^{-1})d.$

Then we see that, if the family of hypersurfaces is in general position, i.e.,
$k=n$, then our result deduces the results of H. Fujimoto and also of M. Ru-S.
Sogome. Of course, compaired to the original form of Cartan-Nochka’s theorem
where the first term in the right hand side of the defect relation inequality
is $(2k-n+1)$, our result is still not yet optimal. Therefore, how to give a
sharp defect relation in this case is an open question.

In the above theorem, letting $\epsilon=1+\epsilon^{\prime}$ with
$\epsilon^{\prime}>0$ and then letting $\epsilon^{\prime}\longrightarrow 0$,
we obtain the following corollary.

######  Corollary 1.2.

With the assumption of Theorem 1.1, we have

$\sum_{j=1}^{q}{\delta}^{[u-1]}_{f}(Q_{j})\leq p(n+1)+1+\dfrac{\rho
u(u-1)}{d},$

where $p=k-n+1,\ u=\bigl{(}^{N+n}_{\ \ n}\bigl{)}\leq
e^{n+2}(dp(n+1)^{2})^{n}$ and $N=(n+1)d(1+p(n+1)^{2}).$

In the last part of this paper, we will apply Theorem 1.1 to give a non-
integrated defect relation of the Gauss map of a regular submanifold of
${\mathbb{C}}^{m}$ (see Theorem 4.2 below).

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

2.1. Counting function. We set
$||z||=\big{(}|z_{1}|^{2}+\dots+|z_{m}|^{2}\big{)}^{1/2}$ for
$z=(z_{1},\dots,z_{m})\in\mathbb{C}^{m}$ and define

$\displaystyle\mathbb{B}^{m}(r)$
$\displaystyle:=\\{z\in\mathbb{C}^{m}:||z||<r\\},$ $\displaystyle S(r)$
$\displaystyle:=\\{z\in\mathbb{C}^{m}:||z||=r\\}\ (0<r<\infty).$

Define

$v_{m-1}(z):=\big{(}dd^{c}||z||^{2}\big{)}^{m-1}\quad\quad\text{and}$
$\sigma_{m}(z):=d^{c}\text{log}||z||^{2}\land\big{(}dd^{c}\text{log}||z||^{2}\big{)}^{m-1}\text{on}\quad\mathbb{C}^{m}\setminus\\{0\\}.$

For a divisor $\nu$ on a ball $\mathbb{B}^{m}(R)$ of $\mathbb{C}^{m}$, with
$R>0$, and for a positive integer $M$ or $M=\infty$, we define the counting
function of $\nu$ by

$\displaystyle\nu^{[M]}(z)=\min\ \\{M,\nu(z)\\},$ $\displaystyle
n(t)=\begin{cases}\int\limits_{|\nu|\,\cap\mathbb{B}^{m}(t)}\nu(z)v_{m-1}&\text{
if }m\geq 2,\\\ \sum\limits_{|z|\leq t}\nu(z)&\text{ if }m=1.\end{cases}$

Similarly, we define $n^{[M]}(t).$

Define

$N(r,r_{0},\nu)=\int\limits_{r_{0}}^{r}\dfrac{n(t)}{t^{2m-1}}dt\quad(0<r_{0}<r<R).$

Similarly, define $N(r,r_{0},\nu^{[M]})$ and denote it by
$N^{[M]}(r,r_{0},\nu)$.

Let $\varphi:\mathbb{C}^{m}\longrightarrow\mathbb{B}^{m}(r)$ be a meromorphic
function. Denote by $\nu_{\varphi}$ the zero divisor of $\varphi$. Define

$N_{\varphi}(r,r_{0})=N(r,r_{0},\nu_{\varphi}),\
N_{\varphi}^{[M]}(r,r_{0})=N^{[M]}(r,r_{0},\nu_{\varphi}).$

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

2.2. Characteristic function and first main theorem. Let
$f:\mathbb{\mathbb{B}}^{m}(R)\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a
meromorphic mapping. For arbitrarily fixed homogeneous coordinates
$(w_{0}:\dots:w_{n})$ on $\mathbb{P}^{n}(\mathbb{C})$, we take a reduced
representation $\tilde{f}=(f_{0},\ldots,f_{n})$, which means that each $f_{i}$
is a holomorphic function on $\mathbb{B}^{m}(R)$ and
$f(z)=\big{(}f_{0}(z):\dots:f_{n}(z)\big{)}$ outside the analytic subset
$\\{f_{0}=\dots=f_{n}=0\\}$ of codimension $\geq 2$. Set
$\|\tilde{f}\|=\big{(}|f_{0}|^{2}+\dots+|f_{n}|^{2}\big{)}^{1/2}$.

The characteristic function of $f$ is defined by

$T_{f}(r,r_{0})=\int_{r_{0}}^{r}\dfrac{dt}{t^{2m-1}}\int\limits_{\mathbb{B}^{m}(t)}f^{*}\Omega\wedge
v^{m-1},\ (0<r_{0}<r<R).$

By Jensen’s formula, we will have

$\displaystyle
T_{f}(r,r_{0})=\int\limits_{S(r)}\log\|f\|\sigma_{m}-\int\limits_{S(r_{0})}\log\|\tilde{f}\|\sigma_{m}+O(1),\text{
(as $r\rightarrow R$)}.$

Let $Q$ be a hypersurface in ${\mathbb{P}}^{n}({\mathbb{C}})$ of degree $d$.
Throughout this paper, we sometimes identify a hypersurface with the defining
polynomial if there is no confusion. Then we may write

$Q(\omega)=\sum_{I\in\mathcal{T}_{d}}a_{I}\omega^{I},$

where $\mathcal{T}_{d}=\\{(i_{0},...,i_{n})\in\mathbb{Z}_{+}^{n+1}\ ;\
i_{0}+\cdots+i_{n}=d\\}$, $\omega=(\omega_{0},...,\omega_{n})$,
$\omega^{I}=\omega_{0}^{i_{0}}...\omega_{n}^{i_{n}}$ with
$I=(i_{0},...,i_{n})\in\mathcal{T}_{d}$ and $a_{I}\ (I\in\mathcal{T}_{d})$ are
constants, not all zeros. In the case $d=1$, we call $Q$ a hyperplane of
${\mathbb{P}}^{n}({\mathbb{C}})$.

The proximity function of $f$ with respect to $Q$, denoted by
$m_{f}(r,r_{0},Q)$, is defined by

$m_{f}(r,r_{0},Q)=\int_{S(r)}\log\frac{||\tilde{f}||^{d}}{|Q(\tilde{f})|}\sigma_{m}-\int_{S(r_{0})}\log\frac{||\tilde{f}||^{d}}{|Q(\tilde{f})|}\sigma_{m},$

where $Q(\tilde{f})=Q(f_{0},...,f_{n})$. This definition is independent of the
choice of the reduced representation of $f$.

We denote by $f^{*}Q$ the pullback of the divisor $Q$ by $f$. We may see that
$f^{*}Q$ identifies with the zero divisor $\nu^{0}_{Q(\tilde{f})}$ of the
function $Q(\tilde{f})$. By Jensen’s formula, we have

$N(r,r_{0},f^{*}Q)=N_{Q(\tilde{f})}(r,r_{0})=\int_{S(r)}\log|Q(\tilde{f})|\sigma_{m}-\int_{S(r_{0})}\log|Q(\tilde{f})|\sigma_{m}.$

Then the first main theorem in Nevanlinna theory for meromorphic mappings and
hypersurfaces is stated as follows.

######  Theorem 2.1 (First Main Theorem).

Let $f:\mathbb{B}^{m}(R)\to{\mathbb{P}}^{n}({\mathbb{C}})$ be a holomorphic
map, and let $Q$ be a hypersurface in ${\mathbb{P}}^{n}({\mathbb{C}})$ of
degree $d$. If $f(\mathbb{C})\not\subset Q$, then for every real number $r$
with $r_{0}<r<R$,

$dT_{f}(r,r_{0})=m_{f}(r,r_{0},Q)+N(r,r_{0},f^{*}Q)+O(1),$

where $O(1)$ is a constant independent of $r$.

If $\lim\limits_{r\rightarrow 1}\sup\dfrac{T(r,r_{0})}{\log 1/(1-r)}=\infty,$
then the Nevanlinna’s defect of $f$ with respect to the hypersurface $Q$
truncated to level $l$ is defined by

$\delta^{[l]}_{f,*}(Q)=1-\lim\mathrm{sup}\dfrac{N^{l}(r,r_{0},f^{*}Q)}{T_{f}(r,r_{0})}.$

There is a fact that

$0\leq\delta^{[l]}_{f}(Q)\leq\delta^{[l]}_{f,*}(Q)\leq 1.$

(See Proposition 2.1 in [10])

2.3. Auxiliary results. Repeating the argument in [5, Proposition 4.5], we
have the following.

######  Proposition 2.2.

Let $F_{0},\ldots,F_{N}$ be meromorphic functions on the ball
$\mathbb{B}^{m}(R_{0})$ in $\mathbb{C}^{m}$ such that
$\\{F_{0},\ldots,F_{N}\\}$ are linearly independent over $\mathbb{C}.$ Then
there exists an admissible set

$\\{\alpha_{i}=(\alpha_{i1},...,\alpha_{im})\\}_{i=0}^{N}\subset\mathbb{Z}^{m}_{+}$

with $|\alpha_{i}|=\sum_{j=1}^{m}|\alpha_{ij}|\leq i\ (0\leq i\leq N)$ such
that the following are satisfied:

(i)
$W_{\alpha_{0},\ldots,\alpha_{N}}(F_{0},\ldots,F_{N}){\mathrel{\mathop{{:=}}\limits^{Def}}}\det{({\mathcal{D}}^{\alpha_{i}}\
F_{j})_{0\leq i,j\leq N}}\not\equiv 0.$

(ii)
$W_{\alpha_{0},\ldots,\alpha_{N}}(hF_{0},\ldots,hF_{N})=h^{N+1}W_{\alpha_{0},\ldots,\alpha_{N}}(F_{0},\ldots,F_{N})$
for any nonzero meromorphic function $h$ on $\mathbb{B}^{m}(R_{0}).$

In [10], M. Ru and S. Sogome gave the following lemma on logarithmic
derivative for the meromorphic mappings of a ball in ${\mathbb{C}}^{m}$ into
${\mathbb{P}}^{n}({\mathbb{C}})$.

######  Proposition 2.3 (see [10, Proposition 3.3]).

Let $L_{0},\ldots,L_{N}$ be linear forms of $N+1$ variables and assume that
they are linearly independent. Let $F$ be a meromorphic mapping of the ball
$\mathbb{B}^{m}(R_{0})\subset{\mathbb{C}}^{m}$ into
${\mathbb{P}}^{N}({\mathbb{C}})$ with a reduced representation
$\tilde{F}=(F_{0},\ldots,F_{N})$ and let $(\alpha_{0},\ldots,\alpha_{N})$ be
an admissible set of $F$. Set $l=|\alpha_{0}|+\cdots+|\alpha_{N}|$ and take
$t,p$ with $0<tl<p<1$. Then, for $0<r_{0}<R_{0},$ there exists a positive
constant $K$ such that for $r_{0}<r<R<R_{0}$,

$\int_{S(r)}\biggl{|}z^{\alpha_{0}+\cdots+\alpha_{N}}\dfrac{W_{\alpha_{0},\ldots,\alpha_{N}}(F_{0},\ldots,F_{N})}{L_{0}(\tilde{F})\ldots
L_{N}(\tilde{F})}\biggl{|}^{t}\sigma_{m}\leq
K\biggl{(}\dfrac{R^{2m-1}}{R-r}T_{F}(R,r_{0})\biggl{)}^{p}.$

Here $z^{\alpha_{i}}=z_{1}^{\alpha_{i1}}...z_{m}^{\alpha_{im}},$ where
$\alpha_{i}=(\alpha_{i1},...,\alpha_{im})\in\mathbb{N}^{m}_{0}$.

##  3\. Non-integrated defect relation for nondegenerate mappings sharing
hypersurfaces in subgeneral position

First of all, we need the following lemma due to [8, 9]. For the sake of
completeness, we also include the proofs.

######  Lemma 3.1 (see [8, Lemma 3.1], [9, Lemma 3.1]).

Let $Q_{1},...,Q_{k+1}$ be hypersurfaces in ${\mathbb{P}}^{n}({\mathbb{C}})$
of the same degree $d\geq 1$, such that

$\left(\bigcap_{i=1}^{k+1}Q_{i}\right)=\emptyset.$

Then there exist $n$ hypersurfaces $P_{2},...,P_{n+1}$ of the forms

$P_{t}=\sum_{j=2}^{k-n+t}c_{tj}Q_{j},\ c_{tj}\in{\mathbb{C}},\ t=2,...,n+1,$

such that $\left(\bigcap_{t=1}^{n+1}P_{t}\right)=\emptyset,$ where
$P_{1}=Q_{1}$.

###### Proof.

Set $P_{1}=Q_{1}$. It is easy to see that

$\dim\left(\bigcap_{i=1}^{t}Q_{i}\right)\leq k-t+1,\ t=k-n+2,...,k+1,$

where $\dim\emptyset=-\infty$.

Step 1. We firstly construct $P_{2}$ as follows. For each irreducible
component $I$ of dimension $n-1$ of $P_{1}$, we put

$V_{1I}=\\{c=(c_{2},...,c_{k-n+2})\in{\mathbb{C}}^{k-n+1}\ ;\ I\subset
Q_{c},\text{ where }Q_{c}=\sum_{j=2}^{k-n+2}c_{j}Q_{j}\\}.$

Here, we also consider the case where $Q_{c}$ may be zero polynomial and it
determines all ${\mathbb{P}}^{n}({\mathbb{C}})$. It easy to see that $V_{1I}$
is a subspace of ${\mathbb{C}}^{k-n+1}$. Since
$\dim\left(\bigcap_{i=0}^{k-n+1}Q_{i}\right)\leq k-2$, there exists $i(1\leq
i\leq k-n+1)$ such that $I\not\subset Q_{i}$. This implies that $V_{1I}$ is a
proper subspace of ${\mathbb{C}}^{k-n+1}$. Since the set of irreducible
components of dimension $k-1$ of $P_{0}$ is finite,

${\mathbb{C}}^{k-n+1}\setminus\bigcup_{I}V_{1I}\neq\emptyset.$

Then, there exists $(c_{12},...,c_{1(k-n+2)})\in{\mathbb{C}}^{k-n+1}$ such
that the hypersurface

$P_{2}=\sum_{j=2}^{k-n+2}c_{1j}Q_{j}$

does not contain any irreducible component of dimension $k-1$ of $P_{1}$. This
implies that $\dim\left(P_{1}\cap P_{2}\right)\leq k-2.$

Step 2. Similarly, for each irreducible component $I^{\prime}$ of dimension
$n-2$ of $\left(P_{1}\cap P_{2}\right)$, put

$V_{2I^{\prime}}=\\{c=(c_{2},...,c_{k-n+3})\in{\mathbb{C}}^{k-n+2}\ ;\
I^{\prime}\subset Q^{\prime}_{c},\text{ where
}Q^{\prime}_{c}=\sum_{j=2}^{k-n+3}c_{j}Q_{j}\\}.$

Hence, $V_{2I^{\prime}}$ is a subspace of ${\mathbb{C}}^{k-n+2}$. Since
$\dim\left(\bigcap_{i=1}^{k-n+3}Q_{i}\right)\leq n-3$, there exists $i,(2\leq
i\leq k-n+3)$ such that $I^{\prime}\not\subset Q_{i}$. Hence $V_{2I^{\prime}}$
is a proper subspace of ${\mathbb{C}}^{k-n+2}$. Since the set of irreducible
components of dimension $n-2$ of $\left(P_{1}\cap P_{2}\right)$ is infinite,

${\mathbb{C}}^{k-n+2}\setminus\bigcup_{I^{\prime}}V_{2I^{\prime}}\neq\emptyset.$

Then, there exists $(c_{22},...,c_{2(N-k+3)})\in{\mathbb{C}}^{k-n+2}$ such
that the hypersurface

$P_{3}=\sum_{j=2}^{k-n+3}c_{2j}Q_{j}$

does not contain any irreducible components of dimension $n-2$ of $P_{1}\cap
P_{2}$. Hence $\dim\left(P_{1}\cap P_{2}\cap P_{3}\right)\leq n-3.$

Repeating again the above steps, after the $n$-th step we get the
hypersurfaces $P_{2},...,P_{n+1}$ satisfying that

$\dim\left(\bigcap_{j=1}^{t}P_{j}\right)\leq n-t.$

In particular, $\left(\bigcap_{j=1}^{n+1}P_{j}\right)=\emptyset.$ The lemma is
proved. ∎

Let $f:M\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a meromorphic mapping
with a reduced representation $\tilde{f}=(f_{0},\ldots,f_{n})$. We define

$Q_{i}(\tilde{f})=\sum_{I\in\mathcal{I}_{d}}a_{iI}f^{I},$

where $f^{I}=f_{0}^{i_{0}}\cdots f_{n}^{i_{n}}$ for $I=(i_{0},...,i_{n})$.
Then we can consider $f^{*}Q_{i}=\nu_{Q_{i}(\tilde{f})}$ as divisors. We now
have the following.

######  Lemma 3.2.

Let $\\{Q_{i}\\}_{i\in R}$ be a family of hypersurfaces in
$\mathbb{P}^{n}(\mathbb{C})$ of the common degree $d$ and let $f$ be a
meromorphic mapping of $\mathbb{C}^{m}$ into $\mathbb{P}^{n}(\mathbb{C})$.
Assume that $\bigcap_{i\in R}Q_{i}=\emptyset$. Then, there exist positive
constants $\alpha$ and $\beta$ such that

$\alpha||\tilde{f}||^{d}\leq\max_{i\in
R}|Q_{i}(\tilde{f})|\leq\beta||\tilde{f}||^{d}.$

###### Proof.

Let $(x_{0}:\cdots:x_{n})$ be homogeneous coordinates of
$\mathbb{P}^{n}(\mathbb{C})$. Assume that each $Q_{i}$ is defined by
$\sum_{I\in\mathcal{I}_{d}}a_{iI}x^{I}=0.$

Set $Q_{i}(x)=\sum_{I\in\mathcal{I}_{d}}a_{iI}x^{I}$ and consider the
following function

$h(x)=\dfrac{\max_{i\in R}|Q_{i}(x)|}{||x||^{d}},$

where $||x||=(\sum_{i=0}^{n}|x_{i}|^{2})^{\frac{1}{2}}$.

Since the function $h$ is positive continuous on
${\mathbb{P}}^{n}({\mathbb{C}}),$ by the compactness of
${\mathbb{P}}^{n}({\mathbb{C}})$, there exist positive constants $\alpha$ and
$\beta$ such that $\alpha=\min_{x\in\mathbb{P}^{n}(\mathbb{C})}h(x)$ and
$\beta=\max_{x\in\mathbb{P}^{n}(\mathbb{C})}h(x)$. Therefore, we have

$\alpha||\tilde{f}||^{d}\leq\max_{i\in
R}|Q_{i}(\tilde{f})|\leq\beta||\tilde{f}||^{d}.$

The lemma is proved. ∎

By Jensen’s formula, we have the following lemma.

######  Lemma 3.3.

Let $\\{L_{i}\\}_{i=1}^{u}$ be a family of hypersurfaces in
$\mathbb{P}^{n}(\mathbb{C})$ of the common degree $d$ and let $f$ be a
meromorphic mapping of $\mathbb{B}^{m}(R_{0})\subset\mathbb{C}^{m}$ into
$\mathbb{P}^{n}(\mathbb{C})$, where $u=\binom{n+d}{n}$. Assume that
$\\{L_{i}\\}_{i=1}^{u}$ are linearly independent. Then, for every
$0<r_{0}<r<R_{0}$, we have

$T_{F}(r,r_{0})=dT_{f}(r,r_{0})+O(1),$

where $F$ is the meromorphic mapping of $\mathbb{B}^{m}(R_{0})$ into
${\mathbb{P}}^{u-1}({\mathbb{C}})$ defined by the representation
$F=(L_{1}(\tilde{f}):\cdots:L_{u}(\tilde{f}))$.

Proof of Theorem 1.1. By using the universal covering if necessary, we may
assume that $M=\mathbb{B}^{m}(1)$.

Replacing $Q_{j}$ by $Q^{\frac{d}{d_{j}}}_{j}\ (j=1,...,q)$ if necessary, we
may assume that $Q_{j}\ (j=1,\ldots,q)$ have the same of the common degree
$d$.

It is easy to see that there is a positive constant $\beta$ such that
$\beta||\tilde{f}||^{d}\geq|Q_{i}(\tilde{f})|$ for every $1\leq i\leq q.$ We
set

$\mathcal{A}=\\{(i_{1},...,i_{k+1})\ ;1\leq i_{j}\leq q,i_{j}\neq i_{t}\
\forall j\neq t\\}.$

For each $I=(i_{1},...,i_{k+1})\in\mathcal{A}$, we denote by
$P_{I1},...,P_{I(n+1)}$ the hypersurfaces obtained in Lemma 3.1 with respect
to the family of hypersurfaces $\\{Q_{i_{1}},...,Q_{i_{k+1}}\\}$. It is easy
to see that there exists a positive constant $B\geq 1$, which is chosen common
for all $I\in\mathcal{A}$, such that

$|P_{It}(\omega)|\leq B\max_{1\leq j\leq k+1-n+t}|Q_{i_{j}}(\omega)|,$

for all $\omega=(\omega_{0},...,\omega_{n})\in{\mathbb{C}}^{n+1}$.

Consider a reduced representation
$\tilde{f}=(f_{0},\ldots,f_{n}):\mathbb{B}^{m}(1)\rightarrow{\mathbb{C}}^{n+1}$
of $f$. For a fixed point
$z\in\mathbb{B}^{m}(1)\setminus\bigcup_{i=1}^{q}Q_{i}(\tilde{f})^{-1}(\\{0\\})$.
We may assume that

$|Q_{i_{1}}(\tilde{f})(z)|\leq|Q_{i_{2}}(\tilde{f})(z)|\leq\cdots\leq|Q_{i_{q}}(\tilde{f})(z)|.$

Since $Q_{i_{1}},\ldots,Q_{i_{q}}$ are in $k-$subgeneral position, by Lemma
3.5, there exists a positive constant $A$, which is chosen common for all $z$
and $(i_{1},...,i_{q})$, such that

$||\tilde{f}(z)||^{d}\leq A\max_{1\leq j\leq
k+1}|Q_{i_{j}}(\tilde{f})(z)|=A|Q_{i_{k+1}}(\tilde{f})(z)|.$

Therefore, we have

$\displaystyle\prod_{i=1}^{q}\dfrac{||\tilde{f}(z)||^{d}}{|Q_{i}(\tilde{f})(z)|}$
$\displaystyle\leq
A^{q-k}\prod_{j=1}^{k}\dfrac{||\tilde{f}(z)||^{d}}{|Q_{i_{j}}(\tilde{f})(z)|}$
$\displaystyle\leq
A^{q-k}B^{n}\dfrac{||\tilde{f}(z)||^{kd}}{\bigl{(}\prod_{j=2}^{k-n+1}|Q_{i_{j}}(\tilde{f})(z)|\bigl{)}\cdot\prod_{j=1}^{n}|P_{Ij}(\tilde{f})(z)|}$
$\displaystyle\leq
c_{1}\dfrac{||\tilde{f}(z)||^{(k-n+1)nd}}{\prod_{j=1}^{n}|P_{Ij}(\tilde{f})(z)|^{k-n+1}},$

where $I=(i_{1},...,i_{k+1})$ and $c_{1}$ is a positive constant, which is
chosen common for all $I\in\mathcal{A}$. The above inequality implies that

(3.4)
$\displaystyle\log\prod_{i=1}^{q}\dfrac{||\tilde{f}(z)||^{d}}{|Q_{i}(\tilde{f})(z)|}\leq\log
c_{1}+(k-n+1)\log\prod_{j=1}^{n}\dfrac{||\tilde{f}(z)||^{d}}{|P_{Ij}(\tilde{f})(z)|}.$

Now, for a positive integer $L$, we denote by $V_{L}$ the vector subspace
of${\mathbb{C}}[x_{0},\ldots,x_{n}]$ which consists of all homogeneous
polynomials of degree $L$ and zero polynomial. We see that $N$ divisible by
$d$. Hence, for each $(i)=(i_{1},\ldots,i_{n})\in\mathbf{N}_{0}^{n}$ with
$\sigma(i)=\sum_{s=1}^{n}i_{s}\leq\frac{N}{d}$, we set

$W^{I}_{(i)}=\sum_{(j)=(j_{1},\ldots,j_{n})\geq(i)}P_{I1}^{j_{1}}\cdots
P_{In}^{j_{n}}\cdot V_{N-d\sigma(j)}.$

Then we see that $W^{I}_{(0,\ldots,0)}=V_{N}$ and $W^{I}_{(i)}\supset
W^{I}_{(j)}$ if $(i)<(j)$ (in the sense of lexicographic order). Therefore,
$W^{I}_{(i)}$ is a filtration of $V_{N}$. We have the following lemma due to
[2].

######  Lemma 3.5.

Let
$(i)=(i_{1},\ldots,i_{n}),(i)^{\prime}=(i_{1}^{\prime},\ldots,i_{n}^{\prime})\in\mathbf{N}^{n}_{0}$.
Suppose that $(i^{\prime})$ follows $(i)$ in the lexicographic ordering and
defined

$m^{I}_{(i)}=\dim\dfrac{W^{I}_{(i)}}{W^{I}_{(i)^{\prime}}}.$

Then, we have $m^{I}_{(i)}=d^{n},$ provided $d\sigma(i)<N-nd$.

We assume that

$V_{N}=W^{I}_{(i)_{1}}\supset W^{I}_{(i)_{2}}\supset\cdots\supset
W^{I}_{(i)_{K}},$

where $(i)_{s}=(i_{1s},...,i_{ns})$, $W^{I}_{(i)_{s+1}}$ follows
$W^{I}_{(i)_{s}}$ in the ordering and $(i)_{K}=(\frac{N}{d},0,\ldots,0)$. We
see that $K$ is the number of $n$-tuples $(i_{1},\ldots,i_{n})$ with
$i_{j}\geq 0$ and $i_{1}+\cdots+i_{n}\leq\frac{N}{d}$. Then we easily estimate
that

$K=\binom{\frac{N}{d}+n}{n}.$

We define $m^{I}_{s}=\dim\frac{W^{I}_{(i)_{s}}}{W^{I}_{(i)_{s+1}}}$ for all
$s=1,\ldots,K-1$ and set $m^{I}_{K}=1$.

Let $u=\dim V_{N}$. From the above filtration, we may choose a basis
$\\{\psi^{I}_{1},\ldots,\psi^{I}_{u}\\}$ of $V_{N}$ such that

$\\{\psi_{u-(m^{I}_{s}+\cdots+m^{I}_{K})+1},\ldots,\psi_{u}\\}$

is a basis of $W^{I}_{(i)_{s}}$. For each $s\in\\{1,\ldots,K\\}$ and
$l\in\\{u-(m^{I}_{s}+\cdots+m^{I}_{K})+1,\ldots,u-(m^{I}_{s+1}+\cdots+m^{I}_{K})\\}$,
we may write

$\psi^{I}_{l}=P_{I1}^{i_{1s}}\ldots P_{In}^{i_{ns}}h_{l},\ \text{ where
}(i_{1s},\ldots,i_{ks})=(i)_{s},h_{l}\in V^{I}_{N-d\sigma(i)_{s}}.$

Then we have

$\displaystyle|\psi^{I}_{l}(\tilde{f})(z)|$
$\displaystyle\leq|P_{I1}(\tilde{f})(z)|^{i_{1s}}\ldots|P_{In}(\tilde{f})(z)|^{i_{ks}}|h_{l}(\tilde{f})(z)|$
$\displaystyle\leq
c_{2}|P_{I1}(\tilde{f})(z)|^{i_{1s}}\ldots|P_{In}(\tilde{f})(z)|^{i_{ks}}||\tilde{f}(z)||^{N-d\sigma(i)_{s}}$
$\displaystyle=c_{2}\left(\dfrac{|P_{I1}(\tilde{f})(z)|}{||\tilde{f}(z)||^{d}}\right)^{i_{1s}}\ldots\left(\dfrac{|P_{In}(\tilde{f})(z)|}{||\tilde{f}(z)||^{d}}\right)^{i_{ks}}||\tilde{f}(z)||^{N},$

where $c_{2}$ is a positive constant independently from $l$, $I$, $f$ and $z$.
This implies that

(3.6)
$\displaystyle\begin{split}\log\prod_{l=1}^{u}|\psi^{I}_{l}(\tilde{f})(z)|&\leq\sum_{s=1}^{K}m^{I}_{s}\left(i_{1s}\log\dfrac{|P_{I1}(\tilde{f})(z)|}{||\tilde{f}(z)||^{d}}+\cdots+i_{ns}\log\dfrac{|P_{In}(\tilde{f})(z)|}{||\tilde{f}(z)||^{d}}\right)\\\
&\ \ \ +uN\log||\tilde{f}(z)||+\log c_{2}.\end{split}$

We fix $\phi_{1},...,\phi_{u}$, a basic of $V_{N}$,
$\psi_{s}^{I}(\tilde{f})=L_{s}^{I}(\tilde{F})$, where $L_{s}^{I}$ are linear
forms and $\tilde{F}=(\phi_{1}(\tilde{f}),\ldots,\phi_{u}(\tilde{f}))$ is a
reduced representation of a meromorphic mapping $F$. We set

$\displaystyle b_{j}^{I}=\sum_{s=1}^{K}m_{s}^{I}i_{js},\ 1\leq j\leq k.$

From (3.6) we have that

$\log\prod_{s=1}^{u}|L_{s}^{I}(\tilde{F})(z)|\leq\log\left(\prod_{j=1}^{n}\biggl{(}\dfrac{|P_{Ij}(\tilde{f})(z)|}{||\tilde{f}(z)||^{d}}\biggl{)}^{b_{j}^{I}}\right)+uN\log||\tilde{f}(z)||+\log
c_{2}.$

We set $b=\min_{j,I}b_{j}^{I}$. Because $f$ is algebraically non degenerate
over ${\mathbb{C}}$, $F$ is linearly non degenerate over ${\mathbb{C}}$. Then
there exists an admissible set
$\alpha=(\alpha_{1},...,\alpha_{u})\in(\mathbb{Z}_{+}^{m})^{u}$, with
$|\alpha_{s}|\leq s-1$, such that

$\displaystyle
W^{\alpha}(\phi_{s}(\tilde{f})):=\det(\mathcal{D}^{\alpha_{i}}(\phi_{s}(\tilde{f})))_{1\leq
i,s\leq u}\not\equiv 0.$

We also have

(3.7)
$\displaystyle\begin{split}\log\frac{||\tilde{f}(z)||^{qdb}|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{\prod_{i=1}^{q}|Q_{i}(\tilde{f})(z)|^{b}}&\leq\log\frac{||\tilde{f}(z)||^{pndb}|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{\prod_{j=1}^{n}|P_{Ij}(\tilde{f})(z)|^{pb}}+O(1)\\\
&\leq\log\frac{||\tilde{f}(z)||^{pd\sum_{j=1}^{n}b_{j}^{I}}|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{\prod_{j=1}^{n}|P_{Ij}(\tilde{f})(z)|^{pb^{I}_{j}}}+O(1)\\\
&\leq\log\frac{||\tilde{f}(z)||^{puN}|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{\prod_{i=1}^{u}|\psi_{i}^{I}(\tilde{f})(z)|^{p}}+O(1)\\\
&\leq\log\frac{||\tilde{f}(z)||^{puN}|W^{\alpha}(\psi^{I}_{s}(\tilde{f}))(z)|^{p}}{\prod_{i=1}^{u}|\psi_{i}^{I}(\tilde{f})(z)|^{p}}+O(1),\end{split}$

where
$W^{\alpha}(\psi^{I}_{s}(\tilde{f}))=\det(\mathcal{D}^{\alpha_{i}}(\psi^{I}_{s}(\tilde{f})))_{1\leq
i,s\leq u}$, $O(1)$ depends only on $N$ and $\\{Q_{i}\\}_{i=1}^{q}$. This
inequality implies that

(3.8) $\displaystyle\log\frac{||\tilde{f}(z)||^{qdb-
puN}|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{(\prod_{i=1}^{q}|Q_{i}(\tilde{f})(z)|^{b})}\leq\log\frac{|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{\prod_{i=1}^{u}|\psi_{i}^{I}(\tilde{f})(z)|^{p}}+O(1),$

for all $z\in{\mathbb{C}}^{m}$ outside a proper analytic subset of
${\mathbb{C}}^{m}$, which is the union of zero sets of functions
$Q_{i}(\tilde{f}),P_{Ij}(\tilde{f})$.

Put
$S_{I}=\dfrac{|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|}{\prod_{i=1}^{u}|\psi_{i}^{I}(\tilde{f})(z)|}$.
Then, there exists a positive constant $K_{0}$ such that, for each
$z\in\mathbb{C}^{m}$,

$\displaystyle\frac{||\tilde{f}(z)||^{qdb-
puN}|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{\prod_{i=1}^{q}|Q_{i}(\tilde{f})(z)|^{b}}\leq
K^{p}_{0}.S^{p}_{I}(z).$

for some $I\subset\\{1,...,q\\}$ with $\sharp I=k+1$.

######  Lemma 3.9.

For $N=(n+1)d+p(n+1)^{3}I(\epsilon^{-1})$ as in the assumption, we have

$\displaystyle\mathrm{(a)}$ $\displaystyle\ \ \
\frac{puN}{db}\leq(k-n+1)(n+1)+\epsilon,$ $\displaystyle\mathrm{(b)}$
$\displaystyle\ \ \ u\leq
e^{n+2}\left(dp(n+1)^{2}I(\epsilon^{-1})\right)^{n}.$

Proof of Lemma. For a real number $x\in[0,\frac{1}{(n+1)^{2}}]$, we have

(3.10)
$\displaystyle\begin{split}(1+x)^{n}&=1+nx+\sum_{i=2}^{n}\binom{n}{i}x^{i}\leq
1+nx+\sum_{i=1}^{2}\frac{n^{i}}{i!(n+1)^{2i-2}}x\\\
&=1+nx+\sum_{i=2}^{n}\frac{1}{i!}x\leq 1+(n+1)x.\end{split}$

We also note that

(3.11)
$\displaystyle\frac{(n+1)d}{N-(n+1)d}=\frac{(n+1)d}{p(n+1)^{3}I(\epsilon^{-1})d}\leq\frac{1}{(n+1)^{2}}.$

Now, we have the following estimates. First,

$u=\binom{N+n}{n}=\frac{(N+1)\cdots(N+n)}{1\cdots n}.$

Second, since the number of nonnegative integer $t$-tuples with summation
$\leq T$ is equal to the number of nonnegative integer $(t+1)$-tuples with
summation exactly equal $T\in\mathbf{Z}$, which is $\bigl{(}^{T+t}_{\ \
t}\bigl{)}$, since the sum below is independent of $j$, we have that

$\displaystyle b_{j}^{I}=$ $\displaystyle\sum_{\sigma(i)\leq
N/d}m^{I}_{(i)}i_{j}\geq\sum_{\sigma(i)\leq N/d-n}m^{I}_{(i)}i_{j}$
$\displaystyle=$ $\displaystyle\sum_{\sigma(i)\leq
N/d-n}d^{n}i_{j}=\frac{d^{n}}{n+1}\sum_{\sigma(i)\leq
N/d-n}\sum_{j=1}^{n+1}i_{j}$ $\displaystyle=$
$\displaystyle\frac{d^{n}}{n+1}\sum_{\sigma(i)\leq
N/d-n}\left(\frac{N}{d}-n\right)=\frac{d^{n}}{(n+1)}\left(\frac{N}{d}-n\right)\binom{N/d}{n}$
$\displaystyle=\frac{d^{n}(N/d)(N/d-1)\cdots(N/d-n-1)(N/d-n)}{1\cdots(n+1)d}$
$\displaystyle=\frac{N(N-d)\cdots(N-(n-1)d)(N-nd)}{(n+1)!d}.$

This implies that

$\displaystyle\frac{puN}{db}\leq$ $\displaystyle
p(n+1)\frac{(N+1)\cdots(N+n)}{(N-d)\cdots(N-nd)}=p(n+1)\prod_{j=1}^{n}\frac{N+j}{N-(n+1)d+jd}$
$\displaystyle\leq$ $\displaystyle
p(n+1)\left(\frac{N}{N-(n+1)d}\right)^{n}\leq
p(n+1)\left(1+\frac{(n+1)d}{N-(n+1)d}\right)^{n}$ $\displaystyle\leq$
$\displaystyle p(n+1)\left(1+(n+1)\frac{(n+1)d}{N-(n+1)d}\right)\ \ \ (*)$
$\displaystyle\leq$ $\displaystyle
p(n+1)\left(1+(n+1)\frac{(n+1)d}{p(n+1)^{3}I(\epsilon^{-1})d}\right)$
$\displaystyle\leq$ $\displaystyle
p(n+1)\left(1+\frac{1}{p(n+1)\epsilon^{-1}}\right)=p(n+1)+\epsilon,$

where the inequality (*) comes from (3.10) and (3.11). Also, one can be
estimated that

$\displaystyle u$ $\displaystyle=\binom{N+n}{n}\leq
e^{n}\left(1+\frac{N}{n}\right)^{n}\leq
e^{n}\left(\frac{n+(n+1)d}{n}+\frac{p(n+1)^{3}I(\epsilon^{-1})d}{n}\right)^{n}$
$\displaystyle=e^{n}(p(n+1)^{2}I(\epsilon^{-1})d)^{n}\left(1+\frac{1}{n}+\frac{n+(n+1)d}{np(n+1)^{2}I(\epsilon^{-1})d}\right)^{n}$
$\displaystyle\leq\left(edp(n+1)^{3}I(\epsilon^{-1})\right)^{n}\cdot\left(1+\frac{1}{n}+\frac{2}{n(n+1)}\right)^{n}$
$\displaystyle\leq\left(edp(n+1)^{3}I(\epsilon^{-1})\right)^{n}\cdot\left(1+\frac{2}{n}\right)^{n}\leq
e^{n+2}\left(dp(n+1)^{2}I(\epsilon^{-1})\right)^{n}.$

The lemma is proved.

######  Claim 3.12.

$\bigl{(}b\sum_{j=1}^{q}\nu_{Q_{j}(\tilde{f})}-p\nu_{W^{\alpha}(\phi_{s}(\tilde{f}))}\bigl{)}\leq
b\sum_{i=1}^{q}\min\\{u-1,\nu_{Q_{j}(\tilde{f})}\\}$.

Fix $z\in{\mathbb{C}}^{m}$, we may assume that

$\nu_{Q_{1}(\tilde{f})}(z)\geq\cdots\geq\nu_{Q_{t}(\tilde{f})}(z)>0=\nu_{Q_{t+1}(\tilde{f})}(z)=\cdots=\nu_{Q_{q}(\tilde{f})}(z),$

where $0\leq t\leq k,$ ($t$ may be zero). We denote by
$\\{P_{1},\ldots,P_{n+1}\\}$, the family of hypersurfaces corresponding to the
family $\\{Q_{1},...,Q_{k+1}\\}$ as in the Lemma 3.1. Then we will see that

$\displaystyle\nu_{P_{1}}(z)$ $\displaystyle=\nu_{Q_{1}}(z),$
$\displaystyle\nu_{P_{i}}(z)$ $\displaystyle\geq\nu_{Q_{k-n+i}}(z).$

Put $I=(1,...,n+1)$ and $M=u-1$. We have

$p\nu_{W^{\alpha}(\phi_{s}(\tilde{f}))}(z)=p\nu_{W^{\alpha}(\psi_{s}^{I}(\tilde{f}))}(z)\geq
p\sum_{s=1}^{u}\max\\{\nu_{\psi_{s}^{I}(\tilde{f})}(z)-M,0\\}.$

For $\psi=P_{1}^{i_{1}}...P_{n}^{i_{n}}h\in\\{\psi_{s}^{I}\\}_{s=1}^{u}$, we
have

$\psi(\tilde{f})(z)=P_{1}^{i_{1}}(\tilde{f})(z)\ldots
P_{n}^{i_{n}}(\tilde{f})(z).h(\tilde{f})(z).$

Hence

$\displaystyle\max\\{\nu_{\psi(\tilde{f})}(z)-M,0\\}\geq$
$\displaystyle\sum_{t=1}^{n}\max\\{\nu_{(P_{t}^{i_{t}}(\tilde{f})}(z)-M,0\\}$
$\displaystyle\geq$
$\displaystyle\sum_{t=1}^{n}i_{t}\max\\{\nu_{P_{t}(\tilde{f})}(z)-M,0\\}.$

This implies that

$\displaystyle p\sum_{s=1}^{u}\max$
$\displaystyle\\{\nu_{\psi_{s}(\tilde{f})}(z)-M,0\\}\geq
p\sum_{(i)}m_{(i)}^{I}\sum_{t=1}^{k}i_{t}\max\\{\nu_{P_{t}(\tilde{f})}(z)-M,0\\}$
$\displaystyle=p\sum_{t=1}^{n}b_{t}^{I}\max\\{\nu_{P_{t}(\tilde{f})}(z)-M,0\\}\geq
p\sum_{t=1}^{n}b\max\\{\nu_{P_{t}(\tilde{f})}(z)-M,0\\}$
$\displaystyle\geq\sum_{t=1}^{K}b\max\\{\nu_{Q_{t}(\tilde{f})}(z)-M,0\\}=\sum_{i=1}^{q}b\max\\{\nu_{Q_{i}(\tilde{f})}(z)-M,0\\}$
$\displaystyle=b\sum_{i=1}^{q}\max\\{(f^{*}Q_{i})(z)-M,0\\}=b\sum_{i=1}^{q}\bigl{(}\nu_{Q_{i}(\tilde{f})}(z)-\min\\{u-1,\nu_{Q_{i}(\tilde{f})}(z)\\}\bigl{)}.$

Hence

$\displaystyle
b\sum_{i=1}^{q}\nu_{Q_{i}(\tilde{f})}(z)-p\nu_{W^{\alpha}(\phi_{i}(\tilde{f}))}(z)$
$\displaystyle\leq b\sum_{i=1}^{q}\min\\{u-1,\nu_{Q_{i}(\tilde{f})}\\}.$

The claim is proved.

Assume that

$\rho\Omega_{f}+\dfrac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log
h^{2}\geq\mathrm{Ric}\omega.$

We now suppose that

$\sum_{j=1}^{q}\delta^{[u-1]}_{f}(Q_{j})>\frac{puN}{db}+\dfrac{\rho
pu(u-1)}{db}.$

Then, for each $j\in\\{1,\ldots,q\\},$ there exist constants $\eta_{j}>0$ and
continuous plurisubharmonic function $\tilde{u}_{j}$ such that
$e^{\tilde{u}_{j}}|\varphi_{j}|\leq||\tilde{f}||^{d\eta_{j}},$ where
$\varphi_{j}$ is a holomorphic function with
$\nu_{\varphi_{j}}=\min\\{u-1,f^{*}Q_{j}\\}$ and

$q-\sum_{j=1}^{q}\eta_{j}>\frac{puN}{db}+\dfrac{\rho pu(u-1)}{db}.$

Put $u_{j}=\tilde{u}_{j}+\log|\varphi_{j}|$, then $u_{j}$ is a
plurisubharmonic and

$e^{u_{j}}\leq||\tilde{f}||^{d\eta_{j}},\ j=1,\ldots,q.$

Let

$v(z)=\log\left|(z^{\alpha_{1}+\cdots+\alpha_{u}})^{p}\dfrac{(W^{\alpha}(\phi_{s}(\tilde{f}))(z))^{p}}{(\prod_{i=1}^{q}Q_{i}(\tilde{f})(z))^{b}}\right|+b\sum_{j=1}^{q}u_{j}(z).$

Therefore, we have the following current inequality

$\displaystyle 2dd^{c}[v]$ $\displaystyle\geq
p[\nu_{W^{\alpha}(\phi_{i}(\tilde{f}))}]-b\sum_{j=1}^{q}[\nu_{Q_{i}(\tilde{f})}]+\sum_{j=1}^{q}2dd^{c}[u_{j}]$
$\displaystyle=p[\nu_{W^{\alpha}(\phi_{i}(\tilde{f}))}]-b\sum_{j=1}^{q}[\nu_{Q_{i}(\tilde{f})}]+b\sum_{j=1}^{q}[\min\\{u-1,\nu_{Q_{i}(\tilde{f})}\\}]\geq
0.$

This implies that $v$ is a plurisubharmonic function on $\mathbb{B}^{m}(1)$.

On the other hand, by the growth condition of $f$, there exists a continuous
plurisubharmonic function $\omega\not\equiv\infty$ on $\mathbb{B}^{m}(1)$ such
that

$\displaystyle e^{\omega}dV\leq||\tilde{f}||^{2\rho}v_{m}$

Set

$t=\dfrac{2\rho}{db(q-\frac{puN}{db}-\sum_{j=1}^{q}\eta_{j})}>0$

and

$\lambda(z)=(z^{\alpha_{1}+\cdots+\alpha_{u}})^{p}\dfrac{\left(W^{\alpha}(\phi_{i}(\tilde{f}))\right)^{p}(z)}{Q_{1}^{b}(\tilde{f})(z)\ldots
Q_{q}^{b}(\tilde{f})(z)}.$

We see that

$\frac{u(u-1)p}{2}t<\frac{u(u-1)p}{2}\cdot\dfrac{2\rho}{2\rho u(u-1)p}=1,$

and the function $\zeta=\omega+tv$ is plurisubharmonic on the Kähler manifold
$M$. Choose a position number $\delta$ such that
$0<\frac{u(u-1)pt}{2}<\delta<1.$ Then, we have

$\displaystyle e^{\zeta}dV$ $\displaystyle=e^{\omega+tv}dV\leq
e^{tv}||\tilde{f}||^{2\rho}v_{m}=|\lambda|^{t}(\prod_{j=1}^{q}e^{tbu_{j}})||\tilde{f}||^{2\rho}v_{m}$
$\displaystyle\leq|\lambda|^{t}||\tilde{f}||^{2\rho+\sum_{j=1}^{q}bdt\eta_{j}}v_{m}=|\lambda|^{t}||\tilde{f}||^{dbt(q-\frac{puN}{db})}v_{m}.$

Integrating both sides of the above inequality over $\mathbb{B}^{m}(1),$ we
have

(3.13)
$\displaystyle\begin{split}\int_{\mathbb{B}^{m}(1)}e^{\zeta}dV&\leq\int_{\mathbb{B}^{m}(1)}|\lambda|^{t}||\tilde{f}||^{t(qdb-
puN)}v_{m}.\\\
&=2m\int_{0}^{1}r^{2m-1}\left(\int_{S(r)}\bigl{(}|\lambda|||\tilde{f}||^{qdb-
puN}\bigl{)}^{t}\sigma_{m}\right)dr\\\ &\leq
2m\int_{0}^{1}r^{2m-1}\left(\int_{S(r)}\sum_{{\mathrel{\mathop{{I\subset\\{1,...,q\\}}}\limits^{\sharp
I=k+1}}}}\bigl{|}(z^{\alpha_{1}+\cdots+\alpha_{u}})K_{0}S_{I}\bigl{|}^{pt}\sigma_{m}\right)dr.\end{split}$

(a) We first consider the case where

$\lim\limits_{r\rightarrow 1}\sup\dfrac{T_{f}(r,r_{0})}{\log 1/(1-r)}<\infty.$

We note that $(\sum_{i=1}^{u}|\alpha_{i}|)pt\leq\frac{u(u-1)p}{2}t<\delta<1$.
Then by Proposition 2.3, there exists a positive constant $K_{1}$ such that,
for every $0<r_{0}<r<r^{\prime}<1,$ we have

$\displaystyle\int_{S(r)}\left|(z^{\alpha_{1}+\cdots+\alpha_{u}})K_{0}S_{I}(z)\right|^{pt}\sigma_{m}\leq
K_{1}\left(\dfrac{r^{\prime
2m-1}}{r^{\prime}-r}dT_{f}(r^{\prime},r_{0})\right)^{\delta}.$

Choosing $r^{\prime}=r+\dfrac{1-r}{eT_{f}(r,r_{0})}$, we get

$T_{f}(r^{\prime},r_{0})\leq 2T_{f}(r,r_{0})$

outside a subset $E\subset[0,1]$ with $\int_{E}\frac{dr}{1-r}<+\infty$. Hence,
the above inequality implies that

$\displaystyle\sum_{{\mathrel{\mathop{{I\subset\\{1,...,q\\}}}\limits^{\sharp
I=k+1}}}}\int_{S(r)}\left|(z^{\alpha_{1}+\cdots+\alpha_{u}})K_{0}S_{I}(z)\right|^{pt}\sigma_{m}\leq\dfrac{K}{(1-r)^{\delta}}\left(\log\frac{1}{1-r}\right)^{\delta}$

for all $z$ outside $E$, where $K$ is a some positive constant. By choosing
$K$ large enough, we may assume that the above inequality holds for all
$z\in\mathbb{B}^{m}(1)$. Then, the inequality (3.13) yields that

$\displaystyle\int_{\mathbb{B}^{m}(1)}e^{\zeta}dV$ $\displaystyle\leq
2m\int_{0}^{1}r^{2m-1}\dfrac{K}{(1-r)^{\delta}}\left(\log\frac{1}{1-r}\right)^{\delta}dr<+\infty$

This contradicts the results of S.T. Yau [13] and L. Karp [6].

Hence, we must have

$\sum_{j=1}^{q}\delta^{[u-1]}_{f}(Q_{j})\leq\frac{puN}{db}+\dfrac{p\rho
u(u-1)}{db}.$

Since $p\leq b$, the above inequality implies that

$\sum_{j=1}^{q}\delta^{[u-1]}_{f}(Q_{j})\leq(k-n+1)(n+1)+\epsilon+\dfrac{\rho
u(u-1)}{d}.$

The theorem is proved in this case.

(b) We now consider the remaining case where

$\lim\limits_{r\rightarrow 1}\sup\dfrac{T(r,r_{0})}{\log 1/(1-r)}=\infty.$

Repeating the argument in the proof of Theorem 1.1, we only need to prove the
following theorem.

######  Theorem 3.14.

With the assumption of Theorem 1.1 and suppose that $M=\mathbb{B}^{m}(R_{0})$.
Then, we have

$(q-p(n+1)-\epsilon)T_{f}(r,r_{0})\leq\sum_{i=1}^{q}\dfrac{1}{d}N^{[u-1]}_{Q_{i}(\tilde{f})}(r)+S(r),$

where $S(r)\leq K(\log^{+}\frac{1}{R_{0}-r}+\log^{+}T_{f}(r,r_{0}))$ for all
$0<r_{0}<r<R_{0}$ outside a set $E\subset[0,R_{0}]$ with
$\int_{E}\frac{dt}{R_{0}-t}<\infty.$

###### Proof.

Repeating the above argument, we have

$\int_{S(r)}\left|(z^{\alpha_{1}+\cdots+\alpha_{u}})^{p}\frac{||\tilde{f}(z)||^{qdb-
puN}|W^{\alpha}(\phi_{s}(\tilde{f}))(z)|^{p}}{\prod_{i=1}^{q}|Q_{i}(\tilde{f})(z)|^{b}}\right|^{t}\sigma_{m}\leq
K_{1}\left(\dfrac{R^{2m-1}}{R-r}dT_{f}(R,r_{0})\right)^{\delta}$

for every $0<r_{0}<r<R<R_{0}$. Using the concativity of the logarithmic
function, we have

(3.15)
$\displaystyle\begin{split}p\int_{S(r)}&\log|(z^{\alpha_{1}+\cdots+\alpha_{u}})|\sigma_{m}+(qdb-
puN)\int_{S(r)}\log||\tilde{f}||\sigma_{m}+p\int_{S(r)}\log|W^{\alpha}(\phi_{s}(\tilde{f}))|\sigma_{m}\\\
&-b\sum_{j=1}^{q}\int_{S(r)}\log|Q_{j}(\tilde{f})|\sigma_{m}\leq
K\left(\log^{+}\dfrac{1}{R_{0}-r}+\log^{+}T_{f}(R,r_{0})\right)\end{split}$

for some positive constant $K$. By the Jensen formula, this inequality implies
that

(3.16) $\displaystyle\begin{split}(qdb-
puN)T_{f}(r,r_{0})&+pN_{W^{\alpha}(\phi_{s}(\tilde{f}))}(r)-b\sum_{i=1}^{q}N_{Q_{i}(\tilde{f})}(r)\\\
&\leq
K\left(\log^{+}\dfrac{1}{R_{0}-r}+\log^{+}T_{f}(R,r_{0})\right)+O(1).\end{split}$

From Claim 3.12, we have

$b\sum_{i=1}^{q}N_{Q_{i}(\tilde{f})}(r)-pN_{W^{\alpha}(\phi_{s}(\tilde{f}))}(r)\leq\sum_{i=1}^{q}N^{[u-1]}_{Q_{i}(\tilde{f})}(r).$

Combining this estimate and (3.16), we get

$\displaystyle\left(q-\frac{puN}{db}\right)T_{f}(r,r_{0})$
$\displaystyle\leq\sum_{i=1}^{q}\frac{1}{d}N^{[u-1]}_{Q_{i}(\tilde{f})}(r)+K\left(\log^{+}\dfrac{1}{R_{0}-r}+\log^{+}T_{f}(R,r_{0})\right)+O(1).$

Since $\frac{puN}{db}\leq p(n+1)+\epsilon$, the above inequality implies that

$\displaystyle(q-p(n+1)-\epsilon)T_{f}(r,r_{0})$
$\displaystyle\leq\sum_{i=1}^{q}\frac{1}{d}N^{[u-1]}_{Q_{i}(\tilde{f})}(r)+K\left(\log^{+}\dfrac{1}{R_{0}-r}+\log^{+}T_{f}(R,r_{0})\right)+O(1).$

Choosing $R=r+\dfrac{1-r}{eT_{f}(r,r_{0})}$, we get

$T_{f}(R,r_{0})\leq 2T_{f}(r,r_{0})$

outside a subset $E\subset[0,1]$ with $\int_{E}\frac{dr}{1-r}<+\infty$. Thus

$(q-p(n+1)-\epsilon)T_{f}(r,r_{0})\leq\sum_{i=1}^{q}\frac{1}{d}N^{[u-1]}_{Q_{i}(\tilde{f})}(r)+K\left(\log^{+}\dfrac{1}{R_{0}-r}+\log^{+}T(r,r_{0})\right)+O(1).$

This implies that

$\sum_{j=1}^{q}\delta^{[u-1]}_{f}(Q_{j})\leq\sum_{j=1}^{q}\delta^{[u-1]}_{f,*}(Q_{j})\leq
p(n+1)+\epsilon.$

The theorem is proved in this case. ∎

##  4\. Value distribution of the Gauss map of a complete regular submanifold
of ${\mathbb{C}}^{m}$

Let $M$ be a connected complex manifold of dimension $m$. Let

$f=(f_{1},\ldots,f_{n}):M\rightarrow{\mathbb{C}}^{n}$

be a regular submanifold of ${\mathbb{C}}^{n}$; namely, $f$ be a holomorphic
map of $M$ into ${\mathbb{C}}^{n}$ such that $\mathrm{rank}d_{p}f=\dim M$ for
every point $p\in M.$ We assign each point $p\in M$ to the tangent space
$T_{p}(M)$ of $M$ at $p$ which may be considered as an $m$-dimensional linear
subspace of $T_{f(p)}({\mathbb{C}}^{n})$. Also, each tangent space
$T_{p}({\mathbb{C}}^{n})$ can be identified with
$T_{0}({\mathbb{C}}^{n})={\mathbb{C}}^{n}$ by a parallel translation. Hence,
each $T_{p}(M)$ is corresponded to a point $G(p)$ in the complex Grassmannian
manifold $G(m,n)$ of all $m$-dimensional linear subspaces of
${\mathbb{C}}^{n}$.

######  Definition 4.1.

The map $G:p\in M\mapsto G(p)\in G(m,n)$ is called the Gauss map of the map
$f:M\rightarrow{\mathbb{C}}^{n}$.

The space $G(m,n)$ is canonically embedded in
${\mathbb{P}}^{N}({\mathbb{C}})={\mathbb{P}}(\bigwedge^{m}{\mathbb{C}}^{n})$,
where $N=\binom{n}{m}-1$. Then we may identify the Gauss map $G$ with a
holomorphic mapping of $M$ into ${\mathbb{P}}^{N}({\mathbb{C}})$ given as
follows: taking holomorphic local coordinates $(z_{1},\ldots,z_{n})$ defined
on an open set $U$, we consider the map

$\bigwedge:=D_{1}f\wedge\cdots\wedge
D_{n}f:U\rightarrow\bigwedge^{m}{\mathbb{C}}^{n}\setminus\\{0\\},$

where

$D_{i}f=(\dfrac{\partial f_{1}}{\partial z_{i}},\cdots,\dfrac{\partial
f_{n}}{\partial z_{i}}).$

Then, locally we have

$G=\pi\circ\bigwedge,$

where
$\pi:{\mathbb{C}}^{N+1}\setminus\\{0\\}\rightarrow{\mathbb{P}}^{N}({\mathbb{C}})$
is the canonical projection map. A regular submanifold $M$ of
${\mathbb{C}}^{m}$ is considered as a Kähler manifold with the metric $\omega$
induced from the standard flat metric on ${\mathbb{C}}^{m}$. We denote by $dV$
the volume form on $M$. For arbitrarily holomorphic coordinates
$z_{1},\ldots,z_{m},$ we see that

$dV=|\bigwedge|^{2}\left(\sqrt{-1}{2}\right)^{m}dz_{1}\wedge
d\bar{z_{1}}\wedge\cdots\wedge dz_{m}\wedge dz_{m},$

where

$|\bigwedge|^{2}=\sum_{1\leq i_{1}<\cdots<i_{m}\leq
n}\dfrac{\partial(f_{i_{1}},...,f_{i_{m}})}{\partial(z_{1},...,z_{m})}^{2}.$

Therefore, for a regular submanifold $f:M\rightarrow{\mathbb{C}}^{m}$, the
Gauss map $G:M\rightarrow{\mathbb{P}}^{N}({\mathbb{C}})$ satisfies the
following growth condition

$\Omega_{G}+dd^{c}\log h^{2}=dd^{c}\log|\bigwedge|^{2}=\mathrm{Ric}(\omega),$

where $h=1$. Then Theorem 1.1 immediately gives us the following.

######  Theorem 4.2.

Let $M$ be a complex manifold of dimension $m$ such that the universal
covering of $M$ is biholomorphic to a ball $\mathbb{B}^{m}(R_{0})\
(0<R_{0}\leq+\infty)$ in ${\mathbb{C}}^{m}.$ Let
$f:M\rightarrow{\mathbb{C}}^{n}$ be a complete regular submanifold. Assume
that the Gauss map $G:M\rightarrow{\mathbb{P}}^{N}({\mathbb{C}})$ is
algebraically non-degenerate, where $N=\binom{n}{m}-1$. Let
$Q_{1},\ldots,Q_{q}$ be $q$ hypersurfaces of degree $d_{j}\ (1\leq j\leq q)$
in $k$-subgeneral position in ${\mathbb{P}}^{N}({\mathbb{C}})$. Let $d$ be the
least common multiple of $d_{i}$’s, i.e., $d=l.c.m.\\{d_{1},\ldots,d_{q}\\}$.
Then, for every $\epsilon>0$ we have

$\sum_{i=1}^{q}\delta_{G}^{[u-1]}(Q_{i})\leq p(N+1)+\epsilon+\frac{\rho
u(u-1)}{d},$

where $p=k-N+1$, $L=(N+1)d+p(N+1)^{3}I(\epsilon^{-1})d$ and
$u=\binom{L+N}{N}\leq 3^{N+2}(dp(N+1)^{2}I(\epsilon^{-1}))^{N}$.

Acknowledgements. This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant number
101.04-2015.03.

## References

  * [1] Z. Chen, M. Ru and Q. Yan, The degenerated second main theorem and Schmidt’s subspace theorem, Sci. China Math. 55 (2012) 1367–1380.
  * [2] P. Corvaja and U. Zannier, On a general Thue’s equation, Amer. J. Math. 126 (2004) 1033–1055.
  * [3] H. Fujimoto, On the Gauss map of a complete minimal surface in ${\mathbf{R}}^{m}$, J. Math. Soc. Japan, 35 (1983), No.2, 279-288.
  * [4] H. Fujimoto, Value distribution of the Gauss maps of complete minimal surfaces in ${\mathbf{R}}^{m}$, J. Math. Soc. Japan, 35 (1983), 663-681.
  * [5] H. Fujimoto, Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into ${\mathbb{P}}^{N_{1}}({\mathbb{C}})\times\cdots\times{\mathbb{P}}^{N_{k}}({\mathbb{C}})$, Japan. J. Math., 11 (1985), No 2, 233-264.
  * [6] L. Karp, Subharmonic functions on real and complex manifolds, Math. Z., 179 (1982), 535-554.
  * [7] J. Noguchi, A note on entire pseudo-holomorphic curves and the proof of Cartan-Nochka’s theorem, Kodai Math. J., 28 (2005), 336-346
  * [8] S. D. Quang, Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces, arXiv:1610.03951 [math.CV].
  * [9] S. D. Quang, Second main theorem for meromorphic mappings with moving hypersurfaces in subgeneral position, Preprint.
  * [10] M. Ru and S. Sogome, Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into ${\mathbb{P}}^{n}({\mathbb{C}})$ intersecting hypersurfaces, Trans. Amer. Math. Soc. 364, No. 3 (2012), 1145-1162.
  * [11] Tran Van Tan and Vu Van Truong, A non-integrated defect relation for meromorphic maps of complete Kähler manifolds into a projective variety intersecting hypersurfaces, Bull. Sci. Math. 136(2012), 111-126.
  * [12] Q. Yan,  Non-integrated defect relation and uniqueness theorem for meromorphic maps of a complete Kähler manifold into $P^{n}(\mathbb{C})$, J. Math. Anal. Appl. 398(2013), 567-581.
  * [13] S.T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana U. Math. J. 25 (1976), 659-670.

Si Duc Quang
1 Department of Mathematics, Hanoi National University of Education,
136-Xuan Thuy, Cau Giay, Hanoi, Vietnam.
2 Thang Long Institute of Mathematics and Applied Sciences,
Nghiem Xuan Yem, Hoang Mai, HaNoi, Vietnam.
E-mail<EMAIL_ADDRESS>
Nguyen Thi Quynh Phuong
Department of Mathematics, Hanoi National University of Education,
136-Xuan Thuy, Cau Giay, Hanoi, Vietnam.
E-mail<EMAIL_ADDRESS>
Nguyen Thi Nhung
Department of Mathematics, Thang Long University,
Nghiem Xuan Yem, Hoang Mai, HaNoi, Vietnam.
E-mail<EMAIL_ADDRESS>