# Some variants of the generalized Borel Theorem
and applications

Dinh Tuan Huynh
Hue Geometry–Algebra Group  Hue Geometry–Algebra Group, Department of
Mathematics, University of Education, Hue University, 34 Le Loi St., Hue City,
Vietnam<EMAIL_ADDRESS>
###### Abstract.

In the first part of this paper, we establish some results around generalized
Borel’s Theorem. As an application, in the second part, we construct example
of smooth surface of degree $d\geq 19$ in $\mathbb{CP}^{3}$ whose complements
is hyperbolically embedded in $\mathbb{CP}^{3}$. This improves the previous
construction of Shirosaki where the degree bound $d=31$ was gave. In the last
part, for a Fermat-Waring type hypersurface $D$ in $\mathbb{CP}^{n}$ defined
by the homogeneous polynomial

$\sum_{i=1}^{m}h_{i}^{d},$

where $m,n,d$ are positive integers with $m\geq 3n-1$ and $d\geq m^{2}-m+1$,
where $h_{i}$ are homogeneous generic linear forms on $\mathbb{C}^{n+1}$, for
a nonconstant holomorphic function
$f\colon\mathbb{C}\rightarrow\mathbb{CP}^{n}$ whose image is not contained in
$\operatorname{supp}D$, we establish a Second Main Theorem type estimate:

$\big{(}d-m(m-1)\big{)}\,T_{f}(r)\leq N_{f}^{[m-1]}(r,D)+S_{f}(r).$

This quantifies the hyperbolicity result due to Shiffman-Zaidenberg and Siu-
Yeung.

###### Key words and phrases:

holomorphic map, algebraic degeneracy, Nevanlinna theory, complex
hyperbolicity, Borel Theorem, Fermat-Warning hypersurface

###### 2010 Mathematics Subject Classification:

32H25, 32H30

##  1\. Introduction

It was conjectured by Kobayashi in 1970 [27] that a general hypersurface $D$
in projective space $\mathbb{CP}^{n}$ of degree $d$ large enough is
hyperbolic. According to Zaidenberg [54], the expected optimal degree bound
should be $d=2n-1$. In the so-called logarithmic case, it is also anticipated
that if $d\geq 2n+1$, then the complement $\mathbb{CP}^{n}\setminus D$ is also
hyperbolic. The subject of hyperbolicity has attracted much attention and
research, partly because it has been believed to be intimately related to
Diophantine Geometry. For instance, Lang conjectured that an algebraic variety
$V_{\mathbb{K}}$ defined over a number field $\mathbb{K}$ can contain only
finitely many $\mathbb{K}$-rational points provided that $V_{\mathbb{C}}$ is
hyperbolic after some base change $\mathbb{K}\hookrightarrow\mathbb{C}$.

Many works have been done during recent decades towards the above conjectures.
Notably, these conjecture were confirmed under the condition that the degree
of $D$ is very high compared with the dimension. In the case of surfaces in
$\mathbb{CP}^{3}$, by studying the entire leaves of foliations on surfaces,
the first proof was given by McQuillan [30] with degree bound $d\geq 36$.
Demailly-El Goul [13] provided a better degree bound $d\geq 21$. Pǎun [34]
employed the technique of using slanted vector field of Siu [46] together with
the work of McQuillan [31] to improve the degree bound to $d\geq 18$.
Subsequently, hyperbolicity of generic three-folds in $\mathbb{CP}^{4}$ was
confirmed [35, 16] with degree bound $d\geq 593$.

In the case of arbitrary dimension $n$, by generalizing the variational
approach of Clemens [9] and Voisin [51], Siu [46] outlined a strategy which
finally led to the proof of Kobayashi’s conjecture [47] with very high degree
bound $d(n)\gg 1$. Many works are influenced by Siu’s program, especially the
significant results of Diverio-Merker-Rousseau [15] on Green-Griffiths’
conjecture (see also [11, 10]). Recently, based on the method of using
Wronskian differential operators [52, 4], Brotbek gave a new proof for
hyperbolicity of high degree generic hypersurface [3]. The explicit degree
bound was given shortly afterwards [14, 12]. Very recently, Bérczi-Karwan [1]
reached the degree bound of polynomial growth by an approach from geometric
invariant theory.

On the other side, in the past five decades, it has been considered a very
challenging problem to construct new examples of hyperbolic hypersurfaces of
low degree in projective spaces, as the motto goes: “the lower degree, the
more difficult the hyperbolicity”. The first example of compact hyperbolic
manifold due to Brody-Green [2] is a surface in $\mathbb{CP}^{3}$ defined by
the equation

$z_{0}^{d}+z_{1}^{d}+z_{2}^{d}+z_{3}^{d}+(\epsilon\,z_{0}z_{1})^{\frac{d}{2}}+(\epsilon\,z_{0}z_{2})^{\frac{d}{2}}=0,$

where $d=2k\geq 50$ and $|\epsilon|$ is small enough. After that several
hyperbolic hypersurfaces in low dimension were constructed to improve the
degree bound [32, 22, 48, 44, 43, 20, 40, 39, 42, 7, 17]. Duval [17]
constructed a sextic hyperbolic surface in $\mathbb{CP}^{3}$, which is the
lowest degree found up to date. Some recent constructions could also reach
this degree bound [8, 18]. It is still not known yet whether there exists any
quintic hyperbolic surface.

Currently, there are two main methods of constructing hyperbolic hypersurfaces
in projective space. The first one is to seek them among the class of
perturbations of Fermat hypersurface. This method mades use some variants of
the generalized Borel Theorem to study the degeneracy of entire holomorphic
curves. The second one is the deformation method introduced by Zaidenberg
[55], whose main idea is to find hyperbolic hypersurfaces in the linear pencil
of hypersurfaces $\Sigma_{\epsilon}=\\{s_{0}+\epsilon s=0\\}$, where
$S_{0}=\\{s_{0}=0\\}$ is a singular hypersurface, $S=\\{s=0\\}$ is a generic
hypersurface and $\epsilon$ is small enough.

The first examples in any dimension were given by Masuda-Noguchi [29] with
very large degree. Explicit constructions with large degree were gave by
Fujimoto [20, 21]. Some constructions with lower degree asymptotic were
provided by Siu-Yeung [48] with $d(n)\geq 16\,(n-1)^{2}$ and by Shiffman-
Zaidenberg [41] with $d(n)\geq 4\,(n-1)^{2}$. Currently, the best asymptotic
degree bound is $d\geq{\textstyle{\big{(}\frac{n+2}{2}\big{)}^{2}}}$, obtained
in [24]. In low dimension $3\leq n\leq 6$, example of hyperbolic hypersurface
of lowest degree bound $d=2n$ was given in [25].

On the other hand, example of smooth algebraic curve in $\mathbb{CP}^{2}$ of
degree $d\geq 5$ whose complement is hyperbolically embedded was constructed
by Zaidenberg [55] using deformation. However, in the logarithmic case, from
dimension $n\geq 3$, this method is not effective. Examples of hyperbolically
embedded hypersurfaces were provided by Masuda-Noguchi [29] in all dimension,
and by Shirosaki [44] in low dimension $n\leq 4$ with better degree bound.
Generalizing the construction of Shirosaki to arbitrary dimension, Shiffman-
Zaidenberg showed the existence of hyperbolically embedded hypersurfaces in
$\mathbb{CP}^{n}$ with all degree $d\geq 4n^{2}-2n+1$.

Our first aim in this paper is to improve the previous degree bound $d\geq 31$
for hyperbolically embedded surfaces in $\mathbb{CP}^{3}$ by Shirosaki [44].
We first prove a Borel type result for holomorphic curves into projective
space avoiding or contained in the hypersurface defined by the homogeneous
polynomials of the form

$\sum_{i=0}^{n}z_{i}^{d-\delta_{i}}Q_{i}.$

Similar results have been given in [32], [22] [50] using meromorphic
connections (see also [53], [49] for another approach via value distribution
theory). Following the arguments of Green, we obtain a stronger degeneracy
statement (see Section 2 for details). As an application, we construct a
smooth algebraic surface in $\mathbb{CP}^{3}$ of degree $d$ for any $d\geq 19$
such that its complement is hyperbolically embedded in $\mathbb{CP}^{3}$.

######  Theorem A.

Let $d\geq 19$ be an integer. There exist some nonzero complex numbers
$a_{0},a_{1},a_{2},a_{3}$ such that the surface $D\subset\mathbb{CP}^{3}$
defined in the homogeneous coordinates $[z_{0}:z_{1}:z_{2}:z_{3}]$ of
$\mathbb{CP}^{3}$ by the homogeneous polynomial

$z_{0}^{d}+z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})$

is smooth, and the complement $\mathbb{CP}^{3}\setminus D$ is hyperbolically
embedded in $\mathbb{CP}^{3}$.

On the other hand, in the quantitative aspect of the hyperbolicity problem,
called value distribution theory or Nevanlinna theory, one studies the
frequency of impacts of entire holomorphic curves into projective space and a
family of hypersurfaces, by means of certain Second Main Theorem type
estimate.

Before entering the details of the next result, we give a brief introduction
to Nevanlinna theory in projective space. For a positive number $r>0$, we
denote by $\Delta_{r}\subset\mathbb{C}$ the disk of radius $r$ centered at the
origin. Fix a truncation level $m\in\mathbb{N}\cup\\{\infty\\}$, for an
effective divisor $E=\sum_{i}\alpha_{i}\,a_{i}$ on $\mathbb{C}$ where
$\alpha_{i}\geq 0$, $a_{i}\in\mathbb{C}$, the $m$-truncated degree of the
divisor $E$ on the disk $\Delta_{r}$ is given by

$n^{[m]}(r,E):=\sum_{a_{i}\in\Delta_{r}}\min\,\\{m,\alpha_{i}\\},$

the truncated counting function at level $m$ of $E$ is then defined by taking
the logarithmic average

$None$
$N^{[m]}(r,E)\,:=\,\int_{1}^{r}\frac{n^{[m]}(t,E)}{t}\,\operatorname{\text{d}}\\!t$

When $m=\infty$, for abbreviation we write $n(t,E)$, $N(r,E)$ for
$n^{[\infty]}(t,E)$, $N^{[\infty]}(r,E)$.

Let $f\colon\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be an entire
holomorphic curve having a reduced representation $f=[f_{0}:\cdots:f_{n}]$ in
the homogeneous coordinates $[z_{0}:\cdots:z_{n}]$ of
$\mathbb{P}^{n}(\mathbb{C})$. Let $D=\\{Q=0\\}$ be a divisor in
$\mathbb{P}^{n}(\mathbb{C})$ defined by a homogeneous polynomial
$Q\in\mathbb{C}[z_{0},\dots,z_{n}]$ of degree $d\geq 1$. If
$f(\mathbb{C})\not\subset D$, then
$f^{*}D=\sum_{a\in\mathbb{C}}\operatorname{ord}_{a}f^{*}Q$ is a divisor on
$\mathbb{C}$. We then define the truncated counting function of $f$ with
respect to $D$ as

$N_{f}^{[m]}(r,D)\,:=\,N^{[m]}\big{(}r,f^{*}D\big{)},$

which measures the frequency of the intersection $f(\mathbb{C})\cap D$. Next,
the proximity function of $f$ associated to the divisor $D$ is given by

$m_{f}(r,D)\,:=\,\int_{0}^{2\pi}\mathrm{log}\,\frac{\big{\|}f(re^{i\theta})\big{\|}_{\max}^{d}\,\|Q\|_{\max}}{\big{|}Q(f)(re^{i\theta})\big{|}}\,\frac{\operatorname{\text{d}}\\!\theta}{2\pi},$

where $\|Q\|_{\max}$ is the maximum absolute value of the coefficients of $Q$
and where

(1.1) $\big{\|}f(z)\big{\|}_{\max}:=\max\\{|f_{0}(z)|,\dots,|f_{n}(z)|\\}.$

Since $\big{|}Q(f)\big{|}\leq\left(\begin{subarray}{c}d+n\\\
n\end{subarray}\right)\,\|Q\|_{\max}\cdot\|f\|_{\max}^{d}$, we see that
$m_{f}(r,D)\geq O(1)$ is bounded from below by some constant. Lastly, the
Cartan order function of $f$ is defined by

$None$
$T_{f}(r):=\frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{log}\,\big{\|}f(re^{i\theta})\big{\|}_{\max}\operatorname{\text{d}}\\!\theta=\int_{1}^{r}\dfrac{\operatorname{\text{d}}\\!t}{t}\int_{|z|<t}f^{*}\omega_{FS}+O(1)$

capturing the growth of the area of the image of the disks under $f$, with
respect to the Fubini–Study metric $\omega_{\operatorname{FS}}$. The
Nevanlinna theory is then established by comparing the above three functions.
It consists of two fundamental theorems (for backgrounds and recent progresses
in Nevanlinna theory, see Noguchi-Winkelmann [33] and Ru [38]). The first one
is a reformulation of the Poisson-Jensen formula.

######  First Main Theorem.

Let $f\colon\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic
curve and let $D$ be a hypersurface of degree $d$ in
$\mathbb{P}^{n}(\mathbb{C})$ such that $f(\mathbb{C})\not\subset D$. Then one
has the estimate

$m_{f}(r,D)+N_{f}(r,D)\,=\,d\,T_{f}(r)+O(1)$

for every $r>1$, whence

(1.2) $N_{f}(r,D)\,\leq\,d\,T_{f}(r)+O(1).$

Hence the First Main Theorem provides an upper bound on the counting function
in term of the order function. The reverse direction, called Second Main
Theorem, is usually much harder, and one often needs to take the sum of the
counting functions of many divisors.

Throughout this paper, for an entire curve $f$, by $S_{f}(r)$, we mean a real
function of $r\in\mathbb{R}^{+}$ such that

$S_{f}(r)\leq O(\mathrm{log}\,(T_{f}(r)))+\epsilon\mathrm{log}\,r$

for every positive constant $\epsilon$ and every $r$ outside of a subset
(depending on $\epsilon$) of finite Lebesgue measure of $\mathbb{R}^{+}$. In
the case where $f$ is rational, we understand that $S_{f}(r)=O(1)$. In any
case, there holds

$\liminf_{r\rightarrow\infty}\dfrac{S_{f}(r)}{T_{f}(r)}=0.$

A family $\\{D_{i}\\}_{1\leq i\leq q}$ of $q\geq n+2$ hypersurfaces in
$\mathbb{CP}^{n}$ is said to be in general position if $\cap_{i\in
I}D_{i}=\varnothing$ for any subset $I\subset\\{1,\dots,q\\}$ of cardinality
$n+1$. For a linearly nondegenerate entire curve
$f:\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ and for a family of $q\geq
n+2$ hyperplanes $\\{H_{i}\\}_{i\,=\,1,\dots,\,q}$ in general position, Cartan
[6] established a second main theorem

(1.3) $(q-n-1)\,T_{f}(r)\,\leq\,\sum_{i=1}^{q}N_{f}^{[n]}(r,H_{i})+S_{f}(r),$

which implies the defect relation

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

In the particular case $n=1$, Cartan recovered the classical Nevanlinna’s
Second Main Theorem. In the collinear case, by a purely potential theoretic
approach, a Second Main Theorem for non-constant holomorphic curves $f$ in
$\mathbb{CP}^{n}$ and family of hypersurfaces $\\{D_{i}\\}_{i=1}^{q}$ in
general position without truncation were given by Eremenko-Sodin [19] which
implies a defect relation bounded by $2n$. Assuming the algebraically
nondegenerate condition for entire curves, Ru [36, 37] obtained a defect
relation bounded by $n+1$ by method from Diophantine approximation. Second
Main Theorem quantifying the logarithmic Green-Griffiths and Kobayashi
conjecture were given in [26], [5], using the logarithmic jet differentials.

Currently, from dimension $n\geq 2$, there are only two results about Second
Main Theorem for non constant holomorphic curve $f$ into $\mathbb{CP}^{n}$.
These are the Second Main Theorems of Eremenko-Sodin [19] and Brotbek-Deng
[5]. In these works, one either needs many targets (at least $2n+1$) or very
large degree (exponential growth compared with the dimension). Our next
purpose in this paper is to seek Second Main Theorem for non-constant entire
holomorphic curves into $\mathbb{CP}^{n}$ and a hypersurface in this space
whose complement is hyperbolically embedded in $\mathbb{CP}^{n}$. We shall
deal with the hypersurface of Fermat-Waring type constructed by Siu-Yeung [48]
and Shiffman-Zaidenberg [41]. Here is the statement of our next result.

######  Theorem B.

Let $m,n,d$ be positive integers with $m\geq 3n-1$ and $d\geq m^{2}-m+1$. Let
$D$ be the Fermat-Waring type hypersurface in $\mathbb{CP}^{n}$ defined by the
homogeneous polynomial

$\sum_{i=1}^{m}h_{i}^{d},$

where $h_{i}$ are generic homogeneous linear forms on $\mathbb{C}^{n+1}$. For
a nonconstant holomorphic function
$f\colon\mathbb{C}\rightarrow\mathbb{CP}^{n}$ whose image is not contained in
$\operatorname{supp}D$, the following Second Main Theorem type estimate holds

$\big{(}d-m(m-1)\big{)}\,T_{f}(r)\leq N_{f}^{[m-1]}(r,D)+S_{f}(r).$

Notably, this implies a defect relation involving the degree of the
hypersurface $D$. It shows that the defect of $f$ with respect to $D$ is small
when the degree of $D$ is large (see [50], [53], [49] for related results).

######  Defect relation.

Let $m,n,d$ be positive integers with $m\geq 3n-1$ and $d\geq m^{2}-m+1$. Let
$D$ be the Fermat-Waring type hypersurface in $\mathbb{CP}^{n}$ defined by the
homogeneous polynomial

$\sum_{i=1}^{m}h_{i}^{d},$

where $h_{i}$ are generic homogeneous linear forms on $\mathbb{C}^{n+1}$. For
a nonconstant holomorphic function
$f\colon\mathbb{C}\rightarrow\mathbb{CP}^{n}$ whose image is not contained in
$\operatorname{supp}D$, the following defect inequality holds

$\delta_{f}^{[m-1]}(D)\leq\dfrac{m(m-1)}{d}.$

Consequently

$\lim_{d\rightarrow\infty}\delta_{f}^{[m-1]}(D)=0.$

From the above defect relation, and in view of the fundamental conjecture for
entire curves of Griffiths, one may expect the following

######  Conjecture.

Let $f\colon\mathbb{CP}^{n}$ be a nonconstant entire holomorphic curve. Let
$D$ be a generic hypersurface of degree $d>2n$. If the image of $f$ is not
contained in $D$, then the following defect relation holds

$\delta_{f}(D)\leq\dfrac{2n}{d}.$

## Acknowledgement

A part of this article was written while the author was visiting Vietnam
Institute for Advanced Study in Mathematics (VIASM).

##  2\. Some variants of generalized Borel’s Theorem

As pointed out by Shiffman in a private conversation with Siu [45], from the
Cartan Second Main Theorem with truncated counting functions, one can deduce a
Second Main Theorem for algebraically nondegenerate holomorphic curves and a
family of Fermat type hypersurfaces. Using this technique, some
generalizations have been done recently for small perturbations of Fermat type
hypersurfaces [53], [49] (see also [50] for a geometric approach via
connections). For later purpose, we present here a slightly modifications of
these results with a simplified proof.

######  Proposition 2.1.

Let $d,n,\delta_{0},\dots,\delta_{n}$ be integer numbers with $\delta_{i}\geq
0$, $n\geq 2$ and $d>n(n+1)+\sum_{i=0}^{n}\delta_{i}$. Let $Q_{i}$ ($0\leq
i\leq n$) be homogeneous polynomials of degree $\delta_{i}$. Suppose that the
family of hypersurfaces $\\{D_{i}\\}_{0\leq i\leq n}$ where
$D_{i}=\\{z_{i}^{d-\delta_{i}}Q_{i}=0\\}$ is in general position in
$\mathbb{CP}^{n}$. Let $D$ be the hypersurface in $\mathbb{CP}^{n}$ defined by
the homogeneous polynomial

$\sum_{i=0}^{n}z_{i}^{d-\delta_{i}}Q_{i}.$

Let $f\colon\mathbb{C}\rightarrow\mathbb{CP}^{n}$ be a non-constant
holomorphic curve. If there exists no nontrivial linear relation among
$(z_{i}^{d-\delta_{i}}Q_{i})\circ f$, then

$\big{[}d-\big{(}n(n+1)+\sum_{i=0}^{n}\delta_{i}\big{)}\big{]}T_{f}(r)\leq
N_{f}^{[n]}(r,D)+S_{f}(r).$

###### Proof.

Since $\\{z_{i}^{d-\delta_{i}}Q_{i}\\}_{0\leq i\leq n}$ is in general
position, for any
$w\in\mathbb{S}_{n+1}:=\\{w\in\mathbb{C}^{n+1}\setminus\\{0\\},\|w\|=1\\}$,
there exists at least one index $i$ with
$(z_{i}^{d-\delta_{i}}Q_{i})(w)\not=0$. Thus, by compactness property of
$\mathbb{S}_{n+1}$, there exist constants $C_{1},C_{2}>0$ such that

$None$ $C_{1}\leq\max_{0\leq i\leq
n}\big{|}(z_{i}^{d-\delta_{i}}Q_{i})(w)\big{|}^{\frac{1}{d}}\leq C_{2}$

Since $z_{i}^{d-\delta_{i}}Q_{i}$ are homogeneous of degree $d$, the above
estimate implies

$None$ $C_{1}\leq\max_{0\leq i\leq
n}\dfrac{\big{|}(z_{i}^{d-\delta_{i}}Q_{i})(w)\big{|}^{\frac{1}{d}}}{\|w\|}\leq
C_{2}$

Now let $f=[f_{0}:\dots:f_{n}]$ be a reduced representation of $f$. Setting

$\displaystyle\pi\colon\mathbb{CP}^{n}$
$\displaystyle\rightarrow\mathbb{CP}^{n},\qquad[z_{0}:\dots:z_{n}]\mapsto[z_{0}^{d-\delta_{0}}Q_{0}:\dots:z_{n}^{d-\delta_{n}}Q_{n}],$
$\displaystyle g=[g_{0}:\dots:g_{n}]\colon\mathbb{C}$
$\displaystyle\rightarrow\mathbb{CP}^{n},\qquad z\mapsto\pi\circ f(z),$

where $g_{i}:=(z_{i}^{d-\delta_{i}}Q_{i})\circ f$. It follows from the above
estimate that

(2.1) $T_{g}(r)=d\,T_{f}(r)+O(1).$

Let $\\{H_{i}\\}_{0\leq i\leq n+1}$ be the family of $n+2$ hyperplanes in
$\mathbb{CP}^{n}$ given by

$\displaystyle H_{i}$
$\displaystyle=\\{z_{i}=0\\}\qquad(0\,\leq\,i\,\leq\,n),$ $\displaystyle
H_{n+1}$ $\displaystyle=\\{\sum_{i=0}^{n}z_{i}=0\\},$

which is in general position. By assumption, the map $g$ is linearly
nondegenerate. Hence, applying Cartan’s Second Main Theorem for $g$ and
$\\{H_{i}\\}$, one receives

(2.2) $T_{g}(r)\leq\sum_{i-0}^{n}N_{g}^{[n]}(r,H_{i})+N_{g}^{[n]}(r,H_{n+1}).$

For $0\leq i\leq n$, the curve $g$ intersects $H_{i}$ if and only if the curve
$f$ intersects $H_{i}$ or $Q_{i}$. Furthermore, if $f$ intersects such
$H_{i}$, then $g$ intersects $H_{i}$ with multiplicity at least $d$. Hence,
from these observations, one gets

$None$ $N_{g}^{[n]}(r,H_{i})\leq N_{f}(r,Q_{i})+\dfrac{n}{d}N_{g}(r,H_{i}),$

which implies

$None$
$N_{g}^{[n]}(r,H_{i})\leq\delta_{i}\,T_{f}(r)+\dfrac{n}{d}T_{g}(H_{i}),$

by the First Main Theorem. Obviously, one also has
$N_{g}^{[n]}(r,H_{n+1})=N_{f}^{[n]}(r,D)$. Combining these facts together with
(2.1), (2.2), we receive the desired estimate. ∎

One can deduce immediately from Proposition 2.1 that, under the same
assumptions therein, if $f$ avoids $D$, then it must be (algebraically)
degenerate. By following the arguments in [28, Example 3.10.22], one can
actually obtain stronger degeneracy result.

######  Generalized Borel’s Theorem (logarithmic case).

Let $d,n,\delta_{0},\dots,\delta_{n}$ be integer numbers with $\delta_{i}\geq
0$, $n\geq 2$ and $d>n(n+1)+\sum_{i=0}^{n}\delta_{i}$. Let $Q_{i}$ ($0\leq
i\leq n$) be homogeneous polynomials of degree $\delta_{i}$. Suppose that the
hypersurfaces $D_{i}=\\{z_{i}^{d-\delta_{i}}Q_{i}=0\\}$ ($0\leq i\leq n+1$)
have empty intersection. Then, for the collection of $n+1$ entire holomorphic
functions $f_{i}$ ($0\leq i\leq n$) such that

$\sum_{i=0}^{n}f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})$

is nowhere vanishing, there is a partition of indexes
$\\{0,\dots,n\\}=\cup_{\alpha=0}^{\ell}I_{\alpha}$ such that the followings
hold

  1. (i)

$f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})\equiv 0$ if and only if
$i\in I_{0}$ (of course the set $I_{0}$ may be empty);

  2. (ii)

The cardinality $|I_{\alpha}|\geq 2$ for every $1\leq\alpha\leq\ell$, with at
most one exception;

  3. (iii)

For each $1\leq\alpha\leq\ell$, for arbitrary indexes $i,j\in I_{\alpha}$,
there exists a constant $c_{ij}\in\mathbb{C}$ such that
$\frac{f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})}{f_{j}^{d-\delta_{j}}Q_{j}(f_{0},f_{1},\dots,f_{n})}=c_{ij}$;

  4. (iv)

$\sum_{i\in
I_{\alpha}}f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})\equiv 0$ for all
$0\leq\alpha\leq\ell$ with one exception.

###### Proof.

Consider the map $f\colon\mathbb{C}\rightarrow\mathbb{CP}^{n}$ given as
$f=[f_{0}:\dots:f_{n}]$. Let the map $g=[g_{0}:\dots:g_{n}]$ and the family of
hyperplanes $\\{H_{i}\\}_{0\leq i\leq n+1}$ be as in the proof of Proposition
2.1. Let $I_{0}=\\{0\leq i\leq n:g_{i}\equiv 0\\}$ and
$J=\\{0,\dots,n\\}\setminus I_{0}$. Suppose that $|I_{0}|=\ell$. Then the
image of $g$ lies in the subspace $H=\cap_{i\in
I_{0}}H_{i}\cong\mathbb{CP}^{m-1-\ell}$. By the same arguments as in the proof
of Proposition 2.1, one deduces that $g(\mathbb{C})$ lies in some hyperplane
of $H$. Next, we follow the arguments as in [28, Example 3.10.21] (see also
[23]). Suppose that $g_{i}$ satisfy the relation

$\sum_{k\in K}a_{k}g_{k}=0,$

where $K\subset J$, $|K|\geq 2$ and $a_{k}\in\mathbb{C}$ are nonzero
constants. We claim that there exist two indexes $i,j\in k$ such that
$g_{i}/g_{j}$ is constant. Indeed, if $|K|=2$, then we have nothing to prove.
Otherwise, we consider the map $g_{K}=[g_{k}]_{[k\in K]},\mathbb{C}\rightarrow
H_{k}\cong\mathbb{CP}^{|K|-2}$, where $H_{K}$ is the hyperplane in $H$ defined
by $\\{\sum_{k\in K}z_{k}=0\\}$, and the $|K|$ hyperplanes $\\{z_{k}=0\\}(k\in
K)$. Using arguments as in the proof of Proposition 2.1, one obtains further
degeneracy. Inductively, the claim is proved.

Let $\sim$ be the equivalence relation on the index set $J$ defined as $i\sim
j$ if and only if $g_{i}/g_{j}$ is constant and let
$\\{I_{1},\dots,I_{\ell}\\}$ be the partition of $J$ by $\sim$. Then for each
$1\leq s\leq\ell$, we pick an index $i_{s}\in I_{s}$ and put
$g_{j}=\ell_{j}g_{i_{s}}$. Set $b_{s}=\sum_{j\in I_{s}}\ell_{j}$, then

$\sum_{i=0}^{n}g_{i}=\sum_{s=1}^{\ell}b_{s}g_{i_{s}}.$

It suffices to prove that the set $M=\\{s:1\leq s\leq\ell:b_{s}\not=0\\}$
consists only one elements. Indeed, if $|M|\geq 2$, then can use arguments as
in Proposition 2.1 to get a nontrivial linear relation among
$\\{g_{s}\\}_{s\in M}$. Then by the above claim, there exist two indexes
$s_{1},s_{2}\in M$ such that $g_{s_{1}}/g_{s_{2}}$ is constant, contradiction.
Hence $M$ is of cardinality $1$. This finishes the proof of the Theorem. ∎

We can reformulate the above statement in a more geometric way as follows.

######  Corollary 2.2.

Keeping the same assumption as in Proposition 2.1. Putting

$\displaystyle\pi\colon\mathbb{CP}^{n}$
$\displaystyle\rightarrow\mathbb{CP}^{n},\qquad[z_{0}:\dots:z_{n}]\mapsto[z_{0}^{d-\delta_{0}}Q_{0}:\dots:z_{n}^{d-\delta_{n}}Q_{n}],$
$\displaystyle g=[g_{0}:\dots:g_{n}]\colon\mathbb{C}$
$\displaystyle\rightarrow\mathbb{CP}^{n},\qquad z\mapsto\pi\circ f(z),$

where $g_{i}:=(z_{i}^{d-\delta_{i}}Q_{i})\circ f$. If
$f\colon\mathbb{C}\rightarrow\mathbb{CP}^{n}\setminus D$ is a non-constant
holomorphic curve avoiding $D$, then the image of the composition map $g$ is
contained in a smaller linear subspace of $\mathbb{CP}^{n}$ of dimension at
most $\big{[}\frac{n}{2}\big{]}$.

###### Proof.

As before, we can find a partition $\\{I_{0},I_{1}\dots,I_{\ell}\\}$ of
$\\{0,1,\dots,n\\}$, all $I_{i}$ ($1\leq i\leq\ell$) have cardinality at least
$2$ with at most one exception, such that

  1. (a)

$g_{i}\equiv 0$ for all $i\in I_{0}$;

  2. (b)

For each $1\leq s\leq\ell$, pick an index $i_{s}\in I_{s}$, then for any $j\in
I_{s}$ there exists a nonzero constant $\mu_{s,j}$ such that
$g_{j}=\mu_{s,j}g_{i_{s}}$.

The image of $g$ lies in the linear subspace defined by the equations

$z_{j}=\mu_{s,j}z_{z_{s}}\quad(1\leq s\leq\ell,\quad j\in
I_{s},\,j\not=i_{s});\qquad z_{j}=0\quad(j\in I_{0}).$

Thus $\\{g_{i}\\}_{0\leq i\leq n}$ must satisfy at least
$|I_{0}|+\sum_{s=1}^{\ell}(|I_{s}|-1)=n+1-\ell$ independent linear relations.
Since $|I_{s}|\geq 2$ for at least $\ell-1$ indexes among
$\\{1,\dots,\ell\\}$, there holds $\ell\leq\bigg{[}\dfrac{n+2}{2}\bigg{]}$,
which implies that the image of $g$ is contained in a linear subspace of
dimension at most
$\bigg{[}\dfrac{n+2}{2}\bigg{]}-1=\bigg{[}\dfrac{n}{2}\bigg{]}$. ∎

Similarly, one can also get analog results in the compact case.

######  Generalized Borel Theorem (compact case).

Let $d,n,\delta_{0},\dots,\delta_{n}$ be integer numbers with $\delta_{i}\geq
0$, $n\geq 2$ and $d>(n-1)(n+1)+\sum_{i=0}^{n}\delta_{i}$. Suppose that the
hypersurfaces $D_{i}=\\{z_{i}^{d-\delta_{i}}Q_{i}=0\\}$ ($0\leq i\leq n+1$)
have empty intersection. Then, for the collection of $n+1$ entire holomorphic
functions $f_{i}$ ($0\leq i\leq n$) such that

$\sum_{i=0}^{n}f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})\equiv 0,$

there is a partition of indexes
$\\{0,\dots,n\\}=\cup_{\alpha=0}^{\ell}I_{\alpha}$ such that the followings
hold

  1. (i)

$f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})\equiv 0$ if and only if
$i\in I_{0}$ (of course the set $I_{0}$ may be empty);

  2. (ii)

The cardinality $|I_{\alpha}|\geq 2$ for every $1\leq\alpha\leq\ell$;

  3. (iii)

For each $1\leq\alpha\leq\ell$, for arbitrary indexes $i,j\in I_{\alpha}$,
there exists a constant $c_{ij}\in\mathbb{C}$ such that
$\frac{f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})}{f_{j}^{d-\delta_{j}}Q_{j}(f_{0},f_{1},\dots,f_{n})}=c_{ij}$;

  4. (iv)

$\sum_{i\in
I_{\alpha}}f_{i}^{d-\delta_{i}}Q_{i}(f_{0},f_{1},\dots,f_{n})\equiv 0$ for all
$0\leq\alpha\leq\ell$.

######  Corollary 2.3.

Keeping the same assumption as above. Putting

$\displaystyle\pi\colon\mathbb{CP}^{n}$
$\displaystyle\rightarrow\mathbb{CP}^{n},\qquad[z_{0}:\dots:z_{n}]\mapsto[z_{0}^{d-\delta_{0}}Q_{0}:\dots:z_{n}^{d-\delta_{n}}Q_{n}],$
$\displaystyle g=[g_{0}:\dots:g_{n}]\colon\mathbb{C}$
$\displaystyle\rightarrow\mathbb{CP}^{n},\qquad z\mapsto\pi\circ f(z),$

where $g_{i}:=(z_{i}^{d-\delta_{i}}Q_{i})\circ f$. If
$f\colon\mathbb{C}\rightarrow D$ is a non-constant holomorphic curve into $D$,
then the image of the composition map $g$ is contained in a smaller linear
subspace of $\mathbb{CP}^{n}$ of dimension at most
$\big{[}\frac{n-1}{2}\big{]}$.

######  Remark 2.4.

In the case where $\delta_{0}=\dots=\delta_{n}=0$, we recover the classical
Borel Theorem [23], [28].

##  3\. Example of smooth hyperbolic surface whose complement is
hyperbolically embedded in $\mathbb{CP}^{3}$

###  3.1. Genus of some plane algebraic curves

We collect some results about the genus of plane algebraic curves (see [22]
for explicit computations). By using the generalized Borel Theorem, these
curves arise naturally as the degeneracy locus of entire holomorphic curves
avoiding the Fermat type surface given in Theorem A.

######  Proposition 3.1.

For arbitrary nonzero constants
$\beta,\epsilon_{1},\epsilon_{2}\in\mathbb{C}\setminus\\{0\\}$, the plane
curve $\mathcal{C}$ defined in inhomogeneous coordinates $(X,Y)$ of
$\mathbb{CP}^{2}$ by the polynomial

$\beta+X^{d-2}(X^{2}+\epsilon_{1}^{2})+Y^{d-2}(y^{2}+\epsilon_{2}^{2})$

has genus $g\geq 2$ when $d\geq 5$.

######  Proposition 3.2.

For arbitrary nonzero constants
$\epsilon_{1},\epsilon_{2}\in\mathbb{C}\setminus\\{0\\}$ with $(\pm
i\epsilon_{1})^{d}+(\pm i\epsilon_{2})^{d}\not=0$, the plane curve
$\mathcal{C}$ defined in inhomogeneous coordinates $(X,Y)$ of
$\mathbb{CP}^{2}$ by the polynomial

$X^{d-2}(X^{2}+\epsilon_{1}^{2})+Y^{d-2}(y^{2}+\epsilon_{2}^{2})$

has genus $g\geq 2$ when $d\geq 4$.

###  3.2. Construction of smooth hyperbolic surface of low degree whose
complement is hyperbolically embedded in $\mathbb{CP}^{3}$

This section is devoted to prove Theorem A. Recalling that
$D\subset\mathbb{CP}^{3}$ is the surface defined by the homogeneous polynomial

$z_{0}^{d}+z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2}).$

Using the generalized logarithmic Borel Theorem obtained in the previous
section, we will prove the hyperbolicity of $\mathbb{CP}^{3}\setminus D$. The
hyperbolicity of $D$ can be treated by the same arguments, using the compact
version of the generalized Borel Theorem. Let
$f\colon\mathbb{C}\rightarrow\mathbb{CP}^{3}\setminus D$ be a holomorphic
function. We need to prove that $f$ is constant. As before we consider the map

$\pi\colon\mathbb{CP}^{3}\rightarrow\mathbb{CP}^{3},[z_{0}:z_{1}:z_{2}:z_{3}]\mapsto[z_{0}^{d}:z_{1}^{d-2}(z_{2}^{2}+a_{0}z_{0}^{2}):z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2}):z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})].$

By the generalized Borel Theorem, the image of $f$ must satisfy two algebraic
equations. Precisely, the image of $g:=\pi\circ
f=[g_{0}:g_{1}:g_{2}:g_{3}]\colon\mathbb{C}\mapsto\mathbb{CP}^{3}$ must lie in
a projective line $L$ which is the intersection of two planes defined by the
linear polynomials of the form $\lambda_{i}z_{i}+\mu_{j}z_{j}=0$. To get
further degeneracy, we need to study in very details all possibilities of the
line $L$. We consider two cases:

  1. (a)

The image of $g$ lies in some coordinate plane;

  2. (b)

The image of $g$ does not lie in any coordinate plane.

Let us consider the case (a). If $g$ lands in another coordinate plane, then
by Proposition 2.1, $g$ must be constant and so is $f$. Thus we can suppose
that the image of $g$ is contained in only one coordinate plane. We treat
separately the following possibilities for the position of the image of $g$:

  1. (a1)

The image of $g$ lies in the plane $z_{0}$;

  2. (a2)

The image of $g$ lies in the plane $z_{i}$ for $1\leq i\leq 2$;

  3. (a3)

The image of $g$ lies in the plane $z_{3}$.

In the case (a1), one has $f_{0}\equiv 0$. Since $g(\mathbb{C})$ does not lie
in any other coordinate plane, we need to consider three circumstances:

  1. (a1.i)

The image of $f$ lies in the plane curve $z_{1}^{d}+z_{2}^{d}=0$ and avoid
$z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})$;

  2. (a1.ii)

The image of $f$ lies in the plane curve
$z_{1}^{d}+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ and avoids
$z_{2}=0$;

  3. (a1.ii’)

The image of $f$ lies in the plane curve
$z_{2}^{d}+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ and avoids
$z_{1}=0$.

Consider the case (a1.i). The curve $f(\mathbb{C})$ lies in some projective
line $z_{1}+\mu z_{2}=0$ where $\mu^{d}=-1$. This line intersects with
$z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ at three distinct
points. This forces $f$ to be constant by the Little Picard Theorem.

Next we consider the case (a1.ii). The plane curve
$z_{1}^{d}+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ in the
inhomogeneous coordinate $(X,Y)$ is given by
$X^{d}+a_{2}X^{2}+a_{3}Y^{2}+1=0$. This curve is smooth and therefore, by the
genus formula, its genus is at least $2$, which forces $f$ to be constant. The
case (a1ii’) could be treated similarly.

Now we consider the case (a2), and without lost of generality, we assume that
$g_{1}\equiv 0$. As before, we have three circumstances:

  1. (a2.i)

The image of $f$ lies in the surface
$z_{0}^{d}+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$ and avoids the surface
$z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$;

  2. (a2.ii)

The image of $f$ lies in the surface
$z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$
and avoids the surface $z_{0}=0$;

  3. (a2.iii)

The image of $f$ lies in the surface
$z_{0}^{d}+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ and avoids
the surface $z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$.

First we treat the case (a2.i). The image of $f$ lies in the planes
$z_{0}=\lambda z_{2}$ and $z_{1}=0$ or $z_{0}=\mu z_{1}$ where $\lambda,\mu$
are some constants satisfying $\lambda^{d}+a_{1}\lambda^{2}+1=0$ and
$\mu^{2}+a=0$. Hence its image lies in a line intersecting with
$z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ at three distinct
points, which implies that $f$ is constant.

For the case (a2.ii), as before, the image of $f$ lies in the plane $z_{1}=0$
or $z_{0}=\mu z_{1}$. In the first case, the image of $f$ also lies in
$z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+z_{3}^{d-2}(a_{3}z_{2}^{2}+z_{3}^{2})=0$.
In inhomogeneous coordinates $(X,Y)$, this curve is given by the equation
$X^{d-2}(X^{2}+a_{1}Y^{2})+a_{3}X^{2}+1=0$. For all but except finite choices
of $a_{1},a_{3}$, this curve is smooth and we can use the genus formula to
conclude. In the second case where $f(\mathbb{C})$ lies in the plane
$z_{0}=\mu z_{1}$, it must also lie in the curve
$z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+z_{3}^{d-2}(-a_{0}a_{2}z_{0}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$.
Again, for all but except finite choices of $a_{i}$, this curve is smooth and
we get the constancy of $f$.

Finally, consider the case (a2iii). The image of $f$ lies in the plane
$z_{1}=0$ or $z_{0}=\mu z_{1}$. Furthermore, it lies also in the plane curve
$z_{0}^{d}+z_{3}^{d-2}(\alpha z_{0}^{2}+a_{3}z_{2}^{2}+z_{3}^{3})=0$, where
$\alpha\in\mathbb{C}$ is a constant. This curve is given in the inhomogeneous
coordinate $(X,Y)$ as $X^{d}+\alpha X^{2}+a_{3}Y^{2}+1=0$, which is of genus
at least $2$, which implies that $f$ is constant.

We now move to the case (a3). As before, we have three sub-cases:

  1. (a3.i)

The image of $f$ lies in the surface
$z_{0}^{d}+z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$ and avoids the surface
$z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$;

  2. (a3.i’)

The image of $f$ lies in the surface
$z_{0}^{d}+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$ and avoids the surface
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$;

  3. (a3.ii)

The image of $f$ lies in the surface
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$
and avoids the surface $z_{0}=0$.

In the case (a3.i) and (a3.i’), the image of $f$ lies in a projective line or
conic and avoids three points, and hence it must be constant. In the case
(a3.ii), we employ Proposition 3.2 to conclude.

Now we pass to the case (b). By the generalized Borel Theorem, we need to
consider the following circumstances:

  1. (b1)

The image of $f$ lies in the surfaces
$z_{0}^{d}+z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$,
$z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+\alpha
z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ for $\alpha\not=1$ and
avoids the surface $z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$;

  2. (b2)

The image of $f$ lies in the surfaces $z_{0}^{d}+\alpha
z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$,
$z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$
for $\alpha\not=1$ and avoids the surface
$z_{0}^{d}+z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$;

  3. (b1’)

The image of $f$ lies in the surfaces
$z_{0}^{d}+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$,
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+\alpha
z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$ for $\alpha\not=1$ and
avoids the surface $z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$;

  4. (b2’)

The image of $f$ lies in the surfaces $z_{0}^{d}+\alpha
z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$,
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$
for $\alpha\not=1$ and avoids the surface
$z_{0}^{d}+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$;

  5. (b3)

The image of $f$ lies in the surfaces
$z_{0}^{d}+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$,
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+\alpha
z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$ for $\alpha\not=1$ and avoids the
surface $z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$;

  6. (b4)

The image of $f$ lies in the surfaces $z_{0}^{d}+\alpha
z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$,
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$
for $\alpha\not=1$ and avoids the surface
$z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$;

  7. (b3’)

The image of $f$ lies in the surfaces $z_{0}^{d}+\alpha
z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$,
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$
for $\alpha\not=1$ and avoids the surface
$z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$;

  8. (b4’)

The image of $f$ lies in the surfaces
$z_{0}^{d}+z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$,
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+\alpha
z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$ for $\alpha\not=1$ and avoids the
surface $z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$;

  9. (b5)

The image of $f$ lies in the curve $z_{0}^{d}=\alpha
z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=\beta
z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$, where
$\alpha,\beta\not=0,1+\alpha^{-1}+\beta^{-1}=0$ and avoids
$z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$;

  10. (b6)

The image of $f$ lies in the curve $z_{0}^{d}=\alpha
z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=\beta
z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})$, where
$\alpha,\beta\not=0,1+\alpha^{-1}+\beta^{-1}=0$ and avoids
$z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$;

  11. (b6’)

The image of $f$ lies in the curve $z_{0}^{d}=\alpha
z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=\beta
z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})$, where
$\alpha,\beta\not=0,1+\alpha^{-1}+\beta^{-1}=0$ and avoids
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=0$;

  12. (b7)

The image of $f$ lies in the curve
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=\alpha
z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=\beta
z_{3}^{d-2}(a_{2}z_{1}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})$, where
$\alpha,\beta\not=0,1+\alpha^{-1}+\beta^{-1}=0$ and avoids $z_{0}=0$.

By symmetry, we only need to consider the case (bi). Details of case-by-case
arguments are given belows.

  1. $\bullet$

In the case (b1), the curve $f$ must lie in the plane $z_{1}=\lambda z_{0}$,
where $\lambda$ depending only on $a_{0}$. Hence $f(\mathbb{C})$ must lie in
the plane curve $z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})+\alpha z_{3}^{d-2}(\mu
z_{0}^{2}+a_{3}z_{2}^{2}+z_{3}^{2})=0$, where $\mu=\lambda^{2}a_{2}$. In the
inhomogeneous coordinates $(X,Y)$, this curve is given by the algebraic
equation $X^{d-2}(X^{2}+a_{1}Y^{2})+\alpha(\mu Y^{2}+a_{3}X^{2}+1)=0$. A
straightforwards computation shows that this curve has at most four ordinary
singularities. Hence this curve is irreducible and has genus at least $2$,
which yields the constancy of $f$.

  2. $\bullet$

The case (b2) can be treated similar as in the case (b1).

  3. $\bullet$

In the case (b3), applying Proposition 3.2, the curve
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})+\alpha
z_{2}^{d-2}(z_{2}^{2}+a_{1}z_{0}^{2})=0$ is either irreducible and has genus
at least $2$, or it has irreducible components with lines of the form
$z_{1}+\lambda z_{3}=0$ and an irreducible component of genus at least $2$.
Therefore we can suppose that the image of $f$ lies in a line of the form
$z_{1}+\lambda z_{3}=0$. This implies that $f(\mathbb{C})$ is contained in the
plane curve $z_{0}^{d}+z_{3}^{d-2}(\mu z_{1}^{2}+z_{3}^{2})=0$. This curve in
the inhomogeneous coordinates is given as $X^{d}+\mu Y^{2}+1=0$.

  4. $\bullet$

In the case (b4), we can apply directly Proposition 3.2 to get conclusion.

  5. $\bullet$

In the case (b5), the image of $f$ lies in a line and omits three distinct
points there. The constancy of $f$ follows from the Little Picard Theorem.

  6. $\bullet$

In the case (b6), the image of $f$ lies in the plane $z_{1}=\lambda z_{0}$ and
the algebraic curve $z_{0}^{d}=\beta z_{3}^{d-2}(\mu
z_{0}^{2}+a_{3}z^{2}+z_{3}^{2})=0$, or $X^{d}=\beta(\mu
X^{2}+a_{3}Y^{2}+a_{3})=0$ in the inhomogeneous coordinates $(X,Y)$. This
curve is irreducible and has at most two ordinary singularities, and hence its
genus is at least $2$, which implies the constancy of $f$.

  7. $\bullet$

In the case (b7), the image of $f$ lies in a plane of the form $z_{1}=\lambda
z_{2}$ for some constant $\lambda$. Therefore it lies also in the plane curve
$z_{1}^{d-2}(z_{1}^{2}+a_{0}z_{0}^{2})=\beta z_{3}^{d-2}(\mu
z_{1}^{2}+z_{3}^{2})$, or $X^{d-2}(X^{2}+a_{0}Y^{2})-\beta(\mu X^{2}+1)=0$ in
the inhomogeneous coordinates $(X,Y)$. Straightforward computation shows that
this curve is irreducible and has genus at least $2$, where concludes the
constancy of $f$.

######  Remark 3.3.

Similar constructions in the compact case were gave in [22], [48]. The
hyperbolic surfaces were constructed in these works are given by algebraic
equations of the form

$z_{0}^{d-2}(z_{0}^{2}+\epsilon_{1}^{2}z_{1}^{d}+\epsilon_{2}^{2}z_{2}^{2})+z_{1}^{d}+z_{2}^{d}+z_{3}^{d}=0,$

where $\epsilon_{1},\epsilon_{2}$ are suitable constants. Note however that
the complement of these surfaces are never hyperbolic. Indeed, the complement
of such surface contains the image of the holomorphic curve
$f\colon\mathbb{C}\rightarrow\mathbb{CP}^{3},z\mapsto[0:e^{\varphi}:\psi:\epsilon\psi]$,
where $\varphi,\psi$ are entire functions and $\epsilon$ is a $d$–root of
$-1$.

##  4\. Holomorphic curves intersecting Fermat-Warning type hypersurface

###  4.1. Counting dimension of some subvarieties of Grassmanians

Let $\operatorname{Gr}_{m,k}$ denote the Grassmannian of complex codimension
$k$ subspaces of $\mathbb{C}^{m}$. Let $Q_{m,k}$ denote the subspace
$0_{k}\times\mathbb{C}^{m-k}\subset\operatorname{Gr}_{m,k}$. For integers
$a,b,c$ with $1\leq a\leq c\leq m-1$ and $1\leq b\leq c\leq a+b$, consider the
set

$\Gamma_{m,a,b,c}:=\\{V\in\operatorname{Gr}_{m,a}:\dim V\cap Q_{m,c}\geq
m-c\\}.$

The following counting dimension result is due to Shiffman-Zaidenberg [41].

######  Lemma 4.1.

$\dim\operatorname{Gr}_{m,a}-\dim\Gamma_{m,a,b,c}=(m-c)(a+b-c).$

###  4.2. Second Main Theorem for nonconstant holomorphic curves into
projective and a Fermat-Warning type hypersurface

Let us now enter the proof of Theorem B. Let $[z_{0}:\dots:z_{n}]$,
$[w_{1}:\dots:w_{m}]$ be homogeneous coordinates of $\mathbb{CP}^{n}$ and
$\mathbb{CP}^{m-1}$, respectively. Let $[f_{0}:f_{1}:\dots:f_{n}]$ be a
reduced representation of $f$. Consider the morphism

$\pi\colon\mathbb{C}\mathbb{P}^{n}\longmapsto\mathbb{C}\mathbb{P}^{m-1},\qquad[z_{0}:z_{1}:\dots:z_{n}]\longmapsto[h_{1}^{d}:h_{2}^{d}:\dots:h_{m}^{d}],$

and put $g=\pi\circ
f\colon\mathbb{C}\longmapsto\mathbb{P}^{m-1},z\longmapsto[g_{1}^{d}:\dots:g_{m}^{d}]$,
where $g_{i}:=h_{i}\circ f$. It follows from the definition that

(4.1) $T_{g}(r)=d\,T_{f}(r)+O(1).$

Let $\\{H_{i}\\}_{1\leq i\leq m+1}$ be the family of $m+1$ hyperplanes in
general position in $\mathbb{CP}^{m-1}$ given by

$\displaystyle H_{i}$ $\displaystyle=\\{w_{i}=0\\}$ $\displaystyle(1\leq i\leq
m),$ $\displaystyle H_{m+1}$ $\displaystyle=\\{\sum_{j=1}^{m}w_{j}=0\\}.$

Set $I_{0}:=\\{i\in\mathbb{N}:1\leq i\leq m,g_{i}\equiv 0\\}$ and assume that
$\ell=|I_{0}|$. Then $0\leq\ell\leq n-1$ and the image of $g$ lies in the
subspace $H=\cap_{i\in I_{0}}H_{i}\cong\mathbb{CP}^{m-1-\ell}$. Denote by
$J=\\{1\leq j\leq m\\}\setminus I_{0}$, and consider the map

$\widetilde{g}\colon\mathbb{C}\mapsto H\cong\mathbb{CP}^{m-1-\ell},\qquad
z\rightarrow[g_{j}^{d}(z)]_{j\in J},$

we then still have

(4.2) $T_{\widetilde{g}}(r)=d\,T_{f}(r)+O(1).$

If $\widetilde{g}$ is linearly nondegenerate, then by applying Cartan’s Second
Main Theorem for the collection of hyperplanes $\\{\widetilde{H_{j}}\\}_{j\in
J\cup\\{m+1\\}}$, where $\widetilde{H_{j}}=H_{j}\cap H$, one obtains

(4.3) $T_{\widetilde{g}}(r)\leq\sum_{j\in
J}N_{\widetilde{g}}^{[m-1-\ell]}(r,\widetilde{H_{j}})+N_{\widetilde{g}}^{[m-1-\ell]}(r,\widetilde{H_{m+1}})+S_{\widetilde{g}}(r).$

For each $j\in J$, since the multiplicity of
$\widetilde{g}^{*}\widetilde{H_{j}}$ is at least $d$ at every point on its
support, one has

$N_{\widetilde{g}}^{[m-1-\ell]}(r,\widetilde{H_{j}})\leq\bigg{(}\dfrac{m-1-\ell}{d}\bigg{)}N_{\widetilde{g}}(r,\widetilde{H_{i}}),$

which implies

$N_{\widetilde{g}}^{[m-1-\ell]}(r,\widetilde{H_{j}})\leq\bigg{(}\dfrac{m-1-\ell}{d}\bigg{)}T_{\widetilde{g}}(r)+O(1),$

by the First Main Theorem. Furthermore, it is clear from the definition that

$N_{\widetilde{g}}^{[m-1-\ell]}(r,\widetilde{H_{m+1}})=N_{f}^{[m-1-\ell]}(r,D).$

Thus it follows from (4.2), (4.3) that

(4.4) $\big{(}d-(m-\ell)(m-\ell-1)\big{)}T_{f}(r)\leq
N_{f}^{[m-1-\ell]}(r,D)+S_{f}(r),$

which implies the desired estimate of the Main Theorem.

Next, consider the case where the image of $g$ lies in some hyperplane of the
subspace $H\cong\mathbb{CP}^{m-1-\ell}$. Suppose that the holomorphic
functions $g_{j},j\in J$ satisfy the equation

(4.5) $\sum_{k\in K}a_{k}g_{k}^{d}=0,$

where $K\subset J$ is some subset of $J$ with $|K|\geq 2$ and where $a_{k}$
are nonzero complex numbers. We first prove the following

######  Claim.

There exist some indexes $i\not=j$ with $i,j\in J$ such that $g_{i}/g_{j}$ is
constant.

Indeed, if $|K|=2$, the claim follows directly from (4.5). Suppose
$|K|=\gamma\geq 3$. Consider the homogeneous coordinates $[w_{k}]_{k\in K}$ of
the linear subspace $\mathbb{C}\mathbb{P}^{\gamma-1}$. Similar as in above,
consider the hyperplane
$H_{K}\cong\mathbb{C}\mathbb{P}^{\gamma-2}\subset\mathbb{C}\mathbb{P}^{\gamma-1}$
defined as $\sum_{k\in I}a_{k}w_{k}=0$ and the holomorphic map

$\mathsf{g}_{K}\colon\mathbb{C}\mapsto
H_{K}\cong\mathbb{C}\mathbb{P}^{\gamma-2},\qquad\mathsf{g}_{K}(z)=[g_{k}^{d}(z)]_{k\in
K}.$

If the image of $\mathsf{g}_{K}$ doesn’t lie in some smaller linear subspace,
then using ramification theorem for $\mathsf{g}_{K}$ and the family of
$\gamma$ hyperplanes $\\{H_{k}\\}_{k\in K}$ in
$H_{K}\cong\mathbb{C}\mathbb{P}^{\gamma-2}$, one obtains

(4.6) $\sum_{k\in K}\bigg{(}1-\dfrac{\gamma-2}{d}\bigg{)}\leq\gamma-1,$

contradiction. Thus $\mathsf{g}_{K}$ is linearly degenerate. Inductively, the
claim is proved.

Going back to the proof of the Theorem, we now follow the arguments in [28,
Example 3.10.21]. Let $\sim$ be the equivalence relation on the index set $J$
defined as $i\sim j$ if and only if $g_{i}/g_{j}$ is constant and let
$\mathcal{I}:=\\{I_{1},\dots,I_{r}\\}$ be the partition of $J$ by $\sim$. For
each $1\leq s\leq r$, suppose that $|I_{s}|=\kappa_{s}$. For convenient, we
put $|I_{0}|=\ell=\kappa_{s}$ and we write
$I_{s}=\\{i_{s,1},\dots,i_{s,\kappa_{s}}\\}$. Then

$None$ $g_{i_{s,j}}=\mu_{s,j}g_{i_{s,1}}$

for some constants $\mu_{s,j}\in\mathbb{C}$. we pick an index $i_{s}\in I_{s}$
and put $g_{j}=\ell_{j}g_{i_{s}}$ for each $j\in I_{s}$. Set
$b_{s}=1+\sum_{j=2}^{\kappa_{s}}\mu_{s,j}^{d}(1\leq s\leq r)$, then

$\sum_{j\in J}g_{j}^{d}=\sum_{s=1}^{r}b_{s}g_{i_{s,1}}^{d}.$

Now, put $M=\\{s:1\leq s\leq r,b_{s}\not=0\\}$. Consider the case where
$|M|\geq n+1$. Similar as in above, one considers the map

$\mathsf{g}_{M}\colon\mathbb{C}\mapsto
H_{M}\cong\mathbb{C}\mathbb{P}^{|M|-1},\qquad\mathsf{g}_{M}(z)=[g_{i_{s,1}}^{d}(z)]_{s\in
M}.$

Since $|M|\geq n+1$, one has $T_{\mathsf{g}_{M}}(r)=d\,T_{f}(r)$. If
$\mathsf{g}_{M}$ is linearly degenerate, by the above claim, one has
$g_{i}/g_{j}$ is constant for some $i,j\in M$, a contradiction. Hence
$\mathsf{g}_{M}$ is linearly nondegenerate, and by the same arguments as in
(4.4), one can use Cartan’Second Main Theorem for $\mathsf{g}_{M}$ and for the
$|M|+1$ hyperplanes $\\{w_{s}=0\\}(s\in M),\\{\sum_{s\in M}b_{s}w_{s}=0\\}$,
and one gets the desired estimate.

Now, consider the case where $|M|\leq n$. For each $s\notin M$, the set
$I_{s}$ contains at least $2$ indexes. The image of $g$ lies in the $r$-plane
$\mathcal{Y}_{\mu}^{\mathcal{I}}$ given by equations

$w_{i_{s,j}}=\mu_{s,j}w_{i_{s,1}}\quad(1\,\leq\,s\,\leq\,r,\,2\,\leq\,j\,\leq\,\kappa_{s});\qquad
w_{i_{0,j}}=0\quad(1\,\leq\,j\,\leq\,\kappa_{0}),$

where $\mathcal{I}=\\{I_{1},\dots,I_{r}\\},\mu=\mu_{s,j}$. Let
$\operatorname{Gr}_{m,k}$ denote the Grassmannian of complex codimension $k$
linear subspace of $\mathbb{C}^{m}$. We need to check that for generic
$V\in\operatorname{Gr}_{m,m-n-1}$, the intersection
$\mathcal{Y}_{\mu}^{\mathcal{I}}\cap\mathbb{P}(V)$ is at most a point for all
such above $(\mathcal{I},\mu)$. Indeed, by Lemma 4.1,
$\mathcal{Y}_{\mu}^{\mathcal{I}}\cap\mathbb{P}(V)$ is either a point or empty,
unless $V$ lies in a subvariety of $\operatorname{Gr}_{m,m-n-1}$ of
codimension

$\alpha=2(m-n-r+1).$

On the other hand, since $b_{s}=1+\sum_{j=2}^{\kappa_{s}}\mu_{s,j}^{d}=0$ for
any $s\not\in M$, the $\mu$-moduli space of the lifting
$\widetilde{\mathcal{Y}}_{\mu}^{\mathcal{I}}$ of
$\mathcal{Y}_{\mu}^{\mathcal{I}}$ to $\mathbb{C}^{m}$ in
$\operatorname{Gr}_{m,m-n-1}$ has dimension

$\beta\leq\sum_{s\in
M}(\kappa_{s}-1)+\sum_{1\,\leq\,s\leq\,r,\,s\,\not\in\,M}(\kappa_{s}-2)=\sum_{s=1}^{r}(\kappa_{s}-2)+|M|=m-\kappa_{0}-2r+|M|\leq
m-2r+n.$

Since $m\geq 3n-1$, one has $\alpha>\beta$. Therefore, for generic
$V\in\operatorname{Gr}_{m,m-n-1}$, the intersection
$\mathcal{Y}_{\mu}^{\mathcal{I}}\cap\mathbb{P}(V)$ is a point or empty for all
$(\mathcal{I},\mu)$. Hence $g$ is constant and so is $f$, a contradiction.
This finishes the proof of Theorem B.

######  Remark 4.2.

To get the hyperbolicity result of the complement, one only needs the
condition $m\geq 2n$. However, to get Second Main Theorem estimate, in the
above proof, we need the comparability between $T_{\mathsf{g}_{M}}(r)$ and
$T_{f}(r)$. For this reason, we must put stronger assumption $m\geq 3n-1$.

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