Study of some holomorphic curves in $\C^3$ and their projection into the complex projectve space $\C P^2$

TL;DR

This paper investigates holomorphic curves in complex three-space that avoid certain hyperplanes and subspaces, demonstrating that their projections into complex projective space can be non-constant, which challenges previous assumptions.

Contribution

It provides new insights into the behavior of holomorphic curves avoiding specific subspaces, showing their projections into projective space are not always constant.

Findings

Projections of certain holomorphic curves into $ ext{CP}^2$ can be non-constant.

Holomorphic curves avoiding four hyperplanes and a real subspace exhibit unexpected projection properties.

Abstract

We study holomorphic curves $f:\C\longrightarrow \C^3$ avoiding four complex hyperplanes and a real subspace of real dimension four or five in $\C^3$. We show that the projection of $f$ into the complex projective space $\C P^2$ is not necessarily constant.

Equations

$$Spa{n}_{\mathbb{R}}(H_i^{\perp },H_j^{\perp },H_k^{\perp })={\mathbb{R}}^6.$$

$$\displaystyle f(\mathbb{C})\bigcap\big{(}\bigcup_{i=1}^{4}H_{i}\bigcup H\big{)}=\emptyset$$

$$f(\mathbb{C})\cap H=\varnothing \Rightarrow \pi (f)(\mathbb{C})\cap \pi (H^{\star })=\varnothing .$$

$$\pi (H^{\star })=\{\left[1:z_2:z_3\right]\in \mathbb{C}{P}^2/a_1+a_2{z}_2+a_3{z}_3=0\}\cup \{[0:1:-\frac{a_2}{a_3}]\}$$

$$=\{[1:z:-\frac{a_1+a_{2}z}{a_3}],z\in \mathbb{C}\}\cup \{[0:1:-\frac{a_2}{a_3}]\}.$$

$$\pi (f)(z)=\left[1:\lambda :-\frac{a_1+a_2\lambda }{a_3}\right].$$

$$\pi (f)(z)=\left[0:1:-\frac{a_2}{a_3}\right].$$

$$\displaystyle\begin{array}[]{ccc}&H_{1}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{1}=0\},\\ &H_{2}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{2}=0\},\\ &H_{3}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{3}=0\}.\\ \end{array}$$

$$\displaystyle\begin{array}[]{ccc}&\pi(H^{\star}_{1})=&\{[0:1:z];~{}z\in\mathbb{C}\cup\{\infty\}\}\cup[0:0:1],\\ &\pi(H^{\star}_{2})=&\{[1:0:z];~{}z\in\mathbb{C}\cup\{\infty\}\}\cup[0:0:1],\\ &\pi(H^{\star}_{3})=&\{[1:z:0];~{}z\in\mathbb{C}\cup\{\infty\}\}\cup[0:1:0].\end{array}$$

$$h_k:=H_k(f),k=1,...,2n.$$

$$\displaystyle\textbf{(S)}\ \left\{\begin{array}[]{ccllll}h_{2}&=&\alpha_{2}h_{1}&\\ h_{3}&=&\alpha_{3}h_{1}&\\ &\vdots&\\ h_{2n}&=&\alpha_{2n}h_{1}\\ \end{array}\right.$$

$$S_1=\{\sigma (1),...,\sigma (n)\},S_2=\{\sigma (n+1),...,\sigma (2n)\}.$$

$$\displaystyle(S_{1})\left\{\begin{array}[]{ccllll}h_{\sigma(2)}&=&\alpha_{2}h_{\sigma(1)}&\\ h_{\sigma(3)}&=&\alpha_{3}h_{\sigma(1)}&\\ &\vdots&\\ h_{\sigma(n)}&=&\alpha_{n}h_{\sigma(1)}\\ \end{array}\right.~{}~{}~{}~{}(S_{2})\left\{\begin{array}[]{ccllll}h_{\sigma(n+1)}&=&\beta_{n+1}h_{\sigma(2n)}&\\ h_{\sigma(n+2)}&=&\beta_{n+2}h_{\sigma(2n)}&\\ &\vdots&\\ h_{\sigma(2n-1)}&=&\beta_{2n-1}h_{\sigma(2n)}\\ \end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ccllll}f(\mathbb{C})&\subset&(\bigcap_{k=2}^{n}H_{k})\bigcap H_{1}&=&\bigcap_{k=1}^{n}H_{k}\\ f(\mathbb{C})&\subset&(\bigcap_{k=n+1}^{2n-1}H_{k})\cap H_{2n}&=&\bigcap_{k=n+1}^{2n}H_{k}\\ \end{array}\right.$$

$$\displaystyle\begin{array}[]{lllc}&H_{1}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{1}=0\},\\ &H_{2}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{2}=0\},\\ &H_{3}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{3}=0\},\\ &H_{4}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{1}+z_{2}+z_{3}=0\},\\ &H_{5}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/a_{1}z_{1}+a_{2}z_{2}+a_{3}z_{3}=0\},~{}~{}~{}~{}a_{j}\in\mathbb{R}\setminus\{0\}~{}~{}~{}\forall~{}j=1,2,3.\\ \end{array}$$

$$f=\left(e^{h_1},e^{h_2},e^{h_3}\right).$$

$$\left[e^{h_1(z)}:e^{h_2(z)}:e^{h_3(z)}\right]=\left[{\omega }_1:{\omega }_2:{\omega }_3\right].$$

$$\left[1:\frac{e^{h_2(z)}}{e^{h_1(z)}}:\frac{e^{h_3(z)}}{e^{h_1(z)}}\right]=\left[1:\frac{{\omega }_2}{{\omega }_1}:\frac{{\omega }_3}{{\omega }_1}\right]$$

$$\displaystyle\left\{\begin{array}[]{cllll}e^{h_{2}(z)-h_{1}(z)}&=&\frac{\omega_{2}}{\omega_{1}}\\ {}{}\hfil\\ e^{h_{3}(z)-h_{1}(z)}&=&\frac{\omega_{3}}{\omega_{1}}\\ \end{array}\right.\Rightarrow\left\{\begin{array}[]{cllll}e^{h_{2}(z)}&=&\frac{\omega_{2}}{\omega_{1}}e^{h_{1}(z)}\\ {}{}\hfil\\ e^{h_{3}(z)}&=&\frac{\omega_{3}}{\omega_{1}}e^{h_{1}(z)}\\ \end{array}\right.$$

$$H_k:=\left\{Z\in {\mathbb{C}}^3/\sum _{i=1}^3{\alpha }_i^k{z}_i=0,{\alpha }_i^k\in \mathbb{C},1⩽k⩽n\right\}.$$

$${\alpha }_1^k+{\alpha }_2^k{c}_2+{\alpha }_3^k{c}_3\ne 0.$$

$$\displaystyle\begin{array}[]{lllc}&H_{1}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{1}=0\},\\ &H_{2}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{2}=0\},\\ &H_{3}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{3}=0\},\\ &H_{4}=&\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/z_{1}+z_{2}+z_{3}=0\},\\ \end{array}$$

$$\displaystyle\begin{array}[]{ccc}&H_{1}^{\perp}=&Span_{\mathbb{R}}\big{[}(1,0,0,0,0,0);(0,1,0,0,0,0)\big{]},\\ &H_{2}^{\perp}=&Span_{\mathbb{R}}\big{[}(0,0,1,0,0,0);(0,0,0,1,0,0)\big{]},\\ &H_{3}^{\perp}=&Span_{\mathbb{R}}\big{[}(0,0,0,0,1,0);(0,0,0,0,0,1)\big{]},\\ &H_{4}^{\perp}=&Span_{\mathbb{R}}\big{[}(1,0,1,0,1,0);(0,1,0,1,0,1)\big{]}.\\ \end{array}$$

$$\displaystyle H=\left\{\begin{array}[]{cllll}X_{1}-X_{2}&=&0\\ {}{}\hfil\\ X_{1}-X_{3}&=&0\\ \end{array}\right.$$

$$f=(e^{f_1},e^{f_2},e^{f_3}).$$

$$g=\left[1:e^{g_2}:e^{g_3}\right],$$

$$\displaystyle\begin{array}[]{cllll}\pi(H^{\star}_{i})&=&\{\left[z_{1}:z_{2}:z_{3}\right]\in\mathbb{C}P^{2}:z_{i}=0\}~{}For~{}j=1,2,3,\\ \pi(H^{\star}_{4})&=&\{\left[z_{1}:z_{2}:z_{3}\right]\in\mathbb{C}P^{2}:z_{1}+z_{2}+z_{3}=0\}.\\ \end{array}$$

$$\begin{array}[]{cllll}\Delta_{12,34}&=&\{\left[z_{1}:z_{2}:z_{3}\right]\in\mathbb{C}P^{2}:z_{1}+z_{2}=0\},\\ \Delta_{13,24}&=&\{\left[z_{1}:z_{2}:z_{3}\right]\in\mathbb{C}P^{2}:z_{2}+z_{3}=0\},\\ \Delta_{14,23}&=&\{\left[z_{1}:z_{2}:z_{3}\right]\in\mathbb{C}P^{2}:z_{1}+z_{3}=0\}.\end{array}$$

$$f=(e^{f_1},-e^{f_1},e^{f_3}).$$

$$Re(e^{f_3})=Re(e^{2f_1})=Re(e^{f_1}{e}^{f_1})=Re(e^{f_1}{)}^2-Im(e^{f_1}{)}^2.$$

Full text

keywords Complex projective space, Holomorphic curves, Kobayashi hyperbolicity

1. Introduction

The classical Picard Theorem [1] (see also [2]) states that every holomorphic map from the complex Euclidean space $\displaystyle\mathbb{C}$ to $\displaystyle\mathbb{C}P^{1}$ that avoids three points, is constant. This Theorem has been extended to higher dimension by M.Green [3] who provided with examples of complex Kobayashi hyperbolic manifolds. We note that if $\displaystyle H_{1},...,H_{m}$ are complex hyperplanes in $\displaystyle\mathbb{C}P^{n}$, then they are said to be in general position if $\displaystyle m\geqslant n+1$ and any $\displaystyle(n+1)$ of these hyperplanes are linearly independent. Let us recall the Green Theorem:

  

Theorem (Green, [3]). Let $\displaystyle C$ be a union of $\displaystyle 2n+1$ complex hyperplanes in general position in $\displaystyle\mathbb{C}P^{n}$. Then, any holomorphic curve $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}P^{n}\setminus C$ is constant.

In particular, for $\displaystyle n=2$, any holomorphic curve $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}P^{2}\setminus C$ is constant, where $\displaystyle C$ is a union of five complex lines in general position in $\displaystyle\mathbb{C}P^{n}$.

As a direct consequence of the Green Theorem, the canonical projection into the complex projective space $\displaystyle\mathbb{C}P^{2}$ of any holomorphic map $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$ which avoids five complex hyperplanes in $\displaystyle\mathbb{C}^{3}$ is constant, since its image avoids the projections of the five complex hyperplanes, which are complex projective lines in general position in $\displaystyle\mathbb{C}P^{2}$ (see Lemma 2.1). Our main goal is to study the projection into $\displaystyle\mathbb{C}P^{2}$ of a holomorphic curve $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$ which avoids four complex hyperplanes in general position in $\displaystyle\mathbb{C}^{3}$ and a real subspace $\displaystyle H$ of real dimension four or five and check if the projection remains constant.

Throughout the paper we identify $\displaystyle\mathbb{R}^{6}$, endowed with its standard complex structure $\displaystyle J_{st}$, to $\displaystyle\mathbb{C}^{3}.$

Definition 1.1**.**

Let $\displaystyle n\geqslant 3$ and let $\displaystyle\mathcal{H}=(H_{1},...,H_{n})$ be a family of real subspaces of $\displaystyle\mathbb{R}^{6}$ such that $\displaystyle codim_{\mathbb{R}}H_{j}=2$ for $\displaystyle j=1,...,n$. Then $\displaystyle\mathcal{H}$ is said to be in general position if for every 3-tuple $\displaystyle(i,j,k)$ of distinct integers $\displaystyle i,j,k\in\{1,...,n\}$,

[TABLE]

Here, if $\displaystyle H$ is a real subspace in $\displaystyle\mathbb{R}^{6}$, then $\displaystyle H^{\perp}$ denotes the orthogonal complement of $\displaystyle H$ with respect to the Euclidean metric.

We first study the case of four real dimensional subspaces in $\displaystyle\mathbb{C}^{3}.$ We have the following

Theorem 1**.**

$\displaystyle(i)$* Let $\displaystyle H_{1},...,H_{n}$ be $\displaystyle n$ complex hyperplanes in $\displaystyle\mathbb{C}^{3}$ in general position ($\displaystyle n\geqslant 5$), then there exists a non constant holomorphic curve $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$ which avoid these $\displaystyle n$ hyperplanes and $\displaystyle\pi(f)$ is constant.

$\displaystyle(ii)$ Let $\displaystyle H_{1},H_{2},H_{3},H_{4}$ be four complex hyperplanes in $\displaystyle\mathbb{C}^{3}$. Then there exists a real subspace $\displaystyle H$ of  $\displaystyle\mathbb{R}^{6}$, of real dimension four, such that $\displaystyle(H,H_{j},H_{k})$ are in general position for all $\displaystyle j\neq k,~{}~{}j,k\in\{1,...,4\}$, and there exists $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$ holomorphic, such that*

[TABLE]

and $\displaystyle\pi(f)$ is non constant.

Remark 1.1**.**

Here $\displaystyle\pi$ denotes the canonical projection from $\displaystyle\mathbb{C}^{3}\setminus\{0\}$ into $\displaystyle\mathbb{C}P^{2}$ and $\displaystyle\pi(f):=\pi\circ f.$ Notice that $\displaystyle\pi(f)$ is well-defined in Theorem 1 (ii) since $\displaystyle f(\mathbb{C})\subset\mathbb{C}^{3}\setminus\{0\}.$ In case (i), according to the Green Theorem and to Lemma 2.1 (see below), $\displaystyle\pi(f)$ is constant.

We study then the case of a subspace in $\displaystyle\mathbb{C}^{3}$ of real dimension five. We have the following:

Theorem 2**.**

*Let $\displaystyle H_{1},H_{2},H_{3},H_{4}$ be four complex hyperplanes in $\displaystyle\mathbb{C}^{3}$ and let $\displaystyle H$ be a real subspace of $\displaystyle\mathbb{R}^{6}$ of real dimension five. Let $\displaystyle\tilde{H}$ be a complex hyperplane of $\displaystyle\mathbb{C}^{3}$ such that $\displaystyle\tilde{H}\subset H$. Then:

$\displaystyle\left(1\right)$ If $\displaystyle(\tilde{H},H_{j},H_{k})$ are in general position for all $\displaystyle j\neq k,~{}~{}j,k\in\{1,...,4\}$, then every holomorphic map $\displaystyle~{}f:\mathbb{C}\rightarrow\mathbb{C}^{3}$ such that $\displaystyle f(\mathbb{C})\bigcap(\bigcup_{i=1}^{4}H_{i}\bigcup H)=\emptyset$ is constant.

$\displaystyle\left(2\right)$ If there exist $\displaystyle H_{j},H_{k}$, $\displaystyle j\neq k,~{}j,k\in\{1,...,4\},$ such that $\displaystyle(\tilde{H},H_{j},H_{k})$ are not in general position, then there exists $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$, holomorphic, such that \displaystyle f(\mathbb{C})\bigcap\big{(}\bigcup_{i=1}^{4}H_{i}\bigcup H\big{)}=\emptyset and $\displaystyle\pi(f)$ is non constant.*

Remark 1.2**.**

$\displaystyle\left(a\right)$* The existence and uniqueness of $\displaystyle\tilde{H}\subset H$ is explained in the proof of Theorem 2.

$\displaystyle\left(b\right)$ The condition ”$\displaystyle(\tilde{H},H_{j},H_{k})$ are not in general position” is equivalent to the condition ”$\displaystyle dim_{\mathbb{R}}Span_{\mathbb{R}}(\tilde{H}^{\perp},H_{j}^{\perp},H_{k}^{\perp})=4$”.

$\displaystyle\left(c\right)$ The fact of considering four complex hyperplanes is an optimal condition (see the end of section two for more details).*

The paper is organized as follows. In the first section, we give some results and properties. In section two, we prove Theorem 1. Finally, in section three, we prove Theorem 2.

2. Preliminaries and properties

In 1972, Fujimoto [4] (see also M.Green[3] and [5]) showed a statement that characterizes the image of a holomorphic map $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}P^{n}$ omitting $\displaystyle(n+p)$ hyperplanes in general position. He proved the following

Theorem (Fujimoto [4], Serge Lang [5] pp 196). Let $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}P^{n}$ be holomorphic. Assume that the image of $\displaystyle f$ lies in the complement of $\displaystyle n+p$ hyperplanes in general position, then this image is contained in a complex projective subspace of complex dimension $\displaystyle\leqslant[n/p].$

The version of the Green Theorem stated in the introduction is a particular case of the previous Theorem, with $\displaystyle p=n+1.$

In the remaining of the paper we will need the following properties satisfied by the canonical projection in $\displaystyle\mathbb{C}P^{2}$ of a holomorphic curve $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$. For $\displaystyle H$ a real subspace of $\displaystyle\mathbb{R}^{6}$, we denote by $\displaystyle H^{\star}$ the set $\displaystyle H\setminus\{0\}.$ Then, we have the following Lemma

Lemma 2.1**.**

Let $\displaystyle\pi:\mathbb{C}^{3}\setminus\{0\}\rightarrow\mathbb{C}P^{2}$ be the canonical projection. Then:

1. (1)

If H is a complex hyperplane in $\displaystyle\mathbb{C}^{3}$, then $\displaystyle\pi(H^{\star})$ is a complex projective line in $\displaystyle\mathbb{C}P^{2}$. 2. (2)

If $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$ is holomorphic and H is a complex hyperplane in $\displaystyle\mathbb{C}^{3}$, then

[TABLE] 3. (3)

If $\displaystyle H_{1},H_{2},H_{3}$ are complex hyperplanes in general position in $\displaystyle\mathbb{C}^{3}$, then $\displaystyle\pi(H^{\star}_{1}),\pi(H^{\star}_{2}),\pi(H^{\star}_{3})$ are in general position in $\displaystyle\mathbb{C}P^{2}$.

Notation: if $\displaystyle Z\in\mathbb{C}P^{2}$, we denote $\displaystyle\left[z_{1}:z_{2}:z_{3}\right]$ its homogeneous coordinates, where

$\displaystyle(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}.$

Proof.

Point (1). We may assume that $\displaystyle H=\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}/a_{1}z_{1}+a_{2}z_{2}+a_{3}z_{3}=0\}$, with $\displaystyle a_{1},a_{2},a_{3}\in\mathbb{C},$ $\displaystyle a_{3}\neq 0.$ Then

[TABLE]

[TABLE]

We notice that $\displaystyle[0:1:-\frac{a_{2}}{a_{3}}]$ corresponds to $\displaystyle[\frac{1}{\infty}:1:-\frac{a_{1}+a_{2}\infty}{a_{3}\infty}].$ Hence $\displaystyle\pi(H^{\star})$ is a projective complex line in $\displaystyle\mathbb{C}P^{2}.$

Point (2). We first notice that $\displaystyle\pi(f)$ is well defined since, by assumption $\displaystyle f(\mathbb{C})\cap H=\emptyset$, which implies that $\displaystyle f(\mathbb{C})\subset\mathbb{C}^{3}\setminus\{0\}.$ Assume now, to get a contradiction, that $\displaystyle\pi(f)(\mathbb{C})\cap\pi(H^{\star})\neq\emptyset$. Then there are two possibilities.

Case $\displaystyle(\alpha).$ There exists $\displaystyle z\in\mathbb{C}$ and there exists $\displaystyle\lambda\in\mathbb{C}$ such that

[TABLE]

Then, there exists $\displaystyle c_{z}\in\mathbb{C}^{*}$ such that \displaystyle f(z)=\big{(}c_{z},\lambda c_{z},-\frac{a_{1}+a_{2}\lambda}{a_{3}}c_{z}\big{)}. In particular $\displaystyle a_{1}f_{1}(z)+a_{2}f_{2}(z)+a_{3}f_{3}(z)=0$, where $\displaystyle f=(f_{1},f_{2},f_{3}).$ Hence, $\displaystyle f(z)\in H.$ This is a contradiction.

Case $\displaystyle(\beta).$ There exists $\displaystyle z\in\mathbb{C}$ such that

[TABLE]

Then, there exists $\displaystyle c_{z}\in\mathbb{C}^{*}$ such that \displaystyle f(z)=\big{(}0,c_{z},-\frac{a_{2}}{a_{3}}c_{z}\big{)} and $\displaystyle a_{1}f_{1}(z)+a_{2}f_{2}(z)+a_{3}f_{3}(z)=0.$ We obtain again that $\displaystyle f(z)\in H:$ this is a contradiction.

Point (3). Since $\displaystyle H_{1},H_{2},H_{3}$ are complex hyperplanes in $\displaystyle\mathbb{C}^{3}$, then there is a linear change of coordinates such that the hyperplanes are defined by equations

[TABLE]

Now by projection into $\displaystyle\mathbb{C}P^{2}$, we get

[TABLE]

Hence $\displaystyle\pi(H^{\star}_{1})\cap\pi(H^{\star}_{2})\cap\pi(H^{\star}_{3})=\emptyset$, meaning that $\displaystyle\pi(H^{\star}_{1}),\pi(H^{\star}_{2}),\pi(H^{\star}_{3})$ are in general position since there is no triple point.

∎

3. Proof of Theorem 1

To prove theorem 1, we need the following Lemma which characterize the image of a holomorphic map $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}P^{n}$ avoiding $\displaystyle 2n$ complex hyperplanes in general position. This precises the result of H.Fujimoto [4], [5] pp 196.

Definition 3.1**.**

Let $\displaystyle H_{1},...,H_{m}$, $\displaystyle m\geqslant 2n$, be hyperplanes of $\displaystyle\mathbb{C}P^{n}$. We call diagonal, a line passing through the two points $\displaystyle\bigcap_{i\in I}H_{i}$ and $\displaystyle\bigcap_{j\in J}H_{j}$, where $\displaystyle card(I)=card(J)=n$ and $\displaystyle I\cap J=\emptyset.$

Lemma 3.1**.**

Let $\displaystyle H_{1},...,H_{2n}$ be $\displaystyle(2n)$ projective hyperplanes in general position in $\displaystyle\mathbb{C}P^{n}$. Then there are $\displaystyle\frac{1}{2}C_{2n}^{n}$ diagonals $\displaystyle\Delta_{1},....,\Delta_{\frac{1}{2}C_{2n}^{n}}$ such that for every holomorphic curve $\displaystyle f:\mathbb{C}\longrightarrow\mathbb{C}P^{n}\setminus\bigcup_{i=1}^{2n}H_{i}$, there exists $\displaystyle k_{f}\in\{1,...,\frac{1}{2}C_{2n}^{n}\}$ such that $\displaystyle f(\mathbb{C})\subset\Delta_{k_{f}}.$

Proof.

The proof is inspired by the Fujimoto Theorem, [5] pp 196.

Let $\displaystyle f:\mathbb{C}\longrightarrow\mathbb{C}P^{n}$ be holomorphic, such that $\displaystyle f(\mathbb{C})\bigcap(\bigcup_{i=1}^{2n}H_{i})=\emptyset$.

Let $\displaystyle L_{1}$,…, $\displaystyle L_{2n}$ be linear forms defining the hyperplanes $\displaystyle H_{1},...,H_{2n}$, namely $\displaystyle H_{k}=L_{k}^{-1}\left(\{0\}\right)$ for $\displaystyle k=1,...,2n$. If $\displaystyle f=\left[f_{1}:...:f_{n+1}\right]$, we denote

[TABLE]

Let $\displaystyle I=\{1,\cdots,2n\}$ be the set of indices and $\displaystyle\sim$ be the equivalence relation defined by $\displaystyle i\sim j$ if $\displaystyle h_{i}/h_{j}$ is constant. We take a partition of the set of indices according to $\displaystyle\sim$. First, we know that the complement of a given class $\displaystyle S$ has at most n elements (see [5] pp 197). Hence $\displaystyle S$ has at least n elements and there are at most two classes.

The case of one class is not possible. In fact, There exists $\displaystyle\alpha_{2},...,\alpha_{2n}\in\mathbb{C}$ such that

[TABLE]

Hence $\displaystyle f(\mathbb{C})\subset(\bigcap_{k=2}^{n+1}H_{k})\bigcap H_{1}=\bigcap_{k=1}^{n+1}H_{k}=\emptyset$, which is impossible. Hence there are exactly two classes $\displaystyle S_{1}$ and $\displaystyle S_{2}$.

 We know that each of the two classes $\displaystyle S_{1},S_{2}$ contains $\displaystyle n$ elements. Then there exists a permutation $\displaystyle\sigma:\{1,...,2n\}\rightarrow\{1,...,2n\}$ such that

[TABLE]

Hence There exists $\displaystyle\alpha_{2},...,\alpha_{n},\beta_{n+1},...,\beta_{2n-1}\in\mathbb{C}$ such that $\displaystyle h_{1},...,h_{2n}$ satisfy the systems:

[TABLE]

Hence

[TABLE]

Then $\displaystyle f(\mathbb{C})\subset\Delta_{\sigma}$, where $\displaystyle\Delta_{\sigma}$ is the unique diagonal (line) passing through the two points $\displaystyle\bigcap_{k=1}^{n}H_{\sigma(k)}$ and $\displaystyle\bigcap_{k=n+1}^{2n}H_{\sigma(k)}$.

Now the two points, and consequently $\displaystyle\Delta_{\sigma}$, are completely determined by $\displaystyle S_{1}=\{\sigma(1),...,\sigma(n)\}$ since $\displaystyle S_{2}$ is automatically fixed once $\displaystyle S_{1}$ is chosen. Hence $\displaystyle\Delta_{\sigma}$ is completely determined by a choice of a partition of $\displaystyle\{1,...,2n\}$ into two subsets, each of them containing $\displaystyle n$ elements. There are exactly $\displaystyle\frac{1}{2}C_{2n}^{n}$ such partitions. This proves the Lemma. ∎

We may prove now Theorem 1.

We denote by $\displaystyle z=(z_{1},z_{2},z_{3})$ the coordinates in $\displaystyle\mathbb{C}^{3}$, where $\displaystyle z_{j}=x_{j}+iy_{j}$, $\displaystyle j=1,2,3$. Hence $\displaystyle(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3})$ denote the coordinates in $\displaystyle\mathbb{R}^{6}.$

  

Point (i). Consider first the case $\displaystyle n=5$. By a linear change of coordinates, we take the hyperplanes $\displaystyle H_{1},H_{2},H_{3},H_{4}$ and $\displaystyle H_{5}$ in standard form defined by the following equations

[TABLE]

By hypothesis \displaystyle f(\mathbb{C})\cap\big{(}\bigcup_{i=1}^{5}H_{i}\big{)}=\emptyset. Then there exists $\displaystyle h_{1},h_{2},h_{3}:\mathbb{C}\rightarrow\mathbb{C}$, holomorphic, such that

[TABLE]

Moreover, since $\displaystyle\pi(f)(\mathbb{C})$ omits $\displaystyle\pi(H_{i})$ for $\displaystyle i=1,...,5$ (see Lemma 2.1) and $\displaystyle\pi\circ f$ is constant by Green (see [3]), there exists $\displaystyle(\omega_{1},\omega_{2},\omega_{3})\neq(0,0,0)$ such that for all $\displaystyle z\in\mathbb{C}$,

[TABLE]

Therefore

[TABLE]

which implies that

[TABLE]

Hence $\displaystyle f=(e^{h_{1}},c_{2}e^{h_{1}},c_{3}e^{h_{1}})$, with $\displaystyle 1+c_{2}+c_{3}\neq 0$, and $\displaystyle f$ is not constant.

Essentially the same type of argument works in general. Let $\displaystyle H_{1},...,H_{n},~{}~{}~{}n\geqslant 5$, be $\displaystyle n$ hyperplanes defined by:

[TABLE]

By hypothesis \displaystyle f(\mathbb{C})\bigcap\big{(}\bigcup_{i=1}^{n}H_{i}\big{)}=\emptyset, then in particular \displaystyle f(\mathbb{C})\cap\big{(}\bigcup_{i=1}^{5}H_{i}\big{)}=\emptyset and $\displaystyle f=(e^{h},c_{2}e^{h},c_{3}e^{h})$ is not constant, where $\displaystyle h$ is holomorphic from $\displaystyle\mathbb{C}$ to $\displaystyle\mathbb{C}$.

Hence, in order that $\displaystyle f$ avoids $\displaystyle H_{1},...,H_{n}$, it is sufficient to choose $\displaystyle c_{2},c_{3}\in\mathbb{C}$ such that for every $\displaystyle k=1,...,n$

[TABLE]

We point out that what preceeds proves more generally that given a countable set of complex hyperplanes in $\displaystyle\mathbb{C}^{3}$ passing through the origin, there exists $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$ not constant and avoiding each hyperplane. This proves Point (i).

Point (ii). Let $\displaystyle H_{1},H_{2},H_{3}$ and $\displaystyle H_{4}$ be four complex hyperplanes in general position in $\displaystyle\mathbb{C}^{3}$. We know that there is a linear change of coordinate such that $\displaystyle H_{1},H_{2},H_{3}$ and $\displaystyle H_{4}$ are defined in standard form by :

[TABLE]

Then

[TABLE]

We pose now

[TABLE]

Then \displaystyle H^{\perp}=Span_{\mathbb{R}}\big{[}(1,0,1,0,0,0);(1,0,0,0,1,0)\big{]}, which of course satisfies the condition $\displaystyle Span_{\mathbb{R}}(H^{\perp},H_{j}^{\perp},H_{k}^{\perp})=\mathbb{R}^{6}$ for all $\displaystyle j\neq k,~{}~{}j,k\in\{1,...,4\}$.

Since $\displaystyle f(\mathbb{C})\bigcap(\bigcup_{i=1}^{4}H_{i})=\emptyset,$ then there exists holomorphic functions $\displaystyle f_{i}:\mathbb{C}\rightarrow\mathbb{C}$, $\displaystyle i=1,2,3$ such that

[TABLE]

Then, by Lemma 2.1 (2), $\displaystyle g:=\pi(f)$ satisfies $\displaystyle g(\mathbb{C})\subset\mathbb{C}P^{2}\setminus\bigcup_{j=1}^{4}\pi(H^{\star}_{j})$. Hence $\displaystyle g$ has the following form

[TABLE]

where $\displaystyle g_{2}=f_{2}-f_{1}$ and $\displaystyle g_{3}=f_{3}-f_{1}$. According to Lemma 3.1 there exists $\displaystyle\frac{1}{2}C^{2}_{4}=3$ diagonals $\displaystyle\Delta_{12,34},\Delta_{13,24},\Delta_{14,23}$ such that $\displaystyle g=\pi(f(\mathbb{C}))$ is contained in one of these diagonals, where $\displaystyle\Delta_{ij,kl}$ is the diagonal line passing through \displaystyle\big{(}\pi(H^{\star}_{i})\bigcap\pi(H^{\star}_{j})\big{)} and \displaystyle\big{(}\pi(H^{\star}_{k})\bigcap\pi(H^{\star}_{l})\big{)}.

We recall that

[TABLE]

Hence $\displaystyle\Delta_{12,34},\Delta_{13,24},\Delta_{14,23}$ are given by

[TABLE]

Suppose that $\displaystyle g(\mathbb{C})$ is contained in $\displaystyle\Delta_{12,34}$, the cases $\displaystyle g(\mathbb{C})\subset\Delta_{13,24}$ or $\displaystyle g(\mathbb{C})\subset\Delta_{14,23}$ being similar. Then

$\displaystyle e^{g_{2}}+1=0\Rightarrow e^{g_{2}}=-1\Rightarrow g=\left[1:-1:e^{g_{3}}\right]$, where $\displaystyle g_{3}=f_{3}-f_{1}$. Hence

[TABLE]

On another hand $\displaystyle f(\mathbb{C})\cap H=\emptyset\Leftrightarrow\forall z\in\mathbb{C}$, \displaystyle\left\{\begin{array}[]{cllll}Re(e^{f_{1}(z)})&\neq&0\\ or\\ Re(e^{f_{1}(z)}-e^{f_{3}(z)})&\neq&0.\\ \end{array}\right. 

We pose $\displaystyle f_{3}=2f_{1}$, then $\displaystyle f=(e^{f_{1}},-e^{f_{1}},e^{2f_{1}})$ avoids $\displaystyle H$. In fact

[TABLE]

Now if $\displaystyle Re(e^{f_{1}(z)})=0$ for some $\displaystyle z\in\mathbb{C}$, then $\displaystyle Im(e^{f_{1}(z)})\neq 0$ and consequently $\displaystyle Re(e^{f_{3}}(z))\neq 0.$ Hence $\displaystyle f(\mathbb{C})\cap H=\emptyset.$

Finally, $\displaystyle\pi(f)=\left[1:-1:e^{f_{1}}\right]$ is not constant and \displaystyle f(\mathbb{C})\bigcap\big{(}\bigcup_{j=1}^{4}H_{j}\bigcup H\big{)}=\emptyset. This concludes the proof of Theorem 1.

4. Proof of Theorem 2

Let $\displaystyle H$ be a real subspace of $\displaystyle\mathbb{C}^{3}$ such that $\displaystyle dim_{\mathbb{R}}H=5$, then $\displaystyle H$ contains a unique complex hyperplane $\displaystyle\tilde{H}$ of $\displaystyle\mathbb{C}^{3}$. Indeed, there exists $\displaystyle(a_{1},b_{1},a_{2},b_{2},a_{3},b_{3})\in\mathbb{R}^{6}\setminus\{0\}$ such that

[TABLE]

Hence $\displaystyle\tilde{H}:=\left\{z\in\mathbb{C}^{3}/~{}\sum_{j=1}^{3}(a_{j}-ib_{j})z_{j}=0\right\}$ is a complex hyperplane in $\displaystyle\mathbb{C}^{3}$, contained in $\displaystyle H$.

Point (1). Assume that $\displaystyle(\tilde{H},H_{j},H_{k})$ are in general position for some $\displaystyle j\neq k$, $\displaystyle j,k\in\{1,...,4\}$. Since $\displaystyle\tilde{H}\subset H$, where $\displaystyle\tilde{H}$ is a complex hyperplane of $\displaystyle\mathbb{C}^{3}$, and

[TABLE]

then it follows from Theorem 1 (i) that there is $\displaystyle(c_{1},c_{2})\in(\mathbb{C}^{*})^{2}$ which satisfies $\displaystyle 1+c_{2}+c_{3}\neq 0$ and there exists $\displaystyle h:\mathbb{C}\rightarrow\mathbb{C}$ holomorphic such that

[TABLE]

On another hand $\displaystyle H:=\{(x_{1},y_{1},...,x_{3},y_{3})\in\mathbb{R}^{6}/~{}\sum_{j=1}^{3}(a_{i}x_{i}+b_{i}y_{i})=0\}.$ By hypothesis $\displaystyle f(\mathbb{C})\cap H=\emptyset$ then for every $\displaystyle z\in\mathbb{C}$ we have,

[TABLE]

Thus, for every $\displaystyle z\in\mathbb{C}$

[TABLE]

We denote

[TABLE]

then

[TABLE]

However $\displaystyle\{(x,y)\in\mathbb{R}^{2}~{}/~{}ax+by=0\}$ is either a real line or $\displaystyle\mathbb{R}^{2}$, depending on the values of $\displaystyle a$ and $\displaystyle b$. Then by the little Picard Theorem $\displaystyle e^{h}$ is constant because it avoids an infinite number of points. Hence $\displaystyle h$ is constant and $\displaystyle f$ is then constant. We point out that the projection of $\displaystyle f$ into $\displaystyle\mathbb{C}P^{2}$ is also constant.

Point (2). Suppose there exists $\displaystyle j\neq k$, $\displaystyle j,k\in\{1,...,4\},$ such that

$\displaystyle dim_{\mathbb{R}}Span_{\mathbb{R}}(\tilde{H}^{\perp},H_{j}^{\perp},H_{k}^{\perp})=4$. Then:

[TABLE]

In fact for all $\displaystyle i=1,...,4$, $\displaystyle dim_{\mathbb{R}}H_{i}^{\perp}=2$, then $\displaystyle dim_{\mathbb{R}}Span_{\mathbb{R}}(H_{i}^{\perp},H_{j}^{\perp})=4$.

Suppose $\displaystyle\tilde{H}^{\perp}\subset Span_{\mathbb{R}}(H_{1}^{\perp},H_{2}^{\perp})$ then there exists $\displaystyle\alpha_{1},\alpha_{2}\in\mathbb{C}$ such that

[TABLE]

Since $\displaystyle f(\mathbb{C})\bigcap(\bigcup_{i=1}^{4}H_{i})=\emptyset$, then by 3

[TABLE]

We take $\displaystyle f_{1}=c,~{}c\in\mathbb{C}\setminus\{0\}$, such that $\displaystyle Re(\alpha_{1}e^{c}-\alpha_{2}e^{c})\neq 0$ and $\displaystyle f_{3}$ not constant. Then

[TABLE]

avoids $\displaystyle\bigcup_{i=1}^{4}H_{i}\bigcup H$, and $\displaystyle\pi(f)$ is not constant. This concludes the proof of Theorem 2. ∎

By the end of the paper, we show the optimality of considering four complex hyperplanes. Let $\displaystyle H_{1},H_{2},H_{3}$ be three complex hyperplanes in $\displaystyle\mathbb{C}^{3}$, then there exists $\displaystyle H$ a real hyperplane in $\displaystyle\mathbb{R}^{6}$ and a complex hyperplane $\displaystyle\tilde{H}$ contained in $\displaystyle H$, \displaystyle\big{(}H_{1},H_{2},H_{3},\tilde{H}\big{)} are in general position, and there exists $\displaystyle f:\mathbb{C}\rightarrow\mathbb{C}^{3}$, holomorphic, such that \displaystyle f(\mathbb{C})\cap\big{(}\bigcup_{j=1}^{3}H_{j}\bigcup H\big{)}=\emptyset and $\displaystyle\pi\circ f$ is not constant. In fact:

We pose $\displaystyle H=\{x_{1}+x_{2}+x_{3}=0\}$ and $\displaystyle\tilde{H}=\{z_{1}+z_{2}+z_{3}=0\}$, which is clearly contained in $\displaystyle H$. Since \displaystyle f(\mathbb{C})\bigcap\big{(}\bigcup_{j=1}^{3}H_{j}\bigcup H\big{)}=\emptyset, then \displaystyle f(\mathbb{C})\bigcap\big{(}\bigcup_{j=1}^{3}H_{j}\bigcup\tilde{H}\big{)}=\emptyset and $\displaystyle f=(e^{f_{1}},e^{f_{2}},e^{f_{3}}).$ Hence

[TABLE]

where $\displaystyle g_{2}=f_{2}-f_{1}$ and $\displaystyle g_{3}=f_{3}-f_{1}$. By lemma 3.1, $\displaystyle g:=\pi(f)$ is contained in one of diagonals $\displaystyle\Delta_{12,34},\Delta_{13,24},\Delta_{14,23}$ (see 2). Suppose $\displaystyle\pi(f)(\mathbb{C})\subset\Delta_{13,24},$ then

[TABLE]

Hence $\displaystyle f=(1,e^{g_{2}},-e^{g_{2}})$ avoids \displaystyle\big{(}\bigcup_{j=1}^{3}H_{j}\bigcup H\big{)} and $\displaystyle\pi(f)$ is not constant.