[Complex Line Bundles I] Line Bundles on $\mathbb{P}^n$

We will study complex line bundle on $\mathbb{P}^n$. As applications, we will prove these statements:

I. Let $m>n$ be two non-negative integers, then any holomorphic map from $\mathbb{P}^m$ to $\mathbb{P}^n$ is constant.

II. Let $S$ be a submanifold of $\mathbb{P}^n$ of codimension 1, then $S$ is algebraic manifold. It is a special case of Chow's theorem, where he proved that any submanifold of $\mathbb{P}^n$ is actually algebraic.

1. Complex line bundles via transition functions and divisors associated with line bundles.

As we already discussed on our previous post about in vector bundles and line bundles, by a complex line bundle $E$, we mean a holomorphic line bundle over a complex manifold $M$. We then know that a complex line bundle represent a class in $H^1(M,\mathscr{O}^*)$ by their transition functions $\{f_{jk}\}$ where $f_{jk}$ satisfied the cocycle conditions.

i. $f_{ii}=1$ on $U_i$.
ii. $f_{ij}=f_{ik}f_{kj}$ on $U_i\cap U_j\cap U_k$
iii. $f_{ij}=f_{ji}^{-1}$ on $U_i\cap U_j$

where $\{U_j\}$ is an open covering of $M$. And $f_{jk}$ is in $\mathscr{O}^*(U_j\cap U_k)$. Conversely, if we have $\{f_{jk}\}$ in $\mathscr{O}^*(U_j\cap U_k)$ satisfying the cocycle condition, we then construct a holomorphic line bundle $E$ over $M$ as follows.

First, let $X=\coprod_j U_j\times \mathbb{C}$, a disjoint union of $U_j\times\mathbb{C}$. We then define the relation $\sim$ on $X$ as follows:
$$(p,v_j)\sim (p,v_k)\text{ if } v_k=f_{jk}(p)v_j \text{ (where } (p,v_j)\in U_j\times\mathbb{C}, (p,v_k)\in U_k\times \mathbb{C})$$
By the cocycle conditions, $\sim$ is an equivalent relation. We then define $E=X/\sim$. It can be seen that $E$ is a holomorphic line bundle over $X$.

Now, we turn to an important construction. Let $D=\sum_\nu m_\nu S_\nu$, where $m_\nu$ are integers and $S_\nu$ are submanifolds of $M$ of codimension 1. Then $D$ is called a divisor on $M$. By sufficient fine covering $\{U_j\}$ of $M$, suppose that the minimal equation of $S_\nu$ on $U_j$ is $f_{\nu j}$. We then define $f_{jk}:=(\frac{f_{\nu j}}{f_{\nu k}})^{m_\nu}$, then it can be seen that $\{f_{jk}\}$ defines a cocycle in $H^1(M,\mathscr{O}^*)$, and hence, determine a unique line bundle, which we denote by $[D]$.

2. Complex line bundles on $\mathbb{P}^n$.

We denote $(\zeta_0:...:\zeta_n)$ as homogeneous coordinate of $\mathbb{P}^n$. Let $H$ be a hyperplane of $\mathbb{P}^n$ defined by $\zeta_0=0$, and $E=[H]$ is a line bundle associated with a divisor $H$. We will give the description for $\Gamma(\mathbb{P}^n,\mathscr{O}(E))$, i.e. the holomorphic sections of $[E]$ over $\mathbb{P}^n$.

First, let $U_j$ be the affine open coverings of $\mathbb{P}^n$. Then the minimal equation for $H$ over each $U_j$ is given by $\zeta_0/\zeta_j$. This yields by our first section that the transition functions of $E$ is given by $e_{jk}=\zeta_k/\zeta_j$. And due to the construction of line bundles from transition functions, we have $(p,v_j)\equiv (p,v_k)$ iff $v_k=v_je_{jk}(p)=v_j\frac{\zeta_k}{\zeta_j}(p)$, where $p\in U_j\cap U_k$.

Now, if $f$ is any homogeneous linear polynomial in variables $\zeta_0,...,\zeta_1$. We define $f_j:=f/\zeta_j$, and $s_j: U_j\to E$ defined by $s_j(p)=(p,f_j(p))$. Then it can be seen that $s_j$ is a holomorphic section on $U_j$ and if $p\in U_j\cap U_k$, we have
$$s_j(p)=(p,f_j(p))=(p,\frac{f}{\zeta_j}(p))=(p,\frac{f}{\zeta_k}\frac{\zeta_k}(p){\zeta_j}(p))=(p,f_k(p))=s_k(p)$$
So, we can glue $\{s_j\}$ to get a holomorphic section $s$ on $\mathbb{P}^n$. So, a homogeneous linear polynomial $f$ defines a global holomorphic section on $\mathscr{O}(E)$ given by  $s(p)=(p,f(p))$, and it can be seen that global section on $\mathscr{O}(E)$ is generated by $\zeta_0,...,\zeta_n$.

Conversely, we will prove that any holomorphic section $s$ of $E$ over $\mathbb{P}^n$ actually comes from a homogeneous linear polynomial. Let $s_j$ be (restricted) holomorphic section of $s$ on $U_j$ given by $s_j(p)=(p,\psi_j(p))$. Then it can be seen that on $U_j\cap U_k$, we have $\psi_j=f_{jk}\psi_k=\psi_k\frac{\zeta_k}{\zeta_j}$. This yields $\frac{\zeta_k}{\zeta_0}\psi_k={\zeta_j}{\zeta_0}\psi_j$ on $U_j\cap U_k\cap U_0$.

Let us denote $\mathbb{C}^n:=\mathbb{P}^n\setminus H$, with coordinate $(z_1,...,z_n)=(\frac{\zeta_1}{\zeta_0},...,\frac{\zeta_n}{\zeta_0})$. Then on $\mathbb{C}^n\cap U_j$, there exists a holomorphic function $\varphi$ such that
$$\zeta_0\varphi(z_1,...,z_n)=\zeta_j\psi_j$$
Let $p$ be any point on $E$, i.e. $p=(0:x_1:...:x_n)$. We assume that $p\in U_j$. Then if we represent the power series locally at a point $(x_1,...,x_n)\in\mathbb{C}^n$. we get
$$\sum_{m\ge 0}\frac{\zeta_0}{\zeta_0^m}\varphi_m(x_1,...,x_n)=\zeta_j\psi_j(p)$$
where $\varphi_m$ denotes the homogeneous part of $\varphi$ of degree $m$. One can see the RHS is a holomorphic in a sufficient small neighborhood of $p$. So, by analytic continuation of the LHS, we have $m\le 1$. This yields $\zeta^0\varphi(\zeta_1/\zeta_0,...,\zeta_n/\zeta_0)=:\Psi$ is a homogeneous linear polynomial. And hence, $\Psi=\zeta_j\psi_j$ on $U_j$, i.e. $\psi_j=\Psi/\zeta_j$ on $U_j$. This yields $s(p)=(p,\Psi(p))$ on $\mathbb{P}^n$. Therefore, any global holomorphic section of $E$ is actually from a homogeneous polynomial of degree 1.

If we denote $E^h$ the holomorphic line bundle on $\mathbb{P}^n$ with transition function $f_{jk}=e_{jk}^h$. Then it can be seen by the proof above that for any $h\ge 0$, any section in $\Gamma(\mathbb{P}^n,\mathscr{O}(E^h))$ actually comes from a homogeneous polynomial of degree $h$. So, in particular, when $h=0$, we obtain the familiar result: any holomorphic function on $\mathbb{P}^n$ is constant, i.e. $\Gamma(\mathbb{P}^n,\mathscr{O})=\mathbb{C}$. In the case $h<0$, then we can take a look on the sum of homogeneous components above, and it yields $\varphi$ must be zero, i.e. $\Gamma(\mathbb{P}^n,\mathscr{O})=0$.

IMPORTANT FACT (will be proved in Part 2). We contemporary accept that $H^1(\mathbb{P}^n,\mathscr{O}^*)\cong \mathbb{Z}$, i.e. the group of line bundle on $\mathbb{P}^n$ is isomorphic to $\mathbb{Z}$, and and it is generated by $E$, where $E$ is the line bundle associated with $[H]$, where $H$ is a hyperplane of $\mathbb{P}^n$.

3. Applications. 

Using the fact that any morphism from a variety $X$ to $\mathbb{P}^n$ is given by $n$ sections $s_1,...,s_n$ of a line bundle $E$ over $X$ such that for all point on $X$, $s_i$'s do not vanish simultaneously, we will prove that any morphism from $\mathbb{P}^m$  to $\mathbb{P}^n$, where $m>n$ is constant.

Because we can embed $\mathbb{P}^n$ to $\mathbb{P}^{m-1}$, so it is sufficient for us to prove when $n=m-1$. Now, if $X=\mathbb{P}^m$, then any morphism from $\mathbb{P}^m$ to $\mathbb{P}^n$ is given by $n$ sections $s_1,...,s_{m-1}$ of a line bundle $E$ over $X$ and the sections do not vanish simultaneously on $\mathbb{P}^m$. If this morphism is not constant, so is $s_i$. We know from Section 2 that any line line bundle on $\mathbb{P}^m$ is of the form $E^h$ (Important fact), and any section on $E$ comes from a homogeneous polynomial of degree $h$. So, we will prove that these $(m-1)$ polynomials simultaneously vanish at a point. Consider the intersection product of $m-1$ times $A^1(\mathbb{P}^m)\circ...\circ A^1(\mathbb{P}^m)$. But then, the Chow's group $A^1(\mathbb{P}^n)$ is generated by a hyperplane. So it is sufficient for us to choose $(m-1)$
 hyperplane on $\mathbb{P}^n$, and prove that they have common points. Choose hyperplane $H_i$
 defined by $\zeta_j=0$, for $j=0,...,m-1$. Then $H_0\cap H_1\cap ...\cap H_{m-1}=(0:...:0:1)$. This is a contradiction. Hence, any morphism from $\mathbb{P}^m$ to $\mathbb{P}^n$, for $m>n$ is constant.

Furthermore, we obtain a special case of the Chow's theorem.

Proposition 3.1. Let $S$ be a submanifold of $\mathbb{P}^n$ of codimension 1, then $[S]=E^h$, where $E$ is defined as above, and $h$ is a positive integer. Furthermore, $S$ is algebraic.

Proof. Let us denote $f_j$ the minimal equation of $S$ on $U_j$. Then we can see $f_j$ is not constant. Also, the transition functions of $[S]$ is given by $\{f_j/f_k\}$. And due to the Important Fact, $[S]=E^h$, for some $h$, i.e. $[f_{jk}]=[e_{jk}^h]\in H^1(M,\mathscr{O}^*)$. So, there is a cochain $\{u_j\}\in C^0(M,\mathscr{O}^*)$, where $u_j\in \mathscr{O}^*(U_j)$, and $f_{jk}=u_j^{-1}e_{jk}^hu_k$ on $U_j\cap U_k$. That means, on $U_j\cap U_k$, we have $\frac{f_j}{f_k}=u_j^{-1}e_{jk}^hu_k$, i.e. $f_ju_j=e_{jk}^hf_ku_k$. So, $\{f_ju_j\}$ defines a global holomorphic section on $\mathscr{O}(E^h)$, and $f_ju_j$ are not constant. Now, because $\Gamma(\mathbb{P}^n,\mathscr{O})=\mathbb{C}$, and $\Gamma(\mathbb{P}^n,\mathscr{O}(E^h))=0$, for $h<0$, we have $h$ is positive.

Now, by our results from Section 2, $f_ju_j=\Psi/\zeta_j^h$, where $\Psi$ is a homogeneous polynomial of degree $h$. And hence, the zero of $f_j$ on $U_j$ is exactly the zero of $\Psi$ on $U_j$. Say another words, $S$ is the zeros of $\Psi$, and $S$ is algebraic. (Q.E.D)