Let
$$
H^d=\mathbb{C}^{n+1}\backslash\{0\}\times\mathbb{C}/\sim
$$
where $(z_0,\cdots,z_n;\zeta)\sim (\lambda z_0,\cdots,\lambda z_n;\lambda^d \zeta)$.
It seems that $H^d$ is a real 2-dimensional vector bundle over $\mathbb{C}P^n$, and I try to find explicit trivializations of this bundle and compute the corresponding transition maps.
But now I just know the differential structure of complex projective space: Let
$$
U_i=\left\{[z] \in \mathbb C P^n \mid x=\left(z_0,\cdots,z_n\right) \in \mathbb C^{n+1}-\{0\}, x_i \neq 0\right\},\\
\varphi_i([z])=\left(z_0 / z_i, \cdots, z_{i-1} / z_i, z_{i+1} / z_i, \cdots, z_n / z_i\right),[z]=\left(z_0, \ldots, z_n\right) \in U_i.
$$
These atlas determine the differential structure of $\mathbb{C}P^n$.
I think it has something to do with the problem I'm trying to solve, but I don't know how to deal with the last coordinate of $H^d$.
In short, my problem is still to find explicit trivializations of this bundle and compute the corresponding transition maps. How can I do it?
Thanks in advance.
Best Answer
Question: "In short, my problem is still to find explicit trivializations of this bundle and compute the corresponding transition maps. How can I do it? Thanks in advance."
Answer: If you are familiar with schemes and invertible sheaves you find an answer in the link below. When you define the linebundle $L(d)$ using the character $\rho_d(z):=z^d$ it follows your rank one vector bundle is (schematically) the scheme $\mathbb{V}(\mathcal{O}(-d))$ where we define
$$L(d):=\mathbb{V}(\mathcal{O}(-d)):=Spec(Sym^*_{\mathcal{O}_{\mathbb{P}^n}}(\mathcal{O}(-d)).$$
It follows $\pi:L(d) \rightarrow \mathbb{P}^n$ is a rank one vector bundle with global sections $H^0(\mathbb{P}^n, \mathcal{O}(d))$.
If you are not familiar with this you get the following transition maps: Let $D(x_i) \subseteq \mathbb{P}^n:=X$ be the standard open set. It follows
$$\mathcal{O}(d)(D(x_i)) \cong k[x_0/x_i,..,x_n/x_i]x_i^d$$
and on the intersection $D(x_ix_j)$ you get the relation
$$ x_i^d=(\frac{x_i}{x_j})^dx_j^d.$$
The transition function for the vector bundle $L(d)$ on the open set $D(x_ix_j)$ is
$$z_i=(\frac{x_j}{x_i})^d z_j.$$
Here is a more detailed explanation:
A description of line bundles on projective spaces, $\mathcal{O}_{\mathbb{P}^n}(m)$ defined using a character of $\mathbb{C}^*$.
Comment: "After reading your answer, I went to understand some line bundles of complex projective space and figured out the general situation. The only thing that puzzles me is what the xi and zi stand for in your last formula? – JingHao Yang"
Response: @JingHaoYang: Projective space $\mathbb{P}^n$ has coordinates $x_0,..,x_n$. Over the open set $U_i:=D(x_i)$, the inverse image $π^{-1}(U_i):=V_i$ has $z_i$ as "local coordinate". The relation $z_i=(x_j/x_i)^dz_j$ is valid on the open set $V_i∩V_j$. There is a nontrivial result from the 1950s saying that the classification of line bundles using "holomorphic functions" equals the classification one gets using "rational functions". The holomorphic vector bundle $L(d)$ is algebraic and may be defined using rational functions.
In algebraic geometry (see Hartshorne, the exercises in CH II) there is an "equivalence of categories" between the category of "finite rank algebraic vector bundles" and the "category of finite rank locally trivial sheaves", and I believe there is a similar "equivalence of categories" for "finite rank holomorphic vector bundles" and "finite rank locally free sheaves". Any complex projective manifold is algebraic, and any holomorphic finite rank vector bundle on a complex manifold is algebraic.
Note moreover: The invertible sheaf $\mathcal{O}(d)$ is the "sheaf of sections" of the morphism $\pi: L(d) \rightarrow \mathbb{P}^n$ in the sense of HH. Ex.II.5.18 - this exercise is worth doing.