Transition matrix for holomorphic vector bundles [closed]

Question

If we are given the local trivialities of a holomorphic vector bundle then by definition we can write down the transition matrix of that vector bundle.

In some very natural situations, we are not given local trivialities of that vector bundle but are given some description for it. The question is how we can write down the transition matrices from what are given. I will illustrate the question by two examples.

Example 1: This is a classical example. Let $J$ be the universal line bundle over $\mathbb{CP}^n$, which is given by $J=\{(z,\theta )\in \mathbb{CP}^n\times \mathbb{C}^{n+1}:~\theta \mbox{ is in the line generated by }z\}$. In this case, we can choose a meromorphic section $s$ for $J$ by $([z_0:z_1:\ldots :z_{n}],(1,z_1/z_0,\ldots ,z_n/z_0))$. The divisor of $s$ is $-z_0$, and hence we have $J=[-H]$.

Example and Question 2: We consider a similar construction. Let $W=\mathbb{CP}^n\times \mathbb{CP}^n-\Delta $ where $\Delta \subset \mathbb{CP}^n\times \mathbb{CP}^n$ is the diagonal. Let $V=(z_1,z_2,\theta )\in W\times \mathbb{C}^{n+1}$ so that $\theta$ is in the plane generated by $z_1$ and $z_2$. Then $V$ is a holomorphic vector bundle of rank $2$ on $W$. Then what is the transition matrix for this vector bundle?

I tried to think about this question off and on but still get stuck. Any help or hint is very helpful.

Addition: My comment after David's comment was not totally correct, so I add this into my question. So in my Questions (either Example 1 or 2) above, you are free to choose any open cover for the base space, and my question is can you write down a trivial for the vector bundle?

Answer 1

Consider the map $f:W \to Gr(2,n+1)$ taking a point $(x,y) \in W$ to the plane generated by $x$ and $y$y. Then the bundle under the question is the pullback of the tautological bundle on the Grassmannian. So, the transition matrix can be written as the pullback of the transition matrix for the tautological bundle on the Grassmannian.

Answer 2

I found the natural local trivialities for the vector bundles in the questions:

For Question 1: We have the following trivialities for the line bundle: Let $U_i=\{z_i:~z_i\not= 0\}$. Then we have an isomorphism $\pi : J_{U_0}\rightarrow U_0\times \mathbb{C}$ by: $\pi (z,\lambda )=(z,\tau )$ where $\lambda =(\lambda _0,\lambda _1,\ldots ,\lambda _n)=\tau (1,z_1/z_0,\ldots ,z_n/z_0)$. Observe that $\tau =\lambda _0$. Similarly we have the other isomorphisms $\pi : J|U_i\rightarrow U_i\times \mathbb{C}$. Note that the transition function for this is $g_{0,1}=z_0/z_1$ which is invertible on $U_0\cap U_1$, and similarly for the other $g_{i,j}$ which shows that $J$ is a holomorphic line bundle as we claimed from beginning.

Now the transition function for the line bundle $H=[z_0=0]$ are $h_{0,1}=1/z_0$ and so on. This shows that $J$ and $H$ are inverse to each other.

For Question 2, we cover $\mathbb{P}^n\times \mathbb{P}^n-\Delta$ by $U_i\times U_j-\Delta$, and then can do like the way in Question 1.