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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?

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closed as too localized by David Roberts, Deane Yang, Torsten Ekedahl, Ryan Budney, Dan Petersen Sep 19 '11 at 8:01

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I fixed the latex. What do you mean by 'the transition matrix' for example 2? The transition functions depend on the choice of open cover. They also depend on a choice of local trivialisation. There is in general no abstract technique. 'Some description' is not a very helpful starting point. –  David Roberts Sep 17 '11 at 23:57
    
@David: Thanks for fixing Latex. For the open cover we can choose for example the usual cover by the sets $U_i=\{z_i\not= 0\}$. You are right, so my question should be this: How can we write a local trivial for the vector bundle in Question 2? The same for Example 1, assuming that we start from zero, not knowing that $J$ is isomorphic to $[-H]$. –  september Sep 18 '11 at 0:05
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I view this question as a good exercise for someone learning the subject. Many of us had to sort this out when we were students, and it was a useful experience in sorting out the definitions. The point is that with your definitions, you can write down a basis of sections on appropriately chosen open sets and then you can figure out how two different bases are related on the intersection of two such open sets. –  Deane Yang Sep 18 '11 at 1:06
    
@Deane: Thank for your answer. I tried to find some natural local triviality for the vector bundles in the examples, but was not successful. After posting this question and seeing your answer, I thought a little more and found the local trivialities. –  september Sep 19 '11 at 0:26

2 Answers 2

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.

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@Sasha. Thank you for your nice answer. Is it easy to compute the transition matrix from using your map? –  september Sep 19 '11 at 0:43

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.

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