3

Let $L$ be tautopological bundle of $\mathbb{C}P^n$ and $L^{-1}$ be its duality. Because $L$ is a subbundle of $\underline{\mathbb{C}}^{n+1}$, $\underline{\mathbb{C}}=L\otimes L^{-1}$ is a subbundle of $\underline{\mathbb{C}}^{n+1}\otimes L^{-1}$. We get an exact sequence $$0\rightarrow\underline{\mathbb{C}}\rightarrow\underline{\mathbb{C}}^{n+1}\otimes L^{-1}\rightarrow Q\rightarrow0$$ $Q$ is the quotient bundle. My questions are

  • How to prove $Q=T\mathbb{C}P^n$

  • Can we use this conclusion to calculate the first chern class: $$c_1(T\mathbb{C}P^n)$$

Any advice is helpful. Thank you.

  • Let $L^\perp$ be the orthogonal bundle of $L$. We will have

    • $T\mathbb{C}P^n=L^{-1}\otimes L^\perp$

    • $\underline{\mathbb{C}}^{n+1}=L\oplus L^\perp$

So we will have $Q=L^{-1}\otimes L^\perp=T\mathbb{C}P^n$.

I am still curious about the question 2. From question 1, we have $$c_1(T\mathbb{C}P^{n})=c_1(\underline{\mathbb{C}}^{n+1}\otimes L^{-1})$$.

Then how do we compute $c_1(\underline{\mathbb{C}}^{n+1}\otimes L^{-1})$?

gaoxinge
  • 4,434
  • 1
    Both are pretty standard exercises. Could you please add something about context: what examples you've seen, what theorems you have available, and/or any ideas you have? Thank you. – Andrew D. Hwang Dec 16 '15 at 11:38
  • @AndrewD.Hwang Thank you for pointing out this. I will improve my questions. – gaoxinge Dec 17 '15 at 10:30

1 Answers1

2

Hint 1: Think about how a point in $\mathbb{C}P^n$ is a line in $\mathbb{C}^{n+1}$. What happens if you move the point a little bit (i.e. the line in $\mathbb{C}^{n+1}$.) Can you parameterize these movements?

Hint: How do you compute the chern class of a sum of vector bundles? How can you apply this to an exact sequence of vector bundles?

Thomas Rot
  • 10,023
  • I have improved my questions, Does my computation follow your answer? Or is there any mistakes in my attempt? Thank you. – gaoxinge Dec 17 '15 at 12:08
  • Have a look at Theorem 14.10 and page $58$ of Milnor and Stasheff. Basically the idea is that all complex lines in the neighborhood of a fixed line $V$ are in correspondence with graphs of linear maps of $V$ to $V^\perp$. You do not need to write down transition functions, but of course you can. – Thomas Rot Dec 17 '15 at 13:59
  • Thank you. It is nice. I am curious about the question 2. We have $\underline{\mathbb{C}}\oplus T\mathbb{C}P^n=\underline{\mathbb{C}}^{n+1}\otimes L^{-1}$. That is to say, $c(\underline{\mathbb{C}})\wedge c(T\mathbb{C}P^n)=c(\underline{\mathbb{C}}^{n+1}\otimes L^{-1})$, i.e. $c(T\mathbb{C}P^n)=c(\underline{\mathbb{C}}^{n+1}\otimes L^{-1})$. Then what can we do? – gaoxinge Dec 18 '15 at 01:07
  • the tensor product is distributive over the whitney sum for line bundles, so you get the chern class of n+1 copies of $L^{-1}$, which you can write as... – Thomas Rot Dec 18 '15 at 06:36