# Chern class of tautological line bundle over the projectivization of a vector bundle

Let $$\mathbb{C}^k\hookrightarrow E\to B$$ be a complex vector bundle. Let $$\mathbb{CP}^{k-1}\hookrightarrow\mathbb{P}(E)\to B$$ be its projectivization. We can consider the tautological line bundle $$L$$ over $$\mathbb{P}(E)$$ which is the line bundle $$L= \{([x],V) \in \mathbb{P}(E)\times E | \ V \in [x] \} \to \mathbb{P}(E)$$ $$([x],V)\mapsto [x].$$ I would like to compute the first Chern class of this line bundle.

In the case when $$E = B\times \mathbb{C}^k$$ is the trivial vector bundle, then $$\mathbb{P}(E) = B\times \mathbb{CP}^{k-1}$$ and the tautological line bundle is $$L = B\times \mathcal{O}(-1)$$ where $$\mathcal{O}(-1)\to \mathbb{CP}^{k-1}$$ is the tautological line bundle. Therefore in this case we get that the first Chern class is given by $$- P.D.([B\times \mathbb{CP}^{k-2}])$$. But what can we say in general?

Motivation: study the normal bundle of $$\mathbb{P}(E)$$ inside $$L$$ in order to understand the blow-up along a submanifold.

• See this MO question. Apr 10, 2019 at 14:20
• I would be really surprised if there were an easy description of the firt chern class for general $E \to B$. I'm happy to be wrong though. Apr 10, 2019 at 17:09

I suspect in your post that $$n = k-1$$. In any case you're off by a sign. After all, it is the first Chern class of $$\mathcal{O}(1)$$ on $$\mathbb{P}^n$$ which is (the Poincare dual of) a hyperplane!
The general case is similar. If you think of the first Chern class as being Poincare dual to a generic non-zero section, then note that you can think of sections of $$\mathcal{O}(1)$$ on $$\mathbb{P}(V)$$ as cutting out a a family of hyperplanes over your base.
$$H^*(\mathbb{P}(V)) = H^*(B)[\zeta]/(\zeta^n + c_1\zeta^{n-1} + \ldots + c_n)$$
Here $$\zeta = c_1(\mathcal{O}(1))$$.