# Chern class of a vector bundle and the associated projective space bundle

I have a very basic question regarding Chern classes. Let $$X$$ be a smooth projective variety and $$\mathcal{E}$$ a vector bundle on it. Let $$\pi:\mathbb{P}(\mathcal{E})\to X$$ denote the projective space bundle over $$X$$ associated to $$\mathcal{E}$$.

How are the Chern classes $$c_i$$ of the vector bundle $$\mathcal{E}$$ on $$X$$ related to the Chern classes $$c_i'$$ of the projective variety $$\mathbb{P}(\mathcal{E})$$, i.e. the Chern classes of the tangent bundle of $$\mathbb{P}(\mathcal{E})$$? My naive hope would be that we simply have $$c_i'=\pi^* c_i$$. Is that true? Is there a good reference?

• The answer is correct, but one can say more. There is a formula for the tangent bundle of a projective bundle that relates it to the pullback of the tangent bundle from below AND the relative tangent bundle. So there is a formula, but the formula makes it clear that what you are asking will almost never hold.
– meh
Apr 13, 2019 at 20:10
• Hi @aginensky ! That sounds interesting, can you maybe state this formula in a new answer or point me to a reference?
– Hans
Apr 15, 2019 at 7:55

The nice relation that you hope for is not true unfortunately. Here is a counter-example: consider the trivial rank 2 vector bundle $$E= \mathcal{O} \oplus \mathcal{O}$$ over $$\mathbb{CP}^{1}$$. Then $$\mathbb{P}(E) \cong \mathbb{CP}^{1} \times \mathbb{CP}^{1}.$$
Now, note that $$c_{2}(E)=0$$ since $$E$$ is a trivial bundle, but $$\int_{\mathbb{P}(E)} c_{2}(T\mathbb{P}(E)) = 4$$, since this is equal to the topological Euler characteristic of $$\mathbb{P}(E)$$ (it is a standard result that the integral of the top Chern class of a complex manifold is equal to the topological Euler characteristic).
In general, I don't think that there will be a nice relation (although I could be wrong) since when $$E$$ has different Chern numbers, the topological type of $$\mathbb{P}(E)$$ will be different in general, so we are not even talking about Chern classes on the same complex manifold. However, you can actually compute some topological invariants of $$\mathbb{P}(E)$$ from the chern classes of $$E$$: see Proposition 15 of Okonek and Van de Ven's "Cubic forms and complex 3-folds".