Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V \rightarrow M$ be the projectivization of $V$.

$\textbf{Question}:$ Is there a formula for $c(T\mathbb{P}V)$, the total Chern class of the Tangent space of $\mathbb{P}V$?

My naive guess would be that it should be $\pi^*(c(TM))(1+c_1(\gamma^*))^{k+1}$, where $\gamma \rightarrow \mathbb{P}V $ is the tautological line bundle over $\mathbb{P}V$. I think my guess is correct if $M$ was just a point, or more generally if $V$ was a trivial bundle. But I do not know if this is correct in general.

The specific case for which I need an answer is when $M:= \mathbb{P}^1 \times \mathbb{P}^1$ and $V:= \mathcal{O}(d_1) \oplus \mathcal{O}(d_2)$.

$\textbf{Added Later}:$ It has been pointed out my guess is wrong in general. The correct answer is $$\pi^*(c(TM))c(\pi^*V \otimes \gamma^*).$$


2 Answers 2


No, your formula is not correct. You have to take into account the Chern classes of $V$. The relative tangent bundle $T_{\mathbb{P}V/M}$ is given by the so-called Euler exact sequence $$0\rightarrow \mathscr{O}_{\mathbb{P}V}\rightarrow \pi ^*V\otimes \gamma^* \rightarrow T_{\mathbb{P}V/M}\rightarrow 0\ ,$$ while $$0\rightarrow T_{\mathbb{P}V/M}\rightarrow T_{\mathbb{P}V}\rightarrow \pi ^*T_M\rightarrow 0\ .$$Putting things together we find $c(T_{\mathbb{P}V})=c(\pi ^*V\otimes \gamma^* )\,\pi ^*c(T_M)$.

Then use the standard formula for $c(\pi ^*V\otimes \gamma^* )$.


For any smooth fiber bundle

$$ F\hookrightarrow P \stackrel{\pi}{\to} M $$

we have a short exact sequence of vector bundles over $P$

$$ 0\to VTP\to TP \to \pi^* TM\to 0, $$

where $VTP$ denotes the vertical tangent bundle defined as the kernel of the differential of $\pi$. If the bundle is holomorphic then the above is a short exact sequence of complex vector bundles and we deduce

$$ c(TP)= c(VTP)\cdot \pi^* c(TM). $$

The classical Euler exact sequence argument shows that when $P=\mathbb{P}(V)$ that $\newcommand{\bC}{\mathbb{C}}$

$$ \gamma^*\otimes \pi^*V \cong \underline{\bC}\oplus VTP, $$

where $\underline{\bC}$ denotes the trivial line bundle. Hence

$$ c(TP)= c(\gamma^*\otimes \pi^*V)\cdot \pi^* c(TP). $$

In Section I.3 of Fulton-Lang Riemann-Roch algebra you can find an explicit formula for $c_k(L\otimes E)$, $L$ line bundle and $E$ vector bundle of rank $m$. More precisely

$$ c_k(L\otimes E)=\sum_{j=1}^k \binom{m-j}{k-j} c_j(E)c_1(L)^{k-j}. $$ Note. The original answer had an error that I have now corrected. (Hat tip to abx).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.