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
$$\pi^*(c(TM))c(\pi^*V \otimes \gamma^*).$$
Best Answer
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 $c(\pi ^*V\otimes \gamma^* )$.