Given a complex vector bundle $V\rightarrow M$, we can form a fibre bundle $\mathbb{P} V\rightarrow M$, where the fiber over each point is the corresponding projective space. In particular consider the space $ \mathbb{P}(T \mathbb{P}^2)$, the projectivized tangent space of $\mathbb{P}^2$. Can this complex manifold be embedded (holomorphically) in some projective space $\mathbb{P}^N$?

More generally if we can embed $M$ in a projective space, does it imply we can embed $\mathbb{P}(V)$ in a projective space?

Everything is over complex numbers and the projective spaces are all complex projective spaces.