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. 



Can one embedd the projectivezed tangent space of CP^2 in a projective space? 
Yes to both questions. In particular, $\mathbb P(T\mathbb P^2)$ is isomorphic to a hyperplane section of the Segre embedding $\mathbb P^2\times\mathbb P^2\hookrightarrow \mathbb P^8$. Indeed, $\mathbb P(T\mathbb P^2)$ is isomorphic to the variety of complete flags in the vector space $\mathbb C^3$. Suppose now that in the embedding of $\mathbb P^2\times\mathbb P^2$ in $\mathbb P^8$, the first copy of $\mathbb P^2$ is projectivization of $\mathbb C^3$ and the second is projectivization of $(\mathbb C^3)^\*$. Then the linear functional on $\mathbb C^3\otimes (\mathbb C^3)^\*$ that defines the hyperplane section in question is the "evaluation" bilinear form on $\mathbb C^3\times(\mathbb C^3)^\*$.


