Let $E \to X$ be a homomorphic vector bundle over a projective variety $X$. Does $\mathbb{P}(E)$ always have holomorphic sections? If not what is the obstruction? 



Does the projectivization of a vector bundle have sections?
I'll assume you're asking about sections of $PE$. The bundle $E$ has Chern classes $c\_i(E)\in H^{2i}(X)$ and thus a Chern polynomial $f\_E(t)=\sum\_{i=0}^nc\_{n-i}(E)t^i$, where $n=\dim(E)$. A section of $PE$ corresponds to a line bundle $L\leq E$ and thus a factorisation $f\_E(t)=f\_L(t)f\_{E/L}(t)$, and thus a root $-c\_1(L)\in H^2(X)$ of $f\_E(t)$. Thus, if $f\_E(t)$ has no roots in $H^2(X)$ then $PE$ does not even have a topological section. One can give similar arguments in the topological category using any complex oriented multiplicative cohomology theory such as $MU^\*(X)$ or $KU^\*(X)$. I believe that you can give similar arguments in the algebraic category using the Chow ring or algebraic $K$-theory, but I will not swear to the technical details of that.


