# Does the projectivization of a vector bundle have sections?

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?

• Are you asking about section of $\mathbb{P}(E)$ (as in the title) or about section of $E$ (as in the question)? Nov 5, 2019 at 11:49

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.