Question

Let $X$ be a complex manifold or variety and $L$ a line bundle on it. Given a rational section $s$ of $L$, we get a divisor $D=Div(s)$ and may recover $L$ as $\mathcal{O}(D)$. What about vector bundles? I think it makes sense to speak about rational section $t$ of a vector bundle $E$, as a section is locally a collection of functions. My question is, is this $t$ useful? For example, can we say anything about Chern classes of $E$ from $t$?

Answer 1

One can recover a bundle on a Riemann surface completely given a rational frame. This goes under the name of Weil uniformization of the moduli of bundles. More precisely, there is a map from the moduli of $G$-bundles on $X$ with a rational frame to $$\prod_{x\in X}' G(\mathbf{C}((t_x)))/G(\mathbf{C}[[t_x]])$$ by sending a section to its Taylor expansion around the zeros and poles. This map is an isomorphism and the moduli space of bundles is obtained by forgetting the frame, i.e. $$\mathrm{Bun}_G(X)\cong G(\mathbf{C}(X))\backslash\prod_{x\in X}' G(\mathbf{C}((t_x)))/G(\mathbf{C}[[t_x]]),$$ where $G(\mathbf{C}(X))$ is the group of rational maps from $X$ to $G$.

I don't think you can get much information from a single section: let's say $E = L_1\oplus L_2$ is a sum of two line bundles. Suppose the section $t$ of $E$ comes from a section of $L_1$, which allows you to compute $c_1(L_1)$. Then neither $c_1(E)=c_1(L_1)+c_2(L_2)$ nor $c_2(E)=c_1(L_1)c_1(L_2)$ are determined by it.