Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $t$?

share|improve this question
add comment

1 Answer

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.

share|improve this answer
    
could you suggest a modern (or expository) account of Weil uniformization? thanks –  Jacob Bell Jan 17 '13 at 16:50
    
Unfortunately, I am not aware of a reference. I would guess the proof essentially follows from the factorization property of the space of $G$-bundles with a rational section known as the Beilinson-Drinfeld Grassmannian. The factorization property of the latter follows from the Beauville-Laszlo theorem. There are many places where it's mentioned, e.g. Beilinson-Drinfeld's "Quantization of the Hitchin system", section 5.3.10 and Frenkel-Ben-Zvi "Vertex algebras and algebraic curves", section 20.3.5. –  Pavel Safronov Jan 17 '13 at 23:42
    
Thank ou for the answer. –  Fermion Jan 19 '13 at 10:13
    
@Muon: see the answer nd comments at mathoverflow.net/questions/112593/… for a modern discussion of this "uniformization". –  user30379 Jan 20 '13 at 5:19
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.