Question

My Friends,

Shocking as this sounds, I believe to have found a typo in Griffiths & Harris.

In the chapter on surfaces, section Rational Surfaces 1, I am trying to read the result that a holomorphic vector bundle over $\mathbb{P}^1_{\mathbb{C}}$ is a sum of invertible bundles.

What is the exact sequence that shows up at the start of his argument? Mine only has 2 terms and involves the fibres of E and H, which doesn't really make sense to me.

Any help would be appreciated.

Answer 1

Let $H$ be the divisor corresponding to the point $x\in \mathbb{P}^1$. Tensoring the exact sequence $$ 0\to O_{\mathbb{P}^1}(-H)\to O_{\mathbb{P}^1}\to O_x\to 0. $$ with $E\otimes H^k$, gives $$ 0\to O_{\mathbb{P}^1}(E\otimes H^{k-1})\to O_{\mathbb{P}^1}(E\otimes H^{k})\to E_x\otimes H_x^{k}\to 0. $$Here GH writes $E_x\otimes H_x^{k}$ for $E\otimes H^k\otimes O_x$, or what is the same, the fiber of $E\otimes O(H^{k})$ over $x$.

Answer 2

I don't have my copy of Griffiths and Harris in front of me, but regarding Grothendieck's theorem, there's a nice elementary argument by Michiel Hazewinkel & Clyde F. Martin. Here is a link to ScienceDirect. It's only five pages, and mostly consists of linear algebra (and really, is closer to 3 pages, ignoring abstract, intro, and white space at the end).