Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going into detail with how it comes up in the paper, I am curious as to how this connection comes about, more or less how to consider formulating a proof about this. I must say I have a very tiny math background(struggling physics dude here) but I mean I can handle math lingo, but I need a gentle response. I am sorry if this is too trivial a question, and apologize in advance. .
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$\begingroup$ Are you asking about the result of Birkhoff-Grothendieck (and, earlier, Del Pezzo - Bertini if one properly interprets the classification of varieties of minimal degree) that every holomorphic vector bundle over the Riemann sphere is isomorphic to a direct sum of holomorphic line bundles? What precisely would you like to know about this result? $\endgroup$– Jason StarrCommented Jan 2, 2014 at 1:22
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$\begingroup$ I would love to know an elementary proof of the result, one that I can sort of understand $\endgroup$– user39066Commented Jan 2, 2014 at 2:04
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$\begingroup$ What is your background? This can be proved complex analytically, but it can also be proved algebraically. $\endgroup$– Jason StarrCommented Jan 2, 2014 at 2:22
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$\begingroup$ Well, to be honest complex analytically I would understand better, but then again an algebraic proof would work for the paper I am kind of writing since there is quite a bit of stuff with hopf algebra already in it. To really answer your question I am mostly self taught in both. Can you proof in one then sketch out the procedure in the other perhaps. I know I am asking for a lot here, . . . $\endgroup$– user39066Commented Jan 2, 2014 at 2:35
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$\begingroup$ An elementary proof can be found in Lemma 7 of Atiyah, M. F. Vector bundles over an elliptic curve. Proc. London Math. Soc. (3) 7 1957 414–452 (this is a variant of the answer by abx). $\endgroup$– Damian RösslerCommented Jan 2, 2014 at 16:01
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1 Answer
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The proof is straightforward if you accept some standard results on cohomology over $\mathbb{P}^1$. Let me try. We want to prove that any vector bundle $E$ on $\mathbb{P}^1$ is a direct sum of line bundles $\mathcal{O}(k)$.
The proof is by induction on the rank of $E$.
Replacing $E$ by $E(k)$ for some integer $k$ we may assume $H^0(E)\neq 0$, $H^0(E(-1))=0$. Let $s\neq 0$ in $H^0(E)$. Then $s$ is nonzero at every point (otherwise $s$ would give a nonzero section of $E(-1)$), so there is an extension
$$0\rightarrow \mathcal{O}\rightarrow E \rightarrow F\rightarrow 0\ .$$
By induction we have $F=\sum \mathcal{O}(k_i)$. Tensor product with $\mathcal{O}(-1)$ + cohomology exact sequence + $H^1(\mathbb{P}^1,\mathcal{O}(-1))=0$ gives
$H^0(\mathbb{P}^1,F(-1))=0$, hence $k_i\leq 0$ for each $i$.
Now you have to accept that such extensions are classified by the vector space $H^1(\underline{\mathrm{Hom}}(F,\mathcal{O}))=H^1(F^*)$. Since $H^1(\mathcal{O}(k))=0$ for $k\geq 0$, this is zero, hence the extension is trivial and you are done.
