Timeline for Holomorphic vector bundles and Swan's theorem
Current License: CC BY-SA 2.5
11 events
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| Dec 15, 2010 at 2:32 | comment | added | Vamsi | Sorry to return to the question after such a long time, but, why is the support of the cokernel finite-dim'l? I mean, let us take line bundles over Stein Riemann surfaces, they are trivial, hence the theorem trivially holds. Now, if I take a complex surface, then why is it that one can find a (transverse) section whose zero set is a finite collection of connected Riemann surfaces? Maybe I have misunderstood the argument. | |
| Aug 21, 2010 at 19:23 | comment | added | BCnrd | Vamsi, for any ringed space $X$, loc. free sheaf $V$ of finite rank, $O_X$-mod $F$, and $i \ge 0$, ${\rm{Ext}}^i(V,F) = {\rm{H}}^i(X, F \otimes V^{\ast})$. This vanishes if $F$ coherent, $X$ Stein, and $i > 0$. To prove finite generation, assume $X$ finite-dim'l (e.g., irreducible). The irreducible components $X_i$ are loc. finite in $X$, so can find $x_i \in X_i$ not in any other $X_j$. By Steinness, if $V$ has rank $n$ can find global sections $s_1,\dots,s_n$ generating $V$ near each $x_i$. Restrict $V$ to support of cokernel of $O_X^n \rightarrow V$, induct on dimension, use Nakayama. QED | |
| Aug 21, 2010 at 19:03 | comment | added | Donu Arapura | Vamsi, yes I should have been more clear. Let $K=ker(f)$. Then $Ext^1(\mathcal{E}, K)= H^1(X,\mathcal{E}^*\otimes K)=0$. The first equality is an algebraic formality, the second is by Cartan B (or is it Gauss ?) | |
| Aug 21, 2010 at 18:53 | comment | added | Vamsi | @ Daniel, I don't know and actually part of my question is that. @ Donu, I am sorry, my background is more in the analytic side, so does $Ext ^1 (\mathcal{E}, ker f) = 0$ follow obviously from Cartan? (I mean is their an injective resolution of Coherent sheaves?). I apologise for the silly question. | |
| Aug 21, 2010 at 18:25 | comment | added | Vamsi | By Hormander, I meant you can use a version of it to extend sections and thus essentially prove a special case of Cartan anyway :). | |
| Aug 21, 2010 at 18:25 | comment | added | Daniel Pomerleano | Sorry lost my train of thought... Is there a holomorphic version too? | |
| Aug 21, 2010 at 18:24 | comment | added | Daniel Pomerleano | I was thinking along the same lines(I thought it was Grauert:)) and my intuition is that it will work because in the algebraic category for an affine scheme, this is the same thing as the statement that a finite projective module is a direct summand of a free module. I'd have to go back and read those papers though. But I also wanted to ask as far as I understand Swan's theorem has to do with C^infinity manifolds. Then the point is basically the same as the affine scheme case. | |
| Aug 21, 2010 at 18:08 | comment | added | Donu Arapura | I think it's Cartan, but yeah I was thinking along those lines. The space of sections would be infinite dimensional, so you need to be careful, but it seems conceivable that a finite set of sections generates. If you can do that, then you be done. (You would get a surjection $f:\mathcal{O}_X^N\to \mathcal{E}$ onto your locally free sheaf, which would split because $Ext^1(\mathcal{E},\ker f)=0$.) | |
| Aug 21, 2010 at 17:57 | comment | added | Vamsi | Thanks. Is it true for Stein manifolds? I mean, by Hormander's theorem, the global sections generate the vector bundle. So, if the global sections are finite dimensional it ought to be true, ought not it? | |
| Aug 21, 2010 at 17:41 | history | edited | Donu Arapura | CC BY-SA 2.5 |
added 124 characters in body
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| Aug 21, 2010 at 17:29 | history | answered | Donu Arapura | CC BY-SA 2.5 |