Timeline for Tensor product of a line bundle with a large multiple of another positive line bundle also positive?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2010 at 10:11 | answer | added | Dmitri Panov | timeline score: 2 | |
| Sep 6, 2010 at 15:55 | comment | added | Dmitri Panov | Laurent, thank you very much! That was a silly mistake of mine | |
| Sep 6, 2010 at 15:12 | comment | added | Laurent Moret-Bailly | @Dmitri: It is not true that line bundles on affine varieties are trivial: if $X$ is a compact Riemann surface and $P$ is a point on $X$, then $\mathrm{Pic}(X\setminus\{P\})$ is the quotient of $\mathrm{Pic}(X)$ by the subgroup generated by $\mathcal{O}_X(P)$. | |
| Sep 6, 2010 at 13:46 | comment | added | Henri | Just take any smooth strictly psh function $\varphi$ on your Stein space, and then $i \partial \bar \partial \varphi$ gives you the metric. For $\mathbb C^2$, you may take $\varphi(z_1,z_2)=\log(1+|z_1|^2+|z_2|^2)$, and you're done. An equivalent way would consist in considering the restriction of the Fubini-Study form of $\mathbb P^2 \mathbb C$ to $\mathbb C^2$. – Henri 0 secs ago | |
| Sep 6, 2010 at 12:34 | comment | added | pinaki | @Dmitri: Could you please explain a bit? E.g. what would be a Kahler metric of positive curvature on the trivial bundle over $\mathbb{C}^2$? | |
| Sep 6, 2010 at 10:17 | comment | added | pinaki | Well - I stated one definition in the second paragraph :) In any case, it is equivalent to the following definition of Griffiths-Harris: a line bundle $\mathcal{L}$ on $X$ is positive iff there exists a hermitian metric on $\mathcal{L}$ with curvature form $\Theta$ such that $\frac{i}{2\pi}\Theta$ is a positive (1,1) form. | |
| Sep 6, 2010 at 9:30 | comment | added | Henri | I guess he means a bundle equipped with some hermitian metric with positive curvature. | |
| Sep 6, 2010 at 9:25 | comment | added | Dmitri Panov | What bundles do you call positive? For example, what is a positive bundle on C^1? | |
| Sep 6, 2010 at 8:59 | history | edited | pinaki |
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| Sep 5, 2010 at 21:31 | history | asked | pinaki | CC BY-SA 2.5 |