Timeline for Holomorphic line bundles on $\mathbb{P}^1$ from gluing data
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Mar 5, 2018 at 10:12 | history | suggested | CommunityBot | CC BY-SA 3.0 |
fixed a typo
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| Mar 5, 2018 at 6:23 | review | Suggested edits | |||
| S Mar 5, 2018 at 10:12 | |||||
| Mar 5, 2018 at 3:36 | vote | accept | David E Speyer | ||
| Mar 2, 2018 at 12:28 | answer | added | Jason Starr | timeline score: 6 | |
| Mar 2, 2018 at 10:54 | comment | added | Jason Starr | Your $g_{i,j}(z)$ are rational functions, i.e., elements in $\mathbb{C}(z)$. Define $U\subset\mathbb{C}\mathbb{P}^1$ to be the complement of $\{ 0\}$, and you can choose $V\subset \mathbb{C}$ to be the complement of the finite set $\Delta$ where either some $g_{i,j}$ is polar, or the determinant of $g_{i,j}$ is nonzero. Your glueing data works just fine for this covering, and it makes clear that the bundle is constructed by an explicit "elementary transform up/down" procedure along $\Delta\cup \{0,\infty\}$ coming from the "elementary divisors" form of $(g_{i,j})$ at each point. | |
| Mar 2, 2018 at 4:52 | comment | added | Sasha | That it is not very direct, but you can use $g$ to compute the cohomology of all twists of the vector bundle (via Cech complex). And knowing cohomology gives you the $d_i$. | |
| Mar 2, 2018 at 2:23 | history | asked | David E Speyer | CC BY-SA 3.0 |