Timeline for Relation between the cohomology group of a curve and the cohomology group of its jacobian
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2021 at 9:47 | vote | accept | Roxana | ||
| Nov 7, 2021 at 8:59 | answer | added | David E Speyer | timeline score: 15 | |
| Nov 6, 2021 at 14:55 | history | edited | Roxana | CC BY-SA 4.0 |
added 55 characters in body
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| Nov 5, 2021 at 20:31 | comment | added | Roxana | Dear @abx thank you for your comment. | |
| Nov 5, 2021 at 20:20 | comment | added | Roxana | Dear @DavidESpeyer In the paper I am reading they define $J_C$ as an abelian variety isomorphic to $A_0(C)$ the group of $0$-cycles of degree zero on $C$ modulo rational equivalence, and the cohomology as something like de Rham cohomology. | |
| Nov 5, 2021 at 16:57 | comment | added | David E Speyer | And welcome to MO! | |
| Nov 5, 2021 at 16:57 | comment | added | David E Speyer | I was going to leave a much friendlier version of this comment and then a student came in. So, trying now: What background are you coming from here? Are you defining $J_C$ complex analytically as something like $H^0(C, \Omega^1)^{\vee}/H_1(C, \mathbb{Z})$, or are you defining it as something like the group of degree $0$ line bundles? Are you defining cohomology as something topological, or as something like de Rham cohomology? | |
| Nov 5, 2021 at 16:33 | review | Close votes | |||
| Nov 10, 2021 at 3:04 | |||||
| Nov 5, 2021 at 16:15 | comment | added | abx | This is indeed very basic. $J_C$ is by definition $H^0(C,K_C)^*/H_1(C,\mathbb{Z})$, so $H_1(J_C,\mathbb{Z})$ is canonically isomorphic to $H_1(C,\mathbb{Z})$, hence $H^1(J_C,\mathbb{C})$ to $H^1(C,\mathbb{C})$. | |
| Nov 5, 2021 at 16:03 | history | asked | Roxana | CC BY-SA 4.0 |