Timeline for Sheaf of Kähler differentials for complex manifold
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 17, 2022 at 21:25 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Mar 16, 2022 at 17:43 | answer | added | Dmitri Pavlov | timeline score: 2 | |
| Mar 16, 2022 at 15:30 | comment | added | abx | For $X$ a complex manifold, you'll find the definition of $\Omega^1_X$ in any introductory book, e.g. Wells (Differential Analysis on Complex Manifolds). Why do you need this algebraic definition of Kähler differentials? | |
| Mar 16, 2022 at 15:09 | comment | added | singularity | yes i think it coincides for algebraic varieties but i need kahler differential for complex manifold and i don't find anything | |
| Mar 16, 2022 at 15:06 | comment | added | Francesco Polizzi | But they are the same for smooth algebraic varieties, right? | |
| Mar 16, 2022 at 15:02 | comment | added | singularity | thanks abx, i find in a lot of papers the notion of kahler differential and they link it with the cotangent bundle, maybe they don't use the same definition ? | |
| Mar 16, 2022 at 14:57 | comment | added | abx | No, $\Omega^1_{X\smallsetminus Sing}$ is not (locally) free of rank $n$. See the discussion in this MO question. For instance, $d(e^z)$ is not proportional to $dz$. | |
| Mar 16, 2022 at 14:56 | comment | added | Henri | Yes it is. You can see this directly by choosing a system of local holomorphic coordinates $(z_i)$ on a neighborhood $U$ of a smooth point $x\in X$. Then, you can see that $\Omega_X^1|_U$ is generated by the $dz_i$'s, with no relations | |
| Mar 16, 2022 at 14:41 | history | asked | singularity | CC BY-SA 4.0 |