Timeline for When Atiyah class and Chern class coincide?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Nov 10, 2022 at 8:23 | vote | accept | Tom | ||
| S Nov 10, 2022 at 8:23 | vote | accept | Tom | ||
| S Nov 10, 2022 at 8:23 | |||||
| S Nov 10, 2022 at 8:23 | vote | accept | Tom | ||
| S Nov 10, 2022 at 8:23 | |||||
| S Nov 10, 2022 at 8:23 | vote | accept | Tom | ||
| S Nov 10, 2022 at 8:23 | |||||
| Nov 4, 2022 at 8:29 | vote | accept | Tom | ||
| S Nov 10, 2022 at 8:23 | |||||
| Oct 31, 2022 at 18:24 | answer | added | David E Speyer | timeline score: 6 | |
| Oct 31, 2022 at 16:21 | comment | added | LSpice | I agree it's confusing, but, if I put $0 \to \mathbb C \hookrightarrow \mathcal O \xrightarrow d \Omega^1$ or $0 \to \mathbb C \hookrightarrow \mathcal O \xrightarrow d \Omega^1 \to 0$ in a diagram, and assert only its commutativity, then the latter is redundant, but surely makes no claim about exactness? | |
| Oct 31, 2022 at 16:15 | history | edited | Martin Sleziak |
the tag (chern-classes) seem suitable here
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| Oct 31, 2022 at 16:01 | answer | added | Misha Verbitsky | timeline score: 5 | |
| Oct 31, 2022 at 15:32 | comment | added | Tom | @MishaVerbitsky, yes, I agree it is not exact, and Atiyah does not assume dimension conditions of the manifold $X$, but induced by the inclusion $Z^{1,0}_d\subset \Omega^1$, the image of the map $H^1(X,\mathcal O^*)\to H^1(X,\Omega^1)$ is represented by a $\bar\partial$-closed (1,1) form is not affected, isn't it? | |
| Oct 31, 2022 at 15:18 | comment | added | Misha Verbitsky | It is not exact, Atiyah is wrong (or maybe at the time $\to 0$ did not mean exactness). Maybe he was thinking about 1-dimensional manifolds. | |
| Oct 29, 2022 at 9:32 | comment | added | Tom | @S.D. but he put $\to 0$ after $\Omega^1$, and wrote $0\to \mathbb C\hookrightarrow \mathcal O\stackrel{d}\to \Omega^1\to 0$, doesn't it mean the exactness? | |
| Oct 29, 2022 at 9:25 | comment | added | S.D. | I wanted to just point out that on page 196, Atiyah just said that it is a commutative diagram of sheaves. Does not say anything about exactness! But anyway, your main question is different. | |
| Oct 29, 2022 at 9:16 | comment | added | Tom | @S.D. It is taken from Atiyah's paper, p.196. Actually, I have the same suspect, I think $\Omega^1$ should be replaced by $Z^{1,0}:=A^{1,0}\cap\ker d$, maybe in the Kähler case (as in Atiyah's paper), $\Omega^1=Z^{1,0}$? so the author takes $\Omega^1$ for $Z^{1,0}$? | |
| Oct 29, 2022 at 9:06 | comment | added | S.D. | Why is the sequence involving $\Omega^1$ exact? | |
| Oct 28, 2022 at 15:42 | history | asked | Tom | CC BY-SA 4.0 |