Timeline for Is every $\omega\in H^0(U,\Omega_U^n)$ representable in $H^n(U,\mathbb{C})$ by an element from $H^0(X,\Omega_X^n(\log D))$?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2020 at 20:00 | comment | added | user2520938 | Do you know how to describe the two components of the form you consider in your answer under this decomposition of $H^1(U,\mathbb{C})$? | |
| Nov 15, 2020 at 19:21 | comment | added | abx | Oh, yes, sorry — I was considering the general case. In the case of the example you are right. | |
| Nov 15, 2020 at 18:09 | vote | accept | user2520938 | ||
| Nov 15, 2020 at 17:38 | comment | added | user2520938 | But isn't (in your example) $H^1(U,\mathbb{C})$ a pure mixed hodge structure of weight $1$? And $V^{1,0}=F^1H^1(U,\mathbb{C})=H^0(X,\Omega_X^1(\log D))$. Then $V^{0,1}=\overline{H^0(X,\Omega_X^1(\log D))}$, and the splitting $H^1(U,\mathbb{C})=V^{0,1}\oplus V^{1,0}$ should give a projection right? | |
| Nov 15, 2020 at 16:01 | comment | added | abx | No, there is no canonical projection from $H^1(U,\mathbb{C})$ to $H^0(X,\Omega ^1_X(\log D))$ — it goes the other way around. There is indeed a canonical map $H^1(U,\mathbb{C})\rightarrow H^1(X,\mathscr{O}_X)$, induced by the morphism of complexes $\Omega ^{\scriptstyle\bullet}_X(\log D)\rightarrow \mathscr{O}_X$. | |
| Nov 15, 2020 at 15:16 | comment | added | user2520938 | Do you know if there is a sense in which we can associate to this $\tilde{\omega}$ its "part" lying in $H^0(X,\Omega_X^1(\log D))$? So is there a canonical projection $H^1(U,\mathbb{C})\to H^0(X,\Omega_X^1(\log D))$ and $H^1(U,\mathbb{C})\to H^1(X,\mathcal{O}_X)$? | |
| Nov 15, 2020 at 14:00 | vote | accept | user2520938 | ||
| Nov 15, 2020 at 18:08 | |||||
| Nov 15, 2020 at 13:27 | history | answered | abx | CC BY-SA 4.0 |