Timeline for Torsion freeness of direct image of structure sheaf?
Current License: CC BY-SA 4.0
13 events
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| Sep 1, 2020 at 21:39 | comment | added | guest0803 | In addition to @Hacon's comment, in [loc. cit. \S 3] Koll\'ar has also pinpointed the torsion free and the torsion part of $R^if_*\mathcal{O}_X$. The torsion free part is the dual of $R^{n-i}f_*\omega_X$ and hence is also reflexive. | |
| Aug 29, 2020 at 2:21 | comment | added | xin fu | @Hacon. Thanks for the comment! | |
| Aug 28, 2020 at 15:23 | comment | added | Hacon | If you strengthen your smoothness conditions, then it is true. Assume that $f$ is is smooth over $U=Y-B$ the complement of a snc divisor. Then $Rf_*\omega _X=\sum R^if_*\omega _X[-i]$ and each $R^if_*\omega _X$ and $R^if_*\mathcal O _X$ is locally free (Koll\'ar's Higher Direct Images II Thm 2.6). Note that the $R^if_*\omega _X$ are always torsion free. | |
| S Aug 27, 2020 at 22:56 | history | suggested | KReiser | CC BY-SA 4.0 |
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| Aug 27, 2020 at 22:24 | review | Suggested edits | |||
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| Aug 27, 2020 at 20:16 | history | edited | xin fu | CC BY-SA 4.0 |
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| Aug 27, 2020 at 20:10 | history | edited | xin fu | CC BY-SA 4.0 |
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| Aug 27, 2020 at 20:03 | comment | added | xin fu | @Anonymous Nice example, thanks! | |
| Aug 27, 2020 at 19:58 | comment | added | Anonymous | Q1: Take a normal surface $Y'$ with non-rational singularity. Let $g:X \to Y'$ be a resolution, and let $f:X \to Y = \mathbf{P}^2$ be the composition of $g$ with a finite map $Y' \to \mathbf{P}^2$. Then $R^1 f_* \mathcal{O}_X \neq 0$ and is supported at finitely many points. | |
| Aug 27, 2020 at 19:45 | history | edited | xin fu | CC BY-SA 4.0 |
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| Aug 27, 2020 at 19:42 | comment | added | xin fu | @abx Thanks.Yes I need $f$ surjective. | |
| Aug 27, 2020 at 19:18 | comment | added | abx | As your questions stand, the answer is no for obvious reasons: just take for $f$ a closed immersion. You probably want $f$ surjective, $X$ connected, ...? | |
| Aug 27, 2020 at 19:01 | history | asked | xin fu | CC BY-SA 4.0 |