Timeline for Direct image of the relative dualizing sheaf
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 10, 2017 at 19:55 | comment | added | user21574 | See 2.3. Proposition, and 3.9. Corollary math1.unice.fr/~hoering/articles/a8-pos-geom.pdf | |
| Jun 10, 2017 at 19:32 | comment | added | user21574 | In general Fujita-Kawamata semi-positivity say that Let $π : X → Y$ be an algebraic fiber space which satisfies Unipotent Reduction Condition. Then $π_∗ω_{X/Y}$ is a locally free sheaf and semi-positive. For example , If $π : X → Y$ is semi-stable family,then the Unipotent Reduction Condition holds automatically | |
| Nov 6, 2015 at 18:06 | comment | added | Tong | @nfdc23 Thank you for your answer. In my case, there could be some multiple fibers. | |
| Nov 6, 2015 at 18:04 | comment | added | Tong | @KarlSchwede Thank you very much for your reference. I will read that. | |
| Nov 6, 2015 at 15:09 | comment | added | nfdc23 | Correction: for my comment above I should have also assumed the geometric fibers to be reduced (or more generally that $O_{X_y}(X_y) = k(y)$ for each $y \in Y$); that was implicit in knowing the fibral trace is an isomorphism. | |
| Nov 6, 2015 at 15:00 | comment | added | Karl Schwede | Some statements along these lines are in section 7 of Kollár-Kovács (click here) but they are probably not quite what you want. | |
| Nov 6, 2015 at 14:55 | comment | added | nfdc23 | For $n=1$ (so ${\rm{R}}^1f_{\ast}$ is right-exact on coherent sheaves), it suffices that $f$ has geometrically connected fibers. Indeed, fibral trace ${\rm{H}}^1(X_y \omega_{X_y/y})\rightarrow k(y)$ is then an isomorphism, so ${\rm{R}}^1f_{\ast}(\omega_{X/Y})$ is locally monogenic by Nakayama. Thus, the base-change compatible trace to $\mathscr{O}_Y$ is a fiberwise isomorphism, so an isomorphism. The formation of ${\rm{R}}^1f_{\ast}(\omega_{X/Y})$ thus commutes with base change to fibers, so $f_{\ast}\omega_{X/Y}$ is a vector bundle compatible with any base change. | |
| Nov 6, 2015 at 14:45 | history | edited | Tong | CC BY-SA 3.0 |
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| Nov 6, 2015 at 14:39 | history | edited | Tong | CC BY-SA 3.0 |
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| Nov 6, 2015 at 14:32 | history | asked | Tong | CC BY-SA 3.0 |