Timeline for Local freeness of direct images
Current License: CC BY-SA 3.0
9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 2, 2011 at 18:09 | answer | added | Karl Schwede | timeline score: 2 | |
| Aug 2, 2011 at 3:10 | comment | added | algori | Karl -- thanks for the references. More specifically, I'm interested in the dual of $\Omega^1_{X/Y}$, which happens to be $\Omega^{r-1}_{X/Y}(k)$. Here r is the relative dimension of $X$ and $k$ is, in the case I'm interested in, a positive number between $1$ and $n−2$ where $n$ is the relative dimension of the projective space over $Y$ where $X$ lives. I know that all direct images of any $\Omega^i_{X/Y}$ are locally free. | |
| Aug 1, 2011 at 16:54 | comment | added | Karl Schwede | algori, maybe you could say exactly what you had in mind then. There are a lot of results about local-freeness for twists of these guys, especially if $i$ in $\Omega_{X/Y}^i$ is the relative dimension. For example, see Theorem 7.9(2) in arxiv.org/PS_cache/arxiv/pdf/0902/0902.0648v4.pdf I think there are also some results along these lines in arxiv.org/PS_cache/arxiv/pdf/1003/1003.2913v3.pdf Finally, since you said you can assume your base is a DVR, you just need these pushdowns to be torsion free see Kollar's papers Higher direct images... for a starting place. | |
| Aug 1, 2011 at 14:07 | comment | added | algori | Karl -- yes, that's the example I had in mind. | |
| Aug 1, 2011 at 13:28 | answer | added | Jason Starr | timeline score: 2 | |
| Aug 1, 2011 at 13:07 | comment | added | Karl Schwede | Can we assume anything special about $\mathcal{F}$. For example, $\mathcal{F} = \Omega_{X/Y}^i$. | |
| Aug 1, 2011 at 13:07 | history | edited | algori | CC BY-SA 3.0 |
added a paragraph
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| Aug 1, 2011 at 10:23 | history | asked | algori | CC BY-SA 3.0 |