Timeline for Isotrivial families with non-zero Kodaira spencer map
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| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2015 at 18:04 | vote | accept | Pancho | ||
| Jul 30, 2015 at 17:31 | answer | added | Jason Starr | timeline score: 6 | |
| Jul 30, 2015 at 14:25 | comment | added | Pancho | @JasonStarr Thank you. I'm convinced now that an isotrivial family of polarized varieties has trivial Kodaira-Spencer map. Could you please post your comments as an answer so that I can accept your solution? | |
| Jul 30, 2015 at 13:34 | comment | added | Jason Starr | A morphism of tosion-free coherent sheaves on a reduced scheme $S$, e.g., your Kodaira-Spencer map $T_S\to R^1 f_* T_{\mathcal{X}/S}$, is identically zero if and only if it is zero on a dense open. | |
| Jul 30, 2015 at 13:30 | comment | added | Pancho | @JasonStarr I agree that the Isom scheme is generically flat (even smooth). But it might fail to be so over $P$. So I think your argument only shows that the family $f$ is locally trivial over some dense open of $S$. (This dense open might not contain the point $P$ though.) Or am I missing something obvious? | |
| Jul 30, 2015 at 12:18 | comment | added | Jason Starr | ... not just generically flat, but smooth (on a dense open of some irreducible component dominating $S'$) because the characteristic is $0$. | |
| Jul 30, 2015 at 11:58 | comment | added | Jason Starr | In addition to Fischer-Grauert, if you are working with projective schemes (which you are), there are also purely algebro-geometric proofs. Over $S'=S\times S$ with its two pullback families, $\mathcal{X}'_1 =\text{pr}_1^*\mathcal{X}$ and $\mathcal{X}'_2 = \text{pr}_2^*\mathcal{X}$, you can form the relative Isom scheme $\text{Isom}_{S'}(\mathcal{X}'_1,\mathcal{X}'_2)\to S'$ using existence of the Hilbert scheme. Your hypothesis implies that this is surjective over $S'$, thus generically flat over $S$. Base change, and then apply your result for a product family. | |
| Jul 30, 2015 at 11:42 | comment | added | diverietti | If all fibers are isomorphic, then by the Fischer-Grauert theorem the family is locally trivial. Then, the Kodaira-Spencer map -being "local"- should be zero, isn't it? | |
| Jul 30, 2015 at 11:10 | review | First posts | |||
| Jul 30, 2015 at 11:52 | |||||
| Jul 30, 2015 at 11:09 | history | asked | Pancho | CC BY-SA 3.0 |