Timeline for When is a holomorphic submersion with isomorphic fibers locally trivial?
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 30, 2019 at 14:08 | comment | added | Georges Elencwajg | Dear @Arrow: no, unfortunately I don't know such a modern reference. | |
| Apr 30, 2019 at 9:33 | comment | added | Arrow | Dear @GeorgesElencwajg, do you know of any modern reference to the theorem by Fischer-Grauert (and perhaps of this generalization)? | |
| Apr 11, 2015 at 20:55 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
deleted 9 characters in body
|
| Aug 24, 2014 at 14:39 | comment | added | Qfwfq | @Andrea: take a non-Zariski-locally-trivial (locally isotrivial) projective fibration; then, by FG, it's locally (in the eucliden topology) analytically trivial. Or did you mean an explicit example? | |
| Feb 27, 2011 at 20:52 | vote | accept | Georges Elencwajg | ||
| Feb 26, 2011 at 23:05 | comment | added | Sándor Kovács | @Andrea: take a look at the addendum to my answer below. The point is that if a family does not have a section, then it cannot be locally trivial. This way you get tons of examples. | |
| Feb 26, 2011 at 23:03 | answer | added | Sándor Kovács | timeline score: 6 | |
| Feb 26, 2011 at 19:32 | answer | added | Jason Starr | timeline score: 5 | |
| Feb 26, 2011 at 18:23 | comment | added | Andrea Ferretti | Uhm... I understand the point. It would be very interesting to see a proper family which is analitically, but not algebraically, locally trivial. | |
| Feb 26, 2011 at 17:49 | comment | added | Qfwfq | In other words, maybe it could happen that any such family is analytically locally trivial but it doesn't necessarily admit an algebraic trivialization. | |
| Feb 26, 2011 at 17:49 | comment | added | Qfwfq | Dear Andrea, I don't know if the existence of a proper algebraically non-locally-trivial family with fixed fibers would contradict the FG theorem. After all, working complex analytically can give more "degrees of freedom" for trivialization. [Not directly related, but: incidentally it happens that all holomorphic line bundles over a punctured elliptic curve are holomorphically trivial, but the algebraic Pic of that variety is nontrivial...]. | |
| Feb 26, 2011 at 17:36 | comment | added | Georges Elencwajg | Dear Angelo: you are absolutely right, the Fischer-Grauert theorem only tackles proper maps, as is stated in my question. As for an example where the analogon becomes false in the non proper case, here is a candidate. Take the open ball $|z_1|^2+ |z_2|^2 \lt 1$ in $\mathbb C^2$and project it onto the unit disk in the $z_1$-plane.This is a submersion with all fibers isomorphic to a disk but I don't think it is locally trivial, because of Poincaré's theorem that a bidisk is not isomorphic to a 2-dimensional ball. This is not a rigorous proof but I think one might transform it into one. | |
| Feb 26, 2011 at 17:19 | comment | added | Georges Elencwajg | Dear Andrea, yes I'm claiming that Fischer and Grauert proved that. I have no opinion on the terminology because I am not used to the adjective "isotrivial" in an analytic context ( although I know Serre introduced the notion "espaces fibrés localement isotriviaux" for algebraic varieties in the 1958 Séminaire Chevalley) | |
| Feb 26, 2011 at 17:11 | answer | added | Sándor Kovács | timeline score: 4 | |
| Feb 26, 2011 at 17:09 | comment | added | Angelo | Dear George, I thought that the result of Fischer and Grauert was for the proper case. My question was if there are known examples in the non-compact case. | |
| Feb 26, 2011 at 16:47 | comment | added | Andrea Ferretti | So you are claiming that every isotrivial family of compact complex manifolds is actually locally trivial? I have to admit I do not have a proof of non-local-triviality for any isotrivial family (over $\mathbb{C}$) I can think of (and foor good reasons, apparently!) but then I wonder why the terminology... I wish I was always just told "here is a fiber bundle with fiber $F$". | |
| Feb 26, 2011 at 16:32 | comment | added | Georges Elencwajg | Dear Angelo: no, there are no such examples because they are excluded by Fischer-Grauert's theorem! It seems this theorem doesn't have the celebrity it deserves, maybe because of the already mentioned relative obscurity of the journal where it was published. And also because of the language barrier: if articles are to be read, they must be written in Modern Latin, i.e. English! | |
| Feb 26, 2011 at 16:23 | comment | added | Georges Elencwajg | Dear Andrea, you write that there "certainly" exist counterexamples to Fischer-Grauert's theorem (I suppose that by isotrivial you mean isomorphic fibers ?). What makes you think so ? You can find that theorem stated in English in Barth, Peters and Van de Ven's classic book Compact Complex Surfaces in Chapter I,Theorem 10.1 on page 29. In a way, I'm happy about your doubts: it shows how unbelievable Fischer-Grauert's theorem is! | |
| Feb 26, 2011 at 15:40 | comment | added | Angelo | I don't know the answer; but are there any know examples where the fibers are all isomorphic, but the family is not locally trivial? | |
| Feb 26, 2011 at 15:19 | comment | added | Andrea Ferretti | I am confused about the Grauert-Fischer theorem, and I don't have a reference at hand (not to mention my german is not in very good shape). Certainly there exist proper maps $\pi \colon X \to S$, with both $X$ and $S$ compact, which are isotrivial on an open subset $U \subset S$, but not locally trivial. It seems to me that the restriction $\pi^{-1}(U) \to U$ is still proper, but does not satisfy the conclusion of the G-F theorem. :-? | |
| Feb 26, 2011 at 14:00 | history | asked | Georges Elencwajg | CC BY-SA 2.5 |