Timeline for Degeneration of relative Hodge-de Rham spectral sequence
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 13, 2018 at 16:47 | comment | added | K.K. | @JürgenBöhm Yes, that is the issue here -- even in a topologically trivial family of complex manifolds, you'll need further assumptions to ensure that all of the fibers are isomorphic as complex manifolds. | |
| Aug 13, 2018 at 16:34 | comment | added | Jürgen Böhm | @Tony You are right, I was somehow thinking the usual result of Ehresmann would "strengthen itself" when one goes to the category of complex manifolds. But there is probably something more subtle involved here, which I am currently not aware of - does it have to do with "variation of complex structure in the family $f:X \to S$"? | |
| Aug 13, 2018 at 16:12 | comment | added | K.K. | Nitpick: Ehresmann's theorem only guarantees that there is a diffeomorphism between $X_s$ and $X_{s'}$, not a biholomorphism. | |
| Aug 12, 2018 at 22:53 | comment | added | Donu Arapura | You're welcome. By the way, a reference for the degeneration is theoreme 5.5 of Deligne, Theoreme de Lefschetz et criteres de degenerescence de suites spectrales | |
| Aug 12, 2018 at 22:45 | comment | added | Jürgen Böhm | @Donu Arapura Thank you very much for taking the time to look at my proof (attempt)! I really had hoped that Q3 would be right, because I want to base an algorithm for explicitly calculating the Gauss-Manin connection in Macaulay2 on it. | |
| Aug 12, 2018 at 22:34 | comment | added | Donu Arapura | Oh, I missed the fact you're assuming $S$ is affine. Then Q3 is OK, but the splitting would be highly unnatural. | |
| Aug 12, 2018 at 21:24 | comment | added | Donu Arapura | I don't have time to check carefully; your basic strategy seems fine initially. However the answer to Q3 is no, it won't split except in trivial cases. | |
| Aug 12, 2018 at 20:47 | history | asked | Jürgen Böhm | CC BY-SA 4.0 |