Timeline for Proper family deformation retracts onto special fiber
Current License: CC BY-SA 3.0
18 events
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| May 29, 2017 at 16:15 | vote | accept | improv305 | ||
| Apr 7, 2017 at 14:39 | answer | added | Misha Verbitsky | timeline score: 8 | |
| Mar 26, 2017 at 6:23 | comment | added | naf | If $X$ is smooth, use the flow mentioned by @VivekShende. In general, by resolution of singularities, one can find $\pi: X' \to X$, a proper birational morphism which is an isomorphism outside the special fibre. One can then define a retraction for $X$ by simply composing the flow on $X'$ with $\pi$. This clearly gives a well defined set-theoretic map and continuity follows from the properness of $\pi$. | |
| Mar 25, 2017 at 18:26 | comment | added | R. van Dobben de Bruyn | @improv305: even though the special fibre may have singularities, the total space can still be smooth. This is already an interesting case that is often studied in algebraic geometry. For example, I believe that in your elliptic curve example, the total space should be smooth (but you should check this). | |
| Mar 25, 2017 at 18:00 | history | edited | improv305 | CC BY-SA 3.0 |
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| Mar 24, 2017 at 1:25 | history | edited | improv305 | CC BY-SA 3.0 |
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| Mar 24, 2017 at 1:24 | comment | added | improv305 | @VivekShende Perhaps I don't understand - isn't the point that the central fiber will have singularities? I'm not sure how such a flow would work in that case. | |
| Mar 21, 2017 at 9:05 | comment | added | Vivek Shende | If X is smooth, choose a metric and then flow by $\nabla |s|^2$ | |
| Mar 21, 2017 at 0:34 | history | edited | improv305 | CC BY-SA 3.0 |
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| Mar 20, 2017 at 14:56 | comment | added | improv305 | @PiotrAchinger Do you mean sections 1.5 and 1.7 from Part 1? If so, can you elaborate? I don't see how to get the deformation retraction from the cited sections. | |
| Mar 19, 2017 at 11:21 | comment | added | Piotr Achinger | I think this can be deduced from sections 1.5 and 1.7 of the book "Stratified Morse theory" by Mark Goresky and Robert MacPherson. | |
| Mar 19, 2017 at 0:47 | history | edited | John Pardon | CC BY-SA 3.0 |
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| Mar 19, 2017 at 0:47 | comment | added | John Pardon | This is true (i.e. $X$ does deformation retract onto the special fiber $s^{-1}(0)$). I unfortunately don't know a reference (and would also be interested in seeing one). | |
| Mar 19, 2017 at 0:10 | history | edited | improv305 | CC BY-SA 3.0 |
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| Mar 19, 2017 at 0:06 | comment | added | improv305 | Thank you for the reference - I am interested in the proof, if anyone knows a reference. | |
| Mar 18, 2017 at 10:56 | comment | added | Piotr Achinger | Deligne (SGA 7 II, Exp. XIII, second paragraph of the introduction) suggests that something slightly stronger might be true. I'm sure it has been worked out since, but I don't know a reference. In any case, we know that in the algebraic case the inclusion of the special fiber induces an isomorphism on cohomology and fundamental groups... | |
| Mar 17, 2017 at 22:33 | review | First posts | |||
| Mar 17, 2017 at 22:47 | |||||
| Mar 17, 2017 at 22:32 | history | asked | improv305 | CC BY-SA 3.0 |