Timeline for Is there a good way to think of vanishing cycles and nearby cycles?
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| Sep 21, 2023 at 3:59 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
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| Oct 22, 2009 at 21:39 | comment | added | Mike Skirvin | Sometimes when people say vanishing cycle or nearby cycle, they mean the functor applied to the constant sheaf. In this case, taking cohomology leads to the description given in the Wikipedia article. In general the nearby cycles, for example, are defined by taking a pull-back, then a (derived) push-forward, and then another pull-back. The vanishing cycles are defined using the triangulated category structure on the derived category as the cone of the so-called specialization map (defined using nothing more than the adjunction between push-forward and pull-back). | |
| Oct 22, 2009 at 21:23 | comment | added | Ilya Nikokoshev | Wikipedia: "In mathematics, vanishing cycles are studied in singularity theory and other part of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber." This sounds like less info than an element of derived category, though I'm not an expert, again. | |
| Oct 22, 2009 at 21:05 | comment | added | Mike Skirvin | As far as I know, vanishing cycles and nearby cycles are always derived, in the sense that they're usually defined as functors from the derived category of constructible sheaves on the total space to the derived category of constructible sheaves on the singular fiber. Furthermore, they're even perverse functors, meaning that they're functors on the underlying categories of perverse sheaves (sometimes they need to be shifted for this to be true, depending on how the functors are defined). | |
| Oct 22, 2009 at 18:18 | history | answered | Ilya Nikokoshev | CC BY-SA 2.5 |