In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber.
The classical, geometric theory of Solomon Lefschetz was recast in purely algebraic terms, in SGA7. This was for the requirements of its application in the context of l-adic cohomology; and eventual application to the Weil conjectures. There the definition uses derived categories, and looks very different. It involves a functor, the nearby cycle functor, with a definition by means of the higher direct image and pullbacks. The vanishing cycle functor then sits in a distinguished triangle with the nearby cycle functor and a more elementary functor. This formulation has been of continuing influence, in particular in D-module theory.
- Given in , for Morse functions.
- Section 3 of Peters, C.A.M. and J.H.M. Steenbrink: Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces, in : Classification of Algebraic Manifolds, K. Ueno ed., Progress inMath. 39, Birkhauser 1983.
- For the étale cohomology version, see the chapter on monodromy in Freitag, E.; Kiehl, Reinhardt (1988), Etale Cohomology and the Weil Conjecture, Berlin: Springer-Verlag, ISBN 978-0-387-12175-8
- Deligne, Pierre; Katz, Nicholas, eds. (1973), Séminaire de Géométrie Algébrique du Bois Marie – 1967–69 – Groupes de monodromie en géométrie algébrique – (SGA 7) – vol. 2, Lecture notes in mathematics 340, Berlin, New York: Springer-Verlag, pp. x+438, see especially Pierre Deligne, Le formalisme des cycles évanescents, SGA7 XIII and XIV.
- Massey, David (2010). "Notes on Perverse Sheaves and Vanishing Cycles". arXiv:math/9908107.
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