Vanishing cycle

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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.

A classical result is the Picard-Lefschetz formula^[1], detailing how the monodromy round the singular fiber acts on the vanishing cycles, by a shear mapping.

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.

[edit] References

^ Given in[1], for Morse functions.

Section 3 of Peters, C.A.M. and J.H.M. Steenbrink: Infinitesmal 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, Rinhardt (1988), Etale Cohomology and the Weil Conjecture, 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, vol. 340, Lecture notes in mathematics, Berlin, New York: Springer-Verlag, pp. x+438 , see especially Pierre Deligne, Le formalisme des cycles évanescents, SGA7 XIII and XIV.