High Quality Content by WIKIPEDIA articles! 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, 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.

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