Vanishing cycle

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 [ Given in [http://eom.springer.de/m/m064700.htm] , for Morse functions.] , 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.

References

*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 Citation | last1=Freitag | first1=E. | last2=Kiehl | first2=Rinhardt | title=Etale Cohomology and the Weil Conjecture | isbn=978-0-387-12175-8 | year=1988
*, see especially Pierre Deligne, "Le formalisme des cycles évanescents", SGA7 XIII and XIV.

External links

* [http://eom.springer.de/V/v096070.htm EoM article]