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, detailing how the monodromy round the singular fiber acts on the vanishing cycles, by a shear.

[edit] 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 the book Weil Conjectures by Freitag and Keihler
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