nLab vanishing cycle

There is a special classical case due to Lefschetz. In SGA 7.II, Deligne proposed the framework of vanishing and nearby cycle functors. Cf. also Milnor fiber.

• Pierre Deligne, Le formalisme des cycles évanescents, Exposés XIII, XIV in:Groupes de monodromie en géométrie algébrique. II. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II). Dirigé par P. Deligne et N. Katz. Lecture Notes in Mathematics 340, Springer-Verlag 1973. x+438 pp, MR354657

• wikipedia vanishing cycle; eom: vanishing cycle, monodromy transformation

• S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1924; Jbuch 50, 663 (reprinted in his Selected papers, Chelsea Publ. 1971, MR299447)

• V. Ginsburg, Characteristic varieties and vanishing cycles, Inv. Math. 84, 327–402 (1986)

• Luc Illusie, Vanishing cycles over general bases after P. Deligne, O. Gabber, G. Laumon and F. Orgogozo, pdf

• MathOverflow: is-there-a-good-way-to-think-of-vanishing-cycles-and-nearby-cycles, higher-vanishing-cycles

• Chris Peters, Tata lectures on motivic aspects of Hodge theory, pdf

• Masaki Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, pdf; in: Algebraic geometry (Tokyo/Kyoto, 1982), 134–142, Lecture Notes in Math. 1016, Springer 1983.

• Mihai Tibăr, Polynomials and vanishing cycles, Cambridge Tracts in Mathematics 170, 2007. xii+253 pp. MR2008m:32048

• A.A. Beilinson, How to glue perverse sheaves, pdf transcription

  http://www.math.harvard.edu/~gaitsgde/grad_2009/Beilinson%20–%20How%20to%20glue%20perverse%20sheaves.pdf).

• Ryan Reich?, Notes on Beilinson’s “How to glue perverse sheaves”, arxiv/1002.1686

• Sam Lichtenstein?, Vanishing cycles for algebraic D-modules, Harvard senior thesis (adv. D. Gaitsgory) March 2009, pdf

• Alexander Beilinson, Dennis Gaitsgory, A corollary of the b-function lemma, arXiv:0810.1504.

• A. Beilinson, J. Bernstein, A proof of Jantzen conjectures, pdf.

• Ofer Gabber, Gersten’s conjecture for some complexes of vanishing cycles, MR1305746, doi

• There is a number of related preprints of D. Massey, cf. here including lecture notes on perverse sheaves and v.c. pdf

• C. Sabbah, Proximité évanescente, Compos. Math. 62, 283-328, 1987.

• Dirk Siersma, Vanishing cycles and special fibres, in: Singularity theory and its applications I_ (Coventry, 1988/1989), 292–301, Springer LNM 1462, 1991, MR92j:32129, doi

• P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp. MR2009f:53143, doi

• A. Dimca, Sheaves in topology, Springer 2004, chap. 4.