nLab
Picard-Lefschetz theory

Idea

Picard-Lefschetz theory is an approach which relates the study of singularities of holomorphic functions to the topology of the domain complex manifold; thus it is a certain analogue of Morse theory in complex geometry. The starting point is the Picard-Lefschetz formula describing monodromy at a critical point. There is also extension to other fields (Deligne and Katz).

References

Related $n$Lab entries include: monodromy, locally constant sheaf, Morse theory, holomorphic function, Riemann-Hilbert problem, vanishing cycle, Painlevé transcendent, Milnor fiber, Gauss-Manin connection, slope filtration, Stokes phenomenon, semiclassical approximation, movable singularity, Fuchsian equation, hypergeometric function

• wikipedia: Picard-Lefschetz theory
• V. A. Vassiliev, Ramified integrals, singularities and lacunas, Kluwer 1994, 289+xvii pp. Russian version: Ветвящиеся интегралы, Moscow 2002, 432 pp.
• P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp. MR2009f:53143, doi
• V. A. Vassiliev, Applied Picard-Lefschetz theory, Mathematical Surveys and Monographs 97. Amer. Math. Soc. 2002. xii+324 pp. MR2003k:32043
• V. A. Vassiliev, Stratified Picard-Lefschetz theory,

  Selecta Math. (N.S.) 1 (1995), no. 3, 597–621, MR96i:32037, doi

• Henryk Żołądek, The monodromy group, Monografie Matematyczne 67, 588 pp. Birkhäuser 2006
• Pierre Deligne, Nicholas Katz, Groupes de monodromie en géométrie algébrique. II, Springer Lect. Notes in Math. 340, 1973, doi, MR0354657

category: analysistopologyalgebraic geometry