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