The Picard-Lefschetz theory of complexified Morse functions

Given a closed manifold N and a self-indexing Morse function f: N --> R with up to four distinct Morse indices, we construct a symplectic Lefschetz fibration pi: E --> C which models the complexification of f on the disk cotangent bundle, f_C : D(T*N) --> C, when f is real analytic. By construction, pi: E --> C comes with an explicit regular fiber M and vanishing spheres V_1,...,V_m in M, one for each critical point of f. Our main result is that (E,pi) is a good model for the complexification (D(T*N),f_C) in the sense that N embeds in E as an exact Lagrangian submanifold, and in addition, pi|N = f and E is homotopy equivalent to N. There are several potential applications in symplectic topology, which we discuss in the introduction.