The Picard-Lefschetz theory of complexified Morse functions

TL;DR

This paper constructs a symplectic Lefschetz fibration modeling the complexification of Morse functions on manifolds, providing explicit fibers and vanishing spheres, with potential applications in symplectic topology.

Contribution

It introduces a method to model the complexification of Morse functions as symplectic Lefschetz fibrations with explicit structures and embeddings.

Findings

Constructed explicit symplectic Lefschetz fibrations for Morse functions

Embedded the original manifold as an exact Lagrangian in the model

Demonstrated homotopy equivalence between the model and the manifold

Abstract

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.