A splitting theorem for good complexifications

Quasiprojective three-manifold groups and complexification of three-manifolds

We characterize the quasiprojective groups that appear as fundamental groups of compact $3$-manifolds (with or without boundary). We also characterize all closed $3$-manifolds that admit good complexifications. These answer questions of Friedl--Suciu, \cite{fs}, and Totaro \cite{tot}

复杂映射的好的实像

Good real images of complex maps

本文证明实映射的好的实扰动与其复化同伦等价的Marar-Mond猜想，并推广到其他维度，给出M-形变应用。

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…