Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry

Abstract

Exploiting the affinity between stable generalized complex structures and symplectic structures, we explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting. We introduce generalized Luttinger surgery and generalized Gluck twist along J-symplectic submanifolds. We also study branched coverings in the generalized complex setting. As an application, we produce stable generalized complex structures on a variety of high-dimensional manifolds. Remarkably, some of them have non-homotopy-equivalent path-connected components of their type change locus.