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

TL;DR

This paper extends symplectic geometric constructions to generalized complex geometry, introducing new surgeries and branched coverings, leading to novel high-dimensional structures with diverse topological features.

Contribution

It introduces generalized Luttinger surgery, generalized Gluck twist, and branched coverings in generalized complex geometry, expanding the toolkit for constructing and analyzing such structures.

Findings

Produced stable generalized complex structures on high-dimensional manifolds.

Found non-homotopy-equivalent components in the type change locus.

Extended symplectic constructions to generalized complex setting.

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 $\mathcal{J}$-symplectic submanifolds. We also export branched coverings to the generalized complex setting. As an application, stable generalized complex structures are produced on a variety of high-dimensional manifolds. Remarkably, some of them have non-homotopy-equivalent path-connected components of their type change locus.