Constructions of generalized complex structures in dimension four

TL;DR

This paper explores the existence and construction of twisted generalized complex structures on four-manifolds, demonstrating their diversity and presence beyond classical complex or symplectic types.

Contribution

It introduces methods to construct twisted generalized complex structures with multiple type change loci on various four-manifolds, expanding known examples.

Findings

Existence of generalized complex structures with multiple type change loci

Construction of structures on both simply and non-simply connected four-manifolds

Structures are neither purely complex nor symplectic

Abstract

Four-manifold theory is employed to study the existence of (twisted) generalized complex structures. It is shown that there exist (twisted) generalized complex structures that have more than one type change loci. In an example-driven fashion, (twisted) generalized complex structures are constructed on a myriad of four-manifolds, both simply and non-simply connected, which are neither complex nor symplectic.