Odd type generalized complex structures on 4-manifolds

TL;DR

This paper characterizes 4-manifolds admitting odd type generalized complex structures as those with transversely holomorphic 2-foliations, and demonstrates their existence on certain circle bundles over Seifert fibered 3-manifolds.

Contribution

It establishes a precise topological criterion for odd type generalized complex structures on 4-manifolds and constructs examples on specific circle bundles.

Findings

A 4-manifold admits odd type generalized complex structures iff it has a transversely holomorphic 2-foliation.

Existence of odd type structures on circle bundles over Seifert fibered 3-manifolds.

Provides a classification linking foliation theory and generalized complex geometry.

Abstract

We prove that a compact smooth 4-manifold admits generalized complex structures of odd type if and only if it has a transversely holomorphic 2-foliation. Consequently, there exist generalized complex structures of odd type on a circle bundle over a closed Seifert fibered 3-manifold.