The basic $dd^{\mathcal{J}}$-lemma

TL;DR

This paper advances the theory of transverse generalized complex structures by establishing equivalent conditions for the basic $dd^{ J}$-lemma, linking it to transverse symplectic structures and Lefschetz map surjectivity, with a new example of such a foliation.

Contribution

It provides new equivalent conditions for the basic $dd^{ J}$-lemma and relates it to transverse symplectic structures and Lefschetz map surjectivity, expanding the understanding of transverse generalized complex structures.

Findings

Established equivalent conditions for the basic $dd^{ J}$-lemma.

Linked the lemma to transverse symplectic structures.

Presented a non-trivial example of a foliation with transverse generalized complex structure.

Abstract

The purpose of this short paper is to further develop the theory of transverse generalized complex structures. We focus on proving some equivalent conditions to the basic $dd^{\mathcal{J}}$ -lemma. We justify our approach by describing the transverse symplectic structure in this language and relating the basic $dd^{\mathcal{J}}$-lemma to the surjectivity of the Lefschetz map. We also present a non-trivial example of a foliation endowed with a transverse generalized complex structure. Transverse generalized complex structures do not rely heavily on the existence of a bundle-like metric, which makes them a convienient tool to study some non-Riemmanian foliations.