The basic dd^J-lemma

2016/09/15 by Paweł Raźny, Razny, Pawel
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1609.04539

openalex publication_date 2016/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^J -lemma. We justify our approach by describing the transverse symplectic structure in this language and relating the basic dd^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.

Citations

• Generalized Complex Geometry

Related

• New aspects of the ddc-lemma
• Hodge theory on transversely symplectic foliations
• Transversal hard Lefschetz theorem on transversely symplectic foliations
• Bi-Lagrangian structures and Teichmüller theory
• Transverse Kähler structures on central foliations of complex manifolds