Quasi-classical generalized CRF structures

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In an earlier paper, we studied manifolds M endowed with a generalized F structure Φ ∈ End (TM⊕ TM) , skew-symmetric with respect to the pairing metric, such that Φ 3+ Φ = 0. Furthermore, if Φ is integrable (in some well-defined sense), Φ is a generalized CRF structure. In the present paper, we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields (A∈ End (TM) , π∈ ∧ 2TM) , where A3+ A= 0 and some relations between A and π hold. We establish the integrability conditions in terms of (A, π). They include the facts that A is a classical CRF structure, π is a Poisson bivector field and imA is a (non)holonomic Poisson submanifold of (M, π). We discuss the case where either kerA or imA is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of imA inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of π, including an associated spectral sequence and a Dolbeault type grading.

Original languageEnglish
Pages (from-to)53-71
Number of pages19
JournalAnnals of Global Analysis and Geometry
Issue number1
StatePublished - 1 Jan 2017

Bibliographical note

Publisher Copyright:
© 2016, Springer Science+Business Media Dordrecht.


  • Generalized CRF structure
  • Generalized complex structure
  • Holomorphic Poisson structure

ASJC Scopus subject areas

  • Analysis
  • Political Science and International Relations
  • Geometry and Topology


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