Quasi-classical generalized CRF structures

In an earlier paper, we studied manifolds M endowed with a generalized F structure Φ ∈ End (TM⊕ T^∗M) , 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.