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Quasi-classical generalized CRF structures

In an earlier paper, we studied manifolds $M$ endowed with a generalized F structure $Φ\in End(TM\oplus T^*M)$, 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 $(A\in End(TM),π\in\wedge^2TM)$ where $A^3+A=0$ 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 $im\,A$ is a (non)holonomic Poisson submanifold of $(M,π)$. We discuss the case where either $ker\,A$ or $im\,A$ is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of $im\,A$ inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of $π$, including an associated spectral sequence and a Dolbeault type grading.