Quasi-classical generalized CRF structures

In an earlier paper, we studied manifolds M endowed with a generalized F structure (Phi in mathrm{End}(TMoplus T^*M)), skew-symmetric with respect to the pairing metric, such that (Phi ^3+Phi =0). Furthermore, if (Phi ) is integrable (in some well-defined sense), (Phi ) 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 ((Ain mathrm{End}(TM),pi in wedge ^2TM)), where (A^3+A=0) and some relations between A and (pi ) hold. We establish the integrability conditions in terms of ((A,pi )). They include the facts that A is a classical CRF structure, (pi ) is a Poisson bivector field and (mathrm{im},A) is a (non)holonomic Poisson submanifold of ((M,pi )). We discuss the case where either (mathrm{ker},A) or (mathrm{im},A) is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of (mathrm{im},A) inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of (pi ), including an associated spectral sequence and a Dolbeault type grading.