# Quasi-classical generalized CRF structures

Research paper by Izu Vaisman

Indexed on: 09 Aug '16Published on: 29 Jul '16Published in: Annals of Global Analysis and Geometry

#### Abstract

Abstract In an earlier paper, we studied manifolds M endowed with a generalized F structure $$\Phi \in \mathrm{End}(TM\oplus T^*M)$$ , 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 $$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$ , 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.AbstractIn an earlier paper, we studied manifolds M endowed with a generalized F structure $$\Phi \in \mathrm{End}(TM\oplus T^*M)$$ , 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 $$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$ , 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.M $$\Phi \in \mathrm{End}(TM\oplus T^*M)$$ $$\Phi \in \mathrm{End}(TM\oplus T^*M)$$ $$\Phi ^3+\Phi =0$$ $$\Phi ^3+\Phi =0$$ $$\Phi$$ $$\Phi$$ $$\Phi$$ $$\Phi$$ $$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$ $$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$ $$A^3+A=0$$ $$A^3+A=0$$A $$\pi$$ $$\pi$$ $$(A,\pi )$$ $$(A,\pi )$$A $$\pi$$ $$\pi$$ $$\mathrm{im}\,A$$ $$\mathrm{im}\,A$$ $$(M,\pi )$$ $$(M,\pi )$$ $$\mathrm{ker}\,A$$ $$\mathrm{ker}\,A$$ $$\mathrm{im}\,A$$ $$\mathrm{im}\,A$$ $$\mathrm{im}\,A$$ $$\mathrm{im}\,A$$ $$\pi$$ $$\pi$$