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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 \(\pi \) \(\pi \) \(\mathrm{im}\,A\) \(\mathrm{im}\,A\) \((M,\pi )\) \((M,\pi )\) \(\mathrm{ker}\,A\) \(\mathrm{ker}\,A\) \(\mathrm{im}\,A\) \(\mathrm{im}\,A\) \(\mathrm{im}\,A\) \(\mathrm{im}\,A\) \(\pi \) \(\pi \)