Quasi-classical generalized CRF structures

TL;DR

This paper explores quasi-classical generalized CRF structures, extending generalized geometry to include tensor fields that generalize holomorphic Poisson structures, and analyzes their integrability, Lie algebroid structures, and Poisson cohomology.

Contribution

It introduces and characterizes quasi-classical generalized CRF structures via tensor pairs (A, π), establishing their integrability conditions and geometric properties.

Findings

A is a classical CRF structure

π is a Poisson bivector field

The dual bundle of im A has a Lie algebroid structure

Abstract

In an earlier paper, we studied manifolds $M$ endowed with a generalized F structure $\Phi\in End(TM\oplus T^*M)$, 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 $(A\in End(TM),\pi\in\wedge^2TM)$ 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 $im\,A$ is a (non)holonomic Poisson submanifold of $(M,\pi)$. 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 $\pi$, including an associated spectral sequence and a Dolbeault type grading.