Connections with Prescribed Curvature via Poisson Geometry

Abstract

In a series of papers, Q. S. Chi, S. A. Merkulov, L. J. Schwachhöfer, M. Cahen and S. Gutt developed a technique for proving the existence of torsion-free connection with prescribed curvature on a G-structure. They noticed that in many cases it was possible to use the desired curvature to deform the Lie-Poisson structure on the semi-direct product of the Lie algebra of G with R^n, obtaining a new Poisson structure. Moreover, this deformed Poisson structure implied the existence of a G-structure with a connection having the desired curvature.However, what they did not notice (and did not have the appropriate language available to do so) was that the deformed Poisson manifold comes with extra symmetries: their cotangent Lie algebroid is a G-structure Algebroid. This additional structure, which has been previously studied by the proponent of this grant in collaboration with R. L. Fernandes, plays a fundamental role in the passage from the local existence results to global results. In fact, the existence of complete connections and the description of their moduli space are controlled by the G-structure groupoid which integrates the cotangent algebroid of the deformed Poisson structure. This groupoid may or may not exist, and the obstructions for its existence is part of the Lie theory for such structures.In this project we propose to study the Lie theory for the cotangent algebroid of the deformed Poisson structure. The main questions we will deal with are: (1) how can we describe the obstructions for integrability in terms of the deforming map (i.e., the prescribed curvature)? (2) In which cases is it possible to obtain explicit integrations for the cotangent algebroid? (3) When are these G-structure groupoids symplectic and what is the role of this symplectic structure in the description of the moduli space of the connections under consideration? The results obtained will be applied to the study of torsion free symplectic connections with special holonomy group. Such connections are the main examples where the local theory has been successfully applied. (AU)