A Hitchin Connection for a large class of families of Kähler Structures

In this paper, we construct a Hitchin connection in a setting, which significantly generalizes the setting covered by the first author in [A5], which in turn was a generalization of the moduli space case covered by Hitchin in his original work on the Hitchin connection [9]. In fact, our construction provides a Hitchin connection which is a partial connection on the space of all compatible complex structures on an arbitrary but fixed prequantizable symplectic manifold, which satisfies a certain Fano-type condition. The subspace of the tangent space to the space of compatible complex structures on which the constructed Hitchin connection is defined is, in fact, of finite codimension, if the symplectic manifold is compact. In a number of examples, including flat symplectic space, symplectic tori and moduli spaces of flat connections for a compact Lie group, we prove that our Hitchin connection is defined in a neighbourhood of the natural families of complex structures compatible with the given symplectic form, which these spaces admits.