A Hitchin Connection for a Large Class of Families of Kähler Structures

AbstractThis chapter presents a Hitchin connection constructed in a setting which significantly generalizes the setting covered by the first author, which, in turn, was a generalization of the moduli space covered in the original work on the Hitchin connection. In fact, this 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 of finite codimension if the symplectic manifold is compact. It also proves uniqueness of the Hitchin connection under a further assumption. A number of examples show that this Hitchin connection is defined in a neighbourhood of the natural families of complex structures compatible with the given symplectic form which these spaces admit.