In this thesis we study aspects of the mathematical formulation of quantization and more specifically geometric quantization. Our main objective is the construction of a Hitchin connection in settings, that generalise the constructions of Andersen in [And12], which again was a generalisation of the original work by Hitchin [Hit90] studying the case of the moduli space of flat connections on a surface.
We review the construction by Andersen and this Author in [AR16], where we succeeded in significantly weakening the so called rigidity condition on the family of complex structures, which was required for Andersens original construction to work. We also include calculations of the curvature in this so-called weakly restricted case.
Afterwards we continue with new work joint with Andersen, where we construct a Hitchin connection for a general family of Kähler structures under certain cohomological conditions. Under similar conditions, we can even state the uniqueness of such a connection, and proof that this condition is also necessary for the existence of such a Hitchin connection. This is at the moment work in progress, which we expect to publish ultimo 2018 [AR18].
Besides stating and proving these results, we introduce the context by going through some basics of complex geometry, quantization and review the original moduli space case studied by Hitchin.