Generalized holomorphic Cartan geometries

This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of Felix Klein, elie Cartan and Charles Ehresmann. Then we present the concept of abranched holomorphic Cartan geometrywhich was introduced by Biswas and Dumitrescu (Int Math Res Not IMRN,2017. 10.1093/imrn/rny003,). It generalizes to higher dimension the notion of a branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries (e.g. all compact complex projective manifolds admit branched holomorphic projective structures). At the same time, this new definition is rigid enough to enable us to classify branched holomorphic Cartan geometries on compact simply connected Calabi-Yau manifolds.