  We introduce the concept of a branched holomorphic Cartan geometry. It
generalizes to higher dimension the definition of 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. We show that all compact complex projective manifolds admit
branched flat holomorphic projective structure. We also give an example of a
non-flat branched holomorphic normal projective structure on a compact complex
surface. It is known that no compact complex surface admits such a structure
with empty branching locus. We prove that non-projective compact simply
connected K\"ahler Calabi-Yau manifolds do not admit branched holomorphic
projective structures. The key ingredient of its proof is the following result
of independent interest: If E is a holomorphic vector bundle over a compact
simply connected K\"ahler Calabi-Yau manifold, and E admits a holomorphic
connection, then E is a trivial holomorphic vector bundle equipped with the
trivial connection.
