  For a representation of a finite group $G$ on a complex vector space $V$ we
determine when a holomorphic $\binom{p}{q}$-tensor field on the principle
stratum of the orbit space $V/G$ can be lifted to a holomorphic $G$-invariant
tensor field on $V$. This extends also to connections. As a consequence we
determine those holomorphic diffeomorphisms on $V/G$ which can be lifted to
orbit preserving holomorphic diffeomorphisms on $V$. This in turn is applied to
characterize complex orbifolds.
