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.
Categories: math.DG math.CV math.RT