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We consider local modifications \(\omega _n+f^*\omega _d\) of the Fubini-Study metric (with associated \((1,1)\)-form \(\omega _n\)) on an open subset \(\Omega \subset \mathbb C\mathbb P^n\) induced by a local holomorphic mapping \(f:\Omega \rightarrow \mathbb P^d\). Our main result is that there are “gaps” in potential dimensions \(m\) such that the modification can be obtained as \(h^*\omega _m\) for some local holomorphic mapping \(h:\Omega \rightarrow \mathbb C\mathbb P^m\). We also consider the case of rational conformal factors.

Local holomorphic isometries of a modified projective space into a standard projective space; rational conformal factors