Local Holomorphic Isometries of a Modified Projective Space into a Standard Projective Space; Rational Conformal Factors

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 $\Omega\subset \bC\bP^n$ induced by a local holomorphic mapping $f\colon \Omega\to \bP^d$. 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 $h\colon \Omega\to \bC\bP^m$. We also consider the case of rational conformal factors.