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

TL;DR

This paper investigates local holomorphic isometries between modified and standard projective spaces, revealing dimension gaps and exploring rational conformal factors in the context of Fubini-Study metric modifications.

Contribution

It identifies dimension gaps for local isometries of modified projective spaces and analyzes the role of rational conformal factors in such mappings.

Findings

Existence of dimension gaps in local isometries

Characterization of modifications as pullbacks of standard metrics

Analysis of rational conformal factors in metric modifications

Abstract

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.