On holomorphic isometries into blow-ups of $\mathbb C^n$

TL;DR

This paper investigates the properties of holomorphic isometries into certain blow-up Kähler-Einstein manifolds, specifically the generalized Burns-Simanca and Eguchi-Hanson manifolds, and establishes their non-relativity to homogeneous bounded domains.

Contribution

It characterizes the holomorphic isometries into these blow-up manifolds and proves their independence from homogeneous bounded domains.

Findings

Holomorphic isometries into the generalized Burns-Simanca and Eguchi-Hanson manifolds are studied.

These manifolds are shown not to be relatives to any homogeneous bounded domain.

Abstract

We study the K\"ahler-Einstein manifolds which admits a holomorphic isometry into either the generalized Burns-Simanca manifold $(\tilde {\mathbb C}^n, g_S)$ or the Eguchi-Hanson manifold $(\tilde {\mathbb C}^2, g_{EH})$. Moreover, we prove that $(\tilde {\mathbb C}^n, g_S)$ and $(\tilde {\mathbb C}^2, g_{EH})$ are not relatives to any homogeneous bounded domain.