Rigidity properties of holomorphic isometries into homogeneous K\"{a}hler manifolds

TL;DR

This paper establishes rigidity results for holomorphic isometries into homogeneous Kähler manifolds, showing certain induced metrics are trivial and clarifying the non-existence of specific relative configurations among these manifolds.

Contribution

It proves new rigidity theorems for holomorphic isometries involving homogeneous Kähler manifolds, extending previous results in the field.

Findings

Kähler-Ricci soliton induced by the homogeneous metric is trivial (Kähler-Einstein).

Certain manifold pairs are not relative in the Kähler product setting.

The results extend previous rigidity theorems in the literature.

Abstract

We prove two rigidity results on holomorphic isometries into homogeneous K\"{a}hler manifolds. The first shows that a K\"{a}hler-Ricci soliton induced by the homogeneous metric of the K\"{a}hler product of a special flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. K\"{a}hler-Einstein. In the second one we prove that: (i) a flat space is not relative to the K\"{a}hler product of a special flag manifold with a homogeneous bounded domain, (ii) a special flag manifold is not relative to the K\"{a}hler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the K\"{a}hler product of a flat space with a special flag manifold. Our theorems strongly extend the results in [4], [5], [12], [13] and [22].