Universal embeddings of flag manifolds and rigidity phenomena

TL;DR

This paper proves a universal holomorphic isometric embedding theorem for flag manifolds, leading to new rigidity results that prevent certain types of shared submanifolds among homogeneous Kähler manifolds, unifying and extending previous rigidity theorems.

Contribution

It introduces a universal embedding theorem for flag manifolds and establishes new rigidity phenomena for holomorphic isometries, extending prior results to weak relatives without restrictions.

Findings

Every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold.

Certain homogeneous Kähler manifolds cannot be weak relatives of each other, including flat spaces, flag manifolds, and bounded domains.

The results unify and extend previous rigidity theorems in the literature.

Abstract

We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of Takeuchi [30] and Nakagawa-Takagi [27]. Using this embedding, we establish new rigidity phenomena for holomorphic isometries between homogeneous K\"ahler manifolds. As a first immediate consequence we show the triviality of a K\"ahler-Ricci soliton submanifod of $C \times \Omega$, where $C$ is a flag manifold and $\Omega$ is a homogeneous bounded domain. Secondly, we show that no \emph{weak-relative} relationship can occur among the fundamental classes of homogeneous K\"ahler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two K\"ahler manifolds are said to be \emph{weak relatives} if they share, up to local isometry, a common K\"ahler submanifold of complex dimension at least two. Our main result precisely shows that if $E$ is (possibly indefinite) flat, $C$ is a flag manifold, and $\Omega$ is a homogeneous bounded domain, then: $E$ is not weak relative to $C\times\Omega$; $C$ is not weak relative to $E\times\Omega$; $\Omega$ is not weak relative to $E\times C$. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from \emph{relatives} to the more flexible notion of \emph{weak relatives} and dispense with the earlier ''special'' restriction on the flag-manifold factor. This result also unifies previous rigidity results from the literature, e.g., [5, 6, 7, 9, 12, 13, 32].