Universal embeddings of flag manifolds and rigidity phenomena

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 Kähler manifolds. As a first immediate consequence we show the triviality of a Kähler-Ricci soliton submanifod of C × Ω, where C is a flag manifold and $Ω$ is a homogeneous bounded domain. Secondly, we show that no weak-relative relationship can occur among the fundamental classes of homogeneous Kähler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two Kähler manifolds are said to be weak relatives if they share, up to local isometry, a common Kähler 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 $Ω$ is a homogeneous bounded domain, then: E is not weak relative to C×Ω; C is not weak relative to E×Ω; $Ω$ is not weak relative to E× C. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from relatives to the more flexible notion of 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].