Universal embeddings of flag manifolds and rigidity phenomena
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916873662103552 |
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| author | Loi, Andrea Mossa, Roberto Zuddas, Fabio |
| author_facet | Loi, Andrea Mossa, Roberto Zuddas, Fabio |
| contents | 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 $C \times Ω$, where $C$ is a flag manifold and $Ω$ is a homogeneous bounded domain. Secondly, we show that no \emph{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 \emph{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 $C\timesΩ$; $C$ is not weak relative to $E\timesΩ$; $Ω$ 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23606 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal embeddings of flag manifolds and rigidity phenomena Loi, Andrea Mossa, Roberto Zuddas, Fabio Differential Geometry 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 $C \times Ω$, where $C$ is a flag manifold and $Ω$ is a homogeneous bounded domain. Secondly, we show that no \emph{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 \emph{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 $C\timesΩ$; $C$ is not weak relative to $E\timesΩ$; $Ω$ 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]. |
| title | Universal embeddings of flag manifolds and rigidity phenomena |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.23606 |