Building on the geometric theory of uniruled projective manifolds by Hwang-Mok, which relies on the study of varieties of minimal rational tangents (VMRTs) from both the algebro-geometric and the differential-geometric perspectives, Mok, Hong-Mok and Hong-Park have studied standard embeddings between rational homogeneous spaces X = G/P of Picard number 1. Denoting by S ⊂ X an arbitrary germ of complex submanifold which inherits from X a geometric structure defined by taking intersections of VMRTs with tangent subspaces and modeled on some rational homogeneous space X0 = G0/P0 of Picard number 1 embedded in X = G/P as a linear section through a standard embedding, we say that (X0, X) is rigid if there always exists some γ ∈ Aut(X) such that S is an open subset of γ(X0). We prove that a pair (X0, X) of sub-diagram type is rigid whenever X0 is nonlinear, which in the Hermitian symmetric case recovers Schubert rigidity for nonlinear smooth Schubert cycles, and which in the general rational homogeneous case goes beyond earlier works dealing with images of holomorphic maps. Our methods apply to uniruled projective manifolds (X,K), for which we introduce a general notion of sub-VMRT structures π : C (S) → S, proving that they are rationally saturated under an auxiliary condition on the intersection C (S) := C (X) ∩ PT (S) and a nondegeneracy condition for substructures expressed in terms of second fundamental forms on VMRTs. Under the additional hypothesis that minimal rational curves are of degree 1 and that distributions spanned by sub-VMRTs are bracket generating, we prove that S extends to a subvariety Z ⊂ X. For its proof, starting with a “Thickening Lemma ” which yields smooth collars around certain standard rational curves, we show that the germ of submanifold (S;x0) and hence the associated germ of sub-VMRT structure on (S;x0) can be propagated along chains of “thickening ” curves issuing from x0, and construct by analytic continuation a projective family of chains of rational curves compactifying the latter family, thereby constructing the projective completion Z of S as its image under

中文翻译：

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Picard 数 $1$ 的有理齐次空间对的刚性和几何子结构在无规则射影流形上的解析延拓

基于 Hwang-Mok 的无规则射影流形的几何理论，该理论依赖于从代数几何和微分几何角度研究的各种极小有理切线 (VMRT)，Mok、Hong-Mok 和 Hong-Park已经研究了有理齐次空间 X = G/P 的 Picard 数 1 之间的标准嵌入。 用 S ⊂ X 表示复杂子流形的任意胚芽，它从 X 继承了一个几何结构，该几何结构通过取 VMRT 与切线子空间的交集而定义，并以一些有理数为模型齐次空间 X0 = G0/P0 的 Picard 数 1 嵌入 X = G/P 作为通过标准嵌入的线性部分，我们说 (X0, X) 是刚性的，如果总是存在一些 γ ∈ Aut(X) 使得S 是 γ(X0) 的开子集。我们证明一对 (X0, 当 X0 是非线性时，子图类型的 X) 是刚性的，在 Hermitian 对称情况下恢复非线性平滑舒伯特循环的舒伯特刚性，并且在一般有理齐次情况下超出了早期处理全纯映射图像的工作。我们的方法适用于无规则射影流形 (X,K)，为此我们引入了子 VMRT 结构 π : C (S) → S 的一般概念，证明它们在交点 C (S) 上的辅助条件下是合理饱和的) := C (X) ∩ PT (S) 和以 VMRT 上的第二基本形式表示的子结构的非简并条件。在最小有理曲线的阶数为 1 并且子 VMRT 跨越的分布是括号生成的附加假设下，我们证明 S 扩展到子变量 Z ⊂ X。为了证明，