Let $G/P$ be a rational homogeneous space (not necessarily irreducible) and $x_0\in G/P$ be the point at which the isotropy group is $P$. The $G$-translates of the orbit $Qx_0$ of a parabolic subgroup $Q\subsetneq G$ such that $P\cap Q$ is parabolic are called $Q$-cycles. We established an extension theorem for local biholomorphisms on $G/P$ that map local pieces of $Q$-cycles into $Q$-cycles. We showed that such maps extend to global biholomorphisms of $G/P$ if $G/P$ is $Q$-cycle-connected, or equivalently, if there does not exist a non-trivial parabolic subgroup containing $P$ and $Q$. Then we applied this to the study of local biholomorphisms preserving the real group orbits on $G/P$ and showed that such a map extend to a global biholomorphism if the real group orbit admits a non-trivial holomorphic cover by the $Q$-cycles. The non-closed boundary orbits of a bounded symmetric domain embedded in its compact dual are examples of such real group orbits. Finally, using the results of Mok-Zhang on Schubert rigidity, we also established a Cartan-Fubini type extension theorem pertaining to $Q$-cycles, saying that if a local biholomorphism preserves the variety of tangent spaces of $Q$-cycles, then it extends to a global biholomorphism when the $Q$-cycles are positive dimensional and $G/P$ is of Picard number 1. This generalizes a well-known theorem of Hwang-Mok on minimal rational curves.

中文翻译：

---

有理齐次空间上关于齐次子空间的局部全纯映射

令 $G/P$ 是一个有理齐次空间（不一定是不可约的），$x_0\in G/P$ 是各向同性群是 $P$ 的点。抛物线子群$Q\subsetneq G$ 的轨道$Qx_0$ 的$G$-平移使得$P\cap Q$ 是抛物线的称为$Q$-循环。我们在 $G/P$ 上建立了局部双全同态的扩展定理，将 $Q$-cycles 的局部部分映射到 $Q$-cycles。我们表明，如果 $G/P$ 是 $Q$-cycle-connected，或者等效地，如果不存在包含 $P$ 和 $ 的非平凡抛物线子群，则此类映射扩展到 $G/P$ 的全局双全同态Q$。然后我们将其应用到在 $G/P$ 上保留实群轨道的局部双全同胚的研究，并表明如果实群轨道允许 $Q$- 的非平凡全纯覆盖，则这样的映射扩展到全局双全同胚循环。嵌入其紧对偶中的有界对称域的非闭合边界轨道就是这种实群轨道的例子。最后，利用 Mok-Zhang 在 Schubert 刚性上的结果，我们还建立了一个关于 $Q$-圈的 Cartan-Fubini 型扩展定理，说如果一个局部双全同胚保留了 $Q$-圈的切空间的多样性，然后它扩展到全局双全同态，当 $Q$-cycles 是正维并且 $G/P$ 是 Picard 数 1 时。这概括了著名的 Hwang-Mok 定理关于最小有理曲线。