# Local holomorphic mappings respecting homogeneous subspaces on rational homogeneous spaces

## TL;DR

This paper proves extension theorems for local holomorphic maps on rational homogeneous spaces that respect certain subspace structures, leading to conditions under which these maps extend globally, with applications to symmetric domains and minimal rational curves.

## Contribution

It introduces new extension criteria for local biholomorphisms respecting $Q$-cycles, generalizing known results and connecting to Schubert rigidity and minimal rational curves.

## Key findings

- Local biholomorphisms respecting $Q$-cycles extend globally under $Q$-cycle-connectedness.
- Maps preserving real group orbits extend if the orbit admits a holomorphic $Q$-cycle cover.
- Extension theorems generalize Hwang-Mok's results on minimal rational curves.

## Abstract

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.