Local holomorphic mappings respecting homogeneous subspaces on rational homogeneous spaces

• Published in: Mathematische annalen 2021-06, Vol.380 (1-2), p.885-909
• Main Authors: Hong, Jaehyun, Ng, Sui-Chung
• Format: Article
• Language: English
• Subjects: Isotropy; Mathematical analysis; Mathematics; Mathematics and Statistics; Orbits; Subgroups; Subspaces; Theorems
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-020-02013-5

Let G / P be a rational homogeneous space (not necessarily irreducible) and x 0 ∈ G / P be the point at which the isotropy group is P . The G -translates of the orbit Q x 0 of a parabolic subgroup Q ⊊ G such that P ∩ 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.