| ABSTRACT |
Let G/P be a rational homogeneous space (not necessarily irreducible) and x0 is an element of 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 subset of such that P boolean AND 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-cyclesen it exten, saying that if a local biholomorphism preserves the variety of tangent spaces of Q-cycles, thds 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. |
