Hyperbolicity of cycle spaces and automorphism groups of flag domains

TL;DR

This paper investigates the complex analytic properties of flag domains associated with real forms of complex semisimple Lie groups, proving hyperbolicity of cycle spaces and characterizing their automorphism groups.

Contribution

It establishes the Kobayashi hyperbolicity of certain cycle space components and provides an explicit description of the automorphism groups of flag domains.

Findings

Cycle space components C_q(D) are Kobayashi hyperbolic.

Automorphism groups of flag domains are mostly Lie groups, often G_0.

Except for holomorphically convex cases, automorphisms are essentially G_0.

Abstract

If G_0 is a real form of a complex semisimple Lie group G and Z is compact G-homogeneous projective algebraic manifold, then G_0 has only finitely many orbits on Z. Complex analytic properties of open G_0-orbits D (flag domains) are studied. Schubert incidence-geometry is used to prove the Kobayashi hyperbolicity of certain cycle space components C_q(D). Using the hyperbolicity of C_q(D) and analyzing the action of Aut(D) on it, an exact description of Aut(D) is given. It is shown that, except in the easily understood case where D is holomorphically convex with a nontrivial Remmert reduction, it is a Lie group acting smoothly as a group of holomorphic transformations on D. With very few exceptions it is just G_0.