Characterization of cycle domains via Kobayashi hyperbolicity
摘要
A real form G of a complex semi-simple Lie group G C has only finitely many orbits in any given G C -flag manifold Z = G C /Q. The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits D generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of D which, with very few well-understood exceptions, are parameterized by the Wolf cycle domains W (D) in G C /K C , where K is a maximal compact subgroup of G. Thus, for the various domains D in the various ambient spaces Z, it is possible to compare the cycle spaces W (D). The main result here is that, with the few exceptions mentioned above, for a fixed real form G all of the cycle spaces W (D) are the same. They are equal to a universal domain AG which is natural from the the point of view of group actions and which, in essence, can be explicitly computed. The essential technical result is that if b is a G-invariant Stein domain which contains AG and which is Kobayashi hyperbolic, then b = AG . The equality of the cycle domains follows from the fact that every W (D) is itself Stein, is hyperbolic, and contains AG .