2002/04/29 by Gregor Fels, Fels, Gregor, Alan Huckleberry +1
Mathematics · #32M06 14C25 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:14C25 #msc:32M06
paper · pdf · doi:10.48550/arxiv.math/0204341
26 pages
arxiv created 2002/04/29 · openalex publication_date 2002/04/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A real form G of a complex semisimple Lie group GC has only finitely many orbits in any given GC-flag manifold Z=GC/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 ΩW(D) in GC/KC, 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 ΩW(D). The main result here is that, with the few exceptions mentioned above, for a fixed real form G all of the cycle spaces ΩW(D) are the same. They are equal to a universal domain ΩAG 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 Ω is a G-invariant Stein domain which contains ΩAG and which is Kobayashi hyperbolic, then Ω=ΩAG. The equality of the cycle domains follows from the fact that every ΩW(D) is itself Stein, is hyperbolic, and contains ΩAG.