Hyperbolicity of cycle spaces and automorphism groups of flag domains

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

Hans Grauert: Mathematician Pur

This article was written on the occasion of Hans Grauert receiving the Cantor Medallion of the Deutsche Mathematische Vereinigung. It is a brief overview of his mathematical contributions and attempts to convey the author's great respect for the man and his science

Karl Stein (1913-2000)

Karl Stein was one of the pillars of the German school of several complex variables. In this article his scientific contributions are outlined in historical perspective

Actions of groups of birationally extendible automorphisms

We study the actions of a Lie group $G$ by birationally extendible automorphisms on a domain $D\subset C^n$. For a large class of such domains defined by polynomial inequalities, all automorphisms are of this type. In the cases 1) $G$ has finitely many components or 2) the degree of the automorphisms is bounded, we prove that the action of $G$ is projectively linearizable, i.e. there exist a linear representation of $G$ on some $C^{N+1}$ and a holomorphic $G$-equivariant embedding $i: D\to P^N$, which is a restriction of a rational mapping. As a corollary we obtain as many rational invariant functions as the dimension of generic orbits allows. A hard copy is available from [email protected]: 30 pages, AmS-TeX- Version 2.1 (amstex.tex

Characterization of cycle domains via Kobayashi hyperbolicity

A real form $G$ of a complex semisimple 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 $\Omega_W(D)$ 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 $\Omega_W(D)$. The main result here is that, with the few exceptions mentioned above, for a fixed real form $G$ all of the cycle spaces $\Omega_W(D)$ are the same. They are equal to a universal domain $\Omega_{AG}$ 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 $\hat \Omega$ is a $G$-invariant Stein domain which contains $\Omega_{AG}$ and which is Kobayashi hyperbolic, then $\hat \Omega =\Omega_{AG}$. The equality of the cycle domains follows from the fact that every $\Omega_W(D)$ is itself Stein, is hyperbolic, and contains $\Omega_{AG}$.Comment: 26 page

On closures of cycle spaces of flag domains

Open orbits D of noncompact real forms G_0 acting on flag manifolds of their semisimple complexifications G are considered. The unique orbit C of a maximal compact subgroup K_0 of G_0 in D can be regarded as a point in the (full) cycle space of D. The group theoretical cycle space is defined to be the connected component containing C of the intersection of the G-orbit of C with the full cycle space of D. The main result of the present article is that the group theoretical cycle space is closed in the full cycle space. In particular, if they have the same dimension, then they are equal. This follows from an analysis of the closure of the Akhiezer-Gindikin domain in any G-equivariant compactification of the affine symmetric space G/K, where K is the complexification of K_0 in G.Comment: 13 page

Classical Symmetries of Complex Manifolds

We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be dimension-theoretically large with respect to the manifold on which it is acting, our classification result states that the manifolds which arise are described precisely as invariant open subsets of certain complex flag manifolds associated to the complexified groups

Infinite-dimensionality of the Automorphism Groups of Homogeneous Stein Manifolds

We show that the group of holomorphic automorphisms of a Stein manifold X of dimension greater than 1 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group

Hans Grauert (1930-2011)

Hans Grauert died in September of 2011. This article reviews his life in mathematics and recalls some detail his major accomplishments.Comment: Written for publication in 2013 in the Jahresber. Deutsch. Math.-Vereinigun

Injectivity of the Double Fibration Transform for Cycle Spaces of Flag Domains

The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D = G_0(z) \subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous holomorphic vector bundle $\mathbb E \to D$. The topic here is the double fibration transform ${\cal P}: H^q(D;{\cal O}(\mathbb E)) \to H^0({\cal M}_D;{\cal O}(\mathbb E'))$ where $q$ is given by the geometry of $D$, ${\cal M}_D$ is the cycle space of $D$, and $\mathbb E' \to {\cal M}_D$ is a certain naturally derived holomorphic vector bundle. Schubert intersection theory is used to show that ${\cal P}$ is injective whenever $\mathbb E$ is sufficiently negative