Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels; Alan Huckleberry

Bulletin de la Société Mathématique de France (2005)

  • Volume: 133, Issue: 1, page 121-144
  • ISSN: 0037-9484

Abstract

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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 Ω 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 Ω 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 Ω W ( D ) are the same. They are equal to a universal domain Ω A G 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 Ω A G and which is Kobayashi hyperbolic, then Ω ^ = Ω A G . The equality of the cycle domains follows from the fact that every Ω W ( D ) is itself Stein, is hyperbolic, and contains Ω A G .

How to cite

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Fels, Gregor, and Huckleberry, Alan. "Characterization of cycle domains via Kobayashi hyperbolicity." Bulletin de la Société Mathématique de France 133.1 (2005): 121-144. <http://eudml.org/doc/272479>.

@article{Fels2005,
abstract = {A real form $G$ of a complex semi-simple Lie group $G^\mathbb \{C\}$ has only finitely many orbits in any given $G^\mathbb \{C\}$-flag manifold $Z=G^\mathbb \{C\}/Q$. 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 $G^\mathbb \{C\}/K^\mathbb \{C\}$, 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 $\widehat\{\Omega \}$ is a $G$-invariant Stein domain which contains $\Omega _\{AG\}$ and which is Kobayashi hyperbolic, then $\widehat\{\Omega \}=\Omega _\{AG\}$. The equality of the cycle domains follows from the fact that every $\Omega _W(D)$ is itself Stein, is hyperbolic, and contains $\Omega _\{AG\}$.},
author = {Fels, Gregor, Huckleberry, Alan},
journal = {Bulletin de la Société Mathématique de France},
keywords = {complex geometry; cycles spaces; Lie groups; Schubert varieties},
language = {eng},
number = {1},
pages = {121-144},
publisher = {Société mathématique de France},
title = {Characterization of cycle domains via Kobayashi hyperbolicity},
url = {http://eudml.org/doc/272479},
volume = {133},
year = {2005},
}

TY - JOUR
AU - Fels, Gregor
AU - Huckleberry, Alan
TI - Characterization of cycle domains via Kobayashi hyperbolicity
JO - Bulletin de la Société Mathématique de France
PY - 2005
PB - Société mathématique de France
VL - 133
IS - 1
SP - 121
EP - 144
AB - A real form $G$ of a complex semi-simple Lie group $G^\mathbb {C}$ has only finitely many orbits in any given $G^\mathbb {C}$-flag manifold $Z=G^\mathbb {C}/Q$. 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 $G^\mathbb {C}/K^\mathbb {C}$, 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 $\widehat{\Omega }$ is a $G$-invariant Stein domain which contains $\Omega _{AG}$ and which is Kobayashi hyperbolic, then $\widehat{\Omega }=\Omega _{AG}$. The equality of the cycle domains follows from the fact that every $\Omega _W(D)$ is itself Stein, is hyperbolic, and contains $\Omega _{AG}$.
LA - eng
KW - complex geometry; cycles spaces; Lie groups; Schubert varieties
UR - http://eudml.org/doc/272479
ER -

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