31 research outputs found
Geometry of invariant domains in complex semi-simple Lie groups
We investigate the joint action of two real forms of a semi-simple complex
Lie group S by left and right multiplication. After analyzing the orbit
structure, we study the CR structure of closed orbits. The main results are an
explicit formula of the Levi form of closed orbits and the determination of the
Levi cone of generic orbits. Finally, we apply these results to prove
q-completeness of certain invariant domains in S.Comment: 20 page
Invariant meromorphic functions on Stein spaces
In this paper we develop fundamental tools and methods to study meromorphic
functions in an equivariant setup. As our main result we construct quotients of
Rosenlicht-type for Stein spaces acted upon holomorphically by
complex-reductive Lie groups and their algebraic subgroups. In particular, we
show that in this setup invariant meromorphic functions separate orbits in
general position. Applications to almost homogeneous spaces and principal orbit
types are given. Furthermore, we use the main result to investigate the
relation between holomorphic and meromorphic invariants for reductive group
actions. As one important step in our proof we obtain a weak equivariant
analogue of Narasimhan's embedding theorem for Stein spaces.Comment: 20 pages, 1 figur
Schottky groups acting on homogeneous rational manifolds
We systematically study Schottky group actions on homogeneous rational
manifolds and find two new families besides those given by Nori's well-known
construction. This yields new examples of non-K\"ahler compact complex
manifolds having free fundamental groups. We then investigate their analytic
and geometric invariants such as the Kodaira and algebraic dimension, the
Picard group and the deformation theory, thus extending results due to
L\'arusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky
construction allows to recover examples of equivariant compactifications of
SL(2,C)/\Gamma for \Gamma a discrete free loxodromic subgroup of SL(2,C),
previously obtained by A. Guillot.Comment: 30 pages; minor modifications, references have been added; to appear
in Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal
Homogeneous K\"ahler and Hamiltonian manifolds
We consider actions of reductive complex Lie groups on K\"ahler
manifolds such that the --action is Hamiltonian and prove then that the
closures of the --orbits are complex-analytic in . This is used to
characterize reductive homogeneous K\"ahler manifolds in terms of their
isotropy subgroups. Moreover we show that such manifolds admit --moment maps
if and only if their isotropy groups are algebraic.Comment: 12 pages. The statement of Theorem 3.5 has been improve
Momentum maps and the K\"ahler property for base spaces of reductive principal bundles
We investigate the complex geometry of total spaces of reductive principal
bundles over compact base spaces and establish a close relation between the
K\"ahler property of the base, momentum maps for the action of a maximal
compact subgroup on the total space, and the K\"ahler property of special
equivariant compactifications. We provide many examples illustrating that the
main result is optimal.Comment: 10 page
Hamiltonian actions of unipotent groups on compact K\"ahler manifolds
We study meromorphic actions of unipotent complex Lie groups on compact
K\"ahler manifolds using moment map techniques. We introduce natural stability
conditions and show that sets of semistable points are Zariski-open and admit
geometric quotients that carry compactifiable K\"ahler structures obtained by
symplectic reduction. The relation of our complex-analytic theory to the work
of Doran--Kirwan regarding the Geometric Invariant Theory of unipotent group
actions on projective varieties is discussed in detail.Comment: v2: 30 pages, final version as accepted by EPIG