9,909 research outputs found
On the automorphism groups of algebraic bounded domains
Let be a bounded domain in . By the theorem of H.~Cartan, the group
of all biholomorphic automorphisms of has a unique structure of a
real Lie group such that the action is real analytic.
This structure is defined by the embedding , , where is arbitrary. Here
we restrict our attention to the class of domains defined by finitely many
polynomial inequalities. The appropriate category for studying automorphism of
such domains is the Nash category. Therefore we consider the subgroup
of all algebraic biholomorphic automorphisms which in
many cases coincides with . Assume that and has a boundary
point where the Levi form is non-degenerate. Our main result is theat the group
carries a unique structure of an affine Nash group such that the
action is Nash. This structure is defined by the
embedding and is
independent of the choice of .Comment: 29 pages, LaTeX, Mathematischen Annalen, to appea
Semicontinuity of the Automorphism Groups of Domains with Rough Boundary
Based on some ideas of Greene and Krantz, we study the semicontinuity of
automorphism groups of domains in one and several complex variables. We show
that semicontinuity fails for domains in \CC^n, , with Lipschitz
boundary, but it holds for domains in \CC^1 with Lipschitz boundary. Using
the same ideas, we develop some other concepts related to mappings of Lipschitz
domains. These include Bergman curvature, stability properties for the Bergman
kernel, and also some ideas about equivariant embeddings
Smooth representations and sheaves
The paper is concerned with `geometrization' of smooth (i.e. with open
stabilizers) representations of the automorphism group of universal domains,
and with the properties of `geometric' representations of such groups. As an
application, we calculate the cohomology groups of several classes of smooth
representations of the automorphism group of an algebraically closed extension
of infinite transcendence degree of an algebraically closed field.Comment: 20 pages, final versio
Modular realizations of hyperbolic Weyl groups
We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group
Modular realizations of hyperbolic Weyl groups
We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group
Towards a Classification of Homogeneous Tube Domains in C^4
We classify the tube domains in C^4 with affinely homogeneous base whose
boundary contains a non-degenerate affinely homogeneous hypersurface. It
follows that these domains are holomorphically homogeneous and amongst them
there are four new examples of unbounded homogeneous domains (that do not have
bounded realisations). These domains lie to either side of a pair of
Levi-indefinite hypersurface. Using the geometry of these two hypersurfaces, we
find the automorphism groups of the domains.Comment: 16 page
The Wong-Rosay type theorem for K\"ahler manifolds
The Wong-Rosay theorem characterizes the strongly pseudoconvex domains of
by their automorphism groups. It has a lot of generalizations to
other kinds of domains (for example, the weakly pseudoconvex domains). However,
most of them are for domains of . In this note, we generalize the
Wong-Rosay theorem to the simply-connected complete K\"{a}hler manifold with a
negative sectional curvature. One aim of this note is to exhibit a Wong-Rosay
type theorem of manifolds with holomorphic non-invariant metrics
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