UNIQUENESS QUESTIONS IN REAL ALGEBRAIC
TRANSFORMATION GROUPS
KARL HEINZ DOVERMANN AND MIKIYA MASUDA
Abstract.
Let
G
be a compact Lie group and
H
a closed subgroup.
We show that the homogeneous space
G/H
has a unique structure as
a real algebraic
G
variety. For a real algebraic
H
variety we show that
the balanced product
G
×
H
X
has the structure of a real algebraic
G

variety, and this structure is uniquely determined by the structure on
X
. On the other hand, suppose that
M
is a closed smooth
G
manifold
with positive dimensional orbit space
M/G
. If
M
has a
G
equivariant
real algebraic model, then it has an uncountable family of birationally
inequivalent such models.
1.
Introduction
Suppose
G
is a compact Lie group.
We like to address the question to
which extent an equivariant real algebraic structure
1
on a compact smooth
G
manifold is unique. There are answers at both extremes. First we consider
a situation in which the structure is unique. The following classical result
serves as a starting point.
Theorem 1.1.
A compact Lie group has the structure of a real linear alge
braic group, and this structure is unique.
The first part of the theorem appears to have been invisioned as a final
conclusion in Chevalley’s book, see [9, p. viii and Chapter VI]. As stated,
the result can be found in [21, p. 246]. Using classical techniques from [9]
and [17] combined with the idea of algebraic quotients from [25], we show
(see Corollary 4.4):
Theorem 1.2.
Let
G
be a compact Lie group and
H
a closed subgroup. The
homogeneous space
G/H
has the structure of a nonsingular real algebraic
G
variety, and this structure is unique. If
K
is another closed subgroup of
G
and
η
:
G/H
→
G/K
is an equivariant map, then
η
is a regular map.
Date
: March 28, 2006.
1991
Mathematics Subject Classification.
Primary 14P25, 57S15; Secondary 57S25.
Key words and phrases.
Real Algebraic Geometry, Quotients, Induction.
1
Precise definitions of some of the terms used in this introduction are given in Section 2.
In particular, real algebraic structures and varieties in this introduction are assumed to
be affine.
1
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2
KARL HEINZ DOVERMANN AND MIKIYA MASUDA
In complex algebraic geometry and transformation groups, the study of
the structure of homogeneous spaces is a standard topic, see [5, Chapter II]
and [18, Chapter IV]. Our proof of Theorem 1.2 also provides a simple real
proof of Theorem 1.1, if one assumes the existence of a faithful representation
of
G
. It would be interesting to have a strictly real proof of the existence of
a faithful representation of a compact Lie group.
Using additional ideas from algebraic geometry and another result of
Schwarz [26], one can show a stronger result.
Theorem 1.3.
Suppose
G
is a compact Lie group,
H
a closed subgroup,
and
X
a real algebraic
H
variety.
Then
G
×
H
X
has the structure of a
real algebraic
G
variety, and this structure is uniquely determined by the
structure of
X
as a real algebraic
H
variety. If
X
is nonsingular, then so
is
G
×
H
X
.
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 Spring '09
 Heiner
 Algebra, Algebraic geometry, Lie group, compact Lie group, Algebraic Quotients, real algebraic Gvariety

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