ALGEBRAIC REALIZATION FOR CYCLIC GROUP
ACTIONS WITH ONE ISOTROPY TYPE
KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN
Abstract.
Suppose
G
is a cyclic group and
M
a closed smooth
G
–
manifold with exactly one isotropy type. We will show that there is a
nonsingular real algebraic
G
–variety
X
which is equivariantly diffeomor
phic to
M
and all
G
–vector bundles over
X
are strongly algebraic.
1.
Introduction
A vector bundle over a variety is said to be strongly algebraic if it is
classified, up to homotopy, by an entire rational map to a Grassmannian
with its canonical algebraic structure. We will catch up with definitions and
background material in Section 3.
Theorem 1.1.
Suppose
G
is a cyclic group and
M
a closed smooth
G
–
manifold with exactly one isotropy type.
Then there is a nonsingular real
algebraic
G
–variety
X
which is equivariantly diffeomorphic to
M
and all
G
–vector bundles over
X
are strongly algebraic.
This theorem is a key ingredient in the proof of
Theorem 1.2.
[15]
Suppose
G
is a cyclic group and
M
a closed smooth
G
–manifold. Then there is a nonsingular real algebraic
G
–variety
X
which
is equivariantly diffeomorphic to
M
and all
G
–vector bundles over
X
are
strongly algebraic.
Our work is motivated by the Nash conjecture [23], which was shown to
hold in [29], [1], and [2].
Early results on algebraic realization are due to
Seifert [24]. Adding a smooth action and vector bundle theory as structure
on the manifold, one has the more general
Conjecture 1.3.
(see [12] and [15])
Let
G
be a compact Lie group and
M
a closed smooth
G
manifold. Then there is a nonsingular real algebraic
G
–variety
X
which is equivariantly diffeomorphic to
M
and all
G
–vector
bundles over
X
are strongly algebraic.
Previously, the conjecture has been proven in the following special cases:
(1)
G
is the trivial group, see [5].
Date
: May 19, 2009.
1991
Mathematics Subject Classification.
Primary 14P25, 57S15; Secondary 57S25.
Key words and phrases.
Algebraic Models, Equivariant Bordism.
1
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KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN
(2)
G
is the product of an odd order group and a 2–torus, see [12].
(3) The action of
G
on
M
is semifree, see [12].
(4) All points of
M
have orbit type
H
, and
NH/H
is of odd order,
see [28].
J. Hanson showed in [17] that for a closed smooth
Z
4
–manifold and one
Z
4
–bundle
ξ
over
M
one can find a nonsingular real algebraic
Z
4
–variety
X
, so that the diffeomorphism pulls
ξ
back to a strongly algebraic bundle
over
X
.
Let us sketch the bigger picture.
Tognol’s important breakthrough in
the discussion of the algebraic realization problem was its reduction to a
bordism problem. The following metatheorem is well supported:
General Principle.
[19, p. 154] If a topological situation is cobordant to
an algebraic situation, then it is isomorphic to an algebraic situation (if you
have the right notion of bordism and do a lot of work).
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 Spring '09
 Heiner
 Algebra, fb, Algebraic geometry, EC ×C Fb, KARL HEINZ DOVERMANN, ARTHUR G. WASSERMAN

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