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ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE
CHRISTOPH GELLHAUS
TILMANN WURZBACHER
Holomorphic group actions with few compact orbits
Annales de la faculté des sciences de Toulouse 6e série, tome 10,
no 2 (2001), p. 293-298
<http://www.numdam.org/item?id=AFST_2001_6_10_2_293_0>
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Holomorphic group actions
with few compact orbits (*)
CHRISTOPH GELLHAUS (1), TILMANN WURZBACHER (2) *
(email: wurzbacher@poncelet.univ-metz.fr,
fax: (33) 03.87.31.52.73)
Annales de la Faculté des Sciences de Toulouse Vol. X, n° 2, 2001
pp. 293-298
RESUME. - Pour une grande classe de varietes compactes complexes,
contenant les images meromorphes des espaces de Kahler compacts, nous
montrons le theoreme suivant: Soit G un groupe de Lie complexe agissant
holomorphiquement sur X tel qu’il y a qu’un nombre fini positif d’orbites
compactes. Alors X est un fibre G x I F, ou G/I est le tore d’Albanese de
X et F est une fibre de l’application d’Albanese. De plus, F est connexe
et son premier nombre de Betti est nul.
ABSTRACT. - For a large class of compact complex manifolds, including
for example meromorphic images of compact Kahler spaces, we prove the
following theorem: Let G be a complex Lie group acting holomorphically
on the manifold X and suppose there is at least one, but only finitely
many compact orbits. Then X is a fibre bundle G x I F, where G/I is the
Albanese torus of X and F is a fibre of the Albanese map. Furthermore
F is connected and has vanishing first Betti number.
1. Introduction
Holomorphic group actions on a compact complex manifold X with an
open set of compact orbits are studied in [GeWu]. There it is proved that X
consists only of compact orbits and is itself of a product structure, reflecting
the single orbit decomposition of Borel and Remmert, provided there are no
(*) > Reçu le 3 novembre 2000, accepte le 9 juillet 2001
(1) Technische Fachhochschule zu Bochum, Fachbereich Elektrotechnik, Herner Str. 45,
D-44787 Bochum, Germany.
(2) Laboratoire de Mathematiques, Université de Metz et C.N.R.S., Ile du Saulcy, F-
57045 Metz cedex 01, France.
* Corresponding author
equivariantly embedded complex tori in the fibres of the Albanese mapping.
This condition especially holds true if X is Kahler.
The authors would like to mention that they learned that this result
was, in the Kahler case, already proved by D. Snow in 1981 ([Sn]). He
used methods, which - in contrast to those of [GeWu] - depend on the fact
that the connected automorphism group of a compact Kahler space acts
compactifiably (cf. [F], [L]). In the algebraic case parts of this result were
obtained by J. Konarski in 1981 ([K]).
Furthermore, Snow showed that the set C of compact orbits of a holo-
morphic action is analytic, if X is Kahler. In this note we consider the case
that C is extremely small, i.e. there are only finitely many compact orbits.
For manifolds in Fujikis class C (see [F]), which contains for example Kahler
and Moishezon manifolds, the following observation can be made:
THEOREM. - Let X be in C, G x X 2014~ X a holomorphic action of a
connected complex Lie group such that there is at least one, but only finitely
many compact orbits. Then the following holds:
X is a fibre bundle G x IF, where G/I is the Albanese torus of X and
F is a fibre of the Albanese map.
Furthermore F is connected and has vanishing first Betti number.
Remark. Looking at a Cn-action as a multidimensional generalization
of a holomorphic dynamical system, the case that I is discrete, corresponds
to the "suspension" of the discrete dynamical system I on F in a continuous
system, the Cn-action on X.
2. Proofs
Let G always denote a connected complex Lie group. If G x X --~ X is
a holomorphic action on a complex space, we refer to X as a holomorphic
G-space. In this situation CG will be the set of compact G-orbits in X. .
For smooth X one has the Albanese map W x : : X --~ Alb X (see [Bl]).
The equivariance of 03C8X implies that there is an induced Lie group homo-
morphism Ax : Auto(X) -7 Auto (Alb X). . The kernel of A x restricted to
the connected component of the identity in Auto(X) is called L(X), the
"linear" automorphisms of X.
Using the compactness of the irreducible components of the space of
analytic cycles in complex spaces in class C resp. Kahler spaces, Fujiki re-
spectively Lieberman prove that L(X) carries the natural structure of a
linear algebraic group (Cor. 5.8 in [F], Thm. 3.12 in [L]). For a subgroup H
of L(X ), H will denote the Zariski closure with respect to this structure.
Consider now a smooth holomorphic G-space X. We may assume that
G acts without ineffectivity. Let I = G n L(X) and I the closure of I.
Furthermore fix a Levi-Malcev decomposition I ° == of I ° .
LEMMA 1. - Let X be a smooth holomorphic G-space in C and x in X .
Assume CG is not empty. Then the following statements are equivalent:
(1) G(x) is compact.
(2) I(x) is compact.
(3) I (x) is compact.
(4) RIo fixes x and ,S’Io (x) is compact.
Proof. - Without loss of generality, we assume throughout this proof
that G acts effectively on X.
(1) ~ (2) Since AutO(Alb X)° = Alb X the isotropy groups of the
induced G-action on Alb X is equal to I for all points in AlbX. Since CG is
not empty, this implies that all G-orbits on Alb X are compact. Thus, G(x)
is compact iff I(x) is compact.
(2) =~ (3) The group of all g in L(X) which stabilize the compact
analytic set is a Zariski closed subgroup of L(X) (Lemma 2.4 in ~F~ ,
Prop. 3.4 in [L]). Hence I stabilizes I {x) and consequently I (x) = I {x) .
(3) ~ (4) Since the radical RIo is a connected solvable subgroup of
L(X) the Borel Fixed Point Theorem for class C (cf. the proof of Prop. 6.9
in [F] resp. [So] in the Kahler case) shows that it has a fixed point on
each component of I (x). Since Rlo is normal in I, it acts trivially on I (x).
Therefore SIo acts transitively on the I-components.
(4) ~ (2) Since G/I = is a subtorus of Alb X, , I /I ° is an abelian
discrete group. Thus the commutator group I’ C I ° and I" c (I° )’. By a
result of Chevalley (Thm. 13 and Thm. 15 of paragraph 14, Chapter II in
[Ch]) (I ° )’ - (I ° )’ and this group is Zariski closed in I.
It follows that the Zariski closure of I" is contained in I. Since I is
Zariski dense in I, the same holds true for I" in ( I )". Therefore ( I )" C I
and a fortiori SI° C 7°. This fact shows that I ° (x) equals ,S’Io (x) . Since I
has only finitely many connected components, the assumptions imply that
I (x) is compact. Obviously the same holds now for I (x) .
The proof of ((2) =~ (3)) shows that indeed I(x) = I (x). C~
COROLLARY 1 (Snow). CG is analytic.
Proof. If the set CG of compact G-orbits is not empty Lemma 1 shows
CG = Cslo n(Fix RIo), Since a fixed point set is obviously analytic it suffices
to consider connected semisimple groups S.
A compact S-orbit in a complex space X in class C is a homogeneous-
rational manifold S/P (Thm. on p. 255 in [F] respectively [BoRe] in the
Kahler case). Thus an arbitrary, but fixed Borel subgroup B of S has non-
empty fixed point set on each compact orbit.
It follows that the map
has image Cs . Factorizing § via
we realize Cs as the image of an analytic space under a proper holomorphic
map. The proper mapping theorem of Remmert (see e.g. [CAS]) yields the
analyticity of Cs . D
COROLLARY 2. - If CG is not empty and A a closed G-invariant ana-
lytic set, then CG intersects A in a non-empty set.
Proof. By Lemma 1 it is enough to show that I, which stabilizes A,
has a compact orbit on A. This follows from the fact that for all x in X,
I(x) is a constructible set with respect to the analytic Zariski topology of X
since I acts meromorphically/compactifiably (Lemma 2.4 in [F], Remark 3.7
in [L]). D
Remark. Lieberman states that G ~ A is Zariski open in its Zariski
closure if G acts compactifiably and A is analytic. Obviously this is wrong
even in the algebraic case. What is meant is that G A contains a Zariski
open subset of its (analytic) Zariski closure in X.
Proof of the Theorem. Without loss of generality we assume that G
acts effectively on X.
Step 1. Fibre bundle structure of the Albanese map.
By assumption CG is not empty. Thus the induced G-action on Alb X
has only compact orbits, all of them isomorphic to G /I, where I = GnL(x). .
Applying Cor. 2 to the analytic sets
it follows that G has at least as many compact orbits in X as in ~~(~). .
Since is connected the assumptions of the theorem imply that
is only one G-orbit, which is a subtorus of Alb X. . Universality of
the Albanese torus implies that W x is surjective.
Denoting by F it is easily checked that the map
is G-equivariant and biholomorphic.
Step 2. The topology of the wx-fibre F.
Stein factorization of ~~~ together with the universality of W x yields the
connectivity of the W x-fibres.
Since I stabilizes F the same follows for the Zariski closure I in L(X). .
By Lemma 1 an I-orbit in F is closed iff the resp. I-orbit is closed iff the
resp. G-orbit is closed. Thus we have only finitely many compact I-orbits
in F. Since I/ Io is finite the same holds true for 1°. Applying Step 1 to the
action I ° x F ~ F, Alb F turns out to be I°-homogeneous.
In [GeWu] it is shown that the Borel Fixed Point Theorem implies that
the restriction morphism
maps L(X)O into L(F) (for F the Albanese fibre as above).
Thus Alb F is homogeneous under a subgroup of L(F), which clearly
says that Alb F reduces to a point. By the equality 1 2b1(Y) = dime (Alb Y)
for Y in class C (Cor. 1.7 in [F]) the first Betti number of F must vanish.
D
Acknowledgements. We would like to thank J. Winkelmann for inter-
esting discussions.
- 298 -
Bibliography
[Bl] BLANCHARD (A.). - Sur les variétés analytiques complexes, Ann. Sci. école
norm. super. 73 (1956), 157-202.
[BoRe] BOREL (A.), REMMERT (R.). 2014 Über kompakte homogene Kählersche Mannig-
faltigkeiten, Math. Ann. 145 (1962), 429-439.
[CAS] GRAUERT (H.), REMMERT (R.). - Coherent Analytic Sheaves, Springer, Berlin
Heidelberg New York Tokyo, 1984.
[Ch] CHEVALLEY (C.). 2014 Théorie des groupes de Lie II. Groupes algébriques, Her-
mann, Paris, 1968.
[F] FUJIKI (A.). - On automorphism groups of compact Kähler manifolds, Invent.
math. 44 (1978), 225-258.
[GeWu] GELLHAUS (C.), WURZBACHER (T.). - Holomorphic group actions with many
compact orbits, Comment. Math. Helvetici 64 (1989), 639-649.
[K] KONARSKI (J.). 2014 Properties of projective orbits of affine algebraic groups, Lect.
Notes Math., Vol. 956, 79-91, Springer, 1982.
[L] LIEBERMAN (D.). - Compactness of the Chow Scheme: Applications to auto-
morphisms and deformations of Kähler manifolds, Lect. Notes Math., Vol. 670,
140-186, Springer, 1978.
[Sn] SNOW (D.). - Transformation groups of compact Kähler spaces, Archiv der
Math. 37 (1981), 364-371.
[So] SOMMESE (A.J.). - Extensions Theorems for reductive group actions on com-
pact Kähler manifolds, Math. Ann. 218 (1975), 107-116.


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