Homogeneous almost complex manifolds and their compact quotients

0.13 MB·

by  Kang-Tae Kim

HOMOGENEOUS ALMOST COMPLEX MANIFOLDS AND
THEIR COMPACT QUOTIENTS
7
1 KANG-TAEKIM,KANG-HYURKLEEANDYOSHIKAZUNAGATA
0
2

n
a
J Abstract. Thispaperinvestigatesthe(non)existenceofcompactquotientsof
thehomogeneousalmost-complexstrongly-pseudoconvexmanifoldsdiscovered
9
andclassifiedbyGaussier-Sukhov[1,2]andK.-H.Lee[3,4].

]
V
C 1. Introduction
h. Let M be an almost complex manifold of real dimension 2m, m 1, with
≥
t an almost complex structure J. It is said to be modeled after bounded strictly
a
m pseudoconvex domains, if the following two properties hold:
[ (1) M is J-holomorphicallyequivalentto a subdomain ofanother almostcom-
plex manifold with its Levi form (cf. Section 2) at every boundary point
1
positive-definite, and
v
(2) M is Kobayashihyperbolic.
3
5 It was generally believed for some time that, if one further assumes for it to be
0
homogeneous,meaningthattheactionbythegroupAut(M,J)ofallJ-holomorphic
2
diffeomorphisms of M into itself is transitive, then the almost complex structure
0
. should be integrable, and consequently—due to the well-known theorem by Wong
1
and Rosay—the complex manifold (M,J) would have to be biholomorphic to the
0
7 unit ball Bm in Cm.
1 While such belief was justified in the case of m 2 by Gaussier and Sukhov
≤
: [1,2],they incontrastshowedthatthereexistsanexampleindicatingthatitisnot
v
i the case if m>2. Then the second named author of this article classified all such
X
manifolds for every m 3 in [3, 4] (cf. Theorem 2.1). It has turned out that there
≥
r are infinitely many such examples, J-holomorphically inequivalent to each other.
a
Uponsuchobservation,therearisesanaturalquestionwhetherthehomogeneous
manifolds (noncompact) obtained here would admit a compact quotient by a dis-
cretesubgroup. Thisquestionwasaskedtothe firstnamedauthorseveraldifferent
times by many prominent mathematicians, from at least 10 years ago, when he
gave lectures explaining this line of research at the institutions including l’E´cole
Polytechniquede PalaiseauofFrance,PekingUniversityofChina,the KoreaInsti-
tute for Advanced Study (Seoul) of Korea, and other places. It is our pleasure to
acknowledge our indebtedness.
2010 Mathematics Subject Classification. 32M05.
Key words and phrases. J-automorphism,compactquotient.
ResearchoftheauthorsissupportedinpartbytheNRFGrant2011-0030044(SRC-GAIA)of
TheRepublicofKorea.
1
2 KANG-TAEKIM,KANG-HYURKLEEANDYOSHIKAZUNAGATA
The purposeofthis articleis,indeed, toprovidethe answer—negative,however.
Deferring the introduction of terminology and necessary definitions to the later
sections, we present our results first.
Theorem 1.1. Let Ω be an open connected subset of an almost complex manifold
(M,J) of real dimension 2m with 2 smooth strictly pseudoconvex boundary. If
C
there exists a discrete subgroup Γ of the automorphism group Aut(Ω,J) such that
Ω/Γ is compact, then (Ω,J) is biholomorphic to the standard unit ball in Cm.
From here on, open connected subsets of a manifold will be called domains, as
usual. Now, notice that Theorem 1.1 implies, according to the discussion above,
the following result concerning the nonexistence of compact quotients.
Theorem1.2. IfΩisahomogeneousdomain with 2 smoothstrictlypseudoconvex
C
boundary in an almost complex manifold (M,J) of real dimension 2m whose J-
structureis non integrable, then (Ω,J) does not admit any compact quotient by any
discrete subgroup of Aut(Ω,J).
We remark that these theorems are significant especially for the case m 3.
≥
2. Model manifolds/domains
For a domain Ω in (M,J), any ∞ smooth diffeomorphism ψ: Ω Ω is called
C →
a J-holomorphic automorphism if J dψ = dψ J. (More generally, a smooth
◦ ◦
map f: (M,J) (M˜,J˜) between two almost complex manifolds is called pseudo-
→
holomorphic or, more precisely, (J,J˜)-holomorphic if J˜ df = df J holds.) Such
◦ ◦
automorphisms form a topological group, denoted by Aut(Ω,J) under the law of
composition endowed with the compact-open topology.
Moreover, if the boundary ∂Ω of Ω is 2 smooth, then the implicit function
C
theorem implies that for every boundary point p there are an open neighborhood
V of M and a 2 function ρ: V R such that Ω V = z V : ρ(z) < 0 and
C → ∩ { ∈ }
that dρ(q)=0 for any q ∂Ω V. Such ρ is called a local defining function.
For a 1-f6ormω, the du∈al J∗∩of J is defined by (J∗ω)(v)=ω(Jv). Then the Levi
form of ρ is defined to be
∗
(v,w):= d(J dρ)(v,Jw).
ρ
L −
Then we say that Ω is strictly J-pseudoconvex at p ∂Ω if (v,v) > 0 for every
ρ
∈ L
nonzero vector v T ∂Ω JT ∂Ω.
p p
∈ ∩
Now write m = n+1 1, let z0,z1,...,zn represent the standard coordinate
functions of Cm = Cn+1, ≥and denote by z′ = (z1,...,zn) the standard coordinate
system of Cn.
For a mapping φ to Cn+1, denote by φ0 = z0 φ and φ′ = z′ φ, and hence
φ=(φ0,φ′). Greekindices α,β,... runfrom1 to n◦andthe summat◦ionconvention
is always assumed: zβ = n zβ, for instance. We also put bar on the
Nαβ β=1Nαβ
indices to denote the complexPconjugation of the corresponding tensor coefficients
such as: Nα¯β¯ =Nαβ, Aαβ¯¯ =Aαβ.
Let H = (z0,z′) C Cn : Rez0+ z′ 2 < 0 be the Siegel half space. But
{ ∈ × k k }
weshallendowanalmostcomplexstructurepotentiallydifferentfromthestandard
integrable one.
HOMOGENEOUS ALMOST COMPLEX MANIFOLDS & COMPACT QUOTIENTS 3
For each n n skew-symmetric matrix = ( ), define the almost complex
αβ
structure JN o×f Cn+1 by N N
∂ ∂ ∂
JN =i∂z0 ⊗dz0+(cid:18)i∂zα +2Nαβzβ∂z¯0(cid:19)⊗dzα
−i∂∂z¯0 ⊗dz¯0+(cid:18)−i∂∂zα¯ +2Nα¯β¯zβ¯∂∂z0(cid:19)⊗dzα¯ .
Note that the almost complex structure JN is integrable if and only if =0.
N
ThenthefollowingcharacterizationofthestrictlyJ-pseudoconvexdomainswith
an automorphism orbit accumulating at a boundary point has been established
earlier, by the second named author of this article:
Theorem 2.1 ([4]). If (Ω,J) is a domain in an almost complex manifold (M,J)
with 2 smooth strictly pseudoconvex boundary point q ∂Ω such that there are
C ∈
a point p Ω and a sequence of J-automorphisms ϕ Aut(Ω,J) satisfying
j
∈ ∈
lim ϕj(p)=q, then (Ω,J) is (J,JN)-biholomorphic to one of (H,JN).
j→∞
Remark 2.2. Every (H,JN) is homogeneous, as one sees in the next section.
3. Automorphisms of (H,JN)
Note that, due to the preceding discussion, the proof of Theorem 1.1 reduces
to demonstrating the (non)existence of the compact quotients for the domains
(H,JN), called the model domains in [3, 4].
There are four types of automorphisms generating the automorphism group of
the model domain (H,JN) for =0:
N 6
(1) R-action: For each s R,
∈
s =(z0+is,z′) Aut(H,JN).
T ∈
(2) Cn-action: For each w′ Cn, let us define
∈
hw′(z′)=−kw′k2−2hz′,w¯′i+i(Nαβzαwβ +Nα¯β¯zα¯wβ¯).
HThereenh(·C,n·,i+is) cthane sbteanemdabredddheedrminittioanApurto(Hdu,cJtNo)fbCyn: hz′,w¯′i = δαβ¯zαwβ¯.
w′ w′ =(z0+hw′(z′),z′+w′) Aut(H,JN),
7−→ H ∈
since w′ v′ = w′+v′ for any w′,v′ Cn.
H ◦H H ∈
We remark in passing that the subgroup generated by w′ and s is in fact
H T
isomorphic to the Heisenberg group.
(3) Dilation: For each t>0, we have
t =(tz0,t1/2z′) Aut(H,JN).
D ∈
(4) Isotropy: Let UN be the set of n×n complex matrices A=(Aαβ) with
δαβ¯ =Aµαδµν¯Aνβ¯¯ and Nαβ =AµαNµνAνβ ,
i.e., I =AtAand =At A. ThenUN canbe realizedasacompactsub-
N N
group of the unitary group U(n) which can be embedded into Aut(H,JN)
via
A A =(z0,Az′) Aut(H,JN).
7−→ R ∈
4 KANG-TAEKIM,KANG-HYURKLEEANDYOSHIKAZUNAGATA
Then we have:
Theorem3.1([4]). For eachφ Aut(H,JN)with =0,thereisauniquechoice
for t>0, s R, w′ Cn and A∈ UN such that N 6
∈ ∈ ∈
φ= s w′ t A = tz0+hw′(z′)+is,t1/2Az′+w′ .
T ◦H ◦D ◦R (cid:16) (cid:17)
4. Discrete subgroups and the limit sets
In this section, we consider only the case =0.
N 6
Definition 4.1. LetH beasubgroupoftheautomorphismgroupAut(H,JN). By
the limit set of H we mean the set Λ(H) of all accumulation points of the orbits
by H. Here, the limit set may contain the points at infinity.
Suppose that Γ is a discrete subgroupof Aut(H,JN). The aim of this section is
toanalyzethelimitsetΛ(Γ),whichwilleventuallyleadustothe proofofTheorem
1.1.
Choose φ Γ and write
ν
∈
φν(z)= tνz0+hwν′(z′)+isν,t1ν/2Aνz′+wν′
(cid:16) (cid:17)
forFsoormpe=tν(>p00,,ps′)ν ∈HR,,wwν′e∈asCsunmaendthAatν φ∈U(pN). (q0,0′) as ν , where q0
ν
is : s R ∈. Since Aν is an element of→the compact gr→ou∞p UN, we ma∈y
{ ∈ }∪{∞}
assume that Aν A∞ in UN. We may assume that tν converges in [0,+ ], and
also s in [ ,+→ ], respectively. Note that w′ can also be assumed to c∞onverge
in theνone-p−o∞int co∞mpactification of Cn. ν
Ctνa→se∞(1a)n:dkwtν−ν′1k/2→wν′∞→. −SAin∞cep′t.1ν/L2eAtνups′+cownsν′id→er0′ ∈ Cn and Aν → A∞, we have
Reφ0(p)=t Rep0 w′ 2 2Re p′,w¯′
ν ν −k νk − h νi
2
=t Rep0 t−1/2w′ 2Re t−1/2p′,t−1/2w¯′ .
ν(cid:18) −(cid:13) ν ν(cid:13) − D ν ν νE(cid:19)
(cid:13) (cid:13)
Since Rep0 t−1/2w′ 2 2Re t−1/2p(cid:13)′,t−1/2w¯(cid:13)′ Rep0 p′ 2 < 0, we have
φ0(p) .−T(cid:13)(cid:13)huνs theνa(cid:13)(cid:13)ccu−mulatiDngνpoint iνs ( ,0ν′E).→ −k k
ν →∞ (cid:13) (cid:13) ∞
Ct aset(>2)0:.wIν′f s→w∞′s6=R0′,.tShiennce t1ν/2Aνp′+wν′ →0′ ∈Cn andAν →A∞, wehave
ν ν
→ → ∈
φν(z)= tνz0+hwν′(z′)+isν,t1ν/2Aνz′+wν′
(cid:16) (cid:17)
−→ φ(z)=(cid:16)tz0+hw∞′ (z′)+is,t1/2A∞z′+w∞′ (cid:17)∈Aut(H,JN).
This is a contradiction to the discreteness of Γ. Hence s so that the
ν
accumulation point is ( ,0′) only. | | → ∞
∞
Case (3): w′ 0′. Supposethatt doesnotconvergeto0. Ifp′ =0′,thent 0
ν → ν 6 ν →
because t1/2A p′+w′ 0′ Cn. Thus we must have p′ =0′ and Rep0 <0.
ν ν ν → ∈
HOMOGENEOUS ALMOST COMPLEX MANIFOLDS & COMPACT QUOTIENTS 5
If t , then
ν
→∞
Reφ0(p)=t Rep0 w′ 2 2Re p′,w¯′ =t Rep0 w′ 2
ν ν −k νk − h νi ν −k νk
and this implies that Reφ0(p) . So the accumulation point is ( ,0′).
ν →−∞ ∞
If t t > 0, by the same argument as in Case (2) s . Thus the
ν ν
accumula→tion point is ( ,0′). | | → ∞
∞
Itremainstoanalyzethecaseoft 0. Forthispurpose,weposethe following
ν
→
Lemma 4.2. Let Γ be a discrete subgroup of Aut(H,JN). If there is φ Γ of the
∈
form φ(z) = tz0+is,t1/2Az′ for some t < 1, then each ψ Γ is of the form
(cid:16) (cid:17) ∈
b s
ψ(z)= az0+ib,a1/2Bz′ with = .
1 a 1 t
(cid:16) (cid:17)
− −
Postponing the proof of this lemma to the end of this section, we continue ana-
lyzing the accumulation points.
Assumethatt 0asν . Thenfixφ suchthatt <1and,consequently,
ν m m
→ →∞
I √t A is invertible. From now on, we simply denote by t = t , s = s ,
m m m m
w′−= wm′ , A = Am and put ζ′ = (I −√tA)−1w′ and γ = Hζ′ ◦Dt◦RA. Thus we
have
γ′(z′)=t1/2Az′+ζ′
(γ−1)′(z′)=t−1/2A−1(z′ ζ′),
−
which immediately implies
(4.1) γ−1 φ γ(z)= tz0+is˜,t1/2Az′ ,
m
◦ ◦ (cid:16) (cid:17)
for some s˜ R. For general integer values of ν, it follows that
∈
(γ−1 φ γ)′(z′)=(γ−1)′ φ′ γ′(z′)
◦ ν ◦ ◦ ν ◦
=t1/2A−1A Az′+t−1/2t1/2A−1A ζ′+t−1/2A−1w′ t−1/2A−1ζ′.
ν ν ν ν ν −
Since γ−1 φ γ is of the same form as φ in Lemma 4.2, we obtain, as far as the
m
element γ−◦1 φ◦ γ of the discrete group γ−1Γγ is concerned, that
ν
◦ ◦
(γ−1 φ γ)′(z′)=t1/2C z′,
◦ ν ◦ ν ν
where C =A−1A A. Now we have
ν ν
(φ )′(z′)=γ′ (γ−1 φ γ)′ (γ−1)′(z)
ν ν
◦ ◦ ◦ ◦
=t1/2AC A−1z′ t1/2AC A−1ζ′+ζ′ ζ′
ν ν − ν ν −→
as ν for any z′ Cn. Since (φ )′(p′) 0′ as ν , it follows that ζ′ = 0,
ν
which→in∞turn implies∈w′ = w′ = 0. Thus φ→(z) = (t→z0∞+is,t1/2Az′) with t < 1.
m m
Applying Lemma 4.2 again for φ=φ , each ψ Γ is of the form
m
∈
ψ(z)= az0+ib,a1/2Bz′
(cid:16) (cid:17)
with
b s
= .
1 a 1 t
− −
Especially,
φ (z)= t z0+is ,t1/2A z′
ν ν ν ν ν
(cid:16) (cid:17)
6 KANG-TAEKIM,KANG-HYURKLEEANDYOSHIKAZUNAGATA
where
s s
ν
= .
1 t 1 t
ν
− −
for any ν. Thus
s(1 t ) s
ν
s = −
ν
1 t −→ 1 t
− −
so
φ (z)= t z0+is ,t1/2A z′ (is/(1 t),0′).
ν (cid:16) ν ν ν ν (cid:17)→ −
Sinces/(1 t)isaninvariantofΓ,theaccumulationpointshouldbe(is/(1 t),0′)
when w′ −0′ and t 0. −
ν → ν →
Proof of Lemma 4.2. Since φ−1(z)= t−1z0 t−1is,t−1/2A−1z′ , we have
−
(cid:0) (cid:1)
φk(z)= tkz0+isσ ,tk/2Akz′
k
(cid:16) (cid:17)
φ−k(z)= t−kz0 ist−kσ ,t−k/2A−kz′
k
(cid:16) − (cid:17)
where φk is the k-th iteration of φ and φ−k =(φk)−1 and
σ =1+t+ +tk−1 .
k
···
Take ψ Γ and let
∈
ψ(z)= az0+hv′(z′)+ib,a1/2Bz′+v′
(cid:16) (cid:17)
Then
φk ψ φ−k(z)
◦ ◦
= az0+i (1 a)sσk+tkb +tkhv′(t−k/2A−kz′),a1/2AkBA−kz′+tk/2Akv′ .
(cid:16) (cid:0) − (cid:1) (cid:17)
Since tkhv′(t−k/2A−kz′) 0 as k , φk ψ φ−k subsequentially converges to
→ →∞ ◦ ◦
some ψ˜ Γ of the form
∈
ψ˜= az0+(1 a)isσ,a1/2ABA−1z′
(cid:16) − (cid:17)
e e
where σ =limσ =1/(1 t). Since Γ is discrete, we conclude that v′ =0 and
k
−
(1 a)sσ +tkb=(1 a)sσ
k
− −
for infinitely many k. Thus
b 1 s
= s(σ σ )=sσ = .
1 a tk − k 1 t
− −
This completes the proof. (cid:3)
HOMOGENEOUS ALMOST COMPLEX MANIFOLDS & COMPACT QUOTIENTS 7
5. Proof of Theorems 1.1 and 1.2
Let Γ be a discrete subgroup of Aut(Ω,J) with Ω/Γ compact. Then there ex-
ists a (finite) boundary point at which the orbit of Γ accumulates. By Theorem
2.1, Ω is biholomorphic to a model (H,JN), say, that is homogeneous and com-
plete Kobayashi hyperbolic. Suppose now that Ω, or equivalently (H,JN), is not
biholomorphic to the standard unit ball of Cm.
Accordingto the argumentsofSection4above,the setΛ(Γ) (C (0,...,0) )
∩ ×{ }
consists of atmost two elements: the extended boundary point ( ,0,...,0), or the
boundary point of type (a,0,...,0) for some a C. Suppose th∞at (H,JN) admits
∈
a compact quotient by a discrete group Γ, say. Then, since (H,JN) is complete
Kobayashihyperbolic, we must have
Λ(Γ) (C (0,...,0) )=∂H (C (0,...,0) )=R (0,...,0) ,
∩ ×{ } ∩ ×{ } ×{ }
whichcannotcoincidewithanytwoelementset. ThisyieldstheproofofTheorems
1.1 as well as 1.2. (cid:3)
References
[1] H.Gaussier;A.Sukhov: WongRosay theoreminalmostcomplexmanifolds,http:
www.arXiv.org: math.CV/0307335.
[2] H.Gaussier;A.Sukhov: Onthegeometryofmodelalmostcomplexmanifoldswithboundary,
Math. Z.254(2006), no.3,567–589.
[3] K.-H.Lee: Domainsinalmostcomplexmanifoldswithanautomorphismorbitaccumulating
atastronglypseudoconvex boundarypoint.MichiganMath. J.54(2006), no.1,179–205.
[4] K.-H. Lee: Strongly pseudoconvex homogeneous domains in almost complex manifolds. J.
Reine Angew. Math.623(2008), 123–160.
[5] J.-P. Rosay: Sur une caractrisation de la boule parmi les domaines de Cn par son groupe
d’automorphismes.(French) Ann. Inst. Fourier (Grenoble) 29(1979), no.4,ix,91–97.
[6] B.Wong: CharacterizationoftheunitballinCn byitsautomorphismgroup.Invent. Math.
41(1977), no.3,253–257.
Kang-Tae Kim and Yoshikazu Nagata: Department of Mathematics and Center for
Geometry and its Applications, Pohang University of Science and Technology, 37673,
The Republicof Korea
E-mail address: [email protected], [email protected]
Kang-Hyurk Lee: Department of Mathematics and Research Institute of Natural
Science, GyeongsangNationalUniversity, Jinju, Gyeongnam,660-701,The Republic of
Korea
E-mail address: [email protected]