The holomorphic symplectic structures on hyper-Kaehler manifolds of type A_{\infty}

0.21 MB·English

by  Kota Hattori

The holomorphic symplectic structures on
hyper-Ka¨hler manifolds of type A
∞
2
Kota Hattori
1
0
2

t
c Graduate School of Mathematical Sciences, University of Tokyo
O
3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

5 [email protected]
1

] Abstract
G
D Hyper-Ka¨hlermanifoldsoftypeA arenoncompactcompleteRicci-flatKa¨hler
∞
. manifolds of complex dimension 2, constructed by Anderson, Kronheimer, LeBrun
h
t [1]andGoto [4]. We studytheholomorphicsymplecticstructurespreservedbythe
a
m naturalC×-actions on thesemanifolds, then show thesufficient andnecessary con-
[ ditions for the existence of C×-equivariant biholomorphisms between two hyper-

Ka¨hler manifolds of type A preserving their holomorphic symplectic structures.
1 ∞
v As a consequence, we can show the existence of a complex manifold of dimension
9
2 on which there is a continuous family of complete Ricci-flat Ka¨hler metrics with
6
8 distinct volume growth.
3
.
0
1 1 Introduction
2
1
: Hyper-K¨ahler manifoldsoftype A were first constructed by Anderson, Kro-
v ∞
i nheimer and LeBrun in [1], as the first example of complete Ricci-flat K¨ahler
X
manifoldswithinfinitetopologicaltype. Here, infinitetopologicaltypemeans
r
a that their homology groups are infinitely generated. After [1], Goto [4, 5] has
succeeded inconstructingthesemanifoldsinanotherway, usinghyper-Ka¨hler
quotient construction. He also constructed the higher dimensional complete
hyper-K¨ahler manifolds with infinite topological type. Some of the topolog-
ical and geometric properties of hyper-K¨ahler manifolds of type A were
∞
studied well in the above papers, and the author studied the volume growth
of the Riemannian metrics in [6]. Then this paper focuses on the complex
geometry on the hyper-K¨ahler manifolds of type A .
∞
1
A hyper-K¨ahler manifold is, by definition, a Riemannian manifold (X,g)
of real dimension 4n equipped with three complex structures I ,I ,I satis-
1 2 3
fying the quaternionic relations I2 = I2 = I2 = I I I = id with respect
1 2 3 1 2 3 −
to all of which the metric g is Ka¨hlerian. Then the holonomy group of g is
a subgroup of Sp(n) and g is Ricci-flat. By the definition, the hyper-Ka¨hler
manifold carry three Ka¨hler forms defined by ω := g(I , ) for i = 1,2,3,
i i
· ·
then we have an non-degenerate closed (2,0)-form ωC = ω3 √ 1ω2, which
− −
is called a holomorphic symplectic form, if (X,I ) is regarded as a complex
1
manifold. Conversely, it is known that (g,I ,I ,I ) are reconstructed from
1 2 3
(ω ,ω ,ω ), thus we call ω = (ω ,ω ,ω ) the hyper-K¨ahler structure over
1 2 3 1 2 3
X. In this paper we regard the hyper-K¨ahler manifold (X,ω) as a complex
manifold with respect to the complex structure I .
1
In[6], the author computed the volume growth ofhyper-Ka¨hler manifolds
of type A . Here, the volume growth of a Riemannian manifold (X,g) is the
∞
asymptotic behavior of the function V (p ,r), which is defined as the volume
g 0
of the geodesic ball of radius r centered at p X. Then the following result
0
∈
was obtained.
Theorem 1.1 ([6]). There exists a C∞ manifold X of dimRX = 4 and a
family of hyper-K¨ahler structures ω(α) on X for 3 < α < 4 which carry
complete hyper-K¨ahler metrics g(α) with
V (p ,r) V (p ,r)
g(α) 0 g(α) 0
0 < liminf limsup < + .
r→+∞ rα ≤ r→+∞ rα ∞
for any p X.
0
∈
The hyper-K¨ahler manifolds (X,ω(α)) in the above theorem are already
constructed in [1][4], and the essential part of Theorem 1.1 is the computa-
tion of the volume growth of these manifolds. In this paper we study the
(α) (α)
holomorphic symplectic structure ω over X. The period of (X,ω ), that
C C
(α)
is the cohomology class determined by ω , is independent of α. Then the
C
(α)
holomorphic symplectic structures ω are expected to be independent of α.
C
We actually obtain the following result.
Theorem 1.2. There exists a complex manifold X of dimCX = 2 and a
family of complete Ricci-flat K¨ahler metrics g(α) on X for 3 < α < 4 with
V (p ,r) V (p ,r)
g(α) 0 g(α) 0
0 < liminf limsup < + .
r→+∞ rα ≤ r→+∞ rα ∞
for any p X.
0
∈
2
Theorem 1.2 will be proved as a corollary of the results in [6] and under-
mentioned Theorem 1.3, which is the main result in this paper.
It is known that there are complete Ricci-flat Ka¨hler metrics over Cn who
do not have the Euclidean volume growth [11]. For example, the Taub-NUT
metrics over C2 are the complete Ricci-flat Ka¨hler metrics whose volume
growth are r3. On the other hand, Theorem 1.2 asserts the existence of a
complex manifold who has a continuous family of complete Ricci-flat Ka¨hler
metrics, whose volume growth also change continuously.
To show that two hyper-K¨ahler quotients are biholomorphic or not, it is
useful to see the GIT quotient construction and study the period. For exam-
ple, Konno [9] has studied the period of holomorphic symplectic structures of
toric hyper-K¨ahler manifolds, that are typical examples of hyper-Ka¨hler quo-
tients, using GIT quotient construction. However, this method is not enough
for studying the case of hyper-K¨ahler manifolds of type A , because these
∞
manifolds are obtained by taking quotients by the action of infinite dimen-
sional Lie groups on infinite dimensional manifolds, then we should develop
other methods to show that (X,ω(α1)) and (X,ω(α2)) are biholomorphic.
C C
In this paper, we consider when two hyper-K¨ahler manifolds of type A
∞
become isomorphic as holomorphic symplectic manifolds. Let (X ,ω ) be
i i
the hyper-K¨ahler manifolds of type A for i = 0,1. Then there are the
∞
natural C×-actions over X preserving their holomorphic symplectic struc-
i
tures ωi,C, and the complex moment maps µi,C : Xi C. Since the complex
→
moment maps are C×-invariant, they define complex valued continuous func-
tions [µi,C] : Xi/C× C on the quotient topological spaces Xi/C×. More-
→
over, the C×-actions define natural partial order structures on the quotient
spaces X /C×. Then we obtain the following result.
i
Theorem 1.3. There exists a C×-equivariant biholomorphic map f : X
0
→
X1 with f∗ω1,C = ω0,C if and only if there is a homeomorphism h : X0/C×
→
X1/C× preserving the order structures and [µ1,C] h [µ0,C] is constant.
◦ −
The above theorem is proven as follows in this paper. Put
X∗ := X p X ; pg = p for all g C×. .
i i\{ ∈ i ∈ }
Then we have an open covering X∗ = Xs where each Xs is biholomor-
i s i i
phic to C× C, consequently biholomoSrphic maps Xs Xs are obtained.
× 0 → 1
Moreover we can show that these biholomorphic maps glue on the intersec-
tions, therefore a biholomorphic map X∗ X∗ is obtained, and it extends
0 → 1
to the biholomorphic map X X by Hartogs’ extension theorem. The
0 1
→
similar way is used already in [11] to show that the complex structure over
R4 given by the Taub-NUT metric is biholomorphic to C2. But in our case,
3
the topological structure of X /C× is so complicated that we should study
i
them precisely.
This paper is organized as follows. First of all we review the results
obtained in [4] in Sections 2 and 3. In Section 2, we construct hyper-Ka¨hler
manifolds of type A by using hyper-K¨ahler quotient constructions and see
∞
that there exists a natural S1-action. In Section 3, we review the another
quotient construction of hyper-K¨ahler manifolds of type A , and see that
∞
the manifolds obtainedin Sections 2 and3 areisomorphic asthe holomorphic
symplectic manifolds.
In Section 4, we study the topological properties of topological quotient
spaces obtained from hyper-K¨ahler manifolds of type A by taking the quo-
∞
tient by C×-action. We can also see that there exist natural partial order
structures.
In Section 5 we construct biholomorphisms between two hyper-Ka¨hler
manifolds of type A , which satisfy the assumption of Theorem 1.3. As
∞
a consequence, we apply Theorem 1.3 for more concrete case, and obtain
Theorem 1.2 and other results in Section 6.
2 Hyper-K¨ahler manifolds of type A
∞
2.1 Hyper-K¨ahler quotient construction
In this section, we review shortly the construction of hyper-Ka¨hler mani-
folds of type A along [4]. Although they can be constructed by Gibbons-
∞
Hawking ansatz [1], we need hyper-K¨ahler quotient construction in [4] for
arguments in Section 4. For more details of construction and basic facts, see
[1][4] or review in Section 2 of [6].
First of all, we describe the definition of hyper-Ka¨hler manifolds.
Definition 2.1. Let (X,g) be a Riemannian manifold of dimension 4n,
I ,I ,I be integrable complex structures on X, and g is a hermitian metric
1 2 3
with respect to each I . Then (X,g,I ,I ,I ) is a hyper-Ka¨hler manifold
i 1 2 3
if (I ,I ,I ) satisfies the relations I2 = I2 = I2 = I I I = 1 and each
1 2 3 1 2 3 1 2 3 −
fundamental 2-form ω := g(I , ), that is, (X,g,I ) is ka¨hlerian.
i i i
· ·
LetH = R Ri Rj Rk = C Cj bequaternionandImH = Ri Rj Rk
⊕ ⊕ ⊕ ⊕ ⊕ ⊕
be its Imaginary part. Then an ImH-valued 2-form ω := iω +jω +kω
1 2 3
∈
Ω2(X) ImH are constructed from the hyper-K¨ahler structure (g,I ,I ,I ).
1 2 3
⊗
Conversely, (g,I ,I ,I ) is reconstructed from ω. Hence we call ω the hyper-
1 2 3
Ka¨hler structure on X instead of (g,I ,I ,I ).
1 2 3
4
To construct hyper-K¨ahler manifolds of type A , we prepare an infinite
∞
countable set I and a parameter space
1
(ImH)I0 := {λ = (λn)n∈I ∈ (ImH)I; 1+ λ < +∞}.
Xn∈I | n|
For a set S, we denote by SI the set of all maps from I to S. An element of
I
x S is written as x = (xn)n∈I. Then we have a Hilbert space
∈
MI := v HI; v 2I < + ,
{ ∈ k k ∞}
where
u,v I := unv¯n, v 2I := v,v I
h i k k h i
Xn∈I
for u,v HI. Here, v¯ H is the quaternionic conjugate of v defined by
n n
∈ ∈
a+bi+cj +dk := a bi cj dk for a,b,c,d R.
Now we fix λ (I−mH)−I, an−d take Λ HI to∈be Λ iΛ = λ . Then we
∈ 0 ∈ n n n
have the following Hilbert manifolds
MΛ := Λ+MI = Λ+v; v MI ,
{ ∈ }
G := g (S1)I; (1+ λ ) 1 g 2 < + , g = 1 ,
λ n n n
{ ∈ | | | − | ∞ }
Xn∈I Yn∈I
g := Lie(G ) = ξ RI; (1+ λ ) ξ 2 < + , ξ = 0 .
λ λ n n n
{ ∈ | | | | ∞ }
Xn∈I Xn∈I
The convergence of g and ξ follows fromthecondition (1+
n∈I n n∈I n n∈I
λ )−1 < + . TheQn G is a HilPbert Lie group whose Lie algebra iPs g . We
n λ λ
| | ∞
can define a right actionofGλ onMΛ by xg := (xngn)n∈I for x MΛ, g Gλ.
∈ ∈
Here the product of x and g is given by regarding S1 as the subset of H
n n
by the natural injections S1 C H. Then G acts on M preserving the
λ Λ
⊂ ⊂
hyper-K¨ahler structure, and we have the hyper-K¨ahler moment map µˆ :
Λ
M ImH g∗ defined by
Λ → ⊗ λ
µˆ (x),ξ := (x ix¯ Λ iΛ )ξ ImH
Λ n n n n n
h i − ∈
Xn∈I
for x M , ξ g . If I is a finite set.
Λ λ
∈ ∈
Since µˆ is G -invariant, then G acts on the inverse image
Λ λ λ
µˆ−1(0) = x M ; x ix λ = x ix λ for all n,m I
Λ { ∈ Λ n n − n m m − m ∈ }
Hence we obtain the quotient space µˆ−1(0)/G which is called the hyper-
Λ λ
Ka¨hler quotient.
5
Definition 2.2. An element λ (ImH)I is generic if λ λ = 0 for all
∈ 0 n − m 6
distinct n,m I.
∈
Theorem 2.3 ([4]). If λ (ImH)I is generic, then µˆ−1(0)/G is a smooth
∈ 0 Λ λ
manifold of real dimension 4, and the hyper-K¨ahler structure on M induces
Λ
a hyper-K¨ahler structure ω on µˆ−1(0)/G .
λ Λ λ
The quotient space µˆ−1(0)/G seems to depend on the choice of Λ HI,
Λ λ ∈
but the induced hyper-K¨ahler structure on µˆ−1(0)/G depends only on λ
Λ λ
from the argument of Section 2 of [6]. Thus we may put
X (λ) := µˆ−1(0)/G
HKQ Λ λ
= x M ;x ix¯ λ is independent of n I /G ,
Λ n n n λ
{ ∈ − ∈ }
and call it hyper-K¨ahler manifold of type A
∞
Recall that we assume that I is infinite. If ♯I = k + 1 < + , then
∞
(X (λ),ω ) becomes an ALE hyper-K¨ahler manifold of type A [3].
HKQ λ k
2.2 S1-actions and moment maps
In [4], an S1-action on X (λ) preserving the hyper-K¨ahler structure de-
HKQ
fined as follows. We denote by [x] µˆ−1(0)/G the equivalence class repre-
∈ Λ λ
sented by x µˆ−1(0). Fix m I and put
∈ Λ ∈
[x]g := [xmg,(xn)n∈I\{m}]
forx = (xm,(xn)n∈I\{m}) ∈ µˆ−Λ1(0)andg ∈ S1. This definitionisindependent
of the choice of m I, and we have the action of S1 onX (λ). The hyper-
HKQ
∈
Ka¨hler moment map µ : X (λ) ImH = R3 is defined by
λ HKQ
→
µ ([x]) := x ix¯ λ ImH.
λ n n n
− ∈
The right handside is independent of the choice of n I since x isanelement
∈
of µˆ−1(0).
Λ
Put
X (λ)∗ := [x] X (λ); x = 0 for all n I ,
HKQ HKQ n
{ ∈ 6 ∈ }
Y := ImH λ ; n I ,
λ n
\{− ∈ }
then we have a principal S1-bundle µ : X (λ)∗ Y , and S1
λ|XHKQ(λ)∗ HKQ → λ
acts on X (λ) X (λ)∗ trivially.
HKQ HKQ
\
Conversely, on the total spaces of some principal S1-bundle over Y ,
λ
hyper-K¨ahler structures preserved by S1-actions are constructed in [1] by
6
Gibbons-Hawking ansatz. It is shown in [4] that each X (λ) is isomor-
HKQ
phic to one of that constructed by Gibbons-Hawking ansatz.
By observing the Gibbons-Hawking construction, it is easy to see that
X (λ) and X (λ′) are isomorphic as hyper-K¨ahler manifolds if λ and
HKQ HKQ
λ′ satisfy one of the following relations; (i) λ′ λ ImH are independent
n − n ∈
of n, (ii) λ′ = λ for some bijective maps a : ImH ImH.
n a(n) →
We can also show easily that X (λ) = X ( λ) by constructing an
HKQ ∼ HKQ
−
isomorphism explicitly.
3 Holomorphic description
In this section we compare hyper-K¨ahler quotients µˆ−1(0)/G with another
Λ λ
kind of quotient spaces µˆ−Λ,1C(0)/GCλ along [4], where µˆΛ,C is the complex
C C
valued component of µˆ and G is the complexification of G .
Λ λ λ
First of all, we complexify the Hilbert Lie group G as follows,
λ
GC := g (C×)I; (1+ λ ) 1 g 2 < + , g = 1 ,
λ { ∈ | n| | − n| ∞ n }
Xn∈I Yn∈I
gC := g C = ξ CI; (1+ λ ) ξ 2 < + , ξ = 0 ,
λ λ ⊗ { ∈ | n| | n| ∞ n }
Xn∈I Xn∈I
where C× = C 0 . Then GC acts smoothly on M , where Λ HI satisfies
\{ } λ Λ ∈
Λ iΛ = λ .
n n n
From now on we write ζ = ζRi ζCk = (ζR,ζC) ImH along the decom-
− ∈
position ImH = Ri Ck. Similarly, we write λ = λRi λCk = (λR,λC) for
λ (ImH)I, where ⊕λR RI and λC CI. The hyper-−K¨ahler moment map
∈ ∈ ∈
µˆΛ is also decomposed into two components as µˆΛ = µˆΛ,R i µˆΛ,C k. Then
µˆΛ,R : MΛ → gλ∗ and µˆΛ,C : MΛ → (gλC)∗ are written as · − ·
µˆΛ,R(z +wj),ξ = ( zn 2 wn 2 λn,R)ξn,
h i | | −| | −
Xn∈I
µˆΛ,C(z +wj),η = (2znwn λn,C)ηn,
h i −
Xn∈I
for z +wj = (zn +wnj)n∈I ∈ MΛ, ξC∈ gλ and η ∈ gλC, where zn,wn ∈ C and
λn = λn,Ri−λn,Ck. Then µˆΛ,C is Gλ invariant.
Definition 3.1. Let t = (tn)n∈I RI. Then z + wj MΛ is t-stable if
∈ ∈
z 2 + w 2 > 0 holds for any n,m I which satisfy t > t .
n m n m
| | | | ∈
Now we put
µˆ−1 (0) := z +wj µˆ−1 (0);z +wj is t stable .
Λ,C t { ∈ Λ,C }
7
Then GC acts on µˆ−1 (0) . If the quotient space X (λ) := µˆ−1 (0) /GC
λ Λ,C t GIT Λ,C λR λ
becomes a smooth manifold, then the standard nowherevanishing (2,0)-form
n∈Idzn ∧ dwn over MΛ induces a holomorphic symplectic form ωλ,C on
PXGIT(λ). Then (XGIT(λ),ωλ,C) depends only on λ, not depends on Λ.
Theorem 3.2 ([4]). Let λ (ImH)I be generic. Then the quotient space
∈ 0
X (λ) becomes a complex manifold of dimension 2.
GIT
For any generic λ, µˆ−1(0) is a subset of µˆ−1 (0) . Then this inclusion
Λ Λ,C λR
induces
φ : X (λ) = µˆ−1(0)/G µˆ−1 (0) /GC = X (λ),
λ HKQ Λ λ → Λ,C λR λ GIT
which is an biholomorphism preserving the holomorphic structure, namely,
φ∗λωλ,C = ωλ,3 −√−1ωλ,2,
where ω = ω i+ω j +ω k is the hyper-K¨ahler structure on X (λ).
λ λ,1 λ,2 λ,3 HKQ
Here, ω √ 1ω is the holomorphic symplectic structures over X (λ)
λ,3 λ,2 HKQ
− −
with respect to the complex structure I . From now on we write
λ,1
(X(λ),ωλ,C) := (XHKQ(λ),ωλ,3 √ 1ωλ,2) = (XGIT(λ),ωλ,C)
− −
if it is not necessary to distinguish them.
In Section 2.2, we have seen that X(λ) has a natural S1-action. Then
by complexifying the action, we have a holomorphic C×-action on X(λ) pre-
serving ωλ,C defined by
[z +wj]g := [zmg +wmg−1,(zn +wn)n∈I\{m}].
It is easy to see that C× acts freely on X(λ)∗ = X (λ)∗, and trivially on
HKQ
X(λ) X(λ)∗.
\
C
4 Topological structure of X(λ)/
×
In the previous section, we obtain C×-action on X(λ). In this section we
will study the topology of the quotient space X(λ)/C× with the quotient
topology.
4.1 The topological space homeomorphic to X(λ)/C
×
Firstofall, wedefineacertainequivalencerelation inImH, whichdepends
λ
on λ (ImH)I, then we show that there exists∼a homeomorphism from
∈ 0
X(λ)/C× to ImH/ .
λ
Put Z := λ ∼ ImH; n I for λ (ImH)I. Then we have a disjoint
λ {− n ∈ ∈ } ∈ 0
union ImH = Y Z .
λ λ
F
8
Definition 4.1. Let λ (ImH)I and η ,η ImH. We write η η if
∈ 0 1 2 ∈ 1 ∼λ 2
they satisfy one of the following conditions; (i) η1 and η2 satisfy η1,C = η2,C
and tη +(1 t)η Y for all t [0,1], (ii) η = η Z .
1 2 λ 1 2 λ
− ∈ ∈ ∈
Now we obtain quotient spaces X(λ)/C× and ImH/ with quotient
λ
∼
topology. Next we construct a homeomorphism between them.
Let µ : X(λ) ImH be the hyper-K¨ahler moment map defined in
λ
→
Section 2. We will show that µ induces a continuous map from X(λ)/C×
λ
to ImH/ by using the following lemma.
λ
∼
Lemma 4.2. Let [z+wj] X(λ)∗ and g C×. Then wehave µ ([z+wj])
λ λ
∈ ∈ ∼
µ ([z +wj]g) and
λ
µλ,R([z+wj]g)
log|g|2 = Z Φλ(t,ζC)dt,
µλ,R([z+wj])
where Φ is defined by
λ
1 1
Φ (ζ) :=
λ
4 ζ +λ
Xn∈I | n|
for ζ Y .
λ
∈
Proof. Take g˜ = (g˜n)n∈I ∈ (C×)I to be n∈I|1−g˜n|2 < ∞ and g = n∈Ig˜n.
Nowwe regardz+wj asanelement ofµˆP−1(0), andsuppose (z g˜ +wQg˜−1j)
Λ n n n n ∈
µˆ−1(0).
Λ
Put ζ = µ ([z +wj]) and η = µ ([z +wj]g). Then we have
λ λ
zn 2 wn 2 = λn,R +ζR, 2znwn = λn,C +ζC,
| | −| |
|zng˜n|2 −|wng˜n−1|2 = λn,R +ηR, 2znwn = λn,C +ηC,
accordingly we have ζC = ηC. Since [z + wj] X(λ)∗, we may suppose
∈
z 2 + w 2 = 0 for all n I. Then g˜ satisfies
n n n
| | | | 6 ∈
g˜ 2 = |η+λn|+ηR +λn,R (if z = 0),
n n
| | ζ +λn +ζR +λn,R 6
| |
g˜ −2 = |η+λn|−(ηR +λn,R) (if w = 0).
n n
| | ζ +λn (ζR +λn,R) 6
| |−
Now we put I±(ζ) := n I; (ζR + λn,R) > 0 . Since g˜n 2 and g˜n −2
{ ∈ ± } | | | |
should be positive, we have η = (ηR,ζC) λ ζ. Then we obtain
∼
Fλ(ηR,ζR,ζC) := log g 2 = log |η+λn|+ηR +λn,R
| | n∈XI+(ζ) |ζ +λn|+ζR +λn,R
ζ +λn (ζR +λn,R)
+ log | |− ,
n∈XI−(ζ) |η +λn|−(ηR +λn,R)
9
where η = ηRi ζCk, then we have log g 2 = Fλ(ηR,ζR,ζC). The function Fλ
− | |
is smooth at (ηR,ζR,ζC) if η,ζ Yλ. Then we have
∈
∂F
λ
= Φλ(ηR,ζC) > 0.
∂ηR
Since Fλ(ζR,ζR,ζC) = 0, we obtain
ηR
log|g|2 = Z Φλ(t,ζC)dt.
ζR
It is obvious that [z + wj] = [z + wj]g if µ ([z + wj]) Z . Then the
λ λ
∈
hyper-K¨ahler moment map µ induces [µ ] : X(λ)/C× ImH/ from
λ λ λ
→ ∼
Lemma 4.2. Since µ is continuous and surjective, [µ ] is also continuous
λ λ
and surjective.
Proposition 4.3. Let λ (ImH)I be generic. Then [µ ] : X(λ)/C×
∈ 0 λ →
ImH/ is a homeomorphism.
λ
∼
Proof. It suffices to show that [µ ] is an injective and open map.
λ
Let [z +wj],[z′ +w′j] X(λ) satisfy µ ([z +wj]) µ ([z′ +w′j]). If
λ λ λ
∈ ∼
µ ([z + wj]) Z , then [z + wj] = [z′ + w′j]. If µ ([z + wj]) Y , then
λ λ λ λ
∈ ∈
µ ([z′ +w′j]) is also an element of Y . If we take g C× to be
λ λ
∈
µλ,R([z′+w′j])
log|g|2 = Z Φλ(t,ζC)dt,
µλ,R([z+wj])
then we have µ ([z+wj]g) = µ ([z′ +w′j]). Since S1 acts on µ−1(ζ) transi-
λ λ λ
tively for all ζ ImH, there exists σ S1 such that [z +wj]gσ = [z′ +w′j].
∈ ∈
Thus the injectivity has been proven.
The openness of [µ ] is easily shown by the elementary argument of gen-
λ
eral topology.
From now on we identify X(λ)/C× with ImH/ by [µ ]. To study
λ λ
∼
the topological properties of X(λ)/C×, we often observe ImH/ for con-
λ
∼
venience.
Nowlet p : X(λ) X(λ)/C× andπ : ImH ImH/ bethequotient
λ λ λ
→ → ∼
maps. Then µλ,C : X(λ) C induces a continuous map [µλ,C] : X(λ)/C×
→ →
C satisfying [µλ,C] pλ = µλ,C. On the other hand, the orthogonal projection
◦
prC : ImH C defined by prC(ζ) := ζC induces a continuous map [prC]λ :
→
ImH/ λ C by [prC]λ πλ = prC. Note that [µλ,C] is identified with [prC]λ
∼ → ◦
by [µλ], that is, [µλ,C] = [prC]λ [µλ].
◦
There exists a natural partial order in ImH/ defined as follows.
λ
∼
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