On the Third Gap for Proper Holomorphic Maps between Balls

0.19 MB·English

by  Xiaojun Huang

On the Third Gap for Proper Holomorphic
2
1 Maps between Balls
0
2

n
Xiaojun Huang Shanyu Ji Wanke Yin
a ∗
J

1
3

]
V 1 Introduction
C
.
h The study of proper holomorphic maps between balls in complex spaces of different dimension
t
a has attracted much attention in the past many years. Among many problems which people
m
have considered along these lines, we mention at least those questions related to therationality
[ for maps with minimum boundary regularity, linearity and semi-linearity, gap phenomenons,

1 degree estimates, as well as, various classifications. This paper is devoted to the study of a
v
gap rigidity problem.
0
4 To start with, we write Bn for the unit ball in the complex space Cn. Write Prop(Bn,BN)
4 for the set of proper holomorphic maps from Bn into BN, Prop (Bn,BN) Prop(Bn,BN) for
6 k ⊂
. those maps that are Ck-smooth up to the boundary, and Rat(Bn,BN) for the space of proper
1
holomorphic rational maps from Bn into BN. We say that F and G Prop(Bn,BN) are
0
∈
2 equivalent if there are automorphisms σ Aut(Bn) and τ Aut(BN) such that F = τ G σ.
1 ∈ ∈ ◦ ◦
For a proper holomorphic map F Prop(Bn,BN), one can always add zero components to
:
v F and then compose it with automor∈phisms from Aut(BN) to produce other proper holomor-
i
X phic maps from Bn into BN′ with N > N. However, maps obtained in this manner have the
′
r
same geometric character as that of the original F and are not regarded as ‘different maps’.
a
A gap rigidity phenomenon for proper holomorphic maps between balls is to ask for which
N, any proper holomorphic map from Bn into BN, which has certain boundary regularity, is
always equivalent to a map of the form (G,0) with G a proper holomorphic map from Bn
into BN′ for N < N. Along these lines, in [HJY], the following gap-rigidity conjecture was
′
proposed:
Conjecture 1.1. (Huang-Ji-Yin [HJY]) Let n 3 be a positive integer. Write K(n) for the
≥
largest positive integer m such that n > m(m + 1)/2. For any k with 1 k K(n), write
≤ ≤
∗Supported in part by DMS-1101481
1
for the collection of integers m such that kn < m < (k +1)n k(k +1)/2. Then, when
k ′ ′
I −
N for a certain 1 k K(n), any proper holomorphic rational map F Rat(Bn,BN)
k
∈ I ≤ ≤ ∈
is equivalent to a map of the form (G,0), where G is a proper holomorphic rational map from
Bn into BN′ with N = kn < N.
′
Notice that = n + 1, ,2n 2 (if n 3), = 2n + 1,2n + 2, ,3n 4 (if
1 2
I { ··· − } ≥ I { ··· − }
n 5), and = 3n+1,3n+2, ,4n 7 (if n 8). For any k K(n) as above, we call
3
≥ I { ··· − } ≥ ≤
the non-empty set of positive integers the k-th gap interval for proper holomorphic maps
k
I
between balls.
It was proved in Theorem 2.8 of [HJY] that for any N ... , there is a map
1 2 K(n)
in Rat(Bn,BN) that is not equivalent to any map of the6∈foIrm∪(IG,∪0) w∪itIh G Rat(Bn,BN′)
∈
for N < N. In particular the following map in Rat(Bn,B3n) is not equivalent to any map of
′
the form (G,0):
F = z , ,z ,λz ,z ,√1 λ2z (z , ,z ,µz , 1 µ2z z) , (1.1)
1 n 2 n 1 n n 1 1 n 1 n n
··· − − − − ··· − −
(cid:0) p (cid:1)
where λ,µ (0,1). Hence, the construction in [HJY] shows that one can only expect the gap
∈
K(n)
phenomenon for N and Conjecture 1.1 actually claims that we do have the gap
∈ ∪k=1 Ik
phenomenonforN K(n) . HereweremarkthatwhenN 1+√1+4nn n3,N K(n) .
∈ ∪k=1 Ik ≥ − 2 ≈ 2 6∈ ∪k=1 Ik
Earlier in [DL], for N n2 2n+ 2, D’Angelo and Lebl constructed a proper holomorphic
≥ −
monomial map from Bn into BN, that is not equivalent to a map of the form (G,0) where G
is a proper rational holomorphic map from Bn into BN′ with N < N.
′
On the other hand, making use of the classifications in Faran [Fa], Huang-Ji [HJ], Hamard
[Ha] and Huang-Ji-Xu [HJX1] (see [HJY] for more detailed discussions on the history and
references on this matter), it is known that, when N with k = 1,2, any rational proper
k
∈ I
holomorphic map from Bn into BN is equivalent to a map of the form (G,0). Conjecture 1.1
thus holds by the work of these people in the case of k = 1,2. Notice that if Conjecture 1.1
holds, then there are approximately √n gap intervals for rational proper maps from Bn into
balls in complex spaces of larger dimensions.
In this paper, we give a proof of the following result which confirms the above conjecture
when k = 3.
Theorem 1.2. Let F Prop (Bn,BN) with 3n < N 4n 7 and n 7. Then F is
3
∈ ≤ − ≥
equivalent to a map of the form (G,0) with G Rat(Bn,B3n). Namely, any proper rational
∈
map F Rat(Bn,BN) with N is equivalent to a map of the form (G,0) where G is a
3
proper r∈ational holomorphic map∈fIrom Bn into BN′ with N = 3n < N.
′
We notice that the dimension N = 3n in Theorem 1.2 is sharp by the map constructed
′
in (1.1). This example also has the property, as we will make more precisely in the end of
2
the next section, that it is the largest possible generic map from Bn into B3n. We also remark
that there are already many rational proper holomorphic maps from Bn into B3n 2 that are
−
not equivalent to any proper polynomial maps by the work in Faran-Huang-Ji-Zhang [FHJZ].
For any F Prop (Bn,BN) with N < n(n+1), by the results proved in Huang [Hu2]
∈ 3 2
and Huang-Ji-Xu [HJX2], F is rational and extends holomorphically across the boundary of
Bn. Hence, for the proof of Theorem 1.2, we can assuem that the map is already rational and
extendsholomorphicallyacrosstheboundary. Therationalityandholomorphicalextendability
for proper holomorphic maps into ballshas previously studied by many mathematicians. Here,
we refer the reader to the work of Forstneric [Fo], Baouendi-Huang-Rothschild [BHR], Mir
[Mir], Huang-Ji-Xu [HJX2] and the references therein.
Moreoevr, for any map F Prop (Bn,BN) with N < n(n + 1)/2, F has the following
2
∈
partial linearity: For a generic point p Bn, there is a unique affine subspace S through p
p
∈
of dimension n κ 2 such that the restriction of F to S is a linear fractional map. In
0 p
− ≥
[Hu2], the first author introduced the concept of geometric rank for F Prop (Bn,BN) to
2
∈
measure the degeneracy of the boundary CR second fundamental form of F. (See Section
2 of the paper for a precise definition). A theorem of [Hu2] states that for N < n(n+1), the
2
just aforementioned κ is precisely the geometric rank of F and thus can be used as a more
0
intuitive but equivalent definition of the geometric rank of the map. When N 4n 7, by
≤ −
a linear algebra formula in [Lemma 3.2, Hu2], κ only takes three different value: 0, 1, or 2.
0
The case with κ = 0 is trivial for the map is then linear. The classification result of maps
0
with κ = 1 in [HJX1] gives, as an easy corollary, Theorem 1.3 for maps with geometric rank
0
1. Hence, to prove Theorem 1.3, one needs only to consider maps with geometric rank κ = 2.
0
In fact, we will prove in this paper the following slightly more general Theorem 1.3.
Theorem 1.3. Let F Prop (Bn,BN) have geometric rank κ = 2. Assume that n 7 and
3 0
∈ ≥
3n N 4n 6. Then F is equivalent to a map of the form (G,0) where G is a rational
prop≤er ho≤lomor−phic map from Bn into BN′ with N = 3n.
′
One has quite a different phenomenon even for rational proper maps between balls when
the codimension gets large. As demonstrated by an early work of Catlin-D’Angelo [CD], one
canarbitrarilyassignpartofthecomponents onceoneallowsthecodimension tobesufficiently
large. The reader is referred to [CD] for the precise statement and many references therein.
Finally, we mention that the two main ingredients for the proof of our main theorem here
include: the normal form obtained in [HJX1] and a basic lemma of the first author [Lemma
3.2, Hu1].
3
2 Notations and Preliminaries
In this section, we set up notation and recall a result established in Huang-Ji-Xu [HJX1].
Write H := (z,w) Cn 1 C : Im(w) > z 2 for the Siegel upper-half space. Similarly,
n −
{ ∈ × | | }
we can define the space Rat(H ,H ), Prop (H ,H ) and Prop(H ,H ). Since the Cayley
n N k n N n N
transformation
2z 1+iw
ρ : H Bn, ρ (z,w) = , (2.1)
n n n
→ (cid:18)1 iw 1 iw(cid:19)
− −
is a biholomorphic mapping between H and Bn, we can identify a map F Prop (Bn,BN)
n ∈ k
or Rat(Bn,BN) with ρ 1 F ρ in the space Prop (H ,H ) or Rat(H ,H ), respectively.
−N ◦ ◦ n k n N n N
Parameterize ∂H by (z,z,u) through the map (z,z,u) (z,u+i z 2). In what follows,
n
→ | |
we will assign the weight of z and u to be 1 and 2, respectively. For a non-negative integer m,
a function h(z,z,u) defined over a small ball U of 0 in ∂H is said to be of quantity o (m)
n wt
if h(tz,tz,t2u) 0 uniformly for (z,u) on any compact subset of U as t( R) 0. We use
tm → ∈ →
| |
the notation h(k) to denote a polynomial h which has weighted degree k. Occasionally, for a
holomorphic function (or map) H(z,w), we write H(z,w) = ∞ H(k,l)(z)wl with H(k,l)(z)
k,l=0
a polynomial of degree k in z. P
Let F = (f,φ,g) = (f,g) = (f , ,f ,φ , ,φ ,g) be a non-constant C2-smooth
1 n 1 1 N n
CRmapfrom∂H into∂H withF(·0·)·= 0.−Forea·c·h· p −M closeto0,wewriteσ0 Aut(H )
n eN ∈ p ∈ n
for the map sending (z,w) to (z +z ,w+w +2i z,z ) and τF Aut(H ) by defining
0 0 h 0i p ∈ N
τF(z ,w ) = (z f(z ,w ),w g(z ,w ) 2i z ,f(z ,w ) ).
p ∗ ∗ ∗ − 0 0 ∗− 0 0 − h ∗ 0 0 i
e e
Then F is equivalent to
F = τF F σ0 = (f ,φ ,g ). (2.2)
p p ◦ ◦ p p p p
Notice that F = F and F (0) = 0. The following is fundamentally important for the
0 p
understanding of the geometric properties of F.
Lemma 2.1 ([ 2, Lemma 5.3, Hu99]): Let F be a C2-smooth CR map from ∂H into
n
§
∂H , 2 n N. For each p ∂H , there is an automorphism τ Aut (H ) such that
N ≤ ≤ ∈ n p∗∗ ∈ 0 N
F := τ F satisfies the following normalization:
p∗∗ p∗∗ ◦ p
i
f = z + a (1)(z)w +o (3), φ = φ (2)(z)+o (2), g = w +o (4), with
p∗∗ 2 ∗p∗ wt ∗p∗ ∗p∗ wt p∗∗ wt
z,a (1)(z) z 2 = φ (2)(z) 2.
h ∗p∗ i| | | ∗p∗ |
4
Definition 2.2 ([Hu2]) Write (p) = 2i(∂2(fp)∗l∗ ) in the above lemma. We
A − ∂zj∂w |0 1≤j,l≤(n−1)
call the rank of the (n 1) (n 1) matrix (p), which we denote by Rk (p), the geometric
F
− × − A
rank of F at p.
Define the geometric rank of F to be κ0(F) = maxp ∂HnRkF(p). Define the geometric
rank of F Prop (Bn,BN) to be the one for the map ρ∈1 F ρ Prop (H ,H ). By
∈ 2 −N ◦ ◦ n ∈ 2 n N
[Hu2], κ (F) depends only on the equivalence class of F and when N < n(n+1), the geometric
0 2
rank κ (F) of F is precisely the κ we mentioned in the introduction. In [HJX1], the authors
0 0
proved the following normalization theorem for maps with degenerate geometric rank, though
only part of it is needed later:
Theorem 2.1. ([HJX1]) Suppose that F Rat(H ,H ) has geometric rank 1 κ n 2
n N 0
∈ ≤ ≤ −
with F(0) = 0. Then there are σ Aut(H ) and τ Aut(H ) such that τ F σ takes the
n N
∈ ∈ ◦ ◦
following form, which is still denoted by F = (f,φ,g) for convenience of notation:
f = κ0 z f (z,w); l κ
l j=1 j l∗j ≤ 0
 f =Pz , for κ +1 j n 1;
j j 0
 φ = µ z z + κ0≤z φ≤ f−or (l,k) ,
 lk lk l k j=1 j ∗lkj ∈ S0
 g = w; P (2.3)

 f (z,w) = δj + iδljµlw +b(1)(z)w +O (4),
l∗j l 2 lj wt
 φφ∗lkj(=z,w)κ0=zOφwt(2)=, O(l,k()3)∈ fSo1r, k = 3, ,N (n+(n 1)+ (n κ ))+2
 3k j=1 j ∗3kj wt ··· − − ··· − 0

Here, for 1P κ n 2, we write = , the index set for all components of φ, where
0 0 1
≤ ≤ − S S ∪S
= (j,l) : 1 j κ ,1 l n 1,j l and = (j,l) : j = κ + 1,κ + 1 l
0 0 1 0 0
S { ≤ ≤ ≤ ≤ − ≤ } S { ≤ ≤
N −n− (2n−κ20−1)κ0}. Also, µjl = √µj +µl for j < l ≤ κ0; and µjl = √µj if j ≤ κ0 < l or
if j = l κ .
0
≤
Finally, we recall the following lemma of the first author in [Hu1], which will play a
fundamental role in our proof:
Lemma 2.2. (Huang, Lemma 3.2 [Hu1]) Let k be a nonnegative integer such that 1 k
≤ ≤
n 2. Assume that a , ,a , b , ,b are germs at 0 Cn 1 of holomorphic functions
1 k 1 k −
− ··· ··· ∈
such that a (0) = 0, b (0) = 0 and
j j
k
a (z)b (z) = A(z,z¯) z 2, for j = 1, ,n 2, (2.4)
i i
| | ··· −
Xi=1
where A(z,z¯) is a germ at 0 Cn 1 of a real analytic function. Then A(z,z¯) = k a (z)b (z)
∈ − i=1 i i
0 P
≡
5
3 Analysis on the Chern-Moser equation
Suppose now that F = (f,φ,g) Rat(H ,H ) satisfies the normalization as in Theorem 2.1
n N
∈
with 1 κ n 2. Write the codimension part φ of the map F as φ := (Φ ,Φ ) with
0 0 1
≤ ≤ −
Φ0 = (φℓk)(ℓ,k)∈S0 and Φ1 = (φℓk)(ℓ,k)∈S1. Write Φ(01,1)(z) = κj=01ejzj with ej ∈ Cκ0n−κ0(κ20+1),
ξ (z) = e Φ(2,0)(z), ξ = (ξ ,...,ξ ). We also write in the foPllowing:
j j · 0 1 κ0
φ(1,1)(z)w = e z w, with e = (e ,eˆ ),
∗j j ∗j j j
X
H = H(i1, ,in)zi1 zin−1win = ∞ H(k,j)(z)wj for H = f or φ.
··· 1 ··· n 1
(i1, X,in−1,in) − kX,j=0
···
Here H(k,j)(z) is a polynomial of degree k in z.
In this section, we demonstrate our basic idea of the proof through an easier case. We
proceed with the following lemma, that will be used later:
[h]
Lemma 3.1. Let (Γ (z)) be some holomorphic functions of z. Let µ µ be as
j 1 j κ0,h=1,2 jl j
≤ ≤
[h]
defined in Theorem 2.1. Suppose that for h = 1,2, (Λ ) are defined as follows:
jℓ (j,ℓ) 0
∈S
[h] [h] [h]
1. µ Λ (z) = 2i(z Γ +z Γ ), j < ℓ κ ,
jℓ jℓ j ℓ ℓ j ≤ 0
[h] [h]
2. µ Λ (z) = 2iz Γ (z), j κ ,
jj jj j j ≤ 0
[h] [h]
3. µ Λ = 2iz Γ (z), j κ < ℓ.
jℓ jℓ ℓ j ≤ 0
Then we have
1 4
Λ[1]Λ[2] =4 z 2 Γ[1]Γ[2] µ z Γ[1] µ z Γ[1]
jℓ jℓ | | µ j j − µ µ (µ +µ ) j j ℓ − ℓ ℓ j
(j,Xℓ)∈S0 (cid:16)jX≤κ0 j (cid:17) 1 2 1 2 j<Xℓ≤κ0(cid:0) (cid:1) (3.1)
[2] [2]
µ z Γ µ z Γ .
· j j ℓ − ℓ ℓ j
(cid:0) (cid:1)
Proof. Making use of the formulas between µ and µ , µ in Theorem 2.1, we get, from a
jℓ j ℓ
6
straightforward computation, the following:
1 z 2 z 2
[1] [2] j [1] [2] ℓ [1] [2]
Λ Λ = | | Γ Γ + | | Γ Γ
4 jℓ jℓ µ j j µ j j
(j,Xℓ)∈S0 1≤Xj≤κ0 j j≤Xκ0<ℓ j
1
[1] [1] [2] [2]
+ (z Γ +z Γ ) (z Γ +z Γ )
µ +µ j ℓ ℓ j · j ℓ ℓ j
j<Xℓ κ0 j ℓ
≤
1 1
= Γ[1]Γ[2] z 2 z 2Γ[1]Γ[2]
µ j j | | − µ | ℓ| j j
(cid:16)jX≤κ0 j (cid:17) ℓ≤κ0X,ℓ6=j≤κ0 j
1
[1] [1] [2] [2]
+ (z Γ +z Γ ) (z Γ +z Γ )).
µ +µ j ℓ ℓ j · j ℓ ℓ j
j<Xℓ κ0 j ℓ
≤
Now the lemma follows from the following elementary identity:
µ µ
j z 2Γ[1]Γ[2] + ℓ z 2Γ[1]Γ[2] 2Re z Γ[2]Γ[1]z
µ | j| ℓ ℓ µ | ℓ| j j − j ℓ j ℓ
ℓ j (cid:0) (cid:1)
1
[1] [1] [2] [2]
= µ z Γ µ z Γ µ z Γ µ z Γ .
µ µ j j ℓ − ℓ ℓ j · j j ℓ − ℓ ℓ j
j ℓ(cid:0) (cid:1) (cid:0) (cid:1)
Next we derive the following formula:
Lemma 3.2.
1 1 1 µ µ 2
Φ(3,0)(z) 2 = ξ (z) 2 z 2 jz ξ ℓz ξ . (3.2)
4| 0 | (cid:16)jX≤κ0 µj| j | (cid:17)| | −j<Xℓ≤κ0 µj +µℓ(cid:12)(cid:12)rµℓ j ℓ −rµj ℓ j(cid:12)(cid:12)
(cid:12) (cid:12)
Proof. Since
Im(g)+ f 2 + φ 2 = 0 over Im(w) = z 2, (3.3)
− | | | | | |
we can consider terms of weighted degree 5 to get
zf(4) +zf(4) +Φ(2)Φ(3) +Φ(3)Φ(2) = 0, over Im(w) = z 2, or (3.4)
0 0 0 0 | |
zf(2,1)(z)(u+i z 2)+zf(2,1)(z)(u+i z 2)+Φ(2)(z) Φ(3,0)(z)+( e z )w
| | | | 0 (cid:18) 0 j j (cid:19)
X
(3,0) (2)
+ Φ (z)+( e z )w Φ (z) 0.
(cid:18) 0 j j (cid:19) 0 ≡
X
7
Here, we know f(4)(z,w) = f(2,1)(z)w by the above mentioned normalization. Collecting terms
of the form zαzβu with α = 1, β = 2, we get
| | | |
zf(2,1)(z)+Φ(2)(z) e z = 0, or
0 j j
X
zf(2,1)(z) = (z ,...,z ) ξ(z). (3.5)
− 1 κ0 ·
Collecting terms of the form zαzβ with α = 3 and β = 2, we get
| | | |
κ0
izf(2,1)(z) z 2 +Φ(2)(z)Φ(3,0)(z)+Φ(2)(z) e z (i z 2) 0. (3.6)
| | 0 0 0 j j | | ≡
Xj=1
We thus get
Φ(2)(z)Φ(3,0)(z) = 2i(z , ,z ) ξ(z) z 2. (3.7)
0 0 1 ··· κ0 · | |
Equivalently, we have
(3,0)
1. µ φ (z) = 2i(z ξ +z ξ ), j < ℓ κ ,
jℓ jℓ j ℓ ℓ j ≤ 0
(3,0)
2. µ φ (z) = 2iz ξ (z), j κ ,
jj jj j j ≤ 0
(3,0)
3. µ φ = 2iz ξ (z), j κ < ℓ.
jℓ jℓ ℓ j ≤ 0
Now Lemma 3.2 follows from Lemma 3.1.
Lemma 3.3. φ(3,0) 2 = A(z,z) z 2 with A(z,z) a real analytic polynomial in (z,z).
| | | |
Proof: Collecting terms of weighted degree 6 in (3.3), we get
zf(5) +zf(5) +Φ(2)Φ(4) +Φ(4)Φ(2) + φ(3) 2 + f(3)(z,w) 2 = 0.
0 0 0 0 | | | |
Collecting terms of the form zαzβ with α = β = 3 and applying the normalization for F,
| | | |
(cid:3)
we easily see the proof (cf., (4.14) below) .
Notice that φ(3,0) 2 = Φ(3,0) 2 + Φ(3,0) 2 and there are κ0(κ0+1) κ negative terms in the
| | | 0 | | 1 | 2 − 0
right hand side of (3.2). Also there are (N (κ +1)n+κ0(κ0+1)) components in Φ . Applying
− 0 2 1
Lemma 2.2 and the D’Angelo Lemma (after a rotation if needed), we immediately get the
following:
Corollary 3.4. If N (κ +2)n κ (κ +1)+κ 2, then
0 0 0 0
≤ − −
2 µ µ 1
Φ(3,0)(z) = jz ξ ℓz ξ ,0 , φ(3,0)(z) 2 = 4 ξ (z) 2 z 2.
1 (cid:16)√µj +µl(cid:0)rµℓ j ℓ−rµj ℓ j(cid:1) ′(cid:17)1≤j<l≤κ0 | | (cid:16)jXκ0 µj| j | (cid:17)| |
≤
8
4 Further applications of the partial linearity and the
Chern-Moser equation
We assume in this section that F = (f,φ,g) Rat(H ,H ) satisfies the normalization as in
n N
∈
(3,0)
Theorem 2.1 with κ = 2. Moreover, by what is done in the last section, we assume that Φ
0 1
has been normalized to take the form as in Corollary 3.4. Namely, the only possible non-zero
(3,0) (3,0)
element in Φ is φ .
1 33
In this section, we prove the following result, which will be crucial for our proof of Theorem
1.3:
Proposition 4.1. Assume F is as in Theorem 2.1 with κ = 2 and N 4n 6. Also, assume
0
(3,0) ≤ −
Φ is normalized as in Corollary 3.4. Then the following holds:
0
(4,0) (4,0)
(1): Φ (z) = (φ (z),0, ,0), where
1 33 ···
2 µ µ
φ(4,0)(z) = 1z η 2z η , η = φ(3,0) e , η = φ(3,0) e .
33 √µ +µ (cid:18)rµ 1 2∗ −rµ 2 1∗(cid:19) 1∗ · ∗1 2∗ · ∗2
1 2 2 1
(2): DαΦ(2,1) span (1,0, ,0),eˆ,eˆ for α = 2.
z 1 ∈ { ··· 1 2} | |
(3): DαΦ(1,2) span eˆ,eˆ for α = 1.
z 1 ∈ { 1 2} | |
Here eˆ,eˆ,e ,e , are defined as at the beginning of the last section, and D is the regular
1 2 ∗1 ∗2
differential operator.
This section is devoted to the proof of the above theorem:
Notice that g = w. By a theorem of the first author in [Hu2], we can assume that for any
ǫ = (ǫ ,ǫ )( C2) 0, there is a unique affine subspace L of codimension two defined by
1 2 ǫ
∈ ≈
equations of the form:
n 1 n 1
− −
z = a (ǫ)z +a (ǫ)w +ǫ , z = b (ǫ)z +b (ǫ)w +ǫ , a (0) = b (0) = 0 (4.1)
1 i i n 1 2 i i n 2 i i
Xi=3 Xi=3
such that F is a linear map on L . Here a ,b are holomorphic functions in ǫ near 0. Hence
ǫ j j
we have
∂2H
= 0 for H = f or φ.
∂w2(cid:12)Lǫ
(cid:12)
Namely, for H(Lǫ) = H in=−31ai(ǫ)zi(cid:12)+an(ǫ)w+ǫ1, in=−31bi(ǫ)zi+bn(ǫ)w+ǫ2,z3,...,zn 1,w ,
−
we have (cid:0)P P (cid:1)
∂2H(L )
ǫ
0 =
∂w2 (cid:12)(cid:12)(ǫ1,ǫ2) (4.2)
∂2H (cid:12) ∂2H ∂2H ∂2H ∂2H ∂2H
= a2 + b2 +2 a b +2 a +2 b + .
(cid:16)∂2z12 n ∂2z22 n ∂z1z2 n n ∂z1∂w n ∂z2∂w n ∂w2(cid:17)(cid:12)(ǫ1,ǫ2,0,...,0)
(cid:12)
(cid:12)
9
(1) (1)
Let a (ǫ) and b (ǫ) be the linear parts in a and b , respectively. Set H = f ,f and φ in
n n n n 1 2
(4.2), respectively. We then get
i i
µ a(1)(ǫ)+f(1,2)(ǫ,0,...,0) = 0, µ b(1)(ǫ)+f(1,2)(ǫ,0,...,0) = 0,
2 1 n 1 2 2 n 2 (4.3)
φ(1,2)(ǫ,0,...,0)+e a(1)(ǫ)+e b(1)(ǫ) = 0.
∗1 n ∗2 n
Notice that by Theorem 2.1, F(1,m)(z) depends only on (z ,z ) for any m. It then follows:
1 2
2i 2i
φ(1,2) = e a(1) e b(1) = f(1,2)e f(1,2)e . (4.4)
− ∗1 n − ∗2 n −µ 1 ∗1 − µ 2 ∗2
1 2
This proves Proposition 4.1 (3). Moreover, we obtain
2i 2i
(1,2) (2,0) (1,2) (2,0) (1,2) (2,0)
Φ Φ = f e Φ + f e Φ
0 · 0 µ 1 1 · 0 µ 2 2 · 0
1 2 (4.5)
2i 2i
= (f(I1+2In)z +f(I2+2In)z )ξ + (f(I1+2In)z +f(I2+2In)z )ξ .
µ 1 1 1 2 1 µ 2 1 2 2 2
1 2
Here and in what follows, write I = (0, ,0,1,0, ,0) Zn, where the non-zero element
j
··· ··· ∈
1 is in the jth-position. From (4.5), we also have
ξ ξ
Φ(I1+2In) Φ(2,0) = 2i( 1f(I1+2In) + 2f(I1+2In))
0 · 0 µ 1 µ 2
1 2 (4.6)
ξ ξ
Φ(I2+2In) Φ(2,0) = 2i( 1f(I2+2In) + 2f(I2+2In)),
0 · 0 µ 1 µ 2
1 2
and the following:
ξ ξ
2i( 1Φ(I1+2In) Φ(2,0) + 2Φ(I2+2In) Φ(2,0))
µ 0 · 0 µ 0 · 0
1 2
4ξ ξ ξ 4ξ ξ ξ
= − 1 ( 1f(I1+2In) + 2f(I1+2In))+ − 2( 1f(I2+2In) + 2f(I2+2In)) (4.7)
µ · µ 1 µ 2 µ µ 1 µ 2
1 1 2 2 1 2
ξ ξ ξ ξ ξ ξ
= 4 1 f(I1+2In) 1 +f(I2+2In) 2 4 2 f(I1+2In) 1 +f(I2+2In) 2 .
− µ 1 µ 1 µ − µ 2 µ 2 µ
1(cid:0) 1 2(cid:1) 2(cid:0) 1 2(cid:1)
Considering terms of weighted degree 6 in the basic equation (3.3), we get
2Re zf(5) +Φ(2)Φ(4) + f(3) 2 + φ(3) 2 = 0. (4.8)
0 0 | | | |
n o
10