K. AZUKAWA
KODAI
MATH. J.
8 (1985),
79—89
AN
INTRINSIC
FIBRE
METRIC
ON THE n-ΎH
SYMMETRIC
TENSOR
POWER
OP THE
TANGENT
BUNDLE
BY
KAZUO
AZUKAWA
0.
Introduction.
Let H(M) be the Hubert space consisting of all square-
integrable
holomorphic m-forms on an m-dimensional complex manifold M. The
Bergman
form K is defined as a specific holomorphic 2m-form on the product
manifold MxM, where M is the conjugate complex manifold of M. Let z—
(z
1
,
•••, z
m
) be a coordinate system with defining domain U
ZJ
and k
z
be the Berg-
man
function
relative
to z, i.e. K(p,
p)=k
z
{p){dz
x
Λ
Λdz^pΛidz
1
Λ
Λdz
m
)
p
,
p^U
z
. In
general,
k
z
^0. In
Kobayashi
[4], the
following
conditions
are con-
sidered
:
(A.I) For
every
p^M,
there
exists
a^H(M)
such
that
a(p)Φθ.
(A.2) For
every
non-zero
tangent
vector
X at p^M,
there
exists
a^H(M)
such
that
a(p)=0
and
X.a(p)Φθ.
Suppose (A.I) holds. Then k
z
>0 for
every
z, and the Bergman pseudo-metric
g, with components
gab—3
α
9
6
.log
k
z
, is defined. Furthermore, the
following
is
known ([4]):
(Ki) g is a metric if and only if (A. 2) holds.
If M
satisfies
(A.I) and (A.2), and if Rated are the components of the hermitian
curvature tensor of the Bergman metric, then the
following
are known ([4]):
(K
2
)
Set
R
aC
bd=Rabcd+gabgcd+gadgcb-
Then ΣRacbdV
a
v
c
v
b
ϋ
d
^ for
every
{V\ -,
V
m
)EΞC
m
.
(K
3
)
RacM=k-\kacba~k-
1
k
ac
k
M
)--k-
2
Y
i
g
U
{k
acl
-- k^k
ac
k{){k
sM
-k^kuk
s
),
where k = k
z
, k
ac
=d
a
d
c
.k, etc., and (g
ts
)=(gab)~
1
>
In
the preceding joint paper [2] with Burbea, conditions
(C
J are defined so that
(C
o
)
(resp. (CO) coincides with (A.I) (resp. (A.2)). Furthermore, under assump-
tion
(C
o
), non-negative functions
μ
Oιn
,
which are biholomorphic invariants, on
the
tangent bundle are introduced.
In
the present paper, we
first
note (Proposition 1.2) that the functions
μ
Qi7l
on
the tangent bundle are, in general, upper semi-continuous, and show (Theo-
rem
2.1) that when M
satisfies
condition (C
o
) there
exists
a unique
fibre
pseudo-
metric
g
(n)
on the n-th symmetric tensor power
S
n
T(M)
of the tangent bundle
Received March 15, 1984
79
80 KAZUO AZUKAWA
T(M)
for n^N such that
(n\)-*μ
o>n
(X)=g^(X
n
,
X
n
),
X^T(M);
in
particular, the pseudo-metric g
a)
coincides with the Bergman one stated before.
In
addition, if
Msatisfies
also (d), •••, (dι-i), then g
(n)
is differentiate (Theorem
2.5), and assertion (K
x
) is generalized as
follows
(Theorem 2.6): g
(n)
is a metric
if and only if (CJ holds. Finally, we consider the curvature of the hermitian
connection
of the hermitian vector bundle
(S
n
T(M),
g
(n)
) in the sense of
Kobayashi and Nomizu [6]. In
view
of Fuks [3], the component
gTbcd
coincides
with R
ab
^ά/A
given
in (K
2
), and (K
2
)
gives
a relationship between the curvature
of g
ω
and the metric g
(2)
. We generalize this relationship to the one between
the
curvature of g
in)
and the metric
g
(n+1)
(Theorem 3.1). The proof of Theo-
rem
3.1 is done by
observing
formula (K
8
) and by the use of a recurrence
formula (Proposition 3.5) for the components of g
{n)
.
1. Preliminaries. Throughout this paper, we are concerned with a
fixed
paracompact
connected complex manifold M of dimension m. The term " coordi-
nate
z" stands for a local holomorphic coordinate system
z—{z
x
,
•••, z
m
) of M
with defining domain U
z
. For simplicity, we set d
z
a
=d/dz
a
(α=l, •••, m), and
dz=dz
1
Λ-~Λdz
m
. For a multi-index Λ=(a
ly
•••, α
n
)eMI(n)={l, •••, m)
n
, set
S
z
A=d
z
ai
•••
d
z
a
n
.
In particular, MI(0)={$4}, and dφ means the identity operator
acting on functions on U
z
. For a constant vector v=(v
1
, •••, v
m
) in C
m
, set
d
z
v=Σ,ΐ=ιV
a
d
z
a.
The powers
(d
z
v
)
n
(n=0, 1, •••) are naturally defined. We denote
by M the conjugate complex manifold of M, and denote by p: M^p^-^p^M the
conjugation. For a coordinate z with defining domain U
z
, we denote by z the
conjugate coordinate of z with defining domain U
z
, i.e. z(p)=z(p) for p^U
z
.
We denote by H{M) the separable Hubert space consisting of all holomorphic
m-forms a on M which
satisfy
\\a\\
2
=(V
ZI
ϊ
m2
/2
m
)\
αΛά<+oo,
and denote by
(,) the hermitian inner product on H(M) corresponding to the norm || ||. There
exists
a unique (2m, 0)-form K, called the
Bergman
form,
on the product manifold
MxM such that K( , p)/dz
p
<=H(M) and a(p)/dz
p
=(a, K( , p)/dzp) for
every
p^M
and a^H(M), where z is a coordinate around p (cf., e.g., [2; Corollary
2.6]). Thus, (ljf, /θ)*iίΓ is an (m, ?n)-form on M. For
every
coordinate z, we
call the function
k
z
={l
M
,
ρ)*K/dz/\d~z on U
z
the
Bergman
function
of Mrelative
to
2τ. That is
K(p,
p)=k
z
(p)dz
p
Λdz
p
, ί (ΞU
Z
.
The
Bergman functions are non-negative (cf., e.g., [2; Proposition 2.7]). It
holds (cf., e.g., [2; Proposition 2.5]) that for
every
multi-index A, the m-form
Kχ(p)=d
Έ
A
.K(-, p)/dz
p
belongs to H{M), and that for
every
a^H(M),
(1.1) 3i.α(/0=(α,
Kl(p))dz
p
.
AN
INTRINSIC
FIBRE METRIC 81
In
particular, if A and B are multi-indices, then
(1.2)
(Ώ(ί),
Ai(ί))=3£9Ϊ.fe,(ί).
Let
neZ+
be a non-negative integer. For
every
p^M, set
where z is a coordinate around £. The subspace H
n
(p) does not depend on the
choice of z. Let
X<ΞT
P
(M)
be a tangent vector at p. For a coordinate z
around
p, represent X as (d$)
p
for some v^C
m
. Then (3|)
π
is a differential
operator
on
U-
Z
=TΓ
Z
,
and
K
z
vn
{p)^{d^
n
.K(-,
p)/dz
p
belongs to H(M). Set
^
n
(*)=max{|
(#{„(/>),
tf)|
2
; a^H
n
(p), \\a\\=l}(dzΛdϊ)
p
.
Then
the (m, m)-form μ
n
(X) does not depend on the representation of X—(d
z
v
)
p
in
terms of z ([2; Proposition 3.7]).
We recall a lemma on a pre-Hilbert space // over C. We denote by
G(x
lt
-" x
n
) the Gramian of a system (#1, •••, x
n
) in //" (especially G(φ)
—
l).
LEMMA
1.1 ([2; Lemma 3.9]). Let (x
u
•••, %
w
) (neZJ ^ α
linearly
inde-
pendent
system
in H, and let x
n+1
^H. Then the maximum of the set
{\(y, ^τz+i)l
2
; y^ {xi,
•••,
Xn}
2
-,
II:v
11=1}
coincides
with
G{x
u
•••,
x
n+1
)/G(x
lt
•••,
x
n
).
Set MII(n)={(α
1
, •••, αJeMI(n) a
1
Sa
2
^
•••
^βj. We denote by ^
TO
—
( ) the cardinality of the set
\J?=oMΠ(/),
and fix a numbering Φ of
UΓ«oMΠ(/)
such that
MΠ(n)={Φ(^
n
_
1
+l),
•••,
Φ(<p
n
)}
For a sequence (j
lf
•••,
y
u
, s, 0 of positive integers, set
(1.3)
By (1.2),
-C
z
(ji,
•••, ju)(p) is the transpose of the Gram matrix of the system
(K
z
ΦUl)
(p),
•••, K
z
φ{ju)
{p)), and
L
2
(;Ί,
•••, y
tt
)(^) is its Gramian.
Now,
let /„,, be the function on
U
2
xC
m
defined by
j"n(@ί)p)=/».,(/>,
v)(dzΛlz)
p
, {p,
v)<ΞU
z
xC
m
.
If {K
Φ(Jl)
(p), •••, K
z
φ
(ju
)(p)} is a maximal linearly independent subset of
{Ώ(ί);
^e\JjWMΠ(y)}, then Lemma 1.1, together with (1.2), implies that
(1.4)
82 KAZUO AZUKAWA
Here
C
Λ
=n \/n,! •••
n
m
!
and v
A
=v
a
ι
•••
v
a
*
(A=(a
u
- ,
fl»)eMII(n),
v=(v\
•••,
v
m
)
<=C
m
),
where
n
v
is
the
cardinality
of
the
set {/e {1, •••, n}
a.y=-v\
{v—\, •••, m).
PROPOSITION
1.2. 77z<?
function
f
n>z
is
upper
semi-continuous
on
U
z
xC
m
.
Proof.
The proof is reduced to the
following
lemma.
LEMMA
1.3. Let f be the
function
on the
power
H
n+1
of a
pre-Hilbert
space
H
over
C
given
by
• , *
n+1
)=max{|(;y,
x
n
+i)\
2
;
y^{Xi,
— , Xn)
L
, 11^11=1}•
Then f is
upper
semi-continuous
on H
n+1
.
Proof.
Let
x°=(x°
lf
•••,
x°
n+1
)^H
n+1
be
fixed,
and let
{xj
(1)
,
— ,
*J
(tt)
}
be a
maximal linearly independent subset
of {x\
f
•••, x°
n
}.
Then
GU
ff(
i),
•••, x
<,(«))
is positive
in a
neighborhood
of x°. So, by
Lemma
1.1 we
have
limsup/(x)rglimsup max{|(^, x
n
+i)\
2
',
y^ {xσω, •••,
^σcw)}
1
, 11^11 —
1}
x->x° x-*x°
=limsup G(x
σ
a),
•••,
x
σ(U
), Λ:
n+
i)/G(Λ:
σα
),
•••,
*
a
(
tt)
)
=/U°).
as desired.
2.
An
intrinsic
fibre
pseudo-metric
on the
holomorphic
vector
bundle
S
n
T(M).
For n^Z+ and
p<=M,
we consider the
following
condition:
(C
n
)p For
every
non-zero vector
(ζ
Λ
)A&Miun)
of dimension ί J, there
exists
a^H
n
(p) such that Σ^dίί
Condition
(C
n
) stands for that (C
n
)
p
hold for all p^M. From (1.1), we reduce
the
following
([2; Lemma 3.4]):
ί Conditions (Cj)
p
(;=0, •••, n) hold if and only if the
(2.1) I set {K
A
(p); A^\J^
0
Mll(j)} is linearly independent,
[ or -Γ
2
(l, •••,
ψ
n
){p)
is positive definite.
Now, suppose M
satisfies
condition (C
o
). Then (1.4) implies that μ
o
(X) —
k
z
(p)(dzΛdz)
p
for
every
X<ΞT
P
(M),
and that k
z
>0 on ί/
β
. So, [0, +oo)-valued
functions
μo
>n
—μn/μo
(neiV) on the holomorphic tangent bundle T(M) are
well
defined. Every function
μ
0>n
is upper semi-continuous on T{M) (by Proposition
1.2) and
satisfies
the
following:
μ
o>n
(ξX)=\ξ\
2n
μ
o
,n(X) for
ZGT(M)
and
£e=C;
therefore (μo
>n
)
1/2n
is an upper semi-continuous Finsler pseudo-metric on M.
Moreover,
μ
0>n
are biholomorphic invariants, i.e. μ
o
,
n
(X)=μo,n(f*X), X^T(M)
for
every
biholomorphic mapping / from M onto another complex manifold ([2;
AN
INTRINSIC
FIBRE
METRIC
83
Proposition 3.2]).
We denote by S
n
T
p
(M) (resp. S
n
T(M)) the n-th symmetric tensor power of
T
P
(M) (resp. T(M)). S
n
T(M) is a holomorphic vector bundle over M, and
{3^; ^4eMII(tt)} forms its local frame on U
z
.
We shall
show
the
following
assertion.
THEOREM
2.1. // a
complex
manifold
M
satisfies
condition
(C
o
),
then
for
every
n^N and p^M
there
exists
a
unique
hermitian
pseudo-inner-product
g
(n)
(-,~) on the
space
S
n
T
p
(M)
such
that
(2.2)
(n\)-*μ
θ!n
(X)=g<
n
\X
n
,
IP), Xt=T
p
{M),
where
X
x
— X,
X
J
=X-X
J
~
1
(the
symmetric
tensor
product).
Furthermore,
the
fibre
pseudo-metric
g
(n)
on S
n
T(M) is
biholomorphic
invariant,
i.e.
g
in)
(Y,
Y) —
g
{n)
(f*Y,
f*Y) for Y^S
n
T(M) and for any
biholomorphic
mapping
f
from
M
onto
another
complex
manifold.
Remark 2.2. The constant (n !)~
2
in the formula (2.2) is chosen so that when
M is the unit
disk
{$ΪΞC
\ξ\ <1} in C the inner product £
(n)
( ,
τ
) on S
n
T
0
(M)
at the origin OeMhas the simplest form,
g
w
{X
n
,
T")=n +
1
for X=(d/dξ)
o
ζΞ
T
0
(M) (cf. [1]).
Proof of Theorem 2.1
(Existence).
Let
{Kφ
{Jl)
(p),
•••,
Kφ
(Ju)
(p)}
be a
maxi-
mal
linearly
independent
subset of
{K
z
A
(p);
A^\J?=?Mϊl(j)}. By (1.4) we have
So,
the
function
g
{n)
{ ,
τ
)
defined
by sesqui-bilinearity and by the
requirement
(2.3)
g
(n)
((d
Φω
)
p
,
(d
Φa)
)
p
)
=(τi
\)-
2
L
2
(j
u
...,
jJipr'k^pr'L^u
•- , Ju s, t)(p)
has the desired property. Thus, the existence is proved.
To complete the
proof,
we prepare two lemmas.
LEMMA
2.3. Let
R=*Σn=oRn
be a commutative, associative, graded algebra
over
C. For every n^N, there exists a linear form F
n
(t
0
, t
lf
•••, t
3n
-i) on C
3n
such
that
(x
n
,
y
n
)n=F
n
(f(l),
f(p), •..,
fip*"-
1
))
for x, y^Ri and for any sesqui-bilinear form (,)
n
on R
n
, where p=z
e
2π
^
/
-
1
^
n
anc
[
)
n
,
(χ+ξy)
n
)n,
Proof.
Since