数理解析研究所講究録
第 819 巻 1993 年 157-166

157

Moduli spaces of holomorphic mappings
into hyperbolically imbedded complex spaces
and hyperbolic fibre spaces (abridged and revised version)
Makoto Suzuki

1.

Introduction.

Classical de Franchis theorem as serts that there are only a finite number of nonconstant holomorphic mappings of a fixed compact Riemann surface $X$ into another fixedcompact Riemann surface of genus greater than one. Since the nonconstant mappings are
regarded as the nontrivial section of the trivial fibre space $(Y\cross X, P_{X}, X)$ , the following
is regarded as a generalization of the above theorem;$Y$

Finiteness theorem for sections. Let be a Riemann surface of finite type. If aholomorphic family $W$ of compact Riemann surfaces with fixed genus greater than one overis non-trivial, then it has only finitely many non-constant holomorphic sections.
$R$

$R$

This has been proved by Manin, Grauert and Miwa independently, and has an impotant
implication in Diophantine problem (it was so-called Mordell’s conjecture over function
fields). In this article, we report the results (cf. [21]) obtained recently about the structureof the moduli spaces of holomorphic mappings considering a higher dimensional analogue in
non-compact case of the above theorems. Noguchi $[14,13]$ and Imayoshi and Shiga [6] gave
another proof of the above finiteness theorem in function theoretic method independently.
In the case of non-compact fibres, Imayoshi and Shiga [6] and Zaidenberg [24] obtained
the finiteness results. Our method is heavily depend on the theory developed by Noguchi[14,15,17] so that we work within the category of hyperbolic geometry. About de Franchis
theorem, many generalizations to higher dimensional cases have already been obtained.
In the case where the target spaces are general type, the finiteness theorem was proved
by Kobayashi and Ochiai [10] and Tsushima [22] (for noncompact case). In the casewhere the target spaces are the quotient spaces of the bounded symmetric domains in thecomplex vector space under some conditions (see \S 2), Sunada [20], Noguchi and Sunada[19], Imayoshi $[4,5]$ and Noguchi [15] obtained the results which contained the one about
the detailed structure of the moduli spaces of holomorphic mappings to such spaces. Under
the assumption that the tangent spaces of the target spaces are negative in some sense,
Kalka, Shiffman and Wong [7] and Urata [23] proved the finiteness theorem for surjective
mappings. In the case of compact hyperbolic K\"ahler manifolds, Horst $[2,3]$ obtained the158
finiteness results. On the other hand, about twenty years ago Lang [11] conjectured thehigher dimensional analogue (in compact hyperbolic case) of the above finiteness theorems
in a standpoint of Diophantine problem, which was recently solved by Noguchi [17](see \S 2
and \S 3 below). Noguchi [17] also conjectured a non-compact version of his theorems, which
is our main theorem.
Letbe a complete hyperbolic complex space. We assume that is hyperbolically
imbedded into an irreducible compact complex space as it’s Zariski open subset. Let $X$
be a Zariski open subset of an irreducible compact complex space. We denote by $Hol(X, Y)$(resp. $Mer_{dom}(X,$ $Y)$ ) the set of all holomorphic (resp. dominant meromorphic) mappingsof $X$ into , where a mapping is said to be dominant if it’s image contains a nonempty
open subset.
$Y$

$Y$

$\overline{Y}$$Y$

Finiteness Theorem for nappings in non-compact case. Let $X$ andabove. Then $Mel_{dom}(X, Y)$ is a finite set.

$Y$

be as

We can also obtain a finiteness theorem of non-constant holomorphic sections and of splitbe as in theting fibre subspaces in the case of non-compact trivial fibre spaces. Let
above Finiteness Theorem and $X$ be a nonsingular Zariski open subset of an irreducible
compact complex space . Let $(Y\cross X, P, X)$ be the trivial hyperbolic fibre space with
the natural projection $P:Y\cross Xarrow X$ . We obtain$Y$$\overline{X}$

Finiteness Theorem for splitting fibre subspaces. Let $X$ and be as above. Then$(Y\cross X, P, X)$ contains only finitely many meromorphically trivial fibre subspaces, andcarries only finitely many holomorphic sections except for constant ones in those bimero$Y$morphic trivializations.
By making use of the above results, we can see the detailed structure of the moduli space$Hol(X, Y)$ .

Throughout this article, we assume that all complex spaces are paracompact andreduced and that all cmplex manifolds are connected. The term (hyperbolic’ is ways
used in the sense of Kobayashi.
$al$

2.

Finiteness of mappings in noncompact case.

Let $X$ be a Zariski open subset of an irreducible compact complex space . Let
be an irreducible complex space. A mapping of $X$ into is said to be dominant if theimage of $X$ by contains a non-empty open subset of . We denote by $Mer_{dom}(X, Y)$(resp. $Hol_{dom}(X,$ $Y)$ ) the set of all dominant meromorphic (resp. holomorphic) mappingsfrom $X$ into . Recently Noguchi [17] proved the following finiteness theorem, which was
conjectured by Lang [11];$\overline{X}$

$Y$

$Y$

$f$

$Y$

$f$

$Y$

Noguchi’s finiteness theorem.

If

$Y$

is compact hyperbolic, then $Mer_{dom}(X, Y)$ is159
a finite set.
First, We prepare for a noncompact version of the above finiteness theorem. Let $X$ be a$:=\tilde{X}-X$
Zariski open subset of a compact complex manifold such that the boundaryof $X$ is a hypersurface with only normal crossings. Let be a connected Zariski open subset
of a compact complex space . Assume that is complete hyperbolic and hyperbolicallyare equipped with compact-open
imbedded into . The spaces $Hol(X, Y)$ andtopology. The extension and convergence theorem of Noguch [15] implies that the extension
mapping $Hol(X, Y)arrow Hol(\tilde{X}, \overline{Y})$ is homeomorphic onto the image of the mapping
and by the natural ’dentification $Hol(X, Y)$ is regarded as the topological subspace of. In fact the following structure theorem due to Noguchi [15] holds:$\tilde{X}$$\partial X$

$Y$

$Y$

$\overline{Y}$$Hol(\tilde{X}, \overline{Y})$$\overline{Y}$$Hol(\tilde{X}, \overline{Y})$Noguchi’s structure theorem. i) The space $Hol(X, Y)$ is a Zariski open subset
where
is the closureof the compact analytic subspaceofand the evaluation mappingof $Hol(X, Y)$ in$\overline{Hol(X,Y)}$$\overline{Hol(X,Y)}$$Hol(\tilde{X}, \overline{Y})$$Hol(\tilde{X}, \overline{Y})$$\Phi:Hol(X, Y)\cross X\ni(f, x)\mapsto f(x)\in Y$

is holomorhpic and extends to a holomorphic mapping$\overline{\Phi}:\overline{Hol(X,Y)}\cross\tilde{X}arrow\overline{Y}$ii) (universality) For a complex space
the natural mapping$T$.

and a holomorphic mapping$\psi$: $T\cross Xarrow Y$ ,$T\ni t\psi(t, \bullet)\in Hol(X, Y)$

is holomorphic.
We set$Hol(k;X, Y):=$ { $f\in Hol(X,$ $Y)$ ; rank $f=k$ },whereis a nonnegative integer. Then we know the following$k$Proposition 2.1 (cf. Noguchi [15]). $Hol(k;X, Y)$ is open and closed in $Hol(X, Y)$ ,hence it carries a structure of complex space. In particular $Hol(n;X, Y)$ is a compact
complex space where $n=\dim Y$ .
For any element $g\in Hol(X, Y)$ , we denote the extension of toby the same letter.
. Put
The next assertion essentially follows from the proof of Noguchi’si)(cf.
structure theorem,
the proof of Theorem 2.8, i) in Noguchi [15], pp. 23-24).
$\tilde{X}$

$g$

$g$

$\partial Y;=\overline{Y}-Y$

Lemma 2.2 (cf. [21]). Let $Z$ be a connected component of $Hol(X, Y)$ Take an$:=\{g\in Z;g^{-}(\partial Y)=\partial X_{0}\}$ .
element $g_{0}\in Z$ and put
and
Then
,

$\partial X_{0}$$Z=Z_{0}$.

$:=g_{0}^{-1}(\partial Y)$$Z_{0}$

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Now our finiteness theorem for mappings of non-compact version is the following. The
use of Lemma 2.2 was pointed out by Professor J. Noguchi and makes the proof of Theorem
2.3 simpler than the original one.

be a complete hyperbolic complex space which isTheorem 2.3 (cf. [21]). Let
hyperbolically imbedded into an irreducible compact complex space and is a Zariski open
subset of Y. Let $X$ be a Zariski open subset of an irreducible compact complex space .
Then $Mer_{dom}(X, Y)$ is finite.
$Y$$\overline{Y}$$\overline{X}$be a resolution
Proof. Assume that $Mer_{dom}(X, Y)$ is not a finite set. Letof singularities due to Hironaka and put $X^{*}:=\alpha^{-1}(X)$ . Then $fo(\alpha|_{X^{*}})\in Mer(X^{*}, Y)$ for$f\in Mer(X, Y)$ . Since
is nonsingular and is hyperbolic, $fo(\alpha|_{X^{*}})\in Hol(X^{*}, Y)$ .
$:=\tilde{X}-X$
Then replacing
by $X$ and putting, we may assume that is acompact complex manifold, $X$ a Zariski open subset of anda hypersurface with only
normal crossings. Assume that $Hol_{dom}(X, Y)$ is not finite. It follows from Proposition 2.1
that $Hol(n;X, Y)$ is a compact complex space with positive dimension where $n=dimY$ .
Take an irreducible component of $Hol(n;X, Y)$ with $\dim Z>0$ and an element $g_{0}\in Z$ .
If we put
, we see from Lemma 2.2 thatfor any $g\in Z$ . Put. Then the subset $Z(x_{0})$ $:=\{\Phi(z, x_{0})\in\overline{Y};z\in Z\}$and take a pointof is a compact hyperbolic complex subspace of , where is the evaluation mapping.
Letbe an irreducible compact hyperbolic complex subspace of containing $Z(x_{0})$ withthe maximum dimension among those subspaces. Take an element $z_{0}\in Z$ at which is: $X_{0}arrow Y$ is proper holomorphic,
nonsingular. Since the mapping
is a compact subvariety in. Let
be the irreducible component of: $X_{0}’arrow Y_{0}$ is surjective. Moreover, the subset
containing . Then the mappingof is an irreducible compact hyperbohc complex subspace containing . Thus
we see that $\Phi(Z\cross X_{0}’)=Y_{0}$ . Because of the finiteness of holomorphic mappings which
map a given point to a given point, together with $\dim Z>0$ , it holds that the mapping$Zarrow Hol_{dom}(X_{0}’, Y_{0})$ is a non-constant mapping. This contradicts Noguchi’s finitenesstheorem, and we complete the proof.
$\overline{X}arrow^{\alpha}\overline{X}$$Y$$X^{*}$$\tilde{X}$$X^{*}$$:=\overline{X},$$\tilde{X}$$\partial X$$\tilde{X}$$\partial X$$Z$$\partial X_{0}$$X_{0}$$:=g_{0}^{-1}(\partial Y)$$g^{-1}(\partial Y)=\partial X_{0}$$:=\tilde{X}-\partial X_{0}$$x_{0}\in X_{0}$$Y$$\overline{Y}$$\Phi$$Y$$Y_{0}$$Z$$(z_{0}|_{X_{0}})^{-1}(Y_{0})$$z_{0}|_{X_{0}}$$X_{0}$$z_{0}|_{X_{0}’}$$x_{0}$$\Phi(Z\cross X_{0}’)$$(z_{0}|_{X_{0}})^{-1}(Y_{0})$$X_{0}’$$Y$$Y_{0}$$\blacksquare$

The following was proved in the case where
Theorem (2.4)).
Corollary 2.4. Let
group of is a finite set.

$Y$

$Y$

is nonsingular by Noguchi (cf. [13],be as in Theorem 2.3. Then the holomorphic automorphism$Y$

Let $D$ be a bounded symmetric domain in the complex vector space and a torsion freearithmetic subgroup of the identity component of the holomorphic automorphism groupof $D$ . Then it is well known that the non-compact quotientis complete hyperbolicand hyperbolically imbedded into it’s Satake compactification (cf. [9]). Thus applying$\Gamma$$D/\Gamma$

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the theorem in this case, we obtain the following, which was first shown by Tsushima [22]within the categoly of general type (see also Noguchi [15]).

Corollary 2.5. Let $X$ be as in Theorem 2.3. Then $Mer_{dom}(X, D/\Gamma)$ is a finite set.

Finiteness of nontrivial sections and of trivial fibre subspaces.

3.

be irreducible compact complex spaces and:a surjectiveLetandholomorphic mapping with connected fibres. Let be a nonsingular Zariski open subset. Put
of and$\overline{R}$$\overline{W}$$\overline{\Pi}$$\overline{W}arrow\overline{R}$$R$$\overline{R}$$\partial R:=\overline{R}-R$$W$$:=\overline{W}|_{R}=\overline{\Pi}^{-1}(R))\Pi$$;=\overline{\Pi}|_{W}$.

$:=\Pi^{-1}(t)$ is irreducible for $t\in R$ . We denote by
Suppose that each fibre
all holomorphic sections of the fiber space $(W, \Pi, R)$ .
$W_{t}$

the set of$\Gamma$

Definition 3.1 (cf. Noguchi [14], \S 1). We call a fibre space $(W, \Pi, R)$ a hyperbolic
for $t\in R$ are hyperbolic. We say that the fibre space
fibre space if all the fibres$(W, \Pi, R)$ is hyperbolically imbedded intoalong
if for any
there areisneighborhoods and of in such that is relatively compact in andhyperbolically imbedded into.
$W_{t}$$(\overline{W}, \overline{\Pi}, \overline{R})$

$U$

$V$

$t$

$\overline{R}$$t\in\partial R$$\partial R$

$V$

$U$

$W|_{U-\partial R}$$\overline{W}|_{V}$Noguchi proved the following global triviality for normal hyperbolic fibre spaces (Noguchi[14], Main Theorem (3.2) and Noguchi [17], Theorem A).

Noguchi’s trividity Theorem for hyperbolic fibre spaces. Let $(W, \Pi, R)$ bea hyperbolic fibre space. Suppose that $(W, \Pi, R)$ is hyperbolically imbedde into a compact

and that $W$ is normal. If there exists a $pointt\in R$ such$that\Gamma(t):=\{s(t)\in W_{t} : s\in\Gamma\}$ is Zariski dense in, then $(W, \Pi, R)$ is holomorphically: $W_{t}\cross Rarrow W$ such that $P=\Pi oF$
trivial, $i.e.$ , there is a biholomorphic mappingwhere $P:W_{t}\cross Rarrow R$ is the natural projection.

fibre space$(\overline{W}, \overline{\Pi}, \overline{R})$along$\partial R$$W_{t}$$F$

Noguchi considered hyperbolic fibre spaces in a more general setting and obtained thefollowing finiteness theorem for sections and for trivial fibre subspaces of a hyperbolic fibre
space, which gave an affirmative answer to the higher dimensional analogue of Mordell’s
conjecture over function fields posed by Lang [11] (cf. Noguchi [17], Theorem and it’scorrection).
$B$

Definition 3.2. We say that a fibre space $(W, \Pi, R)$ is meromorphically trivial if$(W, \Pi, R)$ is bimeromorphically isomorphic to some trivial fibre space over.
$R$

Theorem 3.3 (cf. [21]). Let $(W, \Pi;R)$ be a hyperbolic fibre space. Assume that$(W, \Pi, R)$ is hyperbolically imbedded into some compact fibre spacealong.
$(\overline{W}, \overline{\Pi}, \overline{R})$$\partial R$

162
Then $(W, \Pi, R)$ contains only finitely many meromorphically trivial fibre subspaces withpositive dimensional $fibres_{f}$ and carries only finitely many holomorphic sections except for
constant ones in those bimeromorphic trivializations.

In fact, in the proof it is shown that the normalization of each irreducible fibre subspace$W’$ ofW whose sections are dense in the total space becomes the trivial fibre subspace andthat the normalization of the space of all sections of $W’$ gives the one of each fibre except
for a proper subvariety of .
$R$

Corollary 3.4 (cf. [21]). Let $(W, \Pi, R)$ be as in Theorem 3.3. If there is a point$t\in R$ such that, then the fibre space $(W_{N}, \Pi_{N}, R)$ obtained
is Zariski dense in
by taking the normalization of $W$ is a holomorphically trivial fibre space.
$W_{t}$$\Gamma(t)$Example 3.5. We give an example of the non-normal hyperbolic fibre spaces with
infinitely many sections which are locally nontrivial. The author wishes to thank Professor
T. Ueda for his help in constructing this example.
Let be a compact Riemann surface of genus greater than one. Let be a holomorphic
automorphism of which is not the identity mapping and be the identity mapping of .
Put$\hat{\sigma}(t)=(\sigma(t), t)\in R\cross R$ for $t\in R$$R$$\sigma$$R$$R$$\iota$and$\iota\wedge(t)=(t, t)\in R\cross R$

for $t\in R$ .

$y_{2}\in R\cross R,$
We define an equivalence relation on
as follows: forif and only
$W$
$t\in
R$
if there exists a point
andsuch that. Put $:=R\cross R/\sim$ . Then$W$
we see thatis a complex space and that the projection $\beta:R\cross Rarrow W$ is holomorphic.
Let
be the projection such that $\Pi 0\beta=P_{2}$ on $R\cross R$ where: $R\cross Rarrow R$ is thesecond projection. Then $(W, \Pi, R)$ is a hyperbolic space with compact hyperbolic fibres
and carries infinitely many sections which come from the trivial fibre space $(R\cross R, P_{2}, R)$
through the projection . The fibre space $(W, \Pi, R)$ is locally nontrivial. In fact, supposethat there exists a local triviahzation :where is an open set inandis an irreducible curve. We take the normalizations of the domain and the imageof the localization and consider the lifting of the mapping to the normalizations. Then
we see that generates infinitely many holomorphic automorphisms of . This is absurd
since is compact hyperbolic.
$R\cross R$$y_{1}\sim y_{2}$$y_{1},$$y_{1}=\hat{\sigma}(t)$$y_{2}=\iota^{\wedge}(t)$$\Pi$$P_{2}$$\beta$$W|_{U}arrow^{\underline\simeq}W_{0}\cross U$

$R$

$U$

$\varphi$$W_{0}$$\tilde{\varphi}$$\varphi$$R$$\tilde{\varphi}$$R$

Next we consider about finiteness of trivial fibre subspaces in the case where all thefibres are noncompact. We treat only trivial fibre spaces. Let be an irreducible compact
complex space and $X$ a nonsingular Zariski open subset of . Making use of the idea ofthe proof of Theorem 3.3 and of Theorem 3.3 itself, we get the follwong$\overline{X}$$\overline{X}$Theorem 3.6 (cf. [21], Theorem 4.1). Let $X$ be as above. Let$Y$

be an irreducible

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is hyperbolically imbedded into somecomplete hyperbolic complex space. Suppose thatcompact complex spaceandis Zariski open in Y. Then the trivial fibre space$X,$ $P,$ $X$ ) contains only finitely many meromorphically trivial
fibre subspaces where is
the natural projection ( $i.e$ ., any meromorphically trivial fibre subspace ofisa trivial fibre subspace of one of them) and carries only finitely many holomorphic sections
except for constant ones in those bimeromorphic trivialization.
$Y$

$Y$

$\overline{Y}$$(Y\cross$$P$$(\overline{Y}\cross\overline{X}, P, \overline{X})$Also in the case of non-trivial fibre spaces, under some condition on the imbeddedness of
total space we can prove the finiteness theorem of above type.
Structure of the moduli space $Hol(X, Y)$ .

4.

We can obtain some information about the moduli spaces of holomorphic mappingsin our situation. Let $X$ be a Zariski open subset of a compact complex manifold . We$:=\tilde{X}-X$
is a hypersurface with only normal crossings. Let be a Zariski
assume that
open subset of an irreducible compact complex . Assume that is complete hyperbolicand hyperbolically imbedded into . Let $Z$ be a connected component of $Hol(X, Y)$ .
is a compact complex subspace ofandThen the closure 7 of $Z$ in$Z$ is a Zariski open subset of
by Noguchi’s structure theorem.
$\tilde{X}$$Y$$\partial X$$Y$$\overline{Y}$$\overline{Y}$$Hol(\tilde{X}, \overline{Y})$$Hol(\tilde{X}, \overline{Y})$$\overline{Z}$Proposition 4.1 (Noguchi [15], Miyano and Noguchi [13]).
i)
is complete hyperbolic and hyperbolically imbedded intoii) If is quasi-projective algebraic and carries a projective compactification
is hyperbolically imbedded into , then is quasi-projective.
$Z$$\overline{Z}$

,

$Y$

$Y$

$\overline{Y}$such that$Z$$\overline{Y}$Proposition 4.2. The space $Hol(k;X, Y)$ is compact for $k>\dim\partial Y$ where.

$\partial Y$$:=$$\overline{Y}-Y$

The proof is same as in Noguchi [15], Theorem (3.3), i). We obtain an estimate of thedimension of moduli.

Theorem 4.3. Let
be an irreducible component of $Hol(X, Y)$
non-constant holomorphic mapping, then $\dim Z\leq\dim$ Y–l.
$Z$

,

If

$Z$

contains a

Proof. Since 7 is compact, for any $x\in X$ the extension of the evaluation mapping$\overline{\Phi}(\bullet, x)$is finite. Thus
from the assumption of Z.

$\dim\overline{Z}\leq\dim\overline{Y}$:$\overline{Z}\ni z-z(x)\in\overline{Y}$. The case where $\dim Z=\dim Y$ contradicts Theorem 2.3$\blacksquare$is a noncompact quotient of a bounded symmetric domain in the
In the case wherecomplex vector space by a torsion free arithmetic subgroup of the identity component of
$Y$

164
it’s holomorphic automorphism group, more effective estimates were obtained in Sunada[20], Theorem , and Noguchi [15], Theorem (4.7), (4.10) (see for the compact quotientcase Noguchi and Sunada [19], and Imayoshi $[4,5]$ ).
$B$

Proposition 4.4. Suppose that codim
be an element in $Hol(n-$Let$1;X,$) such that $z_{0}(X)$ is relatively compact in $Y(n=\dim Y)$ . Then $\dim_{z_{0}}Hol(n-$$1;X,$) $=0$ .
$\partial Y\geq 2$ ,$z_{0}$$Y$$Y$

This follows from Proposition 4.2 and Noguchi’s finiteness theorem.

Theorem 4.5 (cf. [21], Corollary 4.2). Take $f\in Hol(X, Y)$ , If $f(X)$ is not relatively
compact in , then the dimension of the irreducible component of $Hol(X, Y)$ which contains
is not greater than the dimension of.
$Y$$\partial Y$$f$

In the case whereY is the quotient space ofabounded symmetric domain byatorsion freearithmetic discrete subgroup of the identity component of the holomorphic automorphismgroup, Theorem 4.5 was obtained in Noguch [15], Theorem 4.7 (iv).
Making use of the theory of harmonic mappings, Miyano and Noguchi proved thefollowing sharper version of the finiteness theorem for mappings under K\"ahler condition.
Theorem 4.6 (cf. [13], Theorem 2.15). Let $X$ be a Zariski open subset of a compact
Kahler manifold and be a quasiprojective algebraic manifold which carries a projective
compactificationsuch thatis hyperbolically imbedded into Y. Suppose thatcarriesa complete Kahler metric whose Riemannian sectional curvatures are non-positive andholomorphic sectional curvatures are negatively bounded away from zero. Then$Y$$\overline{X}$$Y$$\overline{Y}$$Y$$\dim Hol(k;X, Y)\leq\dim Y+k$ .

Note that in the above theorem becomes complete hyperbolic and then $Ho1(k;X, Y)$ has
the complex structure with universal properties. In the case that $X$ andare compact,
algebraic manifolds, Goloff [1] obtained the same result under some different negativity
conditions of . Recentry, Imayoshi [5] proved the same result in the case thatisa Carath\’eodory hyperbolic manifold and that $X$ is a projective algebraic manifold. A
complex manifold
is said to be Carath\’eodory hyperbolic if
has a covering whose
Carath\’eodory pseudo-distance is actually a distance (cf. Kobayashi [8], p. 129).
$Y$

$Y$

$Y$

$Y$

$Y$

$Y$

Conjecture 4.7. Let be a complete hyperbolic complex space which is hyperbolically
imbedded into an irreducible compact complex space
and is a Zariski open subset of .
$X$Let
be a nonsingular Zariski open subset of an irreducible compact complex . Then$Y$$\overline{Y}$$\overline{Y}$$\overline{X}$$\dim Hol(k;X, Y)\leq\dim Y+k$ .

165
Relating to this conjecture, the following seems to be true (cf. [13]).
Conjecture 4.8 Under the same assumptions on $X$ andcomplex space $Hol(k;X, Y)$ is isometrically immersed into
metric.

$Y$$Y$

as in Conjecture 4.7, thewith respect to Kobayashi

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DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, HIROSHmIA UNIVERSITY, HIGASHIHIROSHIMA 724 JAPAN