Hyperbolic Complex Spaces (Grundlehren der mathematischen Wissenschaften)

3 Mappings into Metric Spaces

11

(b) :::} (c). Let rr: X x F ---+ F be the projection. Since CP~I(K x L) = {(x, 1) E K x F; I(x) E L},

we have F K . L = rr(cp~1 (K x L». Since cp~l (K x L) is assumed to be compact, is also compact. (c) :::} (d). A sequence {fn} C F converges to 00 in C(X, Y*) if and only if for given compact subsets K C X and LeY there exists an integer N such that In(K) n L = 0 for n > N. Assume that {fn} does not converge to 00. Then there exist compact subsets K C X and LeY such that h, (K) n L =I 0 for infinitely many n. Hence, In E F K . L for infinitely many n. Since F K •L is compact, a subsequence of {f,,} converges to an element of F K . L C F. (d) :::} (a). Any sequence {f,,} C F converges either to an element of For to the constant map 00. In the latter case, it is compactly divergent. 0 FK,L

A family Fe C(X, Y) is said to be normal if its closure F in C(X, Y) satisfies one of the equivalent conditions in (1.3.5). We prove the following result due to Wu [1]. (\.3.6) Theorem. Let X and Y be as in (1.3.2) and assumejitrther that Y is strongly complete. Then VeX, Y) is a normal family.

Proof Let F = {f,,} c VeX, Y) be a sequence which is not compactly divergent. Then there exist compact subsets K C X and LeY such that for infinitely many h, of F, we have In (K) n L =I 0. By taking a subsequence we may assume that this holds for all In E F. Let 8 be the diameter of K with respect to d x . Then the diameter of In(K) with respect to d y is at most a. Hence, all j,,(K) are contained in the closed a-neighborhood of L. Choose x E K and apply (1.3.3). Then F is relatively compact and contains a subsequence which converges in C(X, Y). 0 (1.3.7) Remark. We note that (1.3.6) is stronger than (1.3.4). In fact, as we see from (1.3.5), !f Fe C(X, Y) is normal, then its closure F is locally compact, (see Wu [I D. For later use we state an immediate consequence of (1.3.2) in the following form. (1.3.8) Theorem. Let X be a locally compact, separable space with a pseudodistance d x . Let Y C Z be locally compact metric spaces with distance functions d y and d z such that d z :::: d y on Y. Then VeX, Y) is relatively compact in C(X, Z) if and only if, for each x E X, the set {f(x); I E Vex, Y)} is relatively compact inZ.

Proof Apply (1.3.2) to the subfamily F

= VeX,

Y)

c

C(X, Y).

o

(1.3.9) Corollary. Let X with d x and Y C Z with d y and d z be as in (1.3.8). IfY is relatively compact in Z, then VeX, Y) is relatively compact in C(X, Z). Let Y be a locally compact metric space with distance dy, and y* = Y U 00 the one-point compactification of Y. If Y is not strongly complete with respect to d y , then we can extend d y to a distance function dy' on y* by setting

12

(1.3.10)

Chapter I. Distance Geometry

dy*(y, 00) = sup{r: (y' E Y; dy(y, y')

:s r}

is compact}.

(1.3.11) Theorem. Let Y be a locally compact metric space with distance d y, and y* = Y U 00 the one-point compact!fication of Y. Let X be a locally compact space with pseudo-distance d x . Then VeX, Y) is relatively compact in C(X, Y*), i.e., every sequence 1" E VeX, Y) has a subsequence which converges to a map [ E C(X, Y*). If Y is strongly complete with respect to d y, then every sequence j~ E VeX, Y) either has a subsequence which converges to a map [ E C(X, Y) or converges to the constant map 00. Proof (i). Consider first the case where Y is strongly complete. By (1.3.6) VeX, Y) is a normal family. By (1.3.5), every sequence 1" E vex, Y) either contains a subsequence which converges in C(X, Y) or is compactly divergent, i.e., converges to the constant map 00. (ii). Assume that Y is not strongly complete. Let Y* be its one-point compactification and d y * the extended distance function, see (1.3.10). Then the assertion follows from (1.3.9). 0 Given a pseudo-distance d y on a topological space Y and a closed subset ,1 of Y, we say that d y is a distance modulo ,1 if d y (y, y') > 0 unless y = y' or y,y' E ,1. (1.3.12) Proposition. Let X and Y be topological spaces with inner pseudodistances d x and d y, respectively. Let ,1 be a (possibly empty) closed subset ofY. Assume that d y is a distance modulo ,1. Let [: X --+ Y be a distance-decreasing mapping. If anyone of the jollowing three conditions is satisfied, then d x is a distance modulo [-I (,1). (1) Given x, x' E X, x =1= x' with [(x) = [(x'), there is a neighborhood V of [(x) in Y such that x and x' are in different connected components of I-I (V); (2) Every x E X has a neighborhood V such that [ is a homeomorphism from V onto an. open set I(V); (3) X is locally compact, and jar every y E Y the inverse image I-I(y) is finite. Proof Let x, x' E X with x =1= x' and x ¢. [ - I (,1). If [(x) =1= I(x'), then dx(x, x') ~ dy(f(x), [(x'» > O. We consider the case where [(x) = [(x'), and set y = [(x) = I(x'). Assume dx(x, x') = O. (1) Let V be a neighborhood of I (x) in Y described in condition (1). Since [(x) ¢. ,1, without loss of generality we may assume that V is an t:-neighborhood of [(x) for some t: > O. Let V be the t:-neighborhood of x in X. Then V contains x' and is finitely arcwise connected by (1.1.7). Let y be a rectifiable curve from x to x' in V. Since [ is distance-decreasing, [ maps V into V, and the curve y lies in [-I (V). This is a contradiction. (2) Let U be a neighborhood of x described in condition (2). Let V = I(V), and apply (1). (3) Let V be an open neighborhood of x with compact closure (; such that (; n I-I (,1) = 0 and I-I(y) n (; = {x}. Then dx(x, aU) ~ dy(y, I(av» > O.

4 Norms and Indicatrices

13

Since any curve from x to x' has length at least dx(x, aU), this is a contradiction. D (1.3.13) Corollary. Let X and Y be topological spaces with an inner pseudodistances d x and dy, respectively. If I: X ---+ Y is a distance-decreasing covering projection and if d y is a complete distance, so is d x . Proof We have only to check the statement concerning the completeness. Assume that Y is complete with respect to d y. Let {xn} be a Cauchy sequence in X. Since I is distance-decreasing, {f (xn)} is also a Cauchy sequence in Y. Let q E Y be the limit point of (f(xn)}. Let V be a 2e-neighborhood of q such that V is homeomorphic to each connected component of I-I (V). Let V' be the eneighborhood of q. Since I(x,,) E V' for n > N, all (Xn}n>N are contained in one of the connected components, say U, of I-I (V). Let p E U be the point such that I(p) = q. Then {XII} converges to p. D

We say that a distance d y modulo ,1 is complete modulo ,1 if for each Cauchy sequence {Yn} in Y with respect to d y , we have one of the following: (a) (Ynl converges to a point q in Y; (b) for every open neighborhood V of ,1 in Y, there exists an integer no such that Yn E V for n > no. (1.3.14) Corollary. Let X and Y be topological spaces with inner pseudo-distances d x and dy, respective(v. Let ,1 be a compact subset oIY. If I: X ---+ Y is a distancedecreasing proper finite-to-one map and if d y is a complete distance modulo ,1, then d x is a complete distance modulo I-I (,1). Proof We have only to check the statement concerning the completeness. Let {x,,} be a Cauchy sequence in X with respect to d x . Consider first the case the Cauchy sequence (f(x lI ) } is not convergent. Given a neighborhood U of I-I (,1) in X, we take a neighborhood V of ,1 such that I-I (V) C U. Let no be an integer such that I(x,,) E V for n > no. Then x" E U for n > no. Next, we consider the case (f(x,,)} converges to a point q in Y. If q E ,1, we argue as in case (b) above. So we assume that q ¢ ,1. We take a compact neighborhood V of q which is disjoint from L1 so that d x is a distance on I-I (V). Let I-I(q) = {PI. ... , pd. Let V be a compact neighborhood of q. Since I-I (V) is compact, a subsequence of {x n } converges to one of {PI, ... , Pk}, say PI. Being a Cauchy sequence, {x,,} must converge to PI. D

4 Norms and Indicatrices The results in this section will be used only in Section 5 of Chapter 3. Let V be an n-dimensional complex vector space, and V* its dual space. Let F be a real nonnegative function defined on a subset of V such that if F is defined at v E V, it is defined at tv for all t E C and (1.4.1)

F(tv)

=

ItIF(u).

14

Chapter 1. Distance Geometry

We allow F to take the value 00. We call such a function F a quasi-norm. We do not assume that F is defined on all of V. Nor do we assume that F is continuous. Let ( 1.4.2)

r (F) = {v

V; F (v) is defined and F (v) ::: I}.

E

Then r(F) is a star-shaped circular subset of V in the sense that if v E reF) then tv E reF) for It I ::: 1. We call r(F) the indicatrix of F. Conversely, given a star-shaped circular subset r of V, there is a unique quasi-norm F such that r = reF). We note that F(vo) = 0 if and only if the complex line CVo = {tvo; t E C} is contained in reF), while F(vo) = 00 if and only if no points of the complex line CVo, except the origin 0, are in reF). A quasi-norm F on V is called pseudo-norm if it satisfies the following convexity condition: ( 1.4.3)

F(u

+ v)

+ F(u)

::: F(u)

u, v

E

V.

This convexity condition is equivalent to the convexity of the indicatrix r. A pseudo-norm F is always continuous on V. A pseudo-norm F is called a norm if 0 < F(v) < 00 for all nonzero v E V. Given a quasi-norm F on V and the corresponding star-shaped circular subset r = reF), we define the dual quasi-norm F* on the dual space V* by ( 1.4.4)

F*V) = sup 1),(v)1

for

A.

E V*.

VEr

Clearly the dual quasi-norm F* is defined everywhere on V*. Whether F satisfies the convexity condition (lA.3) or not, the dual quasi-norm F* always satisfies the convexity condition: (lA.5)

F*(A.

+ f..t)

::: F*(A.)

+ F*(f..t)

A., f..t

E

V*.

The indicatrix r* = r(F*) of the norm F* is not only star-shaped and circular but also convex. It is given also as an intersection of closed circular cylinders: ( 1.4.6)

r* =

nV

E

V*; 1A.(v)l::: l}.

VET

(104.7) Proposition. (1) F*(Je) > O/ar all nonzero ). E V* iland only if F(el) < 00, ... , F(e ll ) < 00 for some basis el, ... , ell of V; (2) F*(J.) < 00 for all A. E V* !fand only if r is bounded; (3) If a quasi-norm F is positive (i.e., 0 < F(v)::: 00 for all nonzero v E V) and satisfies the convexity condition (1.4.3), then F*(Je) < 00 for all X E V*. Proof (1) Let U be the subspace of V spanned by r. Given}. E V*, F*(J,) = 0 if and only if X( v) = 0 for all v E U. Hence, F* (Je) > 0 for all nonzero }, E V* if and only if U = V. On the other hand, U = V if and only if F(el) < 00, ... , F(e n ) < 00 for some basis e), ... , en of V.

4 Norms and Indicatrices

(2) If

r

is bounded so that

r

15

is compact, then

sup 1),(v)1 = m
00.

I.'Er

If r is not bounded, let V1, V2, ••• be a unbounded sequence of points in r. In terms of a basis e1, .... ell of V we write Va = L:7=1 a~ei' Then the sequence a; , a~, ... is unbounded for some i. Define A. E V* by ).(L: x j ej) = xi. Then F*(A) = 00. (3) It suffices to prove that r is bounded. Let V' = (v E V; F(v) < oo}.

Since F satisfies the convexity condition, V' is a vector subspace of V. Clearly, reV'. Since Flv' is a norm on V', the indicatrix r is a bounded subset of V'. D We identify the double dual V** of V, i.e., the dual space of V*, with V in a natural manner. (1.4.8) Proposition. {IF is a quasi-norm on V, then F** = (F*)* is a pseudo-norm on V = V**, and F** ::=: F.

Proof By (1.4.5), F** satisfies the convexity condition and hence is a pseudonorm. Let v E V and ). E V* such that F(v) is defined. From the definition of F* we obtain I)'(v) I ::=: F*(}.)F(v). Hence, sup 1),(v)1 ::=: F(v). i.E i"'

D

From the definition of F** we obtain F**(v) ::=: F(v). Each ). E V* defines a closed circular cylinder

cO.) =

{v E V; IA(V)I::=: I},

which is obviously a convex set. (1.4.9) Lemma. If a closed subset S C V is circular (i.e., invariant under scalar multiplication by any complex number of absolute value I), then its convex hull S is given as an intersection o.lall closed circular cylinders that contain S.

Proof Let S be the intersection of all closed circular cylinders that contain S. Since it is convex, we have S C S. Let Vo E V be a point not contained in S. Then there is a real hyperplane separating Vo from S. In terms of a coordinate system v = (z I , ... , zn), Zk = xk + i / , of V and the dual coordinate system in V*, the real hyperplane may be given in the form: Re(}.(v»

= ~:::akxk - Lbkl =

1,

where A = (a1 + ib\, ... , an + ibn) E V*, and S is in the half-space defined by Re().(v» ::=: 1 while Re().(vo» > 1. Since S is circular, Re(A(cv» :::: I for v E Sand C E C, lei = 1. Hence, iJ,(v) I :::: 1 for v E S. On the other hand,

16

Chapter 1. Distance Geometry > 1. This shows that the circular cylinder CU.) contains S

IA(vo)1 :::: Re(}.(vo»

but not Vo.

D

(104.10) Proposition. Given a quasi-norm F on V, the indicatrix r** of F** is the intersection of all closed circular cylinders that contain r and hence is the convex hull f of r. Proof From F** ::: Fin (1.4.8), we obtain the inclusion r c r**. Let Vo E V, and suppose that there is a circular cylinder CU~) containing but not containing Vo. Then A E r*. Since 1A,(vo)1 > 1, we conclude Va fI. r**.

r D

(1.4.11) Corollary. and F** = F.

If F

is a pseudo-norm on V so that r is convex, then r**

=

r

(1.4.12) Proposition. Let V be an n-dimensional complex vector space. Let F be a quasi-norm on V with indicatrix r, and F** its double-dual with indicatrix r**. Then (1) Every element V E r** is in the simplex determined by the origin of V and at most 2n elements of r, i.e., m V

with

= Ltiui

L

ti:::: 0,

t; :::

I,

Ui E

r,

;=1

where m ::: 2n; (2) Given v E r** and that V=

VI

E

+ ... + Vm

and F(v])

(104.13) Remark. In (2), if F**(v) > such that V = VI

+ ... + Vm

r

> 0, there exist VI, ... , Vm E

and

+ ... + F(v m) :::

°we can find F(vl)

with m ::: 2n such

F**(v)

VI, ... , Vm E

+ ... + F(vm )

:::

+ E. r, (m ::: 2n),

F**(v).

Then by F**(v;) ::: F(Vi) we actually have an equality: F(vl)

+ ... + F(vrn)

= F**(v).

We start the proof with the following lemma due to Caratheodory. (1.4.14) Lemma. Let V be a real vector 5pace, and r a subset containing the origin 0 E V. Let f denote the convex hull of r. Then V E f if and only if v is contained in afinite dimensional simplex having its vertices in r and having as one of its vertices.

°

Proof We first prove Lemma in the special case when r = to, VI, ... , vm }. The proof is by induction on m. Let m

is a finite set, say f so that

m

v=(l-Lt;).O+Ll;V; ;=1

r V E

i=1

with

t;::::O,

Lli:::l.

4 Norms and Indicatrices

17

If VI, ... , Vm are linearly independent, there is nothing to prove. If not, we have a nontrivial relation Sl VI + ... + Sm Vm = O. We may assume that Looking at

LSi::: 0 (by multiplying the equation by -1

if necessary).

we let a increase from 0 until one of t; - as; becomes 0 for the first time, and let a be that value. Without loss of generality we may assume that tm - aSm = O. Then m m m-I V = L tiV; - a LS;v; = L(t; - as;)vi i=1

i=1

;=1

and

m

1'1"1-1

t; - as; ::: 0

and

(t; - as;) = L

L ;=1

m

t; -

;=1

aL

Si :::: I.

;=1

This shows that v is in the convex hull of {O, VI, ... , Vm-I}, thus completing the induction. Now we prove Lemma in the general case. We consider r-dimensional simplices, each with its vertices in r and one of its vertices at O. We denote the union of all such simplices by Sr and set S = Ur Sr. Then S is convex. To prove this, consider a point z on the line segment joining two points V and w of S. Since v is in a simplex with vertices, say 0, VI • ...• Vb and w is in a simplex with vertices, say 0, Vk+I, ... , Vm • the point z is in the convex hull of {O, VI •...• vm }. (The sets {VI ••••• Vk} and {Vk+I, ...• v m } need not be disjoint). Using the finite case of Lemma just proved, we see that z is in S. Hence, f c S. Since Sr C f for all r, we have S C f. Thus, we have S = f. D Proof of (1.4.12). Part (l) clearly follows from Lemma. For any positive real number r, we write r r for the set {rv; V E r}. Similarly, we set r f = {rv; V E f}. Then r f is the convex hull of r r. We prove (2). Given v E r** = f, let r = F**(v). Then v E (r + r:;)f for any r:; > O. By Lemma there exist U I, ... , U m E (r + e)r with m :::: 2n such that m

V = Lt;ui

with

t; > 0

Lt;:::: 1.

;=1

Then L

F(t;u;) = L

tiF(Ui) :::: (r

+ e) Lti

By setting Vi = tiui, we obtain the desired inequality.

:::: (r

+ r:;). D

We shall show that if F**(v) > 0 then the stronger inequality indicated in Remark (1.4.13) holds. Let F**(v) = r > O. Then vErt. By Lemma, there exist ul, ... , u m Err such that v = Ltiu; with t; > 0, Lti:::: 1. The remainder of the argument is the same as in the proof of (2) above.

Chapter 2. Schwarz Lemma and Negative Curvature

1 Schwarz Lemma In this section we prove Ahlfors' generalization of the classical Schwarz lemma in function theory of one complex variable and its variants. For the general theory of intrinsic distances, we need only the classical Schwarz-Pick lemma in the form (2.1.7). Let X be a Riemann surface, i.e., a I-dimensional complex manifold. Let

da 2 = 2A.dzdz be a Hermitian pseudo-metric on X expressed in terms of a local coordinate z, and let w = iAdz Adz be its associated Kahler form. The term pseudo-metric means that da 2 is only positive semidefinite, i.e., A ::: o. We recall the notation dC

= i (d" -

d')

so that dd c

= 2id'd".

To w we asscoiate the Ricci form (2.1.1 )

Ric(w) = -dd c log A = 2Kw,

where . I B2 logA K = ----A BzBz

(2.1.2)

is the (Gaussian) curvature of da 2 • Both Ric(w) and K are defined wherever A is positive. . Let Da denote the open disc of radius a in the Gaussian plane C, Da

=

{z

E

C; Izl < a}.

Then the Poincare metric (2.1.3)

2 dSa

4a 2dzdz = A(a 2 _ IzI2)2'

(A> 0)

20

Chapter 2. Schwarz Lemma and Negative Curvature

on Da is complete and has curvature -A. In the special case where a = 1, the unit disc D j will be denoted D and the Poincare metric ds~ with A = 1 will be denoted ds 2 . Let cp be the Kahler form of the Poincare metric ds 2 • Then Ric(cp) = -2cp, since the curvature K is -1. We prove a generalization of Schwarz-Pick Lemma by Ahlfors [1).

(2.104) Theorem. Let d.l'2 denote the Poincare metric of curvature -Ion the unit disc D. Let da 2 be any Hermitian pseudo-metric on D whose curvature is bounded above by -1. Then

In terms of the Ricci forms Ric(cp) and Ric(w) of ds 2 and da 2 respectively, (2.1.4) may be stated as follows: (2.1.4)'

Ric(w) .:::: -2w

=> w .::::

cpo

As we shall see in the proof, the theorem holds if da 2 is only continuous at zero points of da 2 and is twice differentiable at the points where it is positive (and hence the curvature is defined). This technical point is important in applications. Ahlfors proved the theorem for an upper semicontinuous da 2 with "supporting pseudo-metric". This will be explained later.

Proof Let Da be the disc of radius a with the Poincare metric ds; of curvature -1 given by

We compare this metric with da 2 = 2Mzdz. Let u" be the nonnegative function on D" defined by U a = },//La, i.e., da 2 = u"ds;. The problem is to show that Uj .:::: 1 on D. From the explicit expression for /La, it follows that u" (zo) ~ Uj (zo) at every point Zo E D as a ~ 1. Hence, it suffices to show that U a .:::: 1 on Do for every a < 1. From the explicit expression for /La we see that u" (z) ~ 0 as z approaches the boundary of D". Therefore, u" must attain its maximum in the interior of Da, say at z" E Da. If u" (z,,) = 0, then u" == 0, and there is nothing to prove. So we assume ua(zo) > 0 and calculate the complex Hessian of log U a at Zoo Since U o = A/ /La and since the curvature of ds~ is -1, we have

a2 10g U a

azai

=

a2 log A --- -

azaz

a2 log /La

---=~-

azai

-AK - /La = /La(-uaK - 1).

1 Schwarz Lemma

21

Since the complex Hessian oflog u" must be nonpositive at the maximum point zo, we obtain the inequality -u,,(zo)K(zo) -1 ::'S O. Hence, u,,(zo) ::'S -1/K(zo)::'S 1. Since ua(zo) is the maximum value of u", we have u" ::'S 1 everywhere on D a.

o

(2.1.5) Corollary. Let X be a Riemann surface with a Hermitian pseudo-metric dsi whose curvature (wherever defined) is bounded above by - L Then every holomorphic map I: D -+ X is distance-decreasing, i.e., f*dsi ::'S ds 2 • Proof Set da 2 = f*dsi. Then da 2 is a Hermitian pseudo-metric on D. If we denote the curvature of dsi by Kx, then the curvature of da 2 is given by 1* Kx; this is clear from the definition of the curvature (2.1.2). Now, (2.1.5) follows from (2.1.4). 0 The following classical Schwarz-Pick lemma is immediate from (2.1.5). (2.1.6) Corollary. Let D be the unit disc with the Poincare metric ds 2 . Then every holomorphic map I : D -+ D is distance-decreasing, i.e.,

Comparing the coefficients of dzdz in the inequality in (2.1.6) we can write

1/'(z)1

--~----

<

1

for

----~

I - If(z)i2 - 1 - Iz1 2 '

zED.

This is the usual form of Schwarz-Pick lemma. If I is a holomorphic automorphism of D, then (2.1.6) applied to both I and I-I implies that I is an isometry, i.e., f*ds 2 = ds 2 . Let p denote the Poincare distance on D, i.e., the distance function defined by the Poincare metric ds 2 . We shall find the explicit formula for p. Let 0 < a < 1. If z(t) = x(t) + iy(t), 0 ::'S t ::'S 1, is a curve in D joining the origin 0 E D to a E D, its arc-length I with respect to ds 2 satisfies

1

2(x'(t)2

1

-------=-dt >

1

= >

o

1

+ y'(t)2)t

1 - X(t)2 - y(t)2

2Ix'(t)1

o 1 - x(t)2

-

1" 0

dt 2dx 1- x2

1+a I- a

- - = log ----.

This shows that the ordinary line segment from 0 to a is the shortest path and l+a 1 -a

p(O, a) = log - - .

Since the Poincare metric is invariant under the rotations, we have

22

Chapter 2. Schwarz Lemma and Negative Curvature

I +Ial p(O, a) = log - - = 2 tanh- 1 lal 1 -Ial

for

a

E

D.

Given two points a and b in D, the transformation z-b w = --_I - bz

is an automorphism of D that sends b to invariance of p we obtain pea, b) = log

11 11 -

°and

a to (a - b)/(J - ab). From the

I

I

abl + la - bl a - b = 2 tanh-\ -- . abl - la - bl 1 - ab

However, we shall rarely use this explicit expression for p. The integrated form of (2.1.6) reads as follows: (2.1.7)

p(f(a), feb»~ :5 pea, b)

for

a,b E D.

In order to weaken the differentiability assumption on da 2 in (2.1.4), we first prove the following (2.1.8) Proposition. Let daJ and dar be two smooth Hermitian metrics with curvature Ko and K\ defined in a neighborhood of a point Zo E D. If daJ :5 daf in a neighborhood of Zo with equality holding at zo, then K\ :5 Ko at Zoo Proof Let da? = 2A;dzdz, and u = Aol )0\. As in the proof of (2.1A), we calculate B210g ulBzBz. Then using (2.1.2) we obtain

1 )'\

a2 \og u azBz

= -uKo+ K\.

We evaluate this at ZOo Since u attains the maximum value I at Zo, we have -Ko+K\ atzo. 0

O~

Let da 2 be an upper semicontinuous Hermitian pseudo-metric on D. A pseudoHermitian metric da~ is called a supporting pseudo-metric for da 2 at Zo E D if it is defined and of class C 2 in a neighborhood U of z" and satisfies the following condition: da 2 ~ da,; in U, and with equality at ZOo If da 2 is not smooth, we define its curvature

Kda2

by

(2.1.9)

where the infimum is taken over all supporting pseudo-metrics da} for da 2 at ZOo Then (2.1.4) is generalized to non-smooth metrics, (Ahlfors [1]): (2.1.10) Theorem. Let ds 2 be the Poincare metric on D, and da 2 an upper semicontinuous Hermitian pseudo-metric on D with its curvature::::: -1. Then

1 Schwarz Lemma

23

The proof of (2.1.1 0) is exactly the same as that of (2.104). Since da 2 is upper semicontinuous, the existence of a maximum for the function U a is assured. If Zo is a maximum point for U a = da 2 Ids;, it is also a maximum point for the function Va = da;;lds;; (where da,; is a supporting pseudo-metric for da 2 at zo). Then we have only to calculate the Hessian of log Va instead of log U a • D (2.1.11) Remark. The classical Schwarz lemma asserts also that if an equality holds in (2.1.6) at one point of D, then the equality holds everywhere. This part of Schwarz lemma has been also generalized by Heins [I]. Namely, if an equality holds in (2.1.10) at one point of D, then da 2 = ds 2 everywhere on D. For a finite family of of pseudo-metrics, (2.104) may be generalized (see (2.104),) as follows (Cowen-Griffiths [I]): (2.1.12) Theorem. if a family ofpseudo-metrics dal = 2)'kdzdz, (k = 1, ... , N), on D with Kahler forms Wk = j)'kdZ 1\ dz, has its curvatures collectively bounded by -1 in the sense that then the pseudo-metric da 2 = 2Mzdz,

where

A = C).I .. . )N )\/N

has its curvature bounded by -1 and, consequently, da 2 Proof Let W = iAdz -Ric(w)

1\

s: ds 2 .

dE be the Kahler form of da 2 . Then ddClogA=

~ LddClogAk

1 N LRic(wk)

2i N L

Akdz

1\

~

2 N LWk

dE ~ 2i(A\ ... ).N)\/N dz

1\

' dz = 2w,

showing that the curvature of da 2 is bounded by -1.

D

The following is also a simple application of the maximum principle. (2.1.13) Theorem. Let da 2 be an upper semicontinuous Hermitian pseudometric on D. Let ~ be a holomorphic vector field on D and I~I be the length of~ measured by da 2 . If da 2 has negative curvature in the sense of (2.1.9), then I~ I cannot attain a maximum in the interior of D unless I~ I == O. Proof Assume that I~ I attains a maximum at Zu ED. Without loss of generality we may assume that da 2 is of class C 2 at zoo (In fact, let da,; be as in (2.1.9) and IH the length of ~ measured by da'/;. Then IH attains a maximum at zo.)

24

Chapter 2. Schwarz Lemma and Negative Curvature

Let da 2 = 2Mzdz. If~ = /(olaz), then 1~12 = 2A/ j. Assuming that 1~12 > 0 at z,,' we calculate the Hessian of log I~ 12. Thus

a2log I~ 12

-----::~'--

ozaz

a2 log A

,

= - - - = -AK azaz '

where K is curvature of da;;. The left hand side is nonpositive at right hand side is positive everywhere. This is a contradiction.

z" while the 0

Now we state the Schwarz lemma for Hermitian metrics with a logarithmic singularity at the origin. This will be useful in logarithmic geometry. (2.1.14) Theorem. Let ds 2 be the Poincare metric of curvature -Ion the unit disk. Let da 2 = 2Mzdz be a Hermitian pseudo-metric on the punctured disc D* such that Izl2da 2 becomes upper semicontinuous on D. {fthe curvature of da 2 is bounded above by -lzI 2, then Idda 2 :::: ds 2 • Proof Let d(j2 = IZ12da 2 . Then the curvature

-

I

k

of d(j2 is given by

a2 log 1

K = --:;---_-, A ozoz where ;;. = IdA. This shows that it is related to the curvature K of da by K = IZI2k. Hence, k :::: -1, and (2.1.10) applied to d(j2 yields d(j2 :::: ds 2 •

o

The following theorem of Sibony [4] is a version of Schwarz lemma and will be used to define the Sibony pseudo-distance. (2.1.15) Theorem. Let U be an upper semicontinuous function on D such that (i) U < 1, (ii) logu is subharmonic, (iii) u(O) = 0, and (iv) u(z)/lzI 2 is bounded. Then (I) u(z):::: Izl2 for ZED with equality at some point i= 0 if and only !f u(z) == IZI2; (2) If, in addition, u is of class C 2 in a neighborhood of 0, then a2ulazoz :::: I at 0, with equality if and only if u (z) == Id.

o ::::

Proof (1). Since u attains its minimum at 0, du vanishes at O. Define in D* the function v(z) = u(z)/lzI 2 • Since log v is subharmonic in D*, v is subharmonic in D*. Since v is also bounded, there is a subharmonic extension of v in D. Since IimsuPz-->ao v:::: I, it follows that 1. Hence u(z) :::: IZI2. If u(zo) = Izol2 for some Zo i= 0, then v(zo) = 1. This implies v(z) == 1. (2). Let z = x + iy. Denoting partial differentiation by subscripts, we have

v

v::

Izl 2 :::: u(z)

I

= '2(u xA O)x

2

+ 2u xy (0)xy + Uyy(O)y 2 ) + o(lzl 2 ).

Replacing y by - y and adding the resulting ineqaulity to the above, we obtain u x AO)x 2 +

U yy (O)y2

+ 00z12) :::: 21z12.

2 Negatively Curved Riemann Surfaces

25

Hence, Uz;:(O) =

If U zz (0) = 1, then V(O) =

1

4(uxAO) + Un' (0») ::::

U z: (0)

1.

v == 1.

= 1. Hence,

D

We note that in (2.1.15) if U is of class C 2 in a neighborhood of 0, then u(z)/lzI 2 is automatically bounded in D. For historical comments concerning Schwarz-Pick-Ahlfors lemma, see Royden [10].

2 Negatively Curved Riemann Surfaces The results in this section will be used in Section 7 of Chapter 3. In the preceding section we showed that the Poincare metric on the unit disc D has curvature -1. It is sometimes more convenient to use the upper halfplane in place of the disc. Let H = {w = u

+ iv

E

C;

V

> OJ.

We have a well known correspondence between the unit disc D and the upper halfplane H. It is given by (2.2.1)

i - w z=--ED

i+w

for

wE H.

Pulling back the Poincare metric of D by the correspondence above, we obtain the Poincare metric ds 2 = dwdu)

(2.2.2)

v2

H

of curvature -Ion H. Since it corresponds to the Poincare metric of D, ds~ is also invariant by the automorphisms of H and is complete. Let D* be the punctured disc, i.e., D*

=

{z E C; 0 <

Izl

< l}.

Let p: H -+ D* be the covering projection defined by z = pew) = e27Ciw

for

lJ)

E

H.

Since the Poincare metric ds~ is invariant by the automorphisms of H, in particular, by the covering transformations w t-+ w + n, (n E Z), there is a unique metric ds1. on D* such that p*(ds1.) = ds~. In order to find the explicit form of ds1., we solve z = pew) for w locally in terms of z:

w

=

1 -.Iogz. 27r1

26

Chapter 2. Schwarz Lemma and Negative Curvature

Substituting this into (2.2.2) we obtain the following complete metric d

(2.2.3)

2

SD'

4dzdz = Izl2(log Ijlzl2)2

of curvature - I on D*. Its area element, i.e., its Kahler form, is given by Il-D' =

idz 1\ dz . Izl2(log Iz12)2

Although the origin 0 of D is at infinity with respect to the metric ds1., the area around 0 with respect to Il-D' is bounded. More explicitly, let D~ = {O < Izi < a} with a < I. Then

r

(2.2.4)

lD;

Il-D' < 00.

In fact, when we choose

F

= {w = u + iv E

H; 0:::: u < I}

as a fundamental domain for D*, the subset corresponding to D; is given by Fb = {w E F; v > b} with a suitable positive number b, and its area is given by

r In;

Il-D' =

r dUv~V J Fh

1 b

We consider the twice-punctured plane X = C - to, I} and show that it carries a complete Hermitian metric with curvature K :::: -I. This is obvious if we make use of the fact that the elliptic modular function, usually denoted A., is a covering projection from H onto C - to, I}, the latter being identified with SL(2; Zh\H where SL(2; Zh = {A E SL(2; Z); A == I mod 2}, (see, for example, Ahlfors [3; p.269]). The metric induced from the Poincare metric of H has constant curvature -I. Instead, we shall construct explicitly a complete Hermitian metric dsJe on X = C-{O, I} whose curvature K, although not constant, remains bounded above by -1. More generally, we consider the Riemann sphere minus k points, k :::: 3. Let Z = (zo, z I) be a homogeneous coordinate in PI C, and let z

= zljzO

be its inhomogeneous coordinate. For W = (wo, Wi), we set (Z, W) = zOwo

+ ZIW I •

We use the notaion d C = i(d" - d')

so that

dd c = 2id'd".

2 Negatively Curved Riemann Surfaces

27

Then the associated Kahler fonn of the Fubini-Study metric ds 2 of curvature 1 is given by

cP

= dd c 10g(Z, Z) = dd

. L

log(l

2

+ Izl ) =

2idz /\ dz (l

2 2'

+ Izl )

Given k points Aj = (aJ, a), j = 1, ... , k, in PI C, we shall construct explicitly a positive function p on PI C - {A I, ... , Ad such that (a) p goes to infinity at each of AI,.'" Ak (so that pds 2 becomes a complete metric on PIC - {AI, ... , Ad; (b) the curvature of pds 2 is bounded above by a negative constant; (c) the area of PIC - {AI, ... , Ad with respect to pCP is finite, i.e.,

1

pCP <

00.

P,C-{A, ..... A,}

To construct such a function, we may assume that none of Aj is the point at infinity, i.e., that aJ =1= 0 for all j. We denote the polar of A j by At-Thus, A j-L = (-1 aj , -a-0) j ,

so that (A j , At) = O. Set

a,.' = a'] lao, ] and

n k

(2.2.5) where

p=

1

j=' OJ (l.og c/aj)

2'

is the chordal distance between Z and A defined by

OJ

and c is a large positive number yet to be detennined. Then (a) is clearly satisfied. To prove (b), write

pCP

= 2J.idz /\ dz

with

= (l + Idl-2

),

Ii

+

I laj 12 j=1Iz-ajI2(loge/aj)2

.

Since

a2 log(log c/aj)2 azaz

2 = loge/Oj (1

1

+ Iz12)2 -

the curvature K of pCP is given by

2 1 1 Z 12 (loge/oj? z - aj - 1 + IZl2 '

28

Chapter 2. Schwarz Lemma and Negative Curvature

=

K

I --:;-(1 /,

+ II + III),

where

=

I

=

III E

+ Iz12)2 '

-L

II

Given

k-2 (1

L

(1

+

2 IzI2)2Iogcjoj'

2

1

I

(logcjaj)2 z - aj -

Z

12

1+ Izl2

> 0, we can choose c so large that

Then I + II > 0 for k > 2. Since (III) is non-negative, it follows that K < However,

o.

with a suitable positive constant Cj . Hence,

In order to show that K is bounded away from 0, we have to examine terms in (III). Since the j -th term in III is equal to

we have

1

~(III)

= 2Cj

+ ... ,

where the dots ... indicate the terms which approach 0 as z ~ aj. It follows that K remains away from 0 as z ~ aj. Finally, to prove (c), we consider a neighborhood Uj = liz - ajl < E} of aj with e < 1 and write Then

Hence,

f

p(/J <

U

}

This proves (c).

-

4M

1 1£ 2"

0

de

0

dr

2;rM

< --- < r(logr2)2 - log lie

00.

2 Negatively Curved Riemann Surfaces

29

In summary, we have (2.2.6) Theorem. Let X be PI C minus k points, k ~ 3. Then X admits a complete Hermitian metric with curvature K :::: -1 and finite total area. Such a metric can be obtained by multiplying the Fubini-Study metric of PI C by a function p of the form (2.2.5). The construction of the metric given here is a special case of the volume form constructed by Carlson-Griffiths [1]. Earlier constructions of similar merics for k = 3 are due to Ahlfors [1], Robinson [1] and Grauert-Reckziegel [1]. Such metrics were ,used to give "elementary" proofs (i.e., proofs without recourse to the elliptic modular function) of several important results in function theory including Picard's theorems; see Ahlfors's monograph [4] for further details. Though not smooth, the metric constructed by Ahlfors is more explicit and satisfies the conditions of (2.l.l0). He used it to obtain a good lower bound for the Bloch constant. Every compact Riemann surface X of genus ~ 2 has the upper halfplane H as its universal covering space. Though well known, this is a nontrivial fact. It is obvious from this fact that X carries a Hermitian metric of constant curvature -1. We shall construct a Hermitian metric dsi with curvature K :::: -Ion X in an elementary fashion (i.e., without recourse to the uniformization theorem). We follow Grauert-Reckziegel [I]. Take two linearly independent holomorphic I-forms WI, W2 on X and set dO' 2 = 2(wlwl

In terms of a local coordinate system dO'

2

-

-

+ W2 W2).

z of X, we may write

= 2(f1 II + hh)dzdz,

where

Wi

= [;dz.

Outside the set of common zeros PI, ... , Pk of WI, W2, the pseudo-metric dO' 2 is positive and its Gaussian curvature K(dO' 2 ) is given by

Since WI and Wz are linearly independent, II/his not constant. Since K (dO' 2 ) vanishes exactly where (fIlh), vanishes, K(dO' 2 ) is negative except at finitely many points PHI, ... , Pm of X - {PI, .•. , Pk}. Choose disjoint neighborhoods VI, ... , Vm of these points PI, ... , Pm. We shall show how to modifY dO' 2 in each Vi so that the curvature becomes negative everywhere on X. Using a local coordinate system z in Vi with Z(Pi) = 0, let r be a positive number such that B = {z; Izl < r} C V;. Let B' = {z; Izl < r/2}. Choose a Coo functiona(z,Z) on V; such that (i) 0 :::: a(z, z) :::: 1 on Vi, (ii) a(z, z) = 1 on B', and (iii) a(z, z) = 0 on V; - B. Let c be a constant, 0 < c < 1, and set

30

Chapter 2. Schwarz Lemma and Negative Curvature

where 1 = II iI + Id2 and h = (l + zz)a(z, z). Then the metric dsi = 2gdzdz coincides with da 2 on Vi - B. Since the curvature of da 2 is bounded above by a negative constant on the compact set B - Bf, the curvature of the metric ds~ is strictly negative on B - B' if c is sufficiently small. In order to calculate the curvature of ds~ on B', we set 13 = c and f4 = cz so that 4

dsi = 2

L l.fj l2dzdz

on

B'.

j=l

Its curvature K is given by

K = _ (2:: 1.t;'J2)(2:: Ilkl 2 ) - (2:: J;' ];)(2:: Id{). (L: Ifil 2 )3 Since the vectors

are linearly independent at every point of B', it follows that K is negative everywhere on Bf. By multiplying ds~ with a suitable positive constant, we can make the curvature K bounded above by -1. Tn summary, (2.2.7) Theorem. Every compact Riemann surface X of genus ::>: 2 carries a Hermitian metric ds~ with curvature K :s -I. (2.2.8) Remark. The curvature of the Hennitian metric 2(1 given by 1 K = < o. (I + Iz12)3

+ Izl2)dzdz

on C is

Clearly, the curvature approaches 0 as Izl tends to infinity. As we shall see later, C cannot admit a Hennitian metric with curvature bounded above by a negative constant. If r denotes the geodesic distance from the origin to z with respect to the above metric, then Izl2 ~,J2r for Izllarge so that K ~ -1/(,J2r)3 as r ~ 00, see Remark (3.7.2).

3 Negatively Curved Complex Spaces In this section we shall generalize results of Section 1 to higher dimensional complex spaces. The results will be used in Section 7 of Chapter 3. Let X be a complex space. Let f be a holomorphic map sending a neighborhood U f of the origin 0 in C into X. Let g: U g ~ X be another such map. Then f and g are said to define the same I-jet at 0 if f(O) = g(O) and if they have the same first derivative at O. GeometricaJ1y stated, this means that two holomorphic curves f and g have the same velocity at O. The I-jet represented by f will be

3 Negatively Curved Complex Spaces

31

denoted f'(O). So by definition, 1'(0) = g'(O) if and only if f and g define the same I-jet at O. For each x E X, we call the set TxX of all I-jets f'(O) the tangent cone of X at x, and TX = U, Tx X the tangent cone of X. The tangent cone defined here is in general smaller than the usual tangent cone which consists of tangent vectors to real Coo curves in X. If x is a regular point of X, then t, X coincides with the tangent space Tx X. Let Ox be the ring of germs of holomorphic functions at x EX. Let mx denote its maximal ideal consisting of functions vanishing at x. Then the Zariski cotangent space of X at x, is defined to be rnx/rn;, and its dual space, denoted T, X, is called the Zariski tangent space of X at x. Although T X = UX T, X may not be a fibre bundle if X is singular since the dimension of T.tX may vary with x, it will be still called the Zariski tangent bundle of X. Then we have a natural inclusion TX C T X. In fact, for a holomorphic function q; defined in a neighborhood of x = f(O), set

d

I

(f (O»q; = dz q;(f(z»lz=o·

Then I' (0) may be regarded as a derivation from the algebra of holomorphic functions at x = f(O) into C. If X is nonsingular, then TX coincides with TX. If (z, w) is the coordinate system for C 2 and if X is the curve defined by w 2 = Z3, then the Zariski tangent space at the singular point (0, 0) is the 2-dimensional vector space spanned by a/oz and a/ow. The tangent cone at (0,0) defined here contains only the zero vector. In fact, if I E Hol(D, X) with 1(0) = (0,0), then expressing I by a pair of power series

and using the condition w 2 = Z3, we obtain al = bl = b 2 = O. Hence, the I-jet of I at 0 vanishes. For a systematic account of Zariski tangent spaces, tangent cones and other possible "tangent spaces", see Whitney [I]. A pseudo-length function on X is a real non-negative function F on T X such that F(av) = laIF(v)

for

a E C,

v,av E TX.

We normally assume that F is smooth. However, for applications it is sometimes necessary to consider upper-semi continuous pseudo-length functions. If F(v) > 0 for all nonzero VET X, then we call F a length function on X. We say that a pseudo-length function F is convex if F(v

+ v')

::::: F(v)

+ F(v').

A convex (pseudo-)Iength function is also called a Finsler (pseudo-)rnetric.

32

Chapter 2. Schwarz Lemma and Negative Curvature

Every pseudo-length function F on a complex space X gives rise to an inner pseudo-distance d by (2.3.1 )

d(p, q) = inf y

r

fa

F(y'(t))dt,

where yet), a S t S b, is a piecewise differentiable curve from p to q and y'(t) is the velocity vector of y at y (t). IfF is a length function, then d is a distance function. A Hermitian metric on a complex manifold X defines a length function. However, a length function is much more general than a Hermitian metric or even a Finsler metric. On the other hand, restricted to a holomorphic curve a length function is a Hermitian metric. More precisely, if F is a pseudo-length function on X and if f: D ~ X is a holomorphic map, then f* F defines a Hermitian pseudo-metric on D, i.e., (2.3.2)

f* F2

= 2Mzdz,

where A. is a non-negative function on D. If F is a length function and if f is everywhere non-degenerate, then A is positive and f* F2 defines a Hermitian metric on D. We consider the curvature Kf*F of the pseudo-Hermitian metric f* F2 defined by (2.1.2) (by (2.1.9) if not smooth); it is defined where ;, is positive. Let v E itx, let [v] denote the complex line spanned by v. Given an upper semi continuous pseudo-length function F on X, we define the holomorphic sectional curvature K F ([ v]) in the direction of [v] by (2.3.3) where the supremum is taken over all f E Hol(D, X) such that f(O) = x and [v] is tangent to feD). Let cp: X' ~ X be a holomorphic map from a complex space X' into another complex space X. Given a pseudo-length function F on X, we have the induced pseudo-length function cpo F on X'. From the definition of the curvature we obtain (2.3.4)

for

v

E

ix'

wherever the curvature is defined. In particular, if X' is a complex subspace of X, then the curvature of (X', Fix') is bounded above by the curvature of (X, F). We apply (2.3.4) to a holomorphic map f: D ~ X and (2.1.4) to the Hermitian pseudo-metric da 2 = f* F2. Then we have the following generalization of the Schwarz lemma, (Grauert-Reckziegel [1], Kobayashi [5]). (2.3.5) Theorem. Let F be a pseudo-length/unction on a complex space X. !fits holomorphic sectional curvature is bounded above by -1, then

jor where ds 2 is the Poincare metric of D.

f

E

Hol(D, X),

3 Negatively Curved Complex Spaces

33

We show that for a Hennitian manifold the definition of the holomorphic sectional curvature given here coincides with the usual one. Given a Hennitian metric ds 2 = 2 L7.J=1 gi]dZidi J, on a complex manifold X, the components of its curvature tensor are expressed by

(2.3.6)

a 2 gi]

'"

Ri]kl = - azkail

-agiij ag p ] az k ail'

+ ~ gpq

Then given a unit tangent vector v = LVi (a/az i ), the holomorphic sectional curvature in the direction of v is defined to be

(2.3.7)

' " Ri]kTv i v- j v k-I Hd,l' () V = ~ v .

Let X' be a complex submanifold of X. We choose a local coordinate system zn in such a way that X' is defined by

Zl, ..• ,

= ... = zn = O.

zm+l

so that we may use z1, ... , Zm as a local coodinate system for X'. Then the induced Hennitian metric on X' is given by 2 L;~J=I gi]dz i diJ' We shall compute its curvature R;]kl' Fix a point p E X'. By a linear change of coordinates, we may assume gil = 8ij at p. Then at the point p we have the following equation of Gauss:

(2.3.8)

R' - - = R, ijkl

ag - ag "

n 7

-

IJkl

-

'"

~ p=m+l

~---.!!.!...

az k ai'

for i, j, k, I = I, ... , m. The following proposition is immediate from (2.3.8) (2.3.9) Proposition. Let X' be a complex submanifold of a Hermitian manifold X. Then the holomorphic sectional curvature H~", of X' does not exceed the holomorphic sectional curvature Hds' of X, i.e., for

v E TX'.

Fix a point p EX, and let v E TpX be a unit tangent vector. If X' is a holomorphic curve in X tangent to v at p, then its Gaussian curvature at p is bounded by the holomorphic sectional curvature H d.,2(V) of X in the direction v. We shall show that there is a holomorphic curve tangent to v whose Gaussian curvature at p equals Hds2(V). We start with any holomorphic curve X' tangent to vat p. We may assume that a local coordinate system Zl, .•. , ZIZ with origin p was chosen in such a way that gil = 8ij and X' is given by Z2 = ... = zn = O. From

34

Chapter 2. Schwarz Lemma and Negative Curvature

(2.3.8) we know that if Jgiq/JZ k = 0 at p for q = 2, ... , n, then X' has already desired property. If not, we consider the following coordinate transformation:

where a 2 , ... , a" are constants to be chosen appropriately. Substitute the coordinate transformation above into ds 2 = L g;Jdz i dz.i to express it in terms of Il Th en we 0 btam . d s 2 -- "h - j WIt . h w 1, ... , w. L.. ij-d Wid w, n

h 1q = glq -

L g,qa'w 1. ,=2

It follows that if we set a q = (Jglq/JZ1)p, then (Jh1q/JW1)p = 0, so that the holomorphic curve defined by w 2 = ... = w n = 0 has the desired property. This

proves the following (Wu [5]) (2.3.10) Proposition. For a Hermitian manifold (X, ds 2 ) the holomorphic sectional curvature defined by (2.3.3) coincides with the classical holomorphic sectional curvature defined by (2.3.7). (2.3.11) Remark. We note that for KF the assertion corresponding to (2.3.9) was immediate from the definition (2.3.3), (see (2.3.4». However, (2.3.8) gives much more than (2.3.9). In fact, if

then X' is totally geodesic at p, i.e., the second fundamental form of X' vanishes at p, provided either dim X' = 1 or X is Kahler. From (2.3.5) we obtain (2.3.12) Corollary Let (X, dsi) be a Hermitian man(fold whose holomorphic sectional curvature is bounded above by - 1. Then

for

f

E

Hol(D, X),

where ds 2 is the Poincare metric of D. We extend (2.1.13) to higher dimensional spaces X. (2.3.13) Theorem. Let X be a complex space with an upper semicontinuous pseudolength function F with the property that, at each v E f X with F(v) > 0, F has negative holomorphic curvature. Let ~ be a holomorphic vector field on X. Then F(~) cannot attain a maximum in (the interior qf X) unless F(~) == O.

Proof Assume that F(n attains a positive maximum at x" E X. Choose an imbedding f: D ~ X such that f(O) = x" and ~ is tangent to feD). Then apply (2.1.13) to the pseudo-Hermitian metric f* p2 on D. 0 (2.3.14) Corollary. Let X and P be as in (2.3.13). If X is compact, it admits no holomorphic vector fields.

4 Ricci Forms and Schwarz Lemma for Volume Elements

35

The generalized Schwarz lemma (2.3.5) has been extended to holomorphic maps from a complex manifold M with a length function satisfying certain curvature conditions to a complex manifold X satisfying the condition of (2.3.5), see Yau [3], Chen-Cheng-Lu [I], Royden [7,8], Yang-Chen [I], Q. H. Yu [I], Chen-Yang [1], Matsuura [2], Bums-Krantz [I].

4 Ricci Forms and Schwarz Lemma for Volume Elements The results in this section, which generalize those of Sections I and 2 to volume elements, will not be used until Chapter 7. Let L be a holomorphic line bundle over a complex manifold X with local coordinate system z', ... , zn, and h a pseudo-metric on L, i.e., h is a non-negative smooth function on L such that for

~ E

L,

C E

C.

If h(~) > 0 for all nonzero ~ E L, then h is called a metric on L. To each pseudo-metric h we associate a closed (I,I)-form Ric(h) called the Ricci form as follows. Taking a local non-vanishing holomorphic section ~, we set (2.4.1 )

Ric(h) = -dd' 10gh(O =

-2iaa logh(~) =

2i

L Rjkdz J /\ dz

k,

where (2.4.2)

a2 10gh(n

Rjk = -

azJaz k

•

If ~ is replaced by another non-vanishing holomorphic section TJ = f~ with f holomorphic, then her}) = IfI2h(~) and aa 10gh(TJ) = aa logh(l;), which shows that Ric(h) does not depend on the choice of~. The Ricci form Ric(h) is defined only at the points where h is strictly positive. If h is a metric on L, then Ric(h) is globally defined on X and represents 4rrc, (L), where c, (L) is the first Chern class of L. Sometimes, the Ricci form Ric(h) of a pseudo-metric can be globally defined (even where h vanishes). We say that h has a holomorphic degeneracy if h is locally of the form lal 2q g, where g is strictly positive, a is holomorphic, and q is a rational number. Thus, if ~ is a non-vanishing local holomorphic section of L, then h(~) = lal 2q g, and h(~) vanishes only where a vanishes. In this case, aalogh(l;) = aalogg,

and Ric(h) is defined everywhere on X. In order to generalize (2.1.4) to the case of higher dimension, we consider a pseudo-volume form and the associated Ricci form on a complex manifold X. In terms of a local coordinate system z', ... , zn of X, a pseudo-volume form von X can be locally written as

36

Chapter 2. Schwarz Lemma and Negative Curvature

(2.4.3)

V

"

= V nUdz j

/\

dzJ),

j=l

where V is a (locally defined) nonnegative function. A pseudo-volume form v is a pseudo-metric on the anti-canonical line bundle K- l = /\" T X. If V is positive everywhere, i.e., if v is a metric on the line bundle K- l , then v is a volume form of X. To each v given by (2.4.3) we associate the Ricci form Ric(v) by (2.4.4) where Rjk = -3 2 log V/3z j 3z k • Of course, this is a special case, i.e., L = K- l , of the construction (2.4.1). If X is a Hermitian manifold with fundamental 2-form w and the volume form v = w", then (RjjJ represents the components of the Ricci tensor in the classical sense. Hence the name "Ricci form" for Ric(v). It is, however, important to associate the Ricci form directly to a volume form rather than to a Hermitian metric. If dim X = 1, then (2.4.5)

Ric(v)

= 2Kv,

where K is the Gaussian curvature of X. It is then natural to define K v when dim X = n by the following formula. ( -Ric(v»" Kv = - - - - - -

(2.4.6)

n!(n

+ 1)"v

We say that Ric(v) is negative if the matrix (R jk ) is negative wherever defined (i.e., where v > 0). We say that it is negatively bounded if it is negative and if Kv :s C < 0 wherever defined. If v is negatively bounded, we can always normalize it (by multiplying it with a positive constant) so that Kv :s -l. Let B~ be the ball of radius a in en: B~

= {z =

(Zl, ... ,

Z") E C"; IIzl/2

The unit ball Bf will be denoted Bn. An invariant volume element Jia of

B~

= Iz l l2 + ... + Iz"12

< a 2}.

is given by

(2.4.7) For a = 1, we write Ji for (2.4.8)

{Ll.

A simple calculation shows

Ric( {La ) = -2i(n

+ 1) '~k " J.

and

.5 (a 2 Jk

-

Ilz112) + zj Zk d zj IJz1l2)2

(a2 _

/\

d-Zk ,

4 Ricci Fonns and Schwarz Lemma for Volume Elements

K

(2.4.9) Let

/-la D~

= - (-Ric{/LaW =-1. n!{n + l)nlla

en:

be the polydisc of radius a in D~ = ({Zl, ... , zn) E

(2.4.10)

37

en;

IzII

< a, ... , Iz"l < a}.

The unit polydisc D7 will be denoted D". An invariant volume element Va of D~ is given by

n II

(2.4.11)

Va

= (2a )211

j=1

For a = I, we write

V

for

VI.

idz}!\ dzi (a

2

. 2 2·

-lzll )

By a simple calculation, we obtain 2

n

d · (Va) = - 4·I L a R IC 2 . 2 2} Z . (a -Izll) 1=1

(2.4.12)

!\

j d-Z,

and K

(2.4.13)

(-Ric(v,,»"

v"

------n!(n + I)nva

(n

+ I)n

The following theorem generalizes (2.1.4). (2.4.14) Theorem. Let v be any pseudo-volume form on the unit ball B n with negatively bounded Ricci form Ric( v) and normalized in such a way that K v ::::: -I. Then v ::::: /L.

°

Proof For a, < a < 1, we use the volume form /La on the ball B: defined by (2.4.7). Let U a be the nonnegative function on defined by

B:;

Ua

=

vi /La·

As in the proof of (2.104), U a attains its maximum at some interior point, say Zo of and it suffices to prove ua{zo) ::::: 1. Assuming that Ua(ZO) > 0, we calculate the complex Hessian of logu a at zoo From (2.4.2) we obtain

B:,

2 " aaz log Ua d i d- k dd c Iog U a -- 2·I '~ i az k Z!\ Z

R·IC (/La ) - R·IC (V ) •

-

-

Since the complex Hessian dd c log U a must be nonpositive at the maximum point Zo and since Ric(v) is negative, we have

0< -Ric(v) ::::: -Ric(/La)

at

Zoo

Taking the nth exterior power of this inequality, we obtain at From Kv ::::: -1 = K/-l a' we obtain

zoo

38

Chapter 2. Schwarz Lemma and Negative Curvature at

i.e.,

U a (zo)

zo,

o

:::: 1.

In order to state a corollary to the theorem above, we need to explain the concept of meromorphic map. In general, a meromorphic map f from a complex space X into a complex space Y is a correspondence satisfying the following conditions: (I) For each point x of X, f (x) is a nonempty compact subset of Y; (2) The graph G j = {(x, Y) E X x Y; y E f(x)} is a connected complex subspace of X x Y with dim G t = dim X; (3) There exists a dense subset X* of X such that f(x) is a single point for x E X*.

Let :n:: G f ~ X be the projection defined by :n:(x, y) = x; it is a proper map. Then {x} x f(x) = :n:- 1 (x), and f(x) is a complex subspace of Y. Let E C G f be the set of points where :n: is degenerate, i.e., E = ((x,y) E Gj; dimf(x) > O},

and let S

= :n:(E) = {x

E

X; dim f(x) > OJ.

Then E is a closed complex subspace of codimension 2: I of G f , and S is a closed complex subspace of codimension 2: 2 of X, (see Remmert [1]). It is then clear that f: X - S ~ Y is holomorphic. The subspace S is called the singular locus of f. The graph G r of f is the topological closure of the graph of flx-s, i.e., the closure of {(x, f(x»; x E X - S}. Now, as a consequence of (2.4.14) we have the following generalization of (2.1.5). (2.4.15) Corollary. Let X be an n-dimensional complex manifold with pseudovolume form v such that Ric(v) is negatively bounded and Kv :::: -1. Then every meromorphic map f: B n ~ X is volume-decreasing in the sense that f*v :::: /-t.

f is holomorphic, the proof is the same as that of (2.1.5); simply apply (2.4.14) to f*v. If f is meromorphic, let S C B n be the singular locus of f. Since Ric(v) is negative, the coefficient V of v is plurisubharmonic, (in fact, log V is plurisubharmonic). Hence, the coefficient of f*v is a plurisubharmonic function on B n - S. Since the codimension of S is 2, fOv extends across S. Now (2.4.14) can be applied to f*v. 0 Proof If

(2.4.16) Corollary. Every holomorphic map with respect to /-t, i. e., f* /-t :::: /-t.

f

ofB" into Usellis volume-decreasing

If we want to use the polydisc D n instead of the ball B n as our "model" domain, then in view of (2.4.13) we have to use the "normalized" volume form (n + l) n v rather than v.

4 Ricci Forms and Schwarz Lemma for Volume Elements

39

(2.4.14) and its corollaries can be generalized to more general domains, in particular to symmetric bounded domains, see Kobayashi [6], [7; p. 33] and HahnMitchell [1]. If the domain of a mapping f is a compact manifold rather than a noncompact space such as B", then the argument in (2.4.14) becomes simpler. For example, we have (Kobayashi [6], [7]) (2.4.17) Theorem. Let X and Y be n-dimensional complex manifolds with volume form Vx and pseudo-volume form Vy, respectively. Assume (a) The Ricci forms Ric(vx) and Ric(vy) are negatively bounded; (b) Kvy (y)/ Kvx (x) ::: 1 for x E X and y E Y; (c) X is compact. Then every meromorphic map f: X -+ Y is volume-decreasing in the sense that f*vy ::: Vx. Proof Although f is not holomorphic, f*Vy is a well-defined pseudo-volume form on X; see the proof of (2.4.15). Consider a non-negative function u = f* Vy /vx on X. Since X is compact, it attains its maximum at some point, say Xu EX. To complete the proof, consider ddc log u at Xo and follow the argument in the proof of (2.4.14). 0

if Y

(2.4.18) Corollary. In (2.4.17),

deg f where vol(X)

= Jx Vx

:::

and vol(Y)

is also compact, then

vol(X)/vol(Y),

= j~ Vy.

Proof Jx Vx deg j . = Jx f*Vy < ---Jy Vy ~ Jy Vy .

o If v is a volume element of a compact complex manifold, then its first Chern class c\ (X) is represented by the Ricci form (modulo a constant factor): I

(2.4.19)

C1

.

(X) = -[RIC(Vx )].

4rr

By (2.4.6) {(

lx -

K

) Ux

Vx

=

e-41l")n

n!(n

+ 1)/ C \

eX)" •

Let ax = min(-Kvx) x

and

b x = max(-Kvx).

x

Then (2.4.20)

ax . vol(X) <

~

( ~4rr)n

n!(n

+ l)n Cl (X)n

< b x . vol(X). ~

Similarly, we define ay and by, and we obtain for Y inequalities similar to (2.4.20).

Chapter 2. Schwarz Lemma and Negative Curvature

40

We note that condition (b) in (2.4.17) simply says ayfb x :::: l. If we do not make assumption (b), instead of the inequality f* (Vy) .::: v x we have

f * (Vy)

bx .::: -Vx· ay

From the proof of (2.4.18) and (2.4.20) we see that if X and Yare compact complex manifolds with volume forms Vx and Vy whose associated Ricci forms are negative, then (2.4.21)

bxhyc] (x)n degf.::: --aXayc] (Y)n

As we will see soon, using not only volume forms but Kiihler metrics we can get rid of these constants ax, hx, ay, by from the formula above. But some remarks are in order. The equidimensional Schwarz lemma (2.4.15) in higher dimension was first obtained by Dinghas [1] for holomorphic maps from D n into Einstein-Kahler manifolds X of negative Ricci curvature and was generalized by Chern [2] to maps into Einstein-Hermitian manifolds X of negative Ricci curvature. By considering the Ricci form associated to a pseudo-volume form rather than to a Kahler metric, Kobayashi [6] got rid of the Einstein condition. As we shall see later, for algebraic geometric applications it is essential to get rid of the Einstein condition. The Ricci form associated to a volume form goes back to Koszul, who used it to study homogeneous complex manifolds. However, on a Kahler manifold we can compare its Ricci tensor with the metric tensor. The simplest situation is when both X and Yare compact EinsteinKahler manifolds with negative Ricci tensor. Let
.:::

c]

(X)" /c] (Y)".

Now, if X is a compact complex manifold with negative first Chern class c] (X), then X admits an Einstein-Kahler metric by Aubin [1,2], see also Yau [4].

Hence, (2.4.22) Theorem. Let X and Y be compact complex manifolds with negative first Chern class. Then every meromorphic map f: X ~ Y satisfies

5 Metrics on Jet Bundles

41

5 Metrics on Jet Bundles In this section, Schwarz' lemma (2.3.5) will be generalized to higher order jet metrics. Although the language of jets and jet bundles will be used in Section 9 of Chapter 3, the results of this section will not be used otherwise in the rest of the book. First, we explain the notion of jet and jet bundle introduced by Ehresmann. Let X be an n-dimensional complex space. We consider holomorphic mappings f from domains in C into X, the domain of f depending on f. We say that two such maps f and g osculate to order k at a E C or have the same k-jet at z if they have the same derivatives of order 0, I, ... , k at a: f(a) = g(a), fr(a)

= gr(a),

... , f(k) (a)

= g(k)(a).

The equivalence class thus defined is the k-jet of f at a and is denoted j;U). We call a the source and f(a) the target of the jet. Choosing the origin 0 E C as a source and fixing x EX, we denote the set of k-jets j~U) with f(O) = x by X. If x = f(O) is a nonsingular point of X, in a fixed local coordinate system around x, every k-jet j~U) determines a point of

J;

C kn

Ur(O), f"(O), ... , f(k)(O»,

(2.5.1 )

J;

and this gives a one-to-one correspondence between X and C kn • The isomorkll X ~ C depends on the chosen local coordinate system of X. phism Sometimes, we use k-jets with source at a point a i- o. In such cases, we identify j;Cn with jt(.fc,), where fa is defined by fa(z) = f(z+a). Then j;Cn determines the point

J:

U~(O), f~r(O), ... , fu(k) (0»

= Urea),

f"(a), ... , f(k)(a».

Clearly, J~ X is the tangent cone irx. Even when x is a nonsigular point of X, for k ::: 2, X does not have an intrinsic vector space structure. Nevertheless, X has the origin Ok represented by the constant map to x, and it admits also scalar multiplication which is defined as follows:

J;

J;

(2.5.2)

where

!J-,Cz) = cz.

In terms of coordinates, this may be expressed as follows:

(2.5.3)

c· jtU) = (cf'(O), c 2 1"(0), ... , c k f(k)(O».

Put -<EX

Then Jk X is a fibre space over X; it is a fibre bundle over the regular part X reg of X with fibre C kn • A local coordinate system x = (xl, ... , x") of X reg induces a local coordinate system (x, ~l, ... , ~d, where each ~j has n components (~/, ... , ~n,

42

Chapter 2. Schwarz Lemma and Negative Curvature

see (2.5.1). In particular, if X is nonsingular, JI X is the tangent bundle T X. For k :::: 2, Jk X is not a vector bundle. There is a sequence of natural projections: J~X --+ J;-IX --+ ... --+ J;X.

If x is a nonsingular point of X, each fibre of J~ X --+ J~-I X has an intrinsic affine space structure, and the fibre over the origin Ok-I has a natural vector space structure. This can be seen from the chain rule; if x = (x I, ... ,xn) and y = (yl, ... , y") are two local coordinate systems, then (in symbolic notation omitting indices) dky

ay d k x

dz

ax dz

a 2 y dx dk-1x

- k= - - + - - - - -k + .... k 1 axax dz dZ

-

Hence, the natural projection

makes Jk X rcg into an affine bundle over J k- I Xreg with fibre

en.

Following Green-Griffiths [I] and Grauert [7] we define a jet pseudo-metric F on Jk X to be a continuous non-negative function on Jk X which is smooth

except when zero and is homogeneous of degree 1 in the following sense: (2.5.4)

F(c~)

= IclF(~)

For k = 1, it is a pseudo-length function or pseudo-metric. In terms of a local coordinate system (x, ~I, .•. , ~d, (2.5.5) Every holomorphic map f: D --+ X lifts to a holomorphic map .I.kj. : a

r-+

.l.k(f) =.10.k( J+u ) , a

By pulling back the jet pseudo-metric F by / f*F on D:

a

E

D.

f we obtain a non-negative function

(f* F)(a) = F(j,; f).

This gives rise to a pseudo-Hermitian metric dS}-F := 2(f* F)2dzdz

on D. Its curvature Kf'f-' is given by (see (2.l.2)) (2.5.6)

I d de IogfF=-f * * ( F2 1 del ) iKj* F dz/\dz=-j*F2 d ogF;

it is defined only where f* F is positive. Since f* in (2.5.6) is the pull-back by the differential of / f, it follows that Kj*F(a), a E D, depends only on j,~+1 f,

5 Metrics on Jet Bundles

43

not on f. We define the curvature KF of F to be a function on JkX - {Ok} given by (2.5.7)

KF(~)

= sup Kj'F(a), f

where the supremum is taken over all IE Hol(D, X) such that ~ = j~f. Then (2.5.8)

c =1= O.

If KF :::: -1, then the Ahlfors-Schwarz lemma (2.1.4) states (2.5.9) where ds 2 is the Poincare metric of D. Now (2.3.5) generalizes as follows (see Green-Griffiths [1]): (2.5.10) Theorem. Given a jet pseudo-metric F on the jet bundle Jk X, we define a pseudo-length function Fl: J 1 X -+ R by setting Fl (v)

=

inf

JT(~)=V

F(~),

where the infimum is taken over all ~ E Jk X which are mapped to vET X = J 1X by the projection n: Jk X -+ J 1X. Let Ilull denote the Poincare length ofu E T D. If KF :::: -1, then U

E TD,

IE Hol(D, X).

Proof Since D is homgeneous, we may assume that u is a vector at the origin. By mutiplying u by a suitable constant, we may assume that u = (d/dz)o. Let I E Hol(D, X), and l I: D -+ Jk X its lift. Then

f* F(O) =

F(j~(f»

=

F(f(O), 1'(0), ... , f(k)(O».

Since I*u = (f(0), 1'(0», we have Fl (f*u) :::: have FI (f*u) :::: lIuli.

f* F(O).

Hence, by (2.5.9) we D

For simplicity we shall assume that X is nonsingular in the remainder of this ection. The k-jet bundle Jk X is a generalization of the tangent bundle T X = J 1X. We shall now generalize the cotangent bundle T* X. Given a local coordinate system (xl, ... , xn) in a coordinate neighborhood U, we consider the following symbols: and assign weight p to each of dPx i , ... , dPx n . At each point x be the set of polynomials of weight m in dx i , d 2 xi, ... , d k Xi, (i and set (2.5.11)

D~,m) =

U D~k,m). xeX

E

=

U let Dik •m ) 1, 2 •... , n),

44

Chapter 2. Schwarz Lemma and Negative Curvature

Then D~·m) is a vector bundle over X, defined independently of the coordinate system (x I, ... ,xn). A (local) holomorphic section of this bundle is called a k-jet differential of weight m. For example, a 2-jet differential of weight 4 is locally of the form (2.5.12)

OJ

=

~

· k I L.. aijkl dx idXi dx dx

~ i '2k ~ 2i2' + L.. bijkdx dx i d x + L.. cijd x d Xl,

where aijkl are holomorphic functions, symmetric in all indices while bijk and cij are holomorphic functions symmetric in i and j. Clearly, D~·I) is the cotangent bundle T*X. More generally, (2.5.13) where sm T* X denotes the m-th symmetric tensor power of T* X. Evidently, we have D

(I.m)

x

C D(2.m) C X

...

C D(m.l1l) = D(m+l.m) = X x''',

and the natural multiplication D (k.m) X

D(k.m')

Xx

D(k.m+m')

-+x

.

The differentiation operation d: O(D~,m»

-+ O(D~+I.m+1)

is defined by the Leibniz rule, Every k-jet differential OJ of weight m defines a holomorphic function on the jet bundle Jk X such that (2.5.14) In fact, if OJ is of the form (2.5.12), then for I; = (x, 1;1,1;2)

E

J2 X we have

If sm T* X is very ample for sufficiently large m, then T* X is said to be ample. In such a case X admits a Finsler metric of negative holomorphic curvature, (see Kobayashi [13]). Now we want to generalize this fact to k ~ 2. The scalar multiplication defined by (2.5.2) leads us to the concept of weighted projective space. In general, given positive integers mo, ml, ... , mr, define a scalar multiplication on cr+ 1 by (2.5.15)

C E

This defines the action of the group C* on cr+ 1 variety (2.5.16)

p(mo.m' •.... m r ) = (C r +! -

-

C.

{O}. The resulting quotient

{OD/C*

5 Metrics on Jet Bundles

45

is called the weighted projective space of weights (mo, ml, ... , mr)' If mo = = ... = m" then we obtain an ordinary projective space PrC, A weighted projective space is, in general, a singular variety since the action of C* is not free. However, it can be described as a quotient of PrC by a finite abelian group. Let {j be a primitive mrth root of 1. Then the abelian group Zmo X Zml X •.. X Zm, acts on PrC, with generators being given by ml

(ZO, ..• , Zj, ... , Zr) ~ (zo, ... , {jZj, •.. , Zr).

The quotient of PrC by this group is naturally isomorphic to the weighted projective space P (mo.m I . .... m,) . Thus, if we divide C kn by the action of C* defined by (2.5.2), then we obtain p(n.I,II.2 .... ,n.k), where n . j denotes j repeated n times. We put (2.5.17) Then p(Jk X) is a fibre bundle over X with fibre p(n.l,n.2, .... n.k). Let W C HO(X, D~·m» be a linear space of holomorphic sections of D~·m). Let Wo, WI, ... , WN be a basis for W. Define rp: JkX -+ C N+ I by (2.5.18) Since rp(c~)

=

cmrp(~),

rp induces a map (jJ: p(Jk X) -+ PNC, provided that W has no base locus. We say that W defines an immersion of p(JkX) into PNC ifiP immerses P(JkX) into pNc.

Let w = (wo,

WI, .•. ,

w N ) be the natural coordinate system for C N + I so that W",

= w'"

0

rp

= rp*w"'.

Let dS~N be the Fubini-Study metric for pNc. Its associated Kiihler form C/J PN is given by C/JPN =dd C logllwI1 2 . We may consider dS~N as a pseudo-metric which is degenerate in the fibre direction of the fibering CN+I_{O} -+ pNc. If W immerses p(Jk X) into PNC, then rp*ds~N is a pseudo-metric on Jk X - {Od, which is degenerate only in the fibre direction of the fibering JkX - {Ok} -+ p(JkX). Set

The following is a variant of a theorem in Green-Griffiths [1]. (2.5.19) Theorem. Let X be a compact complex manifold such that a suitable subspace W C HO(X, D~·m» defines an immersion of p(Jk X) into pNc. Let Wo, WI, ..• , WN be a basis for W, and define a jet metric F on Jk X by

Chapter 2. Schwarz Lemma and Negative Curvature

46

•

where the WOI are considered as functions on Jk X. Then the holomorphic sectional curvature K F of F is bounded above by a negative constant. Proof Since KF is a function on JkX - (Od and satisfies the invariance condition (2.5.8), it may be considered as a function on the projective bundle p(Jk X). Since p(JkX) is compact, it suffices to show that KF < 0 on JkX - {Ok}. Let f: D -+ X, and / f: D -+ Jk X be its lift. As before, for (/ f)* we write f*. Thus Since

from (2.5.6) we have (2.5.20) Let ~ E JkX with ~ i= 0, and take f E Hol(D, X) such that ~ = j!f, where a E D. Since F > 0 on Jkx - {Od, f* F(a) > 0 and Kf*F(a) is defined. We want to show that Kf*F(a) < o. From (2.5.20) it is clear that KJ*F .::: O. Suppose Kj*F(a) = O. Then f*t/lJkX = 0 at a, which means that the differential of the map

z I-+}z.k!'

at a E D is vertical (i.e., in the fibre direction) in the fibering Jk X - {Od -+ p(JkX). With respect to a local coordinate system of X, f is given by

j:

j; f

= (f(z);

f'(z), f"(z), ... , f(k)(Z» E X X C kn ,

zED.

Its derivative at a is given by (f'(a); I"(a), f"'(a), ... , fCk+l) (a» E Tf(a)X x ckn.

The condition that it is vertical amounts to the following: f'(a) = 0, (f"(a), f"'(a), ... , f(k+l) (a» ~ c(f'(a), f"(a), ... , f(k)(a».

Since c(f'(a), f"(a), ... , f(k)(a»

= (cf'(a), c 2 f"(a), ... , c k f(k) (a)),

we obtain successively 0= f'(a)

=

/"(a)

= I"'(a) = ... =

t(k+l)(a),

5 Metrics on Jet Bundles

which shows that J; f = O. This is in contradiction to the assumption J! f = ~

47

t- O. o

The following generalization is also a variant of a result in Green-Griffiths [1]. The special case where D~,m) = SkT* X is due to Noguchi [1]. (2.5.21) Corollary. Let X be a compact complex manifold with a very ample line bundle L. Assume that the vector bundle D~,m) ®L -I admits a section for some m. Letao, ... , aN bea basisofHO(X, L), and ro, ... , rq be a basisfor HO(X, D~,m)® L -I). We define a jet pseudo-metric F on Jk X by

Then the curvature K F of F is bounded above by a negative constant. Proof Set Wi.a

= 'Ciaa

E

HO(X, D~·m»), and apply the proof of (2.5.19).

0

The pseudo-metric F constructed in (2.5.21) is degenerate on the base locus of HO(X, D~·m) ® L -I), i.e., the common zeros of ro, ... , 'Cq •

Chapter 3. Intrinsic Distances

1 Two Intrinsic Pseudo-distances Throughout this section we denote the unit disc by D, its Poincare metric by ds 2 , and the Poincare distance by p, (see Chapter 2, Section 1). In order to generalize the Schwarz-Pick lemma to holomorphic mappings between higher dimensional domains, Caratheodory [1],{2] introduced what is now known as the Caratbeodory pseudo-distance. Given a complex space X, let Hol(X, D) denote the set of holomorphic mappings f: X -+ D. CaratModory defined a pseudo-distance Cx on X by setting (3.1.1)

cx(p, q) = sup p(f(p), f(q»

for

p, q EX,

f

where the supremum is taken over all f E Hol(X, D). Since D is homogeneous, it suffices to take the supremum over the subfamily F = {f E Hol(X, D); f(p) = OJ. Applying the Arzela-Ascoli Theorem (1.3.1) and the following proposition (3.l.2) to this family F we see that F is compact. Hence the supremum in (3.l.l) is actually achieved by a mapping belonging to the family F. (3.1.2) Proposition. (1)

If X and Yare two complex spaces,

Cy(f(p), f(q» ::: cx(p, q)

for

then

f E Hol(X, Y) and p, q EX,

that is, f: X -+ Y is distance-decreasing; (2) For X = D, the Caratheodory pseudo-distance CD coincides with the Poincare distance p, i.e., CD =p. Proof (1) is immediate from the definition of the Caratheodory pseudo-distance. The Schwarz-Pick lemma (2.1.7) implies CD::: p while the consideration of the identity transformation of D implies CD ::: p. D Clearly, (3.1.2) may be regarded as 'a generalization of the Schwarz-Pick lemma. Dualizing the construction of the CaratModory pseudo-distance, we define first a function d~ on X x X by setting (3.1.3)

d~(p,

q) = infp(a, b),

50

Chapter 3. Intrinsic Distances

where the infimum is taken over all holomorphic maps I: D ~ X and all pairs of points a, bED such that f(a) = P and I(b) = q. By definition, d~(p, q) = 00 if there is no I E Hol(D, X) such that P, q E feD). Although this function is symmetric, i.e., d~(p, q) = d~(q, p), it may not satisfy the triangular inequality. We define the Kobayashi pseudo-distance d x as the largest pseudo-distance bounded by d~. More explicitly, we construct d x as follows, (Kobayashi [4]). We call an element of Hol(D, X) a holomorphic disc in X. Given two points P, q of X, we consider a chain of holomorphic discs from p to q, that is, a chain of points P = Po, PI,·.·, Pk = q of X, pairs of points ai, b l , •.. , ak, b k of D, and holomorphic mappings 11, ... , /k E Hol(D, X) such that fi(ai) =

and

Pi-I

fi(b i ) =

Pi

for

i = 1, ... , k.

Denoting this chain by ot, we define its length lea) by (3.1.4) and the pseudo-distance d x by

dx(p, q)

(3.l.5)

= infl(a), a

where the infimum is taken over all chains ot of holomorphic discs from P to q. Then (3.1.6) Proposition. (I)

dy(f(p), I(q» (2)

ffX and Yare two complex spaces, then ~

dx(p, q)

for

IE Hol(X, y),

p, q E X;

For the unit disc D, the pseudo-distance d D coincides with p, i.e., d D =p.

The proof of (3.1.6) is similar to that of (3.1.2). Among the pseudo-distances which share the properties stated in (3.1.2) and (3.1.6), Cx and d x represent the two extremes in the following sense. (3.1. 7) Proposition. Let X be a complex space. (1) {(ox is a pseudo-distance such that

p(f(p), I(q»

~

ox(p, q)

for

IE Hol(X, D),

p, q E X,

then cx(p, q) (2)

~

for

ox(p, q)

p, q EX;

If Ox is a pseudo-distance such that 8x (f(a), I(b»

~

pea, b)

for

IE Hol(D, X),

then h(p, q)

~

dx(p, q)

for

p, q EX.

a, bED,

I Two Intrinsic Pseudo-distances

51

Proof We prove only (2) since (1) is even more trivial. As in the definition of dx, let Po, ... , Pk, ai, bl, ... , ak, bk, II, ... , !k be a chain of holomorphic discs from p to q. Then ox(p, q):::: I)X(Pi-l, Pi) = Lox(fi(ai), Ii (b i» :::: LP(a;, bi).

Hence, ox(p, q) :::: infL pea;, b;) = dx(p, q).

o (3.1.8) Corollary. For any complex space X, we have cx(p, q) :::: dx(p, q)

for

p, q EX.

(3.1.9) Theorem. For any complex spaces X and Y, we have dxxy(x, y), (x', y'» = max{dx(x, x'), dy(y, y')}

x, x' EX, y, y' E Y.

Proof Since the projection JT: X x Y --+ X is holomorphic and hence is distancedecreasing, we have dXxY«x, y), (x', y'» ~ dx(x, x').

Similarly, dxxy(x, y), (x', y'» ~ dy(y, y').

Without loss of generality, we may assume that dx(x, x') ::::: dy(y, y'). Then it remains to prove dXxY«x, y), (x', y'» :::: dx(x, x'). Given a chain a of hoI omorphic discs oflength lea) from x to x' in X, we want to construct a chain y of holomorphic discs of length I (y) from (x, y) to (x', y') in X x Y such that ley) :::: lea). By our assumption there is a chain fJ of hoI omorphic discs of length l(fJ) from y to y' in Y such that l(fJ) :::: lea). Let a and fJ consist of x = Xo, ... ,Xk = x' EX; a: ai, ak, a~ E D; II, ... , Ik E Hol(D, X); fJ:

I I

a;, ... ,

y = Yo, ... , Ym = y' E Y; b l , b;, ... , b m , b~ ED; gJ, ... ,gm E Hol(D, Y).

We may assume that k = m and p(ai' a) ~ pCb;, b;) for all i refining these chains as follows. Let a;' be a point on the geodesic from ai to a; so that p(ai, a;) = p(ai, a;')

=

1, ... , k by

+ p(a;', a).

Let xI = Ii (an. Then we obtain a refined chain a ' from x to x' consisting of k + 1 holomorphic discs:

52

Chapter 3. Intrinsic Distances

!

X=XO, ... ,Xi_l,X;,Xi, ... ,Xk=XIEX;

'. a.

, " . aI, aI' ... , ai, a;, a i" • a iI •. ··• ak, aI k E D.

fl •...• j;. j;, ...• fk E Hol(D. X).

We repeat this process a finite number of times and call the resulting chain a' a refinement of the chain a. This process does not really alter the geometric picture of the chain; in particular, lea) = lea'). It is now clear that, after suitable refinements of a and {3, we may assume that k = m and p(ai, a;) ~ p(b i • b;>. By composing fi and gi with suitable automorphisms of D, we may also assume that ai = hi = 0 and that b; are all positive real numbers. Then 0 < b; :s < 1. Let rp;: D ~ D be the multiplication by the positive number h;/a;. We define a holomorphic map hi: D ~ X x Y by

a;,

a;

hi(z) = (fi(Z), gi(rpi(Z)))

for

zED.

Let y be the chain from (x, y) to (x', y') consisting of (x, y)

0,

= (xo, Yo), (XI, Yl), ... , (Xb yd = (x', y')

a;, ... ,0, a~ E D;

E X x Y;

hi, ... , hk E Hol(D, X x Y).

o

Then y possesses the required properties.

The theorem above, proven in a different manner in Royden [2], sharpens the result in Kobayashi [4; p.47]. In particular, (3.1.1 0) Corollary. For a polydise D" = D x ... x D, we have

dD,,«XI, ... , x n), (YI,"" Yn» = max{dD(xi, Yi)}. i

A similar product property for the Caratheodory pseudo-distance, due to Jarnicki and Pflug [4], will be proved in Section 9 of Chapter 4. Here we give a weaker result, which is easy to prove.

(3.1.11) Proposition. Let X and Y be complex spaces. For (x, y), (x', y') we have max{cx(x, x'), ey(y, y')}

E

X X Y,

:s CXxy«X, y), (x', y'» :s cx(x, x') + ey(y, y').

Proof The first inequality is proven in the same way as in the proof of (3.1.9). In order to prove the second inequality, we observe first eXxY«x, y), (x', y» = cx(x, x'). This follows from the distance-decreasing property of the holomorphic maps (x, y) E X X Y ~ X E X and x E X ~ (x, y) E X x Y. Now,

eXxY«x, y), (x', y'))

<

=

eXxY«x, y), (x', y)) + CXXy«x', y), (x', y'» ex(x, x') + ey(y. y').

o

1 Two Intrinsic Pseudo-distances

53

However, for a polydisc D n , we have the result as precise as (3.1.10). (3.1.12) Corollary. For D"

=

D x ... x D, we have

CD,,«XI, ... , XII), (YI,"" Yn»

= max{Cn(Xi, Yi»).

Proof Applying an automorphism to each factor D, we may assume that XI . .. = Xn = 0 and YI, ... , YII are all nonnegative real numbers. Without loss of generality we may assume that Cn (Xi, Yi) :::: Cn (XI, YI) for all i, i.e., Yi :::: YI. Let Ci = Yi /YI so that 0 :::: Ci :::: I. Consider the holomorphic map f: D --+ D n defined by for ZED. fez) = (CIZ, ... , CIIZ) Since

f is distance-decreasing, we have

This combined with the first inequality in (3.1.11) yields the desired equality. D (3.1.13) Proposition. For a complex space X, both Cx and d x are continuous (as mappings/rom X x X to R). Proof Because of (3.1.8) it suffices to prove that d x is continuous. In order to prove the continuity of d x at (p, q) E X x X, let {(Pn, qn)} be a sequence in X x X which converges to (p, q) in the complex space topology of X x X. From the triangular inequality we obtain

Hence the proof is reduced to showing that dx(Pn, p) --+ 0 as Pn --+ p. Let U be a neighborhood of p. Since d x :::: d u in U by (3.1.6), it suffices to show that dU(Pn' p) --+ 0 as Pn --+ pin U. If P is a nonsingular point of X, then we may assume that U is a polydisc Dn, and our assertion follows from (3.1.10). Consider the case where P is a singular point, and assume that there is a positive number,) such that d u (p", p) :::: ,) for all n. Let n: U --+ U be a resolution of the singularity, and {qn} a sequence in U such that n(qn) = PII' Since rr is proper, by taking a subsequence we may assume that (q,,) converges to a point q E U. By continuity, rr(q) = p. Let V be a polydisc neighborhood of q in U. Since n: V --+ U is distance-decreasing and d v is continuous by (3.1.10), we have

as n --+

00,

which is a contradiction.

D

The proof of (3.1.13) given here is due to Barth [3]. It would be of some interest to find a proof that does not rely on resolutions of singularities. Given p, q E X, let ex be a chain of hoiomorphic discs:

54

Chapter 3. Intrinsic Distances

I

p

a:

= Po, PI, ... , Pk = q

E

X;

al,bl,oO.,ak,bkED;

iI, ... , Ik

E

Hol(D, X).

Let C i be the (unique) geodesic from ai to bi in D. Joining II (CI), ... , Ik(Cd consecutively, we obtain a curve from P to q in X. This curve, denoted by lal, will be called the thread of the chain a. It is rectifiable with respect to d x and its length L(laj) in the sense of (1.1.1) is clearly bounded by lea), i.e., L(la!) ~ lea).

(3.l.14)

(3.1.15) Theorem. For any complex space X, the pseudo-distance d x is inner. Proof Let d~ be the inner pseudo-distance induced by d x , (see (1.1.2»: d~(p, q) = infL(y), y

where the infimum is taken over all rectifiable curves y from p to q. Since the inequality d x ~ d~ holds always (see (1.1.3», it suffices to prove d~ ~ d x . We have d~(p, q) ~ infLClaj) ~ infl(a) = dxCp, q), a

a

where the infima are taken over all holomorphic chains a of discs from p to q.

o

In the course of the proof we obtained also the following equality: (3.1.16)

dx(p, q) = inf L(la!), a

where the infimum is taken over all chains a of holomorphic discs from p to q. As we shall see soon, the Caratheodory pseudo-distance c x is not always inner. Since d x is inner, it is without detour by (1.1.6). But (1.1.7) says more: (3.1.17) Corollary. Let X be a complex space. Thenfor every p positive real number E, the open E-ball

E

X andfor every

U(p; E) = {q E X; dx(p, q) < E}

with center p isfinitely arcwise connected. Infact, every q to p by a curve of length < E lying in U (p; E).

E

U (p; E) can bejoined

Following (1.2.2) we define the degeneracy set ,,1(p) for p ,,1(p)

=

{q EX;

dx(p, q) =

E

X by

OJ.

The following is an immediate consequence of (1.2.7). (3.1.18) Corollary. Let X be a complex space and p EX. in particular if X is compact, then ,,1 (p) is connected.

If ,,1 (p) is compact, so

I Two Intrinsic Pseudo-distances

55

In connection with (3.1.17) we note that every q E U(p; e) can be joined to p by a chain a of holomorphic discs of length lea) < e whose thread lal lies in U(p; e). However, a itself may not stay within U(p; e). The following proposition rectifies this situation. (3.1.19) Proposition. Let X be a complex space. Given 0 E X and positive numbers p and e there is a constant C > I such that, for any 8 > 0, every pair of points p, q E U (0; p) can be joined by a chain fJ of holomorphic discs of length I (fJ) < C(dx(p, q) + 8) which lies in U(o; 3p + c). In particular, dU(o;3p+c)(P, q) :::: C . dx(p, q)

for

p, q E U(o, p).

Proof Let r, 0 < r < I, be the positive number determined by dD(O, r) = e so that the disc Dr = {z E C; Izl < r} of radius r has radius e with respect to the Poincare distance d D . We choose C in such a way that d D, (0, a) :::: C . dD(O, a)

for

a

E

Dr / 2 •

In order to show that this C has the desired property, we join p, q E U(o; p) by a chain a of holomorphic discs in X of length lea) < dx(p, q) + 8 < 2p, (where 8 is a small number such that the second inequality holds). Since the length of thread lal is less than 2p, it follows that lal is contained in U(o; 3p). Let k D -'>- X be the i-th holomorphic disc of the chain a sending ai, bi E D to Pi-I, Pi E X. Without loss of generality, we may assume that ai = 0 and Ib;l < r/2. Since Pi-I E U(o; 3p), we have J;(D r ) C U(o; 3p+e). For ifz E Dr. then dD(O, z) < e and dX(Pi-l, fi(Z» = dx(J;(O), J;(z» < e. This shows that we obtain a desired chain fJ by restricting the holomorphic disc fi: D -'>- X to Dr; in order to normalize the smaller disc Dr we set gi(Z)

=

fi(rz)

and let fJ be the chain consisting of g i: D and lies in U(o; 3p + c) and l(fJ)

:s C ·l(a)

for -'>-

ZED,

X. Then fJ is a chain from p to q

< C(dx(p, q)

Since 8 is arbitrary, we obtain the desired inequality.

+ 8). o

(3.l.20) Proposition. Let {Xm} be a monotone increasing sequence of sub domains in a complex space X such that X = U X m . Then (a) (b) Proof (a)

= limcx",(p,q). dx(p, q) = limd xm (p, q).

cx(p,q)

Clearly, we have

Given p, q E X, choose fm E Hol(Xm , D) such that fm(P) = 0 and cxm(p, q) = p(O, fm(q». Then the usual argument (using the Arzela-Ascoli theorem (1.3.1»

56

Chapter 3. Intrinsic Distances

shows that a suitable subsequence of Um} converges to a map f E Hol(X, D). Then limcxm(p, q) = limp(O, fm(q» = p(O, f(q» :'S cx(p, q). (b)

Similarly, we have

Conversely, given p, q E X and 8 > 0, let a be a chain of holomorphic disks from p to q with its length lea) < dx(p, q) + So Let a consist of holomorphic disks fi E Hol(D, X). We shrink each holomorphic disk fi by composing it with the multiplication by r < 1. We set f/'\z) = h(rz). If r is sufficiently close to I, then U/)} defines a chain air) from p to q. Given any 8 > 0, the length l(a(r» < lea) + 8 for r sufficiently close to 1. Since the chain air) is contained in a compact subset K of X, it is contained in Xm for m > mo. Hence, d x ", (p, q) :'S [(air»~ < lea)

+8

< dx(p, q)

+ 28.

o The proposition above is in Hristov [1,3], where he discusses also the situation where X is the limit of a monotone decreasing sequence of complex spaces X m • (3.1.21) Example. For the Gaussian plane C and the punctured plane C* = C-{O}, we have de = 0, de' = 0, Ce = 0, Ce' = O. In fact, given two points p, q E C and an arbitrarily small positive number 8. there is a map f E Hol(D, C) such that f(O) = p and f(8) = q. Hence, dc(p, q) :'S So In order to prove de' = 0, we consider the surjective holomorphic map z E C -+ eZ E C*. Since it is distance-decreasing, de = 0 implies de' = O. The remaining two statements follow from (3.1. 8). The statement Ce = 0 is nothing but Liouville's theorem (that every bounded entire function is constant). Both C and C* are complex Lie groups. Generally we have (3.1.22) Example. For any connected complex Lie group G, we have d G =0

and

Cc

=0.

In fact, given p. q E G, there is a sequence of maps fl' ... , ik E Hol(C, G) such that p E f1 (C), q E fdc) and h(C) n fi+l (C) :f. 0 for i = I, ... , k - 1. (These fi 's are suitable translates of complex I-parameter subgroups of G.) Our assertion follows from (3.1.21) and the fact that the maps h are distance-decreasing. Sitll more generally, (3.1.23) Example. If a connected complex Lie group G acts on a complex space X, then dx(p, q) = 0 for p, q E X belonging to the same G-orbit. In particular,

1 Two Intrinsic Pseudo-distances

57

if X is a complex space on which a complex Lie group G acts with a dense orbit, then dx = and Cx =0.

°

In fact, for a fixed Po E X the mapping G ~ X that sends g is holomorphic. Hence, dx(g(po), g'(po» :s da(g, g') = 0.

E

G to g(po)

E

X

(3.1.24) Example. Let p: en ~ R+ be a nonn (not necessarily the Euclidean nonn), and B = {z E en; p(z) < I) the unit ball for this nonn. Then for

CB(O, z) = dB(O, z) = peO, p(z»

In fact, given Z E B, I(p(z» = z. Then

~

0, define I: D

Z =1=

CB(O, z)

:s dB(O, z) :s

z

E

B.

B by I(t) = tz/ pez) so that

p(O, p(z».

On the other hand, for every z E en, there exists a linear functional )'2: en ~ e such that AAz) = p(z) and IAz(w)l:s pew) for all WEen. Then Az sends B into D and, if z E B, then

:s CB(O, z).

p(O, p(z»

If p is the Euclidean nonn, then B is homogeneous. So in this case, CB and dB

coincide not only at the origin but everywhere. Although d x is always an inner pseudo-distance, ex is not necessarily inner. The following examples are due to Barth [6]. (3.1.25) Example. Fix

°

< r < 1 and

fez,

X = D2 -

w);

°

< s < 1, and consider the Hartogs figure

Izl:s rand Iwl:::: s)

in e 2 . Since every bounded holomorphic function I: X ~ D extends to a holomorphic function j: D2 ~ D with the same bound, C x is the restriction of e D2. Take p = (zo, wo) EX C D2 with Izol > r, Iwol > s, and p(lwol, s) > p(zo, 0), and let q = (-zo, wo) E X. A curve Y = (YI, Y2) joining p to q in X must go around or under the notch in the bidisc. That is, either IYI (t) I :::: r for all t, or IYI (to) I < rand IY2(to)1 < s for some to. In the first case, we have L(y) :::: K > ex(p, q),

where K is the length of a curve (measured in the Poincare metric) from -zo to which stays outside the disc of radius r while ex(p, q) = CD(ZO, -zo) is the geodesic distance from Zo to -Zo in D. In the second case, we have

Zo

L(y)

+ cx(y(to), q)

>

cx(p, y(to»

>

2p(lwol, s) > 2p(zo, 0) = cx(p, q).

:::: p(wo, Y2(tO»

Since p (I Wo I, s) - p (zo, 0) is independent of y, we have L(y) > K' > cx(p, q),

+ P(Y2(tO), wo)

58

Chapter 3. Intrinsic Distances

where K' is a constant independent of y. Thus

ei(p,q)

= infL(y) y

> ex(p,q).

The domain X in the example above is not a domain ofholomorphy. Barth [6] pointed out that the Caratheodory distance of the following domain ofholomorphy constructed by Sibony [I] is not inner. (3.1.26) Example. In order to define a domain X = M(D, V) C C 2, let (an) be a discrete sequence of points in the unit disc D such that every point of the boundary circle aD is the non-tangential limit of a subsequence. Let (I'n) be a sequence of positive real numbers such that A. < 00. We set

Ln n

and V(z) =

e'P(z).

The function cp is negative and subharmonic in D, and the function V (z) is also subharmonic and 0 :::: V(z) < I in D. Since {an} is discrete in D, V(z) is continuous and vanishes only at an. Define X

=

M(D, V)

= fez, w)

E D

x C;

Iwl

<

e-V(zl}

C D x D.

Since the function IwleV(Z) is plurisubharmonic in D x C, the domain X is pseudoconvex and, by Oka's theorem, it is a domain of holomorphy. Sibony has shown that X is a proper subdomain of D2 and every bounded holomorphic function on X extends to a bounded holomorphic function on D2. It follows that ex is the restriction of e D2. Then Barth chooses the sequence {an} in such a way that a] = -az = zo and an I- 0 for all n. Then X is contained in a Hartogs-figure (as in the example above) containing the discs {±zo} x D. Taking wo, p and q as in the example above, we see that Cx is not inner. Then Barth raised the following question: If a bounded domain X is finitely compact with respect to cx, is Cx inner? The following counter-example was found by Vigue [2]. (3.1.27) Example. Let X = fez, w) E c 2 ; Izl + Iwl < 1,lzwl < lj16}. This is a generalized analytic polyhedron (defined in Chapter 4, Section 1). Every generalized analytic polyhedron is finitely compact with respect to its Caratheodory distance. Vigue [2] shows that

cx«O, 0), (z,

z» <

c~((O, 0), (z,

z»

for

1 1 - < Izi < -. 8 4

We note that X is also a Reinhardt domain, i.e., it is invariant under the transformations (z, w) r7 (ei.
I Two Intrinsic Pseudo-distances

59

The following simple I-dimensional example was given by larnicki-Pflug [5]: (3.1.28) Example. For the annulus A =

the Caratheodory distance

{Z CA

E

C;

~

<

Izl

< R},

R> 1,

is not inner.

(3.1.29) Remark. A system which assigns a pseudo-distance 6x to each complex space X is called (by L. A. Harris [1] who considered domains in normed linear spaces) a Schwarz-Pick system if the following conditions are satisfied: (i) it assigns to D the Poincare distance p, (ii) every I E Hol(X, Y) is distance-decreasing, i.e., 6y(f(X), I(x'» ::::: 6x(X, x')

X,X'

E

X.

The Schwarz-Pick system ofCaratheodory pseudo-distances and that of Kobayashi pseudo-distances are the two extremes in the sense of (3.1. 7). An example of Schwarz-Pick system which lies between the Caratheodory pseudodistance and the Kobayashi pseudodistance will be discussed in Section 3 of Chapter 4. (3.1.30) Remark. Demailly-Lempert-Shiffman [I] proved that for a quasi-projective variety X, the pseudo-distance d x can be defined by means of chains of algebraic curves. More precisely, k

dx(p, q)

= infLdC,(Pi-l, Pi), ;=1

where the infimum is taken over all chains of points P = Po, PI, ... , Pk = q and irreducible algebraic curves C I , ... , C k in X such that Po E C I , PI E C I n C2, ... , Pk-I E Ck-I nCb Pk E C k.

(3.1.3 I) Remark. Hahn [3] considers the pseudodistance obtained using only injective holomorphic discs in definition (3.1.5). Clearly, this pseudodistance dominates the Kobayashi pseudodistance. However, Overholt [1] shows that for a domain in cn, n :::: 3, it coincides with the Kobayashi pseudodistance. The same paper briefly summarizes earlier contributions by Minda [I], Vesentini [8], Vigue [10], and 1. H. Zhang [2]. (3.1.32) Remark. Wu [7] introduced an invariant metric related to dx but with more smoothness. See akso Cheung-Kim [1,2]. (3.1.33) Remark. For systematic accounts of intrinsic pseudo-distances for domains in infinite dimensional normed spaces, see L. A. Harris [I], FranzoniVesentini [1], Dineen [I], and Barth [9). (3.1.34) Remark. A construction similar to that of d x defines a pseudonorm on the homology group of X, see Chern-levine-Nirenberg [1], Kr6likowski-Tovar [I], and Zhang [I).

60

Chapter 3. Intrinsic Distances

2 Hyperholicity We shall now study complex spaces X for which the Kobayashi pseudo-distance d x is a distance. A complex space X is said to be hyperbolic if d x is a distance, i.e., if dx(p, q) > 0 for every pair p, q E X with p i- q. A hyperbolic complex space X is said to be complete if it is Cauchy-complete with respect to d x . Since d x is inner by (3.1.15), a complete hyperbolic X is finitely compact with respect

to d x by (1.1.9). Since d x is inner (see (3.1.15» and since every inner distance on a locally compact Hausdorff space X induces the given topology of X (see (1.1.8», we have the following theorem of Barth [3], see also Barth [9]. (3.2.1) Theorem. Let X be a hyperbolic complex space. Then d x defines the topology ofX. (3.2.2) Proposition. Let X be a complex subspace ofa complex space Y. ( 1) If Y is hyperbolic. so is X; (2) IfY is complete hyperbolic and X is closed. X is also complete hyperbolic.

Proof This follows from the fact that the injection i: X hence distance-decreasing.

~

Y is holomorphic and 0

From (3.1.9) we have (3.2.3) Proposition. For complex spaces X and Y. the product X x Y is (complete) hyperbolic if and only if both X and Yare (complete) hyperbolic. (3.2.4) Proposition. Let X and Y be complex spaces, and I: X ~ Y a holomorphic map. Let Y' be a complex subspace of Y. and define X' = I-I Y'. If both X and Y' are complete hyperbolic, so is X'.

Proof Let Of denote the graph of I: X ~ Y; it is a closed complex subspace of X x Y. Let f' be the restriction of I to X', and Of' its graph. Then Gr' = Of n eX x Y'). Hence, Of' is closed in X x Y'. By (3.2.3), X x Y' is complete hyperbolic. By (3.2.2) Of' is complete hyperbolic. Since the projection X x Y' ~ X induces a holomorphic isomorphism from Of' onto X', it follows that X' is complete hyperbolic. 0 In the following proposition, what we have in mind for applications is the situation where X and Xi are all domains in a complex space Y. The proof is immediate from (1.1.11). (3.2.5) Proposition. Let X and Xi, i E I, be complex subspaces ofa complex space Y such that X = ni Xi' If all Xi are complete hyperbolic, so is X.

Although the hyperbolicity is a global concept, we can localize it as follows. (3.2.6) Proposition. Let X be a complex space. If there exist a family of points Prx E X and positive numbers Da such that, for each ct, the Dot-neighborhood

2 Hyperbolicity

61

is hyperbolic and that {Ua } is an open cover of X, then X is hyperbolic. In particular, if for every p E X there is a positive number 8 such that the 8-neighborhood U(p; 8) = {q E X; dx(p, q) < 8} is hyperbolic, then X is hyperbolic. Proof Take positive numbers PDt and there is a constant Ca > 1 such that

such that 3p"

Sa

for

du.(p, q) ::: Ca . dx(p, q) This shows that dx(p, q) is positive for p, q same Ua, clearly dx(p, q) > O.

E

+ Ca

= 8a. By (3.1.19)

p, q E Ua.

Ua, q

1=

p. If p, q are not in the

0

(3.2.7) Proposition. Let X be a complex space. If there is a positive number 8 such that for every p E X the 8-neighborhood U (p; 8) is complete hyperbolic, then X is complete hyperbolic.

Proof By (3.2.6) X is hyperbolic. Let {Pn} be a Cauchy sequence in X. Let p and c be positive number such that 8 = 3p + c. We may assume, by omitting a finite number of points, that dx(Pm, Pn) < p for all m, n. Then by (3.1.19) we have d U (p,;8)(Pm, Pn) ::: C· dx(Pm, Pn), which shows that {Pn} is a Cauchy sequence with respect to d U (Pl;8). Since U(Pl; 8) is complete with respect to d U (p,;8), the sequence converges. 0 Let f: X --+ Y be a holomorphic mapping between complex spaces. Let f*d y = (f-ldy)i be the inner pseudo-distance on X induced from d y by f (see (1.1.12». As we pointed out in Section I of Chapter I, the inequality f* d y ::: d x follows directly from the definition of f*d y. It is reasonable to expect that the equality holds if f is a covering projection. In this connection we prove (3.2.8) Theorem. Let X be a complex space and :n:: X --+ X a covering space of X. Then (1) If P, q E X and p, q E X with :n:(jJ) = p and ir(q) = q, then

dx(p, q) = inJdx(p, q), q

where the infimum is taken over all q E X such that :n:(q) = q; (2) X is (complete) hyperbolic if and only if X is (complete) hyperbolic; (3) If X is hyperbolic, then :n:: (X, d x ) --+ (X, d x ) is a local isometry, and d x = :n:*dx . Proof (1)

Since:n: is holomorphic, we have

dx(p, q) ::: dx(p, q). Let a be a chain of holomorphic discs from p to q. We lift a to a chain a of holomorphic discs in X starting from p. Then a ends at some point q E X with :n:(q) = q, and its length lea) is equal to the length lea) of a. This proves (1).

62

Chapter 3. Intrinsic Distances

(2) Assume that X is hyperbolic. Let p, q E X be such that dx(p, q) = O. Let p E X be such that n(p) = p. By (1) there exists a sequence {q,,} c X such that n(qn) = q and limdx(p, qn) = O. By (3.2.1) {qn} converges to p. Then (n(qn)} converges to p. But n(il,,) = q and hence p = q. Assume that X is complete hyperbolic. If Br +8 is the closed ball of radius r + 8 around p E X and if Br is the closed ball of radius r around p E X, then (I) implies for ,5 > O. Since BrH is compact by assumption (see (1.1.9» and Br is closed, Br is also compact. Hence X is complete. If X is (complete) hyperbolic, so is X by (1.3.13). (3). We prove first that n is a local isometry. Let p E X and p = n(p) E X. Let U be a 2e-neighborhood of p such that U is homeomorphic to each component of n- I (U). We denote the component containing p by U. Let V be the E -neighborhood of p and V = n -I (V). We claim that n maps V isometrically onto V. Let q, rEV and q = n(q), r = nCr) E V. Since dx(q, r) :s dx(q, r) < e, there is a chain of hoi omorphic discs from q to r such that lea) < e. The thread lal of a remains within U since L(lai) :s lea), (see (3.1.14». We lift a to a chain a starting from q in X. Since la I remains within u, la I stays also within U. In particular, the end point of a must be n -I (r) n U = r. This shows that, for every chain a from q to r with lea) < e, there is a chain a from q to with lea) = lea). Hence, dx(q, r) 2: dx(q, r). The opposite inequality is a direct consequence of the fact that n is distance-decreasing. The last statement follows from (1.1.13). 0

un

r

According to Zwonek [3], the infimum in (1) may not be attained in general. Part of (3.2.8) is still valid when n: X -+ X is only a spread, i.e., when every point p E X has a neighborhood U such that n maps U biholomorphically onto the open subset n(U) of X. By (1.3.12) we have (3.2.9) Proposition. A spread X over a hyperbolic complex space X is hyperbolic. From (3.2.2) and (3.2.9) we obtain (3.2.10) Proposition. If a complex space X is holomorphically immersed into a hyperbolic complex :,pace Y, then X is also hyperbolic. The following proposition is of the same character as (3.2.9). From (1.3.12) and (1.3.14) we obtain (3.2.11) Proposition. Let f: X -+ X be a finite-to-one proper holomorphic map. IlX is (complete) hyperbolic, so is X. A complex space X is said to be normal at x if the ring Ox of germs of holomorphic functions at x is integrally closed in its complete ring of quotients, and X is said to be normal if it is normal everywhere. The following more geometric interpretation is useful for us. Let S be the singular locus of X, and Xreg = X - S. Given an open set U C X, a function

2 Hyperbolicity

63

hoI om orphic on Urcg and is bounded on Ureg n K for every compact set K c U is said to be weakly holomorphic on U. Then X is normal at x if and only if every weakly holomorphic function at x extends to a holomorphic function in a small neighborhood of x, see Narasimhan [1; p. 114]. The concept of weakly holomorphic mapping can be defined in terms of local coordinate systems of the target space. A normalization of a complex space X is a pair (X, rr) consisting of a normal complex space X and a surjective holomorphic mapping rr: X ---+ X such that (a) rr is proper and rr-1(x) is finite for every x E X, (b) If S is the singular locus of X, then X - rr -I S is dense in X and rr: X - n- I S ---+ X - S is biholomorphic. The normalization theorem of Oka (see Grauert-Remmert [3], Narasimhan [1; p. 118]) states that every complex space X has a unique (up to an isomorphism) normalization rr: X ---+ X. As an immediate consequence of (3.2.11) we have (3.2.12) Corollary. The normalization X is (complete) hyperbolic.

X ala (complete) hyperbolic complex space

The following example by Kaliman-Zaidenberg [l] shows that the normalization X of a non-hyperbolic complex space X can be hyperbolic. (3.2.13) Example. Let (x, y, u, v) be a coordinate system in C 4 , and X be the affine algebraic surface given by the equations y4 = x4 - I u4

=

y4(v 4

-

I).

°

Its singular locus S is given by y = 0, which implies x4 = 1 and u = while can be arbitrary. Since S consists of complex lines, X is not hyperbolic. Let X be the affine algebraic surface given by

v

y4 = x4 - I

u4 = v 4

-

1.

Then X is nonsingular. Define the map rr: X ---+ X by n(x, y, u, v) = (x, y, yu, v). Then (X, rr) is the normalization of X. Clearly, X is a direct product of two copies of the affine algebraic curve in C 2 defined by y4 = X4 -l. This curve is (complete) hyperbolic. (In fact, its projective completion is a compact Riemann surface of genus 2 and hence hyperbolic by (3.2.8) or by (3.7.3». By suitably compactifying X and X, Kaliman and Zaidenberg obtain also a compact example. We apply (3.2.6) to holomorphic maps other than covering projections. (3.2.14) Theorem. Let rr: X ---+ T be a holomorphic map of complex spaces. For t E T and 8 > 0, we set U(t; 8) = (u E.T; dT(t, u) < 8}. Iffor every point t E T there is a positive number 8 such that n- I (U(t; 8» is hyperbolic, then X is hyperbolic.

64

Chapter 3. Intrinsic Distances

Proof For every p E X, its 8-neighborhood {q E X; dx(p, q) < 8} is contained in Jr-1(U(Jr(p); 8)) because Jr is distance-decreasing. By (3.2.6) X is hyperbolic.

o The following result is due to Eastwood [I]. (3.2.15) Theorem. Let Jr: X -+ T be a holomorphic map of complex spaces. 1fT is (complete) hyperbolic and ifT has an open cover (Ud such that each Jr-1(U i ) is (complete) hyperbolic, then X is (complete) hyperbolic. Proof For each t E T, take 8 > 0 such that U(t,8) C U i for some Vi. Then Jr-1(V(t, 8» is hyperbolic. By (3.2.14) X is hyperbolic. We prove now completeness. Let {PIl 1 be a Cauchy sequence in X. Then (Jr(Pn)} is a Cauchy sequence in T and converges to a point to E T. Take 8 > 0 such that U(t", 8) C Ui for some Ui. Take s > 0 and p > 0 such that 3p +s = 8. Omitting a finite number of Pn, we may assume that

dT(to, Jr(pj) < sand

dx(Pm, Pn) < p.

Let V = (x EX; dX(Pl,X) < s}. Then by (3.1.19) there exists a constant C > 0 such that dvCPm, Pn) ::: C . dx(Pm, Pn) for all m, n, which shows that {Pnl is Cauchy sequence in V with respect to d v . Since V C Jr-1(U;), {Pn} is a Cauchy sequence in Jr-1CVi ) with respect to d,,-,(U')' Since Jr-1(Ui ) is complete hyperbolic, (Pnl converges to a point in Jr-1(Ui ). 0 (3.2.16) Remark. (i) In general, even if T and all Jr- I (t), t X may not be hyperbolic. For example, the domain

X=

fez,

w) E

c 2 ; Izl

< I,

Izwl

< I} -

E T,

are hyperbolic,

(CO, w); Iwl:::: I}

is not hyperbolic. Let T = D = {z; Izl < I} and Jr(z, w) = z. Then each Jr- 1(t) is biholomorphic to a disc. To see that X is not hyperbolic, consider two points P = (0, b) with b i= 0 and q = (0,0). Set PI! = (l/n, b). Then dx(p, q) = limdx(Pn, q). Let all = min{n, .Jn7ibT}. Then the mapping tED -+ (ant/n, anbt) E X maps I/a ll into Pn. Hence, dx(p, q)

= limdx(PII, q)

::: lim dD(l/a ll , 0)

= O.

(ii) However, as we shall see in (3.11.2), ifJr: X -+ T is a proper holomorphic map, hyperbolicity of T and Jr-1(t), t E T, implies hyperbolicity of X. (iii) Let Jr: X -+ T be a holomorphic fibre bundle with fibre F in the sense that every point t E T has an open neighborhood U such that Jr- I (U) is biholomorphic to V x F. If T and F are (complete) hyperbolic, then X is also (complete) hyperbolic by (3.2.3) and (3.2.15), (Kiernan [4]). Conversely, if X is (complete) hyperbolic, so are T and F. In fact, each fibre is (complete) hyperbolic by (3.2.2). To prove (complete) hyperbolicity of T it suffices to show that the bundle is locally flat so that the pullback X of X to

2 Hyperbolicity

65

the universal covering t of T is holomorphically a product t x F. (For if X is (complete) hyperbolic, so is X by (3.2.8). Then T is (complete) hyperbolic by (3.2.3». Thus the proof is reduced to showing the following (see Royden [5] and also (5.4.5) for details): Let f E Hol(D x F, F), and write frey) = f(t. y). Then iffo is an automorphism of F, fr = fofor all t E T. (iv) If n: X ~ T is merely a fibre space, X can be hyperbolic without T being hyperbolic as the following example shows, (Kobayashi [7]). Let X

=

{(z, W) E

c 2;

0 < Id

+ IwI2

< I}

with the natural projection n which assigns to (z, w) homogeneous coordinates (z, w). Then F = D*.

and E

T

=

PIC

X the point of PI C with

(3.2.17) Theorem. Let X be a complete hyperbolic complex space and f a bounded holomorphicfitnction on X. Then the open subspace X'

= {p

E X;

f(p)

i= O} = X - Zero(f)

is complete hyperbolic.

Proof Without loss of generality we may assume that f maps X into the unit disc D. Then apply (3.2.4) with Y = D and Y' = D*. 0 Let Y be a complex space. We say that a complex subspace X C Y is locally complete hyperbolic in Y if every point p of the closure X has neighborhood Vp in Y such that Vp n X is complete hyperbolic. The condition is obviously satisfied by any point p of X. So this is a condition on the boundary points p E

ax

= X-X.

A Cartier divisor A in a complex space Y is a closed complex subspace that is locally defined as the zeros of a single holomorphic function. That is, each point x E A has a neighborhood V in Y such that A n V is defined by one equation f = 0, where f is a holomorphic function on V. (3.2.18) Corollary. Let Y be a complex space and A a Cartier divisor of Y. Then (I) Y - A is locally complete hyperbolic in Y; (2) IfY is (complete) hyperbolic, Y - A is (complete) hyperbolic. Proof Choose a complete hyperbolic Vp such that A n Vp is given as the zeros of a bounded holomorphic function f in Vp and apply (3.2.17) with X = Vp and X' = Vp - A.

0

While removing an analytic subset of codimension I from X can radically change the (pseudo-) distance d x , removing a subset of large codimension does not, in general, change d x . The following theorem is due to Campbell-Ogawa [I] and Campbell-Howard-Ochiai [1]. (3.2.19) Theorem. Let X be a complex man((old and A a closed analytic subset of X of codimension at least 2. Then Hol(D, X - A) is dense in Hol(D, X) in the compact-open topology, and hence

66

Chapter 3. Intrinsic Distances

dX _ A = dx

on

X-A.

As pointed out by Campbell-Howard-Ochiai [I], the first part of (3.2.19) can be generalized as follows:

If A is a closed analytic subset ofcodimension > k in an n-dimensional complex manifold X, then Hol(D k , X - A) is dense in Hol(D k , X). In order to prove (3.2.19), we use the following lemma of Royden [4] which has other applications. (The proof of this lemma will be given in Appendix A of this Chapter.) (3.2.20) Lemma. If f is a holomorphic imbedding of a disc Dr of radius r > 1 into a complex manffold X of dimension n, there exists a holomorphic imbedding cp of the unit polydisc D n into X such that fez) = cp(z, 0, ... , 0, )

for

zED.

Proof of (3.2.19). Let f E Hol(D, X). We want to approximate f by elements of Hol(D, X - A). If we define ff E Hol(D, X) by fr(z) = f(tz), < t < 1, then each ft extends past the boundary aD of D and ft ---+ f as t ---+ 1. Thus, if each ff is in the closure of Hol(D, X - A), so is f. We may therefore assume that f extends to a slightly larger disc Dr, r > 1 and feD) c X. If we define E Hol(D, D x X) by l(z) = (z, f(z», then is an imbedding of D into D x X. Let n: D x X ~ X be the projection. If g E Hol(D, D x (X - A» approximates 1, then n 0 g E Hol(D, X - A) approximates f. We may therefore assume that f imbeds Dr. r > 1 into X. Let cp E Hol(D n , X) be as in (3.2.20), and define B = cp-l (A). If the map j E Hol(D, D/) defined by j(z) = (z, 0, ... ,0) can be approximated by g E Hol(D, D n - B), then the map f = cp 0 j can be approximated by cp 0 g E Hol(D, X - A). The proof of the theorem is now reduced to the case where X is a domain in c n and feD) c X. We consider the map h : D x A ~ C n defined by h(z, a) = fez) - a. Since dim(D x A) < n, h(D x A) is a meager set (i.e., a countable union of nowhere dense subsets) in CIl and, hence, there exists a sequence of points c 1, C2, ... of C" - h (D x A) converging to the origin 0 E C/. Define fm E Hol(D, C") by fm(z) = fez) - Cm. Since feD) c X, there is an integer N such that f",(D) C X for m > N. The sequence {fm, m > N} converges to f in Ho1(D, X), and, by construction, fill (D) c X-A. 0

°

1

1

A few remarks are in order. First, as the following example shows, it is essential in (3.2.19) that X is nonsingular, (Campbell-Ogawa [1]). (3.2.21) Example. Let n: C/+ 1 - {OJ ~ Pn be the natural projection. Let Y C P n be a hyperbolic algebraic manifold, e.g., a nonsingular curve of genus > 1. Let Xc cn+l be the cone over Y, i.e., X = n-1(y) U {OJ. Then d x == 0 since X is a union of lines interesecting at the origin. Let A = {OJ. Then d x -A is nontrivial;

2 Hyperbolicity dX_A(p, q)

~

67

dy(n(p), n(q» > 0

if p, q E X - A do not lie on the same line through the origin. Note that A is the singular locus of X and that by choosing Y to be of large dimension, we can make the codimension of A in X as large as we wish. We state, without proof, a generalization of (3.2.19) by PoletskiI-Shabat [1], see also larnicki-Pflug [10; p. 87]:

(3.2.22) Theorem. Let X be an n-dimensional complex manifold, and A a closed subset with (2n -2)-dimensional Hausdorff measure equal to zero. Then HoI (D, XA) is dense in Hol(D, X) in the compact-open topology, and hence on

d X _ A = dx

X-A.

A generalization of (3.2.19) to k-intrinsic measures (see Section 2 of Chapter 7 for intrinsic measures) was obtained by Kaliman-Zaidenberg [1]. Following Wu [I], we say that a complex space X with a distance function 8 (which is assumed to induce the topology of X) is 8-tight if Hol(D, X) is equicontinuous with respect to 8 and that X is tight if it is 8-tight for some 8. If X is hyperbolic, then it is dx-tight (by (3.1.6)) and hence tight. Conversely (Kiernan [2]), we have

(3.2.23) Theorem. A complex space X is hyperbolic ifand only (fit is tight. Proof Assume that X is 8-tight. Let P and q be two distinct points of X. Let U be an open hyperbolic neighborhood of p such that q ¢. U. Let W be a smaller neighborhood of P, relatively compact in U. Let E > 0 be such that the E-neighborhood V = (x EX; 8(W, x) < E} of W is relatively compact in U. Since Hol(D. X) is an equicontinuous family, there exists a positive number r < 1 such that if IE Hol(D. X) with 1(0) E W, then I(Dr) C V. Let c > 0 be a constant such that dD(O, b) > c·dn,(O, b) for all b E D r / 2 . Let a={p=po,PI, .. ·,Pk=q; al.b1, ... ,akobk; 11 ... ·,/d

be a chain of holomorphic disks from P to q. We may assume that Po, PI, ... , Pi-I E W, Pj ¢. W, al

Then Pi

E

= ... = Uk =

0, bl,···. b k

E

Dr / 2 •

V, and i

lea)

>

j

Ldn(O,bi)~eLdD,(O,bi) i=l

i=1 j

>

c Ldu(Pi-l, Pi) ~ e· du(Po, Pj). i=1

Take c' > 0 such that du(Po, V - W)

~ c'.

Then dx(p, q)

~ lea) ~

ee'.

0

68

Chapter 3. Intrinsic Distances

(3.2.24) Remark. If a complex space X is hyperbolic, then every holomorphic map f of C into X is necessarily constant since

dx(j(a).

feb»~

:::: dda, b) = 0

by (3.1.21). Hence, every holomorphic map f of PIC or a complex torus into a hyperbolic complex space X is constant. As we shall see in (3.6.3) a compact complex space X is hyperbolic if there is no nonconstant holomorphic map of C into X. We say that a complex space X if algebraically hyperbolic if there is no nonconstant holomorphic map of PI C or a complex torus into X. Ballico [1] proved that a generic hypersurface of large degree in Pn + I C is algebraically hyperbolic. It is not known if every algebraically hyperbolic algebraic manifold is hyperbolic. It is important to generalize the concept ofhyperbolicity to allow the Kobayashi distance to be partially degenerate. Let X be a complex space and L1 a closed subset of X. in applications, L1 is usually a closed complex subspace. We say that X is hyperbolic modulo L1 if for every pair of distinct points p, q of X we have dx(p, q) > 0 unless both are contained in L1. Then d x induces a distance function d X / Ll on the quotient space X I L1 in a natural way. We say that X is complete hyperbolic modulo L1 if it is hyperbolic modulo L1 and if for each sequence {Pll} in X which is Cauchy with respect to the pseudodistance d x , we have one of the following: (a) {Pn} converges to a point p in X; (b) for every open neighborhood U of L1 in X, there exists an integer N such that Pn E U for n > N. Clearly, X is complete hyperbolic modulo L1 if and only if the quotient space XIL1 is complete with respect to the distance function d x / Ll . Suppose that L1 and L1' are two closed subsets of X and that X is hyperbolic modulo L1 as well as modulo L1'. Then X is hyperbolic modulo L1 n L1'. So we can speak of the smallest closed subset L1 such that X is hyperbolic modulo L1. Clearly such a closed set is given by

L1x

= the closure of (p

E

X; dx(p. q)

= 0 for some q

=1=

pl.

From (1.1.8) we obtain the following (3.2.25) Theorem. {f a complex space X is hyperbolic modulo a closed subset L1, then (1) for every point P E X - L1 and jor every neighborhood U C X - L1 of p, there exists a a-neighborhood V of p >,vith re!>pect to d x such that V C U; (2) if L1 is compact, on the quotient space XI L1 the pseudo-distance d x induces the quotient topology. The following proposition is a straightforward generalization of (3.2.2). (3.2.26) Proposition. Let X be a complex subspace of a complex space Y.

70

Chapter 3. Intrinsic Distances

(3.2.33) Theorem. Let n: X ---+ X be a .finite-to-one proper holomorphic map. is (complete) hyperbolic modulo a compact subset .1, then X is (complete) hyperbolic modulo n- 1 (.1).

If X

(3.2.34) Remark. Given a complex space X, consider the equivalence relation R defined by the pseudodistance d x , i.e., p is equivalent to q if and only if dx(p, q) = O. Then d x induces a distance on XI R. However, XI R need not carry a complex structure which would make the projection X ---+ XI R holomorphic. Even when XI R admits such a complex structure, the induced distance may not coincides with the intrinsic distance d X / R of the complex space XI R, see Horst [3]. For hyperbolic quotients of homogeneous complex manifolds, see Gilligan [1]. On the degeneracy set .1 for the pseudodistance d x, see Hristov [4, 5, 6, 7] and Adachi-Suzuki [2].

3 Hyperbolic Imbeddings Let Z be a complex space and Y a complex subspace with compact closure Y. We call a point p E Y a hyperbolic point if every neighborhood U of p contains a smaller neighborhood V of p, if c U, such that dy(V

n Y,

Y - U) > O.

We say that Y is hyperbolically imbedded in Z if every point of Y is a hyperbolic point. Clearly, Y is hyperbolically imbedded in Z if and only if, for every pair of distinct points 1', q in Y c Z, there exist neighborhoods VI' and U 4 of p and q in Z such that dy(UI' nY, U q n Y) > O. In the definition above, there is no need to assume that Y is relatively compact. But in applications, Y is almost always a relatively compact open domain in Z. So, unless otherwise stated, we assume that Y is relatively compact in Z. It is clear that a hyperbolically imbedded complex space Y is hyperbolic. The condition of hyperbolic imbedding says that the distance d y (Pn, qll) remains positive when two sequences {I'll} and {q,,} in Y approach two distinct points p and q of the boundary aY = Y- Y. The concept of hyperbolic imbedding was first introduced in Kobayashi [7] to obtain a generalization of the big Picard theorem. The term "hyperbolic imbedding" was first used by Kiernan [6]. We note that a compact hyperbolic complex space is hyperbolically imbedded in itself. The proof of the following proposition is straightforward. (3.3.1) Proposition. If complex spaces Y and Y' are hyperbolically imbedded in Z and Z' respectively, then Y x Y' is hyperbolically imbedded in Z x Z'. The following is obvious. (3.3.2) Proposition. If there is a distance fimction ;) on

Y such that

3 Hyperbolic Imbeddings dy(p, q) ~ 8(p, q)

for

71

p, q E Y,

then Y is hyperbolically imbedded in Z.

Using the concept of length function and the induced distance function (see (2.3.1» we state the converse, (see Kiernan [6] and Kiernan-Kobayashi [2]). (3.3.3) Theorem. Let Y be a relatively compact complex subspace of a complex space Z. Then the following are equivalent: (a) Y is hyperbolically imbedded in Z; (b) Given a lengthfimction F on Z there is a positive constant c such that for

I

E Hol(D, Y).

Proof Assume (a). Ifa constant c in (b) does not exist, then there exist a sequence Un} in Hoi (D, Y) and points {an} in D such that

t,

. n*F2

2 > n· d SD

at

an .

Since D is homogeneous, we may assume that a ll = O. Let e be a unit vector at OED, measured by the Poincare metric. Then the inequality above states

F(dj;,(e»2 > n. Since In (0) E Y and Y is compact, by taking a subsequence we may assume that {f,,(0)} converges to a point p E Y. Let U be a complete hyperbolic neighborhood of p in Z, e.g., a neighborhood biholomorphic to a closed analytic subset of Dm. Assume that there exists a positive number r < 1 such that j;,(D r ) C U for n ~ no. Since j;,(O) belongs to a compact neighborhood of p in U and since U is complete hyperbolic, U;,ID, E Hol(D r , U)} is relatively compact in Hol(D r , U) by (1.3.3) and would have a subsequence which converges in Hol(D r , U). But this is impossible since F(dln(e))2 > n. Thus no such r exists. This means that for each positive integer k, there exist a point Zk E D and an integer Ilk such that IZk I < and f", (Zk) ¢ U. Let Pk = In, (0) and qk = j;" (zd. By taking a subsequence we may assume that {qkl converges to a point q not in U. Since

t

dy(pk. qk) :::: dD(O,

zd

~

0

as

k

~ 00,

this contradicts the assumption that Y is hyperbolically imbedded in Z. Assume (b). Let 8 be the distance function on Z defined by the length function cF. Then 8U(a), feb»~ :::: dD(a, b) for f E Hol(D, Y). By (2) of (3.1.7), 8:::: d y on Y. Given two points p, q E Y, set 2(.1' let Up and Uq be the open balls of 8-radius (.I' around p and q.

= 8(p, q) and D

In (3.2.18) we showed that if Y is the complement of a Cartier divisor in Z, then Y is locally complete hyperbolic in Z. In this connection we prove the following

72

Chapter 3. Intrinsic Distances

(3.3.4) Theorem. Let Y be hyperbolically imbedded in a complex space Z. IfY is locally complete hyperbolic in the sense that every point p E Y has a neighborhood VI' in Z such that VI' n Y is complete hyperbolic, then Y is complete hyperbolic. We start the proof with the following general lemma. (3.3.5) Lemma. Let Y be a complex subspace of a complex space Z. Let P E Y and VI' a neighborhood of p in Z. Given a smaller neighborhood WI' of P such that 8 := dy(Wp n Y, Y - VI') >

°

and a positive constant 8' < 812, there is a constant c > dy(q,q')~c·dv"ny(q,q')

°

such that

for q,q'EWpny with dy(q,q') <8'.

As a consequence, !f a sequence ofpoints in WI' then it is Cauchy with respect to dv"nY.

nY

is Cauchy with respect to d y,

Proof Let q, q' E WpnY such that dy(q, q') < 8'. Consider a chain of hoi omorphic discs a from q to q' oflength lea) < 8' consisting of points q = Yo, YI, ... , Yk = q' in Y, points ai, ... ,ak in D and maps II,.'" fke E Hol(D, Y) such that fi(O) = Yi-i and fi(ai) = Yi. Since q E WI' and lea) < 8', the dy-distance from WI' to Yi-I is less than 8'. Let rand r' be the positive real numbers defined by dD(O, r) = 812 and dD(O, r') = 8' so that r' < r. Then li(Dr) lies in the (812)-neighborhood of Yi-i and hence in VI" Choose a constant c such that dDCO, z) ~ c· dD,(O, z)

Since lea) < 8', we have ai L

E

for

Z E Dr"

Dr' and

d[)(O, Qi) ~ c Ld[),(O, Qi) ~ c Ldv"ny(fiCO), fi(aj» ~ c· dvpny(q, q').

Since this is valid for all holomorphic chains from q to q' of length less than 8', we have

o Proof of (3.3.4). Suppose that Y is not complete hyperbolic. Then there would exist a sequence {Pn} in Y which is Cauchy with respect to d y and such that Pn ~ P ~ Y. Let VI' be a neighborhood of P in Z such that VI' n Y is complete hyperbolic. By (3.3.5) {PI!} is a Cauchy sequence with respect to dv"nY and must 0 converges to a point in VI' n Y. This is a contradiction.

The case where Y is the complement of a Cartier divisor in Z is of particular interest. Combined with (3.2.18) the theorem yields the following (3.3.6) Corollary. Let Z be a compact complex space and A a Cartier divisor in Z. Let Y = Z - A. if Y is hyperbolically imbedded in Z, then Y is complete hyperbolic.

3 Hyperbolic Imbeddings

73

In the corollary above, if Y is not hyperbolically imbedded in Z, then Y need not be complete even when it is hyperbolic. For example, let W be a compact complex manifold of dimension at least 2 with a divisor A such that the complement W - A is hyperbolic. Let Z be the space obtained by blowing up W at a point p of W - A, and B the exceptional divisor obtained from p. Set Y = W - (A U {p}) = Z - (A U B). Then Y is hyperbolic, but not complete by (3.2.19). (3.3.7) Theorem. Let Y he a complex space hyperbolically imbedded in a complex space Z. Let Z' be a complex space with a proper finite map n:: Z' ~ Z, and Y' = n:- I (Y). Then Y' is hyperbolically imbedded in Z'. Proof We know that Y' is hyperbolic by (3.2.11). Let Y' be the closure of Y' in Z'. We extend the distance function dy' of Y' to the closure Y' by setting Z,

z'

E

Y',

where the infimum is taken over all sequences {Ym} and Lv:,} in Y' converging to z and z', respectively. In general, dy' (z, z') can be infinite. Clearly, the extended dy' is lower semicontinuous on Y' x Y'. Since Y is hyperbolically imbedded in Z, dy'(z, z') > 0 if n:(z) =f. n:(z'). Given Z E Y', let L1(z) = {z' E

Y';

dy'(z, z') =

OJ.

The problem is to show that L1(z) is a singleton set {z}. Since L1 (z) c n:- I (n: (z» and since iT-I (iT (z» is a finite set, it suffices to prove that L1(z) is connected. Let U be a compact neighborhood of z containing no other points of L1(z). Let au be the boundary of U. Since dy' is lower semicontinuous, there is a positive 8 such that dyo(z, au n y') C": 8. Suppose L1(z) contains another point z'. Any curve joining z to z' in Y' must go through aU and so has length at least 8. This is a contradiction. D We end this section with simple examples. (3.3.8) Example. Let A be a finite subset of PI C containing at least three points, say 00, 0 and 1. Then Y = PI C - A is hyperbolically imbedded in PI C. It is well known that the disc is the universal covering space of Y. By (3.2.8) Y is complete hyperbolic. (There is also a differential geometric argument to show that Y is complete hyperbolic, see Section 7). Since Y has only isolated boundary points, it is hyperbolically imbedded in PI C. (3.3.9) Example. Let Q be a complete quadrilateral in P2C. That is, Q is the union of six projective lines which pass through a set of four points in general position. We shall show that Y = P2C - Q is complete hyperbolic and hyperbolically imbedded in p 2c. Let {PI, P2, P3, P4} be four points in general position. For i < j, let lij be the (projective) line passing through Pi and Pj, (see Fig. i). Let P be any point of Y = P2C. Without loss of generality we may assume that P 1. [12. If we remove 112 as the line at infinity, we are left with an affine plane C 2 with

74

Chapter 3. Intrinsic Distances

five lines, where Ll3 and 114 are parallel (i.e., meet at infinity) and also parallel, (see Fig. ii), so that

L23

and

L24

are

M:= C 2 - (113 U/14 U[z3 U/ 24 ) ~ (C - {O, I}) x (C - {O, I}). Since p rt. 112 , P is in C 2 • It is clear from (3.3.8) and (3.3.1) that p has neighborhoods V and U with V c U such that dM(V n M, (C 2 - U) n M) > O. Since Y c M and (C 2 - U) n Y = (P2C - U) n Y, we have d y (V

n Y,

(P2C - U)

n Y)

> O.

This proves that Y is hyperbolically imbedded in P2 c. The fact that Y is complete hyperbolic follows from (3.3.4). (3.3.10) Example. From projective plane P2 C we remove four lines, say 10, II, Ib, I;, in general position. The resulting space is not hyperbolic since it contains C*, namely the (projective) line 100 through points 10 n II and I;) n L; with these two points removed, (see Fig. iii). If we remove also 100 as the line at infinity, then we obtain C 2 with two pairs of parallel lines removed, (see Fig. iv). Set Y = P2C - (10 U II U Ib U I; U 100)' Then Y ~ (C - {O, I}) x (C - {O, I}) is complete hyperbolic. We shall show that Y is not hyperbolically imbedded in P2C. If we remove Ib as the line at infinity, we obtain Figure v; I; and 100 are parallel, and 10, II and lex; are concurrent, i.e., meet at one point, say o. Consider another line I passing through 0 and take two points p and q on it, for instance, diametrically opposite to each other with respect to o. We shall show that as we rotate L around () toward 100 , the distance d y (p, q) approaches zero. Let a = I n L; and let r be the distance between 0 and a with respect to the Euclidean metric in C 2 . As [ rotates toward lex;, the point a moves away toward infinity and r tends to infinity. On the line I we take the punctured disc around 0 with radius rand denote it D;. Then dy(p, q) .:::: dD;(p, q). Since df);(p, q) --+ 0 as r --+ 00, we see that dy(p, q) approaches zero as [ rotates toward 100 , This shows that Y is not hyperbolically imbedded in P2 C. (It is not difficult to verify that the condition for hyperbolic imbedding is violated only by the points of l,xJ. On the other hand, the same space Y ~ (C - {O. I}) x (C - {O, I}) can be hyperbolically imbedded in PI C x PI C in a natural manner. We know that the space P2 obtained by blowing up two points of P2C is biholomorphic to the space obtained by blowing up the quadric Q2 = PI C X PI C once. More precisely, by setting one of the coordinates equal to 0, I or 00 in Q2, we obtain two sets of three parallel lines {fo, I, , m} and {/b, L; , m'} in Q2 as in Figure vi. By blowing up the point (00,00) = m n m' of Q2 we obtain with an exceptional curve loc. By blowing down m and m' in we obtain P2 C with [00 as the line at infinity, (see Fig. vii). In terms of homogeneous coordinate systems (zo, Zl, Z2) for P2C and «u D, u l ), (vo, vI» for PIC x PIC, the birational correspondence

02

02

is given by

02

75

3 Hyperbolic Imbeddings

13

114

P3

124

123

/23

124

P4

114

Fig.ii

1]3

112

l' I

Fig.i

I~

10

"-

P

0

II

q 10

I'0

I~

I' I

a

Fig.iii Fig.v

II

10 10

L'I

L'0

Fig.iv II

10 m

m

I'0 l' 1

Fig.vi

m+

~

m'

m'

J

Fig.vii

II

Fig.viii

m_

76

Chapter 3. Intrinsic Distances uo=zo,

UI=ZI,

v l =ZIZ2.

VO=(ZO)2,

Then, in Figure vii, the lines 100 , m and m' are given by 100 = {zo = O},

m = {uo = O},

m' = {vo = O}.

Finally, we shall construct a holomorphic map f: D* x D* -+ Y

which does not extend to a map f: D x D -+ P2C. We may assume without loss of generality that the five lines in P2 C in Figure iii are given by 100 = (zo = O},

10 = (Zl = O},

I~ = {Z2 = O},

Ifwe set

x

=

zllzo, y

=

10={x=O},

We define

II = (zo = Zl},

I; = {Zl = Z2}.

z2lzo, then in Figure v we have 11={x=l},

1~={y=O},

f by x

= tu,

Y

= ult

for

(t, u) E D* x D*.

In terms of the homogeneous coordinate system of P2C, for

f(t, u) = (1, tu, u/t) = (t, t 2 u, u)

Then

I;={x=y}.

f extends to all points of D x

f is given by

(t, u) E D* x D*.

D except the origin (0, 0).

The five lines we deleted from P2 C in the example above are not in general position since 10, II and 100 are concurrent. As we shall see later, when five lines in general position are removed from P2C, the resulting space is hyperbolically imbedded in P2C. The following example due to Kiernan [5] involves not only lines but also curves. (3.3.11) Example. From the projective plane P2 C with homogeneous coordinate system (zo, Zl, Z2) we delete the following three lines 100 ,10, II and two curves 100 =

{zo

= O},

10 = {Zl = O},

m+ = {ZIZ2 = (ZO)2},

II = (Zl = z°}'

m_ = {ZIZ2 = _(ZO)2}.

If we consider 100 as the line at infinity and use the inhomogeneous coordinate system x = Z I I zO, y = Z2 I zo, then the resulting space Y looks like Figure viii, where 10 = {x = A}, IJ = {x = I}, m+ = {xy = I} and m_ = (xy = -I}. Under the mapping cp: (x, y)

E

Y -+ (x, xy) E (C -

to,

ID x (C - (-I, I}),

3 Hyperbolic Imbeddings

77

Y is biholomorphic to (C-{O, I}) x (C-{ -I, I}) and hence is complete hyperbolic. We shall show that Y is not hyperbolically imbedded in P2C. Take any two points on the line 10, say (0,0) and (0, I) in terms of the inhomogeneous coordinate system (x, y). For each positive integer k,

sends the disc Dk = {t

E

C; It I < k} into Y. Ifwe set

then Pk ~ (0,0), qk ~ (0, I) and dy (Pb qd :::: dDk (0, 1) shows that Y is not hyperbolically imbedded in P2C. In this case, we note that the holomorphic map g: D* x D

~

~

0 as k

~ 00.

This

Y C P2 C

defined by get, u)

= (1, t, u/t) = (t, t 2 , u)

does not extend to a map g: D x D except the origin (0,0».

for

~ P2 C.

(t,u) E D* x D

(It extends to all points of D x D

The following example, also due to Kiernan [5], involves transcendental curves. (3.3.12) Example. From the projective plane P2C with homogeneous coordinate system (zo, z 1 , Z2) we delete the three lines loo, lo, II of the example above as well as the following two curves n+, n_: n+ = {Z2

= zOeZO/Zl}

n_

= {Z2

= _zOeZO/ZI}.

In terms of the inhomogeneous coordinate system (x, y) used in the example above, these curves are defined by n+ = {y = e l / x } and n_ = {y = -e l / X }. Then the resulting space Y is biholomorphic to (C - {O, I}) x (C - {-I, I}) under the map 1/1: (x, y) E Y ~ (x, ye- I / x ) E (C - {O, I}) x (C - {-I, I}). So, Y is complete hyperbolic. Using the same sequence of maps ik: Dk ~ Y as in the preceding example, we can show that Y is not hyperbolically imbedded in P2 C. We note that the holomorphic map h: D* ~ Y C P2C defined by h(t)

= 0, t, 2e l /

does not extend to a map h: D

~

l )

for

t E D*

P2 C.

In the preceding section we considered hyperbolicity modulo .1. Similarly, let .1 be a closed subset of a complex space Z. Then we say that a relatively compact complex subspace Y C Z is hyperbolically imbedded modulo .1 in Z if every point P of Y - .1 is a hyperbolic point. This is equivalent to saying that for every pair of distinct points p, q of Y not both contained in .1, there exist neighborhoods

78

Chapter 3. Intrinsic Distances

VI' and Vq of p and q in Z such that dy(VI' n Y, Vq n Y) > O. It is clear that if Y is hyperbolically imbedded modulo L1 in Z, then it is hyperbolic modulo L1 n Y. Generalizing (3.3.3) we have the following (Kiernan-Kobayashi [2])

(3.3.13) Theorem. A relatively compact complex subspace Y ofa complex space Z is hyperbolically imbedded modulo Ll if and only if given a compact neighborhood K of Y in Z and a length function F on K there is a continuous non-negative jUnction cp on K such that (a) (b)

cp is strictly positive on K - Ll, for all

f

E

Hol(D, Y).

Prool Assume that Y is hyperbolically imbedded modulo L1 in Z. The proof of (3.3.3) shows (see also Kiernan-Kobayashi [2]) that given a compact subset L of K - Ll, we can choose a constant c > 0 such that f*(c 2 F) :::: dsb in I-'(L) for all I E Hol(D, Y). Let L, C L2 C ... be an increasing sequence of compact subsets of K - L1 such that K - L1 = U::l L i . Take a corresponding sequence of positive constants Cl ::: C2 ::: .... Let cp be any non-negative continuous function on K such that 0 < cp :::: Ci on L i • The proof of the converse is the same as in (3.3.3). D

Generalizing (3.3.4) we have the following (Kiernan-Kobayashi [2]) (3.3.14) Theorem. Let Y be hyperbolically imbedded modulo L1 in Z. IfY is locally complete hyperbolic in the sense that every p E Y has a neighborhood VI' in Z such that Y n VI' is complete hyperbolic, then Y is complete hyperbolic modulo L1 n Y. Proof We note that every Cauchy sequence with respect to d y is a Cauchy sequence with respect to the pseudo-distance defined by the pseudo-length function cpF of (3.3.13). Hence, if Y is not complete hyperbolic modulo L1 n Y, then there exists a sequence {Pn} in Y which is Cauchy with respect to d y such that Pn --+ p i Y U L1. Let VI' be a neighborhood of p in Z such that VI' n Y is complete hyperbolic. By (3.3.5) {Pll} is Cauchy with respect to dVl'nY and must converges to a point of VI' n Y. This is a contradiction. D

Combined with (3.2.18) the theorem yields the following (3.3.15) Corollary. Let Z be a compact complex space, A a Cartier divisor in Z, and L1 a closed subset of Z. Let Y = Z - A. If Y is hyperbolically imbedded modulo L1 in Z, then Y is complete hyperbolic modulo L1 n Y. (3.3.16) Remarks. We have discussed so far only very elementary examples of hyperbolically imbedded spaces. Nontrivial examples require techniques which will be developed in subsequent sections. At this point we mention the following basic conjecture (cf. Kobayashi [7]). Conjecture 1. (i) A generic hypersur:face of degree:::: 2n + I in Pll C is hyperbolic, and (ii) the complement of such a hypersurface is complete hyperbolic and is hyperbolically imbedded in Pn C.

3 Hyperbolic Imbeddings

79

As we shall see from (3.6.12), parts (i) and (ii) are related. We list several supporting evidence for the conjecture. The classical result in support of (ii) is that the complement of 2n + I hyperplanes in general position in PIlC is hyperbolically imbedded in Pile. This will be proved in Section 10. The first result in support of (i) is the following examples of nonsingular hyperbolic surfaces in P3 C by Brody-Green [1]: (ZO)d + ... + (Z3)d + s(ZOZI)d/2 + t(ZOZ2)d/2

= 0

with an even degree d ~ 50 and generic s, t E C*. This was also the first example of a simply connected compact hyperbolic manifold. Based on the example above, Azukawa-Suzuki [1] constructed the first examples of smooth curves in P2 C with hyperbolically imbedded complements. They proved that the complement of the curve (ZO)d + (ZI)d + (.0 2 )" +£(ZOZI)d/2 +£(ZOZ2)d I 2

=0

in P2 C is hyperbolically imbedded in P2C if either d ~ 30 and £ is a complex number with £2 #- 0,4 or d ~ 14 and £2 = 2. (The curve above is nonsingular if £2 #- 2.) Nadel [I] also obtained similar examples of hyperbolic surfaces in P3C and curves in P2 C with hyperbolically imbedded complements. Zaidenberg [7] has shown that, for each d ~ 5, the set of smooth curves of degree d in P2 C with hyperbolically imbedded complements fonns a nonempty open set in the set of all curves of degree d (in the usual topology, not in the Zariski topology). In order to state more systematic results, let us say that a curve C in P2 C has degree Cd l , ... , dk) if it consists of irreducible components C I , ...• Ck of degrees d l , ••• , dk. Then the complement of a generic curve C of degrees (d l , •.• , dk) in P2C is known to be hyperbolically imbedded in P2 C in the following cases: (i) k ~ 5 with any degree (Babets [3]); (ii) k = 4 with degree (2, 1, 1, 1) (Green [7]); (iii) k = 4 with any degree as long as L d; ~ 5 (Green [2], DethloffSchumacher-Wong [1]); (iv) k = 3 with d l , d2, d 3 ~ 2 (Dethloff-Schumacher-Wong [1, 2]). Masuda and Noguchi [I] have found an algorithm to produce nonsingular hyperbolic hypersurfaces of every degree d ~ den) in PIlC with hyperbolically imbedded complements. However, den) is still very high compared with 2n + 1. A generic surface of degree d ~ 5 in P3 C contains no rational or elliptic curves, which makes such a surface very close to being hyperbolic, see Zaidenberg [12] and Remark (3.2.24). We mention a different kind of hyperbolicity criterion for plane curves. According to Grauert-Peternell [I], the complement of a smooth curve C of genus g ~ 2 in P 2 C is complete hyperbolic and hyerpbolically imbedded in P2C if the dual curve C* of C has only ordinary double points and i cusps with i < 2g - 2.

80

Chapter 3. Intrinsic Distances

We cannot expect to lower the degree 2n+ I in Conjecture I. In fact, Zaidenberg [6] made the following conjecture with strong supporting evidence. Conjecture 2. For every hypersurface A of degree:::: 2n in PIlc' there is a line which meets A only in at most two points. In particular, the complement of A is not hyperbolic. He verified this for a generic hypersurface A of degree :::: 2n. As we shall see later in (3.10.13) a theorem of Snumitsyn implies that the complement of 2n hyperplanes in Pile is never hyperbolic. As another supporting evidence we mention Green [6], where it is shown that the complement of a curve of degree 4 in P2 C is not hyperbolic. As we shall see in (3.10.25), the complement of n + 2 hyperplanes in general position in Pile is hyperbolic modulo the diagonal hyperplanes. Now we state another conjecture, which has much less evidence than the preceding ones. Conjecture 3. (i) A generic hypersurface ofdegree::: n +2 in PnC is hyperbolic modulo a proper algebraic subset, and (ii) the complement ofsuch a hypersur[ace in PnC is complete hyperbolic and hyperbolically imbedded in Pne modulo a proper algebraic subset of pne. There are some results in this direction in terms of the number of irreducible components of the hypersurface rather than in terms of the degree, see Nishino [I], Adachi-Suzuki [1]. We note that Part (i) is a special case of the following general conjecture. Conjecture 4. A projective algebraic manifold of general type is hyperbolic modulo a proper algebraic subset.

4 Relative Intrinsic Pseudo-distance Let Z be a complex space and Y a complex subspace with closure Y. Let d y and d z denote the intrinsic pseudo-distances of Y and Z, respectively. We shall now introduce a relative pseudo-distance d y.z on Y so that Y is hyperbolically imbedded in Z if and only if Y is compact and dy,z is a distance. (Although we call d y.z a pseudo-distance, dy.z(p, q) can be 00 when p and/or q

are on the boundary aY .) Let Fy,z C Hol(D, Z) be the family of holomorphic maps f: D -+ Z such that f~l (Z - Y) is either empty or a singleton. Thus, f E Fy.z maps all of D, with the exception of possibly one point, into Y. The exceptional point is of course mapped into Y. We define a pseudo-distance d y.z on Y in the same way as d z , but using only chains of hoi omorphic discs belonging to Fy,z, Namely, writing lea) for the length of a chain a of holomorphic discs as in (3.1.4), we set (3.4.1)

dy,z(p, q) = infl(a), ct

p,q

E

Y,

4 Relative Intrinsic Pseudo-distance

81

where the infimum is taken over all chains a of holomorphic discs from P to q which belong to Fy,z, (We want to emphasize that chains of points p = Po, PI, ... ,Pk = q we use for chains of holomorphic discs a are points of V. Ifwe take PI, ... , Pk-I only from Y, then we would not get the triangular inequality for dy'z). If P or q is in the boundary of Y, such a chain may not exist. In such a case, dy,z(p, q) is defined to be 00, For example, if Y is a convex bounded domain in en, any holomorphic disc passing through a boundary point of Y goes outside the closure V or is contained in the boundary BY = V - Y, so that dy,p (p, q) = 00 if P is a boundary point of Y. On the other hand, if Y is Zariski-open in Z, any pair of points p, q in V = Z can be joined by a chain of holomorphic discs beloning to Fy,z, so that dy,z(p, q) < 00 for p, q E Z = V, The relative pseudo-distance dy,z is a generalization of the pseudo-distance d y in the sense that Since Hol(D, Y) C Fy'z C Hol(D, Z), we have (3.4,2)

dz

:s d y.z :s dy,

where the second inequality holds on Y while the first is valid on For the punctured disc D* = D - to}, we have (3.4.3)

dD',D

V.

= d D.

The inequality dD',D ::: df) is a special case of (3.4.2). Using the identity map id D E FD'.D as a holomorphic disc joining two points of D yields the opposite inequality. Let Y' C Z' be another pair of complex spaces with V' compact. If f: Z --+ Z' is a holomorphic map such that fey) c Y', then (3.4.4)

dY'.z,(f(p), f(q»

:s dy,z(p, q),

p, q

E Y.

The proof of the following proposition is the same as that of (3.1.7). (3.4.5) Proposition. If8 y is a pseudo-distance on 8y (f(a),

feb»~

:s df)(a, b)

for

V such that

a, bED and

f

E Fy,z,

then 8y (p, q)

:s dy,z(p, q)

for

P, q E

V.

The proof of the following theorem is the same as that of (3.1.9). (3.4.6) Theorem. Let Y C Z and Y' C Z'. Then for p, q E Y and p', q' have dyxY',ZxZ'«p, p'), (q, q'» = max{dy,z(p, q), dy',z'(p', q')}.

E

V' we

82

Chapter 3. Intrinsic Distances

From (3.1.1 0), (3.4.3) and (3.4.6) we obtain (3.4.7) Corollary. For (D*i x D n- k C D n, we have

(3.4.8) Proposition. Let Y C Z. Then (a) d y.z is continuous on Y x Y, and is lower semicontinuous on Y x Y; (b) If Y is the complement of a closed analytic subset A of Z so that Y = Z, then dy,z is continuous on Z x Z. Proof The first assertion in (a) follows from (3.1.13) and (3.4.2). The second follows from the definition of dy,z. In order to prove (b), as in the proof of (3.1.13) it suffices to show that, for every p E Z and for a small neighborhood U of p, dYnu,u(pj, p) ~ 0 as Pj ~ p. Resolving the singularity of (Z, A) we may assume that U = D" and Y n U = (D*)k x Dn-k. Now use (3.4.7), (see the proof of (3.1.13) for details).

(3.4.9) Remark. Since dy,z(p, q) can be infinite when p is in be continuous on Y x Y.

ay, d y. z may not

Making use of (1.1.8), (3.4.8) and the proof of (3.1.15), we can generalize (3.1.15) and (3.2.1) as follows: (3.4.10) Proposition. (a) Let Y C Z such thatdy.z(p,q) > ofor all p,q E Y, p i= q. Then d y.z is an inner distance on Y and defines the topology ofY. (b) If, moreover, Y is the complement of a closed analytic subset A of Z, then dy,z is an inner distance on Z and defines the topology of Z.

(3.4.11) Theorem. Let Y C Z with Y compact. Among the following conditions. (a) and (b) are equivalent, and they imply (c). Thus (a) {} (b)

(a)

=>

(c).

given a length function F on Z, there is a constant c > 0 such that J*(c 2F2) :::: ds~

(b) for p, q E

Y,

p

for all

f

E

Fy.z;

i= q, dy,z(p, q) > 0;

(c)

Y is hyperbolically imbedded in Z.

In (3.6.20) we shall show that (c) is also equivalent to (b) Proof (a) => (b). Let Oz be the distance function defined by the length function cF in (a). Then the restriction of Oz to Y satisfies the condition for Oy in (3.4.5). Hence, (3.4.12)

on

Y.

4 Relative Intrinsic Pseudo-distance

83

(b) =} (a). This implication, due to Joseph-Kwack [I], can be proved in the same way as (3.3.3). If such a constant c does not exists, then there exist a sequence Un} in F y . z and tangent vectors Un E T D such that F(dln(un

» > n· Iunl,

where Iunl is the length of Un with respect to dSb. Then, by Arzela-Ascoli theorem (1.3.1) Un} is not equicontinuous at some point, say a ED. By taking a subsequence of {In} we can find a sequence {an} in D converging to a such that

On the other hand, we have

which contradicts (b). (b) =} (c). Let 8z be the distance function defined by c F in (a). Since 8z ::: d y on Y by (3.4.2) and (3.4.12), Y is hyperbolically imbedded in Z by (3.3.2). 0 (3.4.13) Remark. The inequality (3.4.12) established in the course of the proof is important. Since 8z does not become infinite and defines the topology of Z, it is often better to use 8z in place of dy,z, The pseudo-distance dy,z was introduced in Kobayashi [24], [25] where the equivalence of (b) and (c) was proved. Using chains of hoi omorphic punctured discs, we may define a slightly different pseudo-distance d;. More precisely, We use Hol(D*, Y) with the punctured disk D* equipped with the restriction of the Poincare distance d D = p, (not d D *). Then (3.4.14) In fact, in the definition (3.1.4) of lea) we may assume that points aI, b l , ... , b k are not the origin of D. Then by restricting II, ... , !k to D*, we obtain from each chain of holomorphic discs a chain of holomorphic punctured discs, hence the desired inequality.

ak,

The following proposition is immediate from the definition of d; and (3.4.15) Proposition. Every holomorphic map

I: X --+

d;.

Y between complex spaces

is distance-decreasing with respect to d; and d;.

(3.4.16) Proposition. IfY is a complex subspace of Z with the property that every map

I

E Hol(D*, Y)

extends to a map

j

E Hol(D, Z),

then

Proof If I E Hol(D*, y), then j E Fy,z, Hence, every chain of holomorphic punctured discs in Y gives rise to a chain of holomorphic discs used in the construction of dy,z. Hence, dy,z ::: d;. The remaining inequalities are from (3.4.2) and (3.4.14) 0

84

Chapter 3. Intrinsic Distances

We cannot claim the equality d y.z = d; in (3.4.16) since a chain of holomorphic discs used in the construction of d y.z may not induce a chain of holomorphic punctured discs if some of the connecting points PI, ... , Pk-l are in aY. However, using the natural injection D* -+ D and the the identity map D* -+ D*, we obtain (3.4.17)

and

For (D*)k

X

d~* = d D

on

D*.

Dn-k, our knowledge is less precise:

(3.4.18) The first inequality is a special case of (3.4.16). To prove the second inequality, given P = (al,· .. ,a ll ) and q = Cbj, ... ,bn ) in (D*)k X Dn-k, we consider a chain of points P = Po, PI,···, PIl-l, Pn = q given by Pi = (hI"", hi, ai+I,.·., an).

Then using an obvious chain of holomorphic punctured discs and (3.4.17) we obtain diD")' xDn-k (p, q) S dD(ai, b i ) S n . d D" (p, q).

L

(3.4.19) Proposition. Let Y be a relatively compact complex sub~pace of Z satisfying conditions (a) and (b) qf (3.4.11). Assume that every map f E HoICD*, Y) extends to a map j E Hol(D, Z). Then every map h E Hol«D*)k x Dn-k, Y) extends to a map ii E Hol(D n , Z). As we shall see later in (6.3.7), the assumption on the extendability of f is superfluous; it will be shown there that when Y is hyperbolically imbedded in Z, every f E Hol(D*, Y) extends to j E Hol(D, Z). Proof Set X = D n and X - A = CD*)k X D,,-k. Let 8z be the distance function on Z defined by the length function cF of (3.4.11) so that 8z S d y.z . Given any point Po E A, take a sequence of points Pi E X - A converging to Po. Taking a subsequence we may assume that h (Pi) converges to a point qo E Y. As we remarked in (3.4.13), we use oz which gives the topology of Z. Given a neighborhood V of qo in Z, take an .s-neighborhood W = {q E Z; oz(q,qo) <.s}

such that W c V. Let U be the (.s/4n)-neighborhood of Po in X = Dn with respect to d D ". We have only to show that h(U n (X - A» C V. Let P E un (X - A). Choose m such that Pm E U and 8 z (qo, h(Pm» < .s/2. Then by (3.4.18) d~_A (Pm, p)

S n . dx(Pm, p) S n(dx (Pm, Po)

+ dx(po, p»

< .s/2.

By (3.4.12), (3.4.16) and (3.4.15) liz(h(Pm), h(p»

:s d;Ch(Pm), h(p»

<

dy.z(h(Pm), h(p»

:s

d;_A(Pm,P) <e/2.

4 Relative Intrinsic Pseudo-distance

Hence, 8z (qo, h(p)) <

which proves that h(p) EWe V. Hence, h extends.

E,

85

o

In the proof of (3.4.19), consider more generally a complex space X and a closed complex subspace A and try to extend a map f E Hol(X - A, Y) to a map j E Hol(X, Z). In the proof above we made an essential use of the fact that there is a constant n such that d~_A S n . dx. The existence of such a constant seems to depend on the nature of the singular loci of X and A. Consider the following examples. (3.4.20) Example. Let X

A Z

D =

xD

= fez, w);

Izl

Iwl

< I,

< I},

(0 x D) U (D x 0) U diag(D x D)

=

fez, w) E D

=

PIC

and

x D; zw(z - w) = oJ, Y = PIC - too, 0, I} = C - to, I}.

and f:X - A

Let a, b

E

A, d~_A)

-?

Y

fez, w) = w/z.

to, I}, a =f. b, and t a nonzero complex number. Since f: (X (Y, d;) is distance-decreasing, we have

C-

-?

d~_A «t, at), (t, bt)) ::::: d;U(t, at), f(t, bt))

= d;(a, b)

::::: dy.z(a, b) > 0,

showing that dLA «t, at), (t, bt)) is bounded below by a positive constant independent of t. On the other hand, dx((t, at), (t, bt)) -? 0 as t -? O. Hence, there is no constant c such that d~_A S c· d x . (3.4.21) Example. Let P be a homogeneous polynomial on C 3 , and S be the affine surface defined by P = 0; it is the cone over the projective plane curve C defined by P = O. Let B be the open unit ball around the origin in C 3 . We set X = S n B and let A be the origin of C 3 , i.e., the vertex of the cone S. Let n: X - A - ? C be the natural projection. Assume that C is non-singular and that the degree of P is greater than 3 so that the genus of C is greater than 1. As we shall see in (3.7.3), C is then hyperbolic. We shall show that there is no constant c such that d~_ASc·dx.

Let p and q be linearly independent unit vectors in C 3 that lie in S. Let g, h E Hol(D, X) be defined by g(z)

= zp

and

h(z)

= zq,

zED.

Let {an} be a sequence of points of D converging to O. Using the chain of holomorphic disks consisting of g and h, we see that limn dx(anP, a"q) = o. On the other hand, since the projection n: (X - A, d~_A) -? (C, d~) is distance-decreasing, we have

86

Chapter 3. Intrinsic Distances

Since C is hyperbolic, every map f E Hol(D*, C) extends to a map j E Hol(D, C) as we shall show in (6.3.5). Hence, d~ = de by (3.4.16). Thus, d~_A (aIlP, anq) is bounded below by a positive number dcC1T(p) , 1T(q)) which is independent of n.

5 Infinitesimal Pseudo-metric Fx Given a complex space X, we shall define an intrinsic pseudo-metric, i.e., the infinitesimal form Fx of the Kobayashi pseudo-distance d x. All necessary algebraic results on norms are summarized in Section 4 of Chapter I. Let TxX and Tx* X denote the Zariski tangent space and cotangent space of X at x. Since there are some unresolved technical problems when X has singularities, we often have to assume that X is a complex manifold. We first define a quasinorm F; on the Zariski cotangent space T; X by setting (3.5.1) where II 1* All is the length of the cotangent vector 1* A E T* D measured by the Poincare metric ds 2 of the unit disc D, and the supremum is taken over all f E Hol(D, X) with x E feD). Because of the homogeneity of D, it suffices to take the supremum over all f E Hol(D, X) with f(O) = x. For some A we may have F;(A) = +00. On the other hand, if X is a complex manifold, then we have (3.5.2) In fact, given A i= 0, choose an element f*(u) of T,X such that A(f*(U)) i= o. Then I*A i= 0 and F;(A) ~ 111*;'11 > O. From the defintion (3.5.1) we can easily verify the convexity property for F;: (3.5.3) Let (3.5.4) From (3.5.3) it follows that T<* X is a vector subspace of T<* X. We recall (see (1.4.2)) that the indicatrix of F; at x is the subset rx* of Tx* X defined by (3.5.5) It is convex and circular and is contained in Tx* X. Let ;'1, ... , Ar be a basis for Tx*X, and let F*(A\), ... , F*()"r):::: M. Then by

(3.5.3) for any A. = I:aiA.i with lail :::: 1, we have F;(A.) :::: rM. It follows that F; is bounded on any bounded subset of Tx* X. It follows also that given t: > 0,

5 Infinitesimal Pseudo-metric Fx

87

there is a neighborhood of 0 in ix* X such that F;
I'.<*

(3.5.6) Proposition. The indicatrix

of F; is a compact, convex, circular subset

oJix*x. Let

i\ be the quasi-norm on TxX dual to

(3.5.7)

Fx(v) = F;*(v) = sup j).{v)1

F;, i.e.,

).El~*

Since i.e.,

r;

is compact, the supremum in the definition above is actually attained,

Fx(v) = max IA(v)1

(3.5.8)

;.Er/'

Hence, if (3.5.2) holds, in particular if X is a complex manifold, then (3.5.9)

Fx(v) <

for

00

v

E

TxX.

We also have (see (1.4.5» (3.5.10)

Fx(v

+ Vi) s

Fx(v)

+ Fx(v')

for

v, v'

E

TxX.

Thus, Fx defines a pseudo-norm on TxX. The following proposition corresponds to (3.1.6). (3.5.11) Proposition. (1)

Let f: X -+ Y be a holomorphic mapping between

complex spaces. Then F; U* A)

s

F: ().)

for A E T*Y,

Au*v)

s

Fx(v)

for

v

E

TX;

(2) For the unit disc D, both F1J and FD coincide with the norms defined by the Poincare metric, i.e., F~=ds2. (3.5.12) Proposition. Let X be a complex space. Then

IIrAil

s

F{(A)

for

f

E

Hol(D, X) and A E T* X.

Conversely, if E* is any quasi-norm on Tx* X such that

for all f

E

Hol(D, X) with f(O) = x, then

88

Chapter 3. Intrinsic Distances

Proof. Taking the supremum over

1

of the following inequality

111* All

~ E*(A),

we obtain F;(A)

= sup 111*(A)1I ~ E*(Je).

0

Dualizing (3.5.12) we have the following infinitesimal version of (3.l.7). (3.5.13) Proposition. Let X be a complex space. Then for Conversely,

1

E

Hol(D, X)

E

E

T D.

if E is any quasi-norm on Tx X such that E(f*u) ~ lIu II

for all 1

and u

u E ToD

Hol(D, X) with 1(0) = x, then E**(v) ~ Fx(v)

It should be noted that we are not claiming a stronger inequality E ~

Fx.

Proof. Fix 1 E Hol(D, X) with 1(0) = x and also A E Tx*X. When u varies in ToD and v in TxX, we have sup l(f* Je)(u)1 = sup iA(l*u) I ~

111* All =

lIu 11:0: 1

11"11:0:1

sup

IJe(f*u) I

E(f,u):o:1

sup IA(v)1 = £*0.).

<

E(v):o:1

Thus,

111* All

~ E*(Je)

Using this, we obtain Fx(v)

=

sup IA(v)l:::: F';:(}.):o:l

sup IJe(v)1

= E**(v).

0

E*(}.):o:1

TxX.

At a point x of a complex space X, we consider the tangent cone We recall that Tx X is a subset of the Zariski tangent space consisting of vectors of the form l*(u), where 1 E Hol(D, X) and u E T D. As we remarked in Section 3 of Chapter 2, f< X is in general smaller than what is usually called the tangent cone. If x is a regular point of X, then Tx X = Tx X. The quasi-norm F; on Tx* X defined by (3.5.1) can be obtained as the dual quasi-norm of a more geometric quasi-norm Fx on the tangent cone TxX. For v E TxX, we set (3.5.14)

Fx(v) = inf{lIull; u E T D

and

!*(u) = v},

5 Infinitesimal Pseudo-metric Fx

89

where II U II is the length of the tangent vector U measured by the Poincare metric ds 2 of the unit disc D, and the infimum is taken over all I E Hol(D, X) and u E T D such that I.(u) = v. Because of the homogeneity of D, it suffices to take the infimum over tangent vectors u at the origin 0 of D which are pointing in the positive direction of the real axis. If x is a regular point, then given v E TxX, there is a vector u E T D such that I.(u) = v, so that Fx(v) < 00. However, if x is a singular point and if no such u exists, then we set Fx(v) = 00. The quasi-norm Fx was defined in Kobayashi [4] as a direct infinitesimal analogue of d x, and Fx was introduced in Kobayashi [22] as an improved infinitesimal form of d x , see (3.5.23) below. (3.5.15) Remark. If X is complete hyperbolic and if Fx(vo) < 00, then the infimum in the definition (3.5.14) of Fx(vo) is actually attained, i.e., Fx(vo) = lIuoll for some Uo E ToD and some 10 E Hol(D, X) with lo*(uo) = Vo. To find such an 10, we consider a minimizing sequence

{(Ub !k); Uk

E

ToD, Ik

E

Hol(D, X)}

such that Ih(Uk) = va and Fx(vo) = lim lIukli. Then by passing to a subsequence if necessary, we may assume that {Uk} converges to some vector Uo E ToD. By applying (1.3.3) to the family Ud, we see that a subsequence of Ud converges to a map 10 E Hol(D, X). This proof is valid under the weaker condition that X is taut. The concept of taut complex space will be introduced in Section I of Chapter 5. The following alternate definition of Fx is sometimes useful. Let DR be the disc {z E C; Izl < R} of radius R with Poincare metric ds~ = 4R 2dzdz/(R 2

-

Id)2

= j;ds 2 ,

where jR : DR -+ D is the isomorphism sending z to z/ Rand ds 2 is the Poincare metric of the unit disc D defined in Section 1 of Chapter 2. Let e denote the tangent vector (a/az)o of DR at the origin O. The vector I*(e) E T{(o)X is traditionally denoted 1'(0). From the fact that e has a length 2/ R with respect to ds~, it follows that Fx may be defined as follows: (3.5.16)

.

2

Fx(v) = mf-, R

where the infimum is taken over all positive real numbers R for which there is a holomorphic map I : DR -+ X such that 1'(0) = v. We easily see that Fx defines a quasi-norm on T,X (in the sense of Section 4 of Chapter I). This quasi-norm is defined only on the tangent cone tx c TxX. If x is a regular point, it satisfies the condition (3.5.17)

Fx(v) <

00

The proof of the following proposition is similar to that of (3.1.6) and is straightforward.

90

Chapter 3. Intrinsic Distances

(3.5.18) Proposition. (I)

If X

and Yare two complex spaces, then IE Hol(X, Y)

for

and v E

TX;

For the unit disc D, FD coincides with the Poincare metric, i.e.,

(2)

FA = ds

2•

Corresponding to (2) of (3.1.7), we have (3.5.19) Proposition. Let X be a complex space. Then

I

for

E

Hol(D, X)

and u

E

T D.

Conversely, if E is a quasi-norm function (continuous or otherwise) defined on the tangent cone bundle TX = U TxX such that

I

for

E

Hol(D, X)

and u E T D,

then for

E(v) ::=; Fx(v)

f X.

v E

(3.5.20) Proposition. Let X be a complex space and x E X. Then Fx defined by (3.5.1) is dual to Fx defined by (3.5.14), i.e., F;(},) =

j).(v) I

sup

for},

E

T<*X,

Fx(v)SI

where the supremum is taken over all v E

Tx X with

Fx (v) ::=; 1.

Proof Let HolxCD, X) denote the subset of Hol(D, X) consisting of maps 1(0) = x. From the definition (3.5.14) of Fx we obtain (v E TxX; Fx(v) < I} = {f*u; u E ToD,

lIuli

I with

< 1, IE HolxCD, X)}.

Hence, for any A E Tx* X we have FX(A)

= sup 11/* },II = f

sup 1(/* A)(U) I = sup 1A,(/*u)1 = f.IIulI<1

where the suprema are taken over v E TxX with Fx(v) < 1.

f,liuii
I

Holx(D, X), u

E

sup 1;,(v)l, Fx(v)<1

E

ToD with

lIuli

< 1, and 0

(3.5.21) Corollary. Let Fx* denote the double dual of Fx. Then Fx(v) = Fx*(v)

for

v E

This may be restated in terms of the indicatrices and F;* at x EX: (3.5.22) Corollary. In TxX, fx = Since

rx"

TX.

rx,

fx and rx** of F x,

and fx is the convex hull of rx .

ftx is the double dual of F x, Fx may be characterized as follows:

Fx

5 Infinitesimal Pseudo-metric Fx

91

(3.5.23) Corollary. Fx is the pseudo-length function on T X determined by the following conditions: (a)

Fx'::; Fx

(b)

Fx(v

(c)

+ Vi) :s Fx(v) + Fx(v')

for

v, Vi

E

TxX;

Fx is the largest pseudo-length function satisfying conditions (a) and (b).

As a consequence, (3.5.24)

TX;

on

Fx

may be defined more directly in terms of Fx.

Fx(v) =

inf FX(Vi)

for

v

E

TxX,

V=LVi

where the infimum is taken over all possible ways to write v as a finite sum of vectors Vi in Tx X. The proof of the following proposition is simpler than that of the corresponding proposition (3.1.9). (3.5.25) Proposition. For any complex spaces X and Y we have

= max{Fx(u),

T X, v

TY;

(1)

FxxY(u, v)

Fy(v)}

for

u

(2)

Fxxy(u, v) = max{Fx(u), Fy(v)}

for

u E TX, v E TY.

E

E

Also, corresponding to (1) of (3.2.8) we have (3.5.26) Proposition. Let X be a complex space and rr: X -+ X a covering space of X. Then and Fi = rr* Fx. (3.5.27) Theorem. If X is a nonsingular complex manifold, then Fx and Fx are upper semicontinuous as a function on T X while F; is lower semicontinuous on T*X. Royden [2] has shown that Fx is upper semi continuous. From his result it follows that F; is lower semicontinuous and that Fx is upper semicontinuous. It is possible to prove directly that F; is lower semicontinuous. In any case, the proof makes use of the following extension lemma (Royden [4]); see Appendix A of this chapter. (3.5.28) Lemma. Iff is a holomorphic map of DR, where R > I, into a complex manifold X of dimension n such that its differential f* is nonzero at the origin 0, then there is a holomorphic map cp of the polydisc Dn = D x Dn~l into X such that cp is biholomorphic in some neighborhood of the origin and fez)

= cp(z, 0, ... ,0)

for

ZED.

Proof of Theorem. We shall first prove that F x is upper semicontinuous. Let v E T X. From the definition of F x it follows that, given e > 0, there is a pair, f E Hol(D, X) and u E ToD, such that f.u = v

and

FD(U) < Fx(v)

+ e.

92

Chapter 3. Intrinsic Distances

Let R > 1 and iR: DR ---+ D be the isomorphism defined by iR(Z) = z/ R. By applying (3.5.28) to I 0 iR E Hol(D R , X), we obtain a holomorphic map 'P E Hol(D n , X) which maps a neighborhood U of in D n biholomorphically onto a neighborhood V of 1(0) in X and satisfies

°

10 iR(Z)

for

= 'P(z, 0, ... , 0)

zED.

We choose R > I sufficiently close to I so that FD(Ru) = R· F[)(u) < Fx(v)

+ e.

By imbedding D into D n in an obvious manner (i.e., by sending Z to (z, 0, ... , 0)), we consider the tangent vector Ru E T D as a tangent vector of T Dn. By (3.5.25) we have FDn(Ru)

=

FD(Ru).

Since F D " is continuous by (3.5.25), there exists a neighborhood U' of Ru in T D" such that for u' E U'. FD,,(u') < FD,,(Ru) + e We take U' sufficiently small so that the elements of V' are vectors at points of V. Let v' E 'P*(V'). Combining all these relations, we obtain Fx(v')

=

Fx('P*(u'» S FD"(U') < FD,,(Ru)

+ e = FD(Ru) + e

< Fx(v)

+ 28.

This proves that Fx is upper semicontinuous. In order to prove that Fx is lower semicontinuous, let Ak E T* X be a sequence converging to A E T* X. Assume that F;().. ) < 00. Given any e > 0, there is a map I E Hol(D, X) such that F;(J,) - e < 111*..1.11. By restricting I to a slightly smaller disc, we may assume that I can be extended to a holomorphic map 'P: D x D n - 1 ---+ X so that I(z) = 'P(Z, 0), zED, and the differential 'P* is non-degenerate at (0,0), (see (3.5.28». Then

This shows that Fx is lower semicontinuous at A. Assume that F;(}.) = 00. Given an arbitrarily large number M, there is a map I E Hol(D, X) with an extension 'P as above such that

which shows that F; is lower semi continuous at A. In order to show that frx is upper semicontinuous, we make the following observation. We know that, for each x E X, rx* is compact, (see (3.5.6». Since Fx is lower semicontinuous, it follows that for any compact set K C X the union U xEK T.,* C T* X is also compact. Now, let Vk E Tx,X be a sequence converging to Vo E TxoX. We want to prove the inequality

5 Infinitesimal Pseudo-metric F x

93

Fx (vo) ::: lim sup Fx (Vk).

Assume the contrary. Taking a subsequence, we may assume that Fx(vo) < lim Fx(vd. By (3.5.8), Fx(Vk) = IAk(Vk)1 for some Ak E rx:' Taking a subsequence, we may assume that )'k converges to some ),0 E T~X. Since F; is lower semicontinuous, we have AO E rx:' Then

This is a contradiction. Hence, Fx must be upper semicontinuous.

o

If X is singular, we consider a desingularization ]'[: X ~ X. Let vEt x. As in the nonsingular case, given e > 0 there is a pair I E Hol(D, X) and U E ToD such that I.(u) = v and FD(u) < Fx(v) + e. Assume that we can lift I to j E Hol(D, X) such that I =.]'[ 0 j and set v = j.(u) so that ]'[.(v) = v. Then Fx(v) :::: Fx(v) :::: FD(U) < Fx(v)

+ e.

Since this holds for any e > 0, we obtain Fi(v) = Fx(v). Since Fi is upper semicontinuous, given E > 0 there is a neighborhood V of v in TX such that Fx(v') < Fi(v) + e for all v' E V. Let V = ]'[*(V). For any v' E V choose v' E V such that v' = ]'[*v'. Then Fx(v') :::: Fx(v') < FiCiJ)

+E =

Fx(v)

+ E.

This shows that Fx is upper semicontinuous at v (under the assumption that I can be lifted to j). We consider now the question of lifting I E Hol(D, X) to j E Hol(D, X). Let S be the singular locus of X. (3.5.29) Lemma. A holomorphic map IE Hol(D, X) can be lifted to a holomorphic map j E Hol(D, X) if I(D) ct S. X ~ X is meromorphic with singularity set S. Hence, meromorphic with singularity set I-I (S). Since dim D = 1, the singularities of ]'[-10 I are all removable and ],[-1 0 I is actually holomorphic.

Proof The map ],[-1 0

I: D ~

]'[-1:

X is

o (3.5.30) Theorem. Let X be a complex space with singular locus S. Then Fx and ftx are upper semicontinuous on X - S, and F; is lower semicontinuous on X - S. In jact, Fx is upper semicontinuous at v E i X unless v E is. Proof Given E > 0 there is a pair I E Hol(D, X) and u E ToD such that I.(u) = v and FD(u) < Fx(v) + e. If v f/- is, by (3.5.29) I can be lifted to a map j E Hol(D, X), and Fx is upper semicontinuous at v by the argument above. Let A E Tx* X with x ¢ S. Let I E Hol(D, X) be a map used in the direct proof of lower semicontinuity for Fli:, (see the proof of (3.5.27». Since I(D) ct s, by (3.5.29) f can be lifted to a map J E Hol(D, X). By (3.5.28) we may assume

94

Chapter 3. Intrinsic Distances

1

1

that extends to iP E Hol(D n , X). (If necessary, we first shrink to a slightly smaller disc). Set cp = 7i 0 iP E Hol(D n , X). Now by the same argument as in the proof of (3.5.27) F'X is lower semicontinuous at X From lower semi continuity of F; we obtain upper semicontinuity of Fx as in 0 the proof of (3.5.27). Do Duc Thai [1] also discusses upper semi continuity of F x for complex spaces X with singularities. See also Venturini [6] and Remark (3.5.46). Let X be a nonsingular complex manifold. Since Fx is upper semicontinuous, for any piecewise smooth curve y joining two points, say p and q of X we can define the integral fy Fx . (3.5.31) Theorem. Let X be a complex manifold. Then

dx(p, q)

= infi Fx = infi Fx , y y y y

where the infimum is taken over all piecewise smooth curves y joining p to q. The first equality is due to Royden [2].

Proof Set dx(p, q) = infi Fx, y

y

dx(p, q) = infi Fx· y

y

Since both F~ and F~ agree with the Poincare metric of D (see (3.5.11) and (3.5.18», we know that

do(a, b) = do(a, b) = do(a, b). By integrating (3.5.19) we obtain

dx(f(a), feb»~ ::: do(a, b)

for

f

E

Hol(D, X)

By (3.l.7), we have dx S d x . From (3.5.23) we have dx S

and

a, bED.

dx . Thus

(3.5.32).

y=

We shall first prove the equality dx = d x , which is due to Royden. Let yet), 0 s t ::: 1, be a curve from p to q such that

i

Fx < dx(p, q)

+ c.

Let y'(t) denote the velocity vector of y at time t. Since Fx(y'(t» is upper semicontinuous on [0, 1], it is the limit of a monotone decreasing sequence of continuous functions h n (t) on the interval [0, 1]. By the Lebesgue convergence theorem, we have

5 Infinitesimal Pseudo-metric Fx

95

Let h = hn for a fixed large n so that

i l' Fx <

h(t)dt < dx(p, q)

+ c.

Since h is continuous, it is Riemann integrable, and there is a positive number 8 such that if 0 = to < t, < ... < tk = 1 is any subdivision of [0, 1] with ti - ti-I < 8 and Sl, ... , Sk are points of [0, 1] with Is; - t;l < 8, then k

L h(Si)(ti -

ti-I) < dx(p, q)

+ e.

;=1

Now we shall prove the following "mean value theorem". (3.5.33) Lemma. Let y = y(t), a :::::: t :::::: b, be a C l curve in X. Then given e > 0 and s, a < s < b, there exists 8 > 0 such that for all t in the interval It - sl < 8 we have

+ c)lt -

dx(y(t), yes»~ < (Fx(y'(s»

sl·

Proof of Lemma. From the definition of Fx (see (3.5.16», there exist a disc DR of radius R and a holomorphic map f: DR ---+ X such that (i)

f(O) = yes),

f "(0) =

y (s)

-2 < Fx(Y , (s» R

and

+ e.

Let W be a polydisc neighborhood of yes). We choose 81 > 0 such that yet) E W for It - s I < 81• Since f restricted to the real axis and yare two parametrized curves with the same tangent vector at f(O) = yes), there exists 82 > 0 such that for It - sl < (h we have dw(y(t), f(t - s» < cit - sl.

Since the injection W ---+ X is distance-decreasing, we have (ii)

dx(y(t), f(t -

s» < cit -

sl·

On the other hand, from the distance-decreasing property of f and from the expression for the Poincare metric of DR with curvature -I (see (2.1.3» it follows that there exists 83 > 0 such that dx(f(t - s), f(O» :::::: d DR (t - s, 0) ::::::

(~ + e)it -

sI

for It - sl < h Combining this with (i) we have (iii)

dx(f(t - s), f(O» < (Fx(y'(s»

+ 2c)lt -

sl

for It - sl < 83. By (ii), (iii) and the triangular inequality we have dx(y(t), y(s» < (Fx(y'(s»

+ 3e)lt -

sl,

96

Chapter 3. Intrinsic Distances

thereby proving Lemma (3.5.33). For each S E [0, 1], let 1., denote the interval It - sl < 8 obtained in Lemma (3.5.33). We apply the Lebesgues covering lemma (see, for example, Kelley [1; p. 154]) to the open cover {/.,; s E [0, I]}. (The Lebesgues covering lemma states that, given an open cover U of a compact metric space A, there is a positive number 11 such that the open 11-ball about each point of A is contained in some member ofU). Thus there is 11 > such that ift, t ' E [0, 1] and It - til < 11, then t, t' E I, for some s. Let = to < tl < ... < tk = 1 be a subdivision of [0, 1] with ti - ti-I < 1'}, and choose Sf so that ti-I, ti E Is;. Then

°

°

dx(p, q)

:s L

dx(y(ti-I), yeti»~ < L(h(Si)

+ £)It; -

t;-I1 < dx(p, q)

+ 2£.

This completes the proof of the equality d x = dx . The equality dx = dx has little to do with complex analysis and is a direct consequence of the following theorem in Finsler geometry. (3.5.34) Theorem. Let X be a (real) manifold. Let F: T X ---* R be an upper semicontinuous pseudo-length function on X, and let ft be the convex pseudolength function defined by the property that its indicatrix at each x E X is the convex hull of the indicatrix of F at x. Then the pseudo-distance d defined by F coincides with the pseudo-distance d defined by ft. This theorem has been proved by Busemann and Mayer [1] under the assumption that F is continuous and strictly positive. For the proof of (3.5.34), see 0 Kobayashi [23]. If X is a singular complex space, Fx and ftx may not be upper semicontinuous. However, using the upper integral we can still define dx(p, q) = inf! Fx , y

y

dx(p, q) = inf! y

y

ft x .

Obviously, we have dx(p, q)

:s dx(p, q) :s dx(p, q).

But the question remains whether dx, d x and d x are all equal when X is singular. The proof of (3.2.19) gives also the following infinitesimal analogue of (3.2.19). (3.5.35) Proposition. {{ X is a complex manifold and A is a closed analytic subset of codimension at least 2, then FX-A = Fxlx-A

and

FX-A = Fxlx-A'

The second equality is a direct result of the first.

5 Infinitesimal Pseudo-metric Fx

97

It is not easy to determine F x even for relatively simple domains. The following example is due to Graham-Wu [1].

(3.5.36) Example. Let

X= D xC-

1

fez, w); Izl::: 2' Iwl:::

I},

~ D and q: X ~ C be the projections defined by p(z, w) = z and w. Let Xo = (zo, wo) be a point of X and Vo a nonzero tangent vector

and let p: X q(z, w) at Xo.

=

If Izol < 1/2, then FD(p*(vo» :::: Fx(vo) :::: F 01 / 2 (P*Vo).

If Izol > 1/2 (and hence Iwol < I), then Fx(vo) > O. More precisely, let 8 = dD(zo, zo/2Izol) > 0 and choose r, 0 < r < 1, such that dD(O,r) < 8/2. We shall show

(Since q (xo) = Wo E D, we are considering here q* Vo as a tangent vector of D). The second inequality giving an upper bound for Fx(vo) is trivial since D x D C X. The inequality Fo(p*vo) :::: Fx(vo) is also trivial since p is distancedecreasing. Assuming Fx(vo) < r F D(q*VO) we shall derive a contradiction. From the definition of Fx(vo) there exist U E ToD and f E Hol(D, X) such that f*u = Vo and Fx(vo) :::: FD(u) < rFD(q*vo). If q(f(Dr C D, then FD(qd*(u» :::: FD,(u), and

»

rF/J(q*vo)

= rFD(q*f*(u»

:::: rFD,(u)

=

FD(U),

which is a contradiction. Hence, q(f(D r » ct. D, and there exists a point a E Dr such that q(f(a» ¢. D. Hence, p(f(a» E D 1/ 2 . Then 8 :::: dD(p(f(a», p(xo». On the other hand, do(p(f(a», p(xo» = dD(p(f(a», p(f(O» :::: do(a, 0) <

8/2,

which is also a contradiction. Finally, let Izol = 1/2. Trivially, FD(p*(vo» :::: Fx(vo).

= 0 when p*(vo) = O. F x or Fx is, in general, not continuous.

It is not clear, however, whether Fx(vo)

As the following example shows,

(3.5.37) Example. Let X = {(z, w) E C2 ; Izwl < I}; it is a domain of holomorphy. Consider a vector field ~ = We claim that (i)

(ii)

Fx(~) Fx(~)

=

°

i=- 0

fz - a:'

at at

(a,a) E X, a

(0,0)

E

X.

i=- 0,

98

Chapter 3. Intrinsic Distances

To prove (i), we use the C-action on X defined by t: (z, w)

~

(etz, e-tw),

t

E

C.

The tangent vector to its orbit at (z, w) is given by

a az

a aw

17=Z--W-.

Obviously, Fx (17) = O. Since 17 = a~ at (a, a), we have Fx(O = 0 at (a, a), a =1= O. To see (ii), let f: D --+ X be a holomorphic map with f(O) = (0,0). The map f = (fl, h) is of the form fl (t)

= at + 0(2),

h(t)

= bt + 0(2).

Then fl(t)h(t) = abt 2

+ 0(3),

tED.

Using If1 (t)h(t) I < I and Cauchy's integral formula, we estimate the second derivative of fI12 at t = 0 and obtain labl < 1. In order for the derivative of f at 0 to be in the direction of ~ at (0, 0), the map f must satisfy the condition b = -a. Hence lal 2 < 1. Since

we have

2

Fx(~) 2: ~ > 2.

In certain cases, F x is not only upper semicontinuous but continuous. The following is due to Royden [2]: (3.5.38) Proposition. If a complex space X is complete hyperbolic. then Fx. F; and Fx are continuous.

t

t

Proof In order to prove that Fx is continuous at Vo E X, let Vk E X be a sequence of vectors converging to Va. Since X is complete hyperbolic, by (3.5.15) there exist vectors Uk E ToD and maps ik E Hol(D, X)} such that fh(Uk) = Vk and Fx (Vk) = II Uk II. Since Fx is upper semicontinuous, {Fx (vd} is bounded. Hence, by passing to a subsequence if necessary, we may assume that {Uk} converges to some vector Uo E ToD. By applying (1.3.3) to the family {ik}, we see that a subsequence of Uk} converges to a map 10 E Hol(D, X). Then fo*(uo) = Va and lim Fx(vd = lim IiUkli = Iluoll 2: Fx(vo), showing that Fx is lower semi continuous at Vo. The continuity of F; and Fx follows from (3.5.20) and (3.5.21).

0

5 Infinitesimal Pseudo-metric Fx

99

(3.5.39) Remark. As Royden states and as we remarked in (3.5.15), the proof above is valid under the weaker assumption that X is taut. The concept of taut complex space will be introduced in Section I of Chapter 5. Wright [I] proved that if X is a projective algebraic manifold of general type, then Fx is continuous even when X is not hyperbolic. Although Fx is not very smooth, in the hyperbolic case we can find a smooth length function which can often take place of Fx. (3.5.40) Proposition. If a complex space X is hyperbolic modulo a closed set ..:1 and if £ is a length function on X, then there exists a nonnegative continuous function cp on X which is positive outside ..:1 such that

for

I

E

Hol(D, X).

Proof Let {Um} be an increasing sequence of relatively compact domains in X - ..:1 which exhaust X - ..:1, that is, Um C Um + 1 and U U m = X - ..:1. As in the proof of (3.3.13) we can find a sequence of positive constants Cl ::: C2 ::: ... such that

Take a nonnegative continuous function cp on X such that 0 < cp Then f*(cp2 £2) ::s dsb for I E Hol(D, X).

::s

Cm

on Um. D

(3.5.41) Corollary. If X is hyperbolic modulo ..:1, then given a lengthfonction £ on X, there exists a nonnegative continuous function cp on X such that cp£ ::s Fx and cp > 0 on X - ..:1. We can define also an intrinsic relative pseudo-metric, i.e., the infinitesimal form of the relative pseudo-distance d y.z . Let Y be a complex subspace of a complex space Z. As in Section 4, let Fy,z be the family of holomorphic maps I: D --+ Z such that 1-1 (Z - Y) is either empty or a singleton. Then using the family Fy,z instead of Hol(D, y), we define F; z, Fr,z and Fy,z exactly as in (3.5.1), (3.5.7) and (3.5.14). ' We do not state obvious basic properties of these pseudo-length functions. However, we note that the proof of (3.5.27) shows that if Z is non-singular and Y is the complement of a divisor with no worse than normal crossing singularities, then Fy,z and Fy,z are upper semi-continuous and F; z is lower semi-continuous. We state the folJowing converse to (3.4.11); it st~engthens (3.3.3). The proof wiJI be given in the course of the proof of (3.6.20) (3.5.42) Theorem. Let a complex space Y be hyperbolically imbedded in a complex space Z. Given a length function £ on Z, there is a constant c > 0 such that

for

f

E

Fy,z,

100

Chapter 3. Intrinsic Distances

(3.5.43) Corollary. Let Y be hyperbolically imbedded in Z. Given a length/unction E on Z, there is a constant c > 0 such that cE < fry z on Y. (3.5.44) Remark. For Z = PIC and Y = PIC - {CXl, 0, I}, there is an estimate for Fy.z by Landau, (see Caratheodory [4, vol. 2, p. 198]). In fact, Landau proved that if fez) = alZ + a2z 2 + ... is ho10morphic in Izl < R and if f does not take values 0 or 1 in 0 < Izl < R, then R ::::: 16/1all and that this bound is the best possible. This may be restated as Fy.z «d I dz)o) = 1/8. (3.5.45) Remark. According to the definition of the curvature given by (2.3.3), F x has holomorphic sectional curvature:::: -1, provided X is complete hyperbolic (or taut), see B. Wong [1], Masaaki Suzuki [1] and Royden [9]. Given a nonzero v E Tt X, we want to show that g* F x has curvature:::: -1 for some g E HoI (D, X) such that g(O) = x and v is tangent to g(D). Since X is complete hyperbolic, there is a map g E Hol(D, X) such that Fx(v) = Ilull for some u E ToD with g*u = v, (see the proof of (3.5.38». Then in the inequality g* F; ::::: ds 2 , the equality holds at O. Take a supporting metric do 2 for g* F; at O. Then it is a supporting metric for ds 2 at O. By (2.1.8) it has curvature:::: -1. Hence, g* Fx has curvature:::: -I. (3.5.46). Remark. In order to generalize Royden's result (3.5.31) to singular complex spaces Venturini [6] extended the definition of Fx(~) to vectors ~ of higher order osculation.

6 Brody's Criteria for Hyperbolicity and Applications Theorem (3.6.3) of Brody is the simplest and most useful criterion for hyperbolicity. We give several technical improvements of Brody's criterion and their applications. The following is immediate from (3.1.6) and (3.1.21).

rr

(3.6.1) Proposition. X is a hyperbolic complex space, then every holomorphic map f : C --+ X is constant.

Proof For a, bE C, we have dx(f(a), Hence, f(a) = feb).

feb»~

::::: dda, b) = O.

o

We shall now prove the converse when X is compact. The proof is based on the following Reparametrization Lemma (3.6.2) of Brody [1]. Wu [6] points out that this lemma was first proved by Landau [1; pp. 618-619] in the context of holomorphic functions on the unit disc and then by Za1cman [1] for merom orphic functions on the unit disc. The Poincare metric d s~ of curvature -Ion the disc DR of radius R is given by (see (2.l.3»

6 Brody's Criteria for Hyperbolicity and Applications

In the following we use the metric R2ds~ of curvature the Euclidean metric 4dzdz at the origin O.

-1/ R2, which agrees

101

with

(3.6.2) Lemma. Let X be a complex space with a length Junction F. Given f HoIWR, X), define a Junction

E

U = f*F2/R2ds~ on DR. (fu(O) > c > 0, then there is a map g E Hol(D R , X) such that (a) theJunction g* F2 / R2ds~ is bounded by c on DR and attains the maximum value c at the origin; (b) g = f 0 ILr 0 tp, where tp is a holomorphic automorphism oj DR and ILr is the multiplication by suitable r, 0 < r < I, (i.e., ILr(Z) = rzJor Z E DR)' Proof For t

E

[0, I), define ft

ft(z) = f Set Ut = ft· F2 / R2ds~ = (f

Hol(DR, X) by

0

ILt(Z) = f(tz)

0

ILt)* F2 / R2ds~. Then

IL: f* F2

Ut

E

IL*ds 2

for

= IL:(R2ds~)' ~s~ R = IL;(U)

zE

DR.

t 2(R2 - Iz12)2

(R2 _ Itz12)2 .

Set

From the explicit expression for Ut(z) given above, we see that, for each t E [0, I), Ut(z) approaches zero at the boundary of DR and hence SUP~EDR ut(z) is attained in the interior of DR. It is easy to see that U (t) is continuous in the interval [0,1). Since Ut (0) = U(0)t 2 > ct 2, we have U (t) > c for t sufficiently close to 1. On the other hand, U(O) = O. Thus, c = U(r) for some r E (0, 1). Let Zo E DR be a point where c = SUPzED" u,. (z) is attained. Let rp be a holomorphic automorphism of DR which sends 0 to zoo Then g = f 0 ILr 0 tp possesses all the desired properties.

o In the preceding section (see (3.5.14) and (3.5.16» we defined the pseudolength function F x as an infinitesimal form of d x. Since we need here only its definition and its most basic property (3.5.18), we shall quickly review Fx. Given a point x in a complex space X, the tangent cone txx consists of vectors of the form f.(u), where u E T D and f E Hol(D, X). Then Fx : 1:,X ---+ R is defined by Fx(v) = inf{llull; u E T D and f*(u) = v}, where II u II is the length of u measured by the Poincare metric of D, and the infimum is taken over all u E T D and f E Hol(D, X) such that fAu) = v. Alternatively, Fx may be defined in terms of a fixed vector e = (8/8z)o E Toe and discs DR of varying radius R:

102

Chapter 3. Intrinsic Distances

Fx(v)

.

2

= mf-, R

where the infimum is taken over all positive real numbers R for which there is a holomorphic map f: DR --* X such that f.(e) = v. Let E be any (continuous) pseudo-length function on X such that E :s Fx. Then E(f*u) :s Fx(f.u) :s lIu II for u E T D, f E Hol(D, X). It follows that if 8 denotes the pseudo-distance defined by E, then every holomorphic map f: (D, p) --* (X, 8) is distance-decreasing so that (by (3.1.7))

In particular, if E is a length function so that 8 is a distance, then X is hyperbolic. The same argument shows that if X is a relatively compact complex subspace of a complex space Y and if there is a length function E on Y such that E :s Fx on X, then X is hyperbolically imbedded in Y. It is useful to introduce the concept of complex line following Zaidenberg [2]. Let z denote the natural coordinate system on C. Let X be a complex space, and E a length function defined on X. A nonconstant holomorphic map h: C --* X such that h*E 2 :s Cdzdz

for some constant C > 0 is called a complex line. If h(C) is contained in a compact subset of X, then this condition is independent of E. Let S be a subset (often a domain) in X. We say that a complex line h: C --* X is a limit complex line coming from S if on each disc DR C C of radius R the mapping hlDR is a limit of holomorphic mappings of DR into S. In this case, we have h(C) C S. Every complex line in X is a limit complex line coming from X. We are now in a position to prove the following theorem of Brody [1]. (3.6.3) Theorem. Let X be a compact complex space. there is a complex line h: C --* X.

If X is not hyperbolic,

then

Proof Let E be a length function on X. Assume that X is not hyperbolic. Let Fx be the pseudo-length function defined above. If there is a positive number a such that a . E :s Fx , then X would be hyperbolic as explained above. Hence, there is a sequence of tangent vectors Vn E X such that E (v n ) = I and F x (v n ) < 1/ n. By applying the second definition of F x above, we find an increasing sequence of concentric discs DR" of radius Rn with lim Rn = 00 and a sequence of maps fn E Hol(D R" , X) such that f~(O) = Vn . (By f~(O) we mean df,,(e), where e = (B/Bz)o is the tangent vector of C at the origin 0.) Since the length lIeli n of e measured by the Poincare metric ds~" = 4R~dzdz/(R~ - Id)2 of DR" is equal to 2/ R n , the function Un = fn* E2 / R~ds~n on DR", evaluated at the origin 0, gives

t

6 Brody's Criteria for Hyperbolicity and Applications

103

By applying (3.6.2) to each fn and a constant 0 < c < 1/4, we obtain a sequence of maps gn E Hol(D R" , X) such that (a)

g~E2:::: cR~ds~" on DRn and the equality holds at the origin 0;

By (a), the family of maps gn precise, since

E Hol(D Rn ,

X) is equicontinuous. To be more for

n::: m,

the family Fm = {gill D Rm ' n ::: m} is equicontinuous for each fixed m. Since the family FJ = {gn ID R, } is equicontinuous, the Arzela-Ascoli theorem (1.3.1) implies that we can extract a subsequence which converges to a map hI E Hol(D R1 , X). (We note that this is where we use the compactness of X). Applying the same theorem to the corresponding sequence in F 2 , we extract a subsequence which converges to a map h2 E Hol(D R2 , X). In this way we obtain maps hk E Hol(D Rp X), k = 1,2, ... , such that each hk is an extension of h k - I . Hence, we have a map h E Hol(C, X) which extends all h k . Since g~ E2 at the origin 0 is equal to (c R~ds~ )z=o = 4cdzdz, it follows that "

which shows that h is nonconstant. Since g~E2 :::: cR~dsR2 , in the limit we have "

h* E2 :::: 4cdzdz.

By suitably normalizing h we obtain h* E2 :::: dzdz.

o

(3.6.4) Corollary. Let X be a compact complex space. Given a nonconstant holomorphic map f: C ~ X, there is a complex line h: C ~ X such that h(C) c fCC). Proof Let DR" be an increasing sequence of concentric discs with lim Rn = 00. Let fn be the restriction of f to DR". Moving the origin of C if necessary, we may assume that f is non-degenerate at O. As in the proof of (3.6.3), let e = (a/az)o be the tangent vector of Cat o. Set v = df(e) = 1'(0). Multiplying E by a suitable constant, we may assume that E(v) = 1. We set v" = v for all n. Since Fx(v) :::: Fde) = 0, we are now in a position to apply the proof of (3.6.3). Given a constant c, 0 < c < 1/4, by (3.6.2) we obtain a sequence of maps gn E Hol(D R" , X) satisfying conditions (a) and (b) in the proof of (3.6.3). According to (b) of (3.6.2), gn is of the form

where IIr. is the multiplication by a suitable rn , 0 < rn < 1, while C{Jn is an automorphism of DR". Repeating the argument in the proof of (3.6.3) we obtain a

104

Chapter 3. Intrinsic Distances

complex line h: C --+ X as a limit ofa subsequence of (gn). Clearly, h(C)

c

I(c).

D With very little change in the proof we can extend Brody's criterion (3.6.3) to certain noncompact complex spaces, (Urata [5]). (3.6.5) Theorem. Let Z be a complex space, and Y a relatively compact complex subspace of z. If Y is not hyperbolically imbedded in Z, then there is a limit complex line h: C --+ Z coming from Y so that h(C) C Y. Conversely, if Y is hyperbolically imbedded in Z, then Z contains no limit complex lines coming from Y. Proof Let £ be a length function on Z. Assume that Y is not hyperbolically

imbedded in Z. As we explained earlier (see the paragraph preceding (3.6.3», there is no length function F on Z such that F :::: F y on Y. Hence, there is no positive constant a such that a . E :::: F y on Y. The remainder of the proof is essentially the same as that of (3.6.3). If there is a limit complex line h: C --+ Z coming from Y, then any pair of points p, q E h(C) C Y would violate the condition for Y to be hyperbolic D imbedded in Z. The following example by D. Eisenman and L. Taylor shows that (3.6.3) does not hold for some noncompact manifolds. (3.6.6) Example. The domain X = fez, w)

E

c 2 ; Izl

< I,

Izwl

< J} - {CO, w);

Iwl

~ I}

is not hyperbolic, but there is no non constant holomorphic map : C --+

x.

Proof The mapping h: (z, w):

H- (z, zw) sends X into the unit bidisc and is oneto-one except on the set z = o. If I: C --+ X is holomorphic, then hoI: C --+ Dl is holomorphic and hence constant by Liouville's theorem. It follows that either I is constant or I maps C into the set {CO, w) E X}. But this set is equivalent to the unit disc {w E C; Iwl < I}. Hence, I is constant in either case. Since h is distance-decreasing, we see that dx(p, q) > 0 for p i= q unless both p and q are in the subset (0, w) E X}. We shall show that if p and q are in this subset, then dx(p, q) = O. Let p = (0, b) with b i= and q = (0,0). Set Pn = (I/n, b). Then dx(p, q) = limdx(Pn, q). Let an = min{n, ,In/lbl}. Then the mapping tED --+ (ant In, a"bt) E X maps l/a" into Pn. Hence,

°

which shows that X is not hyperbolic. D In this example, let Z = PI C X PI C be a natural compactification of C 2 . Then the holomorphic mapping h: C --+ X c Z given by h(z) = (0, z) satisfies the inequality h* £2 :::: dzdz with equality at z = 0 (with respect to the product metric £2 coming from the Fubini-Study metric of PI C). Following Lang [3] we can strengthen (3.6.5) also in the following form.

6 Brody's Criteria for Hyperbolicity and Applications

105

(3.6.7) Theorem. Let Z be a complex space, and Y a relatively compact subset of Z. Let {Un} be a decreasing sequence of relatively compact open subsets of Z such that Un = Y. Assume that none of these Un is hyperbolically imbedded in Z. Then for each n we can find a limit complex line h n: C ---+ Z coming from Un such that !he sequence {h n } converges to a complex line h: C ---+ Z with its image i(C) in Y.

n

Proof Let E be a length function Z. If Un is not hyperbolically imbedded in Z, then by (3.6.5) there is a limit complex line hI!: C ---+ Z coming from Un so that hn(C) C Un. Then h;'E2 ::::: Cndzdz with Cn > O. By composing hll with a suitable affine transformation Z f-+ az+h, we may assume that h~E2 ::::: dzdz with equality holding at z = O. Applying Arzela-Ascoli Theorem (l.3.1) to the family {h n } we obtain the desired result. D

Letting Y to be a compact complex subspace of Z in the theorem above, we obtain: (3.6.8) Corollary. Let Z be a complex space, and Y a compact complex subspace of Z. If Y is hyperbolic, there is a relatively compact neighborhood U of Y which is hyperbolically imbedded in Z. As in (1.2.2), for x

E

X consider the degeneracy set

.1(x)

=

(y EX; dx(x, y)

= OJ.

We know (see (1.2.7) and (3.1.18» that .1(x) is connected if X is compact. As an application of (3.6.7) we obtain (3.6.9) Corollary. Let X be a compact complex space, and x E X. !l.1(x) is non-trivial, i.e., il it contains more than one point, then there is a complex line h: C ---+ X such that h(C) C .1 (x). Proof Let Un = (y

E

X; dx(x, y) < lin}.

Then nUn = .1(x). By (3.1.19), for all n we have dUn (x, y) =

0

for

Y

E

.1(x).

In particular, Un is not hyperbolic. By (3.6.7), we have a holomorphic map h: C ---+ X with the stated property. D In order to consider the case where Y is the complement of a hypersurface in Z, we need a generalization of Hurwitz theorem. The classical theorem of Hurwitz states: (3.6.10) Theorem. The limit of a convergent sequence of nowhere vanishing holomorphic functions on a domain vanishes either nowhere or everywhere. This may be extended as follows.

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Chapter 3. Intrinsic Distances

(3.6.11) Theorem. Let Z be a complex space and S = U;:l Si a Cartier divisor in Z where each Si is irreducible. Assume that a sequence {hml C Hol(D, Z - S) converges to a map h E Hol(D, Z). Then h(D) is either in Z - S or in S. More precisely, h(D) lies either in Z - S or in nEI Si - UjEJ Sj, where 1= Ii; h(O) E S;} and J = {j; h(O) ¢ Sjl. Proof Suppose that h(O) E S. Let V be a neighborhood of h(O) in Z such that V n S is defined by a holomorphic function I = where Ii = 0 defines V n Si. Take i such that f; (h (0» = O. Apply the classical theorem of Hurwitz to a sequence of holomorphic functions {f; 0 h m } which are nowhere zero. Its limit Ii 0 h must be identically zero since f; (h (0» = O. Hence h maps D into Si. D

n '/;,

We are now in a position to state the theorem of Green [7] and Howard. (3.6.12) Theorem. Let Z be a compact complex space with a lengthfunction E. Let S be a Cartier divisor in Z, and Y = Z - S. Then Y is complete hyperbolic and hyperbolically imbedded in Z if the following two conditions are satisfied: (a) There are no complex lines in Y; (b) There are no complex lines in S. Proof Suppose that Y is not hyperbolically imbedded in Z. By (3.6.5) there is a limit complex line h E Hol(C, Z) coming from Y. Hence, either h(C) C Y or h(C) C S by the generalized Hurwitz theorem (3.6.11). This is a contradiction. From (3.3.6) we see that Y is complete hyperbolic. D

More precisely, we have (Green [7]) (3.6.13) Theorem. Let Z be a compact complex space with a length function E. Let S be a union of Cartier divisors Sl, ... , Sm. Then Y = Z - S is complete hyperbolic and hyperbolically imbedded in Z (l the following two conditions are satisfied: (a) There are no complex lines in Y; (b) For any partition of indices I U J = {I, 2, ... , m}, there are no complex lines in niEI S; - UjEJ Si'

As we shall see later, the corollary above combined with Borel's Lemma imply that the complement of 2n + 1 hyperplanes in general position in PnC is complete hyperbolic and hyperbolically imbedded in Pile. The following result of Zaidenberg [4, 5] follows from (3.6.13). (3.6.14) Corollary. Let Z be an n-dimensional compact complex manilold and S = U::l Si a divisor with only normal crossing sint"tilarities. Let S(k), 1 :s k :s n, denote the stratum of S consisting of the points of S ofmultiplici(v k, i.e., S(k)

=

Sk _ Sk+l,

Then the domain Y of the strata S(k), I

=

where Sk =

u

Sil

n ... n Silo

Z - S is hyperbolically imbedded in Z ifneither Y nor any I, contains complex lines.

:s k :s n -

6 Brody's Criteria for Hyperbolicity and Applications

107

Proof In view of (3.6.10) it suffices to prove that each limit complex line h: C ---+ Z coming from Y is contained either in Y or in one of the strata S(k). Suppose that h(C) is not contained in Y. By the generalized Hurwitz theorem (3.6.11), for each i either h(C) C Si or h(C) n Si = 0. Without loss of generality we may assume that h(C) c Si for i = I, ... , k and h(C) n Sj = 0 for j = k + I, ... , m. Then h(C) is contained in the stratum S(k). 0

In order to prove a partial converse to (3.6.14) we start with the following variation of Royden's extension lemma (3.A.I) by Zaidenberg [4]. (3.6.15) Lemma. Let Z be a complex mant/old and S C Z a hypersurface. Let Sreg denote the set of regular points of S. Given a holomorphic mapping f: DR ---+ Sreg with R > I, there is a holomorphic map ifJ: D x D -+ Z such that ifJ(Z, 0) = fez)

for

ZED

and

ifJ(D x D*) C Z - S.

Proof As in the proof of (3.A.I), by considering the graph of f we reduced the problem to the case where f is an imbedding. Thus, we have only to show that if f: DR ---+ Sreg is a holomorphic imbedding, there is a holomorphic imbedding ifJ: D x D ---+ Z with the property above. By (3.A.3) the vector bundle TXI!(D R ) over f(D R ) splits: TXI!(D R ) = TSI!(D R ) EB L,

where L is a line bundle over f(D R ) and normal to S. By (3.AA), there is a holomorphic affine connection in a neighborhood of f(DR). Now we repeat the the last of step of the proof of (3.A.2). Namely, we find a small neighborhood B of Din LI!w) ~ D x C such that B = D x D, and we set ifJ = exp lB. 0 (3.6.16) Corollary. Let Z, Sand Sreg be as in (3.6.15). Let r < R. Then every holomorphic mapping f: DR ---+ Srcg can be approximated on Dr by holomorphic mappings of Dr into Z - S. Proof Let /-ir: D R / r ---+ DR be the multiplication by r. Given f: DR ---+ Sreg, apply (3.6.15) to f 0 /-ir: D Rjr ---+ Srcg to obtain a map ifJ: D x D ---+ Z such that (i) ifJ(z, 0) = f(rz) for zED and (ii) ifJ(D x D*) C Z - S. Set 1/I(z, w) = ifJ(z/r, w) for (z, w) E Dr X D. Then 1/I(z, 0) = fez) for ZEDI" Let hE(z) = 1/I(z, s). Then {hE) is an approximation of f on D. 0

(3.6.17) Lemma Let Z be an n-dimensional complex manifold with smooth hypersurfaces SI, ... , Sk, k < n, whose union S = Ui Si is a divisor with normal crossings. Then every holomorphic mapping f: DR ---+ ni Si can be approximated on each disc Dr, r < R, by holomorphic mappings h: Dr ---+ Z - S. Proof The proof is by induction on k. The case k = I is a special case of(3.6.16) where S smooth. Assume that (3.6.17) holds for k - 1. Set s[m] = n~=1 Si.

Then S[kl is a smooth hypersurface in a smooth manifold S[k-I] of dimension n -k+ 1. Let r < p < R. Then by (3.6.16) the mapping fiD. is approximated by mappings g: Dp ---+ S[k-I] - S[k] C Z - Sk. By the induction hypothesis applied

108

Chapter 3. Intrinsic Distances

to z' = Z - Sk, s; = SI - Sk, ... , S{_I = Sk-l - Sk, the mapping glDp is D approximated by mappings h: Dr ---+ Z' - u~~i S; = Z - U~=1 Si. Now we are in a position to prove the following partial converse to (3.6.13). (3.6.18) Theorem. Let Z be a compact complex manifold with smooth hypersurfaces SI, ... , Sm, whose union S = Ui Si is a divisor with normal crossings. If Y = Z - S is hyperbolically imbedded in Z, then (a) there are no complex lines in Y; (b) for any partition of indices I U J = {1, 2, ... , m}, there are no complex lines in niEl Si - UjEJ Sj. Proof If (a) is violated, then Y is not hyperbolic. Assume that (b) is violated. Then there is h E Hol(C, Z) such that h(C) c n~=1 Si - Uj:k+1 Sj. In view of (3.6.5) it suffices to show that this complex line is a limit line coming from Y. For any R > 0 we can find a neighborhood U of h(DR) in Z such that un Sj = '" for j = k + 1, ... , m. Now apply (3.6.17) to the manifold U with hypersurfaces SI n U, ... , Sk n U. Then hlDI/ can be approximated on each disk Dr, r < R, by D holomorphic mappings of Dr into U - U7=1 S; C Z - ur=1 S;.

Similarly, we have the following partial converse to (3.6.14), (due to Zaidenberg [4, 5]). (3.6.19) Theorem. Let Z, S = U;:1 Si, and Y = Z - S be as in (3.6.18).IfY is hyperbolically imbedded in Z, then (a) there are no complex lines in Y; (b) there are no complex lines in any of the strata S(k), 1 :::: k :::: n - l. Proof If (a) is violated, then Y is not hyperbolic. Assuming (b) is violated, let h E Hol(C, Z) be such that h(C) c S(k) = Sk - Sk+I. Since h(C) C Sk, we may assume that h(C) c SI n ... n Sk. Since h(C) n Sk+1 = 0, h(C) does not interesect any of 51 n ... n Sk n 5j for j = k + 1, ... , m. Hence, h(C) n Sj = 0 for j = k + I, ... , m. This implies h(C) C n~=1 Si - Uj:k+1 Sj. This violates (b) of (3.6.18). Hence, Y is not hyperbolically imbedded in Z. D

Given a complex subspace Y C Z, we defined the pseudo-distance d y. z on Y, see (3.4.1). Let Fy,z be the infinitesimal form of d y.z defined in the same way as the infinitesimal form F z of d z , using the subfamily Fy.z C Hol(D, Z), see the end of Section 5. Now, using the argument in the proof of (3.6.3) we shall prove the implication (c) :::} (b) in (3.4.11), thus making all three conditions in (3.4.11) mutually equivalent. (3.6.20) Theorem. If a complex space Y is hyperbolically imbedded in Z, then dy,z(p, q) > Ofor all pairs p, q E Y, p -=f=. q. Proof Let E be any length function on Z. In order to prove the theorem, it suffices to show that there is a positive constant c such that cE :::: Fy,z on Y. Suppose that there is no such constant. Then there exist a sequence of tangent vectors Vn of Y

6 Brody's Criteria for Hyperbolicity and Applications

109

with E(v n) = I, a sequence of hoI omorphic maps In E Fy.z and a sequence of tangent vectors en of D with Poincare length lien II '\i 0 such that dIn (en) = V n. Since D is homogeneous, we may assume that ell is a vector at the origin of D. As in the proof of (3.6.3), we replace the above .f" E Hol(D, Z) by a new In E Hol(D R", Z) with R" ./ 00; instead of using the fixed disc D and varying vectors en, we use varying discs DR" and the fixed tangent vector e = (d/dz)o at the origin. Let be the family of holomorphic maps I: DR" ---f Z such that I-I (Z - Y) is either empty or a singleton. Having replaced (D, en) by (DR", e), R we may assume that In E and dfn(e) = Vn. By applying Brody's lemma (3.6.2) to each .f" and a constant 0 < c < 1/4 we obtain holomorphic maps gn E Hol(D R", Z) such that

F:'z

Fy.z

(a)

g~E2:s cR~ds~" on DR" and the equality holds at the origin 0;

(b)

Image(gn) C Image(.f,,).

Since gIl is of the form gn = In 0 IL rn 0 h,,, where h" is an automorphism of DRn and ILr", (0 < ILr" < 1), is the multiplication by rn, each gn maps all of DR", except possibly one point, into Y. Now, as in the proof of (3.6.3) we shall construct a nonconstant holomorphic map h: C ---f Z to which a suitable subsequence of {gil} converges. In fact, since for

n c:': m,

the family Fm = {gn IDR",; n c:': m} is equicontinuous for each fixed m. Since the family FI = {g,,1 DR,} is equicontinuous, the Arzela-Ascoli theorem implies that we can extract a subsequence which converges to a map hI E HoJ(DR1' Z). (We note that this is where we use the compactness of Y.) Applying the same theorem to the corresponding sequence in F2, we extract a subsequence which converges to a map h2 E HoJ(DR2' Z). In this way we obtain maps hk E Hol(DRk' Z), k = 1,2, ... , such that each hk is an extension of hk-I. Hence, we have a map h E Hol(C, Z) which extends all hk. Since g~E2 at the origin 0 is equal to (cR~ds~)z=o = 4cdzdz, it follows that (h* E 2),,=0

=

lim (g~E2):=O 1l-+:::xJ

= 4cdzdz -I- 0,

which shows that h is nonconstant. Since g~ E2 :s cR~ds~", in the limit we have h* E2

:s 4cdzdz.

By suitably normalizing h we obtain h* E2

:s dzdz

with the equality holding at

z

= o.

We may assume that {gn} itself converges to h. Since h is the limit of {gn}, clearly h(C) C Y. Let p, q be two points of h(C), say p = h(a) and q =

110

Chapter 3. Intrinsic Distances

h(b). Taking a subsequence and suitable points a, bEe we may assume that gn(a), gn(b) E Y. Then limgn(a) = p and lim gil (b) = q. If there is a subsequence of {gn}, still denoted {gil}, such that gn(DRJ c Y,

then d y (gn (a), gll(b»

s

d DR" (a, b) -+ 0

as

n -+

00,

contradicting the assumption that Y is hyperbolically imbedded in Z. If there is no such subsequence, there is a subsequence, again denoted {gil}, such that each gn maps exactly one point, say C n E DR" into the boundary a y. If the set {c n } is unbounded in C, by taking subsequence we may assume that ICn I ? 00. Then we can find R~ < ICn I < Rn such that R~ ? 00. If we denote the restriction of gil to DR;, by g~ so that g:,(DR) c Y, then dy(g~(a), g~(b»

s

d DR;, (a, b) -+ 0

as

n -+

00,

again contradicting the assumption that Y is hyperbolically imbedded in Z. Therefore we assume that {en} is bounded. Taking a subsequence, we may assume that {c ll } converges to, say C E C. We may assume that c is the origin of C. (For each n we consider a disk D( c, R~) CDR" of center C and radius R;, such that R~ / ' 00, and we restrict gn to D(c, R;,).) Now we assume that limen = O. Set

Then g~(an) = gil (a),

Thus,

g:,

g~(bn) = gn(b),

maps the punctured disk D~;,

=

g;,(O) E

ay.

{O < Izl < R~} into Y. Hence,

dy(g~(all)' g~(bn» S d D " (all, bll ) -+ 0

as

11

-+ 00,

R"

contradicting the assumption that Y is hyperbolically imbedded in Z. We note that since an -+ a and b n -+ b, in order to conclude d D " (an, b n) -+ 0 it suffices to R" show dD~(a, b) -+ 0 as R -+ 00. But this can be seen from the expression for the infinitesimal metric . spondmg to d[);,: 2

dSD~ =

dsb.

R

corre-

4dzdz

Izl2(log R2 - log IzI2)2'

which can be obtained from (2.2.3) by replacing z by zj R.

o

We say that the cotangent bundle T* X of a compact complex space X is ample (or the tangent bundle T X is negative in the sense of Grauert) if the zero section O(X) of T X can be blown down to a point, namely there exists a holomorphic map 1T: T X -+ Y onto a complex space Y = T X jO(X) which maps O(X) to a single point, say Yo E Y, and TX - O(X) biholomorphically onto Y - {yo}. This negativity of T X implies the existence of a length function F with negative

6 Brody's Criteria for Hyperbolicity and Applications

111

curvature. As we shall see in (3.7.1), such a space is hyperbolic. Instead of the differential geometric argument just described (see Kobayashi [13]) we present the proof by Urata [5] which uses Brody's criterion. (3.6.21) Theorem. Let X be a compact complex space with ample cotangent bundle. Then X is hyperbolic.

Proof With the notation above, let V be a small (hence hyperbolic) neighborhood of Yo in Y = T XIO(X). We choose a length function E on X in such a way that every tangent vector of E-length :s 1 belongs to Jl'-I(V). Assume that X is not hyperbolic. By (3.6.3) there exists a nonconstant holomorphic map I: C ~ X such that its differential f': C ~ T X has the property that E U ' (z» :s 1 for all z E C. Since V is hyperbolic, the map Jl' 0 f': C ~ V must be constant and Jl'U'(C» = {Yo}· Hence, f' == 0, which implies that I is constant. This is a contradiction. D As another application of Brody's lemma, Urata [5] proved the following (3.6.22) Theorem. Let X be a complex space with a length function E, and G = Aut(X, E) the group of holomorphic isometries. Assume that XIG is compact. Then X is complete hyperbolic ({there is no complex line h: C ~ x.

Proof Let K be a compact subset of X such that G(K) = X. Suppose that X is not hyperbolic. Let e denote the tangent vector (dldz)o of C at O. Let DIl denote the disk of radius n with its Poincare metric ds; = 4n 2 dzdzl (n 2 -Iz 12)2. Then for each n, there exists a holomorphic mapping In: Dn ~ X such that IdJ,.(e)IE > 1. We repeat the argument in the proof of (3 .6.3). By Brody's lemma (3.6.2) and from G(K) = X it follows that for each n there exists a holomorphic map gn: D" ~ X such that (i)

gf/(O) E K,

(ii)

g~ E2

:s n2ds,~,

with equality holding at 0 E Dn. Let 8x be the distance function on X defined by E. From (ii) above we have

8x (g" (0), gn(Z»

l

1z1

:s o

2n2dt

n + Izl

-2--2 = n log - n - t n - Izl

:s 41z1

for Izl .:::: n12. Since XI G is compact, X is complete with respect to 8x. The estimate above shows that for each fixed z E C, the set {gn(Z); n ::: no}, where no ::: 21zl, is relatively compact in X. By (ii) the family {g,,} is also equicontinuous. By Arzela-Ascoli theorem (1.3.1), a subsequence of {gn} converges to a holomorphic map g: C ~ X. By (ii), g* E2 .:::: 4dzdz with equality holding at O. This is a contradiction, showing that X is hyperbolic. Since d x is invariant by G and since G(K) = X with K compact, d x is a complete distance. D (3.6.23) Corollary. A homogeneous Hermitian manifold X with an invariant Hermitian metric ds 2 is complete hyperbolic if there is no complex line h: C ~ X.

112

Chapter 3. Intrinsic Distances

7 Differential Geometric Criteria for Hyperbolicity The results in Chapter 2 yield differential geometric criteria for hyperbolicity. We shall combine them with Brody's criteria explained in the preceding section. (3.7.1) Theorem. Let X be a complex space. I(there is a lengthfimction F with (holomorphic sectional) curvature KF bounded above by a negative constant, then X is hyperbolic. If moreover, F defines a complete distance on X, then X is complete hyperbolic.

Proof By normalizing F we may assume that K F ::::: -1. Let 8 be the distance function on X defined by F. By (2.3.5) every holomorphic map f: (D, p) -+ (X,8) is distance-decreasing. By (2) of (3.1.7), we have 8 ::::: d x . Our assertion now follows. 0 (3.7.2) Remark. Milnor [1] observed that for a Riemann surface the curvature condition of (3.7.1) can be relaxed as follows. Given a Riemann surface X with a Hermitian metric, let r denote the geodesic distance from a fixed point of X. If the curvature K satisfies K(r) ::::: -1/(r 2 10gr) asymptotically, then X is hyperbolic. Remark (2.2.8) shows that the asymptotic condition K (r) ::::: -1/ r3 is too weak to imply hyperbolicity. For a higher dimensional analogue of Milnor's result, see Greene-Wu [2; p. 113, Theorem G']. We know from (2.2.6) that PI C minus at least three points carries a complete Hermitian metric with curvature K ::::: -1 and from (2.2.7) that every compact Riemann surface of genus 2: 2 admits a Hermitian metric with curvature K ::::: -1. Hence, (3.7.3) Corollary. (1) The Riemann sphere PIC minus at least three points is complete hyperbolic. (2) Every compact Riemann surface of genus 2: 2 is complete hyperbolic. We can generalize (3.7.1) to a pseudo-length function F. Considering F as a nonnegative function on T X, we assume that it is continuous everwhere and twice differentiable wherever it is positive so that if F(v) > 0 at VET X then the curvature Kdv) is defined. We say that (X, F) is negatively curved if there is a negative constant c such that KF(v) ::::: c for all VET X for which F(v) > O. In accordance with (2.1.10) we can weaken the conditions on F as follows. (a) F is upper semicontinuous; (b) For each VET X with F (v) > 0 there is a length function Fv , defined and of class C 2 in a neighborhood U C T X of v, such that (i) F 2: Fv in U, (ii) F(v) = Fv(v), and (iii) the curvature of Fv is bounded above by a negative constant c independent of v. Then we say that (X, F) is negatively curved. A point x E X is called a degeneracy point of F if F (v) = 0 for some nonzero v E Tx X. The set of such degeneracy points is called the degeneracy set

7 Differential Geometric Criteria for Hyperbolicity

113

of F. Then the argument in the proof of (3.7.1) yields the following result. (The second statement in the theorem will not be used and can be skipped). (3.7.4) Theorem. Let X be a complex space. If"there is a pseudo-length function F such that (X, F) is negatively curved in the sense defined above, then X is hyperbolic modulo the degeneracy set of F. More generally, if" F is a jet pseudo-metric on the jet bundle Jk X such that K F :::: -I, then X is hyperbolic modulo the degeneracy set of F\, where F\ is the pseudo-length jimction defined in (2.5.10). As we see from (2.2.8), for X to be hyperbolic it is not sufficient that X admits a Hermitian metric with negative hoI om orphic sectional curvature. It is important that the curvature is not only negative but also bounded away from zero. See Remark (3.7.15) for more comments on this point. On the other hand, Brody's criterion can be strengthened if the holomorphic sectional curvature is only non-positive, (Kobayashi [IS]). (3.7.5) Lemma. Let 2ledzdz with 0 :::: A. :::: I be a pseudo-metric on C expressed

in terms of the natural coordinate function z of e. If its Gaussian curvature K is nonpositive at every point where Ie > 0, then ), is constant. Proof The Gaussian curvature K is given by

Since K :::: 0, we have

a2 10g Ie --->0

azaz -

wherever ). > O. This means that log). is a subharmonic function on e. On the other hand, log Ie :::: 0 since Ie :::: 1. But a bounded subharmonic function on C is constant. Hence, A. is constant. 0 (3.7.6) Lemma. Let X be a complex space with a length function £ whose holomorphic sectional curvature K E is nonpositive. If f: C -+ X is a holomorphic map such that f* £2 :::: dzdz with equality holding at some point. then f* £2 = dzdz. i.e., f is an isometric holomorphic immersion with respect to the metric of X defined by E and the Euclidean metric dzdz ofe.

Proof Set f* £2 = Adzdz, and apply (3.7.5).

0

If X is a Hermitian manifold, we can say a little more about fCC). (3.7.7) Lemma. Let M be a Hermitian manifold with metric dsl.t whose holomorphic sectional curvature is nonpositive. If f: C -+ M is a holomorphic map such that f*dsl.t :::: dzdz with equality holding at some point, then f is a totally geodesic, isometric holomorphic immersion.

114

Chapter 3. Intrinsic Distances

Proof By (3.7.6) f is an isometric holomorphic immersion. Since M has nonpositive holomorphic sectional curvature and fCC) is flat, M has vanishing holomorphic sectional curvature in the direction of fCC) by (2.3.9). By Remark (2.3.11), fCC) is totally geodesic in M. 0 We apply Lemmas (3.7.6) and (3.7.7) to (3.6.3), (3.6.5), (3.6.9), (3.6.11) and (3.6.12) to obtain the following results, (3.7.8) through (3.7.12). (3.7.8) Theorem. Let X be a compact complex space with a length function E whose holomorphic sectional curvature is nonpositive. If X is not hyperbolic, there is an isometric holomorphic immersion h: C ~ X. In fact, if x E X and tl the degeneracy set L1 (x) is non-trivial, then we can find such a map h with its image h(C) in L1(x). If, moreover, X is a compact complex subspace of a Hermitian manifold Y with nonpositive holomorphic sectional curvature, such a map h: C ~ Y is totally geodesic. (3.7.9) Theorem. Let Z be a complex space with a length function E whose holomorphic sectional curvature is nonpositive, and Y a relatively compact open subset of z. IfY is not hyperbolically imbedded in Z, then there is an isometric holomorphic immersion h: C ~ Z such that h(C) c Y. If, moreover, Z is a Hermitian manifold with nonpositive holomorphic sectional curvature, such a map h is totally geodesic. (3.7.10) Theorem. Let Z be a compact complex space with a length function E whose holomorphic sectional curvature is nonpositive. Let S be a union of Cartier divisors S 1, ... , Sm. Then Y = Z - S is complete hyperbolic and hyperbolically imbedded in Z if the following two conditions are satisfied: (a) There are no isometric holomorphic immersions h: C ~ Y; (b) For any partition o.l indices 1 U J = {l, 2, ... , m}, there are no isometric holomorphic immersions h: C ~

n

S; -

iEI

U Sj C Z. jEJ

(3.7.11) Theorem. Let Z be a complex space with a length function E whose holomorphic sectional curvature is nonpositive, and Y a relatively compact subset of Z. Let {V n} be a decreasing sequence of relatively compact open neighborhoods of Y such that Vn = Y. If none of these VI! is hyperbolically imbedded in Z, then there is an isometric holomorphic immersion h: C ~ Z such that h(C) C Y. If, moreover, Z is a Hermitian manilold with nonpositive holomorphic sectional curvature, such a map h is totally geodesic.

n

From (3.7.8) we obtain the following result, conjectured by Lang [1] and proved by Green [8]. (3.7.12) Theorem. A closed complex subspace X of a complex torus T is hyperbolic if there is no nonconstant affine map h: C ~ T such that h (C) c X, or equivalently, if X contains no translate of a complex subtorus of T.

7 Differential Geometric Criteria for Hyperbolicity

115

Proof Suppose that X is not hyperbolic. In (3.7.8), let Y = T. Since T admits a flat Hermitian metric, there is a totally geodesic, isometric holomorphic map h: C """"* T such that h(C) c X. By translating h(C) in T, we may assume that H = h(C) is a subgroup of T. We claim that the Zariski closure H of H in T is a subgroup of T, i.e., (H)-I cHand H· H c H, and hence is a complex torus. This follows from the fact that although the operation TxT """"* T, (x, y) f-+ X -I y, is not Zariski continuous, the operations x f-+ x-I and x f-+ xa (with a fixed) are homeomorphisms of T onto itself in the Zariski topology. (See Lang [3; p. 84] for details. We note that the closure in the usual topology would yield merely a real subtorus). 0

Another result, also conjectured by Lang and proved by Green, follows from (3.7.10): (3.7.13) Theorem. Let T be a complex torus and S a complex hypersurface. (( S contains no complex subtorus, then Y = T - S is complete hyperbolic and hyperbolically imbedded in T. Proof In view of (3.7.10) it suffices to show that there is no totally geodesic, isometric holomorphic immersion h: C""""* Y. Assuming that such a map h exists, we shall obtain a contradiction. Let A be the topological closure of h(C) in T; it is a real sub torus of T. First we consider the case A n S = 0. Let {tn} be a convergent sequence of translations in T such that til (A) n S = 0 and (lim tn)(A) n S =f. 0. Choose points an E C such that lim til (h (an» E S. Let U" be the unit disc neighborhood of an in C. Then we have a sequence of holomorphic maps t" 0 h: Un """"* Y such that limtn(h(Un » n Sol 0. By identifying UII with the unit disc D around the origin o in C, we obtain a sequence of convergent holomorphic maps f,,: D """"* Y with the limit map f = lim fn such that feD) n S # 0. By the generalized Hurwitz theorem (3.6.11), we have feD) c S. From the construction of fn and f it is clear that f extends to an affine map C """"* T. Then f(C) must be also contained in S. This contradicts the assumption of the theorem. Next we consider the case A n S # 0. Choose points an E C such that limh(an) E S. Let Un be the unit disc neighborhood of an in C. The remainder of the proof is the same as in the first case; simply let til be the identity transformation in the proof of the first case. 0

(3.7.14) Corollary. Let T be a simple torus, i.e., a complex torus containing no proper complex subtorus. Then (I) Every closed complex subspace of T is hyperbolic; For every complex hypersurface S, its complement T - S is complete (2) hyperbolic and hyperbolically imbedded in T. (3.7.15) Remark. In general, a hyperbolic complex manifold may not admit a negatively curved Hermitian metric, see Demailly [3]. Cheung [1] discusses a special class of hyperbolic manifolds which admit negatively curved Hermitian metrics.

116

Chapter 3. Intrinsic Distances

8 Subvarieties of Quasi Tori In this section we shall study subvarieties of complex tori by purely differential geometric means. For an algebraic geometric approach, we refer the reader to Ueno [1]. First we quickly review basic facts on the second fundamental form and the equations of Gauss-Codazzi for a Kahler submanifold. We follow here Nagano-Smyth [1], who studied minimal submanifolds in real tori. Since a complex submanifold of a Kahler manifold is a minimal submanifold, we can apply their results. Because of complex analyticity in our case, singularities of complex subspaces present no essential difficulty. We start with the Riemannian case. For (3.8.1) through (3.8.6) we refer the reader to Kobayashi-Nomizu [1; voU]. Let M be an (n + p)-dimensional Riemannian manifold with metric g and with covariant differentiation V, and X an n-dimensional submanifold with covariant differentiation V. Given vector fields u, von X, we have the following formula of Gauss: (3.8.1)

Vuv = Vuv

+ a(u, v),

where a: TxX x TxX -+ T/ X is a symmetric bilinear map called the second fundamental form of X eM. If ~ is a section of the normal bundle T.L X, then we have the following formula of Weingarten: (3.8.2) where A~ defines a symmetric linear transformation of T X and V..L defines a connection in the normal bundle T..L X. Then a and A are related by (3.8.3)

g(A~(u), v)

= g(a(u, v),

~).

The connection V of T X combined with the connection V..L of T.L X defines a connection in T* X ® T* X ® T..L X, which we shall denote by V*. In particular, for a, we have (3.8.4)

(V:a)(v, w) = V;(a(v, w» - a(Vu v, w) - a(v, Vuw).

The curvature R of a Riemannian manifold M is a 2-form with values in the endomorphism bundle End(T M). We associate to R a quadrilinear map, also denoted by R, by

Now we state the equation of Gauss: (3.8.5) Theorem. Let Rand R be the Riemannian curvature tensors of M and X, Thenfor any vector fields VI, V2, V3, V4 on X, we have

re~pectively.

R(v" V2, V3, V4) = R(VI, V2, V3, V4) +g(a(vl' V4), a(v2, V3» - g(a(vl, V3), a(v2, V4».

8 Subvarieties of Quasi Tori

117

The equation of Gauss expresses the tangential component of R(v3, V4)V2 in terms of R and a. Its normal component is described by the following equation of Codazzi: (3.8.6) Theorem. For any vector .fields u, v, w of X, the normal component of R(u, v)w is given by (R(u, v)w)1- = (V=a)(v, w) - (V:a)(u, w).

In particular, if M is a space of constant curvature, then (V=a)(v, w) = (V:a)(u, w).

We define the relative nullity space at x to be (3.8.7)

N,

= {u

E T.tX;

a(u, v)

=0

for all

v

E

TxX}.

Let

v U

=

= min dim Nt, XEX

{x E X; dimNx

=

v}.

Then U is an open subset of X, and N = UtEV N x is a subbundle of T(U) of rank v, i.e., a v-dimensional distribution on U. (3.8.8) Lemma. Assume that M is a space of constant curvature and that v > O. Then the distribution N is integrable and each maximal integral submanifold of N is totally geodesic not only in U but also in M.

Proof. Let u, w be sections of the bundle N, and v any vector field on U. Then by (3.8.4), we have (V:a)(u, w) = O.

Hence, (3.8.6) implies a(v, V'IlW) = O. Since this holds for all vector fields v on U, V'uw is a section of N. This shows that N is integrable and totally geodesic in U. Since Vllw

= Vuw + a(u, w) = V'uw

and since Vu w is in N, it follows that N is totally geodesic in M as well.

0

Now we consider the case where M is a Kiihler manifold and X is a complex submanifold. Then the second fundamental form a satisfies the following (see Kobayashi-Nomizu [1; voI.2]): (3.8.9)

a(Ju, v)

= a(u, Jv) =

J(a(u, v)),

where J is the complex structure of M. This implies that the relative nullity space N x is a complex subspace, Le., invariant by J.

118

Chapter 3. Intrinsic Distances From (3.8.5) and (3.8.9) we obtain

(3.8.10)

R(u, Ju, v, Jv) = R(u, Ju, v, Jv)

+ 2g(a(u, v), a(u, v».

Let Sand S be the Ricci tensors of M and X, respectively; they define symmetric bilinear forms on each tangent space. If we choose a local orthonormal basis of T M in such a way that el, ... , en, J el, ... , J en are tangent to X and en+l, ... , en+ p , len+l, ... , le n + p are normal to X, then n+p

L R(e;, lei, u, lu),

S(u, u)

i=l

n

S(u, u)

L R(e;, Je;, u, Ju).

=

;=1

Using (3.8.10) we obtain Il+P

n

S(u, u) = S(u, u) - 2

L g(a(ei, u), aCe;, u»

-

L

R(ek, leb u, Ju).

k=n+1

i=1

From now on, we assume that M is flat. Then n

(3.8.11)

S(u, u) = -2

L g(a(ei, u), a(ei, u»

SO.

i=l

If u E N x , then S(u, u) = 0 by (3.8.1 I). Conversely, if u E TxX and S(u, u) = 0, then aCe;, u) = 0 for i = 1, ... , n by (3.8.11) and a(Jej, u) = 0 for i = I, ... , n by (3.8.9), and hence u E Nt. Thus, (3.8.12)

Nx

=

(u E TxX; S(u, u) = O}.

Since S is symmetric and negative semi-definite, we have (3.8.13)

N x = {u E TxX; S(u, v) = 0

for all

v E TtX}.

This shows that the relative nullity space Nt> defined originally in terms of the second fundamental form a, can be defined in terms of an intrinsic invariant of X, namely the Ricci tensor S. Now, let M be a locally flat Kahler manifold and X a closed complex subspace. We can apply the results above to the regular locus Xreg of X. Let N be the relatively nul1ity distribution defined on an open set U of X reg • Since the relative nullity spaces N x are all complex vector spaces, we denote by v the minimum of the complex dimensions of N x , x E U. For x E U, let N(x) be the maximal integral submanifold through x defined by the distribution N. Let M be a commutative complex Lie group with an invariant flat Kahler metric; M is of the form C n + p / r, where r is a discrete subgroup of C n + p • By

8 Subvarieties of Quasi Tori

119

parallel translation we can compare tangent spaces at different points. This fact makes it possible to define the Gauss map. Let X be an n-dimensional closed complex subspace of M. Let G(n, n + p) be the complex Grassmann manifold of n-dimensional subspaces in C n + p • The Gauss map G: X reg ---+ G(n, n

+ p)

assigns to each regular point x the n-dimensional subspace of the tangent space ToM parallel to TxX. We shall compute the differential dG: TxX -+ TG(x)G(n, n + p). We take a local orthonormal frame field el, ... , en+ p around x such that e1, ... , en are tangent to X reg (and en +1, ... , en + p normal to X reg ). Then Ta(x)G(n, n + p) is identified with the space of complex (n x p)-matrices. Set (3.8.14)

de;

=

n

n+p

j=l

k=n+!

L w! ej + L

a~ek.

I::;i::;n.

The differential dG is identified with (a;). Comparing (3.8.14) with the definition (3.8.1) of the second fundamental form a, we see that n+p

(3.8.15)

a(u, ei)

=

L

a~(u)ek.

k=I!+1

which says that dG can be identified with the second fundamental form. It follows from (3.8.15) that the relative nullity space N x agrees with the kernel of dG at x: (3.8.16)

x E X reg .

From this we immediately obtain (3.8.17) Lemma. let M be a commutative complex Lie group with an invariant flat Kahler metric and X a closed complex subspace. Then each maximal intergal submanifold N(x), x E U is a connected component of a level set of the Gauss map G, and hence it is closed in U. For a different proof on closedness of N(x), see Nagano-Smyth [1]. The use of the Gauss map was suggested by Wu, see Fischer-Wu [I]. Since, by (3.8.8), N(x) is totally geodesic in M = e+ p / r, it extends to X. This extension is a translate of a connected v-dimensional complex subgroup, say M~, of M and is the closure N(x) of N(x) in M. Hence, this subgroup M~ is closed in M. In general, M~ may vary with x. However, it is independent of x if M is a quasi torus in the following sense. A commutative complex Lie group M = C N / r is said to be a quasi torus if it is an extension of a complex torus T by (C*)k. Thus we have an exact sequence of commutative complex Lie groups:

o -+

(C*)k ---+ M ---+ T ---+ O.

120

Chapter 3. Intrinsic Distances

Actually, a connected complex Lie group is automatically commutative if it is an extension of T by (C*)k, see Iitaka [3]. Let el, ... , eN be the natural basis for V = C N . After a linear coordinate change for C N , the quasi lattice r has a Z-basis el, ... , ek, hi, ... , h 2m , where k + m = N with the following property. Let VI = C k and r l be, respectively, the subspace of V and the subgroup of r spanned by el, ... , ek so that (C*)k ;::: V In. Let V2 = V I VI, and b l , ... , b2m be the images of hI, ... ,h2m in V2 so that the group r l generated by bl , •.. ,blm is the lattice for the complex torus T, i.e., T = VlI r 2. (3.8.18) Lemma. (1) Every connected closed complex subgroup M' of a quasi tOntS M is a quasi tOntS, and the quotient group M I M' is also a quasi tOntS; (2) A quasi torus contains only countably many quasi subtori. Proof (1). Set G = (c*l. Each m = (ml, ... , mk) E Zk defines a character Xm of G, i.e., a homomorphism of G into C* by

Conversely, every character of G is of the form Xm. Thus, the set G* of characters of G is a commutative group isomorphic to Zk. By duality, there is a one-to-one correspondence between the subgroups of G* and the closed subgroups of G, one being the annihilator of the other. In particular, every connected closed subgroup of G is isomorphic to (C*)I. Let M' be a connected closed complex subgroup of M. Set H = M' n G, where G = (C*)k as above. Then M'I H is a complex subgroup of MIG = T. To see that M'I H is compact, let K be the maximal compact subgroup of M. (Since M is commutative, there is only one maximal compact subgroup). Since M = GK, it follows that M' = H(K n M'). Let p: M --+ T = MIG be the projection. Since p(K) = T, we have p(K n M') = Mil H, which shows that M' I H is compact. This proves that M' is a quasi torus. It is now obvious that M I M' is also a quasi torus. (2). If M = V Iris a quasi torus and M' = V'I r' is a quasi subtorus, then V' c V and r' c r. Since there are only countably many subgroups r' of rand since V' is determined by r' (i.e., spanned by r'), there are at most countably many quasi subtori of M. 0 (3.8.19) Lemma. Let M be a quasi torus and X a closed complex subspace. Then the subspaces N(x), x E V, are all parallel translates of one quasi subtonts, say M',ofM. Proof By (1) of (3.8.18) M; = N(x) is a quasi subtorus of M, and by (2) of (3.8.18) M~ is independent of x since N(x) depends continuously on x. D

In order to explain the subgroup M', we need the following result of Bochner. (3.8.20) Lemma. An infinitesimal isometry of constant length on a Riemannian manifold with negative semi-dt:finite Ricci tensor S is parallel and satisfies S(v, v) =

o.

8 Subvarieties of Quasi Tori

121

Proof This follows from the formula: ~(g(v,

v» = g(Vv, Vv) - S(v, v),

which is proved, for example, in Yano-Bochner [I] and Kobayashi [8].

0

(3.8.21) Lemma. Let X C M and M' be as in (3.8.19). Then M' is the largest connected subgroup of M leaving X invariant. Proof Since every one-parameter subgroup of M' induces a vector field, say v, tangent to N(x), it leaves X invariant. Conversely, if v is any vector field on M generating a one-parameter subgroup of M leaving X invariant, then it defines an infinitesimal holomorphic isometry of constant length on X reg . By (3.8.20), v must satisfy S(v, v) = 0 on Xreg. By (3.8.12), the one-parameter group generated by v is in M'. 0

By (3.8.19) the relative nullity distribution N on U extends to a parallel distribution N on X reg ; in fact, the vector fields coming from the action of M' are all parallel vector fields on X reg . Hence, the distribution N.L orthogonal to N is also a parallel distribution on X reg . By the theorem of de Rham on the holonomy decomposition of a Riemannian manifold, a neighborhood of each point x E X reg is locally a Riemannian direct product of the foliation defined by N and the orthogonal foliation. In summary, we have (3.8.22) Theorem. Let X be an n-dimensional closed complex subspace of a quasi torus M = C"+p I r with an invariant flat Klihler metric g, and M' be the largest quasi sub torus of M leaving X invariant. Let T; denote the subspace of the tangent space TxX spanned by all vector fields induced by the action of M'. Let S be the Ricci tensorofXreg, and let N x = (v E TxX; S(v, v) = O}. Let Mil = MIM' with projection :rr: M ---* Mil. Then (I) N x = T: on a dense open subset U of Xreg ; (2) X is a principal M'-bundle over :rr(X) = XI M'; (3) The M'-orbits, i.e., the fibers in the above fibering, are the flat factor in the local holonomy decomposition of X reg ; (4). With respect to the induced metric, the Ricci tensor of:rr(X) is negative definite on a dense open subset. (3.8.23) Corollary. Let X C M, M' and S be as above. Then M' = 0 if and only if the Ricci tensor S is negative definite on a dense open subset of X reg . Let M = Cn+p I r, and WI, •.• , w n + p the natural coordinate system of cn+p. Then using family of holomorphic n-forms dw i , /\ ..• /\ dw in , i l < ... < in, on X, we obtain a meromorphic map (3.8.24) which sends x

E

U to a point with homogeneous coordinates

122

Chapter 3. Intrinsic Distances

We recall the definition of the Plucker imbedding P: G(n, n + p) --+ Pre. Given a point of G(n, n + p), i.e., an n-dimensional vector subspace V of C n+ 1 , choose a basis VI, ••. , Vn for V and assign the line in /\ n Cn+p spanned by VI /\ .•• /\ V n , which is an element of the projective space P(/\ n Cn+P). More explicitly, express each Vj by its components Vj = (v), ... , v;,+p) and consider ("!P) numbers

ViI! I

vnin

which are called the Pliicker coordinates of the point V E G(n, n + p). Thus, P maps V to a point of PrC with homogeneous coordinates (vil .. ;n). From the definitions of ct> and P we obtain (3.8.25) Lemma. In terms of the Gauss map G: Xreg --+ G(n, n + p) and the Plucker imbedding P: G(n, n + p) --+ Pre, the map ct> can be written as follows: ct> = Po G.

(3.8.26) Corollary. Let X eM and M' be as in (3.8.22). {f M' = 0, then ct>: X --+ PrC is a meromorphic immersion. Proof Since P is an imbedding and since by (3.8.16) and (3.8.19) G is a meromorphic immersion, ct> = P 0 G is a meTomorphic immersion. D (3.8.27) Corollary. Let X C M = cn+ p / rand M' be as in (3.8.22). Let t*: X --+ M denote the imbedding. If M' = 0, then X has n + 1 holomorphic I-forms WI, ... , Wn+1 that are linear combinations of t*(dw l ), ... , l*(dw n + p ) such that n + I holomorphic n~forms

are linearly independent. Without loss of generality, we may assume that the coordinate system Wi, ... , w n + p for cn+p is such that WI

= l*(dw l ),

... ,Wn+1

= l*(dw n + I ).

This fact (in the torus case) is exactly what Ochiai [2] needed to complete his proof of Bloch conjecture; and it was proved by Kawamata [1]. Proof The meromorphic mapping ct>: X --+ PrC defined in (3.8.24) is a meromorphic immersion by (3.8.26). By taking a suitable projection p: PrC --+ PnC (which is a meromorphic map) we obtain an equidimensional meromorphic immersion po ct>: X --+ PnC. Now, our assertion is obvious from the definition of the map ct>.

D

8 Subvarieties of Quasi Tori

123

From (3.8.26) we recover the fol\owing algebraic geometric result (see Ueno [I; p.74]). For the concept of Kodaira dimension and that of general type, (for details, see Section 4 of Chapter 7). (3.8.28) Corollary. Let T be a complex torus, and X a closed complex subspace. Let T' be the largest connected subgroup ofT leaving X invariant. Then XIT' is of general type, i.e., the Kodaira dimension K(XIT') is equal to dimXIT'. Proof On account of (3.8.22), replacing T by TIT' and X by XIT' we may assume that T' = O. By (3.8.26), cP is a meromorphic immersion. Since cP is defined using only sections of Kx, X is of general type. 0

(3.8.29) Remarks. (1) Our proof shows that in order to prove K (X IT') = dimXIT' it is sufficient to consider KX/T'; there is no need to take higher tensor powers of K X / T '. (2) In (3.8.28) we have K(X) = K(XIT') = dimXIT'.

The second equality is nothing but (3.8.28). By (3.8.16), cP = Po G has rank equal to dimXIT' at x E U. Hence, K(X) :::: dimXIT'. The opposite inequality follows from Iitaka's theorem (Iitaka [1], see also Deno [1; p.74]). (3) From the fact that XIT' is of general type, hence Moishezon, it follows that X I T' is projective algebraic and the smallest complex subtorus of TIT' containing XIT' is an Abelian variety, (see Deno [1; p. 120]). Moreover, if T is an Abelian variety, then there exist finite unramified coverings T and T" of T and Til = TIT', respectively, such that T ;:: T' x T", and accordingly a finite unramified covering X of X splits as a direct product T' x XIT'. If a Hodge metric is used for T, then this splitting is compatible with the holonomy decomposition of de Rham explained above. (4) If X is a closed complex subspace of a quasi torus M, we can derive from (3.8.25) a statement similar to (3.8.28) on the logarithmic Kodaira dimension of XIM'. As an application of (3.8.25) we have a simple proof of the following theorem (see Ueno [1; p.1l7]). (3.8.30) Corollary. A closed complex subspace X of a complex torus T is a translate of a complex subtorus if and only if its geometric genus is 1, i.e., dim rcKx) = 1. Proof Ifthe geometric genus of X is 1, then the map

(3.8.31) Remarks. Again, the logarithmic version of (3.8.30) can be similarly derived if X is closed complex subspace of a quasi torus M. For closed nonsingular complex submanifolds X of a complex torus T, the fibering X ~ XIT' has been studied by Matsushima-Stoll [1], Matsushima [I], Howard-Matsushima [1], and Smyth [1] as well as by Nagano-Smyth [1].

124

Chapter 3. Intrinsic Distances

For structures of complex submanifolds of complex parallelizable manifolds, see Kodama-Sakane [I] and Huckleberry-Winkelmann [I].

9 Theorem of Bloch-Ochiai Using results of the preceding section we shall prove the theorem of Bloch and Ochiai for holomorphic curves in an abelian variety. What we need from Nevanlinna theory is summarized in Appendix B of this chapter. A quasi torus M, which is an extension of a complex torus T by (c*l, is called a quasi-abelian variety or a semi-abelian variety if T is an abelian variety. Consider M as a bundle over T with fiber (C*)k. Compactify M by compactifying each fiber to either (PI C)k or pkc. Then the compactification of M is projective if M is quasi-abelian. The following theorem corresponds to Theorem A in Ochiai [2]. (3.9.1) Theorem. Let X be an n-dimensional closed algebraic subspace ofa quasiabelian variety M = C n + p / r with imbedding t: X ~ M and with natural coordinate system WI, ... , w n + p for c n + p . Assume that the largest connected subgroup M' of M leaving X invariant is trivial. Let

be holomorphic I-forms obtained in (3.8.27). Let f: DR holomorphic map with a lift j: DR ~ C n + p given by

jet)

~

X C

cn+ p / r be a

= (/1 (t), ... , fn+p(t».

Assume that f(D R ) is not contained in the singular locus of X and that f is nondegenerate with respect to WI, ... ,Wn+1 in the sense that f(D R ) does not lie in a divisor of the form n+1

LajWl /\ ... /\ Wj /\ ... /\ Wn+1

= 0,

j=1

Let cp be a rational function on X such that cp 0 f is function on DR. Then cp 0 f is algebraic over thejield ' K = C (fl'

... ,

d~fined

1,'n+l' f"1 " ' " 1," f(lI) n+l"'" 1 ,

..• ,

as a meromorphic I,(n) ) n+l

generated by meromorphic functions

1,'n+l' f"1"'" 1," fen) I,(n) ' f 1,0", n+l"'" 1 "." n+I' Proof Only after (3.9.14) we shall use the assumption that M is quasi-abelian and X is a closed algebraic subspace of M. Until then, M will be just a quasi-torus and X a closed complex subspace of M.

9 Theorem of B1och-Ochiai

125

We consider the k-jet bundle Jk M over a quasi torus M = C n + p / r. Using the natural coordinate system (wi, ... , w n + p ) for C n +p (or more exactly, using the 1forms dw I, ... , dw n+p ), we identify Jk M with the product bundle M x (Cn+p)k, (for jet bundles, see Section 5 of Chapter 2). Let p(k): Jk M

-+ (Cn+p)k

be the natural projection. The natural coordinate system in (C n + p / is denoted by

(3.9.2)

jg

thus, if f E Jk M with p(k) (jJ f) are given by

f given by

Wi

Wi

(t), then the coordinates for

I ::: i ::: n

+ p,

I ::: ex ::: k;

thus the second superscript (ex) in Wi(a) indicates the number of times differentiation is taken. Let X be an n-dimensional closed complex subspace of M with imbedding map l: X -+ M. Let Jk X be the k-jet bundle of X; since X is imbedded in a nonsingular manifold M, it is defined as a complex subspace of Jk M. When X is singular, Jk X is only a fibre space over X. The imbedding t: X -+ M induces an imbedding Let q(k)

=

p(k) 0 l(k): Jk X

-+ (C n + p / .

In a Zariski neighborhood U of some regular point of X, holomorphic I-forms are linearly independent. For otherwise, WI /\ .•. /\ WIl would vanish identically on X, contradicting the fact that n + 1 holomorphic n-forms WI, ... ,Wn

are linearly independent, see (3.8.27). (In the following we shall replace the Zariski open set U by a smaller Zariski open set whenever necessary without explicitly saying so). Since (l*W I, ... , t*w n ) can be taken as a local coordinate system in U, we set Zi = t*w i , i = 1, ... , n. The imbedding t: U -+ M is given by (3.9.3)

W

i

=

{Zi i

F (z I , ... , zn )

i = 1, ... , n, i=n+l, ... ,n+p.

Using the coordinate system (Zl, ... ,zn) (or more exactly, using the I-forms dz l , ••• , dz n ), we identify JkU with U X (Cn)k. We denote the coordinate system in (Cn)k by

126

Chapter 3. Intrinsic Distances Zl(k)

(3.9.4) zn(k)

Then

q(k):

U x (cn)k -+

(Cn+p)k

)

is given by Zl(l)

(3.9.5)

q(k):

C'

Zl(k)

Zl(l)

:

zn

zn(l)

zn(k)

)~

Zl(k)

zn(l)

zn(k)

Qn+l(1)

Qn+l(k)

Qn+p(k)

Qn+p(k)

where Qi(a) is a polynomial in zj(fJ) (1 ~ i ~ n, 1 ~ f3 ~ a), whose coefficients are partial derivatives of Fi of order ~ a in Zl, ... , zn. In order to find Qi(a) explicitly, we have only to calculate derivatives = =

and then to replace daw i jdt a and dfJzj jdt fJ by Qi(a) and zj(fJ), respectively. Thus (3.9.6) Lemma. Ifwe set FJ = 3Fij3z j , FJk = 3 2 F i jBzjBz k , etc., then we can write Qi(a) as follows:

Given a holomorphic map f: DR -+ X, its k-jet to

l

f: DR -+ JkX, sending t

i tk f, is given locally by Zi(a) =

(3.9.7) The map

(3.9.8)

da i _z_ dt a

q(k)

W

0

l

f: Dr

i=I, ... ,n, a=I, ... ,k. '

-+ (Cn+p)k

is given by

ira) _ { daz i jdt a da Fi (Zl (t), ... , zn(t»jdt a

i=I, ... ,n, i n + 1, ... ,n

=

+ p.

We shall now ignore w n+2, ... , w n+ p , and consider only F(ZI, ... , zn) F n + 1 (Zl, ..• , zn) and Q(a) = Qn+1(a) in (3.9.3) and (3.9.5). Thus

9 Theorem of Bloch-Ochiai

127

i = 1, ... , n, i = n + 1,

(3.9.9) and i

(3.9.10)

=

1, ... , n,

i=n+1,

where Fi = BF/Bzi. In stead of q(k) in (3.9.5), we consider q: U x (cn)n

--+ (c+l)n

given by

Zl(l)

(3.9.11)

zn(l) Q(l)

and now (3.9.6) reads as follows: Q(1)

L

FiZi(l)

Q(2)

L

FiZi (2)

+ LFijZi(l)Zi(l)

t

Let E Ho1(D R , X). We calculate the Jacobian det(dq) ofthe equidimensional map q defined by (3.9.11), and we evaluate it at a point f E ru. Clearly,

j:

± det(B Q(a) /Bz k ).

det(dq) =

Evaluating BQ(a) /Bz k at jtn t, and writing z(t) for (3.9.12)

(Zl (t), ... , zn(t)),

k=l,oo.,n.

To see this, we consider for example the case ex = 2.

»)

2 ( BQ(k

Bz

= i,"!

d 2zi Fik dt 2

L

d 2 zi

L h dt 2 =

Hence, det(dq) at

dz i dz j

Fijkdtdt dz i dz j

+ LFki j dtdt

d 2Fk dt 2

•

KU) is given by the Wronskian F'1

(3.9.13)

+L

det(dq)j,"! = ±

(

F"

;1

F(n)

1

F~

F~' ) F~n)

,

we have

128

Chapter 3. Intrinsic Distances

where F/ ct ) = d a Fi(z(t»/dt ct • If this Wronskian is identically equal to zero, then there is a nontrivial linear relation C]

F{ (z(t))

+ ... + c" F,; (z(t» == 0,

or equivalently, (3.9.14) On the other hand, from (3.9.10) we have

Hence, (3.9.14) is equivalent to saying that L(_l)n-i CiW1 !\ ... /\W;!\ ... !\Wn+l vanishes at every point of f(DR)' Since we are assuming that f is non-degenerate with respect to WI, ... , Wn+l, det(q) #- Oat jtn f for some t E DR. Now, we impose the condition that M is quasi-abelian and X is algebraic. Then WI, "',Wn+] are regular rational I-forms on X. From (3.9.10) it follows that i

=

I, ... ,n,

which shows that F\, ... , Fn are rational functions on X. Moreover, with the notation in the proof of (3.8.27) the meromorphic immersion po CP: X --+ Pne is given by (F], ... , Fn , 1). In particular, F], ... , F" form a transcendental basis for the field of rational functions on X. Thus, every rational function cp on X is algebraic over the field C(F], ... , F,,). In a suitable Zariski open set U, the rational functions F], ... , Fn can be taken as a local coordinate system. (whereas Zl, ... ,zn are transcendental functions). Coordinate functions wi(a) given in (3.9.2) are rational functions on Jk M since they are defined by means of rational I-forms dw 1, ••• , dw n + p of M. Similarly, coordinate functions zi(a) given in (3.9.4) are rational functions on Jk X since they are defined by means of rational I-forms d z I , ... , d zn of X. Since dF; = L Fijdz j and since Fi and dz j are rational, each Fij is a rational function on X. Similarly, from d Fij = L Fijkdz k we see that each F;jk is a rational function on X, and so on. Now, we look at the map q given by (3.9.11). We use rational functions F], ... , Fn instead of Zl, .•. , z" as a local coordinate system in U. Since Q(a) in (3.9.11) is a polynomial in rational functions Zi({3), Fi , Fij , •.. with constant coefficients, it follows that q is a rational map. From (3.9.13) and (3.9.14) we see that if the Wronskian (3.9.13) vanishes identically, then f(D R ) would be contained in a proper algebraic subspace of X defined by L Cj F j = O. Thus, det(q) =1= 0 at it" f for some t E DR. It follows that q is a dominant rational map of a finite degree. Thus, the field C(F\, ... , Fn ,

ZI(1), ... , ZI(n), ... , Zn(1), ... , zn(n»

9 Theorem of Bloch-Ochiai

129

is a finite extension of the field ZI(I), ... , Zl(n),

C(Q(l), ... , Q(II),

Pull back these fields by f (i.e., by

... , Z"(l) , ... , Z/(II).

r f). Since

zi(a)(j," f) = !;(a)(t)

by (3.9.7) and since Q(a) (j;'f) = Qn+l(a)(j,n f) = fn(~)I(t)

from the definition of Q(a), we see that the field C(FI

0

f, ... ,

F"

0

f, f(, ... , fi"), ... , f;, ... , fn(n)

is a finite extension of the field K -- C(f'1"'" fen) 1 , ...

,

1.' fl"'"

1.(n)

II'

1.(1) n+l"'"

1.(Il) )

n+l'

This shows that Fl 0 f, ... , F" 0 f are algebraic over K. Let cp be a rational function on X such that cp 0 f is defined as a meromorphic function on DR. Since cp is algebraic over C(Fl, ... , Fn ), cp 0 f is algebraic over C(FI 0 f, ... , fn 0 f) and hence algebraic over K. 0 Now we state the theorem of Bloch [2] and Ochiai [2] in the form generalized by Noguchi [4]. However, we prove it only in the case M is an abelian variety. For the general case of M quasi-abelian, see Noguchi [4]. Our proof follows Ochiai [2] and Noguchi-Ochiai [I]. (3.9.15) Theorem. Let X be an n-dimensional closed algebraic subspace of a quasi-abelian variety M = C"+P I T. Assume that it is not a translate of a quasiabelian subvariety of M. Let regular rational I ~rorms WI, ... , Wn+1 be as in (3.9.1). Then every holomorphic map f: C -+ X has its image fCC) in a proper closed algebraic subspace of X. More precisely, fCC) lies either in the singular locus of X or in a divisor of the form n+l

L ajWl /\ ... /\ Wj /\ ... /\ W +l = 0, n

(ai, ... , an+l) =1= (0, ... ,0).

j=1 Proof We shall use the notation and results from Nevanlinna theory summarized

in Appendix B of this chapter. Let M' be the largest connected subgroup of M leaving X invariant. Considering XIM' C MIM', we assume that M' = 0. Assume that f(C) is contained neither in the singular locus of X nor in any divisor of the form above. Then we are in a position to apply (3.9.1) to f. Take an imbedding j: M -+ PNC, and let £;0, .. . £;N be a homogeneous coordinate system for pNc. Let

130

Chapter 3. Intrinsic Distances f

j

L

€P: C -----+ X -----+ M -----+ PNC.

Restricted to X, ~ i / ~o, (i = I, ... , N), are rational functions on X. We may assume that fCC) is not contained in the hyperplane ~o = 0, and we define meromorphic functions €Pi 0 f, (i = I, ... , N), on C by ~i

€p;(t) = ~o «j

0

to

i = I, ... , N.

f)(t»,

Then €p is given by (1, €PI, ..• , €PN). By (3.B.21) N

T(r, €p) S LT(r, rpi)

+ 0(1).

;=1

On the other hand, from the fact (see (3.9.1» that €Pi are algebraic over the field K generated by fj(a), ex = I, ... , n; j = I, ... , n + 1, we obtain T(r, €Pi) S

o (max T(r,

fj(CI.»),

where the maximum is taken over ex = 1, ... , nand j = I, ... , n + 1. Postponing the proof of (*) to the end we shall continue the proof. Applying (i) of (3.B.25) to we have T(r, fj(CI.» S O(T(r, + O(logr) outside an exceptional set E. Hence, T(r, €p) S o (max T(r, + O(logr) II.

f;,

f;»

f;»

1

The Kiihler form 1/1" of the invariant flat metric on M = Cn+p / r is given by n+p

1/1" = LddClwjl2. 1=1

We consider the order function of to f: C --+ M with respect to 1/1": TIj/(r,

Since /*i*1/I" =

'L7:+;r ddclti mer,

f;) S

L0

12 ,

f) =

l ! 'd -P o P

/*1/1".

De

(3.B.26) implies

!og(O(TIj/(r,

L0

f))

+

O(logr)

II·

f)))

+ O(logr)

II·

Combining all these, we have T(r, rp) S !og(O(TIj/(r,

L0

The order function T (r, rp) was defined by means of the Kiihler form 4> of the Fubini-Study metric of pNc. Now, we make use of compactness of M .. Since 1/1" and j*ct> are comparable, T(r, €p) and TIj/ (r, L 0 f) are ofthe same order. Hence, T(r, €p) ::: 10g(O(T(r, €p)))

+ O(logr)

II.

9 Theorem of Bloch-Ochiai

131

But this contradicts the following fact: O(T(r, rp» = O(T",(r, t

(3.9.16)

0

f) :::: O(r2).

In order to prove (3.9.16), choose i, 1 ::::: i ::::: n and consider its power series expansion: fi(t) = ao

+ I,

such that fi is nonconstant,

+ alt + a2t2 + ....

Then

by (3.B.9)

establishing (3.9.16). The contradiction comes from the assumption that f (C) is not contained in any divisor of the form stated in the theorem. In order to complete the proof we shall now prove (*). The proof for the following lemma of Val iron [I] is taken from Noguchi-Ochiai [I, p.222]. (3.9.17). Lemma. If ao, ... ,ak are entire functions with ao i= meromorphicfunction on C satisfYing the polynomial equation

then

°

and if rp is a

k

T (r, rp) :::::

L T (r, aj) + constant. j=l

Proof Take tEe such that ao(t)

i= 0, fixing it temporarily. Setting

Ai = aiel),

we define a polynomial in w by pew) = Aow k

+ Al W k - I + ... + Ak.

Let th. th, ... , th be the roots of the equation pew) = 0: P(w) = Ao(w - f3l)(w - fJ2)'" (w - 13k).

Since rp(t) is one of the roots, we assume that fJI = rp(t). We have I

2n

121f log IP(eie)lde =

log IAol

0

+

L -2nI 121f log le k

j=l

ilJ -

f3j Ide.

0

Making use of the following formula (which follows from Cauchy's formula): I

2n

121f log le ie 0

fJlde = log+

IfJl.

132

Chapter 3. Intrinsic Distances

we obtain 1 2n

127r log IP(e ili )ld6l

k

log IAol +

0

L log+ IPjl j=1

:::::

log IAol + log JI + IPd 2 -log2.

On the other hand, since k

log IP(eie)1

= log I L

k

A j ei (k- j )81 ::: Llog+

j=O

IAjl + log(k + I),

j=O

we have

Hence, k

log IAol + log+ IPII :::

L log+ IAj 1+ log2 + log(k + 1). j=O

Substituting Aj

= aj(t)

and PI

= cp(t) back into this, we obtain k

log lao(t)1 + log+ IcpU)1 ::: LlogJI + laj(t)12 + const. j=O

Integrating this inequality over the circle Dr yields I

2n

127r log lao(re ili )ld6l + mer, cp) ::: LT(r, k aj) + const. j=O

0

But (3.B.13) applied to the map f(t) = (ao(t), I) = (1, l/aoU»:C ---+ PIC gives _I 2n

(27r log lao(re ili )ld6l = 10

N(r, ao, 0)

+ C.

In order for cp to satisfy the given polynomial equation, every pole of cp must be a zero of ao of at least the same order. Thus, N(r, cp) ::: N(r, ao, 0).

Hence, k

N(r, cp)

+ mer, cp) ::: LT(r, aj) + const. j=O

D (3.9.18). Corollary. (f each ai is a polynomial of entire functions 1j;[, ... , 1j;m and (rcp is a meromorphicfunction on C satisfying the polynomial equation in (3.9.17), then

9 Theorem of Bloch-Ochiai

133

T(r, cp) ::::: C . max T(r, 1/Ij). J

Proof For an entire function 1/Ij, we have N (r, 1/Ij) = 0 since N (r, 1/Ij) in our notation counts poles of 1/Ij. Hence, T (r, 1/Ij) = m (r, 1/Ij)+ constant by the first main theorem. By (3.B.22) and (3.8.23) we have T(r, aj) ::::: where Co, (3.9.17).

CI, ... ,Cm

CI

T(r,

1/11)

+ ... + CmT(r, 1/Im) + Co,

are positive constants. Substitute this into the inequality in 0

We can reformulate (3.9.15) as follows: (3.9.19) Theorem. Let f be a holomorphic map ofC into a quasi-abelian variety M. Then the Zariski closure of fCC) is a translate of a quasi-abelian subvariety ofM. The original theorem of Bloch-Ochiai is stated as follows: (3.9.20) Theorem. Let X be an n-dimensional projective algebraic manifold with irreguialrity, i.e., dim HO(X, .Q 1), greater than n. Then every holomorphic map f: C ~ X has its image in a propoer closed algebraic subset of X.

Proof To derive this from (3.9.15), let a: X apply (3.9.15) to its image £l(X) C Ax.

~ Ax

be the Albanese map. Now 0

(3.9.21) Corollary. A nonsingular algebraic surface with irregularity> 2 is hyperbolic if and only if it has no curves of genus 0 or 1. It is desirable to strengthen the conclusion of (3.9.15) as follows:

Conjecture. Let X and WI, ... , Wn+1 be as in (3.9.15). Then X is hyperbolic modulo its singularity locus and a divisor of the form

n+1

L ajWl

1\ ... 1\

~

1\ ... 1\

Wn+1 = 0,

(ai, ... , an+l) -=f. (0, ... ,0).

j=1

(3.9.22) Remarks. R. Kobayashi [I] gave a proof of (3.9.15) by establishing the second main theorem for holomorphic curves in abelian varieties. A proof of (3.9.19), which is more arithmetic in spirit, was given by McQuillan [I]. For a smooth algebraic surface X of irregularity 2, a theorem similar to (3.9.20) has been obtained by Grant [1] under the assumption that the Albanese variety of X is simple.

134

Chapter 3. Intrinsic Distances

10 Projective Spaces with Hyperplanes Deleted E. Borel [1] observed that the little Picard theorem may be restated in the following form. If two holomorphic functions f and g on C vanish nowhere and satisfY the identity

f + g ==

(3.10.1)

1,

then they are constant. In fact, they omit two values 0 and 1 and hence must be constant by the little Picard theorem. Conversely, if f is an entire function omitting two values 0 and 1, then f and g = 1 - f satisfy the identity above. Now we state Borel's generalization of the little Picard theorem in the following three equivalent forms. (3.10.2) Theorem. (1) Assume that entirefonctions go, gl, ... , gN vanish nowhere on C and satisfY the identity go

+ gl + ... + gN == O.

Partition the index set I = {O, I, ... , N} into subsets la, I = U~=o la, putting two indices i and j in the same subset la if and only ~f gi I gj is constant. Then, for each a we have

(2) Assume that entire junctions fl' ... , fN vanish nowhere on C and satisfY the identity fl + ... + fN == 1. Then at least one of the fi 's is constant; (3) Under the same assumption as in (2), fl, ... , fN are linearly dependent (over C). We shall first show that these three statements are equivalent and then give applications of the theorem. The theorem will be proved in Appendix B of this chapter. For a more direct and shorter proof, see for example Noguchi-Ochiai [1].

Proof of equivalence. In order to derive (1) from (2), we set

so that ho + ... + hp == O. Since each ha is a constant multiple of a function gi with E la, it follows that either ha == 0 or ha vanishes nowehere. Assume that (1) does not hold. Without loss of generality, we may assume that h o, ... , hm, m ::=: 1, vanish nowhere and hm+1 = ... = hp == O. Set fa = -hal ho, 1 :5 a :5 m, so that fl ... + fm == 1. Then one of the fa's, i.e., say fl = -hI / ho, is constant by (2). Since ho is a constant multiple of a function gi with i E 10 and, similarly, hI is a i

10 Projective Spaces with Hyperplanes Deleted

135

constant multiple of a function gj with i E II. This means that gj / gi is constant, contradicting the definition of I a . We shall derive (2) from (3). Since fl,"" fN are linearly dependent, without loss of generality we may assume the following linear relation: cdl

+ ... + cN-dN-1 + iN == o.

By subtracting this identity from fl +···+fN-1 +fN

==

I,

we obtain (l - cl)fl

+ ... + (l

- cN-I)fN-1

== l.

By applying (3) and the same argument to this identity, we obtain a shorter identity. Finally, we end up with the identity cfl == I. FinaIly, we derive (3) from (1). We set fo = -1 so that fo + fl ... + fN == 0, and apply (I) to this identity. Let 10 be the index set that contains O. If I = 10, then the functions fl, ... f N are all constant and hence linearly dependent. If 10 =1= I, then LiEfa fi == 0 for every O! such that Ia =1= 1o, thus yielding a nontrivial linear 0 relation. Now we shall give various applications of Borel's theorem. In treating n + 2 hyperplanes Ho, HI, ... , Hn+1 in general position in Pn, it is most convenient to consider Pn C as a hyperplane in Pn + 1C given by (3.10.3)

in terms of the homogeneous coordinate system wo, wo ... , w n+1 of pn+lc. By a linear change of coordinates, the defining equations for Ho, HI, ... , Hn+1 may be reduced to the following simple forms: (3.10.4)

Ho

=

{wo

= o},

... , Hn

=

{W"

= O},

H n+1 = {w n +1 = OJ.

This gives an equal status to all n + 2 hyperplanes. We partition the index set I = {O, I, ... , n + I} into two disjoint subsets J = Uo, ii, ... , i p } and K = {ko, k" ... , kq }, where p + q = n. The intersection L J = Hjo n ... n Hjp of p + I hyperplanes defines a linear subspace of dimension q - 1 in Pn C, and the intersection L K = Hko n ... n Hkq of the remaining q + 1 hyperplanes defines a linear subspace of dimension p - 1 in pne. Then LJ and LK span a unique hyperplane HJK of pne. If we set (3.10.5)

FJK

=

Lw

j

= -

jEJ

Lwk, kEK

then the defining equation for HJK is given by FJK =

O.

136

Chapter 3. Intrinsic Distances

If J contains only one index, say j, then L] = Hj and L K is empty. So we consider nontrivial partitions I = J U K such that both J and K contain at least two indices and call H1K a diagonal hyperplane of the configuration Ho, HI, ... , Hn+l • For a quadrangle, (i.e., four lines in general positions), Ho, HI, H2, H3 in the projective plane P2C, there are three diagonal lines, (see Figure i in Section 3). Every holomorphic map 1: C --+ cn+ 2 - {O}, composed with the projection JT: C n+2 - {OJ --+ Pn+1C induces a holomorphic map JT 0 1: C --+ Pn+1C. Conversely, every holomorphic map f: C --+ PII + I C is thus obtained. In fact, using an open cover {V,,} ofC, we lift f locally to a holomorphic map 1,,: v" --+ C n +2 _{0}. Then 1" = h" fJ l fJ on v" n V fJ , where h afJ : V" n VfJ --+ C* is holomorphic. Then {h afJ } defines an element of HI (C, 0*) = 0 so that hafJ = A~I )'fJ with ),,,: V" --+ C* holomorphic. Set 1 = 1"),,, = lfJlfJ· We sometimes denote a lift 1 of f by the same symbol f. Let 11+1

f: C

--+

PnC -

U Hi C Pn+IC i=O

be a holomorphic map and 1: C --+ cn+ 2 - (OJ a lift of f. Then n + 2 entire functions (J°(z), fl(z), ... , f"+I(Z» satisfying fO

1 is given by

+ fl + ... + 1"+1 == O.

r+

1 vanish Since f avoids the n + 2 hyperplanes given by (3.10A), fO, fl, ... , nowhere. Partition the index set I = {O, 1, ... , n, n + I} as in (1) of (3.l0.2). If all fi / f j are constant, then f is a constant map. Assume that f is not a constant map. Then the partition I = U~=ola is nontrivial. If 10' contains only one index, say i, then the identity fi = LiEf. fi == 0 obtained in (3.10.2) contradicts the assumption that f misses the hyperplane Hi. Hence, each I" contains at least two indices. Set J = 10 and K = U,,#o la. Then

Lfj==-Lfk==O, jE]

kEK

and the diagonal hyperplane H] K defined by LjE] w j = 0 contains the image f(C). Thus, from the theorem of E. Borel we have derived the following theorem of A. Bloch [1] and H. Cartan [I]. (3.10.6) Theorem Let Ho, HI, ... , Hn+1 be n + 2 hyperplanes in general position in PIlC, n ::: 2. {( f: C --+ PIIC - U" Hex is a nonconstant holomorphic map, then its image lies in one of the diagonal hyperplanes. The following theorem which strengthens (3.10.6) is due to Dufresnoy [1, Theoreme XVI], see also Fujimoto [4, 5], Green [1], Lang [3]. For generalizations to mappings from Ck, k > 1, see Fujimoto [4, 5].

10 Projective Spaces with Hyperplanes Deleted

137

(3.10.7) Theorem.lfa holomorphic map I:C -+ PnC has its image in the complement of n + p hyperplanes HI, ... , H n+ p in general position. then this image is contained in a linear subspace of dimension :::: [n I p].

Proof Let Fi : C"+I -+ C be a linear form defining Hi. Let!: C -+ cn+1 - to} be a lift of f. Set hi = Fi 0 j. Since I misses Hi, the entire function hi vanishes nowhere in C. We partition the index set I = {I, 2, ... , n + p} into a disjoint union I = U~=I 101 by putting two indices i and j into the same la if and only if hilh j is constant. We claim that the complement of any lao, i.e., UIi#o Iii, contains at most n elements. If our claim is false, we would obtain a set 1 with n + 2 indices by taking n + 1 of them from the complement of lao and one from lao' Then we partition 1 in the same way, i.e., 1 = U 1a with 1a = 1 n la, (dropping those 101 that are empty). As we have seen in the proof of (3.10.6) above, each (nonempty) 101 must contain at least two elements. But we know from the construction of 1 that lao contains exactly one index. This contradiction proves our claim that the complement of each lao contains at most n indices. Hence, each 10/, ex = 1, ... , q, contains at least p elements so that pq :::: n + p. Let I' be any subset of I = {I, ... , n + p} consisting of exactly n + 1 elements. Since HI,"" H n + p are in general position, the linear forms F i , i E I', are linearly independent. Write I' = U-, I~, where I~ = I' n 101 , (Some I~ may be empty). Let ka be the cardinality of I~. Each I~, if nonempty, gives rise to a ka - 1 linearly independent equations: Fi - ciFio = 0,

(io, i E I~, i

i= io),

where Ci = hi! hio' Hence, hI, ... , h n+ p satisfy at least kl - 1 + linearly independent equations. But kl - I

+ ... + kq

- 1= n

+I-

q :::: n

+I-

n+ p

-p

=n

... + kq

- I

n

- -. p

o

Hence, I(C) lies in a linear subspace of dimension:::: nip.

In W. W. Chen [I], (3.10.7) is derived from Borel's theorem by a matrix method. (3.10.8) Corollary. If a holomorphic map f: C -+ PIlC misses 2n hyperplanes in general position, then it is a constant map.

+ 1 or more

Using (3.10.8) and (3.6.13) we can strengthen (3.10.8) as follows. (3.10.9) Corollary. The complement of 2n + I or more hyperplanes in general position in PnC is complete hyperbolic and hyperbolically imbedded in pne.

Proof It suffices to consider the case of 2n + 1 hyperplanes in general position. Let HI,"" H 2n + 1 be hyperplanes in general position in PnC, and let X = PIlCl Hi. By (3.10.8), Condition (a) of (3.6.13) is satisfied. Consider a partition of indices {I, 2, ... , 2n + l} = I U 1, and let L/ = niE/ Hi. If I contains k

U;:i

138

Chapter 3. Intrinsic Distances

elements, L 1 is an n - k dimensional linear subspace. The intersections H j n L I, (j E J), are 2n + 1 - k hyperplanes in general position in LI ~ Pn-kC. Since 2n+ I-k > 2(n-k)+ i, by (3.10.8) all holomorphic maps ofC into L1-UEJ Hj 1 are constant, which shows that Condition (b) of (3 .6.i3) is also satisfied. 0

U;:i

l Hi as (3.iO.l0) Remark. Dufresnoy [1] has shown that, for X = PnC in (3.10.9), Hol(D, X) is relatively compact in Hol(D, PnC). We shall show later that, in general, X is hyperbolically imbedded in Y if and only if Hol(D, X) is relatively compact in Hol(D, Y). So we may say that Dufresnoy had a result equivalent to (3.10.9).

In order to consider more general arrangements of hyperplanes, following Zaidenberg [2] we say that a set of hyperplanes HI, ... , HN in PnC is in hyperbolic configuration or satisfies condition (h) for short if each projective line 1 in PnC intersects Ui Hi in at least three points while it is in hyperbolic-imbedding configuration or satisfies condition (hi) for short if each projective line I intersects UH;;zI1 Hi, (i.e.,the union of those Hi that do not contain I), in at least three points. (These conditions (hi) and (h) are called conditions (a) and (b) in Zaidenberg's paper). Ciearly, condition (hi) is stronger than condition (h). Condition (h) is violated if and only if there is a pair of points p, q E PnC such that each Hi passes through either p or q, but not both. Condition (hi) is vioiated if and only if there is a pair of points p, q E PIlC such that each 8; passes through p or q, possibly both. The following theorem is due to Snumitsyn [I]. (3.10.11) Theorem. I{hyperplanes HI, ... , HN in PnC are in hyperbolic configuration, then N :::: 2n + i. We start with the proof of (3.1 0.i2) Lemma. Let HI, ... , H k + l , (2 S k S n), be hyperplanes in PnC such that dim n7=1 Hi = n - k and n~=1 Hi C Hk+1. Then there is an index m, I :::: m S k, such that n;=1.i#m Hi ct. Hk+1. Proof Set E = n;=1 Hi and E j = n7=l.ih Hi. Assume that E j C H k+1 for all j = 1, ... , k. Then Hk+1 n (n~=1 Hi) contains EI+ I , •• " £k, (1 :::: I S k - i). The first condition in Lemma implies £'+1 ct. H'+I for all I = 0, ... , k - I, and hence Hk+1 n (n~=1 Hi) ct. H'+I. So

(ni=1 1

H'+1 n H k + 1 n

(ni=1 I

Hi) =1= Hk+1

n

8;)

and dim(Hk+l

n

(n '+1

i=l

Hence,

(n I

Hi» = dim(Hk+l

n

i=l

Hi» - 1,

1= 0, ... , k - 1.

10 Projective Spaces with Hyperplanes Deleted

n HI) + 1

dim Hk +1 = dim(Hk+1

n- 1

139

2

dim(Hk+1 n (n H;»

+2=

...

+k =

dim(Hk+1

;=1 k

dim(Hk + 1 n (n HJ)

n E) + k

;=1

=

dim E

+k =n -

k

+ k = n.

This contradiction proves Lemma. Proof of (3 .1O.l1). Assuming that N ::s 2n we shall show that HI, ... , H N are not in hyperbolic configuration, that is, we shall produce a pair of points p, q E P"C such that each H; passes through exactly one of these two points. Obviously, it suffices to consider the case N = 2n. If Hi is non empty , it suffices to take p in Hi and q in Pn C Hi. Hi is empty. We renumber HI, ... , H2n in such Hence we shall assume that a way that

n

n

n

n

k

k = 1, ... , n.

for

dimnH; =n-k i=1

n;'=1

We set Qo = Hi; Qo is a point. Renumbering the remaining hyperplanes Hn+l , ••• , H2n, we assume that ro

Qo E

nHi i=1

211

and

Qo'l.

U

Hj

,

(n

::s ro

< 2n).

j=rn+ 1

nJ:ro+1

We set Po = H j • Clearly, Qo 'I. Po and dim Po :::: ro - n :::: O. If Po ct U~~I Hi, then it suffices to take p E Po and q = Qo. Hence assume that Po C U;~I Hi. Then Po C Hio for some io, I ::s io ::s roo We consider two cases: (a) (b)

1.:::: jo ::s n, n + 1 .:::: jo ::s roo

= n, i.e., Po C Hn. We set = I. Then we renumber Hn, ... , H2n (including HI!) so that QI c Hi, (rl :::: n-l), and QI ct Hi, (after this renumbering HI! is now one of Hr1 +1 , ••. , H2n)' We set PI = n~:r,+1 Hi. The subspace Po is, by definition, the intersection of the 2n-ro hyperplanes (called Hro +1 ' ••• , H21l earlier) that do not contain Qo. Since Qo C Q 1 C Hi for i = 1, ... , rl, these 2n - ro hyperplanes are now among 2n - rl hyperplanes H rl + I , .•• , H 2n . In addition, Po was contained also in another hyperplane (called Hn before the last renumbering) in this group. Hence, Po is given as an intersection of 2n - ro + 1 hyperplanes among Hrl +[, ... , H2n. Since PI can be written as the intersection of Po with hyperplanes among Hrl + I , ... , H2n that do not contain Po, we have In case (a) we renumber HI, ... , Hn so that jo

QI =

n7,:i Hi; dim QI

n;!:1

U;:rl+1

140

Chapter 3. Intrinsic Distances

dim PI

>

dim Po - [(2n - rl) - (2n - ro

>

ro - n - ro

+ rl + 1 = rl

-

+ 1)]

(n - 1)

~

O.

Hence, PI # 0. In case (b) we construct PI and QI as follows. Since Qo = Hi = n~~1 Hi c Hio and since dim Qo = 0, Lemma implies that there is an index m, 1 ::: m ::: n, such that Hi
n;'=1

n7=l,ii"m

may assume that m = n. We set QI = n7~i Hi and renumber Hn , ... , H2n so that QI = n~~1 Hi and QI
nf

Hrt+I, ... , H2n)' Set PI = 2r t+ 1 Hi. Since the hyperplane which had index jo before the last renumbering does not contain Q I, it is among Hrt + I, ... , H2n . By its very definition, Hio contains Po. Hence, as in case (a) we obtain

dim PI If PI


~

rl - (n - 1)

~

o.

U~~ I Hi, then it suffices to take p E PI - U;~ I Hi and q E Q I -

U;2

rt +1 Hi' Hence, we assume that PI C U~~I Hi. Then PI C Hit for some jl, I ::: jl ::: rl. Proceeding as above, we construct subspaces P2 and Q2, etc. If this process continues to the k-th step, then we obtain numbers rk ::: rk-I ::: ... ::: ro and subspaces Pk and Qk such that

(I)

Qk

= n7~; Hi,

(2)

Qk

= n;~1

(3)

Pk =

If Pk


U;2rk+

dim Hk

Hi, Qk

nf2rk+

1


= k;

U;2rk+ 1 Hi;

Hi, dim Pk ~ rk - (n - k) ~ O.

U~~I Hi, then it suffices to take p E Pk - U~~I Hi and q E Qk -

1 Hi' If not, the process continues. If it continues to the (n - 1)-st step, we have

2n

Pn -

I

=

n

Hi,

dim PI1 -

1

~ O.

i=2

(Since dim Qn-I = n - 1, QI cannot be contained in any other Hi than HI and rn-I = 2.) Since by assumption Hi is empty, we have Pn- I
n7:1

Kiernan [I] proved that the complement of2n hyperplanes in general position in Pne is not hyperbolic and conjectured that the complement of2n hyperplanes in any position in Pne is never hyperbolic; he verified the conjecture for n ::: 5. His conjecture follows from (3.10.11), Snurnitsyn [I]. More strongly, we have (3.10.13) Theorem. A set of hyperplanes HI, ... , HN in Pne are in hyperbolic configuration if the complement = PnC Hi is hyperbolic. in particular, the complement of2n hyperplanes in PnC is never hyperbolic.

X

n::1

10 Projective Spaces with Hyperplanes Deleted

141

Proof. If HI, ... , HN are not in hyperbolic configuration, there is a pair of points p, q in PIlC such that each H; passes through p or q, but not both. Let I be the projective line passing through p and q. Then X would contain I - {p, q} ~ C· and would not be hyperbolic. D The following question seems to be open. If HI, ... , HN are in hyperbolic configuration, is X hyperbolic? (3.10.14) Remark. We shall give a special configuration of 2n + I hyperplanes in PIlC such that its complement is complete hyperbolic. Using a homogeneous coordinate system (wo, .... w"), define 2n + 1 hyperplanes by wOw l ... w"(wO - wl)(w l - w 2) ... (w n - I - w ll )

= 0.

We shall show, by induction on n, that the complement X of these hyperplanes is biholomorphic to (C - {O, 1})1l. Using w n - I = as the hyperplane at infinity, we introduce the inhomogeneous coordinate system ZO = wO/w n - I , ..• , zn-2 = w n - 2 /w n - I , Z" = w n /w n - I • Then the equation above is written as

°

This equation defines 2n hyperplanes in the affine space CIl. Separating the variable from the others, we rewrite the above equation as equations defining hyperplanes in cn = C n - I xC:

ZI1

The second equation defines two points {O, I} in the last factor C of C n . The first equation defines 2(n - 1) hyperplanes in the factor cn-I. The complement of these 2(n - 1) hyperplanes in en-I is biholomorphic to the complement of the following 2n - I hyperplanes in Pn-IC: wOw l ... w n - I (w o _ WI)(W I - w 2) ... (W"- 2 - w n - I ) = 0,

which, by induction, is biholomorphic to (C - {O, 1))11-1. We shall construct a holomorphic map f: (D*)n -+ X which does not extend to a map f: D" -+ PnC. For (Zl, ... , Zll) E (D*)1l we set

This map extends to D" - {OJ but not through the origin. As we shall see later in (6.3.9), this implies that X is not hyperbolically imbedded in pne. This example generalizes the configuration of Example (3.3.10). Theorem (3.l0.7) and its Corollary (3.10.8) are concerned with hyperplanes in general position and may be regarded as a generalization of the little Picard theorem. For hyperplanes which are not in general position, we have the following result.

142

Chapter 3. Intrinsic Distances

(3.10.15) Theorem. Let HI, ... , H N, (N ~ 3), be distinct hyperplanes which are not in general position in pnc. Then the image of any holomorphic map f: C ~ Pn C - U H; lies in a hyperplane. Proof Let F; be a linear fonn on C"+ I which defines H;. Then there is a nontrivial linear relation: LC;F; =0. Without loss of generality, we may assume that CI, ... , Ck are nonzero and CHI ... = CN = O. Replacing F; by Ci Fi for 1 ~ i S k we may assume that CI ... = Ck = 1 so that FI + ... + Fk = O.

=

We may further assume that this is the shortest linear relation. Since HI, ... Hk are distinct, we have k ~ 3. Let j: C ~ cn+ 1 - to} be a lift of f, and set h; = F; 0 j, (i = 1, ... , k). Then the entire functions hI, ... ,hk vanish nowhere and satisfy the identity hI + ... +h k == O. As in (I) of (3.10.2) we partition I = {l, 2, .. . k} into subsets la, (0' = I, ... , p) so that LiEf. hi == O. Since hi ¥= 0, each Ia contains more than one element. If II = I, then f is a constant map. Otherwise, by (1) of (3.10.2) we have a linear relation LiEf! hi == 0, where II is a proper subset of I and contains more than one element. This implies that the image fCC) is contained in the hyperplane defined by LiEf! Fi = O. D Following Zaidenberg [2] we strengthen (3.10.8) as follows: (3.10.16) Theorem. rf a set of hyperplanes HI, ... , HN in PIlC are in hyperbolic configuration, then every holomorphic map f: C ~ PnC - U~=I Hi is constant. Proof By (3.10.11), N ~ 2n + 1. Applying (3.10.6) (if the given hyperplanes are in general position) or (3.10.15) (if not), we see that the image fCC) lies in a hyperplane, say H. Set iii = H n Hi. Then the set of hyperplanes iII, ... , iI N in H :;:: P,,_I (C) are in hyperbolic configuration. Repeating this process, we find D after n steps that fCC) is O-dimensional. If a set of hyperplanes HI . ... , HN are in hyperbolic configuration, then N 2n

~

+ 1. Now, we have

(3.10.17) Theorem. (1) {fa set o.fhyperplanes HI ..... H N , N ~ 2n + I, in PIlC are in general position, then they are in hyperbolic-imbedding configuration. (2) Ira set orhyperplanes HI, .... H21Z + 1 in PIlC are in hyperbolic-imbedding configuration, then they are in general position. Proof (1) Assume that HI, ... , HN are in general position. If condition (hi) is not satisfied, then there is a pair of points p, q such that each Hi passes through at least one of these points. Then at least n + 1 of HI, ... , HN must have p or q in common. This is a contradiction.

10 Projective Spaces with Hyperplanes Deleted

143

(2) Assume that HI, ... , H2n+1 are not in general position. By renumbering these hyperplanes, we may assume that the last n+ I hyperplanes Hn+l , ••• , H2n+1 have a point, say p, in common. Since Hi is nonempty, it contains a point, say q. Then the pair p, q violates condition (hi). 0

n;'=1

(3.10.18) Theorem. Given a set of hyperplanes HI, ... , HN in Pnc' set X = PnCU{: I Hi. If the given set of hyperplanes is in hyperbolic-imbedding configuration, then X is complete hyperbolic and is hyperbolically imbedded in Pile. Conversely, if X is hyperbolically imbedded in PnC, then the given set of hyperplanes is in hyperbolic-imbedding configuration.

Proof Assume that the given set of hyperplanes satisfies condition (hi). We shall apply (3.6.13). By (3.10.16) condition (a) of (3.6.13) is satisfied, i.e., there are no complex lines in X. For each subset I C {l, ... , N}, put PI = n i EI Hi. Then the set of hyperplanes (Hi = Hj n PI lUI in the projective space PI satisfies condition (hi). By (3.10.16) condition (b) of (3.6.13) is also satisfied, i.e., there are no complex lines in niEI Hi - UUI H j . (If N = 2n + I, this follows also from (3.10.17) and (3.10.9).) Assume that the given set of hyperplanes does not satisfy condition (hi). Then there is a pair of points p, q such that each Hi passes through at least one of these points. Let I be the set of i such that both p and q are in Hi. Let 1 be the projective line passing through p and q. Then 1- {p, q} c Hi - UjjU Hj . (If I = 0, then by niEI Hi we mean Pile.) Then the map exp: C -+ C* ~ I - {p, q} is a complex line in niEI Hi - UUI Hj (with respect to the Fubini-Study metric of PnC since le z 12dzdz/(l + lez 12)2 :'S dzdz). By (3.6.19), X cannotbe hyperbolically 0 imbedded in Pile.

nEI

From (3.10.17) and (3.10.18) we obtain the following converse to (3.10.9) as well as (3.10.9). (The "only if' part is due to Zaidenberg [2]). (3.10.19) Corollary. The complement ofa set 0/2n + 1 hyperplanes in PIlC is hyperbolically imbedded in PIlC if and only if they are in general position. Given a set of hyperplanes HI,"" HN in PnC, set X = PIlC-Uf=1 Hi. Then there are three cases: (i) condition (hi) is satisfied, or equivalently (by (3.10.18» X is hyperbolically imbedded in P" C; (ii) condition (h) is not satisfied, and hence (by (3.10.13» X is not hyperbolic; and (iii) condition (h) is satsified but not condition (hi). This last case is difficult to understand in general. In case (iii) there is a pair of points p, q such that each Hi passes through at least one of these points, and moreover, some Hi passes through both points. However, for n = 2 case (iii) will be completely analyzed in the following example (see Zaidenberg [4]). (3.10.20) Example. Let i l , ... , iN be a set of lines in P2 C satisfying condition (h) but not condition (hi). Then there is a pair of points p, q such that one of the lines, say iN passes through both points and each of the remaining lines iI, ... , i N- I passes through exactly one of the points. Put X = P2C - Ui:1 ii' Suppose that the first j lines pass through p and the next k lines pass through q. Then N = j + k + 1.

144

Chapter 3. Intrinsic Distances

.

If we consider iN as the line at infinity and identify P2 C -IN with C 2 , then the first j lines are mutually parallel in C 2 , and similarly, the next k lines are also mutually parallel. Hence,

x ;: : : (C -

{j points)) x (C - (k points}).

It follows that X is hyperbolic if and only if j, k :::: 2. Together with (3.10.13) and (3.10.17) this answers a question raised by Iitaka [4].

We consider a set of hyperplanes HI, ... , HN in PnC as a point in (p,~C)N, where Pn*C denotes the dual projective space. The symmetric group SN acts on (Pn*C)N in an obvious manner. In order to define the moduli space M(n, N) of all (unordered) sets of distinct N hyperplanes in P"C, we have to first remove all points of (p,;C)N which are fixed by some elements (other than the identity) of SN and then divide by SN. This yields a nonsingular complex manifold of dimension n N. We then have to divide it by the natural action of the projective linear group PGL(n; C).

Clearly, the set of points in M(n, N) not satisfying condition (hi) is closed. Thus, if X - U:':I Hi is hyperbolically imbedded in PIlC, then under a small perturbation of these N hyperplanes, X remains to be hyperbolically imbedded. On the other hand, the set of points in M(n, N) not satisfying condition (h) is not closed. In fact, the set of points of M(n, N) satisfying (h) but not satisfying (hi) is closed. The truncated defect relation of Cartan (3.B.42), or rather one of its consequences (3.B.46), can be applied to prove the following results of Green [2] on Fermat hypersurfaces. (3.10.21) Example. Let w o, Wi, for Pn + I C, and let

... ,

w n + 1 be the homogeneous coordinate system

be the Fermat hypersurface of degree din pn+lc. If d > n(n + 2), then every holomorphic map f: C ~ F(n, d) has its image in a hyperplane section. In fact, its image lies in a linear subspace of dimension:::: [nI2]. To see this, let PnC be the hyperplane in Pll + 1 C defined by

Then under the projection n: (wo, Wi, ... , WIl + I ) ~ «w 0)", (wl)d, ... , (wll+l)d),

the Fermat hypersurface F(n, d) is a covering space of PnC ramified over the hyperplanes Hi = {wi = O}, j = 0, ... , n+ 1. Assume that the image ofnof does not lie in a lower-dimensional linear subspace. If n 0 f does not interesect Hi, the truncated defect 8[n] (n 0 f, Hj ) = 1. If it intersects Hj , it intersects with multiplicity

10 Projective Spaces with Hypetplanes Deleted

at least d. By the argument in the proof of (3.B.46), we have

8[11](71: 0

f, Hj

)

~ 1-

145

J. Hence,

which would imply d :::: n(n + 2). Hence, 71: 0 f satisfies a linear equation L;'!(~ aj w j = 0 in addition to the linear equation L~''!ri w j = 0, so that f = Uo, ... , .f"+ I) satisfies the two homogeneous equations of degree d: n+1

11+1

L:Uj)d j=O

= 0,

L:ajUj)d = j=O

o.

In order to prove the second assertion, we may assume that f j == 0 does not hold for any j. (For, if f j == 0 for some j, the problem is reduced to a lower dimensional case.) We claim that I Ji is constant for some pair (i, j) with i =j:. j. Without loss of generality, we may assume that all+1 = I. Then taking the difference of the two equations above, we have

t

(ao - I)UO)d

+ ... + (an

- l)(r)d =

o.

Replacing each fj by (aj - l)l/d fj yields a Fermat hypersurface of lower dimension. Inductively, we obtain an equation of the form aUO)d + bUI)d = 0, proving our claim. We partition the index set {O, I, ... , n + I} into II, ... , 1m under the equivalence i ~ j if and only if If j is constant. From each Ir we pick ir and set P = bj/i, for j E I r • Then

r

L:(fj)d = cr(t,)d,

where

jE~

and

Cr

=

L:b1, jE4

m

L: CrUi,)d = o. r=1

Unless all the C r are zero, the equation above defines a Fermat hypersurface of dimension:::: m - I. (Set gr = c;/d f i, so that L;~I (g,)d = 0). Then, by the claim above, fip I fi q is constant for some p =j:. q, which is impossible since ip E Ip and iq E I q . Hence, all Cr = o. Thus, L:uj)d = O. jE/,

In particular, every Ir contains at least two indices. The image of f lies in the linear subspace given by the family of hyperplanes wj-bjw i , =0,

JEI" j=j:.i"

r=I, ... ,m.

(3.10.22) Example. We consider the complement of the Fermat hypersurface F(n - I,d) in pnc. If d > n(n + I), then every holomorphic map f:C --+ PnC - F(n - 1, d) has its image in a hyperplane. In fact, its image lies in a linear subspace of dimension:::: [nI2].

146

Chapter 3. Intrinsic Distances

Let w o, ... , wI! be the homogeneous coordinate system for P"C. We consider the n+2 hyperplanes Hj = {w j = o}, j = 0, ... , n, and Pn-IC = {wo+ ... +w n = OJ. Under the projection

rr: (wo, ... , w n ) --+ «wo)'t, ... , (w")d), PIlC is a covering space of P"C ramified over these hyperplanes. Assuming that the image of rr 0 f does not lie in a lower-dimensional linear subspace, apply (3.B.42) to the map rr 0 f. Then we have (n

+ I) (I

-

~) + I

::: n

+ 1,

which implies d ::: n(n + 1). We omit the remainder of the proof, which is similar to that of (3.10.21), see Green [2] for details. In order to strengthen (3.10.6), we consider sequences of holomorphic maps from D into P"C missing n + 2 hyperplanes Ho, HI,.'" H Il +! in general position. We represent P"C by a hyperplane (3.10.3) in Pn+!C and Ho, HI, ... , Hn+1 by (3.10.4). Following Kiernan-Kobayashi [2], we shall draw a geometric consequence (3.10.27) from the following theorem of Cartan [I; p. 58]. (3.10.23) Theorem. Given an infinite sequence li. = (j;o, f} . ... , fj"+I) of~ystems ofn + 2 nowhere-vanishing holomorphic junctions on the unit disc D satisjying the identity fOI. + I. + ... + /. 1 = 0,.

r'

r+

there is a subsequence, still denoted by fiJor which one of the following (a) or (b) holds. (a) The index set I = to, I, ... , n + I} is partitioned into two disjoint subsets J and K, J containing at least two indices and K possihly empty, such that (1) for i, j E i, the sequence Ul!f!L=1.2. converges to a nowherevanishing holomorphic function; (2) for j E i and k E K, the sequence {fi~ / L.=1.2 .... converges to zero;

i1

f!.)/.fiL.=1,2 .... converges to zero. There are two disjoint suhsets I' and I" of I = to, I, ... , n + I}, each

(3) jor j E J, the sequence {(LiE.! (b)

containing at least two indices and having partitions I' = ]'UK' and I" = i"UK" with Properties (1), (2) and (3) of case (a). We note that in case (a) Property (3) is a consequence of (2). However, in case (b), it is independent of (I) and (2). Set n+1 Z = PIlC, y = P"C-

UH;. i=O

If K is empty so that J = I in case (a), then the subsequence {h} converges to a map in Hol(D, Y). If K is singleton, say {n+ I}, in case (a), then the subsequence (J;J converges to a map in Hol(D, H,,+I) C Hol(D, Z), In the remaining cases of

10 Projective Spaces with Hyperplanes Deleted

147

(a) and in case (b), we have a subset J of I containing at least two but' no more than n-l indices such that, for each i E J, the subsequence {(LjEJ 11)lllli.=I.2... converges to zero. Hence, (3.1 0.24) Corollary. Given a sequence {fi.} of maps from D into Y as in the theorem above, there is a subsequence, also denoted by {hI, for which one of the following holds: (a) The subsequence {J;.I converges to a map in Hol(D, Z); (b) There exist a subset J of {O, 1, ... ,n + 1I containing at least two but no more than n - 1 indices such that. for each i E J. the subsequence {(LjEJ I!)I1!.li.=l.2. converges to zero. In case (b) of the corollary above, we have the following convergence for the subsequence: (3.1 0.25)

(3.10.26) Corollary. Let Y and Z be as above. and let .1 be the union of diagonal hyperplanes. Given a sequence of maps J;. E Hol(D, Y). there is a subsequence. also denoted by {j;J. jor which one of the following holds: (a) The subsequence converges in Hol(D, Z); (b) Given a positive r < 1 and a neighborhood U of .1 in Z. there is an integer )'0 such that J;.(D r ) C U for), ~ A.o. In the terminology of Section 1 of Chapter 5, this simply says that Y is tautly imbedded modulo .1 in Z. As we shall show in (5.1.13), this implies the following geometric theorem. (3.10.27) Theorem. The complement ofn + 2 hyperplanes in general position in Pn C is hyperbolically imbedded in Pn C modulo the diagonal hyperplanes. Cartan [1] developed the idea in an earlier paper of Bloch [1] and strengthened the result of Bloch. (There were also some gaps in Bloch's argument). The main difference between their results is that Bloch had to restrict himself to sequences of holomorphic maps I with fixed 1(0) in the complement of .1 = U7'!~ Hi (or at least teO) staying in a compact subset of P"C - .1) whereas Cartan imposed no such condition. Thus, Bloch's result seems to give hyperbolicity modulo .1. As we observed in Kiernan-Kobayashi [2], the full strength of (3.10.23) is yet to be geometrically explained. In Lang [3] the proof of Cartan's theorem (3.10.23) is reproduced. Cartan conjectured something stronger than (3.10.23). However, it has been disproved by Eremenko [2]. For n = 2, Cowen [3] obtained by a direct differential geometric method an explicit lower bound for the infinitesimal intrinsic metric Fy for the complement Y of five lines in general position in P2C, thus establishing hyperbolicity of Y. A similar result was obtained, independently, by Hall [1], who used a B1och-Cartan type method.

Chapter 3. Intrinsic Distances

162

1 1 r

-1

(3.B.9)

2n

-dp

P

0

12IT v(re,8)d() - -

ddcv = - 1

2n

Dp

v(O).

0

We apply (3.B.9) to vet) = log Ilf(t)II. Since If(t)1 2 = L~=o 1J;(t)1 2 > 0, from (3.B.l) and (3.B.2) we obtain (3.B.10)

T(r, f) = -

I

2n

12IT log II f(re i8 )IId() -log IIf(O)II. 0

Integrating

with respect to d() yields T(r, f)

(3.B.11)

+ log Ilf(O)1I

= mer,

f)

+ _1 2n

(2IT log Ifo(reiiJ)ld(). 10

In order to calculate the integral on the right, we recall Jensen's formula: (3.B.12) Theorem. h(t) = ct m + ..., c log Icl = -

1

2n

If h

is a meromorphic function on DR with Laurent expansion < R we have

-# 0, at 0, then for r

12IT log Ih(reiO)ld() -

L

0

r mj log + Lnk log - r - m logr, laj I lilt I

where the aj -# O's are zeros ofh with multiplicity mj in Dr while the f3k -# 0 's are the poles of h with multiplicity nk in Dr. In particular, if 0 is not a zero or pole of h(t), then

log Ih(O)1 = -

1

2n

12IT log Ih(rei°)ld() 0

r r Lmj log + Lnk log-. laj I lf3k I

We apply this formula to the holomorphic function fo(t). If c is the leading coefficient of the Taylor expansion of fo(t) = ctv(O,f) + ... at 0, then loglcl

= -1

2n

12IT 10glfo(reiiJ)ld() 0

r Lv(aj, f)log- - v(O, f)logr. laj I

With (3.B.6) this can be rewritten as (3.B.13)

_I

2n

(2rr log Ifo(re i8 )ld() =

10

N(r, f)

+ log lei.

Now, (3.B.11) reads as follows: (3.B.14)

T(r, f) = mer, f)

+ N(r, f) + log Icl

-log Ilf(O)II.

Since any hyperplane Ha can be considered as the hyperplane at infinity, we have (3.B.15)

T(r, f) = mer, f, a)

+ N(r,

f, a)

+ c.

B Nevanlinna-Cartan Theory

163

This is the first main theorem of Nevanlinna theory. In particular, we have (3.B.16)

T(r, f) 2: N(r, f, a)

+ c.

Suppose that f is defined on all of C. Since r (p) is monotone increasing, it follows that T(r) -i> 00 as r -i> 00. We define the defect 8(f, a) and the truncated defect olml(f, a) of a (or rather Ha) by setting o(f, a) = liminf(1 _ N(r, f, a»), . HOO T(r, f)

s

Then 0 S 8(f, a) S 8Im1 (f, a)

olml(f, a) = lim inf( 1 r--+oo

l. We note that 8(f, a)

Nlml(r

fa»)

". T(r, f)

= 1 if fCC) n Ha = 0.

Consider the case n = l. A meromorphic function ({I on DR can be written as a quotient of two holomorphic functions with no common zeros: ((I(t) = fl (t)/fo(t).

Thus, it is considered as a holomorphic map f E Hol(D R , PI C) with f(t) (fo(t), fl (t». In this case, we often write T(r, ({I), mer, ({I) and N(r, ((I) for T(r, f), mer, f) and N(r, f). On the other hand, we write mer, ({I, 0) and N(r, ({I, 0) for mer, f, cd and N(r, f, CI). We note that N(r, ((I) counts zeros of fo(t), i.e., poles of ({I while N (r, ({I, 0) counts zeros of fl (t), i.e., zeros of ({I. As a special case of (3.B.7), we have (3.B.17)

mer, ({I)

=

_I

(7r 10g.)1 + 1({I(reili)l2d6l.

2n io

Using the notation log+ x = max{O, logx}, mer, ({I) is often given by _I 2n

{27r log+ 1({I(re iH )ld6l,

10

which is asymptotically the same as (3.B.17) since log+ Ixl S log.}1 + Ixl 2 S log+ Ixl + log2. We prefer to use (3.B.17). If ({I is holomorphic, then f(t) = (1, ({I(t» is in a reduced form, and (3.B.lO) becomes (3.B.18)

T(r, ({I)

= - 1 127r log.}l + Icp(re ili )1 2d6l 2n

-log.}l

0

+ 1({I(0)l2.

For each i, 1 S i S n, we consider the meromorphic function ({Ii (t) J;(t)/fo(t) as a holomorphic map into PIC. Since n

1 + l({Ii(t)1 2 S 1 +

L i=1

we have

n n

I({Ii (t)1 2 s

(I

i=1

+ I({Ii (t)1 2 ),

164

Chapter 3. Intrinsic Distances 1Z

(3.B.19)

mer, rpi) :::: mer, f) :::: L mer, rpi). i=1

In order to count n(p, rpi) we reduce (fo(t), fi(t» by factoring out the common zeros of fo(t) and fi(t), and then count the zeros of fo(t). Therefore we have n

(3.B.20)

N (r, rpi) :::: N (r, f) ::::

L N (r, rpi ). i=1

The first main theorem together with (3.B.19) and (3.B.20) yields n

(3.B.21)

T(r, rpi) :::: T(r, f)

+C

:::: LT(r, rpi)

+ C'.

i=1

For two meromorphic functions rp and l/I on DR, the inequality

implies (3.B.22)

mer, rpl/l) :::: mer, rp)

+ mer, l/I),

while the inequality

implies (3.B.23)

mer, rp

+ l/I) :::: mer, rp) + mer, l/I) + constant.

Now we state Nevanlinna's lemma on logarithmic derivative. For its proof, see Nevanlinna [1; pp.63-64], Noguchi-Ochiai [1; pp.225-227], or Lang [3; p. 172]. (3.B.24) Lemma. Let rp be a meromorphicfunction on C. Then

mer, rp'jrp)

=

O(1og+ T(r, rp)

II.

+ logr)

Here, II indicates that the inequality holds outside an exceptional set E of finite Lebesgue measure, i.e., dr < 00.

IE

(3.B.25) Corollary. Let rp be a meromorphic function on C. Then, for the p-th derivative rp(p) we have (i) (ii)

T(r, rp(p» :::: (p

+ I)T(r, rp) + O(log+ T(r, rp) + logr)

mer, rp(P) jrp) ::::

o (1og+ T (r, rp) + log r),

II,

II,

p:::: 0;

p:::: I.

Proof The following proof by induction on p is from Ochiai-Noguchi [1]. While (i) is trivial for p = 0, (ii) for p = 1 is the lemma above. Assume that (i) holds for p - 1 and (ii) for p. Then

B Nevanlinna-Cartan Theory

mer, cp{P)

=

cp{p) ) m ( r, cp . ----;p

<

mer, cp)

:s mer, cp) + m ( r,

165

cp{P) ) ----;p

+ O(log+ T(r, cp) + logr)

II.

Since the order of each pole of cp increases by 1 everytime cp is differentiated, we have (recalling that N (r, cp) in our notation counts poles of cp)

:s (p + l)N(r, cp).

N(r, cp(P) Hence

+ mer, cp{p) + C (p + I)N(r, cp) + mer, cp) + o (1og+ T(r, cp) + logr) (p + l)T(r, cp) + O(log+ T(r, cp) + logr) II,

:s :s :s

T(r, cp{P)

N(r, cp(p)

which proves (i) for p. By (3.B.24) and (i) for p just proved, cp{P+l)) m (r,-cp

=

cp{P+l)cp(P)) m ( r, ~ ----;p

(cp{P+IJ)

(CP{P))

:s m r, cp{p) + m r,----;p + logr) + O(1og+ T(r, cp) + log r)

:s o (\og+ T(r, cpcP) :s O(\og+ T(r, cp) + logr) which proves (ii) for p + 1.

II, D

The following corollary corresponds to Lemma (6.1.29) in Noguchi-Ochiai [I, p.230]. We derive it from (3.B.24). (3.B.26) Corollary. Let cp be a holomorphic function on C. Then mer, cp') Proof Since mer, cp') -I

mer, cp)

41T

:s log

l

a

br

logO

+ O(1ogr)

II.

+ Icp(re i O)1 2)d8

1 +- l - 1 1 1 +

I -log(1

2

dd c lcpl2

D"

:s mer, cp' /cp)+m(r, cp) by (3.B.22), we first estimate mer, cp).

I -logO 2 <

r dpp 1[

la

+ -I

21T I 21T

r I -log+ -dp 2 a P

2rr

a

Icp(re i &)1 2d8)

r

dp a P

dd C lcpl2

(by concavity of log)

+ C)

(by (3.B.9»

Dp

dd C lcpl2

C' .

Dr

On the other hand, an upper bound for mer, cp'/cp) is given by (3.B.24). Hence,

166

Chapter 3. Intrinsic Distances

1 mer, cp') .:::: -log+ 2

1 1 r

0

-dp P

dd C lcpl2

+ o (1og+ T(r, cp) + logr)

Dp

II·

We obtain the desired inequality by replacing T (r, cp) by the following bound

T(r, cp) .:::: - 1 47T

1J!... 1 r

0

d P

dd C lcpl2

+ e".

De

This latter inequality follows from

D Let q :::: n, and ao, ... , aq with components

E

C n + 1 be q

+ 1 unit

vectors in general position

), = O, ... ,q and with the corresponding hyperplanes n

H;.:

Lak =0,

A=O,l, ... ,q.

i=O

Let f(t) = (fo(t), fl(t), ... , fn(t): DR -+

cn+ 1 -

(OJ

be as before. Set n

(3.B.27)

g;.(t) =

LaLtiU),

), = 0, .. . ,q.

i=O

Take n + 1 integers {AO, ... , Jon} from {O, 1, ... , q}, and let {)'n+!,"" Aq} be the complementary set of integers. The Wronskians W(gJ.o(t), ... , gi.,,(t» and W (fo (t), ... , fn (t) are related by W(g;.o'···' g;..) = C(AO, ... , )'n)W(fo, ... , fn),

where C(AO, ... , An) = det(ai)o::::i,j::::n is a nonzero constant since ao, ... , aq are . in general position. Set

(3.B.28)

Then

W*(g;""

, g;.J =

g:·o

g;l

glo

gil

[;;:0

g~~l

gio

g;,\

g;," g;-n

" g;'11 g;-n

B Nevanlinna-Cartan Theory

167

gogl ... gq W(fo, ... , In)

(3.B.29)

Since the right hand side is independent of the choice {)'o, ... , An}, so is the left hand side. We set h=

(3.B.30)

gogl·· .gq W(fo, ... , In)

so that Then

L

Igi.n +1

•••

gi.q 12

=

Ih 12

L

Ic(Ao, ... , An) W*(g.
where the sum is taken over all choices of {AO, ... , )'n} C {O, 1, ... , q}. For simplicity, we set u=

(L Ic(Ao, ... , All) W* (gi.() , ... , gi.JI2)1/2

so that (3.B.31) (3.B.32) Lemma. Let J(t) = (fo(t), (3.B.27). Then IIJ(t)1I 2 (q-n) =

(L 1.fi(t)1

2 )q-n

II (t), ... , fn(t) :S K

L

and go, gl, ... , gq be as in

Igi.n + 1 (t) ... g'
on

DR,

where K is a sufficiently large constant and the last sum is taken over all subsets {),o, ... , )'n} of{O, 1, ... , q}. Proof For each fixed to E DR, we arrange goCto), ... , gq(to) in the increasing order of their absolute values:

Since n+ 1 vectors a.
we have

o Combining (3.B.31) and (3.B.32) yields

168

Chapter 3. Intrinsic Distances

(3.B.33)

(q - n) log

1If11 ::: log Ihl + log u + const.

We integrate each term of (3.B.33) along the circle _1 [21f log II/(re iH )lld8 21f

(3.B.34)

Jo

=

r(r, f)

aDr . By (3.B.1O)

+ log 11/(0)11.

Jensen's formula (3.B.13) implies I

121f

21f

0

-

r

loglh(re iB )ld8::: Lmjloglajl

+ log lei +mlogr,

where e and m are given by the Laurent expansion h(t) = et m are zeros of h in DR with multiplicity mj' By (3.B.6), N(r, h, 0) = Lmj log

+ ... while aj -I- 0

~ + log lei + m logr. la)1

This combined with (3.B.34) yields (3.B.35)

_1 [21f log Ih(re iB )ld8 ::: N(r, h, 0) 21f

Jo

+ log lei.

We shall bound N (r, h, 0) in terms of truncated counting functions

1 r

N[n1(r, gi., 0) =

o

dp (n[n1(p, gi., 0) - v[n1(0, gi., 0»-

p

+ v[n1(0, gA, 0) logr.

Suppose that to is a zero of the meromorphic function h in DR. As in the proof of (3.B.32), let Igi.oCtO) I ::: Igi" (to) I ::: ... ::: Ig;.,/to) I. Then from (*) in the proof of (3.B.32) we know that 0< Igi.,,+1 (to) I :::

... :::

Igi.q (to)!.

Since h is equal to the left hand side of (3.B.29) and since the numerator gi',,+1 ... gi.q does not vanish at to, the order of zero of h at to is equal to the order of pole of W*(g;.o"'" gi.,,) at to. We estimate the order of pole of W*(gi.", ... , gi.,,) at to. Let Mi. be the order of zero of gi. at to. Then the order of pole of g~.k) / gi. at to is equal to k if Mi. :::: k and to Mi if Mi. < k; in particular, it is bounded by min(Mi., k). Hence, the order of pole of W*(gi. o' .••• gi.,,) at to is bounded by L:;'=o min(Mi." n), i.e., by L:7=0 v[nl(to, g'.i)' Thus, vCto, h, 0) :::

n

q

i=O

).=0

L v[n1(to. gi.i' 0) ::: L v[n1(to. gi., 0).

Since this holds at every zero to of h, we have q

(3.B.36)

N(r, h, 0)

.:s L -<=0

N[n1(r, gA, 0).

B Nevanlinna-Cartan Theory

169

8y the very definition of gi., we have

Hence,

N(to, gi., 0)

= N(to,

f, a;J,

and we can rewrite (3.8.36) as q

(3.8.37)

N(r, h, 0)

:s L

N[n1(r, f, a A).

;,=0

Substituting this into (3.8.35) yields 1

(3.8.38)

2Jr

r21f log Ih(re o)ld8 :s {; N[n1(r, f, a;,} + log Ih(O)I·

10

q

i

Finally, we consider the integral of log Iu I along the circle W*(g;,o' ... , g;,J is a polynomial of g;,P) /g;" (A. = 0, ... , q;

have

q

logu 2

:s K + K

L

a

Dr. Since each p = 1, ... , n), we

n

Llog+ Ig;,P) /g;,I,

-<=0 p=1

where K is a constant. Integrating this over the circle aD" we obtain (3.B.39)

-

121f

1

2n

q

n

logu(re io ) < K+K""m(r,g(P)/g;,}. ~~

o

~

i.=0 p=1

Substituting (3.8.34), (3.8.38) and (3.8.39) into the integral of (3.8.33) yields the following fundamental inequality of H. Cartan: q

(3.8.40)

(q - n)T(r, f)

:s L

N[n1(r, f, a;J

+ S(r),

;,=0

where

q

S(r) = K

+K

L

/I

Lm(r, g?) /g;,}.

;,=0 p=1

Assume that f is defined on all of C. Then by (3.8.25), we have mer, gj,P) /g;.) ::: O(log+ T(r, g;J

Since Ig;,(t)1 2

:::

+ logr)

II.

Li la~J . Lj Ijj(tW = IIf(t)1I2, (3.8.10) and (3.8.18) imply T(r, g;J :s T(r, f).

Hence, C3.B.41)

SCr) = O(log+ T(r, f)

+ logr)

II·

170

Chapter 3. Intrinsic Distances

We rewrite Cartan's inequality (3.B.40) as follows: q

~)T(r, f) - N[nJ(r, f, ai.)) ::: (n

+ I)T(r,

f)

+ S(r).

i.=O

Divide this by T(r, f) and let r ~ 00. Since liminfS(r)/T(r, f) = 0 by (3.B.41), we have the truncated defect relation of Cartan: (3.B.42) Theorem. Let ao, ... , a" E C',+I be q + 1 vectors in general position. Then for any holomorphic map f: C ~ PnC that is non-degenerate in the sense that its image is not contained in any linear subsapce, we have q

L 8[Il]U, a/:) ::: n + 1. i.=O

This is of course sharper than the usual defect relation q

LOU, a;)

(3.B.43)

::: n

+ I.

;.=0

We have now a generalization of the little Picard Theorem: (3.B.44) Corollary. If a holomorphic map f: C ~ PIlC misses n in general position, then its image is contained in a hyperplane.

+ 2 hyperplanes

This implies Borel's theorem (3.10.2), which was stated in three equivalent forms. We prove it in the form (3) of (3.10.2). (3.B.45) Corollary. the identity

If entire functions

fl, ... , in vanishing nowhere on C satisfy

II + ... + 1" ==

I,

then they are linearly dependent. Proof Consider the map f: C

Then

~

PnC given by

f misses the following n + 2 hyperplanes which are in general position: ZO

= 0,

Zl

= 0,

... ,

Z"

= 0,

ZO

+ Zl + ... + zn = o.

o

Now Borel's theorem follows from (3.B.44).

(3.B.46) Corollary. Let f: C ~ P"C and ao, ... , a" be as in Theorem (3.B.42). For each A, let Hi. be the hyperplane perpendicular to ai, and mi. = min{v(a.

We set mi. =

00

I, a;);

(f fCC) n H;. = 0. Then

a

E

f-I(H;)}.

B Nevanlinna-Cartan Theory

171

Proof Since

if m; ::: n, it suffices to show

1- -

n

:::

8[11] (f,

a;.)

m'I.

when mI. > n. In this case,

and

Therefore, 1-

NIIl] (r,

j, ai)

T(r, f)

n N (r,

>1-mi.

Now the inequality (*) follows from (3.B.16).

f. aJ

T(r, f)

.

o

(3.B.47) Remark. In (3.B.42) and (3.B.46) we assumed that j is non-degenerate. These results have been generalized by Nochka [1] to the situation where fCC) lies in a lower-dimensional linear subspaces, see also Fujimoto [14].

Chapter 4. Intrinsic Distances for Domains

1 Caratheodory Distance and Its Associated Inner Distance Let X be a complex space. We denote its Caratheodory pseudo-distance by cx, (see (3.1.1», and the induced inner pseudo-distance by c~, (see (1.1.2». While the Kobayashi pseudo-distance d x is always inner (see (3.1.15», the Caratheodory pseudo-distance Cx need not be (see Examples (3.1.25), (3.1.26), (3.1.27) and (3.1.28). Since every inner distance on a locally compact Hausdorff space X induces the given topology of X (see (1.1.8», we obtain (4.1.1) Theorem.

rr c~ is a distance, it induces the complex space topology on X.

In general, even if Cx is a distance, it may not induce the complex space topology of X. The following proposition is a special case of (1.1.10) (due to Hirstov [2]) and strengthens Sibony [2], in which Cx is assumed to be strongly complete. (4.1.2) Proposition. If Cx is a weakly complete distance. it induces the complex 5pace topology on X. The following result is due to Sibony [2]. (4.1.3) Proposition. rl X is a relatively compact domain in a Stein space M, then Cx induces the complex space topology on x. Proof Let H (M) be the Frechet algebra of holomorphic functions on M equipped with the topology of uniform convergence on compact sets. It is known that M is isomorphic to the spectrum of H (M) with weak topology (see, for example, Gunning-Rossi [I]). Therefore, the topology of X is the weakest one that makes fix continuous for every f E H(M). But H(M)lx is contained in the algebra HOO(X) of bounded holomorphic functions, and every f E HOO(X) is continuous with respect to c x. Hence, the ex-topology is at least as fine as the complex space 0 topology of X.

The following general observation is due to Barth [6], see also Barth [9]. (4.1.4) Proposition. [fcy induces the complex space topology on Y and if there is a holomorphic map f: X -+ Y that is a homeomorphism onto its image. then ex induces the complex space topology on X.

174

Chapter 4. Intrinsic Distances for Domains

Prool This follows from the fact that

f

decreases the Caratheodory distance.

o

The first example of a complex space X whose Caratheodory distance Cx does not define the complex space topology was given by Vigue [5]. (4.1.5) Example. In the tri-disc D J C C 3 we consider the following 2-dimensional complex space X = XI U X 2 , where XI

{(x,y,O) E DJ}

X2

{CO, y, z) E {OJ x D x

=

f)J} f) -

f)

x D x {O},

{CO, y, 0);

(O)

X DI/2

I.vl

I

:s 2}

x {O}.

Using the extension of Cx to (D x D x {O}) U( {OJ x D x D), we see that the sequence PII = (0,0, I/n) converges to the origin (0,0,0) with respect to cx. But it does not converges in the complex space topology of X since D I / Ill x D I / 111 X {O}, (1'11 = 1. 2 .... ), form a neighborhood basis for the origin (0,0,0) in X. Later, Hayashi [1] gave an example of an open Riemann surface X such that Cx is a distance but does not induce the topology of X. Making use of Hayashi's construction, Jarnicki-Pflug-Viguc [I], (see also Jarnicki-Pflug [10]), proved the existence of a domain X of holomorphy in C J such that Cx is a distance but does not induce the topology of X. Because of (4.1.1), these provide also examples of complex spaces X with non-inner Caratheodory distance cx. We must be very careful with the topology defined by the Caratheodory distance. For a E X and r > 0, we set B(a: r) = {x E X: cx(a, x) < r}

and

K(a: r) = {x E X: cx(a, x)

:s r}.

Since ex is continuous on X x X, we have always B(a: r) C K(a; x). However. in general, B(a: r) '# K(a: r) even when X is a strongly pseudoconvex bounded domain in C", as shown by Jarnicki-Pflug-Vigue [2], (see also Jarnicki-Pflug [10]). There is a survey by Barth [9] on topologies defined by Caratheodory and other distances including the infinite dimensional case. A complex space X is said to be Caratheodory-hyperbolic or C-hyperbolic if ex is a distance and induces the complex space topology of X. A C-hyperbolic space X is said to be complete (resp. strongly complete) if X is Cauchy complete with respect Cx (resp. if all closed balls with respect to Cx are compact), (see Section 1 of Chapter I). (4.1.6) Proposition. Ira complex space X has a (compli:'fe) C-hyperbolic covering space X. thell it is (complete) hvperho/c. Proof Since hyperbolic.

ex

:::c d x'

X

is (complete) hyperbolic. By (3.2.8) X is (complete)

0

I Carathcodory Distance and Its Associated Inner Distance

175

Every bounded domain in C" is C-hyperbolic and hence hyperbolic. According to Masaaki Suzuki [3], a complete pseudoconvex circular domain must be bounded if it is hyperbolic. A useful criterion for strong completeness can be stated in terms of peak functions. Given a bounded domain X C C" and a boundary point Xo E ax, a holomorphic function I defined in a neighborhood of the closure X is said to be a peak function (resp. weak peak function) for X at Xo if II(x)1 < 1/(xo)1 for all x E X - {xo} (resp. x E X). We may always assume that If(xo)1 = 1. A local (weak) peak function for X at Xo is by definition a (weak) peak function for X n B(xo, r) at xo, where B(xo, r) is an open ball of small radius r about Xo. (4.1.7) Theorem. It' X C C" is a hounded domain such that there is a weak peak fitnction for X at each point 0, we shall show that the closed ball K (0, a) = {x E X: cx(o. x) :::: a} is compact. Choose a positive number b < I such that (z E D: p(f(o). z) :::: u.

f

E

Fl c

(z E D;

Izl::::

b).

Then K(o, a)

f E Hol(X. D)} :::: a, .f E F} f E Fl.

{x EX; p(f(o). f(x» :::: a. C

(x

c

{x E X:

EX: p(f(o), lex»~

If(x)1 :::: h.

The last set is compact since each boundary point Xo of X has a neighborhood which does not meet the set Ix E X: If,,,(x)l:::: bl. Hence, K(o. a) is compact.

o

Let G be a domain in C" and II .... , Jk holomorphic functions defined in G. Let P be a connected component of the open subset of G defined by

Ifl(z)1

< I. ....

l.fk C::) I < I.

If the closure of P in C" is compact and is contained in G, then P is called an analytic polyhedron. It is clear that at each boundary point of P, one of the functions II . .... fk is a weak peak function for P. (4.1.8) Corollary. Every ana~vtic polyhedron P is C-hyperbolic and is complete with respect to Cpo

strong~v

We can apply (4.1.7) to slightly more general domains than analytic polyhedron. Let h j be real analytic functions on G of the form

176

Chapter 4. Intrinsic Distances for Domains ex;

hj =

L

1./)1111 2,

j=l. ... ,k.

11'1=1

where each ./jill is holomorphic in G. Let P be a connected component of the open subset of G defined by

If the closure of P in C" is compact and is contained in G, we call P a generalized analytic polyhedron. Let S be the set of sequences {alii} of complex numbers such that L~=I I. Then, for each j, we have (*)

{z E G; hj(z) < I} =

n

lam l2 =

x

{z

{a",)ES

E

G: I L a lll ./jll1(z)1 < I}. 111=1

In fact, the left hand side of (*) is contained in the right hand side since

I LlIm!iml2 ~ L Ia 12 L Ihml 2 111

m

III

111

=

Ihi l2.

If z belongs to the right hand side, let am = .fjlll(Z.)/(LIII 1f/III(z)1 2)1/2. Then I > I LUIII!illl(z)1 = (L I.fjm(z)1 2 )1/2 = h i (z)I/2.

III

111

showing that z is contained in the left hand side. From (*) we see that at each boundary point of P a function of the form LII1 am lim is a weak peak function for P. Hence

(4.1.9) Corollary. Every generalized anazvtic polyhedron Pis C-hyperbolic and is strongly complete l-vith respect to Cpo (4.1.10) Corollary. Every bounded convex domain X C C" is C-hyperholic and is strongly complete Ivith respect to ex.

Proof For each Xo E iJX there is a complex linear functional
.r

Pflug [3] has shown that every bounded complete pseudoconvex Reinhart domain X is complete with respect to ex. Since Cx ~ d x , if a complex space X is C-hyperbolic and is complete with rcspect to cx, then it is complete hyperbolic. In particular, all domains described in (4. I. 7) through (4.1.10) are complete hyperbolic. But we can say a little more. (4.1.11) Corollary. If X C C" is U bounded domain such that at every boundary point of X there is a local weak peak/imction, then X is complete hyperbolic.

I Carathcodory Distance and Its Associated Inner Distance

177

Prool Clearly, X is hyperbolically imbedded in en. By (4.1.7) every boundary point x E ax has a neighborhood V in C" such that V n X is strongly complete with respect to Cunx. Then vnx is complete hyperbolic. By (3.3.4) X is complete hyperbolic. D (4.1.12) Corollary. Every bounded strong~v pseudoconvex domain X with C 2 boundary is complete hyperbolic.

Prool For every boundary point x E ax there is a neighborhood V and a biholomorphic map f: V --+ f(V) such that f(V n X) is strongly convex. Hence. there is a local peak function for X at x. D (4.1.13) Example. As in (3.1.26), let X = M(D. V) C C 2 be the domain constructed by Sibony. It is a proper subdomain of D2. and as we explained in (3.1.26). Cx is the restriction of e/)2. In particular, ex is not complete. Following Eastwood [1] we shall show that X is complete hyperbolic. Let {X/1} c X be a Cauchy sequence with respect to d x . Set X/1 = (z". wIll E D x D. Then both {ZIl} and {WIl} are Cauchy sequences in D with respect to d D. Let Zo = lim 2" and Wo = lim w". We have to show that (zo, wo) E X. Let {a,,} be the discrete sequence of points in D used in the construction of the domain X, see (3.1.26). Case (i) Zo = for some p. In this case, V(zo) = 0 and Iwl < I = e- Vicu ) for every WED. Hence, (zo. w) E X for all wED. Case (ii) Zo =1= {II' for all p. Let p be a small positive number such that the neighborhood N = {z E D; dD(zo, z) < 4p} contains none of the a" 'so Given a point 0 E X and a positive number 8, we set V(o; 8) = {x E X; dx(o, x) < 8j. Then by discarding a finite number of x," we may assume that all X/1 are in the V(o; p) for some 0 E X; By (3.1.19) there is a positive constant C such that

(I"

d U {o;4p)(p. q) < C· dx(p, q)

for

p, q E V(o; pl.

This shows that {XII} is a Cauchy sequence in V (0: 4p) with respect to d U (():4p)' Let rc be the projection from X to D sending (z. w) to ;:. Since V (0; 4p) C rc-' (N), {x,,} is a Cauchy sequence in rc-'(N) with respect to d;r-I(N)' It suffices therefore to show that rc-' (N) is complete hyperbolic. We may assume that :::0 = 0 so that N = {z E D; Izl < r} for some r. Take

n

a suitable branch of log(z - al') on N so that «z - (lp)/2Y" converges to a holomorphic function y on N. Then V = Iyl. Now we can write rc-'(N) as follows:

"

where the intersection is taken over all complex numbers u of absolute value 1. This shows that rc-'(N) is a generalized analytic polyhedron. By (4.1.9) and (4.1.6) ][-, (N) is complete hyperbolic. (4.1.14) Example. Following Pyatezkii-Shapiro [2] we define Siegel domains of the second kind. Let V be a convex cone in R n containing no entire straight lines. A mapping F: C m x C m --+ C" is said to be V -Hennitian if

178

Chapter 4. Intrinsic Distances for Domains (a)

F(u, v) = F(v, u),

(b) (c) (d)

F(u, v) is F(u, u) E

Lt,

v E CIf;

complex linear in u; if (the closure of V), F(u, u) = 0 only when u = O.

u

E

cm;

The subset S = S( V, F) of Cd'lIl defined by

S = f(z. u)

E

C" xC"; Im(z) - F(u, u)

E V}

is called the Siegel domain of the second kind defined by V and F. If V is a special convex cone given by yl > 0, ... , ylll > 0 for a suitably chosen linearly independent set of linear functionals y I, ... , yin on Rnl, (i.e., for a suitable coordinate system y I , ... , yin), then S (V, F) is biholomorphic to a product of open balls (see Pyatezkii-Shapiro [2] or Kobayashi [7; pp. 29-31 D, and hence is finitely compact with respect to its Caratheodory distance. Since a general convex cone containing no lines is an intersection of (in general, infinitely many) such special convex cones, a general Siegel domain of the second kind S(V, F) is an intersection of domains, each of which is finitely compact with respect to its Caratheodory distance. It follows from (l.1.I 1) that S(V, F) is finitely compact with respect to its Caratheodory distance. In particular, it is complete hyperbolic. (4.1.15) Remark. In section 5, we shall give boundary estimates for both Kobayashi and Caratheodory distances of strongly pseudoconvex domains with smooth boundary and reprove, in particular, (4.1.12). However, it is not known if every bounded pseudoconvex domain with smooth boundary is complete hyperbolic. Let F be a continuous Minkowski function on C/, i.e., a continuous nonnegative function on C" such that F(z) > 0 for nonzero z E CIf and F(tz.) = ItIF(z) for z E C" and t E C. The bounded domain X F = {z. E C/; F(z) < I} is star-shaped and circular. lamicki-Pflug [9] has an example of F such that X F is pseudoconvex but not complete hyperbolic. According to Sibony (see lamicki-Ptlug [l0]), there is a bounded pseudoconvex domain, not complete hyperbolic, whose boundary is smooth (of class eX) except at one point.

2 Infinitesimal Caratheodory Metric We define the infinitesimal form of the Caratheodory pseudo-distance. Given a complex space X, we define a nonnegative function Ex on the tangent bundle T X by setting (4.2.1)

Ex(v) = sup IIf~vll f

for

VET X,

where II f* v II is the length of the tangent vector f* v of D measured by the Poincare metric ds 2 of D, and the supremum is taken over all f E Hol(X, D). This supremum is actually achieved by some f E Hol(X, D) which sends the base point rr(v) E X to 0 E D, (see the comment following the definition of ex in (3.1.1).)

2 Infinitesimal Caratheodory Metric

179

Clearly, Ex satisfies the following convexity condition: (4.2.2)

Ex(v

+ Vi) s

Ex(v)

+ Ex(v ' )

for

v,

T"X.

Vi E

We call Ex the Caratheodory pseudo-metric or Caratheodory pseudo-length; it is a pseudo-length function in the sense of Section 3 of Chapter 2. Corresponding to (3.1.2) we have (4.2.3) Proposition. (I)

{IX and Yare complex spaces, then

E y (j~ v) SEx ( v)

(2)

f

for

E Hol(X, Y)

and

VET X;

For the unit disc D with Poincare metric ds 2 , we have

E;) = ds 2 . Corresponding to (3.1.7) we have (4.2.4) Proposition. ifF is a pseudo-length function (continuous or otherwise) on X such that

Ilf.vll

s

F(v)

for

f

E Hol(X,

D)

and vET X,

then Ex S F. In particular, if Fx and Fx denote the infinitesimal forms of d x defined in (3.5.7), (3.5.14) and (3.5.\6), then (4.2.5)

Ex S Fx S Fx.

(4.2.6) Theorem. The infinitesimal Carathedory pseudo-metric Ex: T X --+ R is continuous.

Proof: Let Vk E T", X be a sequence of vectors converging to v E TpX. As we remarked above, there is a map f E Hol(X, D) such that Ex(v) = Ilf.vll and f(p) = O. Then

Also for each Uk there is a map fk E Hol(X, D) such that Ex(ud fk(pd = O. Taking a convergent subsequence, we set g = lim fie. Then

=

IIfk*vkll and

o As we remarked after (1.1.2), in constructing the inner pseudo-distance c~ from ex, we use only piecewise differentiable, ex-rectifiable curves. Then we have the following result of Reiffen [I, 2]. (As pointed out by lamicki-Ptlug [10], if all ex-rectifiable curves are used, it is not clear whether (4.2.7) is valid or not).

180

Chapter 4. Intrinsic Distances for Domains

(4.2.7) Theorem. Let X be a complex space. The inner pseudo-distance c~ induced by Cx is obtained by integration o/the Carafheodory pseudo-metric Ex. Proof Let 8 be the pseudo-distance obtained by integration of Ex; it is inner (see (2.3.1». By (4.2.1) we have

f E Hol(X, D),

for

vET X.

Integrating this inequality yields 8(p, q)

~

f E Hol(X, D),

for

p(f(p), f(q»

p, q E X.

By the extremal property of Cx (see (3.1.7», we have 15Cp, q) ~ cxCp, q)

for

p, q E X.

Since 8 is inner, we have for

p, q EX.

In order to prove the opposite inequality, we consider a curve y(t) in X. Let v be the velocity vector of yet) at t = to. Let f E HoICX. D) be such that IIf*vll = Ex(v). From 8(y(to), y(t»

~

c~(y(to), y(t» ~ cx(y(to), yet»~

~

p(f(y(to», f(y(t»).

we obtain Ex(v)

= ~

· 15(y(to), y(t» I1m

t~t"

I·

1m

I-t"

It - tol

cxCy(to), yet»~

It - tol

c~(y(to), yet)~

I'

> 1m -'-'--'-----'---

-

1-10

I'

It - tol

p(f(y(to», f(y(t»)

1m -------------> I-t" It - tal

IIf*vll = Ex(v).

Fixing f = a, let Lds) (resp. L,(s» be the arc-length of y from f = a to t = s measured by Ex or, equivalently, by 8 (resp. by Cx or, equivalently, by c~). Then

On the other hand, the length of y from t = to to t = s measured by 8 is at least as long as its length measured by c~. Hence, Le(s) - LE(to) L,(s) - LeCto) cx(y(to), yes»~ -----> > , s - to s - to S - to

Letting s

-+

to, we obtain Ex(v) = L~(to) 2: L~(to) ~ Ex(v).

2 Infinitesimal Caratheodory Metric

This establishes the equality L ~(to) all f, and 8 = c~.

= L~.(to)

for all fo. Hence, L dt)

= Le(t)

181

for 0

For a nonsingular X it can be shown (Franzoni-Vesentini [I]) that Ex is not only continuous but locally Lipschitz. (4.2.8) Theorem. For a complex manifold X, the infinitesimal Caratheodory pseudo-metric Ex is locally Lipschitz. Proof Let 0 E X and fix a local coordinate system in a neighborhood U of o. Using the coordinate system we identify the tangent space TpX at p E U with C/. Thus we write (p.~) E U X c n for v E TpU. Let Bf" be the ball of radius r around 0 with respect to the coordinate. Let p, q E Bf" /2 and 1;, " E C/. Suppose Ex(p, 1;) ~ Ex(q,I;). Then (the suprema being taken over all f E Hol(X, D) such that f(p) = 0)

Ex(p.O - Ex(q, 0

=

<

2f 21 f* (q , 1;) I sup I *(p, ~)I - sup 1 _ If(q)1 2 2

{I f

suP. *(p,

If.(q,I;)1 }

01 - 1 - If(qW 01 -If*(q, 1;)1)

<

2sup{lf*(p,

.::::

2 sup If*(p, 1;) - f*(q.

<

~)I

C

2"llp - qll . 111;11· r

The last inequality follows from Cauchy's integral formula for the second derivative of f. Similarly, using Cauchy's formula for the first derivative of f we obtain IEx(q,~)

- Ex(q, ,,)1':::: Ex(q.1; - ,,)

= sup If*(q. ~ - TJ)I .::::

C'

-II~

r

- TJII.

where the supremum is taken over all f E Hol(X, D) such that f(q) = O. Now D the theorem follows from the above two estimates. (4.2.9) Remark. We shall show that according to the definition of the curvature given by (2.3.3), Ex has holomorphic sectional curvature.:::: -1, see B. Wong [1] and Masaaki Suzuki [I]. Given a nonzero v E TxX, consider maps g E Hol(D. X) such that g(O) = x and v is tangent to g(D). We want to show that g* Ex has curvature.:::: -I for all g. Take a map f E Hol(X, D) such that f(x) = 0 and Ilf.vll = Ex(v), where I I denotes the length defined by the Poincare metric ds 2 of D. Then f* ds 2 .:::: Ei with equality at x. Then g* f* ds 2 .:::: g* Ei with equality at O. Thus, g* f*ds 2 is a supporting metric for g* Ei at O. Since g* f*ds 2 is isometric to ds 2 by fog, it has curvature -I. By (2.1.9), g* Ex has curvature .:::: -I. This is in contrast to the fact (see Remark (3.5.45» that Fx has holomorphic sectional curvature ~ -1 provided X is complete hyperbolic.

182

Chapter 4. Intrinsic Distances for Domains

Given a point P E X. the indicatrix tangent space TpX defined by

rex, p) = {v

(4.2.10)

E

rex, p)

of Ex at p is a domain in the

TpX; Ex(v) < 2}.

If we denote Holp(X. D) = {f E Hol(X, D); f(p) = OJ, then the indicatrix may be described more directly: (4.2.11)

n

rex, p) =

(v

E

TpX; If.vl < I}.

We remark that If.vl is the ordinary absolute value and hence is a half of the Poincare length II f* v II because of the normalization we adopted for the Poincare metric ds 2 in (2.1.3). That is why we defined reX,p) by Ex < 2 in (4.2.10) rather than by Ex < I; this is, of course, a matter of preference. We state simple consequences of the definition of rex. p), (Caratheodory [2]). (4.2.12) Proposition. IlX is a complex subspace ora complex space Y, then

rex. p) c reY, p) Prool By (4.2.2), we have E y

:::

Ex on X.

(4.2.13) Proposition. The indicatrix in TpX.

rex, p)

is convex. Since

for

vETX,aEC,

Ex(av) = laIEx(v)

rex, p)

0

rex, p) is always a convex circular domain

Proof. Either by (4.2.2) or by (4.2.11)

the indicatrix is circular.

p E X.

for

is invariant under multiplication by

cit,

t E R, and hence 0

Conversely, every convex circular domain in C" is the indicatrix some domain X C C" and some point p EX. In fact, we have

rex. p)

for

(4.2.14) Theorem. Let X be a star-shaped circular domain in C". Then under the natural identification ToX ~ cn, the indicatrix rex, 0) is the smallest convex circular domain containing X. In particular, if X is a convex circular domain, then rex, 0) = X. We recall that a domain X C lal ::: 1 implyap E X.

cn

is said to be star-shaped if p E X and

Proof. Let p E X and f E Holu(X, D). Define cp cp(t) = f(tp),

E

Hol(D, D) by

tED.

Then cp(O) = O. Considering p E X C en as a point in TuX under the identification of ToX with en, we have j~(p) = cp'(O). Since Icp'(O) I < 1 by Schwarz lemma, it follows from (4.2.11) that p E rex, 0). Hence, Xc rex, 0).

2 Infinitesimal Caratheodory Metric

183

In order to prove the opposite inclusion when X is convex, let q be a boundary point of X, and H the supporting real hyperplane at q. By applying a complex linear transformation to the coordinate system Zl, .... Z" of cn, we may assume that q has coordinates (1,0, ... ,0), H is the hyperplane x I = I (where z I = X I + i y I) and X lies in the half-space x I < 1. Since X is circular, it follows that, for any real e, (e iA , 0, ... ,0) is a boundary point of X with the support hyperplane x I cos

e+ i

sin e = I

+i

sine < 1.

and that X lies in the half-space xl

Since this holds for all () C q = {(Zl, ... ,Z");

E

cose

R, the domain X is contained in the circular cylinder

Izil < I}. Clearly, X =

n

Cq •

qEaX

Hence, it suffices to show that rex, 0) is contained in Cq . Let f E Hol(X, D) be the restriction to X of the projection (Zl, ... , zn) Let v=

By (4.2.11),

la II

La

i (-;)

3z

E

t-+ Zl.

reX, 0).

0

= If*vl < 1. Hence, (aI, ... , an) E Cq , showing that

rex, 0) c

Cq .

If X is not convex, let K denote the smallest convex circular domain containing X. From X C K, we obtain

Xc r(X,O) c reK,O) = K. Since r(X,O) is a convex circular domain containing X and since K is the smallest such domain, we conclude rex, 0) = K. 0 (4.2.15) Corollary. Let X be a convex circular domain in C". Then Ei is a Hermitian metric if and only if X is affinely isomorphic to a unit ball {z E en; II z II < I}. Proof Assume that E~ is Hermitian, i.e., for

v = (vi) E TpX.

Then r(x,O)

=

{(Zl, ... , Z") E

en;

Lhjk(O)zji < 2}.

By (4.2.14), X = reX, 0).

o

If X and Yare complex spaces and f E Hol(X, Y) with f(p) = q, then the differential f*: TpX -+ TqY maps rex, p) into r(Y, q). In particular, if f is biholomorphic, then f*: rex, p) -+ r(Y, q) is a linear isomorphism. On the indicatrix of the Caratheodory metric, see Tishabaev [I].

184

Chapter 4. Intrinsic Distances for Domains

3 Pseudo-distance Defined by Plurisubharmonic Functions Following Sibony [2], Klimek [1, 2] and Azukawa [2, 3], we shall use plurisubharmonic functions to define a pseudo-distance p x on each complex manifold X. Let X be a complex manifold. Given a point Xo EX, let P x (xo) denote the set of upper semicontinuous functions cp on X such that (i) 0 :S cp < I, (ii) cp(xo) = 0, (iii) log cp is plurisubharmonic, and (iv) with respect to a local coordinate system z = (ZI, ... , zn) with origin at Xo, cp/llzll is bounded in a neighborhood of Xo. Following Klimek [I] we consider the following extremal function: (4.3.1)

;.x(x, xo) = sup{cp(x): cp E Px(xo)}.

Then (4.3.2) Lemma. If f: X -+ Y is a holomorphic map between complex man[{olds and iflj; E Py(f(xo», then lj; 0 f E Px(xo) and ;.y(f(x), f(xo» :S AX(X, xo)· Proof All we have to show is that lj; 0 f satisfies condition (iv). With respect to a local coordinate system z = (z 1 •... , zf/) around Xo and a local coordinate system w = (Wi, .... w m ) around f(xo), we have loglj;(f(x» -log IIz(x) II

= log lj;(f(x»

-log IIw(f(x»

II + log

Ilw(f(x»1I , IIz(x)1I

o

which is bounded in a neighborhood of Xo.

For further properties of the extremal function, see Demailly [1] as well as Klimek [I]. Let p denote the Poincare distance on D, and d~ be the function on X x X defined in (3. J.3). If ;,x (x, xo) < I, then p(Ax (x, xo), 0) is defined. If ;'x (x , xo) = 1, then p(/'x(x, xo), 0) is defined to be 00. With this understanding, we have p(Ax(x, xo). 0) :S d~(x, xo).

(4.3.3)

To prove this, let f E Hol(D, X) be a map passing through both Xo and x. If no such map exists, then d~ (x, xo) = 00 and the claimed inequality holds trivially. If such a map f exists, by composing it with an automorphism of D we may assume that f(O) = Xo and f(a) = x with 0 :S a < I. Let cp E Px(xo), Then by (4.3.2), cp 0 f E PD(O), and by (2.1.15) cp(f(z» :S Izl for zED. In particular, cp(f(a» :S a. Since this holds for all cp E Px(xo), we have i.x(x. xo)

=

Ax(fCa), xo)

= supcp(f(a»

:S a < l.


Hence, p(AX(X, xo), 0) ::::: pea, 0). Taking the infimum over all pairs (j, a) such that f(O) = Xo and f(a) = x, we obtain (4.3.3).

3 Pseudo-distance Defined by Plurisubhannonic Functions

185

Define p~(x, x')

(4.3.4)

= max{p (I.x (x, x'), 0), p(}.X(X', x), On,

x,x' E X.

Although p~ (x. x') is nonnegative and symmetric, it may not satisfy the triangular inequality. Let Px (x, x') be the largest pseudo-distance bounded by p~ (x, x'). More explicitly, taking a chain of points x = xo. xl, ... ,xk = x', we define (Klimek [ID (4.3.5) where the infimum is taken over all chains of points from x to x'. Then Px is a pseudo-distance on X. Since d~(x, x') is symmetric in x, x', (4.3.3) implies p~(x, x') ::: d~(x. x'). Since d x is the largest pseudo-distance bounded by d~, we have Px ::: dx .

If f:

(4.3.6) Proposition. (I)

X

~

Y is a holomorphic map between two complex

manif()lds. then py(f(x), f(x'» (2)

::: px(X, x'),

X,X ' E X;

For the unit disc D. PD coincides with the Poincare distance p;

(3)

Cx ::: Px ::: d x .

This is immediate from (4.3.2). By (I) Px is invariant under biholomorphic automorphisms of X. Since D = {/zl < I} is homogeneous, it suffices to verify Pf)(z,O) = p(z,O). The function Izl belongs to Pf)(O). Hence, AD(Z, 0) ~ Izi. Conversely, let cp E p[)(O). Then by (2.1.15) (applied to cp2), we obtain cp(z) ::: Izl. Hence. ).[)(z. 0) = Izl, which implies

Proof (I)

(2)

p[)(z.O)

= P(l.D(Z. 0), 0) = p(lzl, 0) = p(z. 0).

(3) We already proved the inequality Px ::: d x . The inequality Cx < Px follows from (1) and (2) above and (3.1.7). 0 For a unit ball B

=

{z E C"; IIzlI < I}, we have (see (3.1.24» CB(Z,O)

= PB(Z, 0) = dECz, 0).

Since any two points of B can be interchanged by an automorphism of B, ;.B(Z. z') is symmetric in z and z'. From (4.3.3) and (4.3.4) we obtain p~(z, 0) ::: d~(z, 0). Since d~(z, 0) = dB(z, 0), we have CB(Z, 0) = PB(Z, 0) ::: p~(z. 0) ::: dB(z, 0).

Hence, p(lIzll,O)

which implies

= eB(Z, 0) = p~(z, 0) = P()'B(Z, 0), 0),

186

Chapter 4. Intrinsic Distances for Domains

(4.3.7)

;.B(Z,

0) = Ilzll·

Using this, we prove the following result of Klimek [1]. (4.3.8) Theorem. The extremal/unction ;.x(x, xu) belongs to Px(xo). Proof Clearly, log;.x(x. xo) is plurisubharmonic and nonpositive. By letting X = D and Y = X in (4.3.2), we see that it is strictly negative at x which can be joined to Xo by a holomorphic disc. Since a plurisubhannonic function cannot attain its maximum unless it is a constant function, ;,x (x, xu) cannot take the value I. To see that Ax(x, xo) satisfies condition (iv), consider an open ball U = (lizil < r) defined by a local coordinate system z = (z I, ... , Z") with origin at Xo. By (4.3.2), I.x(x, xo) :::: I.u(x, xo) for x E U. By (4.3.7), Au(X, xo) = IIz(x)lI/r, and

log ;.x (x, xu) - log IIz(x) II

:::: log Au(x, xo) - log IIz(x) II

= -log r.

o Following Azukawa [2, 3], we shall now define also an infinitesimal pseudometric P x . Let v E T,o X. Essentially we want to set Px(v)

= sup{lv(cp)l;

cp E Px(xo}}.

Since cp is not necessarily differentiable, taking a map f(O) = Xo and f'(0) = v, we set (4.3.9)

f

E Hol(D,

X) such that

;,x(f(t), xo) . P x () v = I Imsup . 1-->0.1 >0 t

As in Sibony [2], we may consider the subfamily Q(xo) c P(xo) consisting of those functions cp E P x (xo) which are of class C 2 in a neighborhood of Xo and define an infinitesimal pseudo-metric Qx by (4.3.10)

Qx(v) = sup{lv(cp)l;