Hyperbolic Manifolds ~olornor~hic klappinYs An Introduction (Second Edition)
Hyperbolic Manifolds Holornorphic Mappings qnt-4
a1 IU
An Introduction (Second Edition)
Shoshichi Kobayashi University of California, Berkeley, USA
\b World Scientific N E W JERSEY
. L O N D O N . SINGAPORE . BElJlNG . SHANGHAI
HONG KONG
TAIPEI
CHENNAI
I
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British 1,ibrar-y Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
The first edition was published in 1970 by Marcel Dekker.
HYPERBOLIC MANIFOLDS AND HOLOMORPHIC MAPPINGS (Second Edition) An Introduction Copyright Q 2005 by World Scientific Publishing Co. Pte. Ltd.
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Preface t o the New Edition
I introduced the intrinsic pseudodistance dx in 1967 and published the first edition of this monograph in 1970 and a survey article in the Bulletin of the American Mathematical Society (1976). In the 35 years since the appearance of the first edition, the subject of hyperbolic complex spaces has seen increasing activities. In 1973 the Mathematical Reviews created two new subsections "invariant metrics and pseudodistances" and "hyperbolic complex manifolds" within the section "analytic mappings" (which is now called "holomorphic mappings") . Since 1980 several books on intrinsic pseudodistances and related topics have appeared, each emphasizing ccrtain aspects of the theory:
T. Franzoni and E. Vcsentini, Holomorphic Maps and Invariant Distances, 1980. J. Noguchi and T. Ochiai, Geometric finction Theory zn Several Complex Variables, 1984 (English translation in 1990). S. Lang, Introduction to Complex Hyperbolic Spaces, 1987. M . Abate, Iteration Theory of Holomorphic Maps on Tmt Manifolds, 1989. S. Dinen, The Schwarz Lemma, 1989. M . Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, 1993. In addition, Encyclopaedia of Mathematical Sciences, Vol. 9 (1989), Several Complex Variables, contains the following two chapters: Chapter I11 Invariant Metrics by E. A. ~oletskirand B. V. Shabat, Chapter IV Finiteness Theorems for Holomorphic Maps by M. G. Zaidenberg and V. Ya. Lin.
A recent undergraduate level book by S. G. Krantz "Compler Analysis: The Geometn'c Vzewpoint" (1990) in the Carus Mathematical Monographs Series of the Mathematical Association of America is an elementary introduct.ion to function theory from the viewpoint of hyperbolic analysis. vii
...
VIII
Prrfuce t o the Neu: Edition
In order to give a systenlatic and comprehensive account of the theory of intrinsic pseudodista~lcesarld holoniorphic mappings, in 1998 1 published Hyperbolic Conz.plez Spuces as Volume 318 of Grundlehre~~ der ~~Iathemat~ischen Wissenschaften. However, t,he old book is not subsun~cdby this new book and continues to provide thc easiest. access to the theory, especially for students. Incorrect or no longer relevant statements haw been deleted from the first edition and, in order to bridge the chasm between the first edition and the \:olurni~~ousGrr~ncllehrenbook of more t,llan 450 pages, the post.script has been added. Shoshiclii Kobayashi Berkeley, .J,une 2005
Preface
This book is a develop~~~ent of lectures delivered in Berkeley in the academic year 1968-69. Its object is to give a coherent account of intrinsic pseudodistances on complex manifolds and of their applications to h o l e morphic mappings. The classical Schwarz-Pick lemma states that every holomorphic m a p ping from a unit disk into itself is distance-decreasing with respect t.o the Poincar6 distance. In Chapter 1: we prove Ahlfors' generalization to holomorphic mappings from a unit disk into a negatively pinched Riemann surface and prcscnt some of its applications in the geometric theory of functions. In Chapters I1 and 111: various higher-dimensional gencralizations of the Schnrarz-Pick-Ahlfors lemma arc proved. The mison 82tre of the first three chapters is to provide intcrcsting cxamples for the subsequent, chapters. It is therefore possible for the reader to start from Chapter IV and go back t,o Chapters 1, 11: and 111 only when he must. In Chapter IV, we introduce a ccrta.in pscudodistancc on cvcry con+ plex manifold in an intrinsic manner. A complcx manifold is said t.o be (completely) hyperbolic if this pseudodistance is a (complete). distance. The classical pseudodistance of Carat,hCodory and this new pseudodistance share two basic properties: (1) they agree with t.he PoincarC dist.ance on the unit disk, and (2) every holomorphic mapping is distance-decreasing. Among the pseudodistanccs with these two properties, the Carat.hBodory pseudodistance is the smallest and the new one is the largest. These pseudodistances permit 11s to obtain many results on complex manifolds by a purely metric space-t,opological method. They enable us also to gain a geometric insight into function theoretic results. Elementary properties of these Pseudodistances and of hyperbolic manifolds are given. In Chapter V, we study hololllorphic mappings of a complex manifold into a hyperbolic manifold. In Chapbcr VI, which is, to a large extent, based on hI. H. Kwack's thesis, we give generalizations of the big Picard theorem t o higher-dimensional manifolds. Although there is rrlore than one
way to interpret the big Picard theorem geometrically. we consider it as an exbension theorem for holomorphic mappings . To avoid technical complications associated with complex spaces. we consider only complex manifolds in Chapters IV. V. and VI . In Chapter VII. we indicate how some of the results in these three chapters could be generalized to complex spaces . In Chapter VIII. the relationships between hyperbolic manifolds and minimal models are studied . The generalized big Picard theorems are essentially used here . To a large extent this chapter is based on J . Zumbrunn's thesis . Closely following the constructions of the pseudodistances in Chapter IV. we define in Chapter IX two kinds of intermediate dimensional measures on a complex manifold in a intrinsic manner . These measures have been studied more thoroughly by D . Eisenman in his thesis . Our approach is perhaps a little more differential geometric . At the end of Chaptcr IX. we list a few unsolved problems on hyperbolic manifolds . In preparing my lectures on hyperbolic manifolds. I had numerous useful conversations with H Wu By solving some of the problems listed in the first draft of t.his book. P. Kiernan has helped me to make a number of improvements . I t was through Professor Chern's papers on holomorphic mappings that I was led into this topic . I wish to express my thanks to these mathematicians .
.
.
Shoshichi Kobayashi Berkeley. January 1970
CONTENTS
Preface to the New Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Chapter I Thc Schwarz Lemma and Its Gcncralizations . . . . . . . . . . . 1 1 The Schwarz-Pick Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . I 2 A Generalization by Ahlfors . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 The Gaussian Plane Minus Two Points . . . . . . . . . . . . . . . . . . . 5 4 Schottky's Thcorcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Compact Riemann Surfaces of Genus 2 2 . . . . . . . . . . . . . . . . . 11 6 Holomorphic Mappings from an Annulus into an Annulus . . . . . . . . 12 Chapter I1 Volume Elements and the Schwarz Lemma . . . . . . . . . . . . 1 Volume Elemcnt and ~IssociatcdHcrrrritiar~Forrr~ . . . . . . . . . . . . . 2 Basic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Holomorphic Mappings f : M' -+ M with Compact M' . . . . . . . . . 4 Holomorphic Mappings f : D . . + M , Where D is a Homogeneous Rounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Affinely Homogeneous Siegel Domains of Second Kind . . . . . . . . . . . 6 Symmetric Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . .
25 28 33
Chapter I l l Distance and the Schwarz Lemma . . . . . . . . . . . . . . . . 1 Hermitian Vector Bundles and Curvatures . . . . . . . . . . . . . . . . . 2 The Case Where the Domain is a Disk . . . . . . . . . . . . . . . . . . . 3 The Case Where the Domain is a Polydisk . . . . . . . . . . . . . . . . . 4 The Case Where D is a Symmetric Bounded Domain . . . . . . . . . . .
36 37 40 40 41
Chapter IV Invariant Distances on Complex Manifolds . . . . . . . . . . . 1 An Invariant Pseudodistance . . . . . . . . . . . . . . . . . . . . . . . . 2 Carathbdory Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Completeness with Respect to the Carathbdory Distance . . . . . . . . 4 Hyperbolic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 On Completeness of a n Invariant Distance . . . . . . . . . . . . . . . . .
44 45 49
16 17
19 20
52
56 63
xii
Contents
C;hapt.er V Holornorphic hlappings into Hyperbolic Manifolds . . . . . . . . I The Little Picard Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Automorphism Group of a Hyperbolic Manifold . . . . . . . . . . . 3 Holomorphic Mappings into Hyperbolic Manifol~ls . . . . . . . . . . . . Chapter VI The Big Picard Theorem and Extension of Holoniorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Invariant Distance on t,hc Pilnct,urcd Disk . . . . . . . . . . . . . . 3 Mappings from the Punctured Disk into a fIyperbolic hianifold . . . . . Holomorphic Mappings into Compact Hyperbolic Manifolds . . . . . . . 5 IIolomorphic h,lappir~gsint.o Colnplete Hyperbolic Mar~ifoltls. . . . . . . 6 Holomorphic Mappings int,o Itelatively Compact IIyperbolic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter V11 Generalization to Complex Spaces . . . . . . . . . 1 Complex Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2 Invariant Distances for Complex Spaces . . . . . . . . . . . 3 Extension of Mappings into Hyperbolic Spaces . . . . . . . 4 Normalization of Ilyperbolic C:orr~plexSpaces . . . . . . . . 5 Complcs V-h,lanifolds (Now Called Orbitfolds) . . . . . . . 6 Ir~variar~t 1)istanccs on : V j T . . . . . . . . . . . . . . . . . .
. . . ... . . . . . . . . . .
. . . . .
. . . . . . . . . . . . . .
67 67 67 70
,
-,I 77 .-
r r
78 81 84 85
1
88
Let D be t,hc ope11 unit, disk in thc complex plane C ?i.e.,
I
The Schwarz-Pick Lemma
I
93
93 95 '36 98 . . . . . . . 100 . . . . . . . 100
Let f : D -, D be a llolonlorphic mappirig such t.llat f ( 0 ) = 0. Then the classical Schwarz lerrirna st.ates
If ( z ) l 5 121
for z E
D
I
and C.:hapter VIII Hyperbolic hfanifolds and h.iinima1 h,fodcls . . . . . . . . . . 103 1 Merorrlorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2 Strong hlinimality and llir~irr~al Rludels . . . . . . . . . . . . . . . . . . 101 3 Relative hlinima1it.y . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 Chapter I S Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 1 lnvariar~thicasurts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 . . . . . . . . . . . . . . . 118 2 Intermediatr Uin~erlsior~al-111varia111 Llcas~~res 3 Unsolved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
and t.hc ccluality 1 fl(0)l = 1 or the equal it,^ I f ( s ) l = z # 0 implies f
(i) = ,FZ
with
a t a sirlgle point
1.
-
Now we shall drop the assumption f ( 0 ) = 0. If f : D D is an arbitrary holor~~orpllic mapping, we fix an arbitrarily chose11 point t f D and consider the autornorphisrns g and h of D defined by <+2
sic) = 1 + z< /I(<) =
c
1 Subject Inder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147
=
Izj
-
f
for
(2)
-m<
iE D. for
< E D.
Then the composed mappir~gF = h . f . g is a liolomorphic mapping of D into itself which sends 0 into itself. Since F ( 0 ) = 0 and F'(0) = h l ( f( 2 ) )f'(2)g1(0),we obtain
I
2
Tibe Schurarz Lemma and Its G'eneralzzut~ons
2
Hence,
A Generalization by Ahlfors
3
We shall now consider Theorem 1.1 from a differential geometric viewpoint. If we consider the Kaehler metric ds$ on D given by
1 Iff(z)l < 1 - l f ( z ) 1 * = 1 - I'
I2
then the inequality in Theorem 1.1 may be written as follows:
for z E D.
f '(ds;)
We may conclude the following:
Theorem 1.1 Let f be a h.olomorphic mapping of th.e unit disk D into itself. Then ldf l -<-
1-lfI2
=
Id'
1-1'12
for z E D,
2 5 dsD.
The metric d s i is called the Poincari metric or the Poincari-Bergman metric of D. Now, Theorem 1.1 may be stated as follows: Theorem 1.2 Let D be the open unit disk in C with the Poincari-Bergman metric d s i . Then e v e v holomorphic mapping f : D -+ D is distancedecreasing, i.e., satisfies
and the equality at a single point z implies that f is an automorphism of D . This result, which was derived from the Schwarz lemma, is act,ually equivalent to the Schwarz lemma. In fact, i f f : D + D is a holomorphic mapping such that f ( 0 ) = 0, then by setting z = 0 in the inequality above, we obtain
and the equality at a single point of D implies that f is a n automorphism of D. I f f is an autornorphism of D, t.hen the Schwarz lemma applied t o both implies that f is an isometry. We note that the Guassian curvature of the metric ds$ is equal to -4 everywhere. (In general, the Gar~ssiaricurvature of a metric 2 h d z d z is givcn by - ( l / h ) ( a 2log h/8zBZ).)
f and f
a r ~ dif 1f '(0)l = 1 , then f is all arltomorphism of D. Moreover,
II
2
Hence,
111
-'
A Generalization by Ahlfors
Let Da be the open disk of radius a in C,
1 + lf(0l< log 1 + ICI 1-ICl7
D, = { z E C ;lzl < a).
logl-lf(ol =
Then the metric
which implies 2
l+lf(Ol
2
1 - l f ( 0 1 - I = 1 - If (01= 1 - ICI 1 - ICI It, follows that If (()I S I(]. The equality If (<)I = ICI at a single point implies the equality
i
C#0
for all z lying on the line segment joining 0 and C. By Theorem 1.1, f is an automorphism of D.and hence, f ( 2 ) = E Z for some E with lei = 1. This proves that Theorem 1.1 implies the classical Schwarz lemma.
4a2dzdz ds, = A(a2 - 1 . ~ 1 ~ ) (~ A > 0 ) on D, has Gaussian curvature -A. The following theorem of Ahlfors [I] generalizes the Schwarz lemma.
Theorem 2.1 Let M be a one-dimensional Kaehler manifold with metric ds& whose Gaussian curvature is bounded above by a negative constant -B . Then every holomorphic map f : Da-+ M satisfies
I
I I I
Proof Ltlt
u
Iw tht. r~oi~nca;itivc, fr111(.1io11 011 D,, ilt.fi~lcl~i by
l r j g = ,L at 2,). Sincc. 11 at,t;~.i~~s its nlilx-i~rit~n~ at 2 cvtrq'wl~vre.
\Vc want to prove that 11 5 ,4/H on I),. Although [ L may not att.ai11 its masirnu~nin !the interior o f ) L1, in gcncral. we shall show that we hiive oniy to consider the case ivhere u attains its nlaxi~nuulill D,. I,ct r bc a positive 1111n1l)rrless t h a i ~(1. Let. zo be a11 arbitrary point of D,. Taking r suficicntly closc to a. we nlay assunlc that -0 is in Dr. \Ye derlote t)y (1s: the metric on D,. obtained from ds; by replacing u by I. hilt keeping Fro111 tile exp1ic:it expressii~l~ for d.s&. it is clear that the s;ilrlr i'i)ll~tallt-4. (d.4: j,,, (d.9;;) z , J AS r (1. If n7e tlrfirie ~~onncgativc funct,ion u , on D,. by f'(d.s.:,! = I I , . ~ R : , tlier~u , ( z o ) -+ ui;,,:) as r + o. Hcncc it suffices to I If we write f *(d8?'iI) = I1 dz dZ on D,: then 12 prove rhat 11, 5 .41B ~ I D,.. is hol~ndedon D, (tllr (:losure of D,). 011the othcs hand, the coefficierlt. ~ ~ ' l ~ -4 : f :~2j2x of (its: a~q)ro:i(.hesinfinity a t the boundary of D r . Hence,
-
-
the fi~nction11, defined on D, goes t o zcro at the boundary of Dr. In in D,.. Thc psoblc~nis thus reduced to ~)a.rtic:illitr.Ir,- attair~sits ~r~axirlllllll tlle case where 11 nttiiins its masinlum in [I,. We shall now imposc the additional assumption that u a t t a i ~ its ~ s Inaximum in 11,. say at 2~ E Do. If u ( z o j = 0. then u = 0 a11d therr is nothing . . ro provc. :\ssumc that ~ ( 2 >~0.) Then the mapping $ : D,, :\I is now dcgcncratc in a ncighhorhooci of 20 so that f is a biholornorphic n ~ a pfrom an opcn neighborhood 'C of' ,-;:I onto the open set. f (l.::~ of ill. Idcnt,if?-ing with fil!! by thc map f , wc usc the coordinate systern z of D, c C also as a local coordinate system ill f il,'). If we \\:rit.e = 2h (1: rl on f ( C ' ) . tllcn , f * ( r l ~ ; = ~ ) '2h (1: dZ on C:. If we ~vritt.d s z = 2y d z ( l i . then [L = h l g . The Caussiarl curvature k of the met.ric tls?, ' 2 l i (1.: d? is givcn by
-
['
-
1 8' log h k = . -11 il: 8.:
=
The Gaussian Plane Minus Two Points
111 view of Tlieore~n 2.1 we are naturally interested ill l i ~ ~ d i ~ Hig o11rdirllcr~sioilalIiaehler n~iinifol(lwhose G:ir~ssiiil~ (:~~rv;itlirc is h o ~ l n d ~above (I b s a ncgative const.ant. 11s wc sha.11 set later (scc Sec. .1 of Cheptcr IF'). the Gallssian plane C cannot carry such a mctxic. Thc metric
C has c:ur.vature k = -1 (1 + 1 ~ 1 ' ) ~ . ~vllichis stxictly negative e\:eryu.llert but. is not bou~ldedabove by any negat.ive constant.. If a one-dirnc~lsionalcoi11p1t.smanifold d l carries a I
This shows thal C - (0) does not admit a I<arhlcr metric \vl:hcrse Caussir~n curwt.urc is boundecl above? by a negative constant. C.:onsiclcr now the Gaussii~nplal-ICniinus t.wo points. say *1I= C {I]. 1). If we 11scthe modular function X i 2 ) . we can show that. d l carrics a coml~lcre Icael~lermct,ric with ~iegativeconstant curvature. Let H denote the upper half-plane in C . i . c . . II - { z - s + ,i.y F C ; y > 0 ) . The11 tile nludular X gives a c.overirlg projection X : Ii - C' - {(I.1). If \ye digrtw n f~~iiction little, the group of c l e c : k - t r a r ~ s f i ) r ~ ~ ~is ~ tgive11 i o r ~ s by -
2~9d:d: is giver1 by
since k < - -B 1,). our clssurnption. u.e have
3 - i.f-8 n,n: = - k h - .4y 1 Bfi - .4y. Since log u attains its n~axirniin~ at 20. rhc left-hand sidc in thc illcrluality a b o l - ~is nonpositive at r;" and so is the right-hand side. Hence. .-1/'B 2 A
it. f0110~::, thiit. .4!:'B 1 11
'
Tho C;;tl~ssinncurratrirc -.-I of rhc mcrric d s i
6' !og 11 - log h 3' log g --
3
0
Tllis group. kno\vn ah the cougruer1c.e s r ~ b g r o l ~n p~ o d2. is a norirlal suljgrijujj clon~aini-. of index 6 in tlie modu:a? group S L ' 2 : Z I . arld its fl~nd:in~cnta! given by F in Fig. I . Thr. borindnry of F and the inlagirlary axis zrc mnppcrl irlto rhc real axis by A. 111 Fig. 2. orle sees roughly hon. the iunclarrle~ititl domain F is mapped onro C - (0.1) by A. On the upper half-ylnni H . t h r
'/'he .s'chrr~nrzL ~ m r n nand 11s C;enernlzznrions
G
Proof
(a) ;tild (1)) ;trtl ilnincdiut~cfrom t.hc clcfinit.io~lof I
(d) is a cor1seqrlellc.e of jr:) and [ f 2 K ( f j+ S % ( g ) ] / ( f + Fig. I.
Fig. 2
2
. k l k 2 / ( k l- k 3 ) .
This latter irleqrlality follows from
-(f2k1 Iy 2 k 2 ) / ( f - y)" 5 -klk2 j ( k l /- k 2 ) . ~r~etric: (is3 = 2dz d z j has ~ curvature ~ ~ - A and is invariant by thc group of h o l o ~ ~ ~ o r ptra~lsforrrlat.iorls hic of II. It follo~vsimmediately that this metric irlduces a rrletric: of curvat,ure - A on C - (0, 1). Unfortunat.cly. thc illodular fur~ctiorlX is so complicated that the indr~cedmetric on C - (0, 1) cannot be expressed ill a si~riplefor111 i11 terr~lsof the natural coordinate of C - (0: 1). (For t,he deGnit.ion and basic properties of the modular fi.inction A. see for irlstarice Alilfors [2j and Ford [I].) SVe shall IIOW give a. more element,ary coilstruction of a niet.ric with Gallssiar~curvatl~re5 -3 on C - (0: 1). The constructioii is due to Graiiert function y(2: 2) defined in an opcn and Keckzicgel [I]. Givcn a positive CDC set in C ! xvc dcfinc a rcal-valued functioii K ( y ) defiuetf i11 the samc opcn sct a.s follo\vs:
The defir~itiorlis rliotivated by the fact t,hat K ( g j is the Gaussian curvature of the ~rletric.cis2 = 2y (1; (12. We first prove thc follon4ng:
\Ve set p(z: z j = 2 2 / 3 Q - 3 ( 1 ti. 1 2 ~ 1 ~ ~ ) : where n is a const,anr., 0 C - ( 0 ) . We have
1% set.
Then f is a positive C""' funct.ion on C we have
c K ( c g j = 1<(g) for all positivc nunzhcrs c: f g K ( f g )= f K i f , !+ gK(,g): i f + g ) 2 K if -t g j 2 f 2 K !f j 4 g 2 K ( g ) : I f K ( f ) 5 -kl < 0 and K ( g ) 2 - k 2 < 0.
then.
-
{O, 1). By (b) of Propositior~:j.1.
Since
K [ p ( z .4); n2 5 p ( 2 - 1.5 - 1 ) - 2 ( 1 -t 32")
Proposition 3.1 For positive furrctior~sf and g . ~ r halve : ia) ihi (c) (d)
< a 2 $. Thcn p is a positive CX' function or1
1
for O < I z 5 -
2
and 0 K [ p (-~ 1; 5 - I ) ] I - p!z. 2 ) 2(1 + 3'0j
we obtain
for 0
1
< ,- - 11 5 -. 2
3
The Schu~arzLemma and Its Generalizations
8
On the other hand, for lz - 1I
2 f , we have
The Gaussian Plane Mznus Two Points
9
We set Q(Z,Z) =
1 3'
s ( z , E ) / ~ z ~ ~ ( for ~ o0 ~<~IzJ z 5~ ~)~ for lzl
1 3'
2-
We set, Similarly, for It 1
1
a, ure have Since K [ q ( z ,z)] = -2 for 0 < Izl
It follows that if Izl
2
and lz - 11 2
< f , we obtain
i, then We set
+
If we denote a2/2(1 32a)by k' so that K ( f ) by (d) of Proposition 3.1 we have Since -4a2/(1
+ 32")35 -,'/2(1 + 32a)for 0 < a 5 a, we have finally K [ f(z,5)1 5 -
a2
2(1
+ 32")
for z E C - ( 0 , l ) .
We have shown that the ~rletricds2 = 2f ( z ,E ) dz dZ on C - { 0 , 1 ) has Gaussian curvature K ( f ) 5 -a2/2(1 32a) < 0. But this metric is not complete. Since ds = f l I d z 1 has a singularity of order Izla-' at 0, the point 0 is at a finite distance from any point of C - {0,1). Similarly, for the point 1. It is also easy t o verify that the point at infinity of C is at a finite distance from any point of C - { 0 , 1 ) with respect to the metric ds2. We shall make an adjustment to obtain a complete metric whose Gaussian curvature is bounded above by a negative constant. Choose a real-valued CbOfunction s ( z , E ) on C such that
+
05
s ( z , ~ ) = lforIzI$i 4' 1 1 ~ ( 2Z ,) 5 1 for - I IzI 5 4 3! 1 S ( ZZ, ) = 0 for 121 2 -. 3
2 -kt
on C - (0, I ) , then
2k1 2k' K[g(z,Z)]2 --2 + ck' < --2 + k1 < 0
a)
on the domain {0 < 1 . ~ 1 < u { O < 1% - 11 < $1 U {lzl > 4 ) . In the complementary set which is compact, we use the following estimate (cf. (c) of Proposition 3.1):
K ( g ) = K [ c ( f+ h ) + (1 - c ) f ]
+( f + chI2
I- c 2 ( f + h ) l K [ c ( f h)]+ (1 - c )f 2~K [ ( 1- C ) f ]
- c ( f + h)'K(f I- c ( f
+ h ) + (1 - C )f 2 K ( f )
( f + chI2
+ h ) ' ~( (f ++f hchI2 ) - (1 -c)f2k'
If we take a sufficiently small c, then K ( g ) is bounded above by a negative constant on the compact set we are considering. We have thus constructed a complete metric ds2 = 29 dz dZ on C - (0: 1 ) whose Gaussian curvature K ( g ) is bounded above by a negative constant, say k. Multiplying the metric by a suitable constant. we may assume that k = -4.
The Schwurz Lemma and Its Cenemlirations
10
4
Schottky's Theorem
One of the best known consequer~cesof the generalized Schwarz lemma is the following theorern of Schottky [I]. Theorem 4.1 Given a complex number a with a # O,1 and a real number. r with 0 5 r < 1, there exists a positive number S = S(a,r) ~ ~ l i tthe h following property: Iff is a holomorphic map from the open unit disk D = {z E C; Izl < 1) into C - {0,1) such that f (0) = a , then If (t)l5 S(a, r) for 121 5 r . Proof Lct ds& denote the Poincare metric on D with curvature -4. We set M = C - {0,1) and let d s k be a complete metric with curvature 5 -4; such a metric was constructed in the preceding section. If wc set r' = log(1 r ) / ( l - r), then the set {z E C ;Izl < r) coincides with the set of points in D whose distances from 0 with respect to ds: do not exceed r'. Let N(a; r') be the set of points in M which are at distance r' or less from a with respect to the metric d s b . Since the metric d s k is complete, N ( a ; r f ) is compact and hence is contained in the set { t E C ;Izl 5 S ) for a suitable positive number S. Let f : D -, M be a holomorphic map such that f(0) = a. Since f is distance-decreasing, i.e., f*(ds?!) 5 dsk by Theorem 2.1, it maps the set {z E D ;lzl 2 r ) into N(a,rJ).
4
+
5
Compact Rtemann Surfaces of Genus
22
11
~f la1 < 3, then it suffices to replace log la1 by log 3. On the other hand, the distance from 0 E D to z E D with respect to ds5 is equal to 1 1+121 - log 2 1-121' = C - {0,1) be a holomorphic map. Since it is distanceLet f : D 4 decreasing, we have 1 lflzl A logIf(z)l < -log -log logIf(0)l 2 1 - IzIt ~rovidedthat If(0)l 2 3 and If(z)l 2 3. If lf(0)I < 3 and If(z)l 1 3, it suffices to replace log If (0)I by log 3. In general, we obtain
JZ
where B = Maxilog 3, log If (0)1) and C = &/A. If we wish to estimate the constant C, we have to write down the function s(z, 2 ) in Sec. 3 explicitly. Similar estimates on log f(z) have been obtained by Ostrowski [I], Pfluger [I], and Ahlfors [I]. See also Heins [I]. 5
Compact Riemann Surfaces of Genus
22
In this section we prove the following statement: Since the construction of the metric d s b in the preceding section is fairly explicit, we can give an estimate on $(a, r). Let dsL = 2gdzd.z be the complete metric with curvature 5 -4 constructed on C - {0,1) in the preceding section. From our construction, we see that. g > A2h on C- (0, 1) for a suitable positive number A, where h was defined in Sec. 3. From the definition of h, we see that, in the domain {z E C; 121 2 31, h coincides with 1 IzI2(log1 ~ 1 ' ~ ) ~ If we use the polar coordinate system (r, 0), then we have
Let a and b be two points of C such that 3 5 la1 5 Ibl. Then the distance from a to b with respect to d s L is greater than lbl
Adr
A log log lbl --
Theorem 5.1 Every compact Riemann surface M of genus g 2 2 admits a Kaehler metric whose Gaussian curvature is bounded above by a negative constant.
P~~oof If we make use of the classical result that every compact Riemann surface M of genus g 2 2 is a quotient of the upper hdf-plane H by a discontinuous group r of linear fractional transformations acting freely on H,we see easily that M admits a Kaehler metric of constant negative curvature. We shall give here a more elementary proof following Grauert and Reckziegel [I]. Since the genus of M is g, there are g linearly independent holomorphic 1-forms wl,. . . ,w, on M. We set ds2 = ~ 1 3 + 1 wz&. In terms of a local coordinate system z of M I we may write
+
ds2 = (fifi f 2 f ~ )dz dZ, where w, = f i dz. If p l , .. . ,pk is the set of common zeros of w l and w2, then ds2 is a Kaehler metric M - {pl, . . . ,pk). By a simple calculation we see that the Gaussian
The Schwarz Lemma and Its Generalzzations
12
= -2lflf2 - flf;I2/(flfl
+ f2J2I3.
Since a1 and w:! are lincarlp independent, it follows that f 2 f 0 and f l /f 2 f constant, and consequently K $ 0. Hence there are a t most finitely many points pk+l.. . . ,p,,, in M - {pl, . . . ,pk) where the curvature K vanishes. For each pi, choose a neighborhood V, in such a way that V , n V, = 8 for i # j . In each V , we shall modify the metric so that the Gaussian curvature beco~r~es negative everywhcre on M. We fix i, i = 1> .. . ,m. Choosing a local coordinate system z in V, such that z(pi) = 0, let r be a positive nurnber and
B = ( 2 ; 121 < r ) K: B' = {z; 121 < r / 2 ) . Let a(z, 2 ) be a Cx fiinct.ion on & such that
-
0 <= a(z,Z) 5 1, alB' Lct c be a constant., 0 < c g = f +ch,
1: a ( ( x- B )
= 0.
< 1: and set
where f = f i f l
+ f2fi
and h = a . ( l + z Z ) .
Then the metric y dz dZ coincides with ds2 on V,- B. Since the curvature of ds2 is bounded above by a negative const.ant.on the compact set. B - B', the nict,ric g d z dz has a strictly negat.ive curvature on B - 13' if c is sufficicntly small. From the estirliate oli K ( g ) givcn at t.he end of Sec. 3, we see that t,he curvature of y dzdz is strictly ncgative in B' if c is srifficiently small.
Remark It is known (see, for inst,ance, Springer [I, p. 2701) t,hat. if the gcnus g 2 1, there is no point of Jrl where all holomorphic 1-forms vanish. It is therefore possible to choose wl and w2 in the foregoing proof in such a way that id1S1+ u2Z2 is positive definite e ~ e r j v h e r eon iff. But this does not simplif?: the proof.
6
Holomorphic Mappings from an Annulus into an Annulus
Let A be the annulus in C defined by
A
Holomoqvl~icMappings from an Annulus into an Annullis
13
Let A1 and A2 be two annuli with moduli M Iand M2, respectively:
curvature K is given by
K
6
=
{z E C; 0 < r
< 121< R}.
The nuniber Af = log(R/r) is called the modulus of A. Two annuli with the same modulus are clearly biholomorphic to each other under a suitable homothetic transformation. We may therefore assume that r R = 1.
< l / r k ) , k = 1,2.
Ak = {Z E C ; 0 < r k <
Let f : A1 -t A2 be a holornorphic mapping arid f, : nl ( A l ) -+ n l (A2) the induced homomorphism on the fundamental groups of Al and A2. Let cwk be the generator of n ~ ( A k = ) Z , k = 1,2. Then the degree o f f , denoted by deg f , is defined by f*(al) = (degf)w. For each integer m such that 17nl 2 LV12/11,41,wc define a holomorphic mapping fm : A1 + A2 with deg f = m as follows:
f,(z)
= zm
for z E A1.
From topology we know that any mapping f
: A1
A2 (holomorphic or not) of degree 7n with lnzl 5 II.I2/hIl is homotopic t o f,. In fact, ariy two mappings A1 into A2 with the same degree are homotopic to each other. We know also that for any integer m there is a mapping (not necessarily holomorphic) of A, into A2 with degree m. But we have -+
Theorem 6.1 Let Al a.nd A2 be annuli with moduli 11.11 and .$I2 us above. If f : A1 -, A:! is a holomorphic mapping, then ldeg f 1 2 M2/MI. If M2/M1 is an integer, then. a h,olomorph.ic mapping f : Ar -+ A2 with, degree 7n = f:\.lz/dfl coincides w i t h the nzappirtg f , , up to a rotntiotl.. Proof We consider the band B = {z E C ;-6 < Im z < b ) of width 2h. We know that it is conformally equivalent t o the open unit disk or equivalently t o the upper half-plane H = {w E C; 1m.u~> 0) in C . Indeed, the mapping -z E B -+ ie"z/2b E H is a biholomorphic mapping. The invariant metric ds; of curvature -1 on H is given by du! d,w d s i = - wherc w = u. + Z.U. v2 If we induce this metric to the band B by the holomorphic mapping given above, we obtain the following invariant metric dsi of curvature -1: 7r2 dz dE where z = x i y . 4b2 cos2(7ry/ b ) We consider now a holomorphic mapping p from B onto thc annulus A = {w E C; r < Iwl < l l r ) , r = e-2"b, defined by
+
ds; =
p(z) = e2niz
t
E B.
Then p : B A is a covering projection. We denote by dsi the metric on A induced by ds;. Then the rectangle ( 2 E B; 0 5 x 5 1) is a fundamental
The Schwarz L e m m and Ils Genemlizations
14
6 H o l o m o ~ h l cilfupptnys from nn Annulus znto an Annulus
15
domain for this covering space (B,Alp). The projection p maps the upper Assume that Ah/All is an integer and Iml = h12/A;ll. The11 .tf12(t)has edge, the lower edge, and the two vertical edges of this rectangle onto the arc-length lm~/2bzl= 7i/2b1. Hence, t,hc closed curve w2(t) coiiicides wit,ll inner boundary, the outer boundary of the annulus A, and tohesegment e 2 m ~ i t0, 5 t 5 1, up to a parametrization. Since f is distance-decreasing {w = u iv E A; u > 0, v = O), respectively. It is also clear that p maps and ui2(t) has arc-length ~ 1 2 6 1f, iriaps t u ~ ( tonto ) w2(t) isomctrically. This { z = x + i y ~ B ; O ~ x ~ 1 , y = O ) o n t o t h e u n i t c i r c l e { w ~ A ; ~ w ~ = 1 implies ), that. zua(t) coincides actually with epmTitup to a rotat.ion. It follows which is the generator of the fundamental group r l ( A ) . Consider a curve that f and f m coincide on the unit circle wl(t) of A1 up to a rotation and w(t),O 5 t 5 1, in A which represents the generator of 7rl(A). We may hence they coincide of A1 up t o a rotation. assume that w(0) = w(1) > 0. To compute its length with respect to d s i , we consider a curve z(t) in the rectangle {z E B;O 5 x 5 1) such that Corollary 6.2 Let f be a holomorphic mapping from a anntilus -4 = p[z(t)] = w(t), Re[z(O)] = 0 and Re[z(l)] = 1, and compute the length of { Z E C; r < IzI < 1/r) into itself. Then ezther f is h.omotopic to a constant z(t) with respect to d s i . From the expression of d s i given above, we see map or is of the form f ( 2 ) = e*2Ti(Z+a),tuh,ere a is a real number. 1 = 1, i.e., that the curve w(t) has the least length when it is the circle I~u(t) Theorenl 6.1 is due to Huber [I]. The proof presentcd here is perhaps when z(t) is real for all t. This least length is given by a little more differential gcometric. In connection wit,h the result of this section, see also Schiffcr [I].Jenkins [I], and Landau and Osserman [l, 21.
+
We have shown that the circle w(t) = e2"",0 5 t 5 1, represents the generator a of .rrl(A) and has arc-length 7r/2b with respect to d s i and that any closed curve in A representing a has arc-length 2 n/2b, where the equality holds only when the closed curve coincides with the unit circle up to a parametrization. Similarlv, the curve w(t) = c ~ ~ " ~ <~ t, <-O 1, representing m a E nl(A) has arc-length Jmn/2b1 and any closed curve representing m a has arc-length > lm7r/2bl unless it coincides with w(t) = e 2 m ~ t tup to a parametrization. Given two annuli A1 and Ap, we consider the bands B1 and B:! of widths bl and bz and covering projections pl : B1 + A1 and p2 : B2 + A2 as above. Let f : Al + A2 be a holomorphic mapping. Since B1 is holomorphically equivalent to the unit open disk, we may apply SchwarzPick-Ahlfors lemma (Theorem 2.1) to the map f o pl : B1 + Ap and see that f o pl is distance-decreasing, i.e., (f o pl)*dsi2 5 d s i , . Since pl is a local isometry, we see that f itself is also distance-decreasing, i.e.. f *d.r;* 5 ds;, . We consider the unit circle wl (t) = epn", 0 2 t 2 1, in A1 and its image curve w2(t) = f [u!l (t)], 0 5 t 2 1, in A2. Since wl (t) has arclength 7r/2bl and f is distance-decreasing, wz(t) has arc-length $ 7r/2bl. If deg f = m so that u12(t)represents m a 2 E nl(Ap), where a 2 is the generator of n1(A2),then w2(t) has arc-length 1 lmn/2bp1. Hence, Imn/2b2l 2 n/2bl. that is, Jml 5 b2/bl. Since r k = eCZTbkand h!fk = log(l/r:) = -2logrk for k = 1,2, we have b2/bl = hl2lMl. Hence. [ntl 5 hf2/M1, which proves the first assertion of the theorem.
C H A P T E R I1
Volume Elements and the Schwarz Lemma
1
Volume Element and Associated H e r m i t i a n F o r m
Let M be a complex manifold of complex dimension n and v~ a \rolunlc! element of M. By definition U M is a 2n-form which is positive in the following sense. In terms of a local coordinate system z l , . . . , tn of M we may write E M = inK dzl A dzl A
.
A dzn A dZn,
where K is a positive function. To this volume element vk1 we associate the ~ by 2-form i p defined
where
R,,
=
-a2 log K / B Z8.9, ~
and also the Hermitian form h~ defined by
For the reason given below, we shall call hM the Ricci tensor of IC3 (with respect t o the volume element VA*). E x a m p l e 1 Let M be an n-dimensional Kaehler manifold with metric d s L = C 2gagdta dzP. Thc volume element v~ of M is given by
en, = i n ~ d z A' d ~ Al . . . A d t n A d f n . where
G = det(gaD). Then the Kcci tensor (or rather its components) is given by (sce, for instance, Kobayaslii and Nomizu [I, Vol. 111) R,$
=
-a2 log
G / ~ z 8^ 9 .
18
2
L'olutne Elements nnd the Schu~nrrLemma
Basic Formula
19
What, we have said so far is true also for a Herrnitian n~anifold!provided t,hat, urc usc the Hermitian cor~nection.(For details, see Sec. 1 of Chaptcr 111.)
The closed 2-form ( 1 / 4 n ) ( p ~is known to represent the first Chern class ,l(M) of M (see Chern [2] and Kobayashi and Nornizu [I, Vol. 21).
Example 2 Let :CI bc an 11-dinicnsionnl complcx nlanifold and H the Hilbert space of holomorphic ?a-formsy such that.
2
LI
inZpA
< 33,
The inner product in H is defined by (9.Q) =
/
i n Z p ~ $for p,Q E H.
nf
Let 90:i p 1 , 9 2 , . . . be an orthonormal basis for H (provided that H is nontrivial). Then 2n-form W A . ~defined by
is independent. of t.he choice of an orthonormal basis. If for every point x of Af there is an element cp E H which does 11ot vanish at x, then 2 ! , ~ is a volume element of M and is called the Bergman kernel form. Thc Hermitian form hA., associated with VM is easily seen to be negative semie -hhr is called the Bergman m.etric of definite. If hxf is ~ ~ e g a t i vdefinite, M .Since the Bergman met.ric -h.,q.f is a Kachlcr mctric, it gives rise to a volume element in the usual sense, which will be denoted by v h . We shall denote by h h the Hermitian form associated with v k . ROIII these construct.ions we see that V M , hh1, v k , and h h are all invariant by t,he group of holomorphic transformations of M . If M is homogeneous, i.e., if the group of holomorphic transformations is transitive on h4, then u;M = c v , ~where , c is a constant. It. follows that if M is homogeneous, then h h = hhf. In other words, if g,j and Raj denote the components of the Bergman metric and of its Ricci tensor, the homogeneity of M implies Ras = - g a ~ .If M is a bounded domain in C n with Euclidean coordinate systern z ' , . . . , zn, then H may be identified in a natural manner with the space of square integrable holomorphic functions on M with respect to the ' . .Adzn Adzn. For a bounded domain M of Lebesgue measure indz' ~ d zA. C n , VM is always positive and -hM is always positive-definite. If we write VM = inK(z, E ) dz' A d ~ A' . . . A dzn A dzn, we call K ( z , z) the Bergm.an kernel fvnction of the domain M. For more details, see Kobayashi [l]and Lichnerowicz [l] as well as Bergman [I].For homogeneous M , see Koszul [I],Hano and Kobayashi [I]. and Pyatetzki-Shapiro [I].
Basic Formula
Let M and M' be two complex manifolds of dimension n with volume elements VM and V M I , respectively. Let h~ and hMl be the Ricci tensors rnociated t o VM and V M ~respectively. , Let f : M' + M be a holomorphic mapping and define the ratio u = f *(VM)/vM' by f * ( v ~ = ) u .VM'. Then u is a nonnegative function on Ad'. At a point where u is positive, i.e., a point where f is locally biholomorphic, we want t o calculate the complex Hessian of logu. Let p E M' be such a point and choose a local coordinate system z l , . . . ,zn in a neighborhood of p and a local coordinate system wl,. . . ,wn in a neighborhood of f (p) in such a way that the map f is giver! by wi = z', i = 1,.. .,n. We write
Then
and
a2log u dt" dzB = hMl - f * h M We have proved (Kobayashi [2]).
Theorem 2.1 Let M and M' be complex manifolds of dimension. 71 with , volume elements vnr and VMJ and Ricci tensors hh, and h ~ p respectively. Let f : M' + M be a holomorphic mapping and let u = f *(VM)/VMJ. Then at a point of M' where u is dzflerent from zero, the following formula holds: dz" d,-O = hM' - f *h,,,1: a,P
e l , . . . ,zn is a local coordinate system at the point.
In general, for a Hermitian form h on a complex vector space V we denote by dim+ h (respectively, dim- h) the di~nensionof a maximal dimensional complex subspace of V on which h is positive (negative) definite.
20
Volume Elements and the Schu~arzLemma
Hence h is positive (negative) definite on Cf if and only if dim+ h = dim C' (di111- IL = dim V ) . P r o p o s i t i o n 2.2 Let f : h9' Then wri have
4
A*I?h M ,hhil, and u be as in Th.eorem 2.1.
( 1 ) If dimf hthp > dim+ hnl or dim- hhp < dim- h M everywhere, t h ~ : function .u attains no n.0nrel.a local maximum on MI; (2) If dimt h h f , < dimt h M or dim- hMl > dim- hA,f everywhere, th.e fun.ction v attains no nonzero local minimum on AAfl. P r o o f Assume that u attains a nonzero local maximum at p 6 l\l'. Then the Hermitian matrix log u/i3za is negative semidefinite a t p. By Theorem 2.1, hnt, - f ' h ~is negative semidefinite a t p. Since u ( p ) f 0, f * is nondcgenerate a t p so that dim+ f *hn, ( p ) = dim' h,t4 [ f ( p ) and dim- f * h n f ( p = ) dim- hhr [ f ( p ) ] .Hence,
(a2
aiB)
dirI1' jiM! ( p ) 5 dimt h,bj [ f( p ) ] , dim- h n . (~P ) 2 dim- hl~l[f( p ) ] . This proves ( 1 ) . The proof for ( 2 ) is similar. manifold with Hermitian metric ds2 = R e m a r k If :Id' is a Her~~lit,ian 2 C Y , , ~dzn d ~ ; 'then , thc complex Laplacian U(1ogu.) of log u is defined by
From Theorem 2.1 we obtain
k t f : M'
Holomorphic Mappings f : M i
-+
M be a ilolomorphic mapping. Assume that hi' is compact.
-+
M with Compact
M'
3
Then (1) If dim+ ~ M > J dim' 1 1 or ~ dim- h M , < dim- hnf everywhere, the degenerate, i.e., f*vM = 0; mapping f is even~wh~ere (2) If dim+ hn4) < dim+ hn4 or dim- hnp > dim- hhf everywhere, th,e mapping f is degenerate at some point of M', i.e., f8vM variishes at some point of M'. p r o o f (1) Assuming the contrary, let p bc a point of hl' where u .= f * v M / vattains ~ ~ its maximum. Then u ( p ) # 0. Now, our assertion follows from (1) of Proposition 2.2. (2) Similarly, ( 2 ) follows from (2) of Proposition 2.2. Corollary 3.2 Let f : Al'
-+
M be 0,s in Theorem 3.1. Then
(1) If the Ricci tensor h , ~ is # ezleryu~herepositive definite on M' and if the Ricci tensor h M is nowhere positive definite on M , the mapping f is degenerate everywhere on M ' ; (2) If h~ is everywhere neyative defin.ite on Af and if hhfl is nourlz,ere negative definite on M', f is degenerate everywhere on MI; (3) If hn, is e ~ ~ e r y w hpusiti7.re m definite on A4 u71.d if h , ~ is , nowhere positive definite on M', f is degen.erate somewhere on Af': (4) If hwl is everywhere negative dejin.ite on M' an,d if itna is nowhere negative definite on M . f is degenerate somewhere on M'. Let p~ be the 2-form associated to I ~ A I (see Sec. 1). Let exterior power and define a function rnr by
where R.k,f,is the scalar curvat,ure of M' and Trace ( f * h I wdenotes ) the trace of f * h M with respect t o bhe metric ds2 of h9'. This formula was proved b!' Chern 121. If M' is a Kaehler manifold, then O(1og u ) = ! j ~ ( l uo) ~! where A is the ordinary Laplacian.
3
H o l o m o r p h i c M a p p i n g s f : M'
-+
\Ve shall apply the results of See. 2 to the case where M' is a conlplex marlifold.
CO~PXC'
T h e o r e m 3.1 Let A.1 and 1M' he n-dimensional complez rnarzifold~ uith volum,e elements and v.,p and Ricci tensor h~ and hhlt, r e s p e c t i ~ e i ~ tl,tf
vzf be its ntll
1
pJ;, = rnfZ..tf. that, locally,
where uM = 2°K d r l A d i l A - . . A din A dzn and y,, = 2i For M' we define r ~ inj the same way.
M w i t h C o m p a c t M'
21
RaJdt"
A dP.
Theorem 3-3 k t M and M' he a-dimensional coniple~manifolds with Ohnnc
(a)
R*
VM
and
E I ; ~respectit~e1,ely. A ssuoie
t ensors h~ and h,\.p are negutive definite;
22
Volume Elements and
the Schwarz Lemma
3
(b) r~ ( p ) / r M '(p') 2 1 for p E M and p' E M'; (c) M' is compact.
Then every holomorphic mapping f : M' the sense that ~ * V M / U M 5 I 1.
+M
Since ~
2
Holomorphic Mappings f: M'
* V M V M ~by
Proof We set u = f * v h { / ~ h fas f before. We have
where the last inequality is a consequence of (b). Let p' E M' be a point where u attains its maximum (which may be assumed to be nonzero). To prove that u 5 1, it suffices to show that f*cph/cph, 5 1 at But this follows from the inequalities:
M with Compact M'
23
Theorem 3.3, we obtain deg f
is volume-decreasing in
-+
2 V(M1)/V(M).
Since f is holomorphic, f * v ~is nonnegative, i.e., f * ~ ~ . ~ / 2 V ~0,f land deg f 2 0. Moreover, deg f = 0 only when f * v M = 0, i.e., only when f is degenerate everywhere on M'. Since deg f is always an integer, we may conclude that degf is a nonnegative integer such that deg f 2 V ( M 1 ) / V ( M ) , and deg f = 0 if and only if f is degenerate everywhere on MI. If V ( M i ) < V ( M ) , then degf = 0. This proves (1). If V ( M ' ) = v ( M ) , then f * v 5~V M ~and
Assume that f is not everywhere degenerate. Then
h ~ 5l f * h~ < 0 at. p', where the first inequality is a consequence of Theorem 2.1 and the second inequality follows from (a).
Being an everywhere nondegenerate mapping of a compact manifold into another compact manifold, f : M' -t M is a covering projection. Since deg f = 1,it follows that f is a biholomorphic mapping.
Corollary 3.4 Let M and M' be n,-dimensional Hennitinn Einstein maniCorollary 3.6 Let M be a compact Hemitian Einstein manifold with negfolds with metric dsil and d s L , such that h~ = -ds& (i.e., Re8 = -gas) ative definite Ricci tensor (i.e., RaD= -cy,y 111ithc > 0). Every holomorand h ~ =,-dsil,. If M' is compact, every holomorphic mapping f : M' ,%.I phic mapping f : M --+ A4 is either degenerate everywhere or biholomorphic is volume-decreasing. and isometric. -+
Theorem 3.5 In Theorem 3.3 or in Corollary 3.4, assume furth.er th.at ,%I is also compact. Let V ( M ) a7~dV ( M 1 )denote the total volumes of Man.d M', respectively:
Proof By Theorem 3.5, iff is nondegenerate somewhere, then f is biholomorphic and volume-preserving, If f is volume-preserving. it preserves the Ncci tensor h M and also the metric -(l/c)hhf. The proof of Theorem 3.5 gives also the following:
( 1 ) If V ( M 1 )< V ( M ) , then every holomorphic mapping f : M' A4 is degenerate everywhere on M'; (2) If V(1CI1)= V ( M ) , then every holomorphic mapping f : M' -+ A.1 is either degenerate everywhere on 121' or a volume-preserving biholomorphic mapping. -+
Proof The topological degree deg f of f can be given by 1
deg f = -
Corollary 3.7 Let M and M' be compact Hemitian E~nsteinmanifolds such that h~ = -cdsi, and h M t = - c d s a , with c > 0. Then, for every blomorphic mapping f : M' M , we have -+
deg f
5 V(M1)/V(M).
Remmk In certain cases. the total volumes V ( M ) and V ( M 1 )may be in terms of topological invariants of M and M'. Assume hn, = -cds$ and hMr = - c d s i f , as in Corollary 3.7. Since the first Chcrn classes 'l(M) and c l ( M 1 )are represented by ( 1 / 4 ? r ) i p ~and (1/4r)p,., we see
24
4
Volume Elements and the Schu~arzLemma
.
are represented by (-l)nav,il and ( - l ) " o v ~ , , t,hat cl(.V1)71and cl respectively, where u is a positive constant. Hence, the ratio of the volumes may be expressed as t,he ratio of certain Chcrn numbers, namcly,
Another case that niay be of interest is the case where both h1 and Al' are covered by the same universal covering manifold which is a homogeneous Hermitian manifold. Since M and MI are locally homogeneous and isometric, the integrands in the Gauss-Bonnet formulas for A1 and M 1are thc same constant. Hence,
where x denotes t.he Euler-PoiricarB characteristic. If we apply the last. remark to Rie~riarinsurfaces of genus may conclude the following:
2 2, then \r;p
> genus of Aft 2 2 ,
the71 etteiy h,olomorph.ic mapping f : :\.I1 a biholomo~rphicmapping.
-+
M is either a ~07tsta7itmap
UT
This corollary niay be derived easily from the nonintegrated form of the second main thcorem of h'eva~llinnatheory. -4s a matter of fact, Nevanlinna theory implies not only the iriequality of Corollary 3.7. deg f
5 x(hlJ)/x(M)
for compact Riemann surfaces M and 111'. but also the equality deg f
=
x(N1)/X(.lI)
25
line bundle IC admits nontrivial holomorphic cross sections. its Then every holomorphic mapping f of M into itself which is n0ndegenerat.c ,t some point is a covering projection (see Peters [ I ] ) . This result of Peters is related to Corollary 3.6 as follows. If M is a compact Hermitian manifold with negative definite Ricci tensor [or with negative first. Chern class C ~ ( ~ I , ~ ) ] ~ then the line bundle Kn'for a suitable positive integer m adnlit,s sufficieritly many holomorphic sections to define an imbedding of h l into a projectilre space (Kodaira [I]). It should be noted also that if H 1 ( M ;R) = 0 and cl(M) = 0, then K itself admits a holomorphic scctioli which vanishes nowhere, i.e., K is a trivial bundle. Other related results of Peters will be discussed in Chapter VIII. 4
Holomorphic Mappings f : D -, M, Where D is a Homogeneous Bounded Domain
Although we have a homogeneous bounded domain D in mind, we shall treat D as an abstract complex rnanifold sat,isfying certain conditioris,
Corollary 3.8 ( 1 ) If M n7ld ,%I1 are cornpact Riemann surfaces such thu t genus of M
Holoniorphzc Mappings f : D -+ M
+ n1(M1)/x(.41),
where nl(.lll) is a certain nonnegative integer callcd the stationary index o f f . For details, see MTu [l,Corollaries 3.2 and 3.31. A generalization of thf equality above to higher dimension is not known. We conclude this section by a remark related t o Corollary 3.6. Let Jf be a compact complex manifold such that a suitable positive power Kn' of
Lemma 4.1 Let D and 124 be PL-dimensionalcomplex munifolds with volume elements zlo and t ! respecti~~ely. ~ Let f : D -, M be a ho1omorph.i~ mapping. Assume (a) The Ricci tens0.r~hu and h.4.f are negative definite; (b) ~ M ( Q ) / T D ( ~2) 1 for p E D a.nd q € h l ; (c) The function u = f*t1nr/vl;, tends to zero at the boundary of D in the sense that for every positive ,number a the set { p E D; u ( p ) 2 a ) is compact.
Then f is uolurne-deci.eusing? i.e., u 5 1. See Sec. 3 for t.he definitioll of rA, and rD.
Pmof In the proof of Thcorexn 3.3, the compactness of A l l (which corresponds to D here) was needed only to ensure the existence of a point where U attains its maximurn. Since (c) guarantees the existence of such a point, Lemma 4.1 follows frorn the proof of Theorem 3.3. 4.2 In Lemmu 1.1, f is volume-decreasing if(c) is replaced by the folloun'ng assumptions:
(c') thew exists a sequence
such that
D l c D2 c - .. c D of open submanifolds
Volume Elements and the
26
4 Holomorphic Mappzngs f : D
Schu~arzLemma
(cI.1) Uk D k = D ; (c1.2) each Dk c a k e s a volume element vr, for which (Dk, M , f IDk) sattsfies (a)?(b), and (c) of Lemma 4.1 and moreover lim vk = V D
(pdntulise on D ) .
+
M
27
proof For each 0 < a < 1, the mapping g, : D -+ D, defined by g,(z) = a t is 8 biholomorphic mapping. Hence, g:(b,) = bD. If we write b~ = i n K ( z , 2) d z l A d ~ A'
A dzn A dzn
and
k-rn
Proof Applying Lemma 4.1 to f : Dk
--
AI, wc obtain
then the invariance g;(b,) = bD implies K ( z , Z ) = a2n~ , ( a z U, Z ) for z E D.
It is now clear that
Hence:
b, = b ~ If. we write
f * v M / t l D = lim f * ~ , \ f / v k 5 1. k-a,
We shall now apply Lemma 4.2 to the case where D is a bounded donlair1 in Cn. With respect to the natural coordinate system z l , . . . ,zn in Cn, the Bergman kernel form bD can be written as (see Example 2 in Sec. 1 )
then the function L ( z , Z ) is continuous on D and hence is bounded on the closure 6, c D for each a. Since K(z,f) goes to infinity at the boundary of D by assumption, so does K,(z. 2) at the boundary of D,. Hence, L ( t , ~ ) l K , ( z ,Z) tends to zero at the boundary of Da. Since f * v M / b , = L/Ka, this completes the proof.
where K ( 2 . 5 ) is the Berglnan kernel function of the domain D.
Theorem 4.4 Let D be a bounded domain in C n with the Bergman kernel form b~ and the ~ o l u ~ clement ne vu defined by the Bergmnn metnc. Assume
Lemma 4.3 Let D be a bounded domain i n Cn with the Bergman kcrnel fonn bD. Assume that (i) the Bergman kernel fi~nctionK(z, 2 ) of D tends to infinity at the boundary of D ; (ii) D is starl~kezn the sense that, for each O 5 a < 1, theeset
D, = { a z E Cn: z E D) (z belng considered a s a
zV?~OT)
zs contuzned i n D. If b, denotes the Bergman kernel form of D, for 0 < a < 1, then
(i) the Bergmnn kernel jhnction K of D goes to infinity at the boundary of D ; (ii) D is starlzke; @) the mtio v ~ / zsb a ~constant function o n D .
Let M be an n-dimensional Hermztian manifold with metric d s L = 2C9aB dwa d u o and volume element v~ . Assume (iv) f i e Ricci tensor h w = 2 1Rap d w a d d is negative defizte and the Wociated 2-form P ~ = I 22 C Rad dwC)A d8' satisfies (-l)npG/2nz;p,4 i.e.,
lim ba = bD;
(-1)" det(Rad)/det(gaa)2 1.
a-1
and for uny complex m.anzfold A f o f dzmenszon n w t h volume element ['.II and for any holomorphic mappzng f : D -+ Af, the function f*uhf/ba 0" D , tends to zero at the bound an^ of D,.
2 1,
every
hobrnorphic mapping f : D
2
-
~ * ~ M / V D1.
M is volume-decreasing, t.e.,
Volume
28
Elements and
5 Afinely tlomogenwus Siege1 Domains of Sewnd Kind
the Schwarz Lemma
Proof Lct d s 5 be the Bergman metric of D and hD its Ricci tensor. By (iiij hD is the Ricci tensor of the volume element bD as well as that of VD and as in Sec. 3, then we have hence is equal to -ds$. If we definc
(3) F(u, u) E (the closure of V) for u (4) F(u, u) = 0 only when u = 0.
29
E Cm,
The subset S of Cn+" defined by
S = {(z,u) E Cn x Cm;Imz - F ( u , u ) E V ) Since ( - l ) * ~ -2~ 1~ by (iv), we have rM(q)/rD(p)2 1 f o r p D ~
and
EM.
which verifies (b) of Lemma 4.1. Sirice V D / b u is a constant function on D.tta/b, is also a constant function on D,. It is therefore clear from Lenlnia 4.3 that
is called the Saegel domain of second kind defined by V and F. Before we prove that S is equivalent to a bounded domain in Cn+m, we consider the Siegel domain of second kind in C1+m defined by
-
Im z - lu 1 I 2 - - .- lum12> 0, where a, u l , . . . ,urn are the coordinate functions in C1+m. We prove first this domain is biholomorphic with the unit ball lz0I2
and
+ . . . + 1zml2< 1
in C1trn. We set
f * . L ~/v, , ~ ~ tends to zero at. the boundary of D,, which verifies (c') of Lemma 4.2. Since hD = -desk, hD is negative definite. On the other hand, hhf is negat.ive definite by assumption. This verifies (a) of Lcmma 4.1. Now, Theorem 4.4 follows from Lemma 4.2. R e m a r k If Ji' is a corripact cop~plcxmanifold wit.h ample canonical l i ~ i l b bundle, 1%1 admits an Herrriitian metric satisfying (iv). Let D be a bounded homogeneous domain in AS we remarked in Example 2 of Sec. 1. the ratio vD/bD is a constant filriction on D. By a well-known theorem of Vinberg. Gindikin, and Pvatctzki-Shapiro [I],D is biholomorphic with all affinely homogeneous Siegel domain of second kind. In the next section b v r shall show that every affirlely homogeneous Siegel domain of second kind satisfies (i) and (ii) of Theorem 4.4 a5. well as (iii). Thus. Theorem 4.4 may be applied to every bounded homogeneous dornain D in C n anti, ill particular, to every Hermitian symmetric space D of noilcompact type.
z-2
zO= -
z+i'
z1
2211
=-
z + i'
...,
zm=-
2um t+i'
Then we have 4
1 - C l z k I 2 = -(Im z - lu'12 - . - lurnI2), k=O which proves our assertion. We shall now prove, following Pyateztki-Shapiro, that the Siegel domain S of second kind defined by V and F is equivalent to a domain contained in a product of balls. By a linear change of coordinate system, we may always assume that V is contained in the cone y1 > 0, . . . ,yn > 0. If V is the cone y1 > 0,. . . ,yn > 0, then the components F1(u,u), . . . ,Fn(u,u) are positive semidefinite Hermitian form in u', . . . ,urn. We represent each Fk(u,U ) as a sum of squares of linear forms:
F k ( u , u ) = 1L:l2
+ - . *+ ILtk12.
We define new Hermitian forms P I , . . . ,Pn as follows. We set 5
Affinely Homogeneous Siegel Domains of Second K i n d
Following Pyateztki-Shapiro [I! wc define the Siegel domains of second kind. Let V be a convex cone in R", containing no entire straight lines. A mapping F : C"' x Cn' C n is said to be IT-Hermitianif
-
(1) F ( u , v) = F(v, u ) for u , v L' C m . (2) F ( a u l + bu.2, u ) = a F ( u l , v ) hF(uz,v) for
+
. U ~ : U ~ . ZL' !
C m : a 1). E C .
S
where the prime indicates that the summation is restricted to those L: that are not linear combinations of L:, . . . , Lf, . Then we set
Volume Elements and the Schwarz Lemma
30
where the prime indicates that the summation is restricted to those L: that are not linear combinations of L:, . . . ,L f l ,LT,. . . , L:2. Similarly, we define p4,. .. ,Fn.Let 3 be the domain defined by
5
A f i n e l y Homogeneous Siegel Domains of Second Kind
complex linear tranformation B of C m , the following holds:
A F(u,u) = F(Bu, Bu) for u E Cm Then, for any xo E R" and uo E C m , the affine transformation
Since F ~ ( u U) , 2 F k ( u , u ) for all k, it follows that 3 contains S. We shall show that 3 is equivalent to a product of balls. By (4) in the definition of F , the system of linear equations
has only one solution, u = 0. It follows that the number of the linear forms L: which really appeared in the construction of F1,. . . ,pk is equal to m. By construction, these forms are linearly independent. If we take them as new variables v', . . . ,vm, then the domain 3 is defined by
It follows that 3 is equivalent to a product of balls in C1+"'1, cl+mz-ml , . . . I c l t m - m , - 1 The following proposition is due to Hahn and Mitchell [2].
Proposition 5.1 Let S be the Siegel domain of second kind in Cn+*' defined by V and F . Let t be a jixed point of V and set
Then SI C S z C translation
S3
C
.. . C S , Uk Sk = S and S k c S ~ + For I . each k, the
is an automorphism of the Siegel domain S. If V is homogeneous, i.e., t.lle group of automorphis~nsA of V is transitive, then S is homogeneous under the group of holomorphic transformations (which are f i n e in Cn+m). 111 fact, if yo E V so that (iyo,O) E S and if (zl, ul) is an arbitrary point of S so that 91 = Im zl - F(uI, u1) E V, then the transformation
+ Re 21 + 22 F(Bu, ul) + i F ( u l , ul) Bu + u1
z
-, Az
u
-+
maps (iyo,O) into (zl,ul), where A is an automorphism of V sending yo into yl and B is a linear map of C'" satisfying A F ( u ,u) = F(Bu, Bu). The Siegel domain S of second kind defined by V and F is said to be afinely homogeneous if V is homogeneous. A theorem of Vinberg, Gindikin, and Pyatetzki-Shapiro [I] says that every homogeneous bounded domain in Cn is biholomorphic to an affinely homogeneous Siegel domain of second kind.
Proposition 5.2 Let D be a homogeneous bounded domain in C n and K ( ~ , zits ) Bergman kernel function. Then K(z, 2) goes to infinity at the bunday of D. Proof Assume that the proposition is false. Then there exists a sequence of points 2 1 , ~ 2 ,... in D which converges to a point on the boundary of D such that
where a is a positive constant. Fix a point zo of D. For each k, let auhmorphism of D such that
fk
be an
gives a biholomorphic mapping from S onto Sk. The proof is straightforward and hence is omitted. h r the Siegel domain S of second kind defined by V and F, an automorphism of V is a linear tranformation A of V such that, for a suitable
Let Jk denote the Jacobian of f k at zo ((with respect to the Euclidean -dinate system in Cn). Since the Bergman kernel form of D is invariant
32
6
Volume Elements and the Schwarz Lemma
6
by f k , we obtain
It follows that IJkl
2 b>0
fork = 1.2,3, ...
Consider {fk) as a family of holonlorphic mappings from D into C n which are uniformly bounded. Taking a subsequence if necessary, we may assume that i f k ) converges to a holomorphic mapping f : D -+ Cn uniformly on all conlpact subsets of D. It is well-known that uniform convergence of { f k ) entails uniform convergence of all corresponding partial derivatives of all orders of {fk}. In particular, the Jacobia~iJf (zo) of f at zo satisfies
Hence, f gives a homeon~orphismof a neighborhood U c D of 20 onto the neighborhood f (U) of f ( t o ) in Cn. Since f (20) is a boundary point of D. f (U) is not contained in D. On the other hand, being the liniit of { f k ) ? f maps D into D. This is a contradiction. Proposition 5.2 shows that an affinely homogeneous Siegel domain of the second kind satisfies (i) of Theorem 4.4. Instead of (ii) in Theorern 4.4. wc usc Proposition 5.1. Now, fro~nTheorem 4.4, we obtain
Theorem 5.3 Let D be an n-dimensional complex manifold which is biholomorphic with an afin.ely h,omogeneous Siegel domain of second kind and let dsg be its Bergman metric. Let A4 be an n-dimensional EinsteinKaehler manzfold with metric d s i f , such that its Ricci tensor h M is equ.01 to -dsi,. Then every holomorphic mapping f : D --+ M is vo1um.edecreasing. If we assume the result of Vinberg, Gindkin, and Pyatetzki-Shapiro. then in Theorem 5.3, D can be any homogeneous bounded domain in C n . Dinghas [I] obtained Theorem 5.3 when D is a unit ball in C n . A result similar to Lemma 4.2 was proved by Chern [l] under stronger curvature assumptions. Hahn and Mitchell (11and Kobayashi [2] proved Theorem 5.3 independently when D is a symmetric bounded domain. The generalization to the case where D is a homogeneous bounded domain is due to Hahn and Mitchell [2]. But in both papers Hahn and Mitchell assumed unnecessarily that f is biholomorphic.
Symmetric Bounded Domazn~
33
Symmetric Bounded Domains
We shall first summarize known results on the Bcrgrnan kernel functiolls of the so-called Cartan classical domains. For details we refer the reader to Hua [I],Tashiro [I], arid Hahn and Mitchell [I]. According to E. Cartan [I]: there exist only six types of irreduciblc bounded symmetric domains - four classical types and two exceptional types. The four classical domains RI, RII, Rrrr, and R I are ~ defined as
follows:
RI = {m x n matrices Z sat.isfying I, - ZZ* > 01, Rrr= {syrnmet,ricmatrices Z of order n satisfying In - ZZ* > 0): RIII= {skew-symmetric matrices Z of order n satisfying In- ZZ* > 01, Rrv = { Z = (21,. . - ,zn) E CT';/t%'I2 + 1 - 222' > 0, Izz' I < I), where I, denotes the identity matrix of order m ,Z* is the complcx conjugate of the transposed Z' of Z. and '2 is the transposed of the vcctor 2. For the dorriain Rj, ( j = I, 11, 111, IV), we denote by K,, bj, ds; and v j the Bergman kernel function, the Bergman kernel form, t.he Bergman metric, and the volume element defined by ds:, respectively. The ratio v j / b j will be denoted by c,. 5I.k denote by V ( R , ) the t,otal volu~neof R, with rcspect to the Euclidean measurc of ambient complex Euclidean space. Then
Volume Elements and the Schwarz Lemma
34
1 K I V ( Z , E=) -(1~2'1~ + 1- ~zz')-~. V(Rrv) Trace[(Im- ZZ*)-' dZ(I, - Z*Z)-I dZa], dsf = 2(rn + TL)
6
Symmetric Bounded Domains
For RIV we have
dsfI = 2(n + 1)Trace[(&,- 22")-' dZ(I,, - Z*Z)-' dZ*], dsfIl = 2(n - 1) Trace[(In - 22')-' dZ(I, - 2'2)-' dZ*], ds;v = 4nAdt[A(In - 2'2)
+ (In - zl~)z*z(In- z'z)] dz*,
where A, B , C, and D are real matrices of dimensions 2 x 2, 2 x n, n x 2, and n x n, respectively, satisfying the relations
where A = 1.z.~')~ + 1 - 2.2z1, C,
= (m
+ n)m" V(Rr), + l)n(n")/2 ~ ( R I I ) ,
sl = 2n(n-')/2 (la
[2(n - 111n(n-1)/'2 V(RIII), = (2n)" ~ ( R I v ) .
~ I I I= CIV
If we denote by Gj the automorphism group of the domain Rj, then a. trarlsformation Z E R j -+ U' E R j belonging to Gj is given as follows. For RI we have
-
W
(-42
E
= (AZ
+ B)(BZ + A)-',
where A'B
= B'A,
AA* - BB* = I,.
For RIIr we have
w = ( A Z + B)(-BZ+A)-', where A'B = -BIA,
A*A - BaB = I,,.
E
R j be a holomorphic mapping from the domain
For j = 11,
1 J(2)I2 5 {det(I, - WW*)/det(In - ZZ*))"+' 2 {det(I,
- ZZ*))-("+').
For j = 111, lJ(z)12 For j = IV,
W
-, W
For j = I ,
AC* = BD*, CC* - DD* = -In.
For RII we have
Rj
following inequalities:
+ B)(CZ - D)-',
where A, B, C , D are matrices of dimensions m x m, m x n, n x m, and n x n, respectively, satisfying the relations AA* - BB* =I,,,
Let Z
Rj into itself and let J ( Z ) be its Jacobian. By Theorem 4.4 we have the
5 {det(In - WW*)/det(I, - ZZ*)}~-' 5 {det(In - ZZ*))-("-').
CHAPTER I11
Distance and the Schwarz Lemma
1 Hermitian Vector Bundles and Curvatures
We shall summarize basic local formulas of Hermitian differential geometry. For a fuller treatment, see Kobayashi and Nomizu (1, Vol. 111. Let E be a holomorphic vector bundle over a complex manifold M with fiber C'. We denote by E, the fiber of E a t p E M. A Hemitian fibermetric g assigns to each point p E M a Hermitian inner product g, in E,. A holomorphic vector bundle E with a Hermitian fiber-metric g is called a Hemitian vector bundle. For a local coordinate system
t', . . . ,zn
of
M ,we set
Let el,. . . ,e, be holomorphic local cross sections of E which are everywhere linearly independent. With respect to this holomorphic local frame field, the components g,~ of g are giveu by
gas = g(ea, B p ) for a,p = I , . . . ,T . The components of the Hermitian connection are given by
the other components are set equal to zero. The covariant derivatives
V Z , ~are B defined by
n e mmponents K;ij of the curvature R are defined by
1
Distance and the Schwarz Lemma
38
Hermitian Vector Bundles and Curvatures
39
cross sections of El. If we fix a point p of M, then we may further assume that, e l , . . . ,e, are ortho~lormala t p, i.c., (gaB)p= (g(ea, z,o))~= dap. If we denote the curvature of E' by K&yij,then
Then
If we set K,~ig = g[R(Zi, Z 6 ) ~ 4ea], then If s =
x:=lsuea is an element of EI, c Ep,then
The components of the Ricci curvature are given by This implies the inequality If we set
then
K.- =
"
-
1 a2G G aziazj
1 dG aG +--~2 azaazj
--a2log G aziaj
'
Given an element s = C sae, of Ep, we consider the Hermitian form
at p E M. If K, is positive (respectively, negative) d e h i t e for every nonzero s, the Hermitian vector bundle E is said to have positive (respectively, negative) curvature. If the Hermitian form 2
1K,- dz' dzj
is positive (respectively, negative) definite, then E is said to have positive (respectively, negative) Ricci curvature (or tensor). Let El be a holomorphic subbundle of E with fiber C9. With respect to the Hermitian fiber-metric induced by g, E' is also a Hermitian vector bundle. The theory of Hermitian vector subbundles is essentially the same as that of Riemannian submanifolds. We s h d give the equation of Gauss which relates the curvature of E' with that of E. In choosing local holomorphic cross sections e l , . . . ,e, of E we may assume that e l , . . . ,e9 are
i.e., K, - Ki is a positive semidefinite Hermitian form. If M is a Hermitian manifold and E is the tangent bundle of 111,then the connection considered above is nothing but the classical Hermitian conis nection. If s, t E Ep are unit vectors, then K,(t,f) = what is called the h.olomorphic bisectionul curvuture determined by s and t in Goldberg and Kobayashi [I].In particular, K,(.F,B) is called the hobmorphic sectional curvature determined by s. If M' is a complex sub-manifold of M, then the tangent bundle E' = T(M1) of MI is a Hermitian vector subbundle of EIM'. Then the formula KL 2 K3 above implies a similar inequality for holomorphic bisectional curvature (and hence for holomor~ h i sectional c curvature also). In summary we state
Theorem 1.1 (1) If E is a Hermitian vector bundle over a complex manifold M and E' is a Hemitian vector subbundle of E, then
K:
K , for s E E';
(2) If M' is a complex submanifold of a Hemitian manifold 111, then the holomorphic bisectional ( o r sectional) curvature of M' does not exceed that of M.
II
1
2
4
Distance and the Schwarz Lemma
40
The Case Where the Domain is a Disk
U'e prove a generalization of Theorem 2.1 of Chapter I. Let D, be the open disk { t E C ; Izl < a ) of radius of a and let d s i be the metric defined by
ds, =
4a2dzdf A(a - 1 ~
1 ~ ) ~ '
Theorem 2.1 Let D , be the open disk of radius a with the metric ds: an.d let M be an n-dimensional Hermitian m.anifo1d whose holon>.o~hic section.al curvature is bounded above by a negative constant - B . Then every holonzorphic mappiny f : D, M satisfies
The Case Wherr D is a Symmetric Bounded Domain
41
whose h o l o m o ~ 1 t . sectional i~ c u r v a t u . ~is bounded above by a negative constant -B. Th.en every holomorphic mapping f : D -+ h..l satisfies
Proof Let ( r ~. .,. ,r l ) be an I-tuple of co~llplex numbers such that lriI2 = 1. Let j : D, -,D be the imbedding defined by
-+
Proof We define a fuilction u on D, by setting
From the explicit expression of ds: given in Sec. 2, we see that j is isometric at the origin of D,. Let X be a tangent vector of D a t the origin. For a suitable (rl, . . . , rl) we can find a tangent vector Y of D, at the origin such that j*(Y)= X.Then, for any holomorphic mapping f : D -+ M , we have
f * ( d s L ) = u ds;
l
'1 1
and want to prove u 5 A / B everywhere on D,. As we have shown in the proof of Theorem 2.1 of Chapter I, we may assume that u attains its xnaximurri at a point, say 20, of D,. IfJe want to prove that u 5 A / B at zo. If u(zo) = 0. then u 0 and t,hcrc is nothing to prove. Assume that ~ ( z o>) 0. Then the mapping f : D, -t h1 is nondegenerate in a neighborhood of L~ so that f gives a holoniorphic i~nbeddingof a neighborhood U of ,-o into ,If. By Theorem 1.1, the liolo~norphicsectional curvature of the onedimcnsio~lalcomplex subnianifold f (U) of ,%I is bounded above by -B. Since dim f (U) = 1, the holonlorphic sectional curvature off (U) is nothing but the Gaussian curvature. The rest of the proof is exactly the same as that of Theorem 2.1 of Chapter I.
-
Thcorem 2.1 is essent,ially equivalent to "Aussage 3" in Grauert and Reckziegel [I], in which they assume that the curvature of every onedimensional complex submanifold of M is boundcd above by -B. 3
The Case Where the Domain is a Polydisk
DL
Let D = = D, x . x D, be the direct product of 1 copies of disk D, of . . + ds? in D. Since ds: 11a.c radius a. Let d s 5 bc the product metric d s i constant curvature - A , the holomorphic sectional curvature of dsg varies between -A and - A l l .
+-
Theorem 3.1 Let D = ~ f be, a polydisk of dimension 1 urith metric ds; = ds: . . + ds: and let M be an n-dimensional Hermitian manifold
+
where the inequality in the middle follows from Thcorem 2.1 (applied to f o j : D, -+ M ) and the last, equality follows from t,he fact that j is isometric at the origin of D,. Since D is homogeneous, the inequality 11 f.XII" ( A / B ) I I x ~holds ~ ~ for all tangent irectors ,Y of D. 4
The Case Where D is a Symmetric Bounded Domain
Let D be a symmetric bounded donlain of rank I. With respect to a canonical Hermitian metric, its holomorphic sectional curvature lies between -A and -A/1 for a suitable positive constant A. For every tangent vector X of D,there is a totally geodesic complex submanifold ~ f =, D, x .. . x D, (a polydisk of dimension 1) such that X is tangent to DL. (If we write D = G / K and g = t + p in the usual manner and denote by a a maximal Abelian subalgebra contained in p, then dim a = rank D = 1. Without loss of generality, we may assume that X is an element of a under the usual identification of p with the tangent space of D a t the origin. If J : p p is the complex structure tensor, then the manifold generated by a J a is the desired totally geodesic subnianifold D:. To see this we have to use the fact that, for suitable root vectors X,, and X-,, with i = 1 , . . . ,1, a is spanned X,, X -,,, i = 1,. . . , l . But this is rather technical and will not be discussed any further. We shall show later explicitly what D; is for each of the classical Cartan domains.)
+
+
+
Distance and the Schwarz Lemma
42
4
From Theorem 3.1 we obtain
R~,,= {Z = (21,. . . ,zn) E Cn;
Theorem 4.1 Let D be a bounded symmetric domain with a canonical invuriant metn'c dsg whose holomorphic sectional curvature is bounded below by a negative constant -A. Let M be a Hennitian manifold with m.etric d s i f whose holomorphic sectional curvature is bounded above by a negative constant -B. Then every holomorphic mapping f : D + A4 satisfies
Corollary 4.2 Let D be a symmetric bounded domain with a canonical invariant metric ds; u~hoseholomorphic sectional curvature is bounded below by a negative constant -A. Let hf be a symmetric bounded domain of rank 1 so that its holomorphic sectional curvature lies between -lB and -B. Then every holomorphic ma.pping f : D + M satisfies
Corollary 4.3 Let D be a sym.metric bounded domain of rank 1 so that its holomorphic sectional curvature lies between -lB and -B. Then e v e 9 holornorphic mapping f : D + D satisfies
f'ds:,
5 l ct.9;.
Both Corollaries 4.2 and 4.3 have been obtained by Koranyi [I]. We shall now exhibit for each of the Cartan domains, a totally geodesic polydisk of dimension 1 (= the rank of the domain).
R1= {m x n matrices Z satisfying I - ZZ* > O), rank = min(m, n). o l = { Z = ( z i j ) ; z,, = O f o r i # j),
l=min(m,n).
RI1 = {symmetric matrices Z of order n satisfying I - ZZ' > O), rank = n . Dn = { Z = (zi3); z,, = 0 for i # 3). RIII= {skew-symmetric matrices Z of order n satisfying I - ZZ* > 0): rank = [:n] .
D' = { Z = (z..). a3 ' ti,
=
The Case Where D Is a Symmetric
o except 212 = -221,234
= -243,.
. .),
+
Bounded Domain
I Z Z ' ~ ~ 1 - 222'
> 0, lzzll < 11,
rank = 2.
0 ' = ( ( ~ 1~, 2 ~ 0. .,-0) . E RIV), where the right-hand side is biholomorphically mapped onto D2 by
CHAPTER IV
Invariant Distances on Complex Manifolds
1 An Invariant Pseudodistance Let D denote the open unit disk in the complex plane C and let p be the distance function on D dcfined by the Poincar6-Bergman metric of D. Let M be a complcx manifold. Ure define a pseudodistance d M on M as follows. Given two points p, q E M , we choose poirits p = po, p l , . . . , p k - l , p k = q of M ,points a l , . . . , u k ! b l , . . . ,b k of D l and holomorphic mappings f . . . ,f k of D into M such that fi(ai) = pi-1 and fi(bi) = p, for i = 1,. . . ,k. For each choice of points and mappirigs thus made, we corisider the number
Let d M ( p ?q ) be the infimum of the numbers obtained in this manner for all possible choices. It is an c'wy matter to verify that dnf : M x M -, R is continuous and satisfies the axioms for pseudodistancc:
The most important property of d,zl is given by thc following proposition, the proof of which is trivial.
Proposition 1.1 Let ,%l and N be tz~locomplez manifolds and let f : i l l-, N be a holomorphic mappzng. Then dn.r(p,q ) 2 d : v ( f (PI!f ( q ) ) for P,q E 11.1.
Corollary 1.2 Etlery biholomorphic m.apping f : M
+
N is an isorr~etry.
Le.,
The pseudodistance dnf may be considered as a generalization of the Poincar& Bergman metric for a ini it disk. We have
46
Invariant Distances on Complez Manifolds
Proposition 1.3 For the open unit disk D i n C , d D coincides with the distance Q defined by the Poincark-Bergman metric. Proof By the Schwarz lemma (cf. Theorems 1.2 and 2.1 of Chapter I), every holomorphic mapping f : D -+ D is distance-decreasing with respect to e. From the very definition of dD we have
d ~ @(7), 2 P(P, q ) for P, q E D . Considering the identity transformation of D, we obtain the inequality ~ D ( P9) , 5 e(p,q). The following proposition says that d M is the largest pseudodistance on M for which f : D + M is distance-decreasing. Proposition 1.4 Let M be a complex manifold and d' any pseudodistance on M such that
d'(f (a), f ( b ) ) 2 e(a,6 ) for a, b E D for every holomorphic mapping f : D d~
IP,
+
M . Then
I
An Inirariant Pseudodistance
47
proof We have
where the first inequality follows from the fact that the mappings f : M 4 M x M' and f' : M' -, M x M' defined by f ( x ) = ( x , p l ) and ft(z') = (q,xi)are distance-decreasing and the second inequality is a conmuence of the triangular axiom. The inequality d M , M , ( ( p ,pi), (q,q')) 2 ~ a x [ (p, d q~) ,d ~(p', l q')] follows fiom the fact that the projections M x Mt -+ M and M x M' + M' are both distance-decreasing.
Example 1 If D is the open unit disk in C , then
To prove this assertion we may assume (because of the homogeneity of D ) that p = pi = 0 and d ~ ( 0 , q 1 ) do(O,qr),i.e., Iql 2 Iq'l. Consider the holomorphic xnapping f : D -, D x D defined by f ( z ) = ( z ! (qi/q)z).Since f is distance-decreasing, wc have
q ) 2 dr(p,q) for P , q E M .
Proof Let pO,. . . ,pk, a l , . . . ,ak, b l , . . . ,bk, fir of d M . Then
. . . ,f k be as in the definition
More generally, if D~ denotes the k-dimensional polydisk D x
- . x D, then
This shows that dDk does not coincides with the distance defined by the
Hence,
Bergman metric of D~ unless k = 1. As we see from the definition of the pseudodistance, d ~ 4is defined in a manner similar to the distance function on a bemannian manifold. It. is therefore quite xlatural to expect the following result:
Proposition 1.5 Let M and A4' be two complex manifolds. Then
Proposition 1.6 Let Af be a complex manifold and hi a coverin,.g manifold of M with coverin,g projection 7: : *
, d.w(p, q ) + ~ M P ( P ' ,2~ ~' )M ~ M ~ ( ( P , P( Q' , )4')) 2 Max[dA!4(p,q)!dM1(p',q')] forp,q E M and p1,q' E M i
th.e infimum is taken over all Q E ,
2
Invariant Dzstances on Complez Manzfolds
Proof Since T : A?
-,M
is distance-decreasing, we have
Assunling the strict inequality, let d , (p, ~ q)
+ E < irlf9 dKf( A41,
where is a positive number. By the very definition of dn.f there exist points a l , . . . ,ak, bl, . . . , bk of the unit disk D and holoniorphic mappirlgs fl!. . . ,f k of D into M such that P = fl(ul),
. . , fk-l(bk-I)
fl(b1) = f i ( ~ 2 ) ~ .
= fk(ak):
fk(bk) = q
and
Then ure can lift f l , . . . , f k to holomorphic xnappings f l , . . . in such a way that.
.fk of D into
I
I
r, = f-l ( a l ) ,
fL(b,) = f , + l ( ~ , + ~for ) i = I , . . . . k - 1, for 1 = 1,. . . . k . r u ( j f l = f,
+
I
49
point Q E U lies on the orbit of a complex 1-parameter subgroup of G through p. Hence, for every point q E U there is a holomorphic mapping f : C -+ M whose image contains both p and q. Since f is distancedecreasing d c is trivial, we have dnr(p,q) = 0. To prove that dn.r is trivial for any pair of points on M , we connect the pair by a chain of small open sets U and apply the triangular axiom. In particular, for a compact homogeneous complex manifold M or for A4 = C - {O}, dhl zs trivial. If ,ll is a compact Riemann surface of genus 0 or 1, then dM is trivial. = { z E C"; r < lzl < R} and B = {% E Cn;lzl < = 1, the universal covering space of 111 is the upper half-plane
Example 4 Let M
R). If n
or equivalently the unit disk. In Sec. 6 of Chapter I we showed how t o construct a rnetric of constant negative curvature d s L from the PoincarCBergman metric of the upper half-plane. From Proposition 1.6 it follows that the invariant distance d~ coincides with the distance defined by d s b . For n > 1, I do not. know what dni looks like, although I know that dhf is not complete, since M is not holomorphically complete (scc Sec. 4 of t.his chapter).
2
If we set 4; = fk(bk): then ~ ( 4 ; )= q and dS1 (@.Q) 2 ~ f e(at, = b,).~ Hencc. 0 dnr (jj. Q) < d,vl (p, q) E , which coritradicts our assumption.
CamthCodory Distance
Carathdodory Distance
Let D be the open unit disk in C with the PoincarbBergrnan distance Q. Let M bc a complex xnanifold. The CarathCodoIy pseudodistance c,\f of M is defined by
Remark It is not clear whether the follou~ingis true:
i.e., whether inf4 d , (jj, ~ ij) is really at.taincd by some (1. E x a m p l e 2 Let 121 be a Rcmann surface whose universal coverin,0 1s R unit disk D. c.g., a coxripact Riemann surface of genus 22. R o m Propositions 1.3 and 1.G we see that dnr is thc distance defined by a Kaehler metric of d l with constant curvature
E x a m p l e 3 It can be seen easily from the definition of the pseudodistance d.\r that for a complex Euclidean space C" the pseudodistance d," is tr-i~iul. i.e.! dctl(p.q) = 0 for all p,q E C". h,lore generally, zf M is u complrl 1.' man,ifold on which a complez Lie group G' acts transitiz~ely, then t~iviul.For each point p E ,tf, let li be a neighborhood of p such that
where the supremum is taken with respect t.o the family of holomorphic mappings f : A.f + D ,\ e prove that the pseudodistance dbf defined in Sec. 1 is greater than or equal t.o cn;l. That will in particular imply that. C M ( P , Q ) is finite. Proposition 2.1 For any complez manifold hi. we have
.
A. in the definition of d,ir(p. q), clioose points p = po,pl.. . . Pk-i,pk = q of M and points a l , . . . ,ak, bl,. . . , bk of D and also m a p Pings f l , . . . f k of D into hf such that f,(u,) =p,-1 and f,(b,) = p,. Let f
.
2
Invariant Distances on Corn~lezManifolds
50
51
Camth4odnry Distance
The following proposition says that CM is the smallest pseudodistance -t D is distance-decreasing.
be a holomorphic mapping of M into D. Then
on M for which f : M
pl-oposition 2.5 Let M be a complex manifold and d a n y psezldodista.race on M such that cl(p, q) 2 e(f(PI, f (q)) P, q E M
for every holomorphic mapping f : M -+ D. Then where the first inequality follows from the Schwarz lemma and the second inequality is a consequence of the triangular axiom. Hence, d u i , b,)
2 SUP d f (PI, f (q)) = c n r h 9).
i= 1
It is an easy matter to verify that C satisfies the axioms for pseudodistance: C M ( q) ~ ,2 O!
C M ( q) ~ ,= C
~ :I
x hl
CM
R is continuous and
2 Max[cM(P, q), CMJ(P', q')] for p, q E M and p', q1 E MI.
shares many properties with d ~ . -+
iV
Example 1 If D~ denotes the k-dimensional polydisk D x . - . x D, then ~ D ~ ( ( P ~ , . . . , P ~ .). ,,qk)) ( Y I= ~ .M a x [ c ~ ( ~ i , q i, )=; 1 , . . . , k ] for pi,qI E D.
c n r ( ~ , q )2 c.v(f(p),f ( q ) ) for p:9 C ~1.f.
The proof is similar to the one in Example 1 of Sec. 1. Hence,
The proof is trivial. Corollary 2.3 Every biholomo~phic7nupping f : M
-+
N is an isometry.
2.e.,
The Carathkodory pseudodistance may be also considered as a generalization of the PoincarkBergman metric for D. Proposition 2.4 For the open unit disk D in C , CD coincides with thf? distance ,Q defined by the Poincu7~t-Be~~nan metric.
Example 2 Let M = G / K be a symmetric bounded domain of rank I in 0 (in its natural realization). For a suitable subspace C 1of Cn, M n C1is a polydisk D1 = D x . . . x D and A l can be written as a union of polydisks k(D1),k E K, where k is considered as a unitary transformation of C n . In particular, we have D' C M C D". Since the injections D'
b e c ~ 2l c~ Proof Using the Schwarz lernma for a holomorphic mapping f : D we obtain
1
i
M
Proposition 2.6 Let M and MI be two complex manifolds. Then
+ ~ ~ 4 (rq) 2, ~M(P!?').
Proposition 2.2 Let M and hr be two complex manifolds and let f : M be a holorno~h,~ir: mapping. Th.cn
1
for P,9 E
CM(P,Q) t CM~(P', ql) 2 C M ~ M ~ PI), ( ( P(q,ql)) ,
M ( ~ ~ P ch1 ) ~ (P, (l)
The Carath6odory pseudodistance
-+
5 c'(P, q)
The proof is trivial. The following proposition can be proved in the same way as Proposition 1.5.
k
~ M ( Pq), = illf
CM (P, Q)
-+
11
from the very definition of CD. Considering the identity transformation of D. 0 we obtain the inequality ~ ( pq), 5 c ~ ( pq). ,
@.
2
M and M Dn are distance-decreasing, we cD.. on D ~ From . Example 1 we see that CDL = C D on ~ -+
-+
Hence,
U h g the result in Example 1 of Sec 1, we obtain a similar result for dDi d ~ Given . any two points p, q E M, there is an element of G which
52
Invariant Distances on Complez Manifolds
sends p and q into D[.We may conclude therefore that, for a symmetric bounded domain M, the two distances chf and dhf coincide.
Example 3 Since C M 2 d~ in general, we see froni Example 3 of Sec. 1 that if M is a complex manifold on which a complex Lie group G acts transitively, then cbf is trivial. Example 4 If M is a compact complex manifold, then c~ is trivial. This is clear, since every holomorphic function on M must be a constant function. This shows that there is no analogue of Proposition 1.6 for cnc Example 5 Let 121 = C - A, where A is a finite set of points, or more genersally, let 1LI = Cn - A, where A is an analytic subset of dimension 5 n - 1. Then chf is trivial. A holomorphic mapping f : M -+ D may be considered as a bounded holomorphic function on hf. By the socalled Riemann extension theorem, f can be extended to a holomorphic mapping of C n into D. By Liouville's theorem (or by Example 3), f must be a constant function and hence cnf is trivial.
~
Example 6 Let :\I= ( 2 E Cn;r< It1 < R ) and B = {zE Cn;lzl < R ) . If 12 2 2, th.en C M = CBIMThis follou~sfrom the fact that the envelop of holomorphy of 219 is B, so that every holomorphic mapping from M intcj the unit disk D can be extended to a holornorphic mapping from B into D. On the other liarid. I do riot know w11a.t the Ca.rathbodory distance c , ~ l looks like if n = 1. This is in contrast to the situation in Example 4 of Sec. 1, where dn., is known for n = 1 but not for n 2 2. The Carathbodory distance was introduced i11 Carath6odory [I, 21. For recent results on the CarathCodory distance, see Reiffen [I, 21. 3
Completeness with Respect to the Carath6odory Distance
Let 31 be a complex manifold. Throughout t.1iis section we are interested in the case where the CarathCodory pseudodista~icechf is a distance, i.e., t.hc case where the family of holomorphic mappirigs f : 111 -, D separates t h e points of 91. Since every bounded holomorphic function on ,kl, multiplied by a suitable constant, yiclds a holomorphic rriappi~igof M into the unit disk D. the necessary and sufficient condition that C M be a distance is the following: For any two distinct points p and q of M , there is a bounded holomorphic function f o n hf such th.at f ( p ) # f ( 9 ) . It is very close to
3
Completeness with Respect to the Carathdodory Distance
53
assuming that M is a bounded do~nainin a Stein manifold. In particnlar. gU compact complex manifolds are excluded. In general, we say that a metric space M is complete if, for each point of M and each positive number r , the closed ball of radius r around p is rnmpact subset of 11.1. If M is complete in this sense, then every Cauchy ~equenceof M converges. The converse is true for a Rieman~lianmetric, but not. in general.
proposition 3.1 If M and .hI1 are com.plex man.ifo1ds with complete Carathkodory distance, so is M x M'. Proof This is immediate from Proposition 2.6. Proposition 3.2 A closed complex submanifold MI of a complex manifold M with com.plete Carathe'odory distance is also complete with respect to its Canathkodory distance. Proof This is immediate from the fact that thc injection M' distance-decreasing.
-+
M is
Proposition 3.3 Let 12-I nn.d :\Ii, i E I , be corrbplerr: submanifolds of a corn.plex manifold 'V such. th.at M = :\Ii. If ecl.ch d.l, is complete with respect to its Carath,6odorp distance, so is M .
ni
Proof Since each injection M -, Ill,. is distance-decreasing, the proposition follows fro111 the following trivial lemma.
Lemma Let M and i E I , be subsets of a topological space Xsuch tlmt M = ,%Ii. Let d a71d d, hc distances on M und Mi such that d(p,q ) 2 di(p,q) for p, q E M . If each A l i is complete with respect to di, then 121 is Wmplete with respect to d.
ni
We shall now give a large class of bounded domains which are complete with respect t.o their CarathBodory distances. Let G be a domain in Cn and f l , . . . ,fk holomorphic functions defined in G. Let P be a connected Component of the open subset of G defincd by
Assume that the closure of P is corrlpact and is co~itainedin G. Then P is c a e d an analytic polphedron.
I
I
1 I
Invanant Distances on Complez Manifolds
54
3
Completeness u..ith Respect lo Ube CaratModory Distance
55
Theorem 3.4 An analytic polyhedron P is complete with respect to its Carathkodory distance cp.
So far, we have given sufficient conditions for a manifold to be complet,e with respect to its Carathbodory distance. \.Ye shall give now some necessary
Proof Let F be the set of holomorphic mappings of P into the unit disk D. Let o be a point of P. Given a positive number a, choose a positive number b, 0 < b < 1, such that
Horstmann [I] has shown that a domain in Cnwhich is cornplcte with to its Carathkdory distance is a dom-ain of holomorphy. We shall give a generalization of Horstmann's result. Given a family F of holomorphic functions on a complex marlifold IM and a subset K of M , we set
{z E D; e(fi(o),z) 5 a for i = 1,.. . , k )
c
{z E
ED; Izl 5 b ) .
5 sl~plf(K)Ifor all f E F ) , where sup)f ( K ) ) denotes the supremum of If (q)l for q E K. Then KF is a K F = {p E
Then
2 a ) = {P E P; e(f (01, f (PI) 5 a for f
{P E PI CP(O,P)
E F)
c {p E P; e(fi(o), fi(p)) 2 a for i = 1,. . . ,k ) c {p E P ; Ifi(p)I 5 b for i = I , . . . , k ) . Since {p E P; I fi(p)l is compact.
b for i = 1,. . . , k) is compact, {p E P;cp(o,p)
5 a) 0
In the definition of an analytic polyhedron, we used only a finitely many holomorphic functions f l , .. . ,f k . If we take an arbitrary family of holomorphic functions f,,,a E A, 011 G and assume that the set defined by I fa(t)l < 1. a c A, is open, then we call each connected compone~lt of this open subset of G a generalized unalytic polyhedron. The proof of Theorern 3.4 is valid for the following:
Theorem 3.5 A generalized analytic polyh,edron P is complete with respect to its Camthe'odory distance cp. Let h j be real analytic functions on G of the form
m=l where each fj,, is holomorphic in G. Let S be the set of sequences { a m ) of complex numbers such that Cz=llam[*= 1. Then
M ; If(p)I
closed subset of M containing K and is called the convex hull of K with . respect t o F coincides with K~ respect t o F. The convex hull of ~ r ; with itself. If ~ r ; is. compact for every compact subset K of M, then 11.1 is said to be convex with respect to F. If F' C F , then > KF. Hence, if M is convex with respect t o F' and if F' c F, then M is convex with respect to F. If M is convex with respect to the family of all holomorphic functions on M, then M is said to be holomorphically convex.
KF!
Theorem 3.6 Let M be u complex manifold with Carathebdory distance. Fix a point o of M and let F be the set of bounded holomorphic jiinctions f on A4 such that f (0) = 0. If M is complete with respect to CM, th,en hl is convex with rrspect to F an.d hence is holomorph.ically convex. Proof Let, a be a positive number and B thc closed ball of radius a arourid 0 E M, i.e.,
B = { p E hi;CM(O,P)$ a). Since M is complete, Li is compact. Since every compact subset K of M is contained in B for a sufficiently large a! it suffices to prove that BI;. is compact. We shall actually show that B F = B. BF =
{p E hf;If(p)l
5 sup1f(B)I for f
This shours that a connected component of the open subset of G defined h?' {z E
G; lhl(z)/ < 1 for j = 1.. . . , k )
is a generalized analytic polyhedron. To such a manifold, Theorem 3.5 is
therefore applicablc.
= {P E lll;e(0.f (p)) 5 a ) = B.
Since BF contains
B,we conclude B~ = B.
E F)
56
Invariant Di.qtances on Cornplez Manzfolds
If f is a bounded holomorphic function on M I then, for a suitable positive constant c. c f is a holomorphic mapping of M into the opcn unit disk D. This means that, in Theorem 3.6, we may choose F to be thc family of holomorphic rriappings f : M + D such that f ( 0 ) = 0. It is not clcar if the converse to Theorem 3.6 holds. As we shall see shortly, M need not bc complctc with respect t o C M even if M is holomorphically convex. I t seems, howcvcr, vcry rcasonablc to expect that Ji is complete with respect to if it is convex with respect to t h e family of bounded holomorphic functions. Let. 111be a complex manifold of complex dimension n and A an analytic subset of dirnerlsion 5 7~ - 1. Since every bounded holomorphic functions on M - A can be extended to a bounded holomorphic function on ,I1 by Ricmann's extension theorem, it follows that M - A cannot be convex with respect to the family of bounded holomorphic functions if A i:, nonempty and that cbf coincides with c h , - ~on hrl - A. In this way we obtain many examples of holomorphically convex manifolds which are nor convex with respect to the farnily of bounded holomorplic furictioris and not complete wit,h respect to their CarathEodory distances. The punctureci disk D - ( 0 ) is the simplest exarriple. The question of completeness with respect to the CarathCodory distance may be quite possibly relat,ed to that of complctcness wit,h respect to the Bergman metric (see Bremermann 11and Kobayxhi [I] on the latter question). Since D is the conlpletion of D - ( 0 ) with rcspcct to the Carath6odory distance c u - ~ o and ~ every holomorphic mapping f : D - (0) 4 '11 is distance-decreasing with respect to C D - {0) and C M ! we have Proposition 3.7 If :\.I is a complex m.an.ifold with complete Carath,iodor:i distance, th.en e v e q h,olomorph.ic mapping f from th,epr~nctr~red disk D - (0) into hl can be extended to a h.01om.orph.i~mappin,g of D into A i . 4
Hyperbolic Manifolds
Let 121 be a complex manifold and dxf the pseudodistance defined in Sec. 1 If d.bl is a distance, i.e.. d,tf(p. q) > 0 for p # q, then 12.1is called a hyperbol~~' A hyperbolic manifold dl is said to be complete if it is corriplrte niu~~zfold. with respect to dxr. As in Sec. 3, 121is complctc with respect to d x r if. fo: each point p of .kf and each positive number r. the closed ball of radius 1' around p is a compact subset of ?I/. As we shall see later, for a liy-perbollr manifold .If, this definition is equivalent to the usual definition in tcrms of Ca~lchysequences.
4
Hyperbolic Manzfolds
57
The proofs of the following three propositions are almost identicd to those of Propositions 3.1, 3.2, arld 3.3. proposition 4.1 If M and M' ure (complete) hyperbolic munifolds, so is
Mx
hl'.
proposition 4.2 A (closed) co.mplex submanifold M' of a (complete) hyperbolic manifold M is a (complete) hyperbolic manifold. proposition 4.3 Let M and Mi,i E I , be closed complex submanifolds of complex manifolds N such that hrl = Mi. If each M, i s a (conbplete) hyperbolic manifold, so is M .
n,
F'rom Proposition 2.1 we obtain Proposition 4.4 A complex mun~lfoldM i s (complete) hyperbolic if its Camthe'odory pseudodistance c,bf is u. (complete) distance. Corollary 4.5 Every bounded domain in Cn is h,yperbolic. Every generalized analytic polyhedron is complete hyperbolic. The second staternerit follows from Thcorcm 3.5. The following proposit,ion is immcdiat,~from Proposition 1.4. Proposition 4.6 If a co.rr~.plexn~unifolds!?Ifndnl.its n (complete) distnnce d' for which evely holomorphic muppin,g f of the open unit disk D into ,kI is distance-decreasing, i.e.! d'( f ( a ) ,f ( b ) ) @(a,b) for a , b E D l then it i s (complete) hyperbolic. The results so far have their counterparts for Carathkodory distance. We shall now give results which are proper to thc invariant distance dnr. Theorem 4.7 Let AS be u complex man.ifold and :
4
lni!ariant Distances o n Complex Manifolds
58
be the closed ball of radius r around p = x(@)E 111. By Proposition 1.6, wc have
B,
c T(B,+J)
for 6 > 0.
-
u
-
.
-
A
l
oi
.
-
59
hence in U . It follows that 6 lies in u,. Let $ be the point of U* defined by = Then { C j ) converges to 6.
,~(6).
A complex manifold M is called a spread (domaine eta16 in French) over a complex manifold M with projection x if every point fi E M has a
Since B,+& is compact by assumption and B , is closed, B , must be compact. Hcnce, M is complete. Assurne that M is hyperbolic. Let 6, Q E h? and dg@, Q ) = 0. Since the projection x is distance-decreasing, we have dhf(x(@), x(Q)) = 0 , which implies a($) = a(G). Let u be a neighborhood of @ in M such that x : -+ ~ ( 0 is )a diffeomorphism and x ( U ) is an &-neighborhood of x(j5) with respect to dkI. In particular, 6' does not contain unless @ = 0. Since da(@,Q) = 0 by assumption, there exist points a l , . . . ,ak, bl,. . . ,hk of D and holomorphic mappings f l , . . .,f k of D into M such that p = f i ( a l ) , fi(b,) = fi+l(ai+l)for i = 1 , . . . , k - 1 arid fk(bk) = Q and that ~ ( a bi) . ~ ,< E . Let aibi denote the geodesic arc from a, to bi in D. Joining the curves fl ( a l b l ) ,. . . fk(akbk) in fi,we obtain a curve from fi to @ in M , which will be denoted by 6 . Since x o f l , . . . , T O f k are distanccdecreasing mappings of D into M and al bl, . . . , akbk are geodesics in D. every point of the curve x ( c ) remains in the &-neighborhoodx ( 0 ) of x(j5). Hcnce, the endpoint (I rnust coincide wit,h ji. Assume that M is complete hyperbolic. We shall prove here that iii is complete in the sense that every Cauchy sequence with respect to do is convergent. In the following section, we shall prove that the completeness in this usual sense implies the completeness defined at the beginning of this section. Lct {$,) be a Cauchy sequence in nh. Since the projection x is disbance-decreasing, { x ( p i ) )is a Cauchy sequence in M and hence converges to a point p E M . Let E bc a positive number and U the 2~-neighborhoodof p in hi.Taking E small wc may assume that x induces a homeomorphis~llof each connected component of x - ' ( U ) onto U . Let N be a large integer such that x(p,) is within the E-neighborhood of p for i > N. Then every poi~lt outside U is at least E away fro111n(Pi).Let Ui be the connected component of n - l ( U ) containing pi. We shall show that the &-neighborhood of fi, lies for i > N. Let Q be a point of i l with Q ) < E. We choose in points a l . . . . , a t , b l , . . . bk of D and holomorphic mappings f l , . . . , fk of D k into < , I in the usual manner so that C j = , e ( a j , b j )< E. Denoting by a,hj the geodesic arc horn a, E b j . let c be the curve from @, to ij obtained b r joining f l ( a l b l ) ,. . . ,fk(akbk)in hf.Let C = R(c). From the constructioll of C, it is clear that C is contained in the €-neighborhood of n ( p i ) and
c!=~
Hyperbolic ManiJolds
-
*eighborhood U such that x is a holomorphic diffeomorphism of 0onto the open set ~ ( 0of)M. This is a concept more general than that of covering From the proof of Theorem 4.7 above, we obtain the following: proposition 4.8 A spread M over a hyperbolic manifold M is hyperbolic.
A similar reasoning yields also the following: Proposition 4.9 If a con~plexmanifold M' is holom.orphiually immersed in a hyperbolic manifold M , then M' is also hyperbolic. In Sec. 3 we saw that the punctured disk D - ( 0 ) is not complete with respect t o its Caratheodory distance. But it is complete hyperbolic. In fact, the half-plane { z = x i y E C ;y > 0) is the universal covering space of D - {O), the projection being given by z elZ. We may also consider the unit disk D as the universal coveririg space of D - ( 0 ) . Our assertion follows from Theorem 4.7. More generally, we have
+
+
Theorem 4.10 Let M be a corr~pletehyperbolic manifold and f a bounded holomorphic function on M . Then the open submanifold M' = { p E M ; f (p) # 0) of 1M is also complete hyperbolic. Proof Multiplying f by a suitable constant, we may assume that f is a holomorphic mapping of M into the open unit disk D. We denote by D*the Punctured disk D - ( 0 ) . Let o be a point of M' and let a and b be positive numbers. Since D* is complete hyperbolic, for a given positive number a can choose a small positive riurnber b such that {Z
E
D; jzl
2 b) 3 {t E D*;do- (f( o ) ,2 ) 2 a ) .
We set A = {p E M ; dhf(o,p)
5 a),
B={pEA.i;If(p)lZb).
Since dhf,(0:p)
2 d,\,
( 0 ,p)
A' = { p E M'; d.%p(0,p) 5 a ) , B'={p~M';If(p)Izb).
by Proposition 1.1, arehave
4
Invariant Distances on Complex Manzjolds
60
Since b is positive and 121'
=
{p E hl; f @) B
Since f : A.I'
-t
I
= B'.
D* is distance-decreasing, we have
I
It should be observed that A.1' above can never be complete with respect. to its CarathCodory distance c ~ unless , M = M'. In fact, c ~ =f c~.flhf' (see Sec. 3). T h e o r e m 4.11 A (complete) Herm.itinn manifold M whose holomo7ph,ic sectional curvature is bourtded above by n negative con.stant is (complet~) hyperbolic. Proof Let. ds$ deliole the I'oincark-Bergrnan metric on the open ~ m i r disk D. Let dsi,, be the Hernlitian metric of M. If we multiply d s i f by a suitable positive constant, we have by Theorem 2.1 of Chapter I11 t,lle following inequality for every liolomorphic mapping f : D -+ M .
5 ds;.
If we denote by d' and Q the distance functions on hr1 and D defined by d s a and ds; respectively, then t,his inequalit,y implies that f is distancedecreasing with respect to d' and @, i.e., d'(f (a),f (b)) 2 @(a,b) for a, b € D. 0 Now the theorem follows fro111 Proposition 4.6. Corollary 4.12 The Gaussian. p1an.e minus two points C - {a, b) ,is complete llyperbolzc nzn.n.ifold.
fl
This follows from the result in Scc. 3 of Chapter I and Theorem 1.11, Similarly. the following corollary follows from Theorem 5.1 of Chapter I. Corollary 4.13 Eue~yconlpact Riemartn surface of genus g pact hyperbolic mun,ifold.
Example 1 In P2(C), let LI, L2, L3, and L4 be four (complex) lines in gneral position. Let a = Ll n L2 and b = Lg n L4. Let Lo be the lines through a and b. Then A4 = P2(C) Li is complete hyperbolic. In fact, M is biholomorphic to the direct product of two copies of C - {0,1). To see this, we consider Lo as the line at infinity so that P2(C)- Lo = c2.Then L1 is parallel t o L2, since Ll n L2 = a E Lo. Similarly, Lg is to L4. But L1 and L3 are not parallel, since L1, L2,L3, and L4 are in general position in P2(C). It is now clear that hf = C 2 Li is affmely equivalent to the direct product of two copies of C - {0,1). On the other hand, P2(C) - u;=, Li is not hyperbolic, since it contains Lo {a, b) = PI( C )- {a?b), which is not hyperbolic. Let pj, j = 1, . . . , 4 , be four points in general position in P2(C). Connecting every pair of these points, we obtain six lines L . , j = 0 , 1 , . . .,5. Since P2(C)Lj is contained 4" Lj? it is also hyperbolic. I do not know whether in M = P2(C) - Ui=o it is also complete. For higher-di~nensionalanalogues of this example, see Kiernan [I].
uLo
Since A is compact subset of M by the completelless of M and B is closed in M , the intersectiorl A n B is compact. Since B = B', it follows that An B is in M'. Since A n B is a compact subset of hl' and A' is closed in M', the intersection A' n (A n B) is a compact subset of MI. Since both A and B contain A', it follows that A' coincides with A' n (A n B) and hence is a compact subset of Al', thus proving that 111' is complete hyperbolic.
f *(ds;.[)
61
As a consequence of and also as a generalization of Corollary 4.12 we have the following example (due to P.J. Kiernan).
# 0), we have
A ' c ( p ~ 1 1 l ' ; d ~ - ( f ( o ) , f ( p ) )cI {pEM';If(p)I a) ?b)=B1=B.
1
Hyperbolic Manifolds
2 2 is a corn-
uL1
uLo
The following result is also due to Kiernan j4]. Theorem 4.14 Let E be a ho1omorph.i~fiber bundle over 11.1 with fibre F and projection x. Then E is (complete) hyperbolic if M and F are (complete) hvperbolic.
Proof Assume that h l and F are hyperbolic. Let p,q E E. If x(p) # r(q),then d ~ ( p , q )2 d . t f ( ~ ( p ) , ~ ( q>) )0. Assume ~ ( p =) n(q). Choose a neighborhood U of ~ ( p in) h.1 such that T-'(u) = U x F. Let B, be the ball in M centered at 7i(p) and of radius s with respect to d ~ Denote . by
Dv the disk { z E C ; 121 < r). Choose s > 0 and r > 0 in such a way that & C U a n d d D ( t , 0 ) < s f o r z € D, (where D = D l ) . Thus,if f : D - + E is h ~ l o m o r ~ h and i c f (0) E T-' (B,): then f (D,) c U x F. Choose c > 0 such that d ~ ( 0a), 2 cdD,(O, a ) for all a E Dr12. Let f, : D E be hol0morphic mappings and let ai,bi be points of D such that p = f i ( a l ) , -t
f i ( b ~= ) f2(a2),. . . : fk(bk) = q. By homogeneity of D, we may assume that = 0 for all i. By inserting extra terms in this chain if necessary, we may assume aLso that bi E Dr/2 for all i. We set po = p,pl = f i ( b ~ )-, .. pk = fk(b~) = q. We have two cases to consider. Consider first the case where at Qi
.
I
I I I
Invariant Distances on Complex Manifolds
62
least one of the pi's is not contained in r - l ( B 8 ) . Then it is easy to see
5
U n Co7npleteness of an Invariant
Distance
63
This shows that {q(p,)) is a Cauchy sequence in F , and therefore cp(p,) YO for some yo E F . Clearly, pn (XO,yo) E U x F . Thus +
+
E is complete. i=l
i=l
i= 1
k
2
C dw(r(pi-l), %(pi))2 s. i=l
Consider next the case where all pi's are in r-'(B,).
.:
Then
k
Remark Since F may be considcrcd a.s a closed complex submanifold of E, it follows that if E is (con~plete)hyperbolic, then F is also (complete hyperbolic. On the other hand, M need not be hyperbolic even if both E and F are complete hyperbolic. Let B* = {(z, w ) E C 2 ; 0 < 1 . ~ + 1~ lwI2 < 1). Then B* is a holo~~lorphic fiber bundle over Pl(C) with fiber E C ; 0 < Iz[ < 1). The bundle space B* is hyperbolic and the fiber D* is coinplete hyperbolic, but the base space Pl (C) is not llyperbolic. If we set E = { ( z ! . ~E) C 2 ; z # 0 and 1zI2 IwI2 < I), the E is a fiber bundle over C with fiber D*. This furnishes an example where E is even complete. By a reasoning similar to t,he proof of Theorem 4.14, Kiernan [4] proves that if E is a Hermitian vector bundle over a hyperbolic manifold hl, then the open unit-ball bundle {X E E ; llX 11 < 1) is hyperbolic. We conclude this section with another example of a complete hyperbolic manifold.
LP = { z
k
2 - c C d ~ ( ~ ( ~ i -pi)) l),
2 C ~ F ( P9)',
i= 1
where y : U x F -+ F is the projection. This shows that Thus E is hyperbolic. Assume that M and F are complete hyperbolic. Let {p,) be a Caudiy scquence in E. Then {r(pn)) is a Cauchy sequence in hf and therefore ~ ( y , ) + xo for soIne 2 0 E M . Choose a neighborhood U of xo in ill such that x-l(IJ) = U x F. Choose s > 0,r > 0, and c > 0 as above. Given E > 0 with 2e < s, choose an intcgcr N such that pn E ~ - l ( B , . j and dE(pnrpm) < E for n, 171 > N , We shall show that d ~ ( q ( p , ) ,q(p,)) < E / C for n, m > I\-.We fix 71!7n > N. We choose holomorphic mappings fi : D + E and points b, E D such that and ~ f = dD(O, , bi) < E. We may again assume without loss of generality that bi E DrI2 for all i. Since
+
Theorem 4.15 il Siegel do,main of the secund kind is complete hyperbolic.
Proof In Sec. 5 of Chaptcr I1 we proved that a Siegel domain of the second kind is equivalcnt to a domain contained in a product of balls. From that proof it is not difficult to see that a Siegel domain of the second kind can be mitten as the intersection of (possibly tincountably many) domains, each of which is biholomorphic to a product of balls. But a product of balls is Wmplete hyperbolic. Our assertion follows now from Proposition 4.3. 5
it follows that fi(0) E n-'(Bz,) C x-'(B,). Hence,
On Completeness of an Invariant Distance
we saw in Theorem 4.7 a similarity between the pseudodistance d~ and a Riemannian metric on M. We shall point out here another important hilarity. Given any subset A of a complex manifold M and a positive number r , let U ( A ;r ) be the open set defined by
U ( A ;r ) = {p E M; d~ (p, a) < r for some point a E A).
64
5
Invanant Distances on Complex Martifolds
With this notation, we have Proposition 5.1 Let o be a point of a com.plex manifold M and let T' be positive numbers. Then
T
und
+ U ( O r; + r')
U [ U ( o ; r )T']; = U ( o ; r r').
P r o o f The inclusion U [ U ( o 7.); ; r'] c is true for any pseudodistance and makes use of the triangular axiom only. In order to prove the ix~clusionin the opposite direction, let p E U ( o ; r r') and set d M ( o , p )= r + r' - 3 ~ Then . there are points a,, bi E D and holomorphic mappings f, : D + hl, i = 1,. . . , k, such that
+
fl(a1) = 0 , fi(bi) = fi+l(ai+l) for i = 1 , . . . ,k - 1, fk(bk) = pt
II
t=l Let j be the largest integer, 1 5 j 1-1
j-1
If we set
< r' so that p E U ( q ; r i ) c U [ U ( o ; r ) : 0
In the proof of Theorem 4.7 we promised to show that if a hyperbolic manifold A l is complete in the sense that every Cauchy sequence converges. then the closure u ( o :r ) of Cr(o;T) is colnpact for all o E M and all positive numbers r. Itre shall show that this assertion is true for a larger class of metric spaces. T h e o r e m 5.2 Let M be a locally co7n.pact metric space with distance function d satisfying the equality
U [U(0;r ) ;T'] = U(o;r
+ T')
Lemma U ( o ;r ) is compact if there exists a positive number b such that O@;b) is compact for every p E U ( o ;r ) . proof Since h! is locally compact, there is a positive number s < r such that. B(o; S ) is compact. It suffices to show that if U(o;s ) is compact, so is U[o; . s + ( b / 2 ) ] . L e t p l , p z , .. . b e p o i n t s o f ~ [ o ; s + ( b / 2 ) ] . ~ e t q l , q .z.,bepointsof 8(0; S ) such that d(p,, q,) < $b. Since ~ ( os);is compact, we may assume (by choosing a subsequence if necessary) that ql, qzI.. . converges to some point, say q, of u ( o ;s). Then U ( q ;b) contains all p, for large i . Since U ( q ;b) is compact by assumption, a suitable subsequence of pl?p2, . . . converges to a point p of O(q;b). Since P I , pz, . . . is a sequence in u [ o ;s (b/2)]and c [ o ;s (b/2)] is a closed set, the limit point p lies in u [ o ;s (b/2)].This completes the proof of Lemma.
+
+
The proof of Theorem 5.2 is I I ~ Wreduced to showing that there exists a positive number b such that, b) is compact for all p E Al. Assume the contrary. Then there exists a point pl E M such that U ( p l ; is noncompact. Applying Lemma to $ ) we see that there exists a point p2 E u ( p 1 ; such that c r [ p 2 ; ( 1 / 2 2 )is] noncompact. In this way we obtain a Cauchy sequence p l r p 2 , p : ~. ., . such that pk E u [ p k - l ; ( 1 / 2 ~ - ' ) ] and I?[pk; ( 1 / 2 ~ )is] noncompact. Let p be the limit point of the Cauchy sequence pl,p2,. . . . Since M is locally compact, for a suitable positive c ) is compact. For a sufficiently large k, U ( p k ; ( 1 / 2 ~ )is] number c, a closed set contained i11 O ( p ;c) and hence must be compact. This is a contradiction.
4)
4)
E.
a= 1
then dbf(o,q) < r and dh*(q,p) r'].
proof U'e have only to prove that if M is (Cauchy-) complete, then U ( o ;T ) is compact; the implicat.ion in the opposite direction is trivial.
u(~;
Let ci be the point on the geodesic born uj to bj in D such that
65
for all o E M and all positive numbers T and r'. Then M is complete i n the sense that every Cauchy seq~~ence converges if and only if the closure a ( o ; r ) of U ( O r; ) is compact for all o E M and a.11 positive numbers r .
+
2 k, such that
1 @(a,,bi) + @ ( a jc, j ) = r -
On Completeness of an Invariant Distance
u(~;
Remark Let M be a hyperbolic manifold and 44' its completion with respect to the distance dnf. Unfortunately, h l * need not be locally compact and hence it need not be complete in the strong sense that every closed ball of radius r in hf* is compact. An example of a hyperbolic manifold M such that M* is not locally compact has been found by Kiernan [4].A similar example of a Riemannian manifold has been found by D. B. A. Epstein (see Kobayashi [4]).
! I
! I
CHAPTER V
Holomorphic Mappings into Hyperbolic Manifolds
1 The Little Picard Theorem The classical theorem of Liouville states that every bounded holomorphic function on the entire complex plane C is a constant function. We may prove this using the distance-decreasing property of the Carathbdory distance. Let D be a bounded domain and f : C -+ D a holomorphic mapping. Since the Carathhdory pseudodistance c c for C is trivial and the CarathCodory pseudodistance C D for D is a distance, the distance-decreasing property off implies that f is a constant mapping. The same reasoning applies to the invariant pseudodistance d M . Delete two points a and b from C . We know (see Corollary 4.12 of Chapter IV) that C - {a,b ) is hyperbolic. On the other hand, dc is trivial. Hence, every holomorphic mapping f : C -+ C - {a, b ) is a constant mapping; in other words, every entire function with two lacunary values must be a constant fu~~ction. This is the so-called little Picard theorem. We may now state
Theorem 1.1 Let M' be a complex manifold on which a complex Lie group acts transitively. Let M be a hyperbolic manifold. Then every holomorphic mapping f : M' + M is a constant mapping. The fact that the pseudodistance d ~ for. such a manifold h1' is trivial was established in Sec. 1 of Chapter IV (see Example 3). 2
The Automorphism Group of a Hyperbolic Manifold
The following lemma is due to van Dantzig and van der U'aerden [I]. For the proof, see also Kobayashi and Nomizu [I, pp. 46-50]. h m m a The group I(M) of isometrics of a connected, locally compact metspace M is locally compact with respect to the compact-open topology, and isotropy subgroup I p ( M ) is compact for each p E M . If M is moreover -pact, then I ( M ) is compact.
*
68
Holomorphic Mappings into flyperbolic Manafolds FYom this lemrna, we derive t,he following
Theorem 2.1 The group H(M) of holomorphic transforn~ationsof a hyperbolic manifold M is a Lie transformation group, and its isotropy subgroup H , ( M ) at p E M is co~npact.If M is moreover compact, then H ( M ) is finite. Proof Let I ( M ) t.he group of isonletries of M with respect t o the invariant distance dnt. Since H(A.1) is a closed subgroup of I ( M ) ,it follows from Lemma that H ( M ) is locally compact with respect to the compact-open topology and H,(h.i) is compact for p E hl.By a theorem of Bochner and Montgomery [I], a locally compact group of differentiable transformations of a nlanifold is a Lie transformation group. Hence, H ( M ) is a Lie transformation group, thus proving the first assertion. Assume that M is compact. Another theorem of Bochner and Montgomery [2] states that the group of holornorphic transformations of a cornpact conlplex manifold is a complex Lie transformation group. The second assertion follows from the following theorem. Theorem 2.2 A connected com.plex Lie group of holomorphic transfom.ations acting effectively o n a hyperbolic ma~rifoldM reduces to the iden,tbty element only.
I
Proof Assume the contrary. Then a complex one-parameter subgroup acts effectively axid holomorphically on M. Its universal covering group C act,s essentially effectively on M. For each point p E M, this action defines a holomorphic mapping z E C + z ( p ) E M . By Theorem 1.1, this hole morphic mapping C -+ hf must be a constant mapping. Since the identity element 0 E C maps p into p, every element z of C maps p into p. Since p is an arbitrary point of M , this shows that the action of C on M is trivial. 0 contradicting the assumption.
2
The Autornorphism
Group of a
Hyperbolzc
Manifold
69
We shall explain Theorem 2.2 from a slightly different viewpoint. Let
X be a holomorphic vector field on a hyperbolic manifold M. Although it generates a local one-parameter group of local holo~norphictransforniations of M, it may in general not generate a global one-paranleter group of holomorphic transformations. If it does, we call X a complete vector field. If X is holomorphic, so is J X , where J denotes the complex structure of M.Theorern 2.2 means that if X is complete, then J X cannot be complete. For a bounded domain in C n , Theorems 2.1 and 2.2 have been proved by H. Cartan [I]. (Of course, the second assertion of Theorem 2.1 is meaningless for a bounded domain.) We note that for a bounded domain these results of H. Cartan may be obtained by means of its Carathkodory distance. For a bounded domain, the Bcrgman metric may be also used to prove these results. Unlike the Caratheodory distance, the Bergman metric may be constructed even on some compact complex manifolds, e.g., any algebraic hypersurface of degree d > n 2 in Pn+l(C). For more details, see Kobayashi [l]. It would he of some interest to note here that one does not know if a generic algebraic hypersurface of degree d > n 2 in P,+l(C) is hyperbolic or not (cf. Problem 4 in Sec. 3 of Chapter IX). Theorem 2.2 is essentially equivalent t o Theorem D (or Theorem D') in Wu 121. For a systematic account of the holomorphic transformation group of a complex manifold, scc Kaup [I]. R o m Corollary 4.13 of Chapter IV and Theorem 2.2 we obtain the following result of Schwarz and Kleixi (see Schwarz (11 and PoincarC: (11).
+
+
Corollary 2.4 If 1b1 is a compact Riemann surface of genus g the group H ( M ) of h,olommphic transformations of M is finite.
2 2,
then
Hurwitz [l]proved that the order of H ( M ) does not exceed 84(g - 1). For similar results on algebraic surfaces, see Andreotti [I]. The following result is originally due to Bochner [I],Hawley [I], and Sampson [I].
Corollary 2.5 Let M be a compact complex manifold that has a bounded domain D c C n as a coz~eringmanifold so that ,bf = D l r , where I? is a ptoperly discontinuous group of holomorphic transformations actir1.g freely on D. Then the group H ( M ) of holom.orpl~ictransformations of 1Ci is finite.
Corollary 2.3 Let M be a Hermitian manifold wh.ose holomorphic sectional curvature is bounded above by a negative constant. Then the group H ( M ) of holornorph,ic transfor-mations of .bf is a Lie transformation group. and its isotropy subgroup Hp(hf) is compact for every p E M . No comple:~ Lie transformation group of positive dimension acts nontrivially on A l . If M is moreover compact, then H(M)is a finite group.
Proof Since D is hyperbolic (Corollary 4.5 of Chapter IV), 11.3 is also hyperbolic by Theorem 4.7 of Chaptcr IV. Our assertion now follows from Theorem 2.2.
Proof This is immediate from Theorem 4.11 of Chapter IV and from 0 Theorems 2.1 and 2.2.
It is possible to obtain Corollaries 2.4 and 2.5 from the following result (Kobayashi [5]):If M is n compact con~plesmanifold whose first Chern class
Holomorphic Mappings into Hyperbolic
70
Manifold8
3
cl(M) is negative, then the group H(M) of holomorphic transformations of M is finite. A compact Kaehler manifold M with negative Ricci tensor satisfies cl ( M ) < 0. T h e o r e m 2.6 Let M be a hyperbolic manifold of complex dimension 72. Then the group H(M) of holomorphic transformations of M has dimension 5 271 n2 and thee linear isotropy representation of the isotropy subgroup Hp(M),p E M, is faithful and is contained in U(n).
+
Proof Let Q : H,(M) + GL(n; C ) be the linear isotropy representation. Since Hp(M) is compact by Theorem 2.2, its image e[Hp(M)]is compact, and is contained in U(n) [after a suitable change of basis in the tangent. space Tp(M)]. Since Hp(M) is compact, we can find a Riemannian metric on M invariant by H,(M) so that H,(M) is contained in the group of isometries of M with respect to that Riemannian metric. It follows then that Q is faithful. Hence, dim Hp(hl) 5 dimU(n) = n2. Finally, dim H ( M ) 5 dim A4 dim H , ( M ) 5 271 n2.
+
+
R e m a r k It is not difficult to see that if dim H ( M ) attains its maximunl 2n n2 in Theorem 2.6, then M is biholomorphic to the open unit ball in C". We shall only outline its proof. The assumption din1 H ( M ) = 2n n2 implies that H,(Af) = Lr(7~)for each p E ,'If and H ( M ) is transitive on ,I! Then there is a Herrnitian metric on M which is invariant by H(M). Since the isotropy subgroup Hp(M) = U(n) contains the element -I, there is no nonzero tensor of odd degree (i.e., odd number of indices) at p, which is invariant by Hp(M). In particular, the torsion tensor field of the invariant Herrnitian metric must vanish. Hence, the metric is Kaehlerian. Sincc U(n) is transitive on the unit sphere, the holomorphic sectional curvature of this invariant Kaehler metric is a constant. Since M is hyperbolic, this constant must be negative. Since a homogeneous Riemannian rnanlfold of negative curvature is simply connected (see Kobayashi and Nomizu [l,Vol. 11, p. 105]), M is simply connected. Being a simply connected complete Kaehler manifold of negative constant holomorphic sectional curvature, M is biholomorphic with the open unit ball in C n , (see Kobayashi and Nomizu [ l , Vol. 11, pp. 169-1701).
+
Holomorphic
Mappings znto Hyperbolic Manifolds
71
T h e o r e m 3.1 Let M be a connected, locally compact, separable space with pseudodistance d~ and N a connected, locally compact, complete met& space with distance d ~ The . set Fof distance-decreasing mappings f : M -, N is locally compact with respect to the compact-open topology. In fact, if p is a point of M and K is a compact subset of N, then the subset F(p, K ) = {f E F ;f (p) E K ) of F is compact. Proof Let { f,,) be a sequencc of mappings belonging to F(p, K ) . We shall show that a suitablc subsequence converges to an element of F(p, K ) . We take a countable set {pi) of points which is dense in M. We set Ki = { q E N; ~ N ( Q K ,) I d ~ ( ppi)). , Then Ki is a closed dM(p, pi)-neighborhood of K and hence is compact. Since each f n is distance-decreasing so that. f,(pi) is in the compact set K, for every n. By the standard argument of taking the diagonal subsequence, we can choose a subsequence ifn,) such that { f n k (pi)) converges to some point of Ki for each pi as k t,ends to infinity. By changing the notation, we denote thus subsequence by {f,) so that { f, (pi)) converges for each pi. We now want to show that { fn(q)) converges for each q E M . We have
+
3 Holomorphic Mappings i n t o Hyperbolic Manifolds The following theorem is similar to the result of Van Dantzig and Van der Waerden (see Sec. 2), but. it is easier to prove.
Given any positive number E , we choose pi such that 2 d ~ ( q , p , )< &/2. We , < &/2for n, m > no. choose also an integer no such that d ~ ( f n ( p i )fm(pi)) Then dN(fn(q),frn(q)) < E for n , m > no. This shows that {f,(q)) is a Cauchy sequence for each q. Since N is complete, we may define a mapping f:M-+Nby
f (Q) = n+Jc lim
fn(q).
Since each fn is distance-decreasing, so is f . Since fn(p) E K for each n, it follows that f maps p into K. We shall complete the proof by showing that the convergence f,(q) + f (q) is uniform on every compact subset C of M . Let b > 0 be given. For each q E C! choose an integer n, such that d ~ ( f ~ ( qf )( q, ) ) < 6/4 for n > n,. Let U, be the open 614-neighborhood of 9 in M. Then for any x E U, and n > n,, we have
72
3
Holomorphic Mappings into Hyperbolic Manajolh
Holornorphzc Mappmgs into Hyperbolic Manijolds
73
Now C can be covered by a finite number of U,'s, say Ui = Uqi,i = 1,. . . ,s. It follows that if n > max,{n,), then
P fo # 0. Then dn'( fk)), = kdmf, for all positive integers k. As k goes to infinity, dn'(fk), also goes to infinity, in contradiction to the fact that F,
djv [fn (x),f (x)] < 6 for each z E C.
is compact. This proves (2). Assume Idet,df,l = 1. From ( I ) , it follows that all eigenvdues of df, have absolute value 1. Put df, in Jordan's canonical form. We claini that df, is then in diagonal forni. If it is not., it must have a diagorlal block of this form:
0
Comparirlg this proof with that of Lemma for Theorem 2.1, we find that Theorem 3.1 is easier for the following reasons: In Theorerr1 3.1, -4' is assumed to be complete and there is no need to prove the existence of f-'. As an application of Theore111 3.1 we have
Theorem 3.2 Let M be a com,plle manifold and N a complete hyperbolic manifold. Then the set F of holom.orphic mappings f : M 4 N is locally compact with respect to the compact-open topology. For a point p of M and a compact subset K of N, th.e subset F(p, K) = {f E F;f(p) E K ) of F is The corresponding diagorial block of (df,)"s
then of the form
Proof This follows from Theorem 3.1 and from the fact that if a sequence of holomorphic mappings f, E F converges to a continuous mapping f in the compact-open topology, then f is holomorphic. Theorem 3.3 Let M be a h.yper.bolic man.ifold and o n point of hi. Let f : M -+ AJ be a holomorphic m.o.pping such. that f(o) = o. We denote bl! df, : T,(M) -+ T,(M) the difierential off at o. Then
It follows that the entries kX"-' irrl~~ietliately above the diagonal of ( ~ f , ) ~ diverge to infinity as k gocs to infinity, co~ltradictingthe compactness of F,. Since df, is a diagonal matrix whose diagonal ent,rics have absolute value 1, there is a subsequence {(dfo)ka)of {(df,)" such that (dfo)ka converges to the identity matrix as i goes to infinity. We denote now by F, the set of all l~olornorphicmappings from M into itself leaving the point o fixed. Assume that M is complete. By Theorem 3.2, F, is compact. Then there is a subsequence of {fkl) which converges to a holornorphic mapping h E F,. By change of notations. we may assunle t,hat { f k a ) is this subsequence. Them dh, = linl d(fk'), is the identity transformation of T , ( M ) . BY (2) of the present theorem, h must be the identity transformation of M. Since F, is compact, there is a convergent subsequence of ifki-'). By change of notations, we may assume that { f ki-l) converges to a holomorphic mapping g E F, as i goes to infinity. Then
(1) The ei,qenvah~esof df, have absolute velue 5 1; (2) If df, is the identity linear transformation., then f is the identity transformation of M; (3) If ldetfol = 1, then f is biholomorphic mapping.
Proof Let T be a positil-e ~lurnbersuch that the closed r-neighborhaud B, = {p E M ; dn.,(o,p) 2 r) of o is compact. Let F, denot,e the set of all (holomorphic or not) mappings of B, into itself which leave o fixed and are distance-decreasing with respect to diM.By Theorem 3.1, F, is compact. Let f : M -, M be a holomorphic mapping such that f ( o ) = o. Let X be and eigenvalue of df,. For each positive integer k, the mapping f ', restricted to BT,belongs to F, and its differential (df,)k at o has eigcnvalue Xk. If /XI > 1, then lXkl goes to infinity as k goes to infinity, in contradictioll to the fact that F, is compact. This proves (1). For the sake of siniplicit.y,we denote by dTnf, all partial derivatives of order m at o. We want to show that if df, is the identity transformation of T,(M), then dmf, = 0 for 7n 2 2. Let m be the least integer 2 2 such that
f o y = f o (lim fkz'-')
i
= lim f k l = identity transformation.
Similarly, g o f is the identity transforn~ationof M. This shows that f is a biholomorphic mapping with inverse g. If M is not complet,e, we argue as follows. Let r be a small positive number such that the closed r-neighborhood B, of o is conipact. We denote
74
Holornorphic
Mappinys into Hyperbolic Manifolds
3 Holomorphic
again by F, the set of all ~nappi~igs of B, into itself wliich leave the point 0 fixed and are distance-decreasing with respect to d ~ By. Theorem 3.1, I;;, is compact. \f7e obtain a sequence { fkb) which converges to an element h of F,. Since the convergence is uniform, h is holomorphic in the interior of B, and dho = limd(fki), is the idcntity transformation of T0(M). Frolil the proof of (2) above, we see t.hat h is the identity transformation of B,. Let W be the largest open subset of M with the property that some subsequence of converges to the identity transformation on W. (To prove the existence of W , consider the union W = UWj of all open subsets I.I..', of 11.1 such that on each Wj sonie subsequence of { f "} converges to the identity transformation. A countable number of Wjls already cover W. We consider the corresponding countable number of subsequences of { f and can extract a desired subsequence of for W by the standard argumerlt using the diagonal subsequence.) Without loss of generality, we may assurrle that f k a converges to the identity transformation on W. Since W cont,ains the interior of B,, it is nonempty. Let p E W and U a neighborhood of p with compact closure U.Since lim f k i (p) = p and each f k' is dista11c:edecreasing, there is a. neighborhood V of p such that fkl(V) c U for i 2 i,,. Let F be t,he set of all distance-decreasing mappings from V into By Theorem 3.1, F is compact. We extract a subsequence from {fki} which is convergent on V . Since it has a converge to the idcntity transfor~nationon V n W, it converges to the identity transformation on V. This proves t,liat W is closed and hence W = M. The remainder of the proof is the same as the case where M is complete. 3
Proof Let aD denote the boundary of D, i.e., aD = D - D. Since f(aD x [O,11) is a compact subset of M , there exists a compact neighborhood K of f (aDx [0, 11) in M. Then we can find a point 2, E D such that f (zO,t) E K for 0 5 t 2 1. By Theorem 3.2, the family F(zo, K) of holomorphic mappings D --, M which send zo into K is compact. Since each f t is in F(zo, K ) for 0 5 t < 1, the limit mapping f l must be also in F(zo, K ) . In particular, fl maps D into M , i.e., f ( z , l ) E M for z E D. Let B, be the unit open ball in Cn. Fkom Theorem 3.4 we see that
B,, - (0) (where 0 denotes the origin) is not complete hyperbolic for n 2 2.
fkl
u.
j
(1) for each t E [O,l] the mapping f t : D --t Cndefined by ft(z) = f ( t . t ! is holomorphic and (2) f (z, t) E M unless lzl < 1 and t = 1, maps D x [O, 11 necessarily into :If. For other definitions of pseudoconvexity and their equivalence with this definition, see Bremermann [2] and Lelong [I]. As an application of Theorem 3.2 we shall prove the following theorem.
75
Theorem 3.4 If a domain M in Cn is complete hyperbolic, then it is pseudoconvex.
{fkl}
For a bounded domain, Theorem 3.3 is due to H. Cartan [2, 31 (and also to CarathCodory [3]).Theorem 3.3 has been proved by Kaup [I] under weaker conditions. It has been also proved independently by Wu [2] ur~dcr slightly stronger conditions. Let M be a domain in Cn and D the closed unit disk in C.According to Oka [I], M is said to be pseudoconvex if every continuous mapping f : b x [0, 11 -t Cnsuch that
Mappznys into Hyperbolic Manifolds
I
CHAPTER VI
The Big Picard Theorem and Extension of Holomorphic Mappings
1 Statement of the Problem The classical big Picard theorel11 is usually stated as follows:
I j a function f ( z ) h,olomorph.ic i n the punctured disk 0 < 1%) < R h.us o.n essential sin.g~gulnritpat z = 0, th.en. th,ere is at most one value a (# oc) such, that the equa.tion f (2) = a has orr.1y a finite number of so1utio.n~in Ure disk. We nlay rephrasc the st,atement abovc as follows. I f f (z) = a has only a finite number of solutioi:~in the disk, then it has no solutions in a s~naller disk 0 < lzl < R', R' 5 R. Hence! thc big Picard theorcrn says t.hat if a function f (2) holonro.ryl~.icin the punctu.red disk 0 < (21 < R nlisses two then it l ~ o . sn removable sing.ulu~.ityor a pole ut 2 = 0 , val,ues a, b if ,x,)? i-e., it can be extended to u meromorphic functio71 on the disk lzl < R. In other words, iJ f is a holomorphic m.upping from th.e punctured disk 0 < 121 < R into 111 = P l ( C ) - ( 3 points), then f can. be extended to a holomorphic 7riappin.g from the disk jzl < R into PI ( C ) . Wc consider the follonring problem in this section.
Let Y be a co7n.plen: mnn~ifoldnn.d A I u .submon.ifold which is hyperbolic and relatively compact. Given a holomoryhic m.nppin,,qf from the prin,ctured disk0 < I, - ] < R into !\.I, is it possible to extend it to a holoniorphic m.appin.g from th,e disk 121 < R i n f o Y'? Rrc shall give an afilrnative answer in some special cases. IVe shall consider also the case where Y and 111 are complex spaces and the doniairl is of highcr ditnension. The following example by Kierrlan [A! shows that the answer to the Question above is ir1 general negative. (It is slightly sinlpler than the original example of Kicrnan.) Let Y = P 2 ( C ) with homoger~eouscoordirlate system ( u . v , w ) . Let 121 = ((1, t . , u l ) E P2(C):0 < )z*I < 1, lwl < le'/UI}.Then the mapping
The Big Picani Theorem and Eztension o j Holomorphic Mappings
78
(1, v , w ) -+ ( v ,we1/") defines a biholomorphic equivalence between M and D* x D ,(where D* = { z E C;O < Izl < 1 ) and D = { z E C ; l z l < 1)j. Hence, M is hyperbolic. Let f : D* + M be the mapping defincd by
Then f cannot be extended to a holomorphic mapping from D into P2(Cj. 2
T h e Invariant Distance o n t h e P u n c t u r e d Disk
Let D be the opcn unit disk in C and D* the punctured disk D - (0): i.e., D* = { z E C ; 0 < Izl < 1). Lct H be the upper half-plane { w = u + i v E C; v > 0) in C. The invariant metric ds$ of curvature -1 on H is given by
dw dtZ ds% = -. u2 Lct. p : H
-,
D* be the covering projection defined by
2
The Invariant Distance on the Punctured Di9k
79
proposition 2.1 Let L ( r ) be the arc-length of the circle It( = r < 1 with ~ s p e c tto the invariant distance do. of the punctured unit disk D*.Then L(r) = a/ log(l/r), wh.ere a is a constant. In particular,
Since we want to consider not only complex manifolds but also complex spaces, wc state the following proposition for a metric space M , not just for a hyperbolic manifold d l . Proposition 2.2 Let M be a locally simply connected metric space with distance function d , t ~Let . D* be the punctured w i t disk with the invariant distance d o - . Let f : D* -, M be a distance-decreasing mapping. Assume that there is a sequence of points { z k ) i n D* such that lirnk,, t k = 0 and f ( z k ) converges to a point pg E M . Then for each positive r < 1, f maps the circle lzl = r into a closed curve wh.ich is h.omotopic to zero. Proof Set, rk = 1zkl. Since the closed curves
z = p(.w) = e2niw for w E H. Lct ds:,
be the met.ric on D* defined by
1 tt = e-4X", we obtain easily Since dz = 2 ~ i z d u and
For each positive number r < 1, let L ( r ) denote the arc-length of the circle It1 = r with respcct to dsg.. Then
In the definit,ion of the pseudodistance dhf in Sec. 1 of Chapter IV, the distance Q on D is the one defined by the PoincarbBergman metric da), of D.LVithout loss of generality (i.e., by multiplying a suitable positive tollstant t o the ~netric),we may assume that d s L has curvature - 1 so that is not only biholomorphic but also isometric to D.Then the invariant distance dH of H is the one defincd by ds$. By Proposition 1.6 of Chapter I\', the invariant distance du. coincides with the one defined by ds;. . We state the result of this section in the form convenient for later uses.
are all homotopic to earh othcr. it suffices to prove that, for a sufficiently large integer k, the closed curve f(yk) is homotopic to zero. Let U be a simply connccted neighborhood of po in M and take a s~nallcrneighborhood V such that V c U . Let N be an integer such that f (2,) E V for n > N. Since f is distance-decreasing, Proposition 2.1 implies that the arc-length of the closed curve f (yk) approaches zero as k goes to infinity. Hence, if k > N is sufficiently largc, f ( y k ) is contained in U. Since U is simply connccted. f(yk) is homotopic to zero. Corollary 2.3 Let M be a locally simply connected, compact metric space ~ t distancefinction. h dnr. Let D*be th.e punctured unit disk with the invariant distance d,.. I f f : D* + M is a distance-decreasing mapping, then f maps a circle Izl = r < 1 into a closed curve ,which is homotopic to zero. Let M, D*, and f be as in Proposition 2.2 or Corollary 2.3. Let A? be a covering space of M with projection T . Then f can be lifted to a mapping f :D* + .CI such that f = T o j. T h e o r e m 2.4 Let M be a complex manifold which has a covering rnunifold fi with Camth,Codory distann c ~ Let. D* be the punctured disk Let f : D* -+ A.1 be a holonc.orphzc mapping such that: for (L suitable sequence
80
The Big Pzcard Theorem and Extensao~roJ Holomorphic Mappings
of points zk E D* converging to the origin, f ( z k ) converges to a point po E A9. Then f extends to a holom.orphic mapping of the (complete) disk D into Af .
Proof Since ccf is assumed t o be a distance, dIG is also a dist,ance b!; Proposit,ion 2.1 of Chapter IV and hence is hypcrbolic. By Theorem 4.7 of Chapter IV, ,l,l is also hypcrbolic. Since f : D* -+ M is distanccdecreasing with respect t o do* and d5*, we can lift f to a holomorphic mapping f : D* + ,I? as we have seen above. Then f ( z k ) converges to a poiilt Po E such that x(po)= po, where a is the projection 1G+ M . The mapping f : D* -+ $1 is dist.ance-decreasing with respect to the Carathhodor!; distances c ~ and . c,,>. Since the disk D is the completion of D' with respect t.o the CarathCodory distance c u - (see Sec. 3 of Chapt.cr IV), f can bc extended to a holomorphic rnapping f, : D + ~$1such that f ( 0 ) = yo. It, follou7s that f can be extended t o a holomorphic mapping f : D -+ M such that, f ( 0 ) = W. Corollary 2.5 Let 12.I be a compact complex m.onifold which has a co~:er.in.g rnanifold Kl with. Garath.dodory distance c g . Then every holomorphic m.0.ppin.9 from the pun.ctu.rsd disk D* into M can be exten.ded to a h.olomo~hir: mnppin.g f7.ona the disk D into JV.
Remark Theorem 2.1 can be applied t,o a complex manifold Ad of the form hl/I',where Jif is a bounded domain in C n and r is a propcrly discontinuous group of holomorphic transformations acting freely on ~ i lIf. M = ?1'I/I' is compact, Corollary 2.5 applies. From Theorem 2.4 we obt,ain also the follouring:
3
Mappings from the Purlclured Dask into a Hyperbolic Manifold
81
The results of this section arc due essentially to Huber 121, who obtained of the big Picard theorem in the following form: Let P be a Riemann surface and M a domain of hyperbolic type. Then every holomorphic mapping f : D* + M can be extended to a holomorphic mapping f : D + Y. This follows from Corollary 2.6 and frorn the fact that the boundary of M is contained in another subdomain of Y which is of hyperbolic type. But this last. fact is known only for Riemann surfaces. 8, generalization
3 Mappings from the Punctured Disk into a Hyperbolic Manifold
In this section we shall prove the following theorem of Kwack [I], which generalizes Theorem 2.4. Theorem 3.1 Let M be a hyperbolic man.ifo1d and D* th.e punctured unit disk. Let f : D* + M be a holomorphic rnapping such tha.t, for a suitable sequence of poin.ts zk E D* con.t~ergingto the origin., f ( z k ) converges to a point po E :II.Th,en f extends to a holom.orphic mapping of the unit disk D into M . Corollary 3.2 If i1.l is a com.pact h;ypcrbolic mmzifold! then ezlenJ h.010morphic 7nnppin.g f : D* + 11.1 exten.ds to a h.olomo~phicnropping of D into A l . Proof As in Sec. 2! we set rk = Izkl, y k ( t ) = f ( ~ k e * ~ " ) o? s t s l , k = l , 2 ,....
In other words, y k is the image of the circle
Corollary 2.6 Let M be a complex suhmariijo1d of a complex manzfol(l Y such that M is compact. tlssume th.at .II lias a covering ma7l.ifold ~r!ithCa:mthCodory clista.nce c,~,. Then. ever?/ h.olornorphic mappin.9 f of the punctured disk D* into 111 sntisfies one of the following two conditions:
Jil= rk by f . Let U be a neighborhood of po in M with local coordinate system w', . . . ,w n . \Ve may assume that po is at the origin of this coordinate system. Let E bc a small positive number and let V be the open neighborhood ofpo defined by
(1) f can bc extended to a haolorriorphicmapping of the disk D into A l ; (2) For every neighborhood N o f the boundary B,li = ,\I of d l i71 117.there exists u neighbor-hood U of the origin in the disk Dstrch that f ( U - ( 0 ) ) C ni.
Taking E sufficiently small, we may assurne t.hat C' neighborhood of po defined by
If we set Y = P l ( C )and M = Y - (3 poirlts). thcn Corollary 2.6 the classical big Picard theorem irnr~lediatcly.
The problem is to show that, for a suitable positive r~umberd: the small Punctured disk { z E D*; Izl < 6) is mapped into Cr by f .
.cf
c U. Let W
be the
82
3 Mappings from the Punctured Disk into a Hyperbolic Manifold
The Big Picard Theorem and Extension oj Holomoryhic hlappinqs
Since the diarnet,er of yk approaches zero as k goes to infinity by Proposition 2.1, all but a finite number of yk's are contained in W . Without loss of generality we may assume that all ~ k ' sare in U '. By taking a subsequence of {zk) if necessary, wc may assume also that the sequence. {rk) is monotone decreasing. Considcr the set of integers k such that thr. image of thc annulus rk+l < lzl < r k by f is not entirely contained in W. If this set of integers is finite, then f maps a small punctured disk 0 < lzl < 0 into Iq. Assuming that this set of integers is infinite, 'we shall obtain a contradiction. By taking a subsequence, we may assumc that, for every k, the image of the anllulus rk+, < < rk by f is not entirely contained in W. For each k, let
IzI
be the largest open annulus such that (1) a k into W . We set
< rk < bk and (2) f maps Rk
83
It follows that if k is sufficiently large, then 1f k is sufficiently large, we can find a simply connected open neighborhood Gk of -wl(q)in C such that fl(ok) C Gk and f '(zk) $ Gk.We apply Cauchy's theorem to the holomorphic function l/[wl - fl(tk)]and the &sed curve f '(ak)in G k . Then
This may be rewritten as follows:
where f ll(z)= df l(z)/dt. Similarly, if k is sufficiently large,
On the other hand, the principle of the argument applied t.o the func111 other words, ak is the inner bo~lntfaryof the annulus Rk and rk is thc outer boundary of thc annulus Rk.Fro111 the definition of ak and b k , :t is clear that bot.11 f (ak)and f (rk)are contained in but not in I;t'. By Proposition 2.1, the diameters of f (ak)and f (rk)approach zero as k goes to infinity. By taking a subseqne~iceif necessary, we may assume that t l ~ c sequences { f (ak)) and { f (rk)) converge to points q and q' of ic - I!'. respectively. Since po is in' W and both q arid q' are on the boundary of I!.. t,he points q and q' are distinct from pu. (The point q might. coincide with the point q'.) By taking a new coordinate system around po if iiecess;ir~. we may assumc that
'
Let f (2)= [ f l( z ) . , . . , f n ( ~ )be ] the local expressioti of f on f - (C) D. Then lirn fl(rrk) = u ~ ' ( ~ ) ,
k-ZL:
lirn f 1(rk)= .up1(q'),
k-m
tion fl(z) - fl(tk)defined in a neighborhood of the annulus bounded by the curve r k - ak yields the following equality:
"(')
Rk which is
dz = 2ni(N - P),
where N and P denote the numbers of zeros and poles of fl(z)- fl(zk) in Rk. In the present situation, P = 0 and N 1 1. We have arrived finally at a cont.radiction. From Theorem 3.1 we obtain also the following:
Corollary 3.3 Let h9 he a hyperbolic subnlanifold of a complex man.ijo1d Y such that h? is compact. Th.en every h o l o m o ~ h i cmapping f of the punch ~ disk d D* into Jl satis$es one of the following two conditions: (1) f can be extended to a holomorphic mapping of the disk D into 1\11 (2) For every neighbortiood of N of th,e boundary aM = M - 111 of M in
M ,there ezists a neiyhlorliood U of the origin i n the disk D such that f (U - (0)) c 1v. This generalizes Corollary 2.6.
84
4
5
The Big Picard the or en^ and Extension of Holomorphic Mappings
Holomorphic Mappings into Compact Hyperbolic Manifolds
As an applicat.ion of Theorem 3.1 we shall prove the following result of Kwack [I].
Theorem 4.1 Let M be a compact hyperbolic manifold. Let X be a complp~ man7jold of dimension nz an,d A an analytic subset of X of dimension 5 Tn - 1. Then every holomorphic mapping f from X - -4 into hl can 1f extended to a holomorphic mapping from X into .h1. Proof k1.k shall first show that it suffices to prove the theorem when is a nonsingular complex sublnanifold of X. Let S be the set of singula points of A. Thcn S is an analytic subset. of X and dim S < dim A (see. for instance, Narasimhan [ l ,pp. 56-58]). Since A - S is a non-singular co~rlplcx submanifold of X - S, we extend first f t o a holomorphic mapping from X - S into AI. Since dim S < dim A, we obtain the theorem by ind~lction on the dimension of A. Ftk shall now assume that A is nonsingular. For each point a of A ~ v c want to find a neighborhood U in X such that f (U n (X - A) can hc extended to a holornorphic mapping from W into :If. We may thercforc asslime that X is a polydisk
Holomorphic Mappings into Complete Hyperbolic Manifolds
distance-decreasing, we have
We shall first consider the case z # 0. Since f : D * x Dn1-I holomorphic and hence distance-decreasing, we have
M is
where the equality is a consequence of Proposition 1.5 of Chapter IV. \Vc have now
provided that z # 0. We shall now consider a point (0,t ) E V x W. Choose any a E V different from zero. Since f : D x { t ) -+ M is holomorphic and hence distance-decreasing, we have
Hence,
This proves that f is continuous at (0,O). and that A is contained in thc subset defined by z = 0. For the sake of simplicity, wc denote ( t ' , . . . ,tm-') and ( z , t l , .. . , tm-') by t and (z,t). respectively. For each fixed t E Dn'-'. we have a holomorphic ~xiappingft from t h e punctured disk D* into .hrl defined by f t ( z ) = f ( z . t ) . Applying Corollary 3.2 to each ft, we extend ft t o a holomorphic mapping ft : D -,M and we st.[ f ( 0 ,t ) = f t ( 0 ) . \?re have to prove that this extended mapping f : X 4 is holomorphic. By the Riemann extensiori theorem? it suffices t,o slio\v that f : X -, :\I is conti~luousat every point of A. To prove that f is co1itinuou3 at a E A, we may assume without loss of generality that a is the origin ( 2 , t ) = (0,O). IVe set p = f (0,O) E A1. Let. E bc any positive number. Let V be the E-neighborhood of 0 in D with respect, to the PoincarP distancc i' do. Let W.'be the E-neighborhood of 0 in Dm-' with respect to thc distallcc dD,rt-,. IVe shall show that f maps I/ x W' into the 3~-nei~hborhood of p in :\I with respect t o the dist,ancc dar Let ( z !t ) E V x W . Since t h e restriction of f to D x {O) is holomorphic and since f : D x ( 0 ) -+ df is
-
5
0
Holomorphic Mappings into Complete Hyperbolic Manifolds
In Theorem 4.1, if A is smaller, it suffices to assume that ,2.I is complete hyperbolic. Before we make an exact definition, we prove Proposition 5.1 Let Dn' = { ( z l , . . . , z m ) E Cm; lr31 < 1 for j = 1,. ..,rn) and let A be a. subset of Dm = Dx Dm-' of the form A = ( 0 ) xA', where A' is nowhere dense in D ~ - ' . Then the distance d ~ m -is ~the Rstriction of the distance d o to Dm - A.
...
Proof Let p and q be two points of D m - A. Since the injection Dm- A -, Dm is holomorphic and hence distance-decreasing,, we have d ~ m - A ( q) ~ . 2 d D m ( p ,q ) . To prove the proposition, it suffices to show the opposite inequality for every pair of points (p, q) belonging to a dense subset of ( D m - A) x (Dm - A).
86
The Big Picad Theorem and Eztension of Holomorphic Mappings
Let S be the subset of ( D m - A ) x ( D m - A ) consisting of pairs ( p ,q j for which there exist points a , b E D and a holomorphic mapping f : D -+ D m - A such that dDr..(p,q)= d D ( a ,b), f ( a ) = p, and f (b) = q. If ( p ,q) E 3 , then
It suffices therefore to prove that S is a dense subset. L e t p = ( a 1 , . .. ,am) andq = ( b l , . . . ,bm) bearbitrarypointsof Dm-A. To show that every neighborhood of (p,q ) in ( D m - A ) x ( D m- A ) contains a point of S, we may assume without loss of generality that a', b1 and 0 are mutually distinct, for the set of such pairs is dense in ( D m - A ) x ( D m - ,-I). The distance d o , (p,q ) is equal to the maximum of dD(ai, b j ) , j = 1,. . . ,nr. say d D ( a k , b k )(see Example 1 in Sec. 1 of Chapter IV). We set a = a" and b = bk so that durn ( p ,q) = d u ( a , b). Since d o ( a J ,b j ) 2 d o ( a , b) for j = 1,. . . , m , there exist holomorphic mappings f j : D + D such that f j ( a ) = aj and f j ( b ) = bj for j = I ! . . . ,m. Since a' # b', we may impose the additional condition that fl be injective. Then f T 1 ( O ) is either empty or a single point c E D. If f F 1 ( 0 ) is empty, then the mapping f : D + D'" defined by f ( 2 ) = ( f l ( 2 ) .. . . ,f,,(z)) scnds D into Dm - A , since f l ( z ) never vanishes. In this casc, (p.q) belongs to S. Assunlr c = f ~ ' ( 0 ) Then . thc mapping f : D --. Dnl defined above maps D into Dm - -4if and only if [f2(c).. . . ,fm(c)] is not in A'. I f f ( D ) C Dm - A, then (p,q) belongs to S So, we have only to consider thc casc [f2(c),. . . . fm(c)] E A'. We assert that given a positive number E there exists a positive number 6 such that for any points c' E D ( j = 2 , . . . , m ) with dD(dl,f,(c)) < 6 tllcre exist automorphis~nsh j : D -+ D satisfying
LVe shall first complete the proof of the proposit,ion and then come back to the proof of this assertion. Given E > 0, let 6 > 0 be as above. Since A' is nowhere dense in D"'-'. there exists a point ( c 2 , .. . , c m ) E Dm-I - A' such that d o ( & , f j ( c ) ) < 8 for j = 2 , . . . ,m. Let h j : D + D be as above. We consider tllc points p' = [a1?h2(a2),. . . , h m ( a m ) ]and q' = [bl,h2(b2),. . . , hm(b7")]of Dm and the holornorphic mapping f' : D -+ Dm defined by f l ( t ) = ( f l ( z ) h2( , f 2 ( z ) ) ,. . . , h m ( f m ( 2 ) ) ) Sincc . a': b'! and 0 arc mutually distincrboth p' and q' arc: in Dm - A. Since Ii., : D -+ D is distance-preserving, \x7"
5
-
IIolomorphic Mappings i n l o Corrcplele i1qperbolic Manifolds
87
have d o ( h j ( a J ) h, j ( b j ) )= d u ( a J ,b j ) and hence
durn
q') = ~ a x { d D ( a bj ,J ) ;j = I , . . . ,m } = dgn1(p,q ) = d o ( a ,b).
Clearly, f l ( a ) = p' and f l ( b ) = q'. This shows that (p', q') belongs to S. Since d ~ ( a jh!j ( a J ) )< E , we have d ~ n(tp ,p') < E . Similarly, du7,,( q ,q') < e. Thus, the 5-neighborhood of ( p ,q ) in ( D m - A ) x ( D m - A ) with respect to dDm x durn contains a point (p',q') of 5'. This completes the proof of t.he proposit,ion except for the proof of the assertion made above. To simplify the notations in the assertion above, we denote aj! b3, f j ( c ) , &, and h j by a , b, c, c', and h respectively. Then the assertion we have to prove reads as follows:
Lemma Let a, b,c E D be given. Then for any E > 0 there is a 6 > 0 such that for any c' E D with d o ( c ,c') < 6 there exists an automorphism h : D -,D satisfying
h (,c ) = c', ,
d -u (. a.,Ma)) , , , < E,
and d ~ ( bh(b)) , < E.
In order to prove the lemma, it is more convenient to replace D by the upper half-plane H in C . Give11 E > 0, choose 61 > 0 and 62 > 0 such that
dH ((1, r ( a
+ t ) )< E
and d~ (b:r(b
+ t ) )<
E
for any real numbers t and 1- such that It1 < 61 and Ir - 11 < 62. The set {r(c + t ) ;It1 < 61, 1r - 1I < b2) contains the 6-neighborhood of c for some 6 > 0. Given c' in the &-neighborhoodof c, we can find an autoinorpllism h : H -+ H of the form h ( z ) = r ( z t ) such that h ( c ) = c', d ~ ( ah ,( a ) ) < E and d ~ ( bh(b)) , < E.
+
As an application of Proposition 5.1 we prove
Theorem 5.2 Let ,21 be a complete h.yperbolic manifold. Let X be a complex manifold of dimension m, and let A be a subset which is nou~heredense in an analytic subset, say B, of X with dim B 5 m-1. Then every holomor~ h i cmapping f : X - A -+ M can be extended to a holomorphic mapping X -,h f . Proof As in the proof of Theorem 4.1, we can reduce the proof to the special case where X = Dm = { ( z l , .. . ,znL)E C": !zjl < 1 ) and B is the subset defined by z1 = 0 so that A is of the form A = ( 0 ) x A': where A' is nowhere dense in Dln-'. Sincc f : Dn' - A + M is distance-decreasing, f can be extended to a continuous mapping from the completion of the
88
6
The Rig Picard Ttieorem and Extension of Holontorphic Mappangs
metric space D"' - A into M. By Propositiori 5.1, Dn' is the completion of Dm - A with respect to the distance dDrn-*. By the Riemann extensiol, theorem, the extended co~itinuousmapping f : Dn' Ill is necessarily holomorphic. C
-
Theorem 5.2 contains the following result of Kwack [I], which was proved by a different method. Corollary 5.3 Let M be a complete hyperbolic manifold. Let X he rc complex m.anif01d of dimension m., and let A be a n analytic subset of dimension 2 m - 2. T h e n every holomorphic mapping f : X - A it[ can be extended to a holomorphic mapping X -+
-
H o l o m o ~ h i cMappings into Relatively Compact Iiyperbolic Martzjolds
89
To prove our claim, assume the contrary. By taking a subsequence if necessary, we may assume that each circle yk has a point zk such that f ( t k ) is not in U.Taking again a siibsequence if necessary, we niay assume that the sequence { f (zk)) converges t o a point qo of A?. In view of Theorem 3.1, we have only to considcr the case where qo is in A? - M. Since f (zk) is out,side U , the limit point qo is also outside U . Taking U sufficiently small, we may assurrle that qo is outside U so that Y - is an open neighborhood of go. By (3), there is a smaller neighborhood V of qo such that V f l U = 0 and the dist.ance between A1 n U and M n V with respect to d,tf is a positive number 6 . Then
L%f.
For extension of a holomorphic mapping X - A M where M has a covering space which is a Stein manifold, see Andreotti and Stoll [I]. -4
6
Holomorphic M a p p i n g s i n t o Relatively C o m p a c t Hyperbolic Manifolds
because f (rk) E U and f (zk) E V . On the other hand? the arc-length L(yk) of -yk tends t o zero as k goes to infinity (see Proposition 2.1). We have
This is a contradiction. We have thus shown that there exists an integer N
The following result is the best, solution we can give to the problem st.ated in Scc. 1 at the nionlent.
such that
T h e o r e m 6.1 Let Y be a complex manifold and A1 n complex sub-mu.ni.d(i of Y sutisfyzng the followiriy three condition..^:
Thc rest of the proof goes as in the proof of Thcorcm 3.1. (In order to conform with notations irl the proof of Theorem 3.1, take a neighborhood W of po defined by l~u'I< €12 in terms of a local coordi11at.esystern ul'. . . . ,lo7' in U as in the proof of Theorem 3.1. By taking a subsequence, we ~riayalso assume t,hat f ( ~ k )c Lt' for all k. ilk then proceed as in the proof of Theorem 3.1.)
(1) is hyperbolic; (2) the closure of hf i n Y is compact; (3) given, a point p of 1 q - M and neighborhood U oJ p i n Y. there e.~ist.s a neigh,borh.ood V of p i n Y such th.at V c LTa71.d the distan.ce befu!ern 3.I P (Y - CT) and 111 n V with respect to dkl is positive. Then wery holomorphic mupp.ing f from the plinct7~red disk D ' i i l f u .\I can be extended to a h,olornorphic 7na.pp.ing from th.e disk D into Y . P r o o f Let { r k ) , O < rk < 1, be a monotone decreasing sequencc w i t h lim rk = 0. W e consider { r k ) as a sequence of points in D ' converging 10 is compact, we rnay assume (by taking a subsequence if t,he origin. Since necessary) that { f (rk)) converges to a point po of :I? The.case where po is in LC1 has been already considered (see Theorem 3.1). ilk assume therefore that po E - AI. Let CT be a neighborhood of po in Y. Let 7ik denote the circle :zI = r k in D w .\Cclaim that there exists an integer N such that
lo
f ( ~ k C) Cr for k > N .
f ( ~ k )C
U for k > N .
If nrc set Y = Pl ( C ) and A1 = PI( C ) - (3 point.^). then condit+ions( I ) , (2), and (3) are niet and we obt,ain the classical big Picard t,heoren~. Example 1 Let E' = P 2 ( C )and p,, j = 1,2?3,4, be four points in general Position in P2(C). Drawing a cornplex line through each pair of these four Points, we obtain a cornplete quadrilateral Q = U'i=o Li as in Example 1 of Sec. 4 of Chapt.er IV. Let = P 2 ( C )- Q. Then conditions ( I ) , (2), and (3) are satisfied. While (1) was verified i11 Sec. 4 of Chapter IV, (2) is evident. TOverifS. (3). let p be a point of Q = (;I - M . Without loss of generality, We may assume t.hat p is not on the line Lo, which we shall considcr as the line at irlfi~iityso t,hat P2(C) - Lo = C2. By changing indices if necessary, we may assume that LI and L2 meet at infinity, i.e., are parallel in C 2 and
90
The Big Picard
Thwrem
and Bztension of Holorrrotylrzc Mappings
that LJ and L.4 are parallel in C 2 = P2(C) - LO.Aftcr a suitable liiwal transformation, we may assume that in C 2 L1 = ((0, u1) E C2;UI € C ) , LS = ((2,O) E c 2 ;2 E C),
c2;w E C ) , L.4 = { ( z l l ) E c2; 2 E C),
L2 = ((1,w) E
so that N = c2- (L1 U L2 U Lg U L4) = (C - (0.1)) x ( C - (0,l)). Lct U be a neighborhood of p in P2(C). We may assume that U does not ~rlcet Lo, i.e., U is contained in C 2 = P2(C)- Lo. It is now clear that there exists a neighborhood V of p i11 C 2 such that V c U and the distance betureen N n (C2 - U) and N n V is positive with respect to d ~Since . M = N - L;, the distance between M n [P2(C) - U ] = M n (C2 - U) and M n It7 1s positive with respect to d.tr. This proves our assertion. From Theorem 6.1 we may conclude that every holomorphic mapping f from the punctured disk D* into M = P2(C) - Q can be extended to a holomorphic mapping from the disk D into P2(C), where Q zs a complete quadrilateral. In Example 1 of Sec. 4 of Chapter IV, we constructed a complete hypcrby deleting a certain five lines from P2(C). It is not clcar bolic manifold whether Theorem 6.1 is valid for such a manifold M .
Theorem 6.2 Assume h l and Y satisfy conditions ( I ) , (2), and (3) of Theorem 6.1. Let X be a complex manifold of dimension m and A a locally closed complex submanifold of dirn.ensiorl 5 m.- 1. Then every h,olomorph.ic mappiny f : X - A + 111 can be extended to a holomorphic mapping from X into Y .
1; ,
,
I do not know whether the theorem is valid when A is an analytic subset with singular points.
,*I,
ti, i
Proof We may assume that X = Dm = D x Dm-' and .4 = (0) x D'"-' so t,hat X - A = D*x Dn'-'. I r e den0t.e a point (2: t l , . . . ,tnl-') E DT" b'! (z!t). Given a holomorphic mapping f : D* x Dm-' + h l , the restriction of f to the punctured disk D* x { t ) can be cxtended to a holomorpllic mapping from t,he disk D x { t ) into Y for each fixed t (by Theorem 6.1) 1% have to show that this extended mapping f : Dm -+ Y is continuuu' at every point (0, t) of A. It suffices of course to show that f is continuou" at the origin (0,O) E D x Dm-'. Let p = f(0:O) E Y. If p is in 31. the11 the proof is the sarne as in the proof of Theorem 4.1. Il'e assunle therefore that. p E ,Cl - 11.1.Let Cr be a neighborhood of p in Y defined by lwil < a, i = 1 , . . . n , with respect to a local coordinate syste~li.wl,. . . , u:'' ?
6
Holomorphic Mappings into Relatively Compact Hyperbolic
Manzfolds
91
s o u n d p. Let V be a neighborhood of p described in condition (3). Let b be t.he distance between M n (Y - U )and M fl V with respect to dA.,. Let r be a positive number such that the disk ( ( ~ ~ 0lzl) ;< r ) is mapped into V by f . Let r' be a positive number such that
< r' for i = 1,. . . ,m - 1. If 0 < 121 < r and It'[ < r' for i = 1:. . . !m - 1, then Itil
where the first inequality follows from the distance-decreasing property of f : D* x Dn+l + M and the second inequality follows frorn the distanccdecreasing property of thc inject.ion t E Dm-' + (z,t) E D* x DVL-'. Since f (z, 0) is in V and f (z, t ) is less than r' away frorn f (2, 0), it follows that f (2,t) is in li. By t,hc Ricmann extension theorem f is holomorphic in {(z,t);121 < r and Itil < r ' ) .
Example 2 Let M be the quotient of a symmetric bounded domain by an arithmetically defined discrete g ~ o u pr. Let Y be the Sat-ake compactification of J l . If the action of 'I is free, then 121 arlcl Y satisfy conditions (I), (2), (3) of Theorem 6.1. Even if the action is not free. M and Y satisfy similar coriditions (see Sec. 6 of Chapter VII). Theorem 6.2 applied to this example yields a simple proof of a result of Bore1 (see Kobayashi and Ochiai [I]). Although Y is a co~nplexspace with singularities, we shall see in Chapter VII that all the results in this section hold when Af and Y are complex spaces.
C H A P T E R VII
Generalization to Complex Spaces
1 C o m p l e x Spaces We shall rcvicnr quickly the definition and basic properties of complex
spaces. For details and proofs, ure refer the reader t o Gunning and Rossi [I] and Narasinihan [I]. Let R be an open set in C". A subset U of R is called an analytic set in R if every point a E R has a ncighborhood Na such that U n Ar, is given as the common zeros of functions f l , . . . ,f p holomorphic in Nu, i.e.,
Un 1% = {a E A;,;fl(x)= .. . = fP(x) = 0).
If follows that U is closed in 52, that 52 - li is dense in Q if U # f1 and that R - C: is cor~nectedif R is connected. Let R and R' be two open neighborhoods of a point a E Cn. Let U and U' be analytic sets in R and R', respectively. Ifre write (11, C') (R', U') at a if there exists an open neighborhood R" C R n Q' of the point a such that U n 11" = U' n 0". Clearly. is an equivalence relation. We call an equivalence class an analytic germ at a. We denote the analytic germ defined by (R, U) at a by Ua. For each point a E C n . let On., denote the ring of g e r m of holomorphic
-
functions at a. We denote by 0, the sheaf of germs of holornorphic functions
on c n . Let U be an analytic set in S2 c Cn. At each point a E 0 , a-e denote by the set of germs of holornorphic functions in R vanishing on the germ U,. Then Ja is an ideal of U ,,,. If a 4 U , then JU = On.,. We . = J ( U ) the sheaf of ideals Ja, a E R. of the analytic set li. denote by 7 i.e., ,7= UaEn-7,. It is subsheafof U(R) = U,,IR. Consider the quotient sheaf O(R)IU(U). Its stalk at a E R is given by On,a1J'a7,; in particular. it is zero for a $ U . The restriction of the quotient sheaf O(R)IJ'(U) to U . denoted by Ou, is called the sheaf of g e m s of holomorphic functlon-5 on CT. A section of the sheaf Uv over an open subset V
Ja= J'(lJ,)
I
94
2 Invariant Distances for Compler Spaces
Genemlzration to Complex Spaces
of U is called a ho1omorph.i~function on V . This is equivalent to the following more elernent,ary dcfinition. A continuous, complex-valued function on V is said to be ho1omorph.i~if it is locally the restriction of a function holomorphic is R. An analytic germ U, at a E R is said to be reducible if it is a union of two analytic germs at a, each of which is different from U,. Then all analytic germ U, is irreducible if and only if J(Ua) is a prime ideal of On,,. Every analytic germ U, can be uniquely written (up to order) as a finite union Ua = U,,,, of irreducible analytic germs U,,, such that, for each v , UvSaC U,+, U,,,,. For an analytic set U in R C C n , the local dimension dim, U at u E R is defined as follows. If a $ U, then dim, U = -1. If a E U, t.hen dim, Cr = n - s, where s = codim, U is the dimension of a maximal dimensional linear subspace L c C n through a such that a is an isolated point of L n U. iVe define dim U = MaxaEu dima U. If dima U = k for all a E U, then U is said to be pure k-dimensional. If U is irreducible, then U is pure k-dimensional for some k. Let U be an analytic subset in R C C n . A point a E U is called a regular point of U of dimension k if a has neighborhood N in Q such that U n N is a k-dimensional submanifold of N . A point, u E U is said t.o be sing1ili~7if it is not regular. The set of singular points of U forms an analytic subsct S of 52 such that dim S 2 dim U - 1. A local geomct.ric description of an analytic set is given as follows. Let U be an irreducible analytic set of dimension k in an open set R C (2''. Let a E U. Then there exists a local coordinate system z l , . . . , zn in a neighborhood D n = {Izil < 1; i = 1,.. . ,n ) of a i11 0 with the followin6 properties: (1) The point a is at the origin (0,. . . ,0) and is an isolated point of t h e subset ((0,.. . ,0, zk+',. . . , zn) E U n Dn) of U n Dn; (2) If we set D~ = {(tl,.. . , zk);lzil < 1 for i = 1 , . . . , k), the map x:U~D"-t~~ddenedby
is surjective and proper; (3) There exists an analytic subset S of dimension k - 1 of U n Dn such that S contains all singular points of U n Dn and x : Cr n D n - S " D k - n(S) is a finitely sheeted covering space. [Moreover, T(S) is an analytic subset of D k of dimension 5 k - 1.1
=<
95
Let X he a Hausdorff topological space and 0 = Ox a subshed of the sheaf of germs of cont.inuous funct,ions on X.The pair ( X , 0)i s called a complex space if every point a E X has an open neighborhood tl such that I/ is an analytic set in an open set 52 C C n and O1U is the sheaf of germs of holornorphic functions on the analytic set CT. The sheaf O is called the structure sheuf of X . For any open subset U of X, the conti~iuoussections of O over U are by definition t.he holornorphic functions on U.For t.he sake simplicity, uvcoften denote (X, O) by X. A continuous lnap f : X -+ Y of one col~lplexspace X into anot.her, Y, is said to be holomo.rph~icif f*(Oy,,(,)) c OX.,, for every a E X , i.e., if h o f is a funct,ion holomorphic in a neighborhood of a E X whenever h is a function holomorphic in a neighborhood of f (a) E Y . Since a corriplex space is locally an analytic space, such local concepts as "regular point", "singular point.", and "local dimension" can be defined in an obvious manner. A complex space X is reducible if it can be written as a union of two conlplex spaces, each of which is different from X. For a conlplex space X , its d.imension dim X is defined to be the maximum of its local dimension. If its local dimension is k ever>~vhere,X is said to be pure k-dimensional. If X is irredilciblc, its local dimension is a constant. 2
Invariant Distances for Complex Spaces
Let X be a connected complex space Then we can defii~ethe Carathbodory pseudodistance c . ~and the invariant pscudodistance d s as in Chapter IV. In defining d . ~ nre , have to use the fact that any two points p and q of X can be connected by a chair1 of analytic disks. To establish this fact, it suffices to provc that, given a (singular) point a of an irredurible analytic set U, there is a holon~orphicmapping f of the unit disk D into U such that f ( D ) contains a and also a regular point of U. Assuming that U is a k-dimensional analytic subset in an open set R c C n , nrc take a local coordinate system 2'. . . . ,zn in R satisfying the three conditions (1). (2), and (3) of Sec. 1. Since 7i(S)is an analytic subset of dimension 2 k - 1 in Dk, we may assume that the coordinate system satisfies the following conditions: x(S) n D' = (0). where D' = (zl!O!. . . ,0) c D~
BY (1) we may assume, by taking a smaller neighborhood of a if necessary, . (3): that a is the only point of U which projects upon the origin 0 of D ~ By : K-l(D1) - {a) + D' - (0)is an s-sheeted covering projection, where s
"
96
Generalization to Complez Spaces
is a posit.ive integer. Then it is not hard to see that t,here is a holomorphic mapping f : D -+ T-'(Dl) c U such that
Then f (0) = a and f ('Lu)is a regular point for w E D - (0). A complex space X is said to be hyperbolic if dx is a distancc. A hypm. bolic spacc X is said to be complete if it is complete with respect to ds All the results in Chapter IV can be immediately generalized to conlplex spaces except those which rnake sense only for nonsingular complex manifolds (e.g., Theorcms 3.4, 3.5, and 4.11). Similarly, the results in Chapter V can be also generalized to complex spaces. For some results in Secs. 2 and 3 of Chapter V, the following fact (scc, for instance, Gunning and Rossi [I, p. 158)) plays an essential role The spacc of holomorphic functions on a complex space is co~nplctcin the topology of uniform convergence on compact subsets. For a complex space X , the tangent space T,(X) at x E X can Lie defined as the spacc of derivations on the ring of germs of holomorphlc functions at x (scc, for example, Gunning and Rossi [I, p. 1521). It is then not difficult to gcntralize Theorem 3.3 of Chapter V to a conlplex spacc. 3
Extension of Mappings i n t o Hyperbolic Spaces
Thcorem 3.1 of Chapter VI may be generalized a s follows:
Theorem 3.1 Let Y be a hyperbolic complex space and D* the pzrn.ct,u.~(l unit disk. Let f : D* + Y be a holontorphic m.apping such th.at, fur a. suitable segueme of points zk 6 D* converging to the origin, f (zk) conz:el:qrs to a point yo E Y . Th.en f extends to a holomorphic mappin.9 of the ?init disk D into Y . Corollary 3.2 If Y is a compact hyperbolic complez space, then every holumorphic mapping f : D* -+ Y extends to a holomorphic mapping of into Y . Proof Let U be a neighborhood of yo in Y which is equivalent to a11 analytic subset in an open set R of Cn. Let uy1... . , wn be the coordinate system in Cn. We may assume that yo is the origin of this coordinate system. Then the rest of the proof goes in the same way as in Thcorem 3.1 of Chapter VI. 0
3
Extension of Mappings into Hyperbolic Spaces
97
Theorems 4.1 and 5.2 of Chapter VI may be generalized as follows without any change in their proofs.
Theorem 3.3 Let Y be a compact hyperbolic complex space. Let X be a complex manifold of dimension m and A an analytic subset of X of dimension 5 Tn - 1. Then every holomorphic mapping f : X - A -+ Y a n be &ended to a holomorphic mappings from X into Y. Theorem 3.4 Let Y be a complete hyperbolic complex space. Let X be a complex manifold of dimension m, and let A be a subset which is nowhere dense in an analytic subset, say 13, of X with dim B 5 m - 1. Then e v e y holomorphic mappings f : X - A -+ Y can be exten.ded to a holornorphic mapping of X into Y. Corollary 3.5 Let Y be a complete hyperbolic complex space. Let X be a complex rna.nifold of dimension m, and let A be a analytic subset of dimension 5 m - 2. Then e v e y holomorphic mapping f : X - A -+ Y can be extended to a holomorphic of X into Y. Similarly, Theorems 6.1 and 6.2 of Chapter follows:
VI
may be generalized as
Theorem 3.6 Let Z be a complex space and Y a complex subspace of Z satisfying the following three conditions: ( 1 ) Y is h.yperbolic; ( 2 ) The closure of Y in Z is compact; (3) Given a point p of Y - Y and a neighborhood U of p in Z, there exists a neighborhood V of p i n Z such that V c U and the distance between Y n (Z - U ) and Y n V with respect to d y is positive. Then every holontorphic mapping f from the punctured disk D* into Y can be extended to a holomophic mapping from the whole disk D into Z.
Corollary 3.7 Let Y and Z be as above. Let X be a complex manifold of dimension m and A a locally closed complex submanifold of dimension 5 : m - 1. Th.en every holomorphic mapping f : X - A -+ Y can be extended 1 to a holomorphic mapping from X into Z. We conclude this section by showing that, in theorems of the kind discussed herc, the assumption that X is a nonsingular manifold is essential. kt Y be a projcctive algebraic variety in Pn(C). Let C(Y)be the afline cone of Y ,i.e., the union of all complex lines through the origin of Cn+'
I
i
98
Genemlizutlon
60
representing the points of Y. We recall t.hat Y is said to be projectivc,~,. normal if C(V) is a normal complex space. 1% rnake use of the followirLg theorem (see Lang [l, p. 1431). If Y is u normal projective algebraic ~ m 7 - i c . t ~ . then Y curt be imbedded into some P,(C) i n such a way that Y is projcctkely normal, i.e., C(Y) is normal. In particular, let. Y be a nonsinguIar projective algebraic manifold which is hyperbolic, e.g., a compact Rienialill surface of genus > 1. Then Y is project,ively normal in some P,(C) a~ld C(Y) is nonsingular except, at the origi11. Let n : C(Y) - (0) + Y be the restriction of the natural projectioxl CTL+'- (0) + P,(C). It is clear that x cannot be extended to a holomorphic mapping from C(Y) into Y. This shows that Theoren1 3.3 does riot hold X = C(Y) and A = (0) and hcllcc that t.he cone C(Y) is sirlgular at the origin. We give another example due to D. Eiscnman. Let ,If be a cornpact co~llplex manifold and L a negative line bundle over M . Accordir~gt o Graue1.t [2]?the space L1:l.l obtained from L by collapsing the zero section to a poir~t is a complex spacc. Let f be the projection from L - {zero section) or~to 121. It is clear that f cannot be e ~ t ~ e n d etod a (~ant~inuous) mapping fro111 L / M into 12.I. To obtain a counter-example, all we have to do is t o take ally compact hyperbolic manifold, e.g., a compact Riemann surface of genus 2 L' as Af. This example shows also that if L is a ~~egat.ive line bundle over a cumpact hyperbolic n~anifold,then the point of L/df corresponding to the zc.1.o section of L is a singular point. For if it were a rionsingular point, f would be extended to a mapping from L / M illto ,l,f by Theorem 4.1 of Chapt,er \:I. More generally, we may take a complex vector bundle E which is negative in a certain sense in place of a negative line bundle (see Grauert (21). A similar reasorling shows that if we obta,in a complex space by collapsing a co~nplexsubspace of a l~yperbolicmanifold t,o a point, then the resulting space has a singular poilit. This fact will be taken up in Chapter VIII. 4
4 Normalization of Hyperbolic Complex Spaces
Corr~plezSpaces
99
The normalization theorem of Oka (see Oka 121, Narasimhan [ l , p. 1181) says that every complex space X ha? a unique (up to an iso~norphism) normalization (Y, x) . The following result is due t o Kwack [I]. T h e o r e m 4.1 Let X an.d Y be complex spaces and let f : Y + X be a proper holomorphic mapping such that f-'(a) is finite for every n E X . If X is hyperbolic, so is Y. Corollary 4.2 If (Y, x ) is a normalization of a hyperbolic complex space also hyperbolic.
X,then Y is
P r o o f Since F is distance-decreasing, we have
Let p and q be two distinct points of Y such that f (p) = f ( 9 ) . Let V and W be disjoint open neighborhoods of p and q respectively such that
Assuming d y (p, q) = 0, we shall obtain a contradict.ion. Let {y,,) be a sequence of curves in Y joining p and q such that their lengths L,-(y,,) measured in terms of dy satisfy
Let U, be the closed ball of radius l l n around the point f (p) in X with respect to dx. Since f is distance-decreasing, f (7,)is contained in Uk for k 2 n. In other words,
Normalization of Hyperbolic Complex Spaces
A complex space X is said to be normal at a point a E X if the ring 0 s of germs of holomorphic functions at a is integrally closed in its ring of quotients. If X is normal at every point of X, then X is said to be normal. -4 normaht.ation of a complex space X is a pair (Y, T)consisting of a normal complex spacc Y and a surjective holomorphic xriappir~g7: : Y --, X such t h a t (i) .ii : Y --, X is proper and x-'(a) is finite for every a E X ; (ii) If S is the set of singular points of X, then Y - r-'(S) is dense in 1and x : Y - n-'(S) + X - S is biholomorphic.
Since V and W are disjoint and y, is a curve from p E V to q E U', there exists a point p, on y, which is not in V U W. Since Uk is compact for s a c i e n t l y large k, so is f-'(Uk). Since p, E f-'(Uk) for k 5 n, it follows that the sequence {p,) has a subsequence which converges to a point, say Po, of Y. Since p, $ (VUW) and VU W is open, the point po is not in VLIU'. On the other hand, f (p,) E U,,and n U, = {f (p)). Hence, f (po) = f ( p ) . This implies po E f [f (p)] = (V u W) n f [f (p)] c V u W, contradicting the statement above that po $ V U W.
-'
-'
100
5
Cer~eralazatzonto Complex Spaces
6
Complex V-Manifolds (Now Called Orbifolds)
Let G be a fi11it.e group of linear transforrnations of C n . Then the q ~ o t i ~ ! , ~ space C n / Gis, algebraically, an affine algebraic variety and, analytic all,^.. ;, ~lorlllalco~rlplexspace (see H.Cartan [4]). A complex space X is called :il: n-dimensional conl.plez \'-manifold if every point p of X has a neigllborhoo(i C' which is biholomorphic to a neighborhood of the origin in C 7 ' / G , :~ h r l . ~ Gp is a finite group of linear transformat.ions of Cn (which depends on 1 ; ; . It follows from the rcsult of Cartan that a corllplex 17-~nanifoldis a norl~-ta] co~llplcxspace. Tlle notion of V-nianifold was introduced by Satake (1, 2 ; and has beer1 extensively investigated by Raily [I, 21. Let ill be a complex manifold and G a finite group of holornorphii trarlsforniations of A l Icaving a point o fixed. With respect to a suita!>lc local coordina.tc systenl with origin at o, the action of G is linear. In fz~cr. let H bc thc open ball of radius r around o with respect. to a Riemanliiall l B nlay be considcrecj metric invariant by G , where r is chosen so s ~ n a lthat as a bounded domain in C". Then G rnay be considered as a groul:, B leaving a point o tist?<.i. holomorphic t,ransformations of a bounded dolr~ai~l By a classical theorern of H. Cartan i3] (see also Bochncr and h,Iart,in [ I : . the action of G is linear with respect to a suitable local coordinate systenl. Let :\I be R co~iiplexmanifold and I' a properly discont~inuousgroup holomorphic transformations of 12.1. Since thc isotropy subgroup of T at eaiir point of 31 is finite, the qltotient space iZJ/r is a complex I.,.-nlanifold. 111 partic:ular. if :\I is a hyperbolic marlifold, e.g., a bounded domain in Cn.n~l:! r is a disc:ret,es u t ~ g r o ~ofl pthe group H ( A I ) of holomorphic t r a n s f o r ~ n a t i c ~ ~ ~ ~ of AJ, then ,2I/r is a conlplex V-~rlanifold. In Secs. 4 ant1 5 of Cl~apter\.!I: we discussed the problem of cxtcntiing If to a holomorphic mapping fronl a holornorphic: ~rlappirlgf : X - .4 S into :\I, where S is a complex rnanifold: A is an analytic subset of .\-. and :\I is a hypcrbolic inanifold. As I havc indicatcd in Sec. 3. the r c s u ! ~ ~ of Secs. 4 and 5 of Chaptcr VI can be gcncralizcd to the case where .\I is 14 hypcrbolic complex space. On the ot,hcr hand. the assumption that S is 8 nonsingular complex rnanifold scclns to be rather essential. It is clear t !i:ir the rcsults call be generalized to thc case where ;Y is a complex l.'-manifc~lci. it:
-
6
Invariant Distances on M / r
Let be a cornplex rnanifold (or rnore generally, a coniplex space) and 1' a properly discontinuous group of holomorphir: transformations of 1 1 . set 1' = i\l/I'. In addition to the pseudodistancc d v defined earlier, we can
Invariant Llistances or1 M / r
101
define another pscudodist,arice d; as follows. Lct n : M -+ Y = , l l / r he tlle natural projection. A holornorphic rrlapping f from a complex spacc X illto Y is said to be lifiablc if t,liere exist.s a holomorphic rrlapping f : ,Y ,$I such that f = n 0 f.It is said t o be locally liftable if every point of ,Y h a a neighborhood U such that the restrict.ion of f to C' is liftable. If X is simply connected, then a locally liftable mapping f : X Y is liftable. If the group r acts freely on :\I so that 12.f is a covering space of Y! then every holomorphic mapping f : X -+ Y is locally liftable. In the definition of d y ! we used chains of liolornorphic mappings fro111 the unit disk L) into Y.If we use only locally liftable holomorphic nlappi~lgsfrom U into Y, we obtain a new ~ ' ~ ~ ( l o d i s t a nwhich c c will be derloted by d t . Since D is sirnply connect,ed, to define d;. it suffices to consider only liftable holornorphic mappings from D into Y. In general. we have
-
-
d k (P.Q )
If
r
db
2 C ~(P. Y q)
for p, q E k:
acts frccly on d l so that T : 121 -, Y is a co\:ering pr.ojec:tion. then following ~ ) r o ~ ~ o s i tis i o trivial. n
= d l - . The
Proposition 6.1 Let AI be (1 c o m p k r spncc and I' u properly discon.tirr11ous gr.0u.p of Icolorrrorplric tran.sfo,rn~,ations of d l . Let f be o locally liftclblc h,olomoryhic rnuppC11.gfrom a co77?.pler sl~aceX irl.10 1' = :21/1'. 7'h.cn
The following proposition is a generalization of Proposit.ion 1.6 of Chapter IV. The proof is similar.
Proposition 6.2 Lct 12.I be a corrlp1e.r .space al1.d T n. proprr.ly discontintrous group of holorr~uryllic tmn.?formations of 221. Let x : ,I1 -- Y = ,%Ill' bc the naturul projection. Let 11.q E 1' nnd l j , G E d l snch that ~ ( 4 =) p cc~ld n(p) = q . Tlren
The proof of t h following ~ proposition is sirnilar to that of Theorern 4.7 of Chapter IV.
Proposition 6.3 Let '11. r and Y be (1s ubove. Then d;. is a ( c o m p l e t ~ ) distance zf and onlv ~fA1 z.9 a (complete) hyperbollc space. 1 P., 7f and ordy if dnr IS a (complete) distance.
102
Generalization to Complex Spaces
Many of the results in Chapters IV, V, and VI may be generalized to
M/r and locally liftable holomorphic mappings. We mention one. Theorem 6.4 Let hi be a hyperbolic complex space and I' a properly discontinuous group of holornorphic transforntations. Let f be a locally Eiflublc holomorphic mapping from the punctured disk D* into Y = M/r such that, for a suitable sequence of points zk E D* converging to the origin, f ( z kj converges to a point po E Y . Then f extends to a holomorphic mapping from the unit disk D into Y.
CHAPTER VIII
Hyperbolic Manifolds and Minimal Models
1 Meromorphic Mappings There arc several definit.ions of a meromorphic mapping from a con~plex space into anot,her complex space. Rernrnert [I] and Stoll [2] have made systematic studies on rneror~lorphicmappings. A meromoryhic mappiny f from a complex space X into a complex space Y i11 the sense of R.emrnert is a correspondence satisfying the following conditions: (1) For each point z of X , f (x)is a nonempty compact subset. of Y; (2) The graph rf = {(x?y) E X x Y ;y E f (x)) of f is a connected colnplcx subspace of X x Y with dim rf = dim X; (3) There exists a dense subset X * of X such that f (x) is a single point for each x in X*. Denote by ii the projection from X x Y onto X. Dcnotc by ~rthc rcstriction of ii to the graph rf.Since f ( x ) = T-'(x) = rf n 5-'(z), it, follows that f (x) is a complex subspacc of Y. Let E bc the set of points of rf where x is degenerate, i t . ,
E = {(.qy)
E
r,; dim f (x) > 0 ) .
Let N = x ( E ) = {x E X ;dim f (x)> 0). Then E is a closed complex subspace of codi~nension2 1 of rf and N is a closed complex subspace of codimension 2 2 of X (see Remrnert [I]). It is then clear that f : X - N + Y is a holomorphic mapping. F'rom Corollary 3.5 of Chapter VII we obtain
Theorem 1.1 Let f be a meromorphic mapping from a com.plex manifold X into a complete hyperbolic space Y. Then f is holomorphic. Observe that X is assumed to be a nonsingular manifold.
2
Hyperbolic Manzfolds and Minimal Models
104
Strong Minimalzty and Minimal Models
105
If we 111ake use of Theorem 3.1 of Chapter VI and t.he property of a meromorphic niapping that. f (x) is nonempty, then we can actually show that every merom.orphic mapping f from a complex manifold into a hyperbolic space (complete or not) is h,olomorphic.
Let X and X' be cornplex spaces. We say that X and X' are bimeromoyhic to each other if there exist meromorphic mappings f : X --+ X' and g : X' --r X such that
Remark Lct X ant1 iV be as above. and set X' = X - N . \lie indicate t h ~ proof that any holomorphic niap f ' : X' -t P m C extends to a meromorphlc map f of X into P,C. In order to show that every holo~norphiclinc bundle <'on X' extends to a line bundle on X. we first prove that evcry point p E X has a neighborhood U such that <'is trivial on U' = U n X'. Let U be a small coordinate ball neighborhood of p. Since N is of at least real codimension 1. we have H2(U', Z) = 0. UTc claim also that H1(U', 0 ) = 0, where R is thc sheaf of germs of holomorphic functions. Let w' be a $-closed (0,l)-form on U'. Since codim N > 2, w' extends to a %closed (0, 1)-form w on C: Since H1(U,fl) = 0, we have w = 8 f , which implies ur' = $f. From the exact sequence
We consider the casc where f : X + X ' is holomorphic and hence singlevalued. Let N' = {x' € X'; dimg(xl) > 0) and N = f-l(N'). Then, codim N 2 1 and codim N' 2 2 (sce Sec. 1). The restricted mapping f : X - N + X' - IIr' is biholomorphic. We call
x E g [f (x)] for x E
<
g : (Xf,N') :.
c'
-+
(X', N')
As we remarked at the end of Sec. 1, we may drop the assumption of completeness. But we are primarily interested in compact spaces. This theorem generalizes a result of Igusa [I] (every compact Kaehler manifold with negative constant holomorphic sectional curvature is strongly minimal) and a result of Shioda [I] (a complex manifold which has a bounded domain of Cn as a covering manifold is strongly minimal). The following result implies that every complex torus is also strongly minimal.
<
\.Ye say that a complex space X is strongly minimal if every nlcronlorphiC mapping from a complex space U into X is holomorphic at every sirnpie (i.e., nonsingular) point of U . This is the definition used by Weil [I: p. 27; in showing that every Abelian variety is minimal.
(X,N)
T h e o r e m 2.1 Every comtplletehyperbolic space is strongly minimal.
.
Strong Minimality and Minimal Models
x' E X'.
a contraction. For details on monoidal transformations and contractions we refcr the reader to Moisezon [I]. Given a class of bimeromorphically equivalent complex spaces, a space Xo is called a minimu1 model of the class if, for evcry space X in the class, there is a contraction f : (X, N ) + (Xo. No). It is clear that in the given class thew is at niost one minimal model. It is clear that if X is a strongly rriinimal complex manifold, then it is the mininial modcl in its class of bi~r~eromorplically equivalent complex manifolds. Theorem 1.2 may be restated as follows:
I
2
--+
E f [g(xl)] for
a monoidal transfom.ation with center N' and its inverse
f : (X, N)
we conclude H1(U'. R*) = 0, showing that CT has the desired property. We cover X by such neighborhoods U,, and set U: = U,n X'. Tilt? transition functions gij : U; n Uj + C of <'extends to gij : Lri n U, -+ C , which become a transition function for <. Let. ~ L I O w1 , ? . . . , wn' be a homogeneous co0rdinat.e systcm for P,,,C. V, be the open set in Pn,C defined by w" # 0. We set h,,g = w"/lua 011 V, n Vo.Let q bc the line bundle ovcr P m C defined by transit.ion functions \lap. This bundle admits rn 1 linearly independent sections to, t l , . . . t,,. Given f : X' 4 P,C, set <' = f *q, and .sb, si!. . . ,sk, be the sections of induced by to, t l , . . . ,t,. The map f is given by p I[.s&(p)? s ; ( ~ ).,. . ,s;,(~)],p € X'. We extend <'to a bundle over X and the sections sb. s i , . . . ,s k to sections SO, s l , . . . , s,, of E. Then the map p H is0(p),s l (p): . . . , S, (p)],p E X, scnds X meromorphically to PI,,C .
+
X and x'
; .
T h e o r e m 2.2 If X is a complex space which has a closed complex subspace X of CN as a corenng space, then X is strongly minimal. Proof Let Li be a complex manifold of dimension m. Let f be a meromorphic mapping of U into X. We want to prove t.hat f is a holomorphic
106
2
Hyperbolic Afanifolds and Minimal Models
mapping from U into X. Since the problem is local with rcspect to thr domain U , we niay assume that U is a ball in Cn' and that f is holomolphic on CT - A where A is an analytic subset of dimension 5 m - 2. Then U - A is simply connected so that the holornorphic mapping f : U - -4-+ X can bc lifted to a holornorphic mapping f : U - A -+ % . C c". Since J is given by IV holornorphic functions, it can be extended to a holomorphic mapping j : I;-, C N by Hartogs' theorem. Since 2 is closed, .f maps L1 into X. If we denote by 7i the projection + X ,then T O f is a holomorph~c 0 mapping of li into X which coincides with f on U - A. Theorcrn 2.1 implies that. a compact Kaehler manifold with negative holomorphic scctional curvature is strongly minimal and hence is the minimal model in its class of bimcromorphically equivalent complex manifolds. \lr, shall now show t.hat a compact Kaehler manifold with negative Ricci tensor is the minimal model in its class of bimeromorphically equivalent complex manifolds. Let E be a holomorphic vector bundle over a compact complex 111ariifold hl. Let r ( E ) der~otethe space of holornorphic cross sections of E. If the restriction mapping a E r ( E ) + a ( z ) E E, is surjective for each point. ;r of 1'11, then E is said to have no base points. Assu~ningthat E has no base points. sct
r = dim E,,
k
+ r = di111r ( E ) .
To each point T of M . we assign thc kernel of the restriction mappillp r ( E ) -t E,, which is a k-dimensional subspace of r ( E ) . In this way KC obtain a holomorphic mapping of M into the complex Grassmann manifold Gk,,(C) of k-plancs in the (k T)-dimensional vector space r ( E ) . If this I Gk.,(C) is an imbedding, thcn the bundle E is said to be mapping A1 . very ample. If a (local) holomorphic section of E is a covariant holomorphic tcriwr ficld of A f , then E is called a covanant holomorphic vector bundle over The bundle of complex (p, 0)-forms on .t1 is a covariant holomorphic vector bundle. In particular, the canonical line bundle, i.e., the bundle of ( n . 0 ) forms on A1 (where 11 = dimM), is a covariant holomorphic vector bundle We prove f i s t the following theorern.
+
Theorem 2.3 Let E be a very ample covariant holomorphic vector bundle over a compact complex manifold M. If f : 11.1 + A1 is a meromo~jt,ic mapping which is nondegenerate at some point, then f is biholomorphic In particular, every bimeromorphic napping f of M onto M is biholomorphic.
I
107
Strong Minimality and Minimal Models
Proof Let w E I'(E). Since w is a covariant holomorphic tensor field on MI f induces a covariant holomorphic tensor field f *(w) on A4 - N , where N is an analytic subset of codimension 2 2 in M. Since f *(w)is given, locally, by a systerri of holomorphic functions, f * ( w ) extends to a holomorphic scction of E over h1 by Hartogs' theorem. We shall show that the lincar mapping f*: r(E) r ( E ) is injective. Let U and U' be open scts in M such that f maps U biholomorphically onto U'. If w E r ( E ) and f*(w) = 0, then f8(w) vanishes identically on U and, consequently, w .va~iishesidentically on U'. Since w is holomorphic, it vanishes identically on M . This proves our assertion. Since dimT(E) < CQ by compactness of 1V, it follows that f a : T(E) + r ( E ) is a linear autornorphism. This linear isomorphisnl f * induces an automorphism p of the Grassmann manifold Gk,,(C) of k-planes in I'(E), where r = dim Ez and k r = dim r ( E ) as above. We consider M as a complex submanifold of Gk.,(C). Then f : h1 -, M is the restriction of p-l : Gk.,(C) -,Gk,,(C) to hf and hence is biholomorphic. -+
+
Corollary 2.4 Let E be a very ample covariant holomorphic vector bundle over a com,pact complex manifold M . Then M is the minimal model in its class of bimeromorphically equivalent conlplex spaces. Proof 'et X be a complex space bimeromorphic to A l . Let f and g : M X be meromorphic mappings such that
:X
-
M
-+
x ~ g [ f ( x ) ] f o r z X~
and p ~ f [ g ( p ) ] f o r p E M .
We want to show that. f is holomorphic, i.e., single-valued. By Theorem 2.3, f o g : M + hf is biholomorphic and f [g(p)] = p for p E 'M. Since g : M + X is surject.ive by x E g[f (x)],it follows that f is single-valued.
Example 1 Let M be a compact Kaehler manifold with negative Ricci tensor or, more generally, with negative first Chern class cl(M). From a result of Kodaira [I] it follows that a suitable positive power Km of the canonical line bundle K of M is very ample. (The converse is a triviality.) Theorem 2.3 implics that a mcromorphic mapping of M into itself which is nondegenerate at some point is a biholomorphic mapping of M onto itself. This is a result of Peters [I]. By Corollary 2.4, M is the minimal model in its class of meromorphicaUy equivalent complex spaces. This is due t o P. Kiernan, who has pointed out t o me that Theorem 2.3 implics Corollary 2.4 immediately.
Hyperbolic Manifolds and Minimal Models
108
Example 2 Consider r nonsingular hypersurfaces Vl , . . . ,V, of degree d l , . . . ,d, in P,+,(C) respectively. Suppose that they are in general position so that the intersection M = Vl f l . . - n V, is a non-singular manifold of dimension n. If we denote by h the second cohomology class of M corresponding to a hyperplane section of M ,then a formula on Chern classes (see Hirzcbruch [I, p. 1591) implies cl(M) = ( n r 1 - C:=l d,)h. If 1d, > n r 1, thcn cl (M) < 0 and Theorem 2.3 and Corollary 2.4 apply to M. In particular, a nonsingular hypersurface h l of degree d in P,+ 1 ( C j with d > n + 2 has c l ( h l ) < 0 and is the minimal model in its class of bimerornorphically equivalent complex spaces.
+ +
+ +
3
Relative Minimality
Let X and X' be complex spaccs which are bi~neromorphict.o each otlier X' and g : X' + X such that with merornorphic mappings f : X -+
x ~ ~ [ f ( z )f ]o r x ~ X and x'E f[g(xf)] f o r x ' E X f . In Sec. 2, we defined the notions of monoidal t.ransformation and contraction. If f is holornorphic. i.e., if f : (X, N) -, (XI: IV') is a contraction, t.hcn we write
3 Relative
Minimality
109
and I/, is an (n - 1)-dimcnsional closed complex subspace of M . The set of divisors on M forms an additive group G(M), called the group of dzvisors. Every nieromorphic function f on M defines a divisor ( f ) = ( f ) o - (f)oo, where (f)" is the variety of zeros of f and (f), is the variety of poles of f . The subgroup of G(M) consisting of divisors ( f ) of meromorphic functions f is denoted by Gl(11.l). The factor group A(M) = G(M)/Gl(M) is called the group of 1inea.r equivalence on M . Each divisor C niK of M gives rise to a complex line bundle over M as follows. Let {Ua) be an open cover of M, where each Ua is sufficientlysmall. Let f,i be a holomorphic function on Ua such that U, fl V , is defined by fai = 0. If U, n Ug # 0, we set
We associate to C n,V, the line bundle defined by the transition functions {gag) and obtain a homoniorphism of the group G ( M ) into the group F of line bundlcs over M, whose kernel is exactly G[(M), and hence an isomorphism of A ( M ) = G(M)/Gl(M) into F. It is known that if M is a closed complcx submanifold of a complex projective space (i.e., if M is projective algebraic), then A(M) = F. Given a complex line bllndlc E E F with t,ransition functions {g,,,?) we define a 2-cocycle { c u p , ) of M by
X > X'. The minimal rriodel Xo in a class of bimeromorphically equivalent complex spaces satisfies, by definition, t.hc inequality X > Xo for every complex space X in the class. On the other hand, a space X in a class of bimeronlorphically equivalent complex spaces is said to be relatively min.imnl if thc class contains no space X' (other than X itself) such that X > .Ti' Thc minimal model rnay or may not cxist but is unique (if it exists) i l l each class. On the other hand, relatively mininial models always cxist but are not necessarily unique in cach class of bi~neroniorphicallyequivalc~lt complex manifolds (see Corollary 3.5 below). The following statement is obvious.
This induces a homomorphism F -t H 2 ( h l ;Z), denoted by cl. Thc class el(<) E H 2 ( M ;2 ) is called the first Chern class of <.Let P be the kernel of q : F + H * ( M ;2);it is called the Picard variety attached to hf. In other words, P is the subgroup of F consisting of line bundles with trivial Chern class. Let R denotc the sheaf of germs of holomorphic functions on M . We define a homomorphism from H1(M;R) onto P as follows. Givcn a 1-cocycle {hap} representing an element of H1(A.i; R), we set
~ninimali n its clus,~o! T h e o r e m 3.1 The minimal model is relati~~ely bim.emmorphically equivalent com.plex spaces.
Then {yafi) defines an element of P . We have now an exact sequence
We shall find other sufficient conditions for a complcx manifold to be relatively minimal. For this purpose, we review quickly results of Kodaira and Spcnccr [I]on divisor class groups. A divisor of an n-dimensional cornpact complex manifold M is a finite sum of the form ni4, where n, E
where H1(M;Z) + H1(A!; R) is induced by rn E Z + 2ninz E R. Assunie that .If is a compact Kaehlcr manifold. Thcn the homomor' phism F -+ H * ( M ;Z) maps F onto H1.'(M; Z), where HI.' (A{; Z) is
110
3
Hyperbolic Manifolds and Minimal Models
the subgroup of H 2 ( M ;Z) consisting of elements which are mapped into H1sl (M; C ) under the natural map H2(M;Z) + H ~ ( MC; ) . Thus we have
where Hc denotes the harmonic forrrl representing the class c and denotes the projection onto thc space of (0,l)-forms.
n,,,l
Theorem 3.2 Let M and M' be com.pact complex m.an.ifolds of dim.ension. n , F ( h l ) and F(M1) the groups of line bundles over M an,d ,tI' respectively, and P ( M ) and P(h.1') the Picard varieties attached to d l and ,%I'respectively. Let f : ( M !N) ( M I ,N') be a contraction. 7'h.en f* : F ( M 1 ) + F ( M ) is injective an.d maps P(M1) in.to P ( M ) , brit th,c image f * F ( M 1 ) does n,ot con$tain the line bvndle over M defined by th,c divisor m N for nn8ynonzero integer m .
111
To provc the last assertion of the theorem, let E be the line bundle associated to the divisor m N and let a be a holoniorphic cross sectioll of whose zero set is precisely m N (multiplicity counted). Assume that = f't', where <'is a line bundle over MI. Let a' be the holomorphic cross section of E' over h1' - N' corresponding to oI(12.I - N). We' extend 0' to a holomorphic cross section over ;l.il.Again looking at the zeros of a' as above, we obtain a contradiction unless m = 0.
<
with
Relative Minimality
<
Corollary 3.3 If llil is a compact com,plex manifold with its Pzcard variety P(11.l) = 0 and its second Betti number bz(M) = 0, then M is relatively minimul i n the class of compact complex manifolds bimeromorphic to 11.1. Proof Assuming the contrary, let (hl, N) -+ (M', N') be a contraction. Since P ( M ) = 0, we have P(M1) = 0. From rank F ( M ) = rank[F(M)/P(hI)] 2 bz(M) = 0, we obtain
-+
Proof We first note that N is given as the set of zeros of the Jacobiall of f so that codim N = 1. As we statcd in Sec. 1?a result of Remrnert implies that codim N' 2 2. Let E' be a line bundle over M' and let E = f *<' be the induced linc bundle over M. Assume t.hat is a trivial line bundle. Then admits a holomorphic cross section a which never vanishes on M . Since f : M - N -+ MI- N' is biholomorphic, f * induces an isomorphism betweell ('1 (11.1' - N ' ) and (1 (A1 - N ) . Let rs' be the holomorphic cross section of ,F' over h1'- N' corresponding to the section al(l'b1- N). Since codim N' 2 2 . we can extend a' to a holomorphic section, denoted again by rs'! over Af' by Hartogs' theorem. Since the set of zeros of 0' on M' is either an enipt!' set or an analytic subset of codimension 1 and 111ust be contained in :V'. it must be empty. This shows that a' never vanishes on M' and hence (' is trivial. The commutativity of the diagram
<
<
0
0
-
P(,l)
+
C
+ P(M1) --
F(hf)
-+
T
-
H ~ ( MZ) ;
7
F(*%I1)H 2 ( A i ' ; 2 )
implies that f* maps P(M1) into P(11f).
rank F(-4.i1) = rank F ( M ) = 0. On the other hand, the bundles associated to the divisors r n N , m nontrivial.
# 0, are
Corollary 3.4 If M is a compact Kaehler manifold with dim H1al(M; C) = 1, then M is relatively minimal i n th,e class of compact Kaehler manifolds bimeromorph.ic to h l . Proof The proof is similar to that of Corollary 3.3. Assu~nirigthe contrary, let (M, :lr) + (M', N') be a contract,ion, wherc M' is a compact Kaehler manifold. The following two facts imply the corollary immediately: (1) rank[F(A1)/P(M')]= rank HI,' ( M ' ; Z) 1 1; (2) dim P ( M ) = dim Ho.'(hl; C) = dim H'qO(M;C ) = dim H ~ . ' ( J ~ C ');= dim HO!'[M';C) = dim P('4.i'). Since (1) is a well-known property of a compact Kaehler manifold Aft, we prove only (2). Since :\1 and .If' are compact Kaehler manifolds, we have
H'."(M; C )= thc space of holomorplic 1-forms on N, ~ ' ~ ~ C ( )~=1the ~ space ' ; of holonlorphic 1-forms on M'.
112
Hyperbolic Manzfolds
and
3 Relative
Minimal Models
It suffices to show that the map f * : H1>O(M';C ) H1-'(M; C ) is an isomorphism, where f is the contraction (Ad,AT) 4 ( M ' , iV1). Let w' be a holomorphic 1-form on hf'. If f8w' = 0 on M,then w' = 0 on M' - N' and hence w' = 0 on k1' since codim N' 2 2. Let w be a holomorphic 1-form on Ill. Consider the holomorphic 1-form on hi' - N' which corrcsponds to wI(~21- N ) under f and then extend it to a holornorphic 1-form w' on A l l . Then f *wf = w on 121- N and hence f *w' = w on 111. -+
Corollary 3.5 Let M be a compact (Kaehler) manifold. Then i n the class of com,pact (Kaehler) manifolds bimeromorphic to hl, there is one u~hichis relatively m.in.imu1. P r o o f If M is not relatively minimal, lct (ILI, N ) + (.?.I1,N') be a contraction. By Theorem 3.2, rank F ( M ) > rank F ( M 1 ) . If Af' is not (llIM,N'') be a contraction so that relatively mini~nal,let (A4', N i ) rank F(,Z.ll) > rank F(M1'). Since rank F(A4) is finite, this proccss must stop eventually.
-
As we shall see shortly, a relative rrlirii~nalmodel in the given class nccil not be unique (sce Example 2 below). E x a m p l e 1 Lct d l be a compact Kaehler ma.nifold with posit.ive scctioll,;! curvature (or more generally with posit.ive holomorphic bisectional curvature). I t is known (Bishop and Goldberg [I], Goldberg and Kobaynshi 11:) that dill1 H1.'(hf; C ) = 1. By Corollary 3.4, M is relatively minimal in the class of compact Kaehler manifolds which arc bimeromorphic to M. E x a m p l e 2 Lct hrl be a compact lio~nogeneousKaehler manifold of the form G / H . wherc G is a connected compact scmisinlple Lie group and H is a closed subgroup with 1-dimensional center. Every irreducible cornpact Hcrmitian symmetric space is such a manifold. Fkom the exact homutopy sequcnce of the bundle G -, G / H , it follows that b2(121) = 1. BY Corollary 3.4. A 1 is relatively minimal in the class of compact Kaehler manifolds which are bimeromorphic to 't1. According t o Goto [I], h i = G / H is a rational algebraic manifold and hence is bimeromorphic to a projecti\e spacc.
113
of Sec. 2. Then b2(M) = 1, (see, for example, Hirzebruch [ I , p. 1611). By Corollary 3.4, M is relatively minimal in the class of compact Kaehler manifolds which arc bimero~riorphicto M. In Example 1 of Sec. 2 we saw that if Ill is a compact complex marlifold such that a suitable positive power Kn' of its canonical line bundle K is very ample, then M is the minimal model. We prove now the following: T h e o r e m 3.6 Let M be a compact complex manifold such that a suitable positive power K m of its canonical line bundle K has no base points (i.e., for each point p of M , there is a holomorphic cross section of K m which does not vanish at p ) . Then M is relatively minimal in its class of bimeromorphically equiva1en.t complex manifolds, P r o o f Let f : (Ad,N ) -, (A,!,' IV') be a contraction, where M' is a complex manifold which is bimeromorphic to M. As in t,he proof of Theorem 2.3, we see that f induces a linear isomorphism from r ( K f m )onto r ( K m ) , where K' denotes the canonical line bundle of M': r ( K l r n )t,he space of holomorphic cross sections of Ktm over A4', arld r ( K m ) the space of holomorphic cross sections of Km over A.1. We claim that f is nondegeneratc cvcrynrhere on ,W. In fact,, i f f is degenerate at a point p of M: then f * ( a 1vanishes ) at, p for every w' E r ( K r m ) . Since f * : F(Klm) -+ I'(KTr')is an isomorphis~n, this mcans that every i~ E 1'(K1")vanishes at p: in contradict,ion to the assumption that Ii'" has no bnsc points. Since f is nondcgrnerate everywhere, it is a covering project,ion from M onto M'. On the othcr hand, f is bimeromorphic and hence ~rlustbe biholomorphic. E x a m p l e 4 Let Af be a compact. complex manifold with cl(A3) = 0. If H1(hl, R ) = 0 (nrherc R is the sheaf of germs of holomorphic functions), in particular if M is Kachlcrian and bl(,ll) = 0, then M is relatively minimal in its class of bimcromorphically equivalent compact, conlplex manifolds. To prove our assertion, consider the exact scqucnce 0-Z+R+R'-0, where R* is the sheaf of germs of non vanishing holomorphic functiorls and the homomorphism R + R* is given by exp(2~z.).Then this yields the fdowing exact sequence: -+
E x a m p l e 3 Let 111 be a nonsingular closed hypersurface in P,+I ( C ) with n 2 3. More generally, let ,If = 7/11. . .nV, be a nonsingular complete intersection of closed hypersilrfaces in P,,+,+l(C) with n 2 3 as in Example ?
Minimality
-.
H ~ ( M . O )-+ H ~ ( A ~ , R-+ * H ) ~[,~I,Z)
Since the horno~r~orphism HI(-41, R*) -+ 1f2(A1,2 ) maps a complex line bundle into its Chern class, a complex line bundle E H1(,Zf, $2') with
<
114
Hyperbolic Manifolds and Mzr~irrialM d e l s
cl (0 = 0 comes from H'(A4, R). If we assume that H1(Abf, 0) = 0, then cl(() = 0 implies that is a trivial bundle. Applying this result to tllc
<
canonical line bundle K of A l , we scc that cl (hrl) = -cl(K) = 0 implies that Ii'is trivial provided H1(M,0) = 0. If K is trivial, t,hen M atfrnits a nonvanisliing holomorphic section. Our assertion follows from Theorem 3 6
Example 5 Let A4 = V, n . . . n V, be a complet,e intersection of 11011singular hypersurfaces Vl! . . . , V,. in P,+,+l(C) as in Exalnple 2 of Sec. 2. .4ssurne 7~ 2 2 so that bl(~1.1) = 0. If d l , . . . ,d, are the degrees of Vl,. . . ,V, respectively and satis@ the equality n + r + 1 = dl +. . . id,, then ,?.I is relatively minimal in its class of bimeromorphically equivalent compact. complex manifolds (see Exarrlple 2 of Sec. 2 and Example 4 above). I11 particular, a nonsingular hypers~lrface M of degree n + 2 in Pn+, (C) with n 2 2 is relatively minimal. Compare this with Example 2 of Sec. 2.
CHAPTER IX
Miscellany
1 Invariant Measures
Let M be a topological space with pseudodistance d. If p is a positive real number, then the pdimensional Hausdorffmeasure pp is defined as follows. For a subset E of M ! we set
where 6(E) dcnotcs the diameter of E. If M is a complex manifold, f in Chapter I11 induce the invariant pseudodistances c.<{ and d ~ defined Hausdorff measures on M . Since these pseudodistances do not increase under holomorphic mappings, the Hausdorff measures they dcfinc do not increase under hololnorphic mappings. There are other invariant ways to construct, interrr~ecliatedimensional measures on complex manifolds. For a systematic study of various invariant measures on complex manifolds, we refer the reader to Eisenman [I]. In this section, we shall briefly discuss invariant measures which rxiay be considered as direct generalizations of c,jf and dfil. Let D , be the open unit ball ill Cn. The volume element defined by the Bergman metric of D,, induces a Irieasure on D,,, which will be derloted by p. Theorem 4.4 of Chapter I1 implies that every holomorphic mapping f : D, -+ Dn is measure-decreasing with respect to 11, i.e., p [f (E)]5 p(E) for every Borel set E in D,.
-
Let !\I be an n-dimensional complex manifold. Given a Borel set B in M, choose holomorphic mappings f, : D, A1 and Borel sets E, in Dn for i = 1.2.. . . such that B c U, f,(E,). Then the measure p.tf(B)of B is defined by CL.W(B) = inf C p ( ~ i ) , i
1
Miscellany
116
14f( B ) ] , f where the supremum is taken with respect to the family of holon~orphic mappings f : M -+ D,. From the definitions of these measures, the follo~-ing proposition is evident. 7.4s ( B ) = sup
Proposition 1.6 Let 111 be a complex manzfold and lG a covering man.ifold of M. Then is measure-hyperbolic if c~ndonly if M is. If G , I is a bounded domain in Cn, then y J Q ( B )is positive whenever B is a nonernpty open set in $1. From Propositions 1.3 and 1.6, we obtain
Proposition 1.1 I f f is a holomorphic mapping of a complex manifold !I1 into another complex manifold N of th,e same dimension, then
Corollary 1.7 Let M be a complez manifold which has a bounded dom.ain of Cn as a covering manifold. Then hf is m.easure-hyperbolic.
for every Borel set B i n M.
p~
117
We call a complex manifold M measure-hyperbolic if p A f ( B )is positive for every nonempty open set B in M. In contrast to Theorem 4.7 of Chapter IV, the following proposition is easy.
where the illfinluln is taken over all possible choices for f , and Ei. In analogy as follows. For each Borel set B in to chi we define another measure we set
Corollary 1.2 If f : M
Invariant Measures
1x1 analogy to Theorem 4.11 of Chapter IV, we have
-,N
is biholornorph.ic, then
Theorem 1.8 Let M be nn n-dimensional Hermitian manifold with. Herrnitian metric dsi* = 2 x g i j d u ! i d ~ jand Ricci tensor Rij such theat (Rij) 5 -(cgij) (i.e., the Hemitian m a , t r i ~( R i j + cg,?) is n.egatzve semidefinite) for som,e positive constan.t c. Then M is measure-h.yperbolz'c.
[ f ( B ) ]= P M ( B ) and y ~ [( Bf ) ]= ~ M ( B )
for every BORE set 13 in M, The following two proposit.ions follow from the fact that a holomorphic rnappi~lgof Dn into itself is measure-decreasing with respect. to p.
Proof By llorrr~aliaingthe metric, we may assunie that c = 1. If ds2 = 2 C h,,dz'd? denotcs the Brlgman metric in D,,. its Ricci tensor is give11 by -hti. By Theorell1 4.4 of Chapter 11, every hololllorphic mapping f : D, -, M is volume-decreasing with respect to the volumes defined (is2 and ds;,. If we denote by p' the rrleasurc on M defined by ds&, then Proposit.ion 1.5 iniplies that p l ( B ) 5 paf(B) for every Borcl set B in iZf. Our assertion is now clear.
Proposition 1.3 For every Borel set B zn a complex mu7~zjoldiZi, we h n ~ l e Y M ( B )5 P M ( B ) .
Proposition 1.4 For th.e unit ball D, itself, both /LD, and y ~ , ,coin(:icle with the measure p defined by the Bergman metric. The following proposition is trivial.
Proposition 1.5 Let hd be an n-dimensional complex manifold.
In particular, every compact Herrnitian manifold with ncgative definite Ricci tcnsor is merrsurohyperbolic.
is a m.easure on M such that every holomorphic rnappili!l ( 1 ) If f : D, + ,bl satisfies
Theorem 1.9 A corr~pactcom.plex m.an.ifold urith n.egative first Chem class is measure-hyperbolic.
~ l [ f ( E5 ) ]P ( E ) for euery Borel set E in D,, tlter~p1 5 fi,tl; is a measure on hl such that every holomorph.ic 771ayPi*$ ( 2 ) If f : M i D, satisfies p l ( B ) 2 pif ( B ) ] for every Borel set B in ~ t f : then p'
2 rivf.
Proof Since such a manifold ad111itsa volume element whose Ricci tensor is negative definite, thc theorem follows from Le1111na 4.1 of Chapter 11. I
I
Example 1 Let = V n . . . .n V, be a complete intersection of nonsingular hypersurfaces V I ,. . . , IG of degrees d l . . . . d, in P,+,+l(C) such that ?
118
Miscellany
dl + . . - + d, 2 n + r + 2. Then M is measure-hyperbolic (see Example 2 ill Sec. 2 of Chapter VIII).
2
Intermediate Dimensional-Invariant Measures
119
denoted by vi. (The construction of the volume element from a Riemannian metric can be applied to hf ds2 even when hfds2 is degenerate. At the points where ht ds2 is degenerate? the volume element vi vanishes.) We set
Theorem 1.10 If a complex manifold M is hyperbolic, it is also measurehyperbolic. Proof Let p2, denote the 2n-dimensional Hausdorff measure defined by the distance dM. R o m Proposition 1.5 we obtain PM (B)2 ~ 2 n ( B ) On the other hand, p2*(B) is positive for every noncmpty open set B (see Hlirewicz and Wallman [ l , Chapter VII]).
where the infimum is taken with respect to all possible choices for {Bi, hi, f,). The measure p(B, p), can be infinite. If we cannot find , we set {Bi, hi, f,) satisfying the condition fi o h, = c p l ~ ~then p(B, y),, = oo. We have clearly
2 n, then p(B, p), is finite. In analogy to the measure yhf defined in Sec. 1, we define another k-dimensional measure y(M, cp), as follows. Choose a holomorphic m a p ping f : &.I+ Dm. Let V J be the volume element defined by cp* (f*d s 2 ) ; it is a k-form vanishing at the points where p*(f* ds2) is degenerate. We set If cp(B) is contained in a compact subset of M and if m
Example 2 Let hi' be a measure-hyperbolic complex manifold and let g : (MI, N') --, ( M ,N) be a monoidal transformation (see Sec. 2 of Chapter VIII). Since the contraction f : (M, N ) -t (M', N') is measuredecreasing and, for every nonempty open subset U of M , f (U) contains a nonempty open subset of M', it follows that M is also measure-hyperbolic, This furnishes an example of a compact Kaehler manifold which is measurehyperbolic but is not relatively minimal. It shows also that the converse to Theorem 1.9 docs not hold (see Example 1 in Sec. 2 of Chapter VIII). Example 3 In connection with Theorem 1.9 and Example 2, we mention that an algebraic manifold of general type is measure hyperbolic (see Kobayashi and Ochiai [2]). 2
Intermediate Dimensional-Invariant Measures
Let M be a complex manifold of dimension n. Let B be a real k-dimensional differentiable manifold (with or without boundary) together with a differentiable mapping cp : B --, 1l.i. For each positive integer m, we define a Ic-dimensional measure p(B, cp), of (B, 9 ) as follows. Choose a countable open cover {Bi) of B, a differentiable mapping hi : Bi + Dm (the ope11 unit ball in Cm), and a holomorphic mapping f, : Dm --, M for each i such that fiOhi = plB,. We denote by ds2 the Bergman metric of Dm. Then h: ds2 is a positive semidefinite quadratic differential form on B, and hence induces a degenerate) volux~ieelement, i.e., a nonnegative k-form, on B,, which will be
where the supremum is taken with respect to all possible holomorphic m a p pings f : M -t Dm. Clearly, we have
We shall see (Proposition 2.3) that y ( B ,cp), is finite if p(B) is contained in a compact subset of M. If B is a real submanifold of M and p is the injection of B into M , then we write p(B), and y(B), for p(B, p), and y(B, w ) ~ respectively. , The following proposition is evident. Proposition 2.1 Iff is a holomorphic mapping of a complex manifold M into another complex manifold N and if B is a real k-dimensional manifold with a mapping cp : B M, then
-
~ ( 1 3f,
S P(B, ~
~ ) m
Corollary 2.2 Iff : Af
--, N
) mand
y(B, f
~
)
is biholomorphic, then
5m y(B: ~ ) m .
Proposition 2.3 ~f M is a complex manifold and B is a mapping ip : B M I then
real manifold ruith
-+
= Y (B, cp)? y(B,ip)1 <
Intermediate Dimensional-Invariant Measures
2
Miscellany
120
Proposition 2.5 Let v be a k-dimensional measure on a complex manifold M of dimension n. Assume rn 2 n. ( 1 ) If v(B, f
2 . . . 5 p(B, ~ ) 52 P(B, Y)I.
B with ?i,
Proof I t suffices to prove the following inequality for every pair of posit.ivc integers p and q:
If we make use of the notations {Bi, hi, fi} and f used in the definitions of p ( B , ip), and y ( B , ip), respectively, then we have the following commutative diagram:
121
:
B
$)
o
+
2
for every real k-dimensional manifold : D, --, M , for every real k-dimensional manifold B with
p(B,+),
Dn and for every holomorphic mapping f
then v(B, cp) 5 p(B, cp), cp:B+M; (2) If v(B,ip) 2 y(B, f o cp), for every real k-dimensional manifold B with ip : B -+ llf and for every holomorphic mapping f : M -+ Dn, then v(B, cp) 2 y(B, cp), for every real k-dimensional manifold B with cp:B-+M. Proof ( 1 ) . Let {U,) be a countable open cover of M with biholomorphic mappings fi : Dn -+ Ui. We set Bi = cp-'(Ui) and $i= f,-' o ylBi.horn the assumption on 21 and Proposition 2.4 we obtain
Denote the metrics on D , and D, by dsg and dsi, respectively. Then where vi is the volume element defined by $; ds2. Hence
-
where the last inequality follows from the Schwarz lemma for f o fi : D, D, (see Corollary 4.2 of Chapter 111). On the othcr hand, the volume elements v , and v, are induced, respectively, froill h: (dsi) and ;s*( f ' ds;). Hence, .VJ 5 u$ on Bi so that
Taking the infimums of the both sides with respect to all possible choices for {U,), we obtain easily the desired inequality. Thc proof of (2) is sirnilar. I3
The fol1ov.-ing proposit,io~lfollonrs also fro111thc Schwarz lemma for h u h morphic illappings Dm + Dn and D, -,D , .
Proposition 2.4 Let B be a real rna~lifoldwitlr u mappin,g ip : B -+ D n . Then: for nr 1 n, both p(B19),, and y(B. q),, coincide .with the inteyl-cll otler B of the volume elem,ent defined by 9 ' ds2! where ds2 denotes the Bergman metric of D,. So far, we h a w been using the expression .'k-dimensional rne*ule" rather loosely. If. for each real k-dimensional manifold B with a mapping 9:B d l . s ( B . p) is a nonnegative real number (including the infinit?) and if v(B. 9)= inf C ,r ( B , , q ) . wherc the infimum is taken nrith respect to all cvuntable open covers {B,) of B, then we shall call u a k-dimenaonu1 measure on h l .
-
We shall say that a complex manifold 123 is (k, rn)-hyperbolic if p ( B , p),, is different from zero for cvcry real k-dimensional manifold B with an imbedding ip : B ,Vl. The following proposit.ion is easy to prove. -+
Proposition 2.6 Let ,If be a complex manifold and a c0verin.g manifold of M . Then .Gf is ( k ,m)-hyperbolic if and only if M is. The proof of the follonling corollary is similar to that of Corollary 1.7.
Corollary 2.7 Let 12,l be a complex manifold which has a bounded domain of Cn as a covering space. Then 12.I is (k, m)-hyperbolic. More generally, we have the following theorem, the proof of which is essentially contained in the proof of Theorem 1.9.
Theorem 2.8 Every hyperbolic complex manifold is (k, m)-hyperbolic.
122
2
Miscellany
We shall now define norms in the kth homotopy group nk(M! 20) of a complex manifold M .Let a be an element of nk(M, xu) and let cp : sk nl be a mapping representing a. We set -+
?(a), = inf dsk, ~)m,
p(a)m = inf p ( s k ,p),,
Intermediate
Dimensional-Invariant Mensures
123
Similarly, we can define norms (or rather, pseudonorms) in the homology groups of a complex manifold. Both p(B, cp), and y(B, cp), can be defined when B is piecewise differentiable, e.g., a simplex. We can therefore define ~ ( s ) , and y(s), when s is a differentiable singular simplex of M. If c = nisi is a differentiable singular chain, we define
xi
where the infimums are taken with respect to all cp representing 0.Sirice S k is compact, p(a),, is finite if rn is not less than the complex dimension of M. It is easy to verify
P C )=
i
i
m
7(c)m =
i
C lni17(si),. i
If a E Hk(hl, Z), then we define for a, a' E nk(M,xo).If k = 1, then a since nl (M, xo) need not be Abelian. It is easy to see that
+ o' should be replaced by aa': where the infimums are taken with respect to all cycles c representing the homology class a . If B is a complex manifold and cp : B M is a holomorphic mapping in the definition of p(B, cp), given at the beginning of this section, it is natural to modify the definition of p(B, p), by considering only holomorphic hi. We obtain a slightly different measure p(B, cp):,, in this way. If B is a complex space, we can still define p(B, cp);, by ignoring the singular locus of B. The singular locus has a lower dimension anyway. In particular, if V is a complex subspace of M , then we can define p ( V ) k . If B is of complex dimension r and if y ( B ) is contained in a compact subset of 111, then p(B, cp)A is finite for m 2 r . -+
and
for every a E r k ( M ,xO) and for every holomorphic mapping f of M into another complex manifold M'. Example 1 Let ll1 = {z E C"; 0 < r < Ilzll < 1). Then 7r2n-1(?tl, xo) = 2. Let a be the generator of ~ ~ , - ~ ( h xr lo.) We claim that y(a), is the volume of the hypersphere {z E C n ; llzll = r) with respect to the Bergman metric of the unit ball Dn provided that n 2 2. Let p : S2"-I -, Ill be a mapping representing a. Let f : M Dn be a holomorphic mapping and t l j the volume eleme~itof S2n-1defined by cp* f ds2, where ds2 is the Bergnlarl metric of D,. Since n 2_ 2. f can be extended to a holomorphic mapping of D, into D,. Since the extended mapping f : Dn -+ D, is distancedecreasing by the Schwarz lemma, it follows that y*f *ds2 cp* ds2. If denote by v the volume element of S2"-' defined by cp* ds2, then -+
<=
Assume that M is a Hermitian manifold with metric dsi,, whose holomorphic sectional curvature is bounded above by a negative constant. We shall normalize d s L in such a way that every holomorphic mapping f : Dm A4 is distance-decreasing, i.e., f d s k 5 ds2. This is possible by Theorem 4.1 of Chapter 111. In defining p(B, cp)&, we choose a countD, and able open cover {Bi) of B and holomorphic mappings hi : Bi f; : Dm M such that fi 0 hi = cpl~,.Then -+
-+
-+
Let S2n-1(r) denote the sphere {z E C n ; llzll = r). Considering the mapping z € S2,-l -+ ~p(z)/I(cp(x)II€ S2n-1(r), we see easily that the integral v is greater than the volume of S2n-1(r). On the other hand, we can find p such that this integral is arbitrarily close to the volume of S2"-'(r). This completes the proof of our claim. This example shows that y ( ~ t ) mis nontrivial sometimes.
ss2,,-,
It follows that the volume element vi on B, constructed from hf ds2 is bounded below by the volume element obtained from p*dsL. This implies that p(B, p)& is bounded below by the volume of B with respect to the volume element constructed from a positive semidefinite form y* ds&. In particular, if V is a complex subspace of M, then p(V)k is bounded below
3
Unsolved Problems
125
Unsolved Problems
by the volume of V with respect to the metric induced by dsL. Hcnce,
3
Proposition 2.9 Let M be a Hermitiart manifold with holomorphic sectional cun~aturebounded above by a negative con.stant. Then p(V)& > 0 for every complex subspace V of positive dimension.
In order t o state some of the unsolved problems on hyperbolic manifolds, we int.roduce notions closely related to that of hyperbolic manifold. Let hrl and N be complex spaces and denote by Hol(N, M) the set of holomorphic mappings from N into M. A sequcnce fi E Hol(Nl M) is said t o be compactly dit1ergen.t if given any compact K in N and conlpact K' in M , there exists j such that f i ( K ) n K' = 0 for all i 2 j. Fix a metric Q on M which induces its topology. Hol(N, ,%I) is said t o be normal if every sequence in Hol(N. M) contains a subsequence which is either uniformly convergent on compact sets or compactly divergent. According to \flu [2], M is said t o be taut if Hol(N, 111) is normal for evcry N. If Hol(N, 1l.f) is equicont.inuous for every N with respect to some metric Q on M , thcn M is said to be tight. Me shall say that a co~nplexspace hl is Carath60dory-hyperbolic or C-hyperbolic for short (respectively, complete C-hyperbolic) if there is a covering spacc hTT of A1 whose Carathbodory pseudodistance c i f is a distance (complcte distance). \fTc say that a complex space M admits no complex line if there is no holomorphic mapping from C into M other than thc const,ant maps. Thesc various coI1cept.s are related in the following manner:
Let M be a compact con~plexmanifold. Every r-dimensional closcd complex subspace V of 111 is a 2r-dimensional cycle. A cycle of the for111 c = C n,V,, where each V , is a closcd complex subspace of dimension r: is called an analytic r-cycle. We say that. C niK is positive if all nils are positivc. By considering ordy those elements of H2,(M; Z) which can be rcprescntcd by analytic r-cycles, we obtain a subgroup H i ( M ; Z) of Hzr(M;Z). For cr E HC(A;l; Z), we set
where the infimums arc taken with respect to all analytic r-cycles c C niV, representing a.
-
Theorem 2.10 Let M be a compact Knehler manifold with negative holom,orphic sectional curvature. Ifa E HC(M; Z) can Ire represented by a positive analytic r-cycle, then. ~(a.)',, is positive. Proof Let c = C niK be any analytic cycle representing a. Lct w be the Kaeliler 2-form of ,+I. Thc volume of I.: with respect to d s L is given by the integral JK w'. This integral docs not exceed p(II,)& (if ds:, is normalized as in the proof of Proposition 2.9). Hcnce,
completeC-hyperbolic 2!
coniplete hypcrbolic 3
1
taut Since wT is closed, the integral w' depends only on a and not on c. Since cr can be represented by a positive analytic cycle, this integral is positive. It follows that p(c)k is bounded below by a positive constant which depends only on a . 0 I t seems difficult to find criteria for ~ ( a ) , to be positive. It would be of interest to investigate relationships betwccn the pseudonorms defined above and the pseudonorm defined by Chern, Levine, and Nirenberg (sfe Nirenberg [I], Chern, Levirle, and Kirenberg 111). I suspcct that their pseudonorm for H k ( d i ; Z) lics between the two pseudonorms I iritroduc~d here and is close to y(a),,, (see Sec. 3).
1
1'
1"
C-hyperbolic 2'
If
hyperbolic 3'
$
no complex line
I
tight
The irnplications (3), (3'): and (1") have bccn proved by Kiernan [2] and Eisenrnan [2] independently. The other implications are eithcr trivial or have been proved in Chapt.er IV. -4bounded domain which is not a domain of holorriorphy provides an example t o show that the converses to ( I ) , (l'), and (1") are not truc. D. Eisenniarl and L. Taylor have shown that t.he converse to (4) is not t,ruc by the following examplc. Let
The mapping h : ( 2 , w) -+ (z, zw) maps Af int.0 the unit bidisk, and is one-to-one except on the set t = 0. Let f : C -, ,V1 be a holomorpllic mapping. Thcn h o f is a constant map. It follows that either f is a constant map or f maps C into the set ((0,w ) E Ill}. But this set is equivalent
126
3
Miscellany
to {w E C ; lull < 1) which is hyperbolic. Hence f is a constant map in either case. To see that M is not hyperbolic, let p = (0, b) E M with b # 0. We shall show that the pseudodistance dbf(O, p) between the origin 0 and p is zero. Set p, = [ ( l l n ) , b]. Then dbr(O, p) = limdn((0, p,). Let a, = min[n, (n/lb1)1/2]. Then the mapping t E D -, (a,t/n, a,,bt)M maps l / a n into p,. This shows that l i m d ~ ( 0pn) , 5 l i m d ~ ( 0l/an) , = 0. The following example by Kaup shows that the converse to (2') is not true. Consider a principal bundle C 2 - (0) over Pl(C) and let L be the associated line bundle. L may be also obtained from C2 by blowing up the origin. Let B be the set of elements of L of length less than 1 with respect to the natural Hermitian fiber metric. B may be obtained from the unit ball in C 2 by blowing up the origin. It is a unit disk bundle over P,(C). Choose three points in P l ( C ) and remove the closed disk of radius from the fibers over these three points. Then the resulting space is a simply connected hyperbolic manifold which is not C-hyperbolic. It would be natural to ask the following questions.
4
Problem 1 Is every taut manifold complete hyperbolic? (I believe that thc answer is probably affirmative.) For each positive integer n, let.
Unsolved Problems
127
defined by f ( t )= (t, (-l)'ldt, 0,. . . , 0 ) is nonconstant. This implies that, if zo,zl,. . . , z, denotes the homogerleous coordinate system for P,(C), then P,(C) minus the hypersurface ( ~ 1 ) .~. - (an)d = 0 admits a complex line for any degree d. But this hypersurface is rather special.
+ +
Problem 3 Let M be P,(C) minus a "generic" hypersurface of high degree d. Is hl hyperbolic? We may further ask questions similar to those in Problem 2. Problem 4 Is a generic hypersurface of large degree in Pn(C) hyperbolic? Is it a t least without complex line? The surface ( . z ~=) ~ ( ~ 2+ ) (~. z ~ ) ~ contains a rational curve
+
Let X be a compact complex space and I11 a complex space admitting an analytic subset of a bounded domain in C n as a covering space. Let a E X. If f and g are holomorphic mappings from X into M such that f (a)= y (a) and f, (u) = g. (u) for all u E T I(X,a), then f = g (Borel and Narasimhan [I]).Actually, Borel and Narasimhitrl proved a little more.
Problem 5 It is probably possible to extend the result of Borel and Narasimhan to the case where M is C-hyperbolic. Will it be also possible to exter~dit to the case whcrc Af is hyperbolic? Problem 6 Is a con~pactC-hyperbolic or hyperbolic manifold projectivealgcbraic? In connection with the problem above, we pose
Let M be the manifold obtained from P,(C) by deletion of e(n) hyperplanes in general position. Wu [3] proved that M admits no complex line. The first ) 10, ~ ( 5=) 13. On the other few values of p are: p(2) = 5, p(3) = 7, ~ ( 4 = hand, Kiernan [I] proved that if M is the manifold obtained from P,(C) by deletion of only 2n hyperplanes in general position, then M admits a complex line and hence is not hyperbolic. Hence, in Mru's result, e(n) is the best possible for n = 2,3. But it may not be the best possible for n 2 4.
Problem 2 Delete from P,(C) p(n) hyperplanes in general position. Is the resulting manifold M hyperbolic? Is it possible to extend every holomorphic mapping from the punctured disk D* into M to a holomorphic mapping from D into Pn(C)? (For this question, see Theorem 6.1 of Chapter VI.) 1s it also possible to lower e(n) t o 2n + l? Let M be C n minus the hypersurface ( ~ 1 +) .~. . + ( z , ) ~ + 1 = 0. As Kiernan [I] pointed out, the holomorphic mapping f : C 111
Problem 7 Classify the two-dimensional compact hyperbolic manifolds. Problem 8 Investigat,e the structures of a homogeneous hyperbolic manifold. Is there any other than the homogeneous bounded do~nainsin Cn? If M = G / H is a homogeneous hyperbolic manifold, then hi! is noncompact by Theorem 2.1' of Chapter V and G has no nondiscrete center by Theorem 2.2 of Chapter V. (If a holomorphic vector field Z on M generates a one-parameter group belonging to t.he center of G , then iZ will also generate a global one-parameter group of transformations of A4 in contradiction to Theorem 2.2 of Chapter V unless Z = 0.) Problem 9 Extend Theorem 6.2 of Chapter VI to the case where A is an analytic subset with singular points. We shall now elaborate on the statement made concerning the intrinsic seminorms of Chern, Levine, and Nirenberg in Sec. 2. Let u be a real C2
128
Mzscellany
function on a cotnplex manifold M. We define a real k-form, 1 2 k 2 dim M, wk(u) as follows:
5 2n =
wZp-I (u) = dCuA (ddcu)~-l, u2P(u) = du
dCuA (ddcu)P-l,
Postscript
where dC= i(8 - a). We denote by Fk = F k ( M )the set of real C 2 functions u satisfying the following three conditions: (i) 0 < u < 1; (ii) u is plurisubharmonic, i.e., (a2u/azjafk) is positive semidefinite; (iii) d[wk(,u)]= 0.
For t.he assertions made in this postscript without reference, the reader is advised to look up the Grundlehren book, Hyperbolic Complex Spaces.
Let B be a real k-dimensional differentiable manifold and p : B differentiable mapping. We define vk(B, 9 ) by
1
+
-4.1 a
Then t1k is a k-dimensional measure on M in the sense defined in Sec. 2. If f is a holomorphic mapping from M into another complex manifold, then
c so that P r o b l e m 10 Docs there exist a (universal) cor~sta~lt
y ( B ,9 ) n
5 c %k(B,P) 5 P ( B ,9 ) n
for k-dimensional B and mappings 9 : B -+ Ma?In view of Proposition 2.5 it suffices to prove these inequalities in the special case where M is t,he ope11 unit ball D, in Cn. To prove the second inequality, it would be necessary to obtain an est.imate similar to the one proved by Chern, Lcvine. and Kirenberg [I]. Their basic inequality says that if K is an open subset wit11 compact closure in D,: then (lujI2
+ [any minor of {ujk)l)dV 2 C
JK
with C a fixed constant independent of u. Here u j = au/az3, u3i = d 2 u / a t a~z k , and dV represents the usual volume element in C".
d M and
CM
I took it for granted that when a complex manifold M is hyperbolic, t,he distance function dh, defines the manifold topology of M. However, this is not completely trivial. For a Ftienlarl~lianmanifold it takes also a little argument to see that the Rie~nanniandistance defines the manifold topology. Both d M and the Riemannian distance belong to a class of distance functions called "inner distance". It is t.he property of an inner distance that it defines the manifold topology. On the other hand, the Carathkodory distance chf may not define the manifold topology. In Sec. 2 of Chapter IV, we showed that for a polydisk '2.I = Dn, cht coincides with d,w. It is a deep result of Le~npertthat for any geometrically convex domain in Cn, CM coincides with dM. As for Example 1 in Sec. 1 of Chapter IV, we have generally
for (p,pl),(q, q') E X x XI. Although a similar formula holds also for the CarathCodory pseudodistance, its proof by Jarnicki and Pflug is much harder. 2
Hyperbolicity with P a r t i a l Degeneracy
In this book? we mainly considered two extreme cases with respect to the intrinsic pseudodistance d x of a complex manifold X, namely t,he hyperbolic case and a few examples of X such as C and C* for which dx is identically zero. However, there are important non-hyperbolic complex manifolds X with non-trivial d x .
Postscript
130
+
The complement of 2n 1 hyperplanes in general position in the complex projective space P n C is known to be hyperbolic. Let X be the complement of only n 2 hyperplanes in general position in P,C. Then X is not hyperbolic. However, d x (p, q) vanishes only when p and q belong to thc so-called diagonal hyperplanes. I elaborate on this when n = 2. The reader is advised to draw the configuration while reading the following explanation. The complement of 5 lines in general position in the projective plane P 2 C is hyperbolic. (A line P 2 C is a projective line PIC.) Let X be the complement of 4 lines Li, i = 1,2,3,4 in general posit.ion in P2C. The configuration has 6 vertices, ptj = Li n Lj, 1 i < j 4. Consider, for examples, the line Ll2 through the vertices pl2 and p34. AS a line in X , LIZ has two points missing, namely pl2 and p34 since they are 011 the deleted lines. Since the projective line LIZ minus two points is biholomorphic to C * = C - (01, the distance dx (p, q) = 0 for any two points p, q E X n L12, (see Exanlple 3 in Sec. 1 of Chapter IV). The same holds for the line L13 connecting pl3, pz4 and the line L1., connecting p14, p23. The threc lines LIZ,L13 and Llq are called the diagonal lines of the given configuration. It is now clear that if p, q E LIZU L ~UJLid, then dx(p, q) = 0. It is a non-trivial fact that dx(p, q) = 0 only for such a pair p, q. This result can be generalized to all n. The example above leads us naturally to the following definition. A complex space X is said to be hyperbolic modulo a closed subset A if dx(p, q) > 0 for distinct points p, q unless both p and q are in A. Although it is generally believed that a generic compact complex space is hyperbolic, concretely given compact complex spaces are often hyperbolic modulo a proper closed complex subspace A.
+
<
3
<
Hyperbolicity Criteria
In Theorem 4.11 of Chapter IV, we gave a differential geometric criterion for hyperbolicity in terms of negative holomorphic sectional curvature. Although this helps us to understand the geometric meaning of hyperbolicity, it is not practical to use in concrete examples. The most powerful criterion for hyperbolicity is Brody's criterion. -A complex space is said to be Brody-hyperbolic if it admits no nonconstarlt holomorphic maps from C . Every hyperbolic complex space is triviall!. Brody-hyperbolic (see Theorem 1.1 of Chapter V). An example of a domai~l in C 2 Sec. 3 of Chapter IX shows that the converse is not true in general. The theorem of Brody says that for a compact complex space the converse holds.
4 Hyperbolic Imbedding
131
Brody's theorem, combined with Nevanlinna-Cartan theory of value distributions, seems to offer the best approach to the hyperbolicity question. I conjectured that a generic hypersurface of P n C of large degree (2 2n 1) is hyperbolic and its complement is hyperbolically imbedded in P,C. There have been a number of papers written in the past 30 years on this conjecture. I mention here only the paper of H. F'ujimoto, from which the interested reader should be able to trace back other papers; Some constructions of hyperbolic hypersurfaces in P n ( C ) in Advanced Study Pure Math. 42, Complex Analysis in Several Variables, 109-114, Math. Soc. Japan. It is an open problem to extend the theorem of Brody to the situation where the space X is hyperbolic modulo a subspace.
+
4
Hyperbolic Imbedding
The big Picard theorem is considered the pinnacle of the classical function theory. Theorem 6.1 of Chapter VI generalizes the big Picard theorem to higher dimension as extension theorems for holomorphic maps. The terminology hyperbolic imbedding did not exist at the time of writing of thc first edit.ion. However, the three conditions in Theorem 6.1 arc precisely the conditions for hyperbolic imbeddcdness. We define hyperbolic imbedding in the way currently stated. Let Z be a complex space and Y a complex subspace with compact closure p. We say that Y is hyperbolically imbedded in Z if for any two distinct points p,q E Y there exist neighborhoods Up and Uq of p and q such that dy(Up n Y, Uq n Y) > 0. In applications, Z is usually compact, Y is the complement of a closed complex subspace of Z, (so that the boundary of Y is precisely this closed complex subspace). For example, in the case of the classical big Picard theorem, Z = P I C and Y is the complement of three points, say oc, 0 and 1. Then Theorem 6.1 reads as follows: If Y is hype~bolicallyimbedded in Z, every holomorphic mapping f from the punctured disk D* znto Y extends to a holomorphic mapping f from the disk D to 2. There are many other ways of defining hyperbolic imbedding, including the one which relates it Montel's theory of normal families. Here, we give a definition in terms of the relative pseudodistance d y , ~ . Let FY,zc Hol(D, Z ) be the family of holomorphic maps f : D + Z such maps all that f - ' ( 2 - Y) is either empty or a singleton. Thus, f E FYsz of D ,with the exception of possibly one point, into Y. The exceptional point
is of course mapped into the boundary d Y . Making use of holomorphic disks we define a pseudodistance dy.z on Y.Since belonging to FYsz,
we have
where the first inequality holds on 7 arid tlie second inequality is valid on Y. Then Y i s hyperbolically imbedded in Z if and only i f dy,z(p, q ) > 0 for a11 distinct pairs p, q E y . Brody's criterion has been extended to a criterion for hypcrbolic imbeddcdncss by Green, Zaidenberg and others.
5
Intrinsic Infinitesimal Pseudometrics
In Sec. 1 of Chapt,cr 1: we defined t.he Poiricare rnetric ds$ of the unit disk D. The distance function p on D is the integrated form of ds$ and, conversely: ds: is the infinit.esima1 form of p. The intrinsic pseudodistance dx is a generalization of p. Then there is naturally a corresponding gcncralization of ds&: i.e., t,he i~lfillitesimalfor111 of d d X .\Vc dcn0t.c it. by Fs; actually, F i is the generalization of ds:. The advantage of dx over FAY is that d . ~ can be defined for singular complex spaccs and infinite dinlcnsional spaces while there are some technical problems in defining Fx for spaccs other than nori-singular complex manifolds. It. is riot a trivial mat.ter to rigorously prove that d x is the integrated form of Fay,arid this was done by Royden. Hc used it also prove that on a Teich~niillerspace T tlie Teichmiillcr distancc coincides with dT. It is an important result with many applications. 6
8 Principle for the Conslruction of d M
Postscript
132
7
133
Finiteness Theorems
In Chapt,cr V, we proved that the automorphism group of a compact hyperbolic coniplex nianifold is finit,e. This is a beginning of morc general finiteness theorems on Hol(X, Y) based on distance-decreasing property of holomorphic mappings. Various finiteness tlieorerns for holomorphic maps and sections, have been obtained by Noguchi and others. Some of the finiteness theorenis are motivated by the Mordell conjecture over function fields, proved by Grauert and Manin, and also by Lang's conjectures in Diophantine geometry. 8
Principle for the Construction of dM
The construction of d.4f is based on a very general and simple principle, which may be used to construct other quantities. Let C be the category of complcx manifolds with holomorphic rr~nppings as morphisms. The category C contains a "model object" D with PoincarC distance p. For any object of C. i.e., any complcx manifold ,+Ithere arc many morphisms from D to M to connect any pair of points of M . That is all we needed to construct d,tf. So the same construction works for thc category of cornplex spaces with sirigularities and also the category of almost complex rrianifolds. Wc may e \ ~ consider ! thc category of ir~finitcdimensional conlplex spares. In Chapter IX we considered the intrinsic measures or volume elcments. This is also based on the same principle with the unit ball with invariant volume element as model object. Compact measure hyperbolic manifolds are closely related to compact complcx manifolds of general type. We can apply the same principle to real projective structures. Let P be the category of real manifolds with projective connections and P' thc subcategory of manifolds with (flat) projective structure. the morphisms bcing projective maps. -4s model object, we take the open interval I = (-1.1). \Ire define the distancc function p in I by
Self-Mappings
Although we discussed autornorphisms of a hyperbolic manifold in Chnpter V, we have not considered more general self-mappings. Abate's book mentioned in Preface presents a comprehensive account on iterates of holomorphic self-maps in hyperbolic complcx spaces. We mention herc also J. E. Fornaess' Dynamics i n Several Complex Variables, CBhIS 87 (1996). Amer. Math. Soc., which gives applications of the intrinsic distance arid hyperbolicity to complex dynamics.
where (-1.1; a: b) dcnot,es the cross ratio of -1: l ? a ,b. This is an analog of the PoincarC distancc. The interested reader is referred t o my paper Projectiz~elyinva.riant distances for a f i n e and projective structures, Banach Center Publ. 12, Warsaw 1984. 127-152. So far, we have not found such a construction in the conformal differential geometry.
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Summary of Notations
We summariae only those notations that are used most frequently throughorit this monograph.
The field of complex numbers. Vector space of n-tuples of complex numbers (zl, . . . ,zn). Open unit disk in C , i.e., {z E C ; lzl < 1). Punctured disk D - (0). Polydisk D x - - . x D (n times). Open unit ball in Cn,i.e., {(t',. . . ,tn);CIzjI2 < 1). Complex projective space of dimension n. Poincark distance (non-Euclidean distance) in D. the pseudot1istanc.e of M defined in Sec. 1 of Chapter IV. the CarathCodory pseudodistance of M defined in Sec. 2 of Chapter IV.
Author Index
Numbers after names show the pages on which their papers are quoted.
A Ahlfors, L. V., 3, 6, 11 Andreotti, A., 69, 88
Baily, W. L., Jr., 100 Bergrnan, S., 18 Bishop, R. L., 112 Bochner, S., 68! 69, 100 Borel, A., 127 Bremermann, 11. J., 56, 74
C Carathkdory, C., 52, 74 Cartan, E., 33 Cartan, H., 69, 74, 100 Chern, S. S., 19, 20, 32, 124, 128
Grauert, H., 6, 11, 40, 98 Gunning, R. C., 93, 96
Hahn, K. T., 30, 32, 33 IIano, J., 18 IIawley, N. S., 69 Heins, M., 11 Hirxebruch, F., 108, 115 Horstmann, H., 55 IIua, L. K., 33 Huber, H., 15, 81 Hurcwicz, W., 118 Hurwitz, A., 69
Igusa, J., 105
D Dinghas, A., 32 E Eisenman, 13.: 115, 125
F Ford, I..,
J Jenkins, J. A , , 15
H.,6
Gindikin, S. G., 28, 31 Coldbcrg, S. I., 39, 112 Goto. XI., 115
Kaup, W., 69, 74 Kiernan, P. J., 61, 63, 65, 77, 125, 126 Kobayashi, S., 17, 18, 19: 32, 37, 39: 56. 65, 67, 69, 70, 91, 112, 118 Kodaira, K., 25, 107, 108 Koranyi, A,, 42 Koszul, J. I>., 18 Kwack, M. H., 81, 84! 88, 99
146
Author Indez
Landau, H. J., 15, Lang, S., 98 Lelong, P., 74 Levinc, H., 124, 128 Lichnerowicz., A., 18
Reckziegel, H., 6, 11, 40, Reiffen, H. J., 52 Remmert, R., 103 Rossi, H., 93, 96
Subject Index
S
M Martin, W. T., 100 Mitchell, J., 30, 32, 33 Moisezon, B. G., 105 Montgomery, D., 68
Narasimhan, R., 84, 93, 99, 127 Nirenberg, L., 124, 128 Nomizu, K., 17, 19, 37, 70
0 Ochiai, T., 91, 118 Oka, K., 74, 99 Osserman, R., 15 Ostrowski, A., 11
Peters, K., 25, 107, Pfluger, A , , 11 PoincarB, H., 69 Pyatetzki-Shapiro, I. I., 18, 28, 31
Sarnpson, J. II., 69 Satake, I., 100 Schiffer, hl., 15 Schottky, F., 10 Schwarz, H. A., 69 Shioda, T., 105 Spencer, D., C., 108 Springer, G., 12 Stoll, W., 88, 103
Tashiro, Y., 33
A ample, 106 analytic germ, 93 analytic polyhedron, 53 generalized, 54 analytic set, 93 reducible, 94
B
v van Danzig, D., 67 van der Waerden, B. L., 67 Vinherg, E. B., 28, 31
Wallman, H., 118 Weil, A,, 104 Wu, H., 24, 69, 74, 125, 126
base point, 106 Bergman kcrnel form, 18 Bergman kernel function. 18 Bergman metric, 18 big Picard t.l~ec>rerr~: 7i bimeromorphic, 105
C C-hyperbolic, 125 CarathQdory hyperbolic, 125 CarathQdory pseudodistance, 49 compactly convergent, 125 complete metric space. 53 complex space, 95 complex V-manifold, 100 contraction, 105 convex hull, 55 convex with respect to F, 55 covariant vector bundle, 106
dimension of a complex space, 95 divisor, 108 group of -s, 109
generalized analytic polyhedron, 54 group of divisors, 109 group of linear equivalence, 109
Hausdorff measure, 115 Hermitian fiber metric, 37 Hermitian vector bundle, 37 holomorphic bisectional curvature, 39 l~olomorphicsectional cilrvatiire, 39 holomorphically convex, 55 hyperbolic, 56 C-, 125 (k, m)-, 121 measure-, 117
k-dimensional measure, 120 (k, m)-hyperbolic, 121
liftable (mapping), 101 locally, 101 linear equivalence, 109 group of, 109 little Picard theorem, 67 local dimension, 94 locally liftable, 101
Subject Index
R measure hyperbolic, 117 mcromorphic mapping. 103 minimal, 104, 108 relatively, 108 strongly, 104 minimal model, 105 monoidal transformation. 106
negative curvature, 38 negative Ricci curvature, 38 no complex line, 125 normal complex space, 98 normal family, 129 normal point, 98 normalization, 98
Picard theorem? 67, 77 big, 77 little, 67 Poincark (-Bergman) metric, 3 positive curvature, 38 positive Ricci 38 projcctivcly normal, 98 pseudoconvex, 74 pseudodistance d,t4, 45 pure k-dimensional, 94
reducible, 94, 95 analytic set, 94 complex space, 95 regular point, 94 relatively minimal, 108 Iticci tensor, 17
Schottky's theorem, 10 Schwara lemma, 1 sheaf of germs of holomorphic functions, 93 Siegel domain of second kind, 28 affinely homogeneous, 31 automorphism of, 31 singular point, 94 spread, 59 strongly minimal, 104 structure sheaf. 95
T taut, 125 tight, 125
