SHOSHICHI KOBAYASHI
Volurnf' )18
GtundlC'hrtn dC'. IIlI,hC'm:&lIl(hen
WI$won.$.(h,fltn A ~fltJ. of
ComprCOhC""Vl'SludlC'. In M :lllhrmallo
HYPERBOLIC COMPLEX SPACES
Springer
Grundlehren der mathematischen Wissenschaften 318 A Series of Comprehensive Studies in Mathematics
Editors
S. S. Chern B. Eckmann P. de la Harpe H. Hironaka F. Hirzebruch N. Hitchin L. Hormander M.A. Knus A. Kupiainen J. Lannes G. Lebeau M. Ratner D. Serre Ya.G. Sinai N. J. A. Sloane J.Tits M. Waldschmidt S. Watanabe Managing Editors
M. Berger J. Coates S. R. S. Varadhan
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Shoshichi Kobayashi
Hyperbolic Complex Spaces With 8 Figures
II~~II
Springer
Shoshichi Kobayashi Department of Mathematics University of California Berkeley, CA 94720 USA email:
[email protected] Library of Congress CataloginginPublication Data Kobayashi, Shoshichi, 1932Hyperbolic complex spaces / Shoshichi Kobayashi. p. cm.  (Grundlehren der mathematischen Wissenschaften, ISSN 00727830; 318) Includes bibliographical references and index. ISBN 3540635343 (hardcover: alk. paper) 1. Analytic spaces. 2. Holomorphic functions. 3. Distance geometry. I. Title. II. Series. QA331.K716 1998 515'.94dc21 9816060 CIP
Mathematics Subject Classification (1991): 32H20, 32H15, 32H25 (primary) 32H35, 32Ho2, 32H04, 32H30 (secondary)
ISSN 00727830 ISBN 3540635343 SpringerVerlag Berlin Heidelberg New¥ork This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. © SpringerVerlag Berlin Heidelberg 1998 Printed in Germany Cover design: MetaDesign plus GmbH, Berlin Typesetting: Typeset in LATEX by the author and reformatted by Kurt Mattes, Heidelberg, using a Springer TEX macropackage SPIN: 10573657 41/31435 4 3 2 1 0 Printed on acidfree paper
Table of Contents
Introduction
IX
Chapter 1. Distance Geometry 2 3 4
Pseudodistances Degeneracy of Inner Pseudodistances Mappings into Metric Spaces Norms and Indicatrices
Chapter 2. Schwarz Lemma and Negative Curvature 2 3 4 5
Schwarz Lemma . . . . . . . . . . Negatively Curved Riemann Surfaces Negatively Curved Complex Spaces Ricci Forms and Schwarz Lemma for Volume Elements Metrics in Jet Bundles
7
8 13
19 19 25 30
35 41
Chapter 3. Intrinsic Distances
49
1 2 3 4 5 6 7 8 9 10 11 A B
49
Two Intrinsic Pseudodistances Hyperbolicity ....... . Hyperbolic Imbeddings Relative Intrinsic Pseudodistance Infinitesimal Pseudometric Fx Brody's Criteria for Hyperbolicity and Applications Differential Geometric Criteria for Hyperbolicity Subvarieties of Quasi Tori ....... . ....... . Theorem of BlochOchiai Projective Spaces with Hyperplanes Deleted Deformations and Hyperbolicity Royden's Extension Lemma NevanlinnaCartan Theory
Chapter 4. Intrinsic Distances for Domains 2
Caratheodory Distance and Its Associated Inner Distance Infinitesimal Caratheodory Metric . . . . . . . . . . .
60 70 80
86 100 Il2 Il6 124 134 148
153 159 173 173 178
VI
3 4 5 6 7 8 9 10 A
Table of Contents
Pseudodistance Defined by Plurisubharmonic Functions Holomorphic Completeness . . . . . . Strongly Pseudoconvex Domains Extremal Discs and Complex Geodesics Extremal Problems and Extremal Discs Intrinsic Distances on Convex Domains Product Property for the Caratheodory Distance Bergman Metric Pseudoconvexity
184 187 192 202 206 215 221 224 234
Chapter 5. Holomorphic Maps into Hyperbolic Spaces
239
I 2 3 4 5
239
Normality, Tautness and Hyperbolicity Taut Domains . . . . . . . . . . . Spaces of Holomorphic Mappings Automorphisms of Hyperbolic Complex Spaces Selfmappings of Hyperbolic Complex Spaces
Chapter 6. Extension and Finiteness Theorems 2 3 4 5 6 7 8 9 A
The Classical Big Picard Theorem ..... Extension through Subsets of Large Codimension Generalized Big Picard Theorems and Applications Moduli of Maps into Hyperboically Imbedded Spaces Hyperbolic and Hyperbolically Imbedded Fibre Spaces Surjective Maps to Hyperbolic Spaces ...... . Holomorphic Maps into Spaces of Nonpositive Curvature Holomorphic Maps into Quotients of Symmetric Domains Finiteness Theorems for Sections of Hyperbolic Fiber Spaces Complex Finsler Vector Bundles ........... .
251 256 262 268 277 277 279 282
290 295
302 313 323
329 335
Chapter 7. Manifolds of General Type
343
I 2 3 4 5 6 7
343 353 360 365
Intrinsic Volume Forms ..... Intrinsic Measures . . . . . Pseudoampleness and Ldimension Measure Hyperbolicity and Manifolds of General Type Extension of Maps into Manifolds of General Type Dominant Maps to Manifolds of General Type Effective Finiteness Theorems on Dominant Maps
Chapter 8. Value Distributions 1 2 3 4
Grassmann Algebra Associated Curves Contact Functions First Main Theorem
370 376
382 393 393
397 402 407
Introduction
A Riemann surface is said to be elliptic, parabolic or hyperbolic according as its universal covering space is the Riemann sphere PI C, the finite plane C or the unit disc D. Most Riemann surfaces are hyperbolic. In particular, all compact Riemann surfaces of genus::: 2 are hyperbolic. From differential geometric view points, the type of a Riemann surface can be characterized by the sign of the curvature of the natural metric it carries. Namely, elliptic with curvature +1, parabolic with curvature 0, and hyperbolic with curvature I. In the compact case, algebraic geometrically, these three types can be characterized by the Kodaira dimensions, I, 0 and +1. We are interested in complex spaces of hyperbolic type since they represent the general case. There are several ways to extend the concept of hyperbolicity to higher dimensional complex spaces. Our hyperbolicity is based on the existence of a certain intrinsic distance, and this intrinsic distance was originally introduced to generalize Schwarz' lemma to higher dimensional complex spaces. Schwarz' lemma, reformulated by Pick, says that every holomorphic map from a unit disc D of C into itself is distancedecreasing with respect to the Poincare distance p, and is at the heart of geometric function theory. This lemma has been generalized to higher dimensional complex spaces in various ways. In 1926 Carath6odory defined an intrinsic pseudodistance Cx for a domain in en by setting cx(p, q) = sup p(f(p), f(q)), where the supremum is taken over all holomorphic maps f from X to D. If X is a bounded domain, Cx becomes a distance. The Carath6odory distance CD of the unit disc D coincides with the Poincare distance p, and any holomorphic map f: (X, cx) ~ (Y, Cy) between two complex spaces X and Y is distancedecreasing. In 1938, Ahlfors generalized Schwarz' lemma as a comparison theorem between the Poincare metric ds 2 of D (normalized so that the curvature is I) and any Hermitian pseudometric da 2 of curvature.::=:: Ion D; Ahlfors's version states da 2 .: =: ds 2 • So if X is any Riemann surface with Hermitian pseudometric dsi of curvature.::=:: 1, then every holomorphic map f: (D, ds 2 ) ~ (X, dsi) is metricdecreasing, i.e., f*dsl .: =: ds 2 . In defining the Caratheodory distance of X, one considers the family Hol(X, D) of holomorphic maps into D. But Ahlfors's generalization suggests that it would be more natural to consider the family Hol(D, X) of holomorphic maps from D. Thus, in 1967 I introduced a new intrinsic pseudodistance d x by dualiz
X
Introduction
ing the Caratheodory's construction. It enjoys the same basic properties of the Caratheodory pseudodistance. Namely, the intrinsic pseudodistance df) for the unit disc coincides with the Poincare distance p, and any holomorphic map f: (X, d x ) + (Y, d y) is distancedecreasing. In fact, d x can be characterized as the largest pseudodistance such that all holomorphic maps (D, p) + (X, d x ) are distancedecreasing. In particular, d x is larger than or equal to ex, but is strictly larger in many cases. This makes d x more useful than cx. For example, if X is a compact complex space, ex == 0 since there are no nonconstant holomorphic functions on X. However, if X is a compact Riemann surface of genus ~ 2, then d x coincides with the distance function coming from the natural Hermitian metric of curvature I. Therefore, as a generalization of the notion of Riemann surface of hyperbolic type, I named a complex space hyperbolic if this intrinsic pseudodistance is actually a distance. (I prefer to put the adjective hyperbolic before "complex space" since a "complex hyperbolic space" would often mean a complex space form, i.e., a Kahler manifold of constant negative holomorphic sectional curvature, which is a very special example of hyperbolic complex space.) This intrinsic pseudodistance has found a number of applications, including generalizations of Picard's theorems. The little Picard theorem says that an entire function missing two values, say 0, I, must be constant. This can be best understood in terms of hyperbolicity. Since de == and C  {O, I} is hyperbolic, the distance decreasing property of a holomorphic map f: C + C  to, I} immediately implies the little Picard theorem. The hard part in this argument is to show that C  {O, I} is hyperbolic. In general, the most satisfactory way of showing that a complex space X admits no nonconstant entire map f: C + X is by proving that X is hyperbolic. Since the hyperbolicity is defined only in terms of holomorphic maps from the disc, this is in accord with Andre Bloch's principle that a result on entire maps f: C + X should be derived from results on holomorphic maps from the disc into X. This principle was ennounced as "Nihil est in in/inito quod non prius fileri! infinito", (see Bloch [4; p.2]). I introduced the intrinsic pseudodistance d x in 1967 and published a little monograph Hyperbolic Manifolds and H%morphic Mappings in 1970. This was followed by a long survey article in the Bulletin of the American Mathematical Society (1976). With increasing activities on hyperbolic complex analysis and geometry, in 1973 the Mathematical Reviews created two new subsections "invariant metrics and pseudodistances" and "hyperbolic complex manifolds" within the section "ana~vtic mappings" (which is now called "holomorphic mappings"). Since 1980 several books on intrinsic pseudodistances and related topics have appeared, each emphasizing certain aspects of the theory:
°


T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances,
1980. J. Noguchi and T. Ochiai, Geometric Function Theory in Several Complex Variables, 1984 (English translation in 1990). S. Lang, Introduction to Complex Hyperbolic Spaces, 1987. M. Abate, Iteration Theory of Holomorphic Maps on Taut Man!folds, 1989.
Introduction
XI
S. Dinen, The Schwarz Lemma, 1989. M. larnicki and P. Pflug, Invariant Distances and Metrics in Complex Ana(vsis, 1993. In addition, Encyclopaedia of Mathematical Sciences, vol. 9 (1989), Several Complex Variables, contains the following two chapters: Chapter III "Invariant Metrics" by E. A. Poletskii and 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 "Complex Analysis: the Geometric Viel,l'point" (1990) in the Carus Mathematical Monographs series ofthe Mathematical Association of America is an elementary introduction to function theory from the viewpoint of hyperbolic analysis. Our aim here is to give a systematic and comprehensive account of the theory of intrinsic pseudodistances and applications to holomorphic mappings. Since some of the basic results on holomorphic mappings make use of only metric properties of spaces together with the distancedecreasing property of maps, in Chapter I we assemble such results under the heading of Distance Geometry. The reader should skip Chapter I on the first reading. It would be best to return to Chapter I when it becomes necessary. In Section I of Chapter 2, we prove Ahlfors' generalization of the SchwarzPick lemma. In the rcmainder of the chapter we present further generalizations as well as applications. The reader is again advised to read only up to Theorem (2.1.10) and then to proceed to Chapter 3. The rest of the chapter can be best understood if it is read when it becomes necessary. Chapter 3, which forms a core of this book, is concerned with basic properties of hyperbolic and hyperbolically imbedded complex spaces and hypcrbolicity criteria. The concept of hyperbolic imbedding, introduced in my 1970 monograph to generalize thc big Picard theorem, is extensively studied. This concept is also essential in generalizing Montel's theory of normal families in Chaptcr 5. Differential geometric critcria for hyperbolicity in terms of negative curvature are often diffcult to use. The most powerful criterion for hyperbolicity is the socalled Brody hyperbolicity. A complex space is said to be Brodyhyperbolic if it admits no nonconstant holomorphic maps from C. Every hyperbolic complex space is trivially Brodyhyperbolic. The theorem of Brody says that every compact Brodyhyperbolic complex space is hyperbolic. This may be regarded as the converse to the Bloch principle. Brody's critcrion has been extended to a criterion for hyperbolic imbeddedness by Green, Zaidenberg and others. Brody's theorem, combined with NevanlinnaCartan theory of value distributions, seems to offer the best approach to the hyperbolicity question. The part of the NevanlinnaCartan theory we need for hyperbolicity criteria is summarized in a fairly selfcontained manner in Appendix B. A complex space X is said to be hyperbolic modulo a closed subset Ll if dx(p, q) > 0 for distinct points p, q unless both p and q are in Ll. Although it is generally believed that a generic compact complex space is hyperbolic, concretely
XII
Introduction
given compact complex spaces are often hyperbolic modulo a proper closed complex subspace Ll. The theorem of Brody does not extend to this situation. To find a usable criterion for hyperbolicity modulo a subspace is an important open problem. Results in Sections I through 7 are basic and are used throughout the book. However, Sections 8 through II may be regarded as special topics. Chapter 4 is more or less independent from the rest of the book. Using intrinsic distances, we compare various completeness concepts for domains in e". We shall see that for strongly pseudoconvex domains with smooth boundary the Carathcodory and Kobayashi distances have the same boundary behavior. Sections 6 and 7 on extremal discs will be followed by Lempert's theorem that the two distances actually coincide for convex domains. Although some of the results are valid for domains in a complex Banach space, the reader interested in the infinite dimensional case should consult the books by FranzoniVesentini and by Dinen mentioned above. Geometry and analysis of bounded domains, particularly those of pseudoconvex domains are vast and rich areas. However, we have confined ourselves to the area directly touching the intrinsic distances. Our discussion on basic properties of the Bergman metric in Section lOis more for the sake of comparison with the Caratheodory distance and the Kobayashi distance. It would take another book to give a good account of the Bergman metric. Main applications of the intrinsic pseudodistances are to holomorphic mappings, and in Chapter 5 we study first those results that depend largely on the distancedecreasing property of mappings. As in the work of GrauertReekziegel, Kaup and Wu, the emphasis in hyperbolic complex analysis was initially laid on normal families of holomorphic mappings. Section I may be regarded as a generalization of Montel's theory of normal families to holomorphic maps into higher dimensional spaces. For a satisfactory generalization of Montel's theory, the concept of hyperbolicity is not sufficient. The essence of MonteI's theory is crystalized in the equivalence of hyperbolic imbeddedness and taut imbeddedness. While Section I is concerned with topological properties (e.g. compactness or relative compactness) of spaces of holomorphie maps HoI(X, Y) into hyperbolic or hyperbolically imbedded spaces Y, Section 3 treats complex structures of Hol(X. Y). Although we discuss in Section 5 Abate's results on iterates of holomorphic selfmaps in hyperbolic complex spaces, we shall not go into a currently active area of complex dynamics in several variables. 1. E. Fornaess' "Dynamics in Several Complex Variables" (1996) in the CBMS series of the American Mathematical Society presents applications of the intrinsic distance and hyperbolicity to complex dynamics. The big Picard theorcm is the pinnacle of the elassical function theory. In Chapter 6, first we generalize the big Picard theorem to higher dimensional spaces as extension theorems for holomorphic maps. We combine these extension theorems with results of Chapter 5 to obtain finer structure theorems for Hol(X, Y). Various finiteness theorems for holomorphic maps and sections, largely due to
Introduction
XIII
Noguchi, are proved in Sections 6 and 9. These finiteness theorems originate in the Mordcll conjecture over function fields, proved by Grauert and Manin, and in Lang's conjectures in Diophantine geometry. In Chapter 7 we consider the intrinsic measures or volume elements; they are useful in studying equidimensional holomorphic mappings. However, the condition of measure hyperbolicity is often not strong enough to yield interesting results. Most of the results in this chapter make use of the algebraic geometric condition of "general typc", which is a little stronger than measure hyperbolicity. Generalizations of these results to the measure hyperbolic case are mostly open problems at this moment. In fact, it is not known if every measure hyperbolic compact complex space is of general type. Chapter 8 is concerned with value distributions for holomorphic curves (mainly in Pn C). We follow here largely Chern [4], CowenGriffiths [I], Shabat [I] and Fujimoto [13]. We shall not go into value distrubutions for holomorphic maps from em with 111 > I. In proving the theorem of BlochOchiai in Section 9 of Chapter 3 we made use of part of the classical Nevanlinna theory. In Section 10 of Chapter 3 we used E. Borel's theorem which generalizes the little Picard theorem to a system of entire functions. Although the proof of Borel's theorem does not require Nevanlinna theory, we prefer to derive it as a consequence of the general defect relation. An important application of the intrinsic distance we were not able to include in this book is the theorem of Royden on the Teichmiiller metric. For this we refer the reader to the book of F. P. Gardiner Teichmiiller Theory and Quadratic DifTerentials, (1987). During the preparation of this manuscript, I was partially supported by the Japan Society for Promotion of Sciences (in 1990 at Hokkaido University) and the Alexander von Humbold Foundation (in 1992 and 1993 at Technische Universitat in Berlin). I was a guest also at International Christian University in Tokyo, University of Tokyo, Seoul National University, Postech in Pohang, Academia Sinica in Taipei and Keio University. I would like to express my gratitude for hospitality to these institutions and my hosts at these institutions  Professors Haruo Suzuki, Udo Simon, Masakiti Kinukawa, Takushiro Ochiai, HongJong Kim, KangTae Kim, Shishyr Roan, and Yoshiaki Maeda. I would like to thank also the following mathematicians for their critical comments made on part of the manuscript at various stages of the preparation  Professors Marco Abate, Kazuo Azukawa, Akio Kodama, Myung He Kwack, and Junjiro Noguchi. While working on the manuscript I could not help reminiscing good old days of scissors and paste. The computer has kept me revising the manuscript for thc past two years. Berkeley, January 1998
S. Kobayashi
Chapter 1. Distance Geometry
1 Pseudodistances Let X be a set. A pseudodistance d on X is a function on X x X with values in the nonnegative real numbers satisfying the following axioms:
Dl D2 D3
d(p, q) = 0 if p = q; d(p, q) = d(q, p), d(p, r) ::: d(p, q) + d(q, r),
(symmetry axiom); (triangular inequality).
A pseudodistance d is called a distance if it satisfies, in addition to D2 and D3, the following Dl'
d(p, q)
=0
if and only if p
= q.
A pseudodistance d on X induces a topology on X in a natural manner. This topology is Hausdorff if and only if d is a distance. In our applications, X will be a complex space. In such a case, X has two topologies, the complex space topology and the dtopology, i.e., the topology induced by d. We consider only arcwise connected Hausdorff topological spaces and assume the following axiom. D4.
If X is a topological space, then d: X x X
~
R is continuous.
In other words, the topology of X is at least as fine as the dtopology. For any of the pseudodistances we construct in Chapter 3, D4 is satisfied. Let X be a topological space with a pseudodistance d. Given a curve yet), a ::: t ::: b, in X, the length L(y) of y is defined by k
(1.1.1)
L(y) = sup Ld(y(tiI), y(ti», ;=1
where the supremum is taken over all partitions a = to < tl < ... < tk = b of the interval [a, b]. A curve y is said to be rectifiable if its length L (y) is finite. The space (X, d) is said to be finitely arcwise connected if every pair of points p, q of X can be joined by a rectifiable curve. Then we define a new pseudodistance d i , called the inner pseudodistance induced by d, by setting
2
Chapter I. Distance Geometry di(p,q)=infL(y),
(l.L2)
where the infimum is taken over all drectifiable curves y joining p and q. When X is a complex space, we modify the definition above by taking the infimum over all drectifiable, piecewise differentiable (e l ) curves y joining p and q. This modification will remove technical difficulties which would arise when we discuss the Caratheodory pseudodistance. The results in this section are valid with this modified definition of d i as long as maps between complex spaces are assumed to be holomorphic. From the definition of d i it follows immediately that (1.1.3)
d(p, q) :::
d' (p, q)
for
p, q
E
X.
The following proposition is in Rinow [1; p. 120]. (1.1.4) Proposition. Let (X, d) be finitely arcwise connected. Then for all rectifiable curves where
Li
y,
is the length defined by the induced inner pseudodistance d i .
Proof Consider a partition a = to < tl < ... < tk = b for the curve y(t), a ::: t ::: b. Let Yj be the portion of y corresponding to the interval tj _I ::: t :::: tj. Then k
L(y)
=
L
L(Yj)·
j=l
From the definition of d i , we obtain
Hence, k
Ldi(y(tj_l), y(tj»:::: L(y). j=1
Taking the supremum of the left hand side over all partitions of the interval [a, b], we obtain The reverse inequality is obvious from d :::: d i .
o
(1.1.5) Corollary. Let (X, d) be finitely arcwise connected. Then (d')i
= di .
We say that a pseudodistance d is inner if d i = d. The corollary above shows that the induced inner pseudodistance d i is inner, thus justifying our terminology. Let (X, d) be finitely arcwise connected. It is said to be without detour (ohne Umweg in Rinow [1]) if for every point p E X and for every positive number E,
1 Pseudodistances
3
there exists a positive number 8 such that every point q E X with d(p, q) < a can be joined to p by a rectifiable curve y of length L(y) < e. Since d :s d i , the topology induced by d i is, in general, finer than the one induced by d. However, we have (Rinow [I; p.119]) (1.1.6) Proposition. Let (X. d) befinitely arcwise connected. Then d and d i define the same topology on X if and only il(X, d) is without detour.
Proof It is clear from the definition above that (X. d) is without detour if and only if every eneighborhood of p with respect to d; contains some 8neighborhood of p with respect to d. 0 If d is inner, i.e., d; = d, then it is without detour by (1.1.6). But in this case we can say a little more; if d (p, q) < e, then q can be joined to p by a rectifiable curve y oflength L(y) < e. Hence, (1.1. 7) Proposition. If d is inner, then for every p E X and for every positive real number e, the open eball U(p; e) = (q E X;_ d(p, q) < e} with center p is finitely arcwise connected. In fact, every q E U (p; e) can be joined to p by a curve y of length L(y) < e lying in U(p; e).
For
Let .d be a closed subset of a topological space X with a pseudodistance d. a > 0, the aneighborhood of .d is defined to be
U(.d; a) =
UU(p; a). pELl
Clearly, U(.d; 8) is an open neighborhood of .d. However, given an open neighborhood V of .d, there mayor may not exist a aneighborhood U(.d; 8) contained in V. Let {.d;} be a family of mutually disjoint closed subsets in X. We say that d is a nondegenerate outside {.d;} if d(p, q) > 0 unless p = q or P. q E .d; for some i. By collapsing each .d; into a single point and denoting it [.dd, we obtain a quotient space X/{.d;}. The pseudodistance d on X induces a distance, denoted also d, on X / {.dd. There are two topologies on X / {.d;}, namely the quotient topology induced from the given topology of X and the metric topology defined by d. The former is finer. But we can say a little more. (1.1.8) Proposition. Let X be locally compact and d an inner pseudodistance on X. Let {.d;} be a family of mutually disjoint closed subsets of X such that d is nondegenerate outside {.d;}. Let .d = U .di. Then (1) for every point p E X .d andfor every neighborhood U C X .d of p, there exists a aneighborhood V of p with respect to d such that V C U; (2) if each {.d;} is compact, the dtopology on X/{.d;} coincides with the quotient topology of X/{.d;}; (3) if d is an inner distance (i.e . .d is empty), it induces the given topology ofX. We note that (1) is stronger than the assertion that dl x  Ll defines the given (relative) topology on X  ,1. It says that both the quotient topology and the
4
Chapter 1. Distance Geometry
dtopology give the same neighborhood systems for every point [p] E X I (~d which is not of the type [~d, i.e., for every point [p] coming from p E X  ~. However, the quotient topology gives a finer neighborhood system for the point [~d than the dtopology. In (2) we show that if Ll; is compact, both topologies give the same neighborhood system for [~;]. Proof We follow the argument of Barth [3] who proved (3).
(1) Since X is locally compact, there is a relatively compact open neighborhood W of p in X with W cU. Let 8 be the minimum value of the positive continuous function d (p, .) on the compact set aW = HI  W, and let V = (q E X; d(p, q) < 8). Then V n aW = 0. Since V is connected by (1.1.7) and pEW n V, we have V eWe U. (2) If~; is compact, we can find a relatively compact neighborhood W of ~;, and the rest of the argument is the same as in (I). (3) This is a special case of (2) where ~ is empty. D Let d be a distance function on X. We say that (X, d) is Cauchy complete or simply complete if every Cauchy sequence (with respect to d) converges. If every closed ball B(o; r) = (p E X; d(o, p) :s r} with a E X and r > 0 is compact, then (X, d) is said to be strongly complete or finitely compact. Following Hristov [2] we say that (X, d) is weakly complete if for every point 0 E X there is an r > 0 (which depends on 0) such that B(o; r) is compact. It is clear that if X is locally compact and if d induces the given topology of X, then (X, d) is weakly complete. We shall later show the converse, see (1.1.10). ( 1.1.9) Proposition. Let d be a distance on a locally compact space X. (1) Then we have the following implications: strongly complete => complete => weakly complete. (2)
lfd is inner, then completeness implies strong completeness.
(I) complete => weakly complete. Assume that (X, d) is not weakly complete. Then there is a point 0 E X such that for every r > 0 the ball B(o: r) is not compact. For each natural number n, take a sequence of points {Pllj}~l in B(o: lin) without accumulation points. We note that all sequences of of the type {qll = Pllj,,}~l' where ill are arbitrary natural numbers, are Cauchy and converges to o. Fix a compact neighborhood U of o. For each fixed i, let N j be the smallest integer such that Pllj E U for all n > N j (so PNjj ¢ U). We put A = SUPj N j :s 00. If A = 00, then there is a subsequence {j (}')}~l of {j }~1 such that Nj(i.) / 00. Then PNj,;,.ili.) ¢ U contradicts the fact that the sequence {Pl/)('Ijli.d~l converges to o. If A < 00, take II> A. Then {PI/j}~l are in a compact set un B(o; lin) and must have an accumulation point in B(o; lin), which is also a contradiction. (2) complete => strongly complete when d is inner. Since d is an inner distance, it induces the given topology of X, see (1.1.8). Pro~f
1 Pseudodistances
5
Lemma. B(o; r) is compact if there is a positive number b such that B(p; b) E B(o; r).
is compact for every P
Proof of Lemma. It suffices to show that if such a positive number b exists and if 8(0; .1') is compact, then 8(0; s + ~) is compact. Let PI, P2,." be points of B(o; s + ~). Since d is inner, we can find points qi E 8(0; .1') such that d(Pi, qi) < 3b/4. Since 8(0; .1') is compact, we may assume (by taking a subsequence) that ql, q2, ... converges to some point q E B(o; .1'). Then B(q; b) contains all Pi for large i. Since B(q; b) is compact, a suitable subsequence of PI, P2,'" converges to a point P of B(q; b). Since B(o; s + ~) is closed, P is in B(o, s + ~). This completes the proof of Lemma.
In order to complete the proof of (2) we shall show that there is a positive number b such that 8(p; b) is compact for all p E X. Assume the contrary. Then there is a point PI E X such that B(PI; ~) is noncompact. Applying lemma to B(PI; ~) we see that there is a point P2 E 8(PI; ~) such that B(P2; is noncompact. In this way we obtain a Cauchy sequence PI, P2, ... such that Pk E B(Pkl; 2L1 ) with noncompact 8(Pk; dr). Let P be the limit point of this Cauchy sequence. Let a be a positive number such that B(p; a) is compact. For a sufficiently large k, B(Pk; dr) is a closed set contained in B(p; a) and hence must be compact. This is a contradiction. D
fr)
It should be pointed out that when (X, d) is not (Cauchy) complete, its completion X* with respect to d need not be strongly complete. This is due to the fact that X* need not be locally compact even if X is. The following is due to Hristov [2].
(1.1.10) Proposition. Let X be locally compact and d a distance limction on X. If (X, d) is weakly complete, then d induces the given topology of X.
Proof Assume that the given topology is strictly finer than the dtopology. Then there exist a point P E X and an open neighborhood U of p in the given topology such that for every c > 0 there is a point q E B(p; c) with q ¢. U. Since (X, d) is weakly complete, there is an co > 0 such that B(p; co) is compact. For c = coin, we have q" E B(p; Fo/n) C B(p; FO),
q" ¢. U.
Since B(p; co) is compact, taking a subsequence we may assume that {ql1} converges to a point q in B(p; c). Since we are assuming that d is continuous in the given topology, we have d(p, q)
= limd(p, q,,) = o.
Hence, P = q = lim qn, in contradiction to ql1 ¢. U.
D
The following proposition is obvious. (1.1.11) Proposition. Let X and Xi, i E I, be subsets o/a Hausdorffspace Y such that X = ni Xi. Let d and d i be distances on X and Xi such that d(p, q) :::
6
Chapter I. Distance Geometry
di(p,q)jor p,q E X.lfeach Xi isfinitelycompact (resp. complete) with respect to di , then X isfinitely compact (re~p. complete) with respect to d. Proof For 0 E X and r > 0, let B(o; r) = {p E X; d(o, p) ::s r} and Bi(o; r) = {p E Xi; di(o, p) ::s r}. If each Xi is finitely compact with respect to d i , then K := Bi(o; r) is a compact subset of Xi = X. Since B(o; r) C Bi (0; r) for all i, B(o; r) is a closed subset of K, and hence is compact. This proves that X is finitely compact with respect to d. The proof for the second statement is even more trivial. 0
ni
ni
Let f: X + Y be a continuous map between topological spaces. Given a pseudodistance d on Y, we can define the induced pseudodistance f1d on X by setting (fId)(p, q) = d(f(p), f(q) for p, q E X. Even if d is inner, f1d needs not be inner. We define the induced inner pseudodistance j* d by It is not hard to see that j* d may be defined directly in the following manner. For two points p, q E X, let y be a curve from p to q. Let L(f(y» be the length
of the curve fey) with respect to d. Then (1.1.12)
(f*d)(p, q) = infL(f(y»,
where the infimum is taken over all curves y connecting p to q. From the construction of the induced inner pseudodistance the following is evident. dy(f(p). f(q)) :s (f*d y )(p, q) for p, q E X. (1.1.13) Proposition. Let (X, d x ) and (y, d y) be topological .Ipaces with inner pseudodistances. If a continuous map f: X + Y has the property that every point p E X has a neighborhood U such that dy(f(q), fer)) = dx(q, r)
for q, r
E
U,
then d x = f*d y. Proof Let p, q
E
X, and y(t), a
dx(p,q)
::s t ::s b, be a curve from
inf L(y) y
=
p to q. Then
infsup I>X(y(tid, y(ti» y
infsup Ldy(f(y(tiI», f(y(ti») y
inf L(f(y» = j*dy(p, q), y
where the infimum is taken over all curves y from p to q while the supremum is taken over all partitions a = to < tl < ... < tk = b. D
2 Degeneracy of Inner Pseudodistances
7
2 Degeneracy of Inner Pseudodistances Let X be a topological space and d a pseudodistance. We define an equivalence relation n c X x X by (1.2.1)
n=
{(p, q)
E
X
X
X; d(p. q) = OJ
and obtain a quotient space X* = Xjn with a naturally induced distancc dO. We define the degeneracy set for p by (1.2.2)
L1(p)
=
(q EX; d(p, q) = OJ.
By collapsing L1(p) into a single point p*, we obtain X*. We can easily verify the following (1.2.3) Proposition_ If d is an inner pseudodistance on X, then d* is an inner distance on X*. The quotient topology on X* is, in general, finer than the metric topology defined by d*. If X* is locally compact, the two topologies coincide by (1.1.8). However, local compactness of X does not, in general, guarantee that of X*. We say that a pseudodistance d has compact degeneracy if L1(p) is compact for every p E X. (1.2.4) Proposition. If X is a locally compact space with a pseudodistance d with compact degeneracy, then X* is locally compact. (1.2.5) Corollary. {f X is a local~v compact ~pace with an inner pseudodistance d with compact degeneraLY, then the induced innerdistance d* defines the quotient topology of X*. (1.2.6) Proposition. If X is a locally compact space with an inner pseudodistance d with compact degeneracy, then the natural projection j: X + X* is a proper map. Proof Let K c X* be compact and let PI! E jI K be an infinite sequence. Then 1(qo) = L1(qo) a subsequence of (J(Pn») converges to a point qij E K. Since l is compact, it has a compact ncighborhood B(qo; 8) = (B(qo; 8». Then a subsequence of {PI!} converges to a point of B(qo: 8). 0
r
r
In spite of (1.1. 7), L1 (p) may not be connected in general. However, compactness of L1 (p) again guarantees connectedness. (1.2.7) Proposition. Let d he an inner pseudodistance on a locally compact space X. If L1(p) is compact, then L1(p) is connected. Proof Suppose that L1(p) is not connected, and let L10 be the component containing p, and let U be a compact neighborhood of L10 which meets no other components of L1(p). Let au be the boundary of U, and let 8 be the distance between p and au with respect to d. Since au is compact and does not meet L1(p), the distance 8 is positive. Let q be any point of L1(p) not in L10. Any curve y joining
8
Chapter 1. Distance Geometry
p to q must go through the boundary contradiction.
au,
and so has length at least 8. This is a 0
The proof of the following proposition is straightforward. (1.2.8) Proposition. Given two topological spaces X and Y with pseudodistances d x and dy, respectively, d~fine a pseudodistance dxxY on X x Y by
dxxY«x, y), (x', y'» = max{dx{x, x'), dy(y, y')} for
(x,y),(x',y')
E X x
Y.
Then (X x Y)* =X* x Y*.
(1.2.9) Proposition. Let (X, d x ) and (Y, d y ) be two topological spaces with pseudodistances. Iff: X + Y is a distancedecreasing map, it induces a distancedecreasing map f: X* + y* benveen the induced metric spaces.
3 Mappings into Metric Spaces Some of the results on holomorphic maps are direct consequences of purely topological results on maps into metric spaces. In this section we shall collect such topological results which will be used later. Given two topological spaces X and Y, we denote by C(X, Y) the space of all continuous maps f: X + Y equipped with the compactopen topology. If Y is a metric space, then the compactopen topology coincides with the topology of uniform convergence on compact sets. Let Y be a metric space with distance function dy. Let F c C(X, Y). The family F is said to be equicontinuous at x E X if for every E: > 0 there exists a neighborhood U of x such that dy(f(x). f(x'» < E for all x' E U and all f E F. ( 1.3.1) ArzeIaAscoli Theorem. Let X be a locally compact, separable space and Y a locally compact metric space with distance function dy. Then a family :F c C(X, Y) is relatively compact in C(X, Y) (i.e., every sequence of maps /" E F contains a subsequence which converges to some map f E C(X, Y) un(formly on every compact subset of X) (f and only if (a) :F is equicontinuous at every point x E X; (b) for every' x E X, the set U(x); f E F} is relatively compact in Y.
Proof Assume that F is relatively compact. If (a) does not hold, there would exist an E > 0, a sequence x" + x and a sequence 1;, E F such that dy(f,,(x), fn(x ll » 2: E. If f E C(X, Y) is the limit to which a subsequence of Un} converges, then we would have dy(f(x), f(x» 2: E, which is a contradiction. In order to prove (b), consider a sequence {fn(x); In E F) in Y. Choose a subsequence Un.} of {f,,} that converges to some element f of C(X, Y). Then Un. (x)} converges to f (x).
3 Mappings into Metric Spaces
9
Assume (a) and (b). Let {xd be a dense sequence of points in X. Given a sequence {fn} in F, we are going to extract a subsequence which converges at all points Xk. By (b), we can find an array of indices nIl O. We consider the case where [(x) = [(x'), and set y = [(x) = I(x'). Assume dx(x, x') = O. (1) Let V be a neighborhood of I (x) in Y described in condition (1). Since [(x) ¢. ,1, without loss of generality we may assume that V is an t:neighborhood of [(x) for some t: > O. Let V be the t:neighborhood of x in X. Then V contains x' and is finitely arcwise connected by (1.1.7). Let y be a rectifiable curve from x to x' in V. Since [ is distancedecreasing, [ maps V into V, and the curve y lies in [I (V). This is a contradiction. (2) Let U be a neighborhood of x described in condition (2). Let V = I(V), and apply (1). (3) Let V be an open neighborhood of x with compact closure (; such that (; n II (,1) = 0 and II(y) n (; = {x}. Then dx(x, aU) ~ dy(y, I(av» > O.
4 Norms and Indicatrices
13
Since any curve from x to x' has length at least dx(x, aU), this is a contradiction. D (1.3.13) Corollary. Let X and Y be topological spaces with an inner pseudodistances d x and dy, respectively. If I: X + Y is a distancedecreasing covering projection and if d y is a complete distance, so is d x . Proof We have only to check the statement concerning the completeness. Assume that Y is complete with respect to d y. Let {xn} be a Cauchy sequence in X. Since I is distancedecreasing, {f (xn)} is also a Cauchy sequence in Y. Let q E Y be the limit point of (f(xn)}. Let V be a 2eneighborhood of q such that V is homeomorphic to each connected component of II (V). Let V' be the eneighborhood of q. Since I(x,,) E V' for n > N, all (Xn}n>N are contained in one of the connected components, say U, of II (V). Let p E U be the point such that I(p) = q. Then {XII} converges to p. D
We say that a distance d y modulo ,1 is complete modulo ,1 if for each Cauchy sequence {Yn} in Y with respect to d y , we have one of the following: (a) (Ynl converges to a point q in Y; (b) for every open neighborhood V of ,1 in Y, there exists an integer no such that Yn E V for n > no. (1.3.14) Corollary. Let X and Y be topological spaces with inner pseudodistances d x and dy, respective(v. Let ,1 be a compact subset oIY. If I: X + Y is a distancedecreasing proper finitetoone map and if d y is a complete distance modulo ,1, then d x is a complete distance modulo II (,1). Proof We have only to check the statement concerning the completeness. Let {x,,} be a Cauchy sequence in X with respect to d x . Consider first the case the Cauchy sequence (f(x lI ) } is not convergent. Given a neighborhood U of II (,1) in X, we take a neighborhood V of ,1 such that II (V) C U. Let no be an integer such that I(x,,) E V for n > no. Then x" E U for n > no. Next, we consider the case (f(x,,)} converges to a point q in Y. If q E ,1, we argue as in case (b) above. So we assume that q ¢ ,1. We take a compact neighborhood V of q which is disjoint from L1 so that d x is a distance on II (V). Let II(q) = {PI. ... , pd. Let V be a compact neighborhood of q. Since II (V) is compact, a subsequence of {x n } converges to one of {PI, ... , Pk}, say PI. Being a Cauchy sequence, {x,,} must converge to PI. D
4 Norms and Indicatrices The results in this section will be used only in Section 5 of Chapter 3. Let V be an ndimensional complex vector space, and V* its dual space. Let F be a real nonnegative function defined on a subset of V such that if F is defined at v E V, it is defined at tv for all t E C and (1.4.1)
F(tv)
=
ItIF(u).
14
Chapter 1. Distance Geometry
We allow F to take the value 00. We call such a function F a quasinorm. We do not assume that F is defined on all of V. Nor do we assume that F is continuous. Let ( 1.4.2)
r (F) = {v
V; F (v) is defined and F (v) ::: I}.
E
Then r(F) is a starshaped circular subset of V in the sense that if v E reF) then tv E reF) for It I ::: 1. We call r(F) the indicatrix of F. Conversely, given a starshaped circular subset r of V, there is a unique quasinorm F such that r = reF). We note that F(vo) = 0 if and only if the complex line CVo = {tvo; t E C} is contained in reF), while F(vo) = 00 if and only if no points of the complex line CVo, except the origin 0, are in reF). A quasinorm F on V is called pseudonorm if it satisfies the following convexity condition: ( 1.4.3)
F(u
+ v)
+ F(u)
::: F(u)
u, v
E
V.
This convexity condition is equivalent to the convexity of the indicatrix r. A pseudonorm F is always continuous on V. A pseudonorm F is called a norm if 0 < F(v) < 00 for all nonzero v E V. Given a quasinorm F on V and the corresponding starshaped circular subset r = reF), we define the dual quasinorm F* on the dual space V* by ( 1.4.4)
F*V) = sup 1),(v)1
for
A.
E V*.
VEr
Clearly the dual quasinorm F* is defined everywhere on V*. Whether F satisfies the convexity condition (lA.3) or not, the dual quasinorm F* always satisfies the convexity condition: (lA.5)
F*(A.
+ f..t)
::: F*(A.)
+ F*(f..t)
A., f..t
E
V*.
The indicatrix r* = r(F*) of the norm F* is not only starshaped and circular but also convex. It is given also as an intersection of closed circular cylinders: ( 1.4.6)
r* =
nV
E
V*; 1A.(v)l::: l}.
VET
(104.7) Proposition. (1) F*(Je) > O/ar all nonzero ). E V* iland only if F(el) < 00, ... , F(e ll ) < 00 for some basis el, ... , ell of V; (2) F*(J.) < 00 for all A. E V* !fand only if r is bounded; (3) If a quasinorm F is positive (i.e., 0 < F(v)::: 00 for all nonzero v E V) and satisfies the convexity condition (1.4.3), then F*(Je) < 00 for all X E V*. Proof (1) Let U be the subspace of V spanned by r. Given}. E V*, F*(J,) = 0 if and only if X( v) = 0 for all v E U. Hence, F* (Je) > 0 for all nonzero }, E V* if and only if U = V. On the other hand, U = V if and only if F(el) < 00, ... , F(e n ) < 00 for some basis e), ... , en of V.
4 Norms and Indicatrices
(2) If
r
is bounded so that
r
15
is compact, then
sup 1),(v)1 = m O. By Lemma there exist U I, ... , U m E (r + e)r with m :::: 2n such that m
V = Lt;ui
with
t; > 0
Lt;:::: 1.
;=1
Then L
F(t;u;) = L
tiF(Ui) :::: (r
+ e) Lti
By setting Vi = tiui, we obtain the desired inequality.
:::: (r
+ r:;). D
We shall show that if F**(v) > 0 then the stronger inequality indicated in Remark (1.4.13) holds. Let F**(v) = r > O. Then vErt. By Lemma, there exist ul, ... , u m Err such that v = Ltiu; with t; > 0, Lti:::: 1. The remainder of the argument is the same as in the proof of (2) above.
Chapter 2. Schwarz Lemma and Negative Curvature
1 Schwarz Lemma In this section we prove Ahlfors' generalization of the classical Schwarz lemma in function theory of one complex variable and its variants. For the general theory of intrinsic distances, we need only the classical SchwarzPick lemma in the form (2.1.7). Let X be a Riemann surface, i.e., a Idimensional complex manifold. Let
da 2 = 2A.dzdz be a Hermitian pseudometric on X expressed in terms of a local coordinate z, and let w = iAdz Adz be its associated Kahler form. The term pseudometric means that da 2 is only positive semidefinite, i.e., A ::: o. We recall the notation dC
= i (d" 
d')
so that dd c
= 2id'd".
To w we asscoiate the Ricci form (2.1.1 )
Ric(w) = dd c log A = 2Kw,
where . I B2 logA K = A BzBz
(2.1.2)
is the (Gaussian) curvature of da 2 • Both Ric(w) and K are defined wherever A is positive. . Let Da denote the open disc of radius a in the Gaussian plane C, Da
=
{z
E
C; Izl < a}.
Then the Poincare metric (2.1.3)
2 dSa
4a 2dzdz = A(a 2 _ IzI2)2'
(A> 0)
20
Chapter 2. Schwarz Lemma and Negative Curvature
on Da is complete and has curvature A. In the special case where a = 1, the unit disc D j will be denoted D and the Poincare metric ds~ with A = 1 will be denoted ds 2 . Let cp be the Kahler form of the Poincare metric ds 2 • Then Ric(cp) = 2cp, since the curvature K is 1. We prove a generalization of SchwarzPick Lemma by Ahlfors [1).
(2.104) Theorem. Let d.l'2 denote the Poincare metric of curvature Ion the unit disc D. Let da 2 be any Hermitian pseudometric on D whose curvature is bounded above by 1. Then
In terms of the Ricci forms Ric(cp) and Ric(w) of ds 2 and da 2 respectively, (2.1.4) may be stated as follows: (2.1.4)'
Ric(w) .:::: 2w
=> w .::::
cpo
As we shall see in the proof, the theorem holds if da 2 is only continuous at zero points of da 2 and is twice differentiable at the points where it is positive (and hence the curvature is defined). This technical point is important in applications. Ahlfors proved the theorem for an upper semicontinuous da 2 with "supporting pseudometric". This will be explained later.
Proof Let Da be the disc of radius a with the Poincare metric ds; of curvature 1 given by
We compare this metric with da 2 = 2Mzdz. Let u" be the nonnegative function on D" defined by U a = },//La, i.e., da 2 = u"ds;. The problem is to show that Uj .:::: 1 on D. From the explicit expression for /La, it follows that u" (zo) ~ Uj (zo) at every point Zo E D as a ~ 1. Hence, it suffices to show that U a .:::: 1 on Do for every a < 1. From the explicit expression for /La we see that u" (z) ~ 0 as z approaches the boundary of D". Therefore, u" must attain its maximum in the interior of Da, say at z" E Da. If u" (z,,) = 0, then u" == 0, and there is nothing to prove. So we assume ua(zo) > 0 and calculate the complex Hessian of log U a at Zoo Since U o = A/ /La and since the curvature of ds~ is 1, we have
a2 10g U a
azai
=
a2 log A  
azaz
a2 log /La
=~
azai
AK  /La = /La(uaK  1).
1 Schwarz Lemma
21
Since the complex Hessian oflog u" must be nonpositive at the maximum point zo, we obtain the inequality u,,(zo)K(zo) 1 ::'S O. Hence, u,,(zo) ::'S 1/K(zo)::'S 1. Since ua(zo) is the maximum value of u", we have u" ::'S 1 everywhere on D a.
o
(2.1.5) Corollary. Let X be a Riemann surface with a Hermitian pseudometric dsi whose curvature (wherever defined) is bounded above by  L Then every holomorphic map I: D + X is distancedecreasing, i.e., f*dsi ::'S ds 2 • Proof Set da 2 = f*dsi. Then da 2 is a Hermitian pseudometric on D. If we denote the curvature of dsi by Kx, then the curvature of da 2 is given by 1* Kx; this is clear from the definition of the curvature (2.1.2). Now, (2.1.5) follows from (2.1.4). 0 The following classical SchwarzPick lemma is immediate from (2.1.5). (2.1.6) Corollary. Let D be the unit disc with the Poincare metric ds 2 . Then every holomorphic map I : D + D is distancedecreasing, i.e.,
Comparing the coefficients of dzdz in the inequality in (2.1.6) we can write
1/'(z)1
~
1
= >
o
1
+ y'(t)2)t
1  X(t)2  y(t)2
2Ix'(t)1
o 1  x(t)2

1" 0
dt 2dx 1 x2
1+a I a
  = log .
This shows that the ordinary line segment from 0 to a is the shortest path and l+a 1 a
p(O, a) = log   .
Since the Poincare metric is invariant under the rotations, we have
22
Chapter 2. Schwarz Lemma and Negative Curvature
I +Ial p(O, a) = log   = 2 tanh 1 lal 1 Ial
for
a
E
D.
Given two points a and b in D, the transformation zb w = _I  bz
is an automorphism of D that sends b to invariance of p we obtain pea, b) = log
11 11 
°and
a to (a  b)/(J  ab). From the
I
I
abl + la  bl a  b = 2 tanh\  . abl  la  bl 1  ab
However, we shall rarely use this explicit expression for p. The integrated form of (2.1.6) reads as follows: (2.1.7)
p(f(a), feb»~ :5 pea, b)
for
a,b E D.
In order to weaken the differentiability assumption on da 2 in (2.1.4), we first prove the following (2.1.8) Proposition. Let daJ and dar be two smooth Hermitian metrics with curvature Ko and K\ defined in a neighborhood of a point Zo E D. If daJ :5 daf in a neighborhood of Zo with equality holding at zo, then K\ :5 Ko at Zoo Proof Let da? = 2A;dzdz, and u = Aol )0\. As in the proof of (2.1A), we calculate B210g ulBzBz. Then using (2.1.2) we obtain
1 )'\
a2 \og u azBz
= uKo+ K\.
We evaluate this at ZOo Since u attains the maximum value I at Zo, we have Ko+K\ atzo. 0
O~
Let da 2 be an upper semicontinuous Hermitian pseudometric on D. A pseudoHermitian metric da~ is called a supporting pseudometric for da 2 at Zo E D if it is defined and of class C 2 in a neighborhood U of z" and satisfies the following condition: da 2 ~ da,; in U, and with equality at ZOo If da 2 is not smooth, we define its curvature
Kda2
by
(2.1.9)
where the infimum is taken over all supporting pseudometrics da} for da 2 at ZOo Then (2.1.4) is generalized to nonsmooth metrics, (Ahlfors [1]): (2.1.10) Theorem. Let ds 2 be the Poincare metric on D, and da 2 an upper semicontinuous Hermitian pseudometric on D with its curvature::::: 1. Then
1 Schwarz Lemma
23
The proof of (2.1.1 0) is exactly the same as that of (2.104). Since da 2 is upper semicontinuous, the existence of a maximum for the function U a is assured. If Zo is a maximum point for U a = da 2 Ids;, it is also a maximum point for the function Va = da;;lds;; (where da,; is a supporting pseudometric for da 2 at zo). Then we have only to calculate the Hessian of log Va instead of log U a • D (2.1.11) Remark. The classical Schwarz lemma asserts also that if an equality holds in (2.1.6) at one point of D, then the equality holds everywhere. This part of Schwarz lemma has been also generalized by Heins [I]. Namely, if an equality holds in (2.1.10) at one point of D, then da 2 = ds 2 everywhere on D. For a finite family of of pseudometrics, (2.104) may be generalized (see (2.104),) as follows (CowenGriffiths [I]): (2.1.12) Theorem. if a family ofpseudometrics dal = 2)'kdzdz, (k = 1, ... , N), on D with Kahler forms Wk = j)'kdZ 1\ dz, has its curvatures collectively bounded by 1 in the sense that then the pseudometric da 2 = 2Mzdz,
where
A = C).I .. . )N )\/N
has its curvature bounded by 1 and, consequently, da 2 Proof Let W = iAdz Ric(w)
1\
s: ds 2 .
dE be the Kahler form of da 2 . Then ddClogA=
~ LddClogAk
1 N LRic(wk)
2i N L
Akdz
1\
~
2 N LWk
dE ~ 2i(A\ ... ).N)\/N dz
1\
' dz = 2w,
showing that the curvature of da 2 is bounded by 1.
D
The following is also a simple application of the maximum principle. (2.1.13) Theorem. Let da 2 be an upper semicontinuous Hermitian pseudometric on D. Let ~ be a holomorphic vector field on D and I~I be the length of~ measured by da 2 . If da 2 has negative curvature in the sense of (2.1.9), then I~ I cannot attain a maximum in the interior of D unless I~ I == O. Proof Assume that I~ I attains a maximum at Zu ED. Without loss of generality we may assume that da 2 is of class C 2 at zoo (In fact, let da,; be as in (2.1.9) and IH the length of ~ measured by da'/;. Then IH attains a maximum at zo.)
24
Chapter 2. Schwarz Lemma and Negative Curvature
Let da 2 = 2Mzdz. If~ = /(olaz), then 1~12 = 2A/ j. Assuming that 1~12 > 0 at z,,' we calculate the Hessian of log I~ 12. Thus
a2log I~ 12
::~'
ozaz
a2 log A
,
=    = AK azaz '
where K is curvature of da;;. The left hand side is nonpositive at right hand side is positive everywhere. This is a contradiction.
z" while the 0
Now we state the Schwarz lemma for Hermitian metrics with a logarithmic singularity at the origin. This will be useful in logarithmic geometry. (2.1.14) Theorem. Let ds 2 be the Poincare metric of curvature Ion the unit disk. Let da 2 = 2Mzdz be a Hermitian pseudometric on the punctured disc D* such that Izl2da 2 becomes upper semicontinuous on D. {fthe curvature of da 2 is bounded above by lzI 2, then Idda 2 :::: ds 2 • Proof Let d(j2 = IZ12da 2 . Then the curvature

I
k
of d(j2 is given by
a2 log 1
K = :;_, A ozoz where ;;. = IdA. This shows that it is related to the curvature K of da by K = IZI2k. Hence, k :::: 1, and (2.1.10) applied to d(j2 yields d(j2 :::: ds 2 •
o
The following theorem of Sibony [4] is a version of Schwarz lemma and will be used to define the Sibony pseudodistance. (2.1.15) Theorem. Let U be an upper semicontinuous function on D such that (i) U < 1, (ii) logu is subharmonic, (iii) u(O) = 0, and (iv) u(z)/lzI 2 is bounded. Then (I) u(z):::: Izl2 for ZED with equality at some point i= 0 if and only !f u(z) == IZI2; (2) If, in addition, u is of class C 2 in a neighborhood of 0, then a2ulazoz :::: I at 0, with equality if and only if u (z) == Id.
o ::::
Proof (1). Since u attains its minimum at 0, du vanishes at O. Define in D* the function v(z) = u(z)/lzI 2 • Since log v is subharmonic in D*, v is subharmonic in D*. Since v is also bounded, there is a subharmonic extension of v in D. Since IimsuPz>ao v:::: I, it follows that 1. Hence u(z) :::: IZI2. If u(zo) = Izol2 for some Zo i= 0, then v(zo) = 1. This implies v(z) == 1. (2). Let z = x + iy. Denoting partial differentiation by subscripts, we have
v
v::
Izl 2 :::: u(z)
I
= '2(u xA O)x
2
+ 2u xy (0)xy + Uyy(O)y 2 ) + o(lzl 2 ).
Replacing y by  y and adding the resulting ineqaulity to the above, we obtain u x AO)x 2 +
U yy (O)y2
+ 00z12) :::: 21z12.
2 Negatively Curved Riemann Surfaces
25
Hence, Uz;:(O) =
If U zz (0) = 1, then V(O) =
1
4(uxAO) + Un' (0») ::::
U z: (0)
1.
v == 1.
= 1. Hence,
D
We note that in (2.1.15) if U is of class C 2 in a neighborhood of 0, then u(z)/lzI 2 is automatically bounded in D. For historical comments concerning SchwarzPickAhlfors lemma, see Royden [10].
2 Negatively Curved Riemann Surfaces The results in this section will be used in Section 7 of Chapter 3. In the preceding section we showed that the Poincare metric on the unit disc D has curvature 1. It is sometimes more convenient to use the upper halfplane in place of the disc. Let H = {w = u
+ iv
E
C;
V
> OJ.
We have a well known correspondence between the unit disc D and the upper halfplane H. It is given by (2.2.1)
i  w z=ED
i+w
for
wE H.
Pulling back the Poincare metric of D by the correspondence above, we obtain the Poincare metric ds 2 = dwdu)
(2.2.2)
v2
H
of curvature Ion H. Since it corresponds to the Poincare metric of D, ds~ is also invariant by the automorphisms of H and is complete. Let D* be the punctured disc, i.e., D*
=
{z E C; 0
b} with a suitable positive number b, and its area is given by
r In;
IlD' =
r dUv~V J Fh
1 b
We consider the twicepunctured plane X = C  to, I} and show that it carries a complete Hermitian metric with curvature K :::: I. This is obvious if we make use of the fact that the elliptic modular function, usually denoted A., is a covering projection from H onto C  to, I}, the latter being identified with SL(2; Zh\H where SL(2; Zh = {A E SL(2; Z); A == I mod 2}, (see, for example, Ahlfors [3; p.269]). The metric induced from the Poincare metric of H has constant curvature I. Instead, we shall construct explicitly a complete Hermitian metric dsJe on X = C{O, I} whose curvature K, although not constant, remains bounded above by 1. More generally, we consider the Riemann sphere minus k points, k :::: 3. Let Z = (zo, z I) be a homogeneous coordinate in PI C, and let z
= zljzO
be its inhomogeneous coordinate. For W = (wo, Wi), we set (Z, W) = zOwo
+ ZIW I •
We use the notaion d C = i(d"  d')
so that
dd c = 2id'd".
2 Negatively Curved Riemann Surfaces
27
Then the associated Kahler fonn of the FubiniStudy metric ds 2 of curvature 1 is given by
cP
= dd c 10g(Z, Z) = dd
. L
log(l
2
+ Izl ) =
2idz /\ dz (l
2 2'
+ Izl )
Given k points Aj = (aJ, a), j = 1, ... , k, in PI C, we shall construct explicitly a positive function p on PI C  {A I, ... , Ad such that (a) p goes to infinity at each of AI,.'" Ak (so that pds 2 becomes a complete metric on PIC  {AI, ... , Ad; (b) the curvature of pds 2 is bounded above by a negative constant; (c) the area of PIC  {AI, ... , Ad with respect to pCP is finite, i.e.,
1
pCP
: 2 carries a Hermitian metric ds~ with curvature K :s I. (2.2.8) Remark. The curvature of the Hennitian metric 2(1 given by 1 K = < o. (I + Iz12)3
+ Izl2)dzdz
on C is
Clearly, the curvature approaches 0 as Izl tends to infinity. As we shall see later, C cannot admit a Hennitian metric with curvature bounded above by a negative constant. If r denotes the geodesic distance from the origin to z with respect to the above metric, then Izl2 ~,J2r for Izllarge so that K ~ 1/(,J2r)3 as r ~ 00, see Remark (3.7.2).
3 Negatively Curved Complex Spaces In this section we shall generalize results of Section 1 to higher dimensional complex spaces. The results will be used in Section 7 of Chapter 3. Let X be a complex space. Let f be a holomorphic map sending a neighborhood U f of the origin 0 in C into X. Let g: U g ~ X be another such map. Then f and g are said to define the same Ijet at 0 if f(O) = g(O) and if they have the same first derivative at O. GeometricaJ1y stated, this means that two holomorphic curves f and g have the same velocity at O. The Ijet represented by f will be
3 Negatively Curved Complex Spaces
31
denoted f'(O). So by definition, 1'(0) = g'(O) if and only if f and g define the same Ijet at O. For each x E X, we call the set TxX of all Ijets f'(O) the tangent cone of X at x, and TX = U, Tx X the tangent cone of X. The tangent cone defined here is in general smaller than the usual tangent cone which consists of tangent vectors to real Coo curves in X. If x is a regular point of X, then t, X coincides with the tangent space Tx X. Let Ox be the ring of germs of holomorphic functions at x EX. Let mx denote its maximal ideal consisting of functions vanishing at x. Then the Zariski cotangent space of X at x, is defined to be rnx/rn;, and its dual space, denoted T, X, is called the Zariski tangent space of X at x. Although T X = UX T, X may not be a fibre bundle if X is singular since the dimension of T.tX may vary with x, it will be still called the Zariski tangent bundle of X. Then we have a natural inclusion TX C T X. In fact, for a holomorphic function q; defined in a neighborhood of x = f(O), set
d
I
(f (O»q; = dz q;(f(z»lz=o·
Then I' (0) may be regarded as a derivation from the algebra of holomorphic functions at x = f(O) into C. If X is nonsingular, then TX coincides with TX. If (z, w) is the coordinate system for C 2 and if X is the curve defined by w 2 = Z3, then the Zariski tangent space at the singular point (0, 0) is the 2dimensional vector space spanned by a/oz and a/ow. The tangent cone at (0,0) defined here contains only the zero vector. In fact, if I E Hol(D, X) with 1(0) = (0,0), then expressing I by a pair of power series
and using the condition w 2 = Z3, we obtain al = bl = b 2 = O. Hence, the Ijet of I at 0 vanishes. For a systematic account of Zariski tangent spaces, tangent cones and other possible "tangent spaces", see Whitney [I]. A pseudolength function on X is a real nonnegative function F on T X such that F(av) = laIF(v)
for
a E C,
v,av E TX.
We normally assume that F is smooth. However, for applications it is sometimes necessary to consider uppersemi continuous pseudolength functions. If F(v) > 0 for all nonzero VET X, then we call F a length function on X. We say that a pseudolength function F is convex if F(v
+ v')
::::: F(v)
+ F(v').
A convex (pseudo)Iength function is also called a Finsler (pseudo)rnetric.
32
Chapter 2. Schwarz Lemma and Negative Curvature
Every pseudolength function F on a complex space X gives rise to an inner pseudodistance d by (2.3.1 )
d(p, q) = inf y
r
fa
F(y'(t))dt,
where yet), a S t S b, is a piecewise differentiable curve from p to q and y'(t) is the velocity vector of y at y (t). IfF is a length function, then d is a distance function. A Hermitian metric on a complex manifold X defines a length function. However, a length function is much more general than a Hermitian metric or even a Finsler metric. On the other hand, restricted to a holomorphic curve a length function is a Hermitian metric. More precisely, if F is a pseudolength function on X and if f: D ~ X is a holomorphic map, then f* F defines a Hermitian pseudometric on D, i.e., (2.3.2)
f* F2
= 2Mzdz,
where A. is a nonnegative function on D. If F is a length function and if f is everywhere nondegenerate, then A is positive and f* F2 defines a Hermitian metric on D. We consider the curvature Kf*F of the pseudoHermitian metric f* F2 defined by (2.1.2) (by (2.1.9) if not smooth); it is defined where ;, is positive. Let v E itx, let [v] denote the complex line spanned by v. Given an upper semi continuous pseudolength function F on X, we define the holomorphic sectional curvature K F ([ v]) in the direction of [v] by (2.3.3) where the supremum is taken over all f E Hol(D, X) such that f(O) = x and [v] is tangent to feD). Let cp: X' ~ X be a holomorphic map from a complex space X' into another complex space X. Given a pseudolength function F on X, we have the induced pseudolength function cpo F on X'. From the definition of the curvature we obtain (2.3.4)
for
v
E
ix'
wherever the curvature is defined. In particular, if X' is a complex subspace of X, then the curvature of (X', Fix') is bounded above by the curvature of (X, F). We apply (2.3.4) to a holomorphic map f: D ~ X and (2.1.4) to the Hermitian pseudometric da 2 = f* F2. Then we have the following generalization of the Schwarz lemma, (GrauertReckziegel [1], Kobayashi [5]). (2.3.5) Theorem. Let F be a pseudolength/unction on a complex space X. !fits holomorphic sectional curvature is bounded above by 1, then
jor where ds 2 is the Poincare metric of D.
f
E
Hol(D, X),
3 Negatively Curved Complex Spaces
33
We show that for a Hennitian manifold the definition of the holomorphic sectional curvature given here coincides with the usual one. Given a Hennitian metric ds 2 = 2 L7.J=1 gi]dZidi J, on a complex manifold X, the components of its curvature tensor are expressed by
(2.3.6)
a 2 gi]
'"
Ri]kl =  azkail
agiij ag p ] az k ail'
+ ~ gpq
Then given a unit tangent vector v = LVi (a/az i ), the holomorphic sectional curvature in the direction of v is defined to be
(2.3.7)
' " Ri]kTv i v j v kI Hd,l' () V = ~ v .
Let X' be a complex submanifold of X. We choose a local coordinate system zn in such a way that X' is defined by
Zl, ..• ,
= ... = zn = O.
zm+l
so that we may use z1, ... , Zm as a local coodinate system for X'. Then the induced Hennitian metric on X' is given by 2 L;~J=I gi]dz i diJ' We shall compute its curvature R;]kl' Fix a point p E X'. By a linear change of coordinates, we may assume gil = 8ij at p. Then at the point p we have the following equation of Gauss:
(2.3.8)
R'   = R, ijkl
ag  ag "
n 7

IJkl

'"
~ p=m+l
~.!!.!...
az k ai'
for i, j, k, I = I, ... , m. The following proposition is immediate from (2.3.8) (2.3.9) Proposition. Let X' be a complex submanifold of a Hermitian manifold X. Then the holomorphic sectional curvature H~", of X' does not exceed the holomorphic sectional curvature Hds' of X, i.e., for
v E TX'.
Fix a point p EX, and let v E TpX be a unit tangent vector. If X' is a holomorphic curve in X tangent to v at p, then its Gaussian curvature at p is bounded by the holomorphic sectional curvature H d.,2(V) of X in the direction v. We shall show that there is a holomorphic curve tangent to v whose Gaussian curvature at p equals Hds2(V). We start with any holomorphic curve X' tangent to vat p. We may assume that a local coordinate system Zl, .•. , ZIZ with origin p was chosen in such a way that gil = 8ij and X' is given by Z2 = ... = zn = O. From
34
Chapter 2. Schwarz Lemma and Negative Curvature
(2.3.8) we know that if Jgiq/JZ k = 0 at p for q = 2, ... , n, then X' has already desired property. If not, we consider the following coordinate transformation:
where a 2 , ... , a" are constants to be chosen appropriately. Substitute the coordinate transformation above into ds 2 = L g;Jdz i dz.i to express it in terms of Il Th en we 0 btam . d s 2  "h  j WIt . h w 1, ... , w. L.. ijd Wid w, n
h 1q = glq 
L g,qa'w 1. ,=2
It follows that if we set a q = (Jglq/JZ1)p, then (Jh1q/JW1)p = 0, so that the holomorphic curve defined by w 2 = ... = w n = 0 has the desired property. This
proves the following (Wu [5]) (2.3.10) Proposition. For a Hermitian manifold (X, ds 2 ) the holomorphic sectional curvature defined by (2.3.3) coincides with the classical holomorphic sectional curvature defined by (2.3.7). (2.3.11) Remark. We note that for KF the assertion corresponding to (2.3.9) was immediate from the definition (2.3.3), (see (2.3.4». However, (2.3.8) gives much more than (2.3.9). In fact, if
then X' is totally geodesic at p, i.e., the second fundamental form of X' vanishes at p, provided either dim X' = 1 or X is Kahler. From (2.3.5) we obtain (2.3.12) Corollary Let (X, dsi) be a Hermitian man(fold whose holomorphic sectional curvature is bounded above by  1. Then
for
f
E
Hol(D, X),
where ds 2 is the Poincare metric of D. We extend (2.1.13) to higher dimensional spaces X. (2.3.13) Theorem. Let X be a complex space with an upper semicontinuous pseudolength function F with the property that, at each v E f X with F(v) > 0, F has negative holomorphic curvature. Let ~ be a holomorphic vector field on X. Then F(~) cannot attain a maximum in (the interior qf X) unless F(~) == O.
Proof Assume that F(n attains a positive maximum at x" E X. Choose an imbedding f: D ~ X such that f(O) = x" and ~ is tangent to feD). Then apply (2.1.13) to the pseudoHermitian metric f* p2 on D. 0 (2.3.14) Corollary. Let X and P be as in (2.3.13). If X is compact, it admits no holomorphic vector fields.
4 Ricci Forms and Schwarz Lemma for Volume Elements
35
The generalized Schwarz lemma (2.3.5) has been extended to holomorphic maps from a complex manifold M with a length function satisfying certain curvature conditions to a complex manifold X satisfying the condition of (2.3.5), see Yau [3], ChenChengLu [I], Royden [7,8], YangChen [I], Q. H. Yu [I], ChenYang [1], Matsuura [2], BumsKrantz [I].
4 Ricci Forms and Schwarz Lemma for Volume Elements The results in this section, which generalize those of Sections I and 2 to volume elements, will not be used until Chapter 7. Let L be a holomorphic line bundle over a complex manifold X with local coordinate system z', ... , zn, and h a pseudometric on L, i.e., h is a nonnegative smooth function on L such that for
~ E
L,
C E
C.
If h(~) > 0 for all nonzero ~ E L, then h is called a metric on L. To each pseudometric h we associate a closed (I,I)form Ric(h) called the Ricci form as follows. Taking a local nonvanishing holomorphic section ~, we set (2.4.1 )
Ric(h) = dd' 10gh(O =
2iaa logh(~) =
2i
L Rjkdz J /\ dz
k,
where (2.4.2)
a2 10gh(n
Rjk = 
azJaz k
•
If ~ is replaced by another nonvanishing holomorphic section TJ = f~ with f holomorphic, then her}) = IfI2h(~) and aa 10gh(TJ) = aa logh(l;), which shows that Ric(h) does not depend on the choice of~. The Ricci form Ric(h) is defined only at the points where h is strictly positive. If h is a metric on L, then Ric(h) is globally defined on X and represents 4rrc, (L), where c, (L) is the first Chern class of L. Sometimes, the Ricci form Ric(h) of a pseudometric can be globally defined (even where h vanishes). We say that h has a holomorphic degeneracy if h is locally of the form lal 2q g, where g is strictly positive, a is holomorphic, and q is a rational number. Thus, if ~ is a nonvanishing local holomorphic section of L, then h(~) = lal 2q g, and h(~) vanishes only where a vanishes. In this case, aalogh(l;) = aalogg,
and Ric(h) is defined everywhere on X. In order to generalize (2.1.4) to the case of higher dimension, we consider a pseudovolume form and the associated Ricci form on a complex manifold X. In terms of a local coordinate system z', ... , zn of X, a pseudovolume form von X can be locally written as
36
Chapter 2. Schwarz Lemma and Negative Curvature
(2.4.3)
V
"
= V nUdz j
/\
dzJ),
j=l
where V is a (locally defined) nonnegative function. A pseudovolume form v is a pseudometric on the anticanonical line bundle K l = /\" T X. If V is positive everywhere, i.e., if v is a metric on the line bundle K l , then v is a volume form of X. To each v given by (2.4.3) we associate the Ricci form Ric(v) by (2.4.4) where Rjk = 3 2 log V/3z j 3z k • Of course, this is a special case, i.e., L = K l , of the construction (2.4.1). If X is a Hermitian manifold with fundamental 2form w and the volume form v = w", then (RjjJ represents the components of the Ricci tensor in the classical sense. Hence the name "Ricci form" for Ric(v). It is, however, important to associate the Ricci form directly to a volume form rather than to a Hermitian metric. If dim X = 1, then (2.4.5)
Ric(v)
= 2Kv,
where K is the Gaussian curvature of X. It is then natural to define K v when dim X = n by the following formula. ( Ric(v»" Kv =      
(2.4.6)
n!(n
+ 1)"v
We say that Ric(v) is negative if the matrix (R jk ) is negative wherever defined (i.e., where v > 0). We say that it is negatively bounded if it is negative and if Kv :s C < 0 wherever defined. If v is negatively bounded, we can always normalize it (by multiplying it with a positive constant) so that Kv :s l. Let B~ be the ball of radius a in en: B~
= {z =
(Zl, ... ,
Z") E C"; IIzl/2
The unit ball Bf will be denoted Bn. An invariant volume element Jia of
B~
= Iz l l2 + ... + Iz"12
< a 2}.
is given by
(2.4.7) For a = 1, we write Ji for (2.4.8)
{Ll.
A simple calculation shows
Ric( {La ) = 2i(n
+ 1) '~k " J.
and
.5 (a 2 Jk

Ilz112) + zj Zk d zj IJz1l2)2
(a2 _
/\
dZk ,
4 Ricci Fonns and Schwarz Lemma for Volume Elements
K
(2.4.9) Let
/la D~
=  (Ric{/LaW =1. n!{n + l)nlla
en:
be the polydisc of radius a in D~ = ({Zl, ... , zn) E
(2.4.10)
37
en;
IzII
< a, ... , Iz"l < a}.
The unit polydisc D7 will be denoted D". An invariant volume element Va of D~ is given by
n II
(2.4.11)
Va
= (2a )211
j=1
For a = I, we write
V
for
VI.
idz}!\ dzi (a
2
. 2 2·
lzll )
By a simple calculation, we obtain 2
n
d · (Va) =  4·I L a R IC 2 . 2 2} Z . (a Izll) 1=1
(2.4.12)
!\
j dZ,
and K
(2.4.13)
(Ric(v,,»"
v"
n!(n + I)nva
(n
+ I)n
The following theorem generalizes (2.1.4). (2.4.14) Theorem. Let v be any pseudovolume form on the unit ball B n with negatively bounded Ricci form Ric( v) and normalized in such a way that K v ::::: I. Then v ::::: /L.
°
Proof For a, < a < 1, we use the volume form /La on the ball B: defined by (2.4.7). Let U a be the nonnegative function on defined by
B:;
Ua
=
vi /La·
As in the proof of (2.104), U a attains its maximum at some interior point, say Zo of and it suffices to prove ua{zo) ::::: 1. Assuming that Ua(ZO) > 0, we calculate the complex Hessian of logu a at zoo From (2.4.2) we obtain
B:,
2 " aaz log Ua d i d k dd c Iog U a  2·I '~ i az k Z!\ Z
R·IC (/La )  R·IC (V ) •


Since the complex Hessian dd c log U a must be nonpositive at the maximum point Zo and since Ric(v) is negative, we have
0< Ric(v) ::::: Ric(/La)
at
Zoo
Taking the nth exterior power of this inequality, we obtain at From Kv ::::: 1 = K/l a' we obtain
zoo
38
Chapter 2. Schwarz Lemma and Negative Curvature at
i.e.,
U a (zo)
zo,
o
:::: 1.
In order to state a corollary to the theorem above, we need to explain the concept of meromorphic map. In general, a meromorphic map f from a complex space X into a complex space Y is a correspondence satisfying the following conditions: (I) For each point x of X, f (x) is a nonempty compact subset of Y; (2) The graph G j = {(x, Y) E X x Y; y E f(x)} is a connected complex subspace of X x Y with dim G t = dim X; (3) There exists a dense subset X* of X such that f(x) is a single point for x E X*.
Let :n:: G f ~ X be the projection defined by :n:(x, y) = x; it is a proper map. Then {x} x f(x) = :n: 1 (x), and f(x) is a complex subspace of Y. Let E C G f be the set of points where :n: is degenerate, i.e., E = ((x,y) E Gj; dimf(x) > O},
and let S
= :n:(E) = {x
E
X; dim f(x) > OJ.
Then E is a closed complex subspace of codimension 2: I of G f , and S is a closed complex subspace of codimension 2: 2 of X, (see Remmert [1]). It is then clear that f: X  S ~ Y is holomorphic. The subspace S is called the singular locus of f. The graph G r of f is the topological closure of the graph of flxs, i.e., the closure of {(x, f(x»; x E X  S}. Now, as a consequence of (2.4.14) we have the following generalization of (2.1.5). (2.4.15) Corollary. Let X be an ndimensional complex manifold with pseudovolume form v such that Ric(v) is negatively bounded and Kv :::: 1. Then every meromorphic map f: B n ~ X is volumedecreasing in the sense that f*v :::: /t.
f is holomorphic, the proof is the same as that of (2.1.5); simply apply (2.4.14) to f*v. If f is meromorphic, let S C B n be the singular locus of f. Since Ric(v) is negative, the coefficient V of v is plurisubharmonic, (in fact, log V is plurisubharmonic). Hence, the coefficient of f*v is a plurisubharmonic function on B n  S. Since the codimension of S is 2, fOv extends across S. Now (2.4.14) can be applied to f*v. 0 Proof If
(2.4.16) Corollary. Every holomorphic map with respect to /t, i. e., f* /t :::: /t.
f
ofB" into Usellis volumedecreasing
If we want to use the polydisc D n instead of the ball B n as our "model" domain, then in view of (2.4.13) we have to use the "normalized" volume form (n + l) n v rather than v.
4 Ricci Forms and Schwarz Lemma for Volume Elements
39
(2.4.14) and its corollaries can be generalized to more general domains, in particular to symmetric bounded domains, see Kobayashi [6], [7; p. 33] and HahnMitchell [1]. If the domain of a mapping f is a compact manifold rather than a noncompact space such as B", then the argument in (2.4.14) becomes simpler. For example, we have (Kobayashi [6], [7]) (2.4.17) Theorem. Let X and Y be ndimensional complex manifolds with volume form Vx and pseudovolume form Vy, respectively. Assume (a) The Ricci forms Ric(vx) and Ric(vy) are negatively bounded; (b) Kvy (y)/ Kvx (x) ::: 1 for x E X and y E Y; (c) X is compact. Then every meromorphic map f: X + Y is volumedecreasing in the sense that f*vy ::: Vx. Proof Although f is not holomorphic, f*Vy is a welldefined pseudovolume form on X; see the proof of (2.4.15). Consider a nonnegative function u = f* Vy /vx on X. Since X is compact, it attains its maximum at some point, say Xu EX. To complete the proof, consider ddc log u at Xo and follow the argument in the proof of (2.4.14). 0
if Y
(2.4.18) Corollary. In (2.4.17),
deg f where vol(X)
= Jx Vx
:::
and vol(Y)
is also compact, then
vol(X)/vol(Y),
= j~ Vy.
Proof Jx Vx deg j . = Jx f*Vy < Jy Vy ~ Jy Vy .
o If v is a volume element of a compact complex manifold, then its first Chern class c\ (X) is represented by the Ricci form (modulo a constant factor): I
(2.4.19)
C1
.
(X) = [RIC(Vx )].
4rr
By (2.4.6) {(
lx 
K
) Ux
Vx
=
e41l")n
n!(n
+ 1)/ C \
eX)" •
Let ax = min(Kvx) x
and
b x = max(Kvx).
x
Then (2.4.20)
ax . vol(X)
 X be the ith holomorphic disc of the chain a sending ai, bi E D to PiI, Pi E X. Without loss of generality, we may assume that ai = 0 and Ib;l < r/2. Since PiI E U(o; 3p), we have J;(D r ) C U(o; 3p+e). For ifz E Dr. then dD(O, z) < e and dX(Pil, fi(Z» = dx(J;(O), J;(z» < e. This shows that we obtain a desired chain fJ by restricting the holomorphic disc fi: D '> X to Dr; in order to normalize the smaller disc Dr we set gi(Z)
=
fi(rz)
and let fJ be the chain consisting of g i: D and lies in U(o; 3p + c) and l(fJ)
:s C ·l(a)
for '>
ZED,
X. Then fJ is a chain from p to q
< C(dx(p, q)
Since 8 is arbitrary, we obtain the desired inequality.
+ 8). o
(3.l.20) Proposition. Let {Xm} be a monotone increasing sequence of sub domains in a complex space X such that X = U X m . Then (a) (b) Proof (a)
= limcx",(p,q). dx(p, q) = limd xm (p, q).
cx(p,q)
Clearly, we have
Given p, q E X, choose fm E Hol(Xm , D) such that fm(P) = 0 and cxm(p, q) = p(O, fm(q». Then the usual argument (using the ArzelaAscoli theorem (1.3.1»
56
Chapter 3. Intrinsic Distances
shows that a suitable subsequence of Um} converges to a map f E Hol(X, D). Then limcxm(p, q) = limp(O, fm(q» = p(O, f(q» :'S cx(p, q). (b)
Similarly, we have
Conversely, given p, q E X and 8 > 0, let a be a chain of holomorphic disks from p to q with its length lea) < dx(p, q) + So Let a consist of holomorphic disks fi E Hol(D, X). We shrink each holomorphic disk fi by composing it with the multiplication by r < 1. We set f/'\z) = h(rz). If r is sufficiently close to I, then U/)} defines a chain air) from p to q. Given any 8 > 0, the length l(a(r» < lea) + 8 for r sufficiently close to 1. Since the chain air) is contained in a compact subset K of X, it is contained in Xm for m > mo. Hence, d x ", (p, q) :'S [(air»~ < lea)
+8
< dx(p, q)
+ 28.
o The proposition above is in Hristov [1,3], where he discusses also the situation where X is the limit of a monotone decreasing sequence of complex spaces X m • (3.1.21) Example. For the Gaussian plane C and the punctured plane C* = C{O}, we have de = 0, de' = 0, Ce = 0, Ce' = O. In fact, given two points p, q E C and an arbitrarily small positive number 8. there is a map f E Hol(D, C) such that f(O) = p and f(8) = q. Hence, dc(p, q) :'S So In order to prove de' = 0, we consider the surjective holomorphic map z E C + eZ E C*. Since it is distancedecreasing, de = 0 implies de' = O. The remaining two statements follow from (3.1. 8). The statement Ce = 0 is nothing but Liouville's theorem (that every bounded entire function is constant). Both C and C* are complex Lie groups. Generally we have (3.1.22) Example. For any connected complex Lie group G, we have d G =0
and
Cc
=0.
In fact, given p. q E G, there is a sequence of maps fl' ... , ik E Hol(C, G) such that p E f1 (C), q E fdc) and h(C) n fi+l (C) :f. 0 for i = I, ... , k  1. (These fi 's are suitable translates of complex Iparameter subgroups of G.) Our assertion follows from (3.1.21) and the fact that the maps h are distancedecreasing. Sitll more generally, (3.1.23) Example. If a connected complex Lie group G acts on a complex space X, then dx(p, q) = 0 for p, q E X belonging to the same Gorbit. In particular,
1 Two Intrinsic Pseudodistances
57
if X is a complex space on which a complex Lie group G acts with a dense orbit, then dx = and Cx =0.
°
In fact, for a fixed Po E X the mapping G ~ X that sends g is holomorphic. Hence, dx(g(po), g'(po» :s da(g, g') = 0.
E
G to g(po)
E
X
(3.1.24) Example. Let p: en ~ R+ be a nonn (not necessarily the Euclidean nonn), and B = {z E en; p(z) < I) the unit ball for this nonn. Then for
CB(O, z) = dB(O, z) = peO, p(z»
In fact, given Z E B, I(p(z» = z. Then
~
0, define I: D
Z =1=
CB(O, z)
:s dB(O, z) :s
z
E
B.
B by I(t) = tz/ pez) so that
p(O, p(z».
On the other hand, for every z E en, there exists a linear functional )'2: en ~ e such that AAz) = p(z) and IAz(w)l:s pew) for all WEen. Then Az sends B into D and, if z E B, then
:s CB(O, z).
p(O, p(z»
If p is the Euclidean nonn, then B is homogeneous. So in this case, CB and dB
coincide not only at the origin but everywhere. Although d x is always an inner pseudodistance, ex is not necessarily inner. The following examples are due to Barth [6]. (3.1.25) Example. Fix
°
< r < 1 and
fez,
X = D2 
w);
°
< s < 1, and consider the Hartogs figure
Izl:s rand Iwl:::: s)
in e 2 . Since every bounded holomorphic function I: X ~ D extends to a holomorphic function j: D2 ~ D with the same bound, C x is the restriction of e D2. Take p = (zo, wo) EX C D2 with Izol > r, Iwol > s, and p(lwol, s) > p(zo, 0), and let q = (zo, wo) E X. A curve Y = (YI, Y2) joining p to q in X must go around or under the notch in the bidisc. That is, either IYI (t) I :::: r for all t, or IYI (to) I < rand IY2(to)1 < s for some to. In the first case, we have L(y) :::: K > ex(p, q),
where K is the length of a curve (measured in the Poincare metric) from zo to which stays outside the disc of radius r while ex(p, q) = CD(ZO, zo) is the geodesic distance from Zo to Zo in D. In the second case, we have
Zo
L(y)
+ cx(y(to), q)
>
cx(p, y(to»
>
2p(lwol, s) > 2p(zo, 0) = cx(p, q).
:::: p(wo, Y2(tO»
Since p (I Wo I, s)  p (zo, 0) is independent of y, we have L(y) > K' > cx(p, q),
+ P(Y2(tO), wo)
58
Chapter 3. Intrinsic Distances
where K' is a constant independent of y. Thus
ei(p,q)
= infL(y) y
> ex(p,q).
The domain X in the example above is not a domain ofholomorphy. Barth [6] pointed out that the Caratheodory distance of the following domain ofholomorphy constructed by Sibony [I] is not inner. (3.1.26) Example. In order to define a domain X = M(D, V) C C 2, let (an) be a discrete sequence of points in the unit disc D such that every point of the boundary circle aD is the nontangential limit of a subsequence. Let (I'n) be a sequence of positive real numbers such that A. < 00. We set
Ln n
and V(z) =
e'P(z).
The function cp is negative and subharmonic in D, and the function V (z) is also subharmonic and 0 :::: V(z) < I in D. Since {an} is discrete in D, V(z) is continuous and vanishes only at an. Define X
=
M(D, V)
= fez, w)
E D
x C;
Iwl
0 for every pair p, q E X with p i q. A hyperbolic complex space X is said to be complete if it is Cauchycomplete with respect to d x . Since d x is inner by (3.1.15), a complete hyperbolic X is finitely compact with respect
to d x by (1.1.9). Since d x is inner (see (3.1.15» and since every inner distance on a locally compact Hausdorff space X induces the given topology of X (see (1.1.8», we have the following theorem of Barth [3], see also Barth [9]. (3.2.1) Theorem. Let X be a hyperbolic complex space. Then d x defines the topology ofX. (3.2.2) Proposition. Let X be a complex subspace ofa complex space Y. ( 1) If Y is hyperbolic. so is X; (2) IfY is complete hyperbolic and X is closed. X is also complete hyperbolic.
Proof This follows from the fact that the injection i: X hence distancedecreasing.
~
Y is holomorphic and 0
From (3.1.9) we have (3.2.3) Proposition. For complex spaces X and Y. the product X x Y is (complete) hyperbolic if and only if both X and Yare (complete) hyperbolic. (3.2.4) Proposition. Let X and Y be complex spaces, and I: X ~ Y a holomorphic map. Let Y' be a complex subspace of Y. and define X' = II Y'. If both X and Y' are complete hyperbolic, so is X'.
Proof Let Of denote the graph of I: X ~ Y; it is a closed complex subspace of X x Y. Let f' be the restriction of I to X', and Of' its graph. Then Gr' = Of n eX x Y'). Hence, Of' is closed in X x Y'. By (3.2.3), X x Y' is complete hyperbolic. By (3.2.2) Of' is complete hyperbolic. Since the projection X x Y' ~ X induces a holomorphic isomorphism from Of' onto X', it follows that X' is complete hyperbolic. 0 In the following proposition, what we have in mind for applications is the situation where X and Xi are all domains in a complex space Y. The proof is immediate from (1.1.11). (3.2.5) Proposition. Let X and Xi, i E I, be complex subspaces ofa complex space Y such that X = ni Xi' If all Xi are complete hyperbolic, so is X.
Although the hyperbolicity is a global concept, we can localize it as follows. (3.2.6) Proposition. Let X be a complex space. If there exist a family of points Prx E X and positive numbers Da such that, for each ct, the Dotneighborhood
2 Hyperbolicity
61
is hyperbolic and that {Ua } is an open cover of X, then X is hyperbolic. In particular, if for every p E X there is a positive number 8 such that the 8neighborhood U(p; 8) = {q E X; dx(p, q) < 8} is hyperbolic, then X is hyperbolic. Proof Take positive numbers PDt and there is a constant Ca > 1 such that
such that 3p"
Sa
for
du.(p, q) ::: Ca . dx(p, q) This shows that dx(p, q) is positive for p, q same Ua, clearly dx(p, q) > O.
E
+ Ca
= 8a. By (3.1.19)
p, q E Ua.
Ua, q
1=
p. If p, q are not in the
0
(3.2.7) Proposition. Let X be a complex space. If there is a positive number 8 such that for every p E X the 8neighborhood U (p; 8) is complete hyperbolic, then X is complete hyperbolic.
Proof By (3.2.6) X is hyperbolic. Let {Pn} be a Cauchy sequence in X. Let p and c be positive number such that 8 = 3p + c. We may assume, by omitting a finite number of points, that dx(Pm, Pn) < p for all m, n. Then by (3.1.19) we have d U (p,;8)(Pm, Pn) ::: C· dx(Pm, Pn), which shows that {Pn} is a Cauchy sequence with respect to d U (Pl;8). Since U(Pl; 8) is complete with respect to d U (p,;8), the sequence converges. 0 Let f: X + Y be a holomorphic mapping between complex spaces. Let f*d y = (fldy)i be the inner pseudodistance on X induced from d y by f (see (1.1.12». As we pointed out in Section I of Chapter I, the inequality f* d y ::: d x follows directly from the definition of f*d y. It is reasonable to expect that the equality holds if f is a covering projection. In this connection we prove (3.2.8) Theorem. Let X be a complex space and :n:: X + X a covering space of X. Then (1) If P, q E X and p, q E X with :n:(jJ) = p and ir(q) = q, then
dx(p, q) = inJdx(p, q), q
where the infimum is taken over all q E X such that :n:(q) = q; (2) X is (complete) hyperbolic if and only if X is (complete) hyperbolic; (3) If X is hyperbolic, then :n:: (X, d x ) + (X, d x ) is a local isometry, and d x = :n:*dx . Proof (1)
Since:n: is holomorphic, we have
dx(p, q) ::: dx(p, q). Let a be a chain of holomorphic discs from p to q. We lift a to a chain a of holomorphic discs in X starting from p. Then a ends at some point q E X with :n:(q) = q, and its length lea) is equal to the length lea) of a. This proves (1).
62
Chapter 3. Intrinsic Distances
(2) Assume that X is hyperbolic. Let p, q E X be such that dx(p, q) = O. Let p E X be such that n(p) = p. By (1) there exists a sequence {q,,} c X such that n(qn) = q and limdx(p, qn) = O. By (3.2.1) {qn} converges to p. Then (n(qn)} converges to p. But n(il,,) = q and hence p = q. Assume that X is complete hyperbolic. If Br +8 is the closed ball of radius r + 8 around p E X and if Br is the closed ball of radius r around p E X, then (I) implies for ,5 > O. Since BrH is compact by assumption (see (1.1.9» and Br is closed, Br is also compact. Hence X is complete. If X is (complete) hyperbolic, so is X by (1.3.13). (3). We prove first that n is a local isometry. Let p E X and p = n(p) E X. Let U be a 2eneighborhood of p such that U is homeomorphic to each component of n I (U). We denote the component containing p by U. Let V be the E neighborhood of p and V = n I (V). We claim that n maps V isometrically onto V. Let q, rEV and q = n(q), r = nCr) E V. Since dx(q, r) :s dx(q, r) < e, there is a chain of hoi omorphic discs from q to r such that lea) < e. The thread lal of a remains within U since L(lai) :s lea), (see (3.1.14». We lift a to a chain a starting from q in X. Since la I remains within u, la I stays also within U. In particular, the end point of a must be n I (r) n U = r. This shows that, for every chain a from q to r with lea) < e, there is a chain a from q to with lea) = lea). Hence, dx(q, r) 2: dx(q, r). The opposite inequality is a direct consequence of the fact that n is distancedecreasing. The last statement follows from (1.1.13). 0
un
r
According to Zwonek [3], the infimum in (1) may not be attained in general. Part of (3.2.8) is still valid when n: X + X is only a spread, i.e., when every point p E X has a neighborhood U such that n maps U biholomorphically onto the open subset n(U) of X. By (1.3.12) we have (3.2.9) Proposition. A spread X over a hyperbolic complex space X is hyperbolic. From (3.2.2) and (3.2.9) we obtain (3.2.10) Proposition. If a complex space X is holomorphically immersed into a hyperbolic complex :,pace Y, then X is also hyperbolic. The following proposition is of the same character as (3.2.9). From (1.3.12) and (1.3.14) we obtain (3.2.11) Proposition. Let f: X + X be a finitetoone proper holomorphic map. IlX is (complete) hyperbolic, so is X. A complex space X is said to be normal at x if the ring Ox of germs of holomorphic functions at x is integrally closed in its complete ring of quotients, and X is said to be normal if it is normal everywhere. The following more geometric interpretation is useful for us. Let S be the singular locus of X, and Xreg = X  S. Given an open set U C X, a function
2 Hyperbolicity
63
hoI om orphic on Urcg and is bounded on Ureg n K for every compact set K c U is said to be weakly holomorphic on U. Then X is normal at x if and only if every weakly holomorphic function at x extends to a holomorphic function in a small neighborhood of x, see Narasimhan [1; p. 114]. The concept of weakly holomorphic mapping can be defined in terms of local coordinate systems of the target space. A normalization of a complex space X is a pair (X, rr) consisting of a normal complex space X and a surjective holomorphic mapping rr: X + X such that (a) rr is proper and rr1(x) is finite for every x E X, (b) If S is the singular locus of X, then X  rr I S is dense in X and rr: X  n I S + X  S is biholomorphic. The normalization theorem of Oka (see GrauertRemmert [3], Narasimhan [1; p. 118]) states that every complex space X has a unique (up to an isomorphism) normalization rr: X + X. As an immediate consequence of (3.2.11) we have (3.2.12) Corollary. The normalization X is (complete) hyperbolic.
X ala (complete) hyperbolic complex space
The following example by KalimanZaidenberg [l] shows that the normalization X of a nonhyperbolic complex space X can be hyperbolic. (3.2.13) Example. Let (x, y, u, v) be a coordinate system in C 4 , and X be the affine algebraic surface given by the equations y4 = x4  I u4
=
y4(v 4

I).
°
Its singular locus S is given by y = 0, which implies x4 = 1 and u = while can be arbitrary. Since S consists of complex lines, X is not hyperbolic. Let X be the affine algebraic surface given by
v
y4 = x4  I
u4 = v 4

1.
Then X is nonsingular. Define the map rr: X + X by n(x, y, u, v) = (x, y, yu, v). Then (X, rr) is the normalization of X. Clearly, X is a direct product of two copies of the affine algebraic curve in C 2 defined by y4 = X4 l. This curve is (complete) hyperbolic. (In fact, its projective completion is a compact Riemann surface of genus 2 and hence hyperbolic by (3.2.8) or by (3.7.3». By suitably compactifying X and X, Kaliman and Zaidenberg obtain also a compact example. We apply (3.2.6) to holomorphic maps other than covering projections. (3.2.14) Theorem. Let rr: X + T be a holomorphic map of complex spaces. For t E T and 8 > 0, we set U(t; 8) = (u E.T; dT(t, u) < 8}. Iffor every point t E T there is a positive number 8 such that n I (U(t; 8» is hyperbolic, then X is hyperbolic.
64
Chapter 3. Intrinsic Distances
Proof For every p E X, its 8neighborhood {q E X; dx(p, q) < 8} is contained in Jr1(U(Jr(p); 8)) because Jr is distancedecreasing. By (3.2.6) X is hyperbolic.
o The following result is due to Eastwood [I]. (3.2.15) Theorem. Let Jr: X + T be a holomorphic map of complex spaces. 1fT is (complete) hyperbolic and ifT has an open cover (Ud such that each Jr1(U i ) is (complete) hyperbolic, then X is (complete) hyperbolic. Proof For each t E T, take 8 > 0 such that U(t,8) C U i for some Vi. Then Jr1(V(t, 8» is hyperbolic. By (3.2.14) X is hyperbolic. We prove now completeness. Let {PIl 1 be a Cauchy sequence in X. Then (Jr(Pn)} is a Cauchy sequence in T and converges to a point to E T. Take 8 > 0 such that U(t", 8) C Ui for some Ui. Take s > 0 and p > 0 such that 3p +s = 8. Omitting a finite number of Pn, we may assume that
dT(to, Jr(pj) < sand
dx(Pm, Pn) < p.
Let V = (x EX; dX(Pl,X) < s}. Then by (3.1.19) there exists a constant C > 0 such that dvCPm, Pn) ::: C . dx(Pm, Pn) for all m, n, which shows that {Pnl is Cauchy sequence in V with respect to d v . Since V C Jr1(U;), {Pn} is a Cauchy sequence in Jr1CVi ) with respect to d,,,(U')' Since Jr1(Ui ) is complete hyperbolic, (Pnl converges to a point in Jr1(Ui ). 0 (3.2.16) Remark. (i) In general, even if T and all Jr I (t), t X may not be hyperbolic. For example, the domain
X=
fez,
w) E
c 2 ; Izl
< I,
Izwl
< I} 
E T,
are hyperbolic,
(CO, w); Iwl:::: I}
is not hyperbolic. Let T = D = {z; Izl < I} and Jr(z, w) = z. Then each Jr 1(t) is biholomorphic to a disc. To see that X is not hyperbolic, consider two points P = (0, b) with b i= 0 and q = (0,0). Set PI! = (l/n, b). Then dx(p, q) = limdx(Pn, q). Let all = min{n, .Jn7ibT}. Then the mapping tED + (ant/n, anbt) E X maps I/a ll into Pn. Hence, dx(p, q)
= limdx(PII, q)
::: lim dD(l/a ll , 0)
= O.
(ii) However, as we shall see in (3.11.2), ifJr: X + T is a proper holomorphic map, hyperbolicity of T and Jr1(t), t E T, implies hyperbolicity of X. (iii) Let Jr: X + T be a holomorphic fibre bundle with fibre F in the sense that every point t E T has an open neighborhood U such that Jr I (U) is biholomorphic to V x F. If T and F are (complete) hyperbolic, then X is also (complete) hyperbolic by (3.2.3) and (3.2.15), (Kiernan [4]). Conversely, if X is (complete) hyperbolic, so are T and F. In fact, each fibre is (complete) hyperbolic by (3.2.2). To prove (complete) hyperbolicity of T it suffices to show that the bundle is locally flat so that the pullback X of X to
2 Hyperbolicity
65
the universal covering t of T is holomorphically a product t x F. (For if X is (complete) hyperbolic, so is X by (3.2.8). Then T is (complete) hyperbolic by (3.2.3». Thus the proof is reduced to showing the following (see Royden [5] and also (5.4.5) for details): Let f E Hol(D x F, F), and write frey) = f(t. y). Then iffo is an automorphism of F, fr = fofor all t E T. (iv) If n: X ~ T is merely a fibre space, X can be hyperbolic without T being hyperbolic as the following example shows, (Kobayashi [7]). Let X
=
{(z, W) E
c 2;
0 < Id
+ IwI2
< I}
with the natural projection n which assigns to (z, w) homogeneous coordinates (z, w). Then F = D*.
and E
T
=
PIC
X the point of PI C with
(3.2.17) Theorem. Let X be a complete hyperbolic complex space and f a bounded holomorphicfitnction on X. Then the open subspace X'
= {p
E X;
f(p)
i= O} = X  Zero(f)
is complete hyperbolic.
Proof Without loss of generality we may assume that f maps X into the unit disc D. Then apply (3.2.4) with Y = D and Y' = D*. 0 Let Y be a complex space. We say that a complex subspace X C Y is locally complete hyperbolic in Y if every point p of the closure X has neighborhood Vp in Y such that Vp n X is complete hyperbolic. The condition is obviously satisfied by any point p of X. So this is a condition on the boundary points p E
ax
= XX.
A Cartier divisor A in a complex space Y is a closed complex subspace that is locally defined as the zeros of a single holomorphic function. That is, each point x E A has a neighborhood V in Y such that A n V is defined by one equation f = 0, where f is a holomorphic function on V. (3.2.18) Corollary. Let Y be a complex space and A a Cartier divisor of Y. Then (I) Y  A is locally complete hyperbolic in Y; (2) IfY is (complete) hyperbolic, Y  A is (complete) hyperbolic. Proof Choose a complete hyperbolic Vp such that A n Vp is given as the zeros of a bounded holomorphic function f in Vp and apply (3.2.17) with X = Vp and X' = Vp  A.
0
While removing an analytic subset of codimension I from X can radically change the (pseudo) distance d x , removing a subset of large codimension does not, in general, change d x . The following theorem is due to CampbellOgawa [I] and CampbellHowardOchiai [1]. (3.2.19) Theorem. Let X be a complex man((old and A a closed analytic subset of X of codimension at least 2. Then Hol(D, X  A) is dense in Hol(D, X) in the compactopen topology, and hence
66
Chapter 3. Intrinsic Distances
dX _ A = dx
on
XA.
As pointed out by CampbellHowardOchiai [I], the first part of (3.2.19) can be generalized as follows:
If A is a closed analytic subset ofcodimension > k in an ndimensional complex manifold X, then Hol(D k , X  A) is dense in Hol(D k , X). In order to prove (3.2.19), we use the following lemma of Royden [4] which has other applications. (The proof of this lemma will be given in Appendix A of this Chapter.) (3.2.20) Lemma. If f is a holomorphic imbedding of a disc Dr of radius r > 1 into a complex manffold X of dimension n, there exists a holomorphic imbedding cp of the unit polydisc D n into X such that fez) = cp(z, 0, ... , 0, )
for
zED.
Proof of (3.2.19). Let f E Hol(D, X). We want to approximate f by elements of Hol(D, X  A). If we define ff E Hol(D, X) by fr(z) = f(tz), < t < 1, then each ft extends past the boundary aD of D and ft + f as t + 1. Thus, if each ff is in the closure of Hol(D, X  A), so is f. We may therefore assume that f extends to a slightly larger disc Dr, r > 1 and feD) c X. If we define E Hol(D, D x X) by l(z) = (z, f(z», then is an imbedding of D into D x X. Let n: D x X ~ X be the projection. If g E Hol(D, D x (X  A» approximates 1, then n 0 g E Hol(D, X  A) approximates f. We may therefore assume that f imbeds Dr. r > 1 into X. Let cp E Hol(D n , X) be as in (3.2.20), and define B = cpl (A). If the map j E Hol(D, D/) defined by j(z) = (z, 0, ... ,0) can be approximated by g E Hol(D, D n  B), then the map f = cp 0 j can be approximated by cp 0 g E Hol(D, X  A). The proof of the theorem is now reduced to the case where X is a domain in c n and feD) c X. We consider the map h : D x A ~ C n defined by h(z, a) = fez)  a. Since dim(D x A) < n, h(D x A) is a meager set (i.e., a countable union of nowhere dense subsets) in CIl and, hence, there exists a sequence of points c 1, C2, ... of C"  h (D x A) converging to the origin 0 E C/. Define fm E Hol(D, C") by fm(z) = fez)  Cm. Since feD) c X, there is an integer N such that f",(D) C X for m > N. The sequence {fm, m > N} converges to f in Ho1(D, X), and, by construction, fill (D) c XA. 0
°
1
1
A few remarks are in order. First, as the following example shows, it is essential in (3.2.19) that X is nonsingular, (CampbellOgawa [1]). (3.2.21) Example. Let n: C/+ 1  {OJ ~ Pn be the natural projection. Let Y C P n be a hyperbolic algebraic manifold, e.g., a nonsingular curve of genus > 1. Let Xc cn+l be the cone over Y, i.e., X = n1(y) U {OJ. Then d x == 0 since X is a union of lines interesecting at the origin. Let A = {OJ. Then d x A is nontrivial;
2 Hyperbolicity dX_A(p, q)
~
67
dy(n(p), n(q» > 0
if p, q E X  A do not lie on the same line through the origin. Note that A is the singular locus of X and that by choosing Y to be of large dimension, we can make the codimension of A in X as large as we wish. We state, without proof, a generalization of (3.2.19) by PoletskiIShabat [1], see also larnickiPflug [10; p. 87]:
(3.2.22) Theorem. Let X be an ndimensional complex manifold, and A a closed subset with (2n 2)dimensional Hausdorff measure equal to zero. Then HoI (D, XA) is dense in Hol(D, X) in the compactopen topology, and hence on
d X _ A = dx
XA.
A generalization of (3.2.19) to kintrinsic measures (see Section 2 of Chapter 7 for intrinsic measures) was obtained by KalimanZaidenberg [1]. Following Wu [I], we say that a complex space X with a distance function 8 (which is assumed to induce the topology of X) is 8tight if Hol(D, X) is equicontinuous with respect to 8 and that X is tight if it is 8tight for some 8. If X is hyperbolic, then it is dxtight (by (3.1.6)) and hence tight. Conversely (Kiernan [2]), we have
(3.2.23) Theorem. A complex space X is hyperbolic ifand only (fit is tight. Proof Assume that X is 8tight. Let P and q be two distinct points of X. Let U be an open hyperbolic neighborhood of p such that q ¢. U. Let W be a smaller neighborhood of P, relatively compact in U. Let E > 0 be such that the Eneighborhood V = (x EX; 8(W, x) < E} of W is relatively compact in U. Since Hol(D. X) is an equicontinuous family, there exists a positive number r < 1 such that if IE Hol(D. X) with 1(0) E W, then I(Dr) C V. Let c > 0 be a constant such that dD(O, b) > c·dn,(O, b) for all b E D r / 2 . Let a={p=po,PI, .. ·,Pk=q; al.b1, ... ,akobk; 11 ... ·,/d
be a chain of holomorphic disks from P to q. We may assume that Po, PI, ... , PiI E W, Pj ¢. W, al
Then Pi
E
= ... = Uk =
0, bl,···. b k
E
Dr / 2 •
V, and i
lea)
>
j
Ldn(O,bi)~eLdD,(O,bi) i=l
i=1 j
>
c Ldu(Pil, Pi) ~ e· du(Po, Pj). i=1
Take c' > 0 such that du(Po, V  W)
~ c'.
Then dx(p, q)
~ lea) ~
ee'.
0
68
Chapter 3. Intrinsic Distances
(3.2.24) Remark. If a complex space X is hyperbolic, then every holomorphic map f of C into X is necessarily constant since
dx(j(a).
feb»~
:::: dda, b) = 0
by (3.1.21). Hence, every holomorphic map f of PIC or a complex torus into a hyperbolic complex space X is constant. As we shall see in (3.6.3) a compact complex space X is hyperbolic if there is no nonconstant holomorphic map of C into X. We say that a complex space X if algebraically hyperbolic if there is no nonconstant holomorphic map of PI C or a complex torus into X. Ballico [1] proved that a generic hypersurface of large degree in Pn + I C is algebraically hyperbolic. It is not known if every algebraically hyperbolic algebraic manifold is hyperbolic. It is important to generalize the concept ofhyperbolicity to allow the Kobayashi distance to be partially degenerate. Let X be a complex space and L1 a closed subset of X. in applications, L1 is usually a closed complex subspace. We say that X is hyperbolic modulo L1 if for every pair of distinct points p, q of X we have dx(p, q) > 0 unless both are contained in L1. Then d x induces a distance function d X / Ll on the quotient space X I L1 in a natural way. We say that X is complete hyperbolic modulo L1 if it is hyperbolic modulo L1 and if for each sequence {Pll} in X which is Cauchy with respect to the pseudodistance d x , we have one of the following: (a) {Pn} converges to a point p in X; (b) for every open neighborhood U of L1 in X, there exists an integer N such that Pn E U for n > N. Clearly, X is complete hyperbolic modulo L1 if and only if the quotient space XIL1 is complete with respect to the distance function d x / Ll . Suppose that L1 and L1' are two closed subsets of X and that X is hyperbolic modulo L1 as well as modulo L1'. Then X is hyperbolic modulo L1 n L1'. So we can speak of the smallest closed subset L1 such that X is hyperbolic modulo L1. Clearly such a closed set is given by
L1x
= the closure of (p
E
X; dx(p. q)
= 0 for some q
=1=
pl.
From (1.1.8) we obtain the following (3.2.25) Theorem. {f a complex space X is hyperbolic modulo a closed subset L1, then (1) for every point P E X  L1 and jor every neighborhood U C X  L1 of p, there exists a aneighborhood V of p >,vith re!>pect to d x such that V C U; (2) if L1 is compact, on the quotient space XI L1 the pseudodistance d x induces the quotient topology. The following proposition is a straightforward generalization of (3.2.2). (3.2.26) Proposition. Let X be a complex subspace of a complex space Y.
70
Chapter 3. Intrinsic Distances
(3.2.33) Theorem. Let n: X + X be a .finitetoone proper holomorphic map. is (complete) hyperbolic modulo a compact subset .1, then X is (complete) hyperbolic modulo n 1 (.1).
If X
(3.2.34) Remark. Given a complex space X, consider the equivalence relation R defined by the pseudodistance d x , i.e., p is equivalent to q if and only if dx(p, q) = O. Then d x induces a distance on XI R. However, XI R need not carry a complex structure which would make the projection X + XI R holomorphic. Even when XI R admits such a complex structure, the induced distance may not coincides with the intrinsic distance d X / R of the complex space XI R, see Horst [3]. For hyperbolic quotients of homogeneous complex manifolds, see Gilligan [1]. On the degeneracy set .1 for the pseudodistance d x, see Hristov [4, 5, 6, 7] and AdachiSuzuki [2].
3 Hyperbolic Imbeddings Let Z be a complex space and Y a complex subspace with compact closure Y. We call a point p E Y a hyperbolic point if every neighborhood U of p contains a smaller neighborhood V of p, if c U, such that dy(V
n Y,
Y  U) > O.
We say that Y is hyperbolically imbedded in Z if every point of Y is a hyperbolic point. Clearly, Y is hyperbolically imbedded in Z if and only if, for every pair of distinct points 1', q in Y c Z, there exist neighborhoods VI' and U 4 of p and q in Z such that dy(UI' nY, U q n Y) > O. In the definition above, there is no need to assume that Y is relatively compact. But in applications, Y is almost always a relatively compact open domain in Z. So, unless otherwise stated, we assume that Y is relatively compact in Z. It is clear that a hyperbolically imbedded complex space Y is hyperbolic. The condition of hyperbolic imbedding says that the distance d y (Pn, qll) remains positive when two sequences {I'll} and {q,,} in Y approach two distinct points p and q of the boundary aY = Y Y. The concept of hyperbolic imbedding was first introduced in Kobayashi [7] to obtain a generalization of the big Picard theorem. The term "hyperbolic imbedding" was first used by Kiernan [6]. We note that a compact hyperbolic complex space is hyperbolically imbedded in itself. The proof of the following proposition is straightforward. (3.3.1) Proposition. If complex spaces Y and Y' are hyperbolically imbedded in Z and Z' respectively, then Y x Y' is hyperbolically imbedded in Z x Z'. The following is obvious. (3.3.2) Proposition. If there is a distance fimction ;) on
Y such that
3 Hyperbolic Imbeddings dy(p, q) ~ 8(p, q)
for
71
p, q E Y,
then Y is hyperbolically imbedded in Z.
Using the concept of length function and the induced distance function (see (2.3.1» we state the converse, (see Kiernan [6] and KiernanKobayashi [2]). (3.3.3) Theorem. Let Y be a relatively compact complex subspace of a complex space Z. Then the following are equivalent: (a) Y is hyperbolically imbedded in Z; (b) Given a lengthfimction F on Z there is a positive constant c such that for
I
E Hol(D, Y).
Proof Assume (a). Ifa constant c in (b) does not exist, then there exist a sequence Un} in Hoi (D, Y) and points {an} in D such that
t,
. n*F2
2 > n· d SD
at
an .
Since D is homogeneous, we may assume that a ll = O. Let e be a unit vector at OED, measured by the Poincare metric. Then the inequality above states
F(dj;,(e»2 > n. Since In (0) E Y and Y is compact, by taking a subsequence we may assume that {f,,(0)} converges to a point p E Y. Let U be a complete hyperbolic neighborhood of p in Z, e.g., a neighborhood biholomorphic to a closed analytic subset of Dm. Assume that there exists a positive number r < 1 such that j;,(D r ) C U for n ~ no. Since j;,(O) belongs to a compact neighborhood of p in U and since U is complete hyperbolic, U;,ID, E Hol(D r , U)} is relatively compact in Hol(D r , U) by (1.3.3) and would have a subsequence which converges in Hol(D r , U). But this is impossible since F(dln(e))2 > n. Thus no such r exists. This means that for each positive integer k, there exist a point Zk E D and an integer Ilk such that IZk I < and f", (Zk) ¢ U. Let Pk = In, (0) and qk = j;" (zd. By taking a subsequence we may assume that {qkl converges to a point q not in U. Since
t
dy(pk. qk) :::: dD(O,
zd
~
0
as
k
~ 00,
this contradicts the assumption that Y is hyperbolically imbedded in Z. Assume (b). Let 8 be the distance function on Z defined by the length function cF. Then 8U(a), feb»~ :::: dD(a, b) for f E Hol(D, Y). By (2) of (3.1.7), 8:::: d y on Y. Given two points p, q E Y, set 2(.1' let Up and Uq be the open balls of 8radius (.I' around p and q.
= 8(p, q) and D
In (3.2.18) we showed that if Y is the complement of a Cartier divisor in Z, then Y is locally complete hyperbolic in Z. In this connection we prove the following
72
Chapter 3. Intrinsic Distances
(3.3.4) Theorem. Let Y be hyperbolically imbedded in a complex space Z. IfY is locally complete hyperbolic in the sense that every point p E Y has a neighborhood VI' in Z such that VI' n Y is complete hyperbolic, then Y is complete hyperbolic. We start the proof with the following general lemma. (3.3.5) Lemma. Let Y be a complex subspace of a complex space Z. Let P E Y and VI' a neighborhood of p in Z. Given a smaller neighborhood WI' of P such that 8 := dy(Wp n Y, Y  VI') >
°
and a positive constant 8' < 812, there is a constant c > dy(q,q')~c·dv"ny(q,q')
°
such that
for q,q'EWpny with dy(q,q') 0,
5 Infinitesimal Pseudometric Fx
87
there is a neighborhood of 0 in ix* X such that F; 0 such that yet) E W for It  s I < 81• Since f restricted to the real axis and yare two parametrized curves with the same tangent vector at f(O) = yes), there exists 82 > 0 such that for It  sl < (h we have dw(y(t), f(t  s» < cit  sl.
Since the injection W + X is distancedecreasing, we have (ii)
dx(y(t), f(t 
s» < cit 
sl·
On the other hand, from the distancedecreasing property of f and from the expression for the Poincare metric of DR with curvature I (see (2.1.3» it follows that there exists 83 > 0 such that dx(f(t  s), f(O» :::::: d DR (t  s, 0) ::::::
(~ + e)it 
sI
for It  sl < h Combining this with (i) we have (iii)
dx(f(t  s), f(O» < (Fx(y'(s»
+ 2c)lt 
sl
for It  sl < 83. By (ii), (iii) and the triangular inequality we have dx(y(t), y(s» < (Fx(y'(s»
+ 3e)lt 
sl,
96
Chapter 3. Intrinsic Distances
thereby proving Lemma (3.5.33). For each S E [0, 1], let 1., denote the interval It  sl < 8 obtained in Lemma (3.5.33). We apply the Lebesgues covering lemma (see, for example, Kelley [1; p. 154]) to the open cover {/.,; s E [0, I]}. (The Lebesgues covering lemma states that, given an open cover U of a compact metric space A, there is a positive number 11 such that the open 11ball about each point of A is contained in some member ofU). Thus there is 11 > such that ift, t ' E [0, 1] and It  til < 11, then t, t' E I, for some s. Let = to < tl < ... < tk = 1 be a subdivision of [0, 1] with ti  tiI < 1'}, and choose Sf so that tiI, ti E Is;. Then
°
°
dx(p, q)
:s L
dx(y(tiI), yeti»~ < L(h(Si)
+ £)It; 
t;I1 < dx(p, q)
+ 2£.
This completes the proof of the equality d x = dx . The equality dx = dx has little to do with complex analysis and is a direct consequence of the following theorem in Finsler geometry. (3.5.34) Theorem. Let X be a (real) manifold. Let F: T X * R be an upper semicontinuous pseudolength function on X, and let ft be the convex pseudolength function defined by the property that its indicatrix at each x E X is the convex hull of the indicatrix of F at x. Then the pseudodistance d defined by F coincides with the pseudodistance d defined by ft. This theorem has been proved by Busemann and Mayer [1] under the assumption that F is continuous and strictly positive. For the proof of (3.5.34), see 0 Kobayashi [23]. If X is a singular complex space, Fx and ftx may not be upper semicontinuous. However, using the upper integral we can still define dx(p, q) = inf! Fx , y
y
dx(p, q) = inf! y
y
ft x .
Obviously, we have dx(p, q)
:s dx(p, q) :s dx(p, q).
But the question remains whether dx, d x and d x are all equal when X is singular. The proof of (3.2.19) gives also the following infinitesimal analogue of (3.2.19). (3.5.35) Proposition. {{ X is a complex manifold and A is a closed analytic subset of codimension at least 2, then FXA = FxlxA
and
FXA = FxlxA'
The second equality is a direct result of the first.
5 Infinitesimal Pseudometric Fx
97
It is not easy to determine F x even for relatively simple domains. The following example is due to GrahamWu [1].
(3.5.36) Example. Let
X= D xC
1
fez, w); Izl::: 2' Iwl:::
I},
~ D and q: X ~ C be the projections defined by p(z, w) = z and w. Let Xo = (zo, wo) be a point of X and Vo a nonzero tangent vector
and let p: X q(z, w) at Xo.
=
If Izol < 1/2, then FD(p*(vo» :::: Fx(vo) :::: F 01 / 2 (P*Vo).
If Izol > 1/2 (and hence Iwol < I), then Fx(vo) > O. More precisely, let 8 = dD(zo, zo/2Izol) > 0 and choose r, 0 < r < 1, such that dD(O,r) < 8/2. We shall show
(Since q (xo) = Wo E D, we are considering here q* Vo as a tangent vector of D). The second inequality giving an upper bound for Fx(vo) is trivial since D x D C X. The inequality Fo(p*vo) :::: Fx(vo) is also trivial since p is distancedecreasing. Assuming Fx(vo) < r F D(q*VO) we shall derive a contradiction. From the definition of Fx(vo) there exist U E ToD and f E Hol(D, X) such that f*u = Vo and Fx(vo) :::: FD(u) < rFD(q*vo). If q(f(Dr C D, then FD(qd*(u» :::: FD,(u), and
»
rF/J(q*vo)
= rFD(q*f*(u»
:::: rFD,(u)
=
FD(U),
which is a contradiction. Hence, q(f(D r » ct. D, and there exists a point a E Dr such that q(f(a» ¢. D. Hence, p(f(a» E D 1/ 2 . Then 8 :::: dD(p(f(a», p(xo». On the other hand, do(p(f(a», p(xo» = dD(p(f(a», p(f(O» :::: do(a, 0)
2.
In certain cases, F x is not only upper semicontinuous but continuous. The following is due to Royden [2]: (3.5.38) Proposition. If a complex space X is complete hyperbolic. then Fx. F; and Fx are continuous.
t
t
Proof In order to prove that Fx is continuous at Vo E X, let Vk E X be a sequence of vectors converging to Va. Since X is complete hyperbolic, by (3.5.15) there exist vectors Uk E ToD and maps ik E Hol(D, X)} such that fh(Uk) = Vk and Fx (Vk) = II Uk II. Since Fx is upper semicontinuous, {Fx (vd} is bounded. Hence, by passing to a subsequence if necessary, we may assume that {Uk} converges to some vector Uo E ToD. By applying (1.3.3) to the family {ik}, we see that a subsequence of Uk} converges to a map 10 E Hol(D, X). Then fo*(uo) = Va and lim Fx(vd = lim IiUkli = Iluoll 2: Fx(vo), showing that Fx is lower semi continuous at Vo. The continuity of F; and Fx follows from (3.5.20) and (3.5.21).
0
5 Infinitesimal Pseudometric Fx
99
(3.5.39) Remark. As Royden states and as we remarked in (3.5.15), the proof above is valid under the weaker assumption that X is taut. The concept of taut complex space will be introduced in Section I of Chapter 5. Wright [I] proved that if X is a projective algebraic manifold of general type, then Fx is continuous even when X is not hyperbolic. Although Fx is not very smooth, in the hyperbolic case we can find a smooth length function which can often take place of Fx. (3.5.40) Proposition. If a complex space X is hyperbolic modulo a closed set ..:1 and if £ is a length function on X, then there exists a nonnegative continuous function cp on X which is positive outside ..:1 such that
for
I
E
Hol(D, X).
Proof Let {Um} be an increasing sequence of relatively compact domains in X  ..:1 which exhaust X  ..:1, that is, Um C Um + 1 and U U m = X  ..:1. As in the proof of (3.3.13) we can find a sequence of positive constants Cl ::: C2 ::: ... such that
Take a nonnegative continuous function cp on X such that 0 < cp Then f*(cp2 £2) ::s dsb for I E Hol(D, X).
::s
Cm
on Um. D
(3.5.41) Corollary. If X is hyperbolic modulo ..:1, then given a lengthfonction £ on X, there exists a nonnegative continuous function cp on X such that cp£ ::s Fx and cp > 0 on X  ..:1. We can define also an intrinsic relative pseudometric, i.e., the infinitesimal form of the relative pseudodistance d y.z . Let Y be a complex subspace of a complex space Z. As in Section 4, let Fy,z be the family of holomorphic maps I: D + Z such that 11 (Z  Y) is either empty or a singleton. Then using the family Fy,z instead of Hol(D, y), we define F; z, Fr,z and Fy,z exactly as in (3.5.1), (3.5.7) and (3.5.14). ' We do not state obvious basic properties of these pseudolength functions. However, we note that the proof of (3.5.27) shows that if Z is nonsingular and Y is the complement of a divisor with no worse than normal crossing singularities, then Fy,z and Fy,z are upper semicontinuous and F; z is lower semicontinuous. We state the folJowing converse to (3.4.11); it st~engthens (3.3.3). The proof wiJI be given in the course of the proof of (3.6.20) (3.5.42) Theorem. Let a complex space Y be hyperbolically imbedded in a complex space Z. Given a length function £ on Z, there is a constant c > 0 such that
for
f
E
Fy,z,
100
Chapter 3. Intrinsic Distances
(3.5.43) Corollary. Let Y be hyperbolically imbedded in Z. Given a length/unction E on Z, there is a constant c > 0 such that cE < fry z on Y. (3.5.44) Remark. For Z = PIC and Y = PIC  {CXl, 0, I}, there is an estimate for Fy.z by Landau, (see Caratheodory [4, vol. 2, p. 198]). In fact, Landau proved that if fez) = alZ + a2z 2 + ... is ho10morphic in Izl < R and if f does not take values 0 or 1 in 0 < Izl < R, then R ::::: 16/1all and that this bound is the best possible. This may be restated as Fy.z «d I dz)o) = 1/8. (3.5.45) Remark. According to the definition of the curvature given by (2.3.3), F x has holomorphic sectional curvature:::: 1, provided X is complete hyperbolic (or taut), see B. Wong [1], Masaaki Suzuki [1] and Royden [9]. Given a nonzero v E Tt X, we want to show that g* F x has curvature:::: 1 for some g E HoI (D, X) such that g(O) = x and v is tangent to g(D). Since X is complete hyperbolic, there is a map g E Hol(D, X) such that Fx(v) = Ilull for some u E ToD with g*u = v, (see the proof of (3.5.38». Then in the inequality g* F; ::::: ds 2 , the equality holds at O. Take a supporting metric do 2 for g* F; at O. Then it is a supporting metric for ds 2 at O. By (2.1.8) it has curvature:::: 1. Hence, g* Fx has curvature:::: I. (3.5.46). Remark. In order to generalize Royden's result (3.5.31) to singular complex spaces Venturini [6] extended the definition of Fx(~) to vectors ~ of higher order osculation.
6 Brody's Criteria for Hyperbolicity and Applications Theorem (3.6.3) of Brody is the simplest and most useful criterion for hyperbolicity. We give several technical improvements of Brody's criterion and their applications. The following is immediate from (3.1.6) and (3.1.21).
rr
(3.6.1) Proposition. X is a hyperbolic complex space, then every holomorphic map f : C + X is constant.
Proof For a, bE C, we have dx(f(a), Hence, f(a) = feb).
feb»~
::::: dda, b) = O.
o
We shall now prove the converse when X is compact. The proof is based on the following Reparametrization Lemma (3.6.2) of Brody [1]. Wu [6] points out that this lemma was first proved by Landau [1; pp. 618619] in the context of holomorphic functions on the unit disc and then by Za1cman [1] for merom orphic functions on the unit disc. The Poincare metric d s~ of curvature Ion the disc DR of radius R is given by (see (2.l.3»
6 Brody's Criteria for Hyperbolicity and Applications
In the following we use the metric R2ds~ of curvature the Euclidean metric 4dzdz at the origin O.
1/ R2, which agrees
101
with
(3.6.2) Lemma. Let X be a complex space with a length Junction F. Given f HoIWR, X), define a Junction
E
U = f*F2/R2ds~ on DR. (fu(O) > c > 0, then there is a map g E Hol(D R , X) such that (a) theJunction g* F2 / R2ds~ is bounded by c on DR and attains the maximum value c at the origin; (b) g = f 0 ILr 0 tp, where tp is a holomorphic automorphism oj DR and ILr is the multiplication by suitable r, 0 < r < I, (i.e., ILr(Z) = rzJor Z E DR)' Proof For t
E
[0, I), define ft
ft(z) = f Set Ut = ft· F2 / R2ds~ = (f
Hol(DR, X) by
0
ILt(Z) = f(tz)
0
ILt)* F2 / R2ds~. Then
IL: f* F2
Ut
E
IL*ds 2
for
= IL:(R2ds~)' ~s~ R = IL;(U)
zE
DR.
t 2(R2  Iz12)2
(R2 _ Itz12)2 .
Set
From the explicit expression for Ut(z) given above, we see that, for each t E [0, I), Ut(z) approaches zero at the boundary of DR and hence SUP~EDR ut(z) is attained in the interior of DR. It is easy to see that U (t) is continuous in the interval [0,1). Since Ut (0) = U(0)t 2 > ct 2, we have U (t) > c for t sufficiently close to 1. On the other hand, U(O) = O. Thus, c = U(r) for some r E (0, 1). Let Zo E DR be a point where c = SUPzED" u,. (z) is attained. Let rp be a holomorphic automorphism of DR which sends 0 to zoo Then g = f 0 ILr 0 tp possesses all the desired properties.
o In the preceding section (see (3.5.14) and (3.5.16» we defined the pseudolength function F x as an infinitesimal form of d x. Since we need here only its definition and its most basic property (3.5.18), we shall quickly review Fx. Given a point x in a complex space X, the tangent cone txx consists of vectors of the form f.(u), where u E T D and f E Hol(D, X). Then Fx : 1:,X + R is defined by Fx(v) = inf{llull; u E T D and f*(u) = v}, where II u II is the length of u measured by the Poincare metric of D, and the infimum is taken over all u E T D and f E Hol(D, X) such that fAu) = v. Alternatively, Fx may be defined in terms of a fixed vector e = (8/8z)o E Toe and discs DR of varying radius R:
102
Chapter 3. Intrinsic Distances
Fx(v)
.
2
= mf, R
where the infimum is taken over all positive real numbers R for which there is a holomorphic map f: DR * X such that f.(e) = v. Let E be any (continuous) pseudolength function on X such that E :s Fx. Then E(f*u) :s Fx(f.u) :s lIu II for u E T D, f E Hol(D, X). It follows that if 8 denotes the pseudodistance defined by E, then every holomorphic map f: (D, p) * (X, 8) is distancedecreasing so that (by (3.1.7))
In particular, if E is a length function so that 8 is a distance, then X is hyperbolic. The same argument shows that if X is a relatively compact complex subspace of a complex space Y and if there is a length function E on Y such that E :s Fx on X, then X is hyperbolically imbedded in Y. It is useful to introduce the concept of complex line following Zaidenberg [2]. Let z denote the natural coordinate system on C. Let X be a complex space, and E a length function defined on X. A nonconstant holomorphic map h: C * X such that h*E 2 :s Cdzdz
for some constant C > 0 is called a complex line. If h(C) is contained in a compact subset of X, then this condition is independent of E. Let S be a subset (often a domain) in X. We say that a complex line h: C * X is a limit complex line coming from S if on each disc DR C C of radius R the mapping hlDR is a limit of holomorphic mappings of DR into S. In this case, we have h(C) C S. Every complex line in X is a limit complex line coming from X. We are now in a position to prove the following theorem of Brody [1]. (3.6.3) Theorem. Let X be a compact complex space. there is a complex line h: C * X.
If X is not hyperbolic,
then
Proof Let E be a length function on X. Assume that X is not hyperbolic. Let Fx be the pseudolength function defined above. If there is a positive number a such that a . E :s Fx , then X would be hyperbolic as explained above. Hence, there is a sequence of tangent vectors Vn E X such that E (v n ) = I and F x (v n ) < 1/ n. By applying the second definition of F x above, we find an increasing sequence of concentric discs DR" of radius Rn with lim Rn = 00 and a sequence of maps fn E Hol(D R" , X) such that f~(O) = Vn . (By f~(O) we mean df,,(e), where e = (B/Bz)o is the tangent vector of C at the origin 0.) Since the length lIeli n of e measured by the Poincare metric ds~" = 4R~dzdz/(R~  Id)2 of DR" is equal to 2/ R n , the function Un = fn* E2 / R~ds~n on DR", evaluated at the origin 0, gives
t
6 Brody's Criteria for Hyperbolicity and Applications
103
By applying (3.6.2) to each fn and a constant 0 < c < 1/4, we obtain a sequence of maps gn E Hol(D R" , X) such that (a)
g~E2:::: cR~ds~" on DRn and the equality holds at the origin 0;
By (a), the family of maps gn precise, since
E Hol(D Rn ,
X) is equicontinuous. To be more for
n::: m,
the family Fm = {gill D Rm ' n ::: m} is equicontinuous for each fixed m. Since the family FJ = {gn ID R, } is equicontinuous, the ArzelaAscoli theorem (1.3.1) implies that we can extract a subsequence which converges to a map hI E Hol(D R1 , X). (We note that this is where we use the compactness of X). Applying the same theorem to the corresponding sequence in F 2 , we extract a subsequence which converges to a map h2 E Hol(D R2 , X). In this way we obtain maps hk E Hol(D Rp X), k = 1,2, ... , such that each hk is an extension of h k  I . Hence, we have a map h E Hol(C, X) which extends all h k . Since g~ E2 at the origin 0 is equal to (c R~ds~ )z=o = 4cdzdz, it follows that "
which shows that h is nonconstant. Since g~E2 :::: cR~dsR2 , in the limit we have "
h* E2 :::: 4cdzdz.
By suitably normalizing h we obtain h* E2 :::: dzdz.
o
(3.6.4) Corollary. Let X be a compact complex space. Given a nonconstant holomorphic map f: C ~ X, there is a complex line h: C ~ X such that h(C) c fCC). Proof Let DR" be an increasing sequence of concentric discs with lim Rn = 00. Let fn be the restriction of f to DR". Moving the origin of C if necessary, we may assume that f is nondegenerate at O. As in the proof of (3.6.3), let e = (a/az)o be the tangent vector of Cat o. Set v = df(e) = 1'(0). Multiplying E by a suitable constant, we may assume that E(v) = 1. We set v" = v for all n. Since Fx(v) :::: Fde) = 0, we are now in a position to apply the proof of (3.6.3). Given a constant c, 0 < c < 1/4, by (3.6.2) we obtain a sequence of maps gn E Hol(D R" , X) satisfying conditions (a) and (b) in the proof of (3.6.3). According to (b) of (3.6.2), gn is of the form
where IIr. is the multiplication by a suitable rn , 0 < rn < 1, while C{Jn is an automorphism of DR". Repeating the argument in the proof of (3.6.3) we obtain a
104
Chapter 3. Intrinsic Distances
complex line h: C + X as a limit ofa subsequence of (gn). Clearly, h(C)
c
I(c).
D With very little change in the proof we can extend Brody's criterion (3.6.3) to certain noncompact complex spaces, (Urata [5]). (3.6.5) Theorem. Let Z be a complex space, and Y a relatively compact complex subspace of z. If Y is not hyperbolically imbedded in Z, then there is a limit complex line h: C + Z coming from Y so that h(C) C Y. Conversely, if Y is hyperbolically imbedded in Z, then Z contains no limit complex lines coming from Y. Proof Let £ be a length function on Z. Assume that Y is not hyperbolically
imbedded in Z. As we explained earlier (see the paragraph preceding (3.6.3», there is no length function F on Z such that F :::: F y on Y. Hence, there is no positive constant a such that a . E :::: F y on Y. The remainder of the proof is essentially the same as that of (3.6.3). If there is a limit complex line h: C + Z coming from Y, then any pair of points p, q E h(C) C Y would violate the condition for Y to be hyperbolic D imbedded in Z. The following example by D. Eisenman and L. Taylor shows that (3.6.3) does not hold for some noncompact manifolds. (3.6.6) Example. The domain X = fez, w)
E
c 2 ; Izl
< I,
Izwl
< J}  {CO, w);
Iwl
~ I}
is not hyperbolic, but there is no non constant holomorphic map : C +
x.
Proof The mapping h: (z, w):
H (z, zw) sends X into the unit bidisc and is onetoone except on the set z = o. If I: C + X is holomorphic, then hoI: C + Dl is holomorphic and hence constant by Liouville's theorem. It follows that either I is constant or I maps C into the set {CO, w) E X}. But this set is equivalent to the unit disc {w E C; Iwl < I}. Hence, I is constant in either case. Since h is distancedecreasing, we see that dx(p, q) > 0 for p i= q unless both p and q are in the subset (0, w) E X}. We shall show that if p and q are in this subset, then dx(p, q) = O. Let p = (0, b) with b i= and q = (0,0). Set Pn = (I/n, b). Then dx(p, q) = limdx(Pn, q). Let an = min{n, ,In/lbl}. Then the mapping tED + (ant In, a"bt) E X maps l/a" into Pn. Hence,
°
which shows that X is not hyperbolic. D In this example, let Z = PI C X PI C be a natural compactification of C 2 . Then the holomorphic mapping h: C + X c Z given by h(z) = (0, z) satisfies the inequality h* £2 :::: dzdz with equality at z = 0 (with respect to the product metric £2 coming from the FubiniStudy metric of PI C). Following Lang [3] we can strengthen (3.6.5) also in the following form.
6 Brody's Criteria for Hyperbolicity and Applications
105
(3.6.7) Theorem. Let Z be a complex space, and Y a relatively compact subset of Z. Let {Un} be a decreasing sequence of relatively compact open subsets of Z such that Un = Y. Assume that none of these Un is hyperbolically imbedded in Z. Then for each n we can find a limit complex line h n: C + Z coming from Un such that !he sequence {h n } converges to a complex line h: C + Z with its image i(C) in Y.
n
Proof Let E be a length function Z. If Un is not hyperbolically imbedded in Z, then by (3.6.5) there is a limit complex line hI!: C + Z coming from Un so that hn(C) C Un. Then h;'E2 ::::: Cndzdz with Cn > O. By composing hll with a suitable affine transformation Z f+ az+h, we may assume that h~E2 ::::: dzdz with equality holding at z = O. Applying ArzelaAscoli Theorem (l.3.1) to the family {h n } we obtain the desired result. D
Letting Y to be a compact complex subspace of Z in the theorem above, we obtain: (3.6.8) Corollary. Let Z be a complex space, and Y a compact complex subspace of Z. If Y is hyperbolic, there is a relatively compact neighborhood U of Y which is hyperbolically imbedded in Z. As in (1.2.2), for x
E
X consider the degeneracy set
.1(x)
=
(y EX; dx(x, y)
= OJ.
We know (see (1.2.7) and (3.1.18» that .1(x) is connected if X is compact. As an application of (3.6.7) we obtain (3.6.9) Corollary. Let X be a compact complex space, and x E X. !l.1(x) is nontrivial, i.e., il it contains more than one point, then there is a complex line h: C + X such that h(C) C .1 (x). Proof Let Un = (y
E
X; dx(x, y) < lin}.
Then nUn = .1(x). By (3.1.19), for all n we have dUn (x, y) =
0
for
Y
E
.1(x).
In particular, Un is not hyperbolic. By (3.6.7), we have a holomorphic map h: C + X with the stated property. D In order to consider the case where Y is the complement of a hypersurface in Z, we need a generalization of Hurwitz theorem. The classical theorem of Hurwitz states: (3.6.10) Theorem. The limit of a convergent sequence of nowhere vanishing holomorphic functions on a domain vanishes either nowhere or everywhere. This may be extended as follows.
106
Chapter 3. Intrinsic Distances
(3.6.11) Theorem. Let Z be a complex space and S = U;:l Si a Cartier divisor in Z where each Si is irreducible. Assume that a sequence {hml C Hol(D, Z  S) converges to a map h E Hol(D, Z). Then h(D) is either in Z  S or in S. More precisely, h(D) lies either in Z  S or in nEI Si  UjEJ Sj, where 1= Ii; h(O) E S;} and J = {j; h(O) ¢ Sjl. Proof Suppose that h(O) E S. Let V be a neighborhood of h(O) in Z such that V n S is defined by a holomorphic function I = where Ii = 0 defines V n Si. Take i such that f; (h (0» = O. Apply the classical theorem of Hurwitz to a sequence of holomorphic functions {f; 0 h m } which are nowhere zero. Its limit Ii 0 h must be identically zero since f; (h (0» = O. Hence h maps D into Si. D
n '/;,
We are now in a position to state the theorem of Green [7] and Howard. (3.6.12) Theorem. Let Z be a compact complex space with a lengthfunction E. Let S be a Cartier divisor in Z, and Y = Z  S. Then Y is complete hyperbolic and hyperbolically imbedded in Z if the following two conditions are satisfied: (a) There are no complex lines in Y; (b) There are no complex lines in S. Proof Suppose that Y is not hyperbolically imbedded in Z. By (3.6.5) there is a limit complex line h E Hol(C, Z) coming from Y. Hence, either h(C) C Y or h(C) C S by the generalized Hurwitz theorem (3.6.11). This is a contradiction. From (3.3.6) we see that Y is complete hyperbolic. D
More precisely, we have (Green [7]) (3.6.13) Theorem. Let Z be a compact complex space with a length function E. Let S be a union of Cartier divisors Sl, ... , Sm. Then Y = Z  S is complete hyperbolic and hyperbolically imbedded in Z (l the following two conditions are satisfied: (a) There are no complex lines in Y; (b) For any partition of indices I U J = {I, 2, ... , m}, there are no complex lines in niEI S;  UjEJ Si'
As we shall see later, the corollary above combined with Borel's Lemma imply that the complement of 2n + 1 hyperplanes in general position in PnC is complete hyperbolic and hyperbolically imbedded in Pile. The following result of Zaidenberg [4, 5] follows from (3.6.13). (3.6.14) Corollary. Let Z be an ndimensional compact complex manilold and S = U::l Si a divisor with only normal crossing sint"tilarities. Let S(k), 1 :s k :s n, denote the stratum of S consisting of the points of S ofmultiplici(v k, i.e., S(k)
=
Sk _ Sk+l,
Then the domain Y of the strata S(k), I
=
where Sk =
u
Sil
n ... n Silo
Z  S is hyperbolically imbedded in Z ifneither Y nor any I, contains complex lines.
:s k :s n 
6 Brody's Criteria for Hyperbolicity and Applications
107
Proof In view of (3.6.10) it suffices to prove that each limit complex line h: C + Z coming from Y is contained either in Y or in one of the strata S(k). Suppose that h(C) is not contained in Y. By the generalized Hurwitz theorem (3.6.11), for each i either h(C) C Si or h(C) n Si = 0. Without loss of generality we may assume that h(C) c Si for i = I, ... , k and h(C) n Sj = 0 for j = k + I, ... , m. Then h(C) is contained in the stratum S(k). 0
In order to prove a partial converse to (3.6.14) we start with the following variation of Royden's extension lemma (3.A.I) by Zaidenberg [4]. (3.6.15) Lemma. Let Z be a complex mant/old and S C Z a hypersurface. Let Sreg denote the set of regular points of S. Given a holomorphic mapping f: DR + Sreg with R > I, there is a holomorphic map ifJ: D x D + Z such that ifJ(Z, 0) = fez)
for
ZED
and
ifJ(D x D*) C Z  S.
Proof As in the proof of (3.A.I), by considering the graph of f we reduced the problem to the case where f is an imbedding. Thus, we have only to show that if f: DR + Sreg is a holomorphic imbedding, there is a holomorphic imbedding ifJ: D x D + Z with the property above. By (3.A.3) the vector bundle TXI!(D R ) over f(D R ) splits: TXI!(D R ) = TSI!(D R ) EB L,
where L is a line bundle over f(D R ) and normal to S. By (3.AA), there is a holomorphic affine connection in a neighborhood of f(DR). Now we repeat the the last of step of the proof of (3.A.2). Namely, we find a small neighborhood B of Din LI!w) ~ D x C such that B = D x D, and we set ifJ = exp lB. 0 (3.6.16) Corollary. Let Z, Sand Sreg be as in (3.6.15). Let r < R. Then every holomorphic mapping f: DR + Srcg can be approximated on Dr by holomorphic mappings of Dr into Z  S. Proof Let /ir: D R / r + DR be the multiplication by r. Given f: DR + Sreg, apply (3.6.15) to f 0 /ir: D Rjr + Srcg to obtain a map ifJ: D x D + Z such that (i) ifJ(z, 0) = f(rz) for zED and (ii) ifJ(D x D*) C Z  S. Set 1/I(z, w) = ifJ(z/r, w) for (z, w) E Dr X D. Then 1/I(z, 0) = fez) for ZEDI" Let hE(z) = 1/I(z, s). Then {hE) is an approximation of f on D. 0
(3.6.17) Lemma Let Z be an ndimensional complex manifold with smooth hypersurfaces SI, ... , Sk, k < n, whose union S = Ui Si is a divisor with normal crossings. Then every holomorphic mapping f: DR + ni Si can be approximated on each disc Dr, r < R, by holomorphic mappings h: Dr + Z  S. Proof The proof is by induction on k. The case k = I is a special case of(3.6.16) where S smooth. Assume that (3.6.17) holds for k  1. Set s[m] = n~=1 Si.
Then S[kl is a smooth hypersurface in a smooth manifold S[kI] of dimension n k+ 1. Let r < p < R. Then by (3.6.16) the mapping fiD. is approximated by mappings g: Dp + S[kI]  S[k] C Z  Sk. By the induction hypothesis applied
108
Chapter 3. Intrinsic Distances
to z' = Z  Sk, s; = SI  Sk, ... , S{_I = Skl  Sk, the mapping glDp is D approximated by mappings h: Dr + Z'  u~~i S; = Z  U~=1 Si. Now we are in a position to prove the following partial converse to (3.6.13). (3.6.18) Theorem. Let Z be a compact complex manifold with smooth hypersurfaces SI, ... , Sm, whose union S = Ui Si is a divisor with normal crossings. If Y = Z  S is hyperbolically imbedded in Z, then (a) there are no complex lines in Y; (b) for any partition of indices I U J = {1, 2, ... , m}, there are no complex lines in niEl Si  UjEJ Sj. Proof If (a) is violated, then Y is not hyperbolic. Assume that (b) is violated. Then there is h E Hol(C, Z) such that h(C) c n~=1 Si  Uj:k+1 Sj. In view of (3.6.5) it suffices to show that this complex line is a limit line coming from Y. For any R > 0 we can find a neighborhood U of h(DR) in Z such that un Sj = '" for j = k + 1, ... , m. Now apply (3.6.17) to the manifold U with hypersurfaces SI n U, ... , Sk n U. Then hlDI/ can be approximated on each disk Dr, r < R, by D holomorphic mappings of Dr into U  U7=1 S; C Z  ur=1 S;.
Similarly, we have the following partial converse to (3.6.14), (due to Zaidenberg [4, 5]). (3.6.19) Theorem. Let Z, S = U;:1 Si, and Y = Z  S be as in (3.6.18).IfY is hyperbolically imbedded in Z, then (a) there are no complex lines in Y; (b) there are no complex lines in any of the strata S(k), 1 :::: k :::: n  l. Proof If (a) is violated, then Y is not hyperbolic. Assuming (b) is violated, let h E Hol(C, Z) be such that h(C) c S(k) = Sk  Sk+I. Since h(C) C Sk, we may assume that h(C) c SI n ... n Sk. Since h(C) n Sk+1 = 0, h(C) does not interesect any of 51 n ... n Sk n 5j for j = k + 1, ... , m. Hence, h(C) n Sj = 0 for j = k + I, ... , m. This implies h(C) C n~=1 Si  Uj:k+1 Sj. This violates (b) of (3.6.18). Hence, Y is not hyperbolically imbedded in Z. D
Given a complex subspace Y C Z, we defined the pseudodistance d y. z on Y, see (3.4.1). Let Fy,z be the infinitesimal form of d y.z defined in the same way as the infinitesimal form F z of d z , using the subfamily Fy.z C Hol(D, Z), see the end of Section 5. Now, using the argument in the proof of (3.6.3) we shall prove the implication (c) :::} (b) in (3.4.11), thus making all three conditions in (3.4.11) mutually equivalent. (3.6.20) Theorem. If a complex space Y is hyperbolically imbedded in Z, then dy,z(p, q) > Ofor all pairs p, q E Y, p =f=. q. Proof Let E be any length function on Z. In order to prove the theorem, it suffices to show that there is a positive constant c such that cE :::: Fy,z on Y. Suppose that there is no such constant. Then there exist a sequence of tangent vectors Vn of Y
6 Brody's Criteria for Hyperbolicity and Applications
109
with E(v n) = I, a sequence of hoI omorphic maps In E Fy.z and a sequence of tangent vectors en of D with Poincare length lien II '\i 0 such that dIn (en) = V n. Since D is homogeneous, we may assume that ell is a vector at the origin of D. As in the proof of (3.6.3), we replace the above .f" E Hol(D, Z) by a new In E Hol(D R", Z) with R" ./ 00; instead of using the fixed disc D and varying vectors en, we use varying discs DR" and the fixed tangent vector e = (d/dz)o at the origin. Let be the family of holomorphic maps I: DR" f Z such that II (Z  Y) is either empty or a singleton. Having replaced (D, en) by (DR", e), R we may assume that In E and dfn(e) = Vn. By applying Brody's lemma (3.6.2) to each .f" and a constant 0 < c < 1/4 we obtain holomorphic maps gn E Hol(D R", Z) such that
F:'z
Fy.z
(a)
g~E2:s cR~ds~" on DR" and the equality holds at the origin 0;
(b)
Image(gn) C Image(.f,,).
Since gIl is of the form gn = In 0 IL rn 0 h,,, where h" is an automorphism of DRn and ILr", (0 < ILr" < 1), is the multiplication by rn, each gn maps all of DR", except possibly one point, into Y. Now, as in the proof of (3.6.3) we shall construct a nonconstant holomorphic map h: C f Z to which a suitable subsequence of {gil} converges. In fact, since for
n c:': m,
the family Fm = {gn IDR",; n c:': m} is equicontinuous for each fixed m. Since the family FI = {g,,1 DR,} is equicontinuous, the ArzelaAscoli theorem implies that we can extract a subsequence which converges to a map hI E HoJ(DR1' Z). (We note that this is where we use the compactness of Y.) Applying the same theorem to the corresponding sequence in F2, we extract a subsequence which converges to a map h2 E HoJ(DR2' Z). In this way we obtain maps hk E Hol(DRk' Z), k = 1,2, ... , such that each hk is an extension of hkI. Hence, we have a map h E Hol(C, Z) which extends all hk. Since g~E2 at the origin 0 is equal to (cR~ds~)z=o = 4cdzdz, it follows that (h* E 2),,=0
=
lim (g~E2):=O 1l+:::xJ
= 4cdzdz I 0,
which shows that h is nonconstant. Since g~ E2 :s cR~ds~", in the limit we have h* E2
:s 4cdzdz.
By suitably normalizing h we obtain h* E2
:s dzdz
with the equality holding at
z
= o.
We may assume that {gn} itself converges to h. Since h is the limit of {gn}, clearly h(C) C Y. Let p, q be two points of h(C), say p = h(a) and q =
110
Chapter 3. Intrinsic Distances
h(b). Taking a subsequence and suitable points a, bEe we may assume that gn(a), gn(b) E Y. Then limgn(a) = p and lim gil (b) = q. If there is a subsequence of {gn}, still denoted {gil}, such that gn(DRJ c Y,
then d y (gn (a), gll(b»
s
d DR" (a, b) + 0
as
n +
00,
contradicting the assumption that Y is hyperbolically imbedded in Z. If there is no such subsequence, there is a subsequence, again denoted {gil}, such that each gn maps exactly one point, say C n E DR" into the boundary a y. If the set {c n } is unbounded in C, by taking subsequence we may assume that ICn I ? 00. Then we can find R~ < ICn I < Rn such that R~ ? 00. If we denote the restriction of gil to DR;, by g~ so that g:,(DR) c Y, then dy(g~(a), g~(b»
s
d DR;, (a, b) + 0
as
n +
00,
again contradicting the assumption that Y is hyperbolically imbedded in Z. Therefore we assume that {en} is bounded. Taking a subsequence, we may assume that {c ll } converges to, say C E C. We may assume that c is the origin of C. (For each n we consider a disk D( c, R~) CDR" of center C and radius R;, such that R~ / ' 00, and we restrict gn to D(c, R;,).) Now we assume that limen = O. Set
Then g~(an) = gil (a),
Thus,
g:,
g~(bn) = gn(b),
maps the punctured disk D~;,
=
g;,(O) E
ay.
{O < Izl < R~} into Y. Hence,
dy(g~(all)' g~(bn» S d D " (all, bll ) + 0
as
11
+ 00,
R"
contradicting the assumption that Y is hyperbolically imbedded in Z. We note that since an + a and b n + b, in order to conclude d D " (an, b n) + 0 it suffices to R" show dD~(a, b) + 0 as R + 00. But this can be seen from the expression for the infinitesimal metric . spondmg to d[);,: 2
dSD~ =
dsb.
R
corre
4dzdz
Izl2(log R2  log IzI2)2'
which can be obtained from (2.2.3) by replacing z by zj R.
o
We say that the cotangent bundle T* X of a compact complex space X is ample (or the tangent bundle T X is negative in the sense of Grauert) if the zero section O(X) of T X can be blown down to a point, namely there exists a holomorphic map 1T: T X + Y onto a complex space Y = T X jO(X) which maps O(X) to a single point, say Yo E Y, and TX  O(X) biholomorphically onto Y  {yo}. This negativity of T X implies the existence of a length function F with negative
6 Brody's Criteria for Hyperbolicity and Applications
111
curvature. As we shall see in (3.7.1), such a space is hyperbolic. Instead of the differential geometric argument just described (see Kobayashi [13]) we present the proof by Urata [5] which uses Brody's criterion. (3.6.21) Theorem. Let X be a compact complex space with ample cotangent bundle. Then X is hyperbolic.
Proof With the notation above, let V be a small (hence hyperbolic) neighborhood of Yo in Y = T XIO(X). We choose a length function E on X in such a way that every tangent vector of Elength :s 1 belongs to Jl'I(V). Assume that X is not hyperbolic. By (3.6.3) there exists a nonconstant holomorphic map I: C ~ X such that its differential f': C ~ T X has the property that E U ' (z» :s 1 for all z E C. Since V is hyperbolic, the map Jl' 0 f': C ~ V must be constant and Jl'U'(C» = {Yo}· Hence, f' == 0, which implies that I is constant. This is a contradiction. D As another application of Brody's lemma, Urata [5] proved the following (3.6.22) Theorem. Let X be a complex space with a length function E, and G = Aut(X, E) the group of holomorphic isometries. Assume that XIG is compact. Then X is complete hyperbolic ({there is no complex line h: C ~ x.
Proof Let K be a compact subset of X such that G(K) = X. Suppose that X is not hyperbolic. Let e denote the tangent vector (dldz)o of C at O. Let DIl denote the disk of radius n with its Poincare metric ds; = 4n 2 dzdzl (n 2 Iz 12)2. Then for each n, there exists a holomorphic mapping In: Dn ~ X such that IdJ,.(e)IE > 1. We repeat the argument in the proof of (3 .6.3). By Brody's lemma (3.6.2) and from G(K) = X it follows that for each n there exists a holomorphic map gn: D" ~ X such that (i)
gf/(O) E K,
(ii)
g~ E2
:s n2ds,~,
with equality holding at 0 E Dn. Let 8x be the distance function on X defined by E. From (ii) above we have
8x (g" (0), gn(Z»
l
1z1
:s o
2n2dt
n + Izl
22 = n log  n  t n  Izl
:s 41z1
for Izl .:::: n12. Since XI G is compact, X is complete with respect to 8x. The estimate above shows that for each fixed z E C, the set {gn(Z); n ::: no}, where no ::: 21zl, is relatively compact in X. By (ii) the family {g,,} is also equicontinuous. By ArzelaAscoli theorem (1.3.1), a subsequence of {gn} converges to a holomorphic map g: C ~ X. By (ii), g* E2 .:::: 4dzdz with equality holding at O. This is a contradiction, showing that X is hyperbolic. Since d x is invariant by G and since G(K) = X with K compact, d x is a complete distance. D (3.6.23) Corollary. A homogeneous Hermitian manifold X with an invariant Hermitian metric ds 2 is complete hyperbolic if there is no complex line h: C ~ X.
112
Chapter 3. Intrinsic Distances
7 Differential Geometric Criteria for Hyperbolicity The results in Chapter 2 yield differential geometric criteria for hyperbolicity. We shall combine them with Brody's criteria explained in the preceding section. (3.7.1) Theorem. Let X be a complex space. I(there is a lengthfimction F with (holomorphic sectional) curvature KF bounded above by a negative constant, then X is hyperbolic. If moreover, F defines a complete distance on X, then X is complete hyperbolic.
Proof By normalizing F we may assume that K F ::::: 1. Let 8 be the distance function on X defined by F. By (2.3.5) every holomorphic map f: (D, p) + (X,8) is distancedecreasing. By (2) of (3.1.7), we have 8 ::::: d x . Our assertion now follows. 0 (3.7.2) Remark. Milnor [1] observed that for a Riemann surface the curvature condition of (3.7.1) can be relaxed as follows. Given a Riemann surface X with a Hermitian metric, let r denote the geodesic distance from a fixed point of X. If the curvature K satisfies K(r) ::::: 1/(r 2 10gr) asymptotically, then X is hyperbolic. Remark (2.2.8) shows that the asymptotic condition K (r) ::::: 1/ r3 is too weak to imply hyperbolicity. For a higher dimensional analogue of Milnor's result, see GreeneWu [2; p. 113, Theorem G']. We know from (2.2.6) that PI C minus at least three points carries a complete Hermitian metric with curvature K ::::: 1 and from (2.2.7) that every compact Riemann surface of genus 2: 2 admits a Hermitian metric with curvature K ::::: 1. Hence, (3.7.3) Corollary. (1) The Riemann sphere PIC minus at least three points is complete hyperbolic. (2) Every compact Riemann surface of genus 2: 2 is complete hyperbolic. We can generalize (3.7.1) to a pseudolength function F. Considering F as a nonnegative function on T X, we assume that it is continuous everwhere and twice differentiable wherever it is positive so that if F(v) > 0 at VET X then the curvature Kdv) is defined. We say that (X, F) is negatively curved if there is a negative constant c such that KF(v) ::::: c for all VET X for which F(v) > O. In accordance with (2.1.10) we can weaken the conditions on F as follows. (a) F is upper semicontinuous; (b) For each VET X with F (v) > 0 there is a length function Fv , defined and of class C 2 in a neighborhood U C T X of v, such that (i) F 2: Fv in U, (ii) F(v) = Fv(v), and (iii) the curvature of Fv is bounded above by a negative constant c independent of v. Then we say that (X, F) is negatively curved. A point x E X is called a degeneracy point of F if F (v) = 0 for some nonzero v E Tx X. The set of such degeneracy points is called the degeneracy set
7 Differential Geometric Criteria for Hyperbolicity
113
of F. Then the argument in the proof of (3.7.1) yields the following result. (The second statement in the theorem will not be used and can be skipped). (3.7.4) Theorem. Let X be a complex space. If"there is a pseudolength function F such that (X, F) is negatively curved in the sense defined above, then X is hyperbolic modulo the degeneracy set of F. More generally, if" F is a jet pseudometric on the jet bundle Jk X such that K F :::: I, then X is hyperbolic modulo the degeneracy set of F\, where F\ is the pseudolength jimction defined in (2.5.10). As we see from (2.2.8), for X to be hyperbolic it is not sufficient that X admits a Hermitian metric with negative hoI om orphic sectional curvature. It is important that the curvature is not only negative but also bounded away from zero. See Remark (3.7.15) for more comments on this point. On the other hand, Brody's criterion can be strengthened if the holomorphic sectional curvature is only nonpositive, (Kobayashi [IS]). (3.7.5) Lemma. Let 2ledzdz with 0 :::: A. :::: I be a pseudometric on C expressed
in terms of the natural coordinate function z of e. If its Gaussian curvature K is nonpositive at every point where Ie > 0, then ), is constant. Proof The Gaussian curvature K is given by
Since K :::: 0, we have
a2 10g Ie >0
azaz 
wherever ). > O. This means that log). is a subharmonic function on e. On the other hand, log Ie :::: 0 since Ie :::: 1. But a bounded subharmonic function on C is constant. Hence, A. is constant. 0 (3.7.6) Lemma. Let X be a complex space with a length function £ whose holomorphic sectional curvature K E is nonpositive. If f: C + X is a holomorphic map such that f* £2 :::: dzdz with equality holding at some point. then f* £2 = dzdz. i.e., f is an isometric holomorphic immersion with respect to the metric of X defined by E and the Euclidean metric dzdz ofe.
Proof Set f* £2 = Adzdz, and apply (3.7.5).
0
If X is a Hermitian manifold, we can say a little more about fCC). (3.7.7) Lemma. Let M be a Hermitian manifold with metric dsl.t whose holomorphic sectional curvature is nonpositive. If f: C + M is a holomorphic map such that f*dsl.t :::: dzdz with equality holding at some point, then f is a totally geodesic, isometric holomorphic immersion.
114
Chapter 3. Intrinsic Distances
Proof By (3.7.6) f is an isometric holomorphic immersion. Since M has nonpositive holomorphic sectional curvature and fCC) is flat, M has vanishing holomorphic sectional curvature in the direction of fCC) by (2.3.9). By Remark (2.3.11), fCC) is totally geodesic in M. 0 We apply Lemmas (3.7.6) and (3.7.7) to (3.6.3), (3.6.5), (3.6.9), (3.6.11) and (3.6.12) to obtain the following results, (3.7.8) through (3.7.12). (3.7.8) Theorem. Let X be a compact complex space with a length function E whose holomorphic sectional curvature is nonpositive. If X is not hyperbolic, there is an isometric holomorphic immersion h: C ~ X. In fact, if x E X and tl the degeneracy set L1 (x) is nontrivial, then we can find such a map h with its image h(C) in L1(x). If, moreover, X is a compact complex subspace of a Hermitian manifold Y with nonpositive holomorphic sectional curvature, such a map h: C ~ Y is totally geodesic. (3.7.9) Theorem. Let Z be a complex space with a length function E whose holomorphic sectional curvature is nonpositive, and Y a relatively compact open subset of z. IfY is not hyperbolically imbedded in Z, then there is an isometric holomorphic immersion h: C ~ Z such that h(C) c Y. If, moreover, Z is a Hermitian manifold with nonpositive holomorphic sectional curvature, such a map h is totally geodesic. (3.7.10) Theorem. Let Z be a compact complex space with a length function E whose holomorphic sectional curvature is nonpositive. Let S be a union of Cartier divisors S 1, ... , Sm. Then Y = Z  S is complete hyperbolic and hyperbolically imbedded in Z if the following two conditions are satisfied: (a) There are no isometric holomorphic immersions h: C ~ Y; (b) For any partition o.l indices 1 U J = {l, 2, ... , m}, there are no isometric holomorphic immersions h: C ~
n
S; 
iEI
U Sj C Z. jEJ
(3.7.11) Theorem. Let Z be a complex space with a length function E whose holomorphic sectional curvature is nonpositive, and Y a relatively compact subset of Z. Let {V n} be a decreasing sequence of relatively compact open neighborhoods of Y such that Vn = Y. If none of these VI! is hyperbolically imbedded in Z, then there is an isometric holomorphic immersion h: C ~ Z such that h(C) C Y. If, moreover, Z is a Hermitian manilold with nonpositive holomorphic sectional curvature, such a map h is totally geodesic.
n
From (3.7.8) we obtain the following result, conjectured by Lang [1] and proved by Green [8]. (3.7.12) Theorem. A closed complex subspace X of a complex torus T is hyperbolic if there is no nonconstant affine map h: C ~ T such that h (C) c X, or equivalently, if X contains no translate of a complex subtorus of T.
7 Differential Geometric Criteria for Hyperbolicity
115
Proof Suppose that X is not hyperbolic. In (3.7.8), let Y = T. Since T admits a flat Hermitian metric, there is a totally geodesic, isometric holomorphic map h: C """"* T such that h(C) c X. By translating h(C) in T, we may assume that H = h(C) is a subgroup of T. We claim that the Zariski closure H of H in T is a subgroup of T, i.e., (H)I cHand H· H c H, and hence is a complex torus. This follows from the fact that although the operation TxT """"* T, (x, y) f+ X I y, is not Zariski continuous, the operations x f+ xI and x f+ xa (with a fixed) are homeomorphisms of T onto itself in the Zariski topology. (See Lang [3; p. 84] for details. We note that the closure in the usual topology would yield merely a real subtorus). 0
Another result, also conjectured by Lang and proved by Green, follows from (3.7.10): (3.7.13) Theorem. Let T be a complex torus and S a complex hypersurface. (( S contains no complex subtorus, then Y = T  S is complete hyperbolic and hyperbolically imbedded in T. Proof In view of (3.7.10) it suffices to show that there is no totally geodesic, isometric holomorphic immersion h: C""""* Y. Assuming that such a map h exists, we shall obtain a contradiction. Let A be the topological closure of h(C) in T; it is a real sub torus of T. First we consider the case A n S = 0. Let {tn} be a convergent sequence of translations in T such that til (A) n S = 0 and (lim tn)(A) n S =f. 0. Choose points an E C such that lim til (h (an» E S. Let U" be the unit disc neighborhood of an in C. Then we have a sequence of holomorphic maps t" 0 h: Un """"* Y such that limtn(h(Un » n Sol 0. By identifying UII with the unit disc D around the origin o in C, we obtain a sequence of convergent holomorphic maps f,,: D """"* Y with the limit map f = lim fn such that feD) n S # 0. By the generalized Hurwitz theorem (3.6.11), we have feD) c S. From the construction of fn and f it is clear that f extends to an affine map C """"* T. Then f(C) must be also contained in S. This contradicts the assumption of the theorem. Next we consider the case A n S # 0. Choose points an E C such that limh(an) E S. Let Un be the unit disc neighborhood of an in C. The remainder of the proof is the same as in the first case; simply let til be the identity transformation in the proof of the first case. 0
(3.7.14) Corollary. Let T be a simple torus, i.e., a complex torus containing no proper complex subtorus. Then (I) Every closed complex subspace of T is hyperbolic; For every complex hypersurface S, its complement T  S is complete (2) hyperbolic and hyperbolically imbedded in T. (3.7.15) Remark. In general, a hyperbolic complex manifold may not admit a negatively curved Hermitian metric, see Demailly [3]. Cheung [1] discusses a special class of hyperbolic manifolds which admit negatively curved Hermitian metrics.
116
Chapter 3. Intrinsic Distances
8 Subvarieties of Quasi Tori In this section we shall study subvarieties of complex tori by purely differential geometric means. For an algebraic geometric approach, we refer the reader to Ueno [1]. First we quickly review basic facts on the second fundamental form and the equations of GaussCodazzi for a Kahler submanifold. We follow here NaganoSmyth [1], who studied minimal submanifolds in real tori. Since a complex submanifold of a Kahler manifold is a minimal submanifold, we can apply their results. Because of complex analyticity in our case, singularities of complex subspaces present no essential difficulty. We start with the Riemannian case. For (3.8.1) through (3.8.6) we refer the reader to KobayashiNomizu [1; voU]. Let M be an (n + p)dimensional Riemannian manifold with metric g and with covariant differentiation V, and X an ndimensional submanifold with covariant differentiation V. Given vector fields u, von X, we have the following formula of Gauss: (3.8.1)
Vuv = Vuv
+ a(u, v),
where a: TxX x TxX + T/ X is a symmetric bilinear map called the second fundamental form of X eM. If ~ is a section of the normal bundle T.L X, then we have the following formula of Weingarten: (3.8.2) where A~ defines a symmetric linear transformation of T X and V..L defines a connection in the normal bundle T..L X. Then a and A are related by (3.8.3)
g(A~(u), v)
= g(a(u, v),
~).
The connection V of T X combined with the connection V..L of T.L X defines a connection in T* X ® T* X ® T..L X, which we shall denote by V*. In particular, for a, we have (3.8.4)
(V:a)(v, w) = V;(a(v, w»  a(Vu v, w)  a(v, Vuw).
The curvature R of a Riemannian manifold M is a 2form with values in the endomorphism bundle End(T M). We associate to R a quadrilinear map, also denoted by R, by
Now we state the equation of Gauss: (3.8.5) Theorem. Let Rand R be the Riemannian curvature tensors of M and X, Thenfor any vector fields VI, V2, V3, V4 on X, we have
re~pectively.
R(v" V2, V3, V4) = R(VI, V2, V3, V4) +g(a(vl' V4), a(v2, V3»  g(a(vl, V3), a(v2, V4».
8 Subvarieties of Quasi Tori
117
The equation of Gauss expresses the tangential component of R(v3, V4)V2 in terms of R and a. Its normal component is described by the following equation of Codazzi: (3.8.6) Theorem. For any vector .fields u, v, w of X, the normal component of R(u, v)w is given by (R(u, v)w)1 = (V=a)(v, w)  (V:a)(u, w).
In particular, if M is a space of constant curvature, then (V=a)(v, w) = (V:a)(u, w).
We define the relative nullity space at x to be (3.8.7)
N,
= {u
E T.tX;
a(u, v)
=0
for all
v
E
TxX}.
Let
v U
=
= min dim Nt, XEX
{x E X; dimNx
=
v}.
Then U is an open subset of X, and N = UtEV N x is a subbundle of T(U) of rank v, i.e., a vdimensional distribution on U. (3.8.8) Lemma. Assume that M is a space of constant curvature and that v > O. Then the distribution N is integrable and each maximal integral submanifold of N is totally geodesic not only in U but also in M.
Proof. Let u, w be sections of the bundle N, and v any vector field on U. Then by (3.8.4), we have (V:a)(u, w) = O.
Hence, (3.8.6) implies a(v, V'IlW) = O. Since this holds for all vector fields v on U, V'uw is a section of N. This shows that N is integrable and totally geodesic in U. Since Vllw
= Vuw + a(u, w) = V'uw
and since Vu w is in N, it follows that N is totally geodesic in M as well.
0
Now we consider the case where M is a Kiihler manifold and X is a complex submanifold. Then the second fundamental form a satisfies the following (see KobayashiNomizu [1; voI.2]): (3.8.9)
a(Ju, v)
= a(u, Jv) =
J(a(u, v)),
where J is the complex structure of M. This implies that the relative nullity space N x is a complex subspace, Le., invariant by J.
118
Chapter 3. Intrinsic Distances From (3.8.5) and (3.8.9) we obtain
(3.8.10)
R(u, Ju, v, Jv) = R(u, Ju, v, Jv)
+ 2g(a(u, v), a(u, v».
Let Sand S be the Ricci tensors of M and X, respectively; they define symmetric bilinear forms on each tangent space. If we choose a local orthonormal basis of T M in such a way that el, ... , en, J el, ... , J en are tangent to X and en+l, ... , en+ p , len+l, ... , le n + p are normal to X, then n+p
L R(e;, lei, u, lu),
S(u, u)
i=l
n
S(u, u)
L R(e;, Je;, u, Ju).
=
;=1
Using (3.8.10) we obtain Il+P
n
S(u, u) = S(u, u)  2
L g(a(ei, u), aCe;, u»

L
R(ek, leb u, Ju).
k=n+1
i=1
From now on, we assume that M is flat. Then n
(3.8.11)
S(u, u) = 2
L g(a(ei, u), a(ei, u»
SO.
i=l
If u E N x , then S(u, u) = 0 by (3.8.1 I). Conversely, if u E TxX and S(u, u) = 0, then aCe;, u) = 0 for i = 1, ... , n by (3.8.11) and a(Jej, u) = 0 for i = I, ... , n by (3.8.9), and hence u E Nt. Thus, (3.8.12)
Nx
=
(u E TxX; S(u, u) = O}.
Since S is symmetric and negative semidefinite, we have (3.8.13)
N x = {u E TxX; S(u, v) = 0
for all
v E TtX}.
This shows that the relative nullity space Nt> defined originally in terms of the second fundamental form a, can be defined in terms of an intrinsic invariant of X, namely the Ricci tensor S. Now, let M be a locally flat Kahler manifold and X a closed complex subspace. We can apply the results above to the regular locus Xreg of X. Let N be the relatively nul1ity distribution defined on an open set U of X reg • Since the relative nullity spaces N x are all complex vector spaces, we denote by v the minimum of the complex dimensions of N x , x E U. For x E U, let N(x) be the maximal integral submanifold through x defined by the distribution N. Let M be a commutative complex Lie group with an invariant flat Kahler metric; M is of the form C n + p / r, where r is a discrete subgroup of C n + p • By
8 Subvarieties of Quasi Tori
119
parallel translation we can compare tangent spaces at different points. This fact makes it possible to define the Gauss map. Let X be an ndimensional closed complex subspace of M. Let G(n, n + p) be the complex Grassmann manifold of ndimensional subspaces in C n + p • The Gauss map G: X reg + G(n, n
+ p)
assigns to each regular point x the ndimensional subspace of the tangent space ToM parallel to TxX. We shall compute the differential dG: TxX + TG(x)G(n, n + p). We take a local orthonormal frame field el, ... , en+ p around x such that e1, ... , en are tangent to X reg (and en +1, ... , en + p normal to X reg ). Then Ta(x)G(n, n + p) is identified with the space of complex (n x p)matrices. Set (3.8.14)
de;
=
n
n+p
j=l
k=n+!
L w! ej + L
a~ek.
I::;i::;n.
The differential dG is identified with (a;). Comparing (3.8.14) with the definition (3.8.1) of the second fundamental form a, we see that n+p
(3.8.15)
a(u, ei)
=
L
a~(u)ek.
k=I!+1
which says that dG can be identified with the second fundamental form. It follows from (3.8.15) that the relative nullity space N x agrees with the kernel of dG at x: (3.8.16)
x E X reg .
From this we immediately obtain (3.8.17) Lemma. let M be a commutative complex Lie group with an invariant flat Kahler metric and X a closed complex subspace. Then each maximal intergal submanifold N(x), x E U is a connected component of a level set of the Gauss map G, and hence it is closed in U. For a different proof on closedness of N(x), see NaganoSmyth [1]. The use of the Gauss map was suggested by Wu, see FischerWu [I]. Since, by (3.8.8), N(x) is totally geodesic in M = e+ p / r, it extends to X. This extension is a translate of a connected vdimensional complex subgroup, say M~, of M and is the closure N(x) of N(x) in M. Hence, this subgroup M~ is closed in M. In general, M~ may vary with x. However, it is independent of x if M is a quasi torus in the following sense. A commutative complex Lie group M = C N / r is said to be a quasi torus if it is an extension of a complex torus T by (C*)k. Thus we have an exact sequence of commutative complex Lie groups:
o +
(C*)k + M + T + O.
120
Chapter 3. Intrinsic Distances
Actually, a connected complex Lie group is automatically commutative if it is an extension of T by (C*)k, see Iitaka [3]. Let el, ... , eN be the natural basis for V = C N . After a linear coordinate change for C N , the quasi lattice r has a Zbasis el, ... , ek, hi, ... , h 2m , where k + m = N with the following property. Let VI = C k and r l be, respectively, the subspace of V and the subgroup of r spanned by el, ... , ek so that (C*)k ;::: V In. Let V2 = V I VI, and b l , ... , b2m be the images of hI, ... ,h2m in V2 so that the group r l generated by bl , •.. ,blm is the lattice for the complex torus T, i.e., T = VlI r 2. (3.8.18) Lemma. (1) Every connected closed complex subgroup M' of a quasi tOntS M is a quasi tOntS, and the quotient group M I M' is also a quasi tOntS; (2) A quasi torus contains only countably many quasi subtori. Proof (1). Set G = (c*l. Each m = (ml, ... , mk) E Zk defines a character Xm of G, i.e., a homomorphism of G into C* by
Conversely, every character of G is of the form Xm. Thus, the set G* of characters of G is a commutative group isomorphic to Zk. By duality, there is a onetoone correspondence between the subgroups of G* and the closed subgroups of G, one being the annihilator of the other. In particular, every connected closed subgroup of G is isomorphic to (C*)I. Let M' be a connected closed complex subgroup of M. Set H = M' n G, where G = (C*)k as above. Then M'I H is a complex subgroup of MIG = T. To see that M'I H is compact, let K be the maximal compact subgroup of M. (Since M is commutative, there is only one maximal compact subgroup). Since M = GK, it follows that M' = H(K n M'). Let p: M + T = MIG be the projection. Since p(K) = T, we have p(K n M') = Mil H, which shows that M' I H is compact. This proves that M' is a quasi torus. It is now obvious that M I M' is also a quasi torus. (2). If M = V Iris a quasi torus and M' = V'I r' is a quasi subtorus, then V' c V and r' c r. Since there are only countably many subgroups r' of rand since V' is determined by r' (i.e., spanned by r'), there are at most countably many quasi subtori of M. 0 (3.8.19) Lemma. Let M be a quasi torus and X a closed complex subspace. Then the subspaces N(x), x E V, are all parallel translates of one quasi subtonts, say M',ofM. Proof By (1) of (3.8.18) M; = N(x) is a quasi subtorus of M, and by (2) of (3.8.18) M~ is independent of x since N(x) depends continuously on x. D
In order to explain the subgroup M', we need the following result of Bochner. (3.8.20) Lemma. An infinitesimal isometry of constant length on a Riemannian manifold with negative semidt:finite Ricci tensor S is parallel and satisfies S(v, v) =
o.
8 Subvarieties of Quasi Tori
121
Proof This follows from the formula: ~(g(v,
v» = g(Vv, Vv)  S(v, v),
which is proved, for example, in YanoBochner [I] and Kobayashi [8].
0
(3.8.21) Lemma. Let X C M and M' be as in (3.8.19). Then M' is the largest connected subgroup of M leaving X invariant. Proof Since every oneparameter subgroup of M' induces a vector field, say v, tangent to N(x), it leaves X invariant. Conversely, if v is any vector field on M generating a oneparameter subgroup of M leaving X invariant, then it defines an infinitesimal holomorphic isometry of constant length on X reg . By (3.8.20), v must satisfy S(v, v) = 0 on Xreg. By (3.8.12), the oneparameter group generated by v is in M'. 0
By (3.8.19) the relative nullity distribution N on U extends to a parallel distribution N on X reg ; in fact, the vector fields coming from the action of M' are all parallel vector fields on X reg . Hence, the distribution N.L orthogonal to N is also a parallel distribution on X reg . By the theorem of de Rham on the holonomy decomposition of a Riemannian manifold, a neighborhood of each point x E X reg is locally a Riemannian direct product of the foliation defined by N and the orthogonal foliation. In summary, we have (3.8.22) Theorem. Let X be an ndimensional closed complex subspace of a quasi torus M = C"+p I r with an invariant flat Klihler metric g, and M' be the largest quasi sub torus of M leaving X invariant. Let T; denote the subspace of the tangent space TxX spanned by all vector fields induced by the action of M'. Let S be the Ricci tensorofXreg, and let N x = (v E TxX; S(v, v) = O}. Let Mil = MIM' with projection :rr: M * Mil. Then (I) N x = T: on a dense open subset U of Xreg ; (2) X is a principal M'bundle over :rr(X) = XI M'; (3) The M'orbits, i.e., the fibers in the above fibering, are the flat factor in the local holonomy decomposition of X reg ; (4). With respect to the induced metric, the Ricci tensor of:rr(X) is negative definite on a dense open subset. (3.8.23) Corollary. Let X C M, M' and S be as above. Then M' = 0 if and only if the Ricci tensor S is negative definite on a dense open subset of X reg . Let M = Cn+p I r, and WI, •.• , w n + p the natural coordinate system of cn+p. Then using family of holomorphic nforms dw i , /\ ..• /\ dw in , i l < ... < in, on X, we obtain a meromorphic map (3.8.24) which sends x
E
U to a point with homogeneous coordinates
122
Chapter 3. Intrinsic Distances
We recall the definition of the Plucker imbedding P: G(n, n + p) + Pre. Given a point of G(n, n + p), i.e., an ndimensional vector subspace V of C n+ 1 , choose a basis VI, ••. , Vn for V and assign the line in /\ n Cn+p spanned by VI /\ .•• /\ V n , which is an element of the projective space P(/\ n Cn+P). More explicitly, express each Vj by its components Vj = (v), ... , v;,+p) and consider ("!P) numbers
ViI! I
vnin
which are called the Pliicker coordinates of the point V E G(n, n + p). Thus, P maps V to a point of PrC with homogeneous coordinates (vil .. ;n). From the definitions of ct> and P we obtain (3.8.25) Lemma. In terms of the Gauss map G: Xreg + G(n, n + p) and the Plucker imbedding P: G(n, n + p) + Pre, the map ct> can be written as follows: ct> = Po G.
(3.8.26) Corollary. Let X eM and M' be as in (3.8.22). {f M' = 0, then ct>: X + PrC is a meromorphic immersion. Proof Since P is an imbedding and since by (3.8.16) and (3.8.19) G is a meromorphic immersion, ct> = P 0 G is a meTomorphic immersion. D (3.8.27) Corollary. Let X C M = cn+ p / rand M' be as in (3.8.22). Let t*: X + M denote the imbedding. If M' = 0, then X has n + 1 holomorphic Iforms WI, ... , Wn+1 that are linear combinations of t*(dw l ), ... , l*(dw n + p ) such that n + I holomorphic n~forms
are linearly independent. Without loss of generality, we may assume that the coordinate system Wi, ... , w n + p for cn+p is such that WI
= l*(dw l ),
... ,Wn+1
= l*(dw n + I ).
This fact (in the torus case) is exactly what Ochiai [2] needed to complete his proof of Bloch conjecture; and it was proved by Kawamata [1]. Proof The meromorphic mapping ct>: X + PrC defined in (3.8.24) is a meromorphic immersion by (3.8.26). By taking a suitable projection p: PrC + PnC (which is a meromorphic map) we obtain an equidimensional meromorphic immersion po ct>: X + PnC. Now, our assertion is obvious from the definition of the map ct>.
D
8 Subvarieties of Quasi Tori
123
From (3.8.26) we recover the fol\owing algebraic geometric result (see Ueno [I; p.74]). For the concept of Kodaira dimension and that of general type, (for details, see Section 4 of Chapter 7). (3.8.28) Corollary. Let T be a complex torus, and X a closed complex subspace. Let T' be the largest connected subgroup ofT leaving X invariant. Then XIT' is of general type, i.e., the Kodaira dimension K(XIT') is equal to dimXIT'. Proof On account of (3.8.22), replacing T by TIT' and X by XIT' we may assume that T' = O. By (3.8.26), cP is a meromorphic immersion. Since cP is defined using only sections of Kx, X is of general type. 0
(3.8.29) Remarks. (1) Our proof shows that in order to prove K (X IT') = dimXIT' it is sufficient to consider KX/T'; there is no need to take higher tensor powers of K X / T '. (2) In (3.8.28) we have K(X) = K(XIT') = dimXIT'.
The second equality is nothing but (3.8.28). By (3.8.16), cP = Po G has rank equal to dimXIT' at x E U. Hence, K(X) :::: dimXIT'. The opposite inequality follows from Iitaka's theorem (Iitaka [1], see also Deno [1; p.74]). (3) From the fact that XIT' is of general type, hence Moishezon, it follows that X I T' is projective algebraic and the smallest complex subtorus of TIT' containing XIT' is an Abelian variety, (see Deno [1; p. 120]). Moreover, if T is an Abelian variety, then there exist finite unramified coverings T and T" of T and Til = TIT', respectively, such that T ;:: T' x T", and accordingly a finite unramified covering X of X splits as a direct product T' x XIT'. If a Hodge metric is used for T, then this splitting is compatible with the holonomy decomposition of de Rham explained above. (4) If X is a closed complex subspace of a quasi torus M, we can derive from (3.8.25) a statement similar to (3.8.28) on the logarithmic Kodaira dimension of XIM'. As an application of (3.8.25) we have a simple proof of the following theorem (see Ueno [1; p.1l7]). (3.8.30) Corollary. A closed complex subspace X of a complex torus T is a translate of a complex subtorus if and only if its geometric genus is 1, i.e., dim rcKx) = 1. Proof Ifthe geometric genus of X is 1, then the map
of the FubiniStudy metric of pNc. Now, we make use of compactness of M .. Since 1/1" and j*ct> are comparable, T(r, €p) and TIj/ (r, L 0 f) are ofthe same order. Hence, T(r, €p) ::: 10g(O(T(r, €p)))
+ O(logr)
II.
9 Theorem of BlochOchiai
131
But this contradicts the following fact: O(T(r, rp» = O(T",(r, t
(3.9.16)
0
f) :::: O(r2).
In order to prove (3.9.16), choose i, 1 ::::: i ::::: n and consider its power series expansion: fi(t) = ao
+ I,
such that fi is nonconstant,
+ alt + a2t2 + ....
Then
by (3.B.9)
establishing (3.9.16). The contradiction comes from the assumption that f (C) is not contained in any divisor of the form stated in the theorem. In order to complete the proof we shall now prove (*). The proof for the following lemma of Val iron [I] is taken from NoguchiOchiai [I, p.222]. (3.9.17). Lemma. If ao, ... ,ak are entire functions with ao i= meromorphicfunction on C satisfYing the polynomial equation
then
°
and if rp is a
k
T (r, rp) :::::
L T (r, aj) + constant. j=l
Proof Take tEe such that ao(t)
i= 0, fixing it temporarily. Setting
Ai = aiel),
we define a polynomial in w by pew) = Aow k
+ Al W k  I + ... + Ak.
Let th. th, ... , th be the roots of the equation pew) = 0: P(w) = Ao(w  f3l)(w  fJ2)'" (w  13k).
Since rp(t) is one of the roots, we assume that fJI = rp(t). We have I
2n
121f log IP(eie)lde =
log IAol
0
+
L 2nI 121f log le k
j=l
ilJ 
f3j Ide.
0
Making use of the following formula (which follows from Cauchy's formula): I
2n
121f log le ie 0
fJlde = log+
IfJl.
132
Chapter 3. Intrinsic Distances
we obtain 1 2n
127r log IP(e ili )ld6l
k
log IAol +
0
L log+ IPjl j=1
:::::
log IAol + log JI + IPd 2 log2.
On the other hand, since k
log IP(eie)1
= log I L
k
A j ei (k j )81 ::: Llog+
j=O
IAjl + log(k + I),
j=O
we have
Hence, k
log IAol + log+ IPII :::
L log+ IAj 1+ log2 + log(k + 1). j=O
Substituting Aj
= aj(t)
and PI
= cp(t) back into this, we obtain k
log lao(t)1 + log+ IcpU)1 ::: LlogJI + laj(t)12 + const. j=O
Integrating this inequality over the circle Dr yields I
2n
127r log lao(re ili )ld6l + mer, cp) ::: LT(r, k aj) + const. j=O
0
But (3.B.13) applied to the map f(t) = (ao(t), I) = (1, l/aoU»:C + PIC gives _I 2n
(27r log lao(re ili )ld6l = 10
N(r, ao, 0)
+ C.
In order for cp to satisfy the given polynomial equation, every pole of cp must be a zero of ao of at least the same order. Thus, N(r, cp) ::: N(r, ao, 0).
Hence, k
N(r, cp)
+ mer, cp) ::: LT(r, aj) + const. j=O
D (3.9.18). Corollary. (f each ai is a polynomial of entire functions 1j;[, ... , 1j;m and (rcp is a meromorphicfunction on C satisfying the polynomial equation in (3.9.17), then
9 Theorem of BlochOchiai
133
T(r, cp) ::::: C . max T(r, 1/Ij). J
Proof For an entire function 1/Ij, we have N (r, 1/Ij) = 0 since N (r, 1/Ij) in our notation counts poles of 1/Ij. Hence, T (r, 1/Ij) = m (r, 1/Ij)+ constant by the first main theorem. By (3.B.22) and (3.8.23) we have T(r, aj) ::::: where Co, (3.9.17).
CI, ... ,Cm
CI
T(r,
1/11)
+ ... + CmT(r, 1/Im) + Co,
are positive constants. Substitute this into the inequality in 0
We can reformulate (3.9.15) as follows: (3.9.19) Theorem. Let f be a holomorphic map ofC into a quasiabelian variety M. Then the Zariski closure of fCC) is a translate of a quasiabelian subvariety ofM. The original theorem of BlochOchiai is stated as follows: (3.9.20) Theorem. Let X be an ndimensional projective algebraic manifold with irreguialrity, i.e., dim HO(X, .Q 1), greater than n. Then every holomorphic map f: C ~ X has its image in a propoer closed algebraic subset of X.
Proof To derive this from (3.9.15), let a: X apply (3.9.15) to its image £l(X) C Ax.
~ Ax
be the Albanese map. Now 0
(3.9.21) Corollary. A nonsingular algebraic surface with irregularity> 2 is hyperbolic if and only if it has no curves of genus 0 or 1. It is desirable to strengthen the conclusion of (3.9.15) as follows:
Conjecture. Let X and WI, ... , Wn+1 be as in (3.9.15). Then X is hyperbolic modulo its singularity locus and a divisor of the form
n+1
L ajWl
1\ ... 1\
~
1\ ... 1\
Wn+1 = 0,
(ai, ... , an+l) =f. (0, ... ,0).
j=1
(3.9.22) Remarks. R. Kobayashi [I] gave a proof of (3.9.15) by establishing the second main theorem for holomorphic curves in abelian varieties. A proof of (3.9.19), which is more arithmetic in spirit, was given by McQuillan [I]. For a smooth algebraic surface X of irregularity 2, a theorem similar to (3.9.20) has been obtained by Grant [1] under the assumption that the Albanese variety of X is simple.
134
Chapter 3. Intrinsic Distances
10 Projective Spaces with Hyperplanes Deleted E. Borel [1] observed that the little Picard theorem may be restated in the following form. If two holomorphic functions f and g on C vanish nowhere and satisfY the identity
f + g ==
(3.10.1)
1,
then they are constant. In fact, they omit two values 0 and 1 and hence must be constant by the little Picard theorem. Conversely, if f is an entire function omitting two values 0 and 1, then f and g = 1  f satisfy the identity above. Now we state Borel's generalization of the little Picard theorem in the following three equivalent forms. (3.10.2) Theorem. (1) Assume that entirefonctions go, gl, ... , gN vanish nowhere on C and satisfY the identity go
+ gl + ... + gN == O.
Partition the index set I = {O, I, ... , N} into subsets la, I = U~=o la, putting two indices i and j in the same subset la if and only ~f gi I gj is constant. Then, for each a we have
(2) Assume that entire junctions fl' ... , fN vanish nowhere on C and satisfY the identity fl + ... + fN == 1. Then at least one of the fi 's is constant; (3) Under the same assumption as in (2), fl, ... , fN are linearly dependent (over C). We shall first show that these three statements are equivalent and then give applications of the theorem. The theorem will be proved in Appendix B of this chapter. For a more direct and shorter proof, see for example NoguchiOchiai [1].
Proof of equivalence. In order to derive (1) from (2), we set
so that ho + ... + hp == O. Since each ha is a constant multiple of a function gi with E la, it follows that either ha == 0 or ha vanishes nowehere. Assume that (1) does not hold. Without loss of generality, we may assume that h o, ... , hm, m ::=: 1, vanish nowhere and hm+1 = ... = hp == O. Set fa = hal ho, 1 :5 a :5 m, so that fl ... + fm == 1. Then one of the fa's, i.e., say fl = hI / ho, is constant by (2). Since ho is a constant multiple of a function gi with i E 10 and, similarly, hI is a i
10 Projective Spaces with Hyperplanes Deleted
135
constant multiple of a function gj with i E II. This means that gj / gi is constant, contradicting the definition of I a . We shall derive (2) from (3). Since fl,"" fN are linearly dependent, without loss of generality we may assume the following linear relation: cdl
+ ... + cNdN1 + iN == o.
By subtracting this identity from fl +···+fN1 +fN
==
I,
we obtain (l  cl)fl
+ ... + (l
 cNI)fN1
== l.
By applying (3) and the same argument to this identity, we obtain a shorter identity. Finally, we end up with the identity cfl == I. FinaIly, we derive (3) from (1). We set fo = 1 so that fo + fl ... + fN == 0, and apply (I) to this identity. Let 10 be the index set that contains O. If I = 10, then the functions fl, ... f N are all constant and hence linearly dependent. If 10 =1= I, then LiEfa fi == 0 for every O! such that Ia =1= 1o, thus yielding a nontrivial linear 0 relation. Now we shall give various applications of Borel's theorem. In treating n + 2 hyperplanes Ho, HI, ... , Hn+1 in general position in Pn, it is most convenient to consider Pn C as a hyperplane in Pn + 1C given by (3.10.3)
in terms of the homogeneous coordinate system wo, wo ... , w n+1 of pn+lc. By a linear change of coordinates, the defining equations for Ho, HI, ... , Hn+1 may be reduced to the following simple forms: (3.10.4)
Ho
=
{wo
= o},
... , Hn
=
{W"
= O},
H n+1 = {w n +1 = OJ.
This gives an equal status to all n + 2 hyperplanes. We partition the index set I = {O, I, ... , n + I} into two disjoint subsets J = Uo, ii, ... , i p } and K = {ko, k" ... , kq }, where p + q = n. The intersection L J = Hjo n ... n Hjp of p + I hyperplanes defines a linear subspace of dimension q  1 in Pn C, and the intersection L K = Hko n ... n Hkq of the remaining q + 1 hyperplanes defines a linear subspace of dimension p  1 in pne. Then LJ and LK span a unique hyperplane HJK of pne. If we set (3.10.5)
FJK
=
Lw
j
= 
jEJ
Lwk, kEK
then the defining equation for HJK is given by FJK =
O.
136
Chapter 3. Intrinsic Distances
If J contains only one index, say j, then L] = Hj and L K is empty. So we consider nontrivial partitions I = J U K such that both J and K contain at least two indices and call H1K a diagonal hyperplane of the configuration Ho, HI, ... , Hn+l • For a quadrangle, (i.e., four lines in general positions), Ho, HI, H2, H3 in the projective plane P2C, there are three diagonal lines, (see Figure i in Section 3). Every holomorphic map 1: C + cn+ 2  {O}, composed with the projection JT: C n+2  {OJ + Pn+1C induces a holomorphic map JT 0 1: C + Pn+1C. Conversely, every holomorphic map f: C + PII + I C is thus obtained. In fact, using an open cover {V,,} ofC, we lift f locally to a holomorphic map 1,,: v" + C n +2 _{0}. Then 1" = h" fJ l fJ on v" n V fJ , where h afJ : V" n VfJ + C* is holomorphic. Then {h afJ } defines an element of HI (C, 0*) = 0 so that hafJ = A~I )'fJ with ),,,: V" + C* holomorphic. Set 1 = 1"),,, = lfJlfJ· We sometimes denote a lift 1 of f by the same symbol f. Let 11+1
f: C
+
PnC 
U Hi C Pn+IC i=O
be a holomorphic map and 1: C + cn+ 2  (OJ a lift of f. Then n + 2 entire functions (J°(z), fl(z), ... , f"+I(Z» satisfying fO
1 is given by
+ fl + ... + 1"+1 == O.
r+
1 vanish Since f avoids the n + 2 hyperplanes given by (3.10A), fO, fl, ... , nowhere. Partition the index set I = {O, 1, ... , n, n + I} as in (1) of (3.l0.2). If all fi / f j are constant, then f is a constant map. Assume that f is not a constant map. Then the partition I = U~=ola is nontrivial. If 10' contains only one index, say i, then the identity fi = LiEf. fi == 0 obtained in (3.10.2) contradicts the assumption that f misses the hyperplane Hi. Hence, each I" contains at least two indices. Set J = 10 and K = U,,#o la. Then
Lfj==Lfk==O, jE]
kEK
and the diagonal hyperplane H] K defined by LjE] w j = 0 contains the image f(C). Thus, from the theorem of E. Borel we have derived the following theorem of A. Bloch [1] and H. Cartan [I]. (3.10.6) Theorem Let Ho, HI, ... , Hn+1 be n + 2 hyperplanes in general position in PIlC, n ::: 2. {( f: C + PIIC  U" Hex is a nonconstant holomorphic map, then its image lies in one of the diagonal hyperplanes. The following theorem which strengthens (3.10.6) is due to Dufresnoy [1, Theoreme XVI], see also Fujimoto [4, 5], Green [1], Lang [3]. For generalizations to mappings from Ck, k > 1, see Fujimoto [4, 5].
10 Projective Spaces with Hyperplanes Deleted
137
(3.10.7) Theorem.lfa holomorphic map I:C + PnC has its image in the complement of n + p hyperplanes HI, ... , H n+ p in general position. then this image is contained in a linear subspace of dimension :::: [n I p].
Proof Let Fi : C"+I + C be a linear form defining Hi. Let!: C + cn+1  to} be a lift of f. Set hi = Fi 0 j. Since I misses Hi, the entire function hi vanishes nowhere in C. We partition the index set I = {I, 2, ... , n + p} into a disjoint union I = U~=I 101 by putting two indices i and j into the same la if and only if hilh j is constant. We claim that the complement of any lao, i.e., UIi#o Iii, contains at most n elements. If our claim is false, we would obtain a set 1 with n + 2 indices by taking n + 1 of them from the complement of lao and one from lao' Then we partition 1 in the same way, i.e., 1 = U 1a with 1a = 1 n la, (dropping those 101 that are empty). As we have seen in the proof of (3.10.6) above, each (nonempty) 101 must contain at least two elements. But we know from the construction of 1 that lao contains exactly one index. This contradiction proves our claim that the complement of each lao contains at most n indices. Hence, each 10/, ex = 1, ... , q, contains at least p elements so that pq :::: n + p. Let I' be any subset of I = {I, ... , n + p} consisting of exactly n + 1 elements. Since HI,"" H n + p are in general position, the linear forms F i , i E I', are linearly independent. Write I' = U, I~, where I~ = I' n 101 , (Some I~ may be empty). Let ka be the cardinality of I~. Each I~, if nonempty, gives rise to a ka  1 linearly independent equations: Fi  ciFio = 0,
(io, i E I~, i
i= io),
where Ci = hi! hio' Hence, hI, ... , h n+ p satisfy at least kl  1 + linearly independent equations. But kl  I
+ ... + kq
 1= n
+I
q :::: n
+I
n+ p
p
=n
... + kq
 I
n
 . p
o
Hence, I(C) lies in a linear subspace of dimension:::: nip.
In W. W. Chen [I], (3.10.7) is derived from Borel's theorem by a matrix method. (3.10.8) Corollary. If a holomorphic map f: C + PIlC misses 2n hyperplanes in general position, then it is a constant map.
+ 1 or more
Using (3.10.8) and (3.6.13) we can strengthen (3.10.8) as follows. (3.10.9) Corollary. The complement of 2n + I or more hyperplanes in general position in PnC is complete hyperbolic and hyperbolically imbedded in pne.
Proof It suffices to consider the case of 2n + 1 hyperplanes in general position. Let HI,"" H 2n + 1 be hyperplanes in general position in PnC, and let X = PIlCl Hi. By (3.10.8), Condition (a) of (3.6.13) is satisfied. Consider a partition of indices {I, 2, ... , 2n + l} = I U 1, and let L/ = niE/ Hi. If I contains k
U;:i
138
Chapter 3. Intrinsic Distances
elements, L 1 is an n  k dimensional linear subspace. The intersections H j n L I, (j E J), are 2n + 1  k hyperplanes in general position in LI ~ PnkC. Since 2n+ Ik > 2(nk)+ i, by (3.10.8) all holomorphic maps ofC into L1UEJ Hj 1 are constant, which shows that Condition (b) of (3 .6.i3) is also satisfied. 0
U;:i
l Hi as (3.iO.l0) Remark. Dufresnoy [1] has shown that, for X = PnC in (3.10.9), Hol(D, X) is relatively compact in Hol(D, PnC). We shall show later that, in general, X is hyperbolically imbedded in Y if and only if Hol(D, X) is relatively compact in Hol(D, Y). So we may say that Dufresnoy had a result equivalent to (3.10.9).
In order to consider more general arrangements of hyperplanes, following Zaidenberg [2] we say that a set of hyperplanes HI, ... , HN in PnC is in hyperbolic configuration or satisfies condition (h) for short if each projective line 1 in PnC intersects Ui Hi in at least three points while it is in hyperbolicimbedding configuration or satisfies condition (hi) for short if each projective line I intersects UH;;zI1 Hi, (i.e.,the union of those Hi that do not contain I), in at least three points. (These conditions (hi) and (h) are called conditions (a) and (b) in Zaidenberg's paper). Ciearly, condition (hi) is stronger than condition (h). Condition (h) is violated if and only if there is a pair of points p, q E PnC such that each Hi passes through either p or q, but not both. Condition (hi) is vioiated if and only if there is a pair of points p, q E PIlC such that each 8; passes through p or q, possibly both. The following theorem is due to Snumitsyn [I]. (3.10.11) Theorem. I{hyperplanes HI, ... , HN in PnC are in hyperbolic configuration, then N :::: 2n + i. We start with the proof of (3.1 0.i2) Lemma. Let HI, ... , H k + l , (2 S k S n), be hyperplanes in PnC such that dim n7=1 Hi = n  k and n~=1 Hi C Hk+1. Then there is an index m, I :::: m S k, such that n;=1.i#m Hi ct. Hk+1. Proof Set E = n;=1 Hi and E j = n7=l.ih Hi. Assume that E j C H k+1 for all j = 1, ... , k. Then Hk+1 n (n~=1 Hi) contains EI+ I , •• " £k, (1 :::: I S k  i). The first condition in Lemma implies £'+1 ct. H'+I for all I = 0, ... , k  I, and hence Hk+1 n (n~=1 Hi) ct. H'+I. So
(ni=1 1
H'+1 n H k + 1 n
(ni=1 I
Hi) =1= Hk+1
n
8;)
and dim(Hk+l
n
(n '+1
i=l
Hence,
(n I
Hi» = dim(Hk+l
n
i=l
Hi»  1,
1= 0, ... , k  1.
10 Projective Spaces with Hyperplanes Deleted
n HI) + 1
dim Hk +1 = dim(Hk+1
n 1
139
2
dim(Hk+1 n (n H;»
+2=
...
+k =
dim(Hk+1
;=1 k
dim(Hk + 1 n (n HJ)
n E) + k
;=1
=
dim E
+k =n 
k
+ k = n.
This contradiction proves Lemma. Proof of (3 .1O.l1). Assuming that N ::s 2n we shall show that HI, ... , H N are not in hyperbolic configuration, that is, we shall produce a pair of points p, q E P"C such that each H; passes through exactly one of these two points. Obviously, it suffices to consider the case N = 2n. If Hi is non empty , it suffices to take p in Hi and q in Pn C Hi. Hi is empty. We renumber HI, ... , H2n in such Hence we shall assume that a way that
n
n
n
n
k
k = 1, ... , n.
for
dimnH; =nk i=1
n;'=1
We set Qo = Hi; Qo is a point. Renumbering the remaining hyperplanes Hn+l , ••• , H2n, we assume that ro
Qo E
nHi i=1
211
and
Qo'l.
U
Hj
,
(n
::s ro
< 2n).
j=rn+ 1
nJ:ro+1
We set Po = H j • Clearly, Qo 'I. Po and dim Po :::: ro  n :::: O. If Po ct U~~I Hi, then it suffices to take p E Po and q = Qo. Hence assume that Po C U;~I Hi. Then Po C Hio for some io, I ::s io ::s roo We consider two cases: (a) (b)
1.:::: jo ::s n, n + 1 .:::: jo ::s roo
= n, i.e., Po C Hn. We set = I. Then we renumber Hn, ... , H2n (including HI!) so that QI c Hi, (rl :::: nl), and QI ct Hi, (after this renumbering HI! is now one of Hr1 +1 , ••. , H2n)' We set PI = n~:r,+1 Hi. The subspace Po is, by definition, the intersection of the 2nro hyperplanes (called Hro +1 ' ••• , H21l earlier) that do not contain Qo. Since Qo C Q 1 C Hi for i = 1, ... , rl, these 2n  ro hyperplanes are now among 2n  rl hyperplanes H rl + I , .•• , H 2n . In addition, Po was contained also in another hyperplane (called Hn before the last renumbering) in this group. Hence, Po is given as an intersection of 2n  ro + 1 hyperplanes among Hrl +[, ... , H2n. Since PI can be written as the intersection of Po with hyperplanes among Hrl + I , ... , H2n that do not contain Po, we have In case (a) we renumber HI, ... , Hn so that jo
QI =
n7,:i Hi; dim QI
n;!:1
U;:rl+1
140
Chapter 3. Intrinsic Distances
dim PI
>
dim Po  [(2n  rl)  (2n  ro
>
ro  n  ro
+ rl + 1 = rl

+ 1)]
(n  1)
~
O.
Hence, PI # 0. In case (b) we construct PI and QI as follows. Since Qo = Hi = n~~1 Hi c Hio and since dim Qo = 0, Lemma implies that there is an index m, 1 ::: m ::: n, such that Hi n(n + 2), then every holomorphic map f: C ~ F(n, d) has its image in a hyperplane section. In fact, its image lies in a linear subspace of dimension:::: [nI2]. To see this, let PnC be the hyperplane in Pll + 1 C defined by
Then under the projection n: (wo, Wi, ... , WIl + I ) ~ «w 0)", (wl)d, ... , (wll+l)d),
the Fermat hypersurface F(n, d) is a covering space of PnC ramified over the hyperplanes Hi = {wi = O}, j = 0, ... , n+ 1. Assume that the image ofnof does not lie in a lowerdimensional linear subspace. If n 0 f does not interesect Hi, the truncated defect 8[n] (n 0 f, Hj ) = 1. If it intersects Hj , it intersects with multiplicity
10 Projective Spaces with Hypetplanes Deleted
at least d. By the argument in the proof of (3.B.46), we have
8[11](71: 0
f, Hj
)
~ 1
145
J. Hence,
which would imply d :::: n(n + 2). Hence, 71: 0 f satisfies a linear equation L;'!(~ aj w j = 0 in addition to the linear equation L~''!ri w j = 0, so that f = Uo, ... , .f"+ I) satisfies the two homogeneous equations of degree d: n+1
11+1
L:Uj)d j=O
= 0,
L:ajUj)d = j=O
o.
In order to prove the second assertion, we may assume that f j == 0 does not hold for any j. (For, if f j == 0 for some j, the problem is reduced to a lower dimensional case.) We claim that I Ji is constant for some pair (i, j) with i =j:. j. Without loss of generality, we may assume that all+1 = I. Then taking the difference of the two equations above, we have
t
(ao  I)UO)d
+ ... + (an
 l)(r)d =
o.
Replacing each fj by (aj  l)l/d fj yields a Fermat hypersurface of lower dimension. Inductively, we obtain an equation of the form aUO)d + bUI)d = 0, proving our claim. We partition the index set {O, I, ... , n + I} into II, ... , 1m under the equivalence i ~ j if and only if If j is constant. From each Ir we pick ir and set P = bj/i, for j E I r • Then
r
L:(fj)d = cr(t,)d,
where
jE~
and
Cr
=
L:b1, jE4
m
L: CrUi,)d = o. r=1
Unless all the C r are zero, the equation above defines a Fermat hypersurface of dimension:::: m  I. (Set gr = c;/d f i, so that L;~I (g,)d = 0). Then, by the claim above, fip I fi q is constant for some p =j:. q, which is impossible since ip E Ip and iq E I q . Hence, all Cr = o. Thus, L:uj)d = O. jE/,
In particular, every Ir contains at least two indices. The image of f lies in the linear subspace given by the family of hyperplanes wjbjw i , =0,
JEI" j=j:.i"
r=I, ... ,m.
(3.10.22) Example. We consider the complement of the Fermat hypersurface F(n  I,d) in pnc. If d > n(n + I), then every holomorphic map f:C + PnC  F(n  1, d) has its image in a hyperplane. In fact, its image lies in a linear subspace of dimension:::: [nI2].
146
Chapter 3. Intrinsic Distances
Let w o, ... , wI! be the homogeneous coordinate system for P"C. We consider the n+2 hyperplanes Hj = {w j = o}, j = 0, ... , n, and PnIC = {wo+ ... +w n = OJ. Under the projection
rr: (wo, ... , w n ) + «wo)'t, ... , (w")d), PIlC is a covering space of P"C ramified over these hyperplanes. Assuming that the image of rr 0 f does not lie in a lowerdimensional linear subspace, apply (3.B.42) to the map rr 0 f. Then we have (n
+ I) (I

~) + I
::: n
+ 1,
which implies d ::: n(n + 1). We omit the remainder of the proof, which is similar to that of (3.10.21), see Green [2] for details. In order to strengthen (3.10.6), we consider sequences of holomorphic maps from D into P"C missing n + 2 hyperplanes Ho, HI,.'" H Il +! in general position. We represent P"C by a hyperplane (3.10.3) in Pn+!C and Ho, HI, ... , Hn+1 by (3.10.4). Following KiernanKobayashi [2], we shall draw a geometric consequence (3.10.27) from the following theorem of Cartan [I; p. 58]. (3.10.23) Theorem. Given an infinite sequence li. = (j;o, f} . ... , fj"+I) of~ystems ofn + 2 nowherevanishing holomorphic junctions on the unit disc D satisjying the identity fOI. + I. + ... + /. 1 = 0,.
r'
r+
there is a subsequence, still denoted by fiJor which one of the following (a) or (b) holds. (a) The index set I = to, I, ... , n + I} is partitioned into two disjoint subsets J and K, J containing at least two indices and K possihly empty, such that (1) for i, j E i, the sequence Ul!f!L=1.2. converges to a nowherevanishing holomorphic function; (2) for j E i and k E K, the sequence {fi~ / L.=1.2 .... converges to zero;
i1
f!.)/.fiL.=1,2 .... converges to zero. There are two disjoint suhsets I' and I" of I = to, I, ... , n + I}, each
(3) jor j E J, the sequence {(LiE.! (b)
containing at least two indices and having partitions I' = ]'UK' and I" = i"UK" with Properties (1), (2) and (3) of case (a). We note that in case (a) Property (3) is a consequence of (2). However, in case (b), it is independent of (I) and (2). Set n+1 Z = PIlC, y = P"C
UH;. i=O
If K is empty so that J = I in case (a), then the subsequence {h} converges to a map in Hol(D, Y). If K is singleton, say {n+ I}, in case (a), then the subsequence (J;J converges to a map in Hol(D, H,,+I) C Hol(D, Z), In the remaining cases of
10 Projective Spaces with Hyperplanes Deleted
147
(a) and in case (b), we have a subset J of I containing at least two but' no more than nl indices such that, for each i E J, the subsequence {(LjEJ 11)lllli.=I.2... converges to zero. Hence, (3.1 0.24) Corollary. Given a sequence {fi.} of maps from D into Y as in the theorem above, there is a subsequence, also denoted by {hI, for which one of the following holds: (a) The subsequence {J;.I converges to a map in Hol(D, Z); (b) There exist a subset J of {O, 1, ... ,n + 1I containing at least two but no more than n  1 indices such that. for each i E J. the subsequence {(LjEJ I!)I1!.li.=l.2. converges to zero. In case (b) of the corollary above, we have the following convergence for the subsequence: (3.1 0.25)
(3.10.26) Corollary. Let Y and Z be as above. and let .1 be the union of diagonal hyperplanes. Given a sequence of maps J;. E Hol(D, Y). there is a subsequence. also denoted by {j;J. jor which one of the following holds: (a) The subsequence converges in Hol(D, Z); (b) Given a positive r < 1 and a neighborhood U of .1 in Z. there is an integer )'0 such that J;.(D r ) C U for), ~ A.o. In the terminology of Section 1 of Chapter 5, this simply says that Y is tautly imbedded modulo .1 in Z. As we shall show in (5.1.13), this implies the following geometric theorem. (3.10.27) Theorem. The complement ofn + 2 hyperplanes in general position in Pn C is hyperbolically imbedded in Pn C modulo the diagonal hyperplanes. Cartan [1] developed the idea in an earlier paper of Bloch [1] and strengthened the result of Bloch. (There were also some gaps in Bloch's argument). The main difference between their results is that Bloch had to restrict himself to sequences of holomorphic maps I with fixed 1(0) in the complement of .1 = U7'!~ Hi (or at least teO) staying in a compact subset of P"C  .1) whereas Cartan imposed no such condition. Thus, Bloch's result seems to give hyperbolicity modulo .1. As we observed in KiernanKobayashi [2], the full strength of (3.10.23) is yet to be geometrically explained. In Lang [3] the proof of Cartan's theorem (3.10.23) is reproduced. Cartan conjectured something stronger than (3.10.23). However, it has been disproved by Eremenko [2]. For n = 2, Cowen [3] obtained by a direct differential geometric method an explicit lower bound for the infinitesimal intrinsic metric Fy for the complement Y of five lines in general position in P2C, thus establishing hyperbolicity of Y. A similar result was obtained, independently, by Hall [1], who used a B1ochCartan type method.
Chapter 3. Intrinsic Distances
162
1 1 r
1
(3.B.9)
2n
dp
P
0
12IT v(re,8)d()  
ddcv =  1
2n
Dp
v(O).
0
We apply (3.B.9) to vet) = log Ilf(t)II. Since If(t)1 2 = L~=o 1J;(t)1 2 > 0, from (3.B.l) and (3.B.2) we obtain (3.B.10)
T(r, f) = 
I
2n
12IT log II f(re i8 )IId() log IIf(O)II. 0
Integrating
with respect to d() yields T(r, f)
(3.B.11)
+ log Ilf(O)1I
= mer,
f)
+ _1 2n
(2IT log Ifo(reiiJ)ld(). 10
In order to calculate the integral on the right, we recall Jensen's formula: (3.B.12) Theorem. h(t) = ct m + ..., c log Icl = 
1
2n
If h
is a meromorphic function on DR with Laurent expansion < R we have
# 0, at 0, then for r
12IT log Ih(reiO)ld() 
L
0
r mj log + Lnk log  r  m logr, laj I lilt I
where the aj # O's are zeros ofh with multiplicity mj in Dr while the f3k # 0 's are the poles of h with multiplicity nk in Dr. In particular, if 0 is not a zero or pole of h(t), then
log Ih(O)1 = 
1
2n
12IT log Ih(rei°)ld() 0
r r Lmj log + Lnk log. laj I lf3k I
We apply this formula to the holomorphic function fo(t). If c is the leading coefficient of the Taylor expansion of fo(t) = ctv(O,f) + ... at 0, then loglcl
= 1
2n
12IT 10glfo(reiiJ)ld() 0
r Lv(aj, f)log  v(O, f)logr. laj I
With (3.B.6) this can be rewritten as (3.B.13)
_I
2n
(2rr log Ifo(re i8 )ld() =
10
N(r, f)
+ log lei.
Now, (3.B.11) reads as follows: (3.B.14)
T(r, f) = mer, f)
+ N(r, f) + log Icl
log Ilf(O)II.
Since any hyperplane Ha can be considered as the hyperplane at infinity, we have (3.B.15)
T(r, f) = mer, f, a)
+ N(r,
f, a)
+ c.
B NevanlinnaCartan Theory
163
This is the first main theorem of Nevanlinna theory. In particular, we have (3.B.16)
T(r, f) 2: N(r, f, a)
+ c.
Suppose that f is defined on all of C. Since r (p) is monotone increasing, it follows that T(r) i> 00 as r i> 00. We define the defect 8(f, a) and the truncated defect olml(f, a) of a (or rather Ha) by setting o(f, a) = liminf(1 _ N(r, f, a»), . HOO T(r, f)
s
Then 0 S 8(f, a) S 8Im1 (f, a)
olml(f, a) = lim inf( 1 r+oo
l. We note that 8(f, a)
Nlml(r
fa»)
". T(r, f)
= 1 if fCC) n Ha = 0.
Consider the case n = l. A meromorphic function ({I on DR can be written as a quotient of two holomorphic functions with no common zeros: ((I(t) = fl (t)/fo(t).
Thus, it is considered as a holomorphic map f E Hol(D R , PI C) with f(t) (fo(t), fl (t». In this case, we often write T(r, ({I), mer, ({I) and N(r, ((I) for T(r, f), mer, f) and N(r, f). On the other hand, we write mer, ({I, 0) and N(r, ({I, 0) for mer, f, cd and N(r, f, CI). We note that N(r, ((I) counts zeros of fo(t), i.e., poles of ({I while N (r, ({I, 0) counts zeros of fl (t), i.e., zeros of ({I. As a special case of (3.B.7), we have (3.B.17)
mer, ({I)
=
_I
(7r 10g.)1 + 1({I(reili)l2d6l.
2n io
Using the notation log+ x = max{O, logx}, mer, ({I) is often given by _I 2n
{27r log+ 1({I(re iH )ld6l,
10
which is asymptotically the same as (3.B.17) since log+ Ixl S log.}1 + Ixl 2 S log+ Ixl + log2. We prefer to use (3.B.17). If ({I is holomorphic, then f(t) = (1, ({I(t» is in a reduced form, and (3.B.lO) becomes (3.B.18)
T(r, ({I)
=  1 127r log.}l + Icp(re ili )1 2d6l 2n
log.}l
0
+ 1({I(0)l2.
For each i, 1 S i S n, we consider the meromorphic function ({Ii (t) J;(t)/fo(t) as a holomorphic map into PIC. Since n
1 + l({Ii(t)1 2 S 1 +
L i=1
we have
n n
I({Ii (t)1 2 s
(I
i=1
+ I({Ii (t)1 2 ),
164
Chapter 3. Intrinsic Distances 1Z
(3.B.19)
mer, rpi) :::: mer, f) :::: L mer, rpi). i=1
In order to count n(p, rpi) we reduce (fo(t), fi(t» by factoring out the common zeros of fo(t) and fi(t), and then count the zeros of fo(t). Therefore we have n
(3.B.20)
N (r, rpi) :::: N (r, f) ::::
L N (r, rpi ). i=1
The first main theorem together with (3.B.19) and (3.B.20) yields n
(3.B.21)
T(r, rpi) :::: T(r, f)
+C
:::: LT(r, rpi)
+ C'.
i=1
For two meromorphic functions rp and l/I on DR, the inequality
implies (3.B.22)
mer, rpl/l) :::: mer, rp)
+ mer, l/I),
while the inequality
implies (3.B.23)
mer, rp
+ l/I) :::: mer, rp) + mer, l/I) + constant.
Now we state Nevanlinna's lemma on logarithmic derivative. For its proof, see Nevanlinna [1; pp.6364], NoguchiOchiai [1; pp.225227], or Lang [3; p. 172]. (3.B.24) Lemma. Let rp be a meromorphicfunction on C. Then
mer, rp'jrp)
=
O(1og+ T(r, rp)
II.
+ logr)
Here, II indicates that the inequality holds outside an exceptional set E of finite Lebesgue measure, i.e., dr < 00.
IE
(3.B.25) Corollary. Let rp be a meromorphic function on C. Then, for the pth derivative rp(p) we have (i) (ii)
T(r, rp(p» :::: (p
+ I)T(r, rp) + O(log+ T(r, rp) + logr)
mer, rp(P) jrp) ::::
o (1og+ T (r, rp) + log r),
II,
II,
p:::: 0;
p:::: I.
Proof The following proof by induction on p is from OchiaiNoguchi [1]. While (i) is trivial for p = 0, (ii) for p = 1 is the lemma above. Assume that (i) holds for p  1 and (ii) for p. Then
B NevanlinnaCartan Theory
mer, cp{P)
=
cp{p) ) m ( r, cp . ;p