Kunihiko Kodaira, Volume I: Collected Works 9781400869855

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Kunihiko KODAIRA:

Collected Works Vo!.1

Kunihiko KODAIRA:

Collected Works Vol. I

Iwanami Shoten, Publishers and Princeton University Press

1975

Copyright © 1975 by Princeton University Press Published by Princeton University Press, Princeton and London, and Iwanami Shoten, Publishers, Tokyo ALL RIGHTS RESERVED

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Preface Kunihiko Kodaira was born in Tokyo on March 16, 1915. He first studied mathematics and subsequently theoretical physics at the University of Tokyo, where his early years were influenced by S. Iyanaga. Until 1943, Kodaira was interested in several fields of mathematics including topology, Hilbert space, Haar measure, Lie groups, and almost periodic functions. Al­ though largely isolated from western countries during the war years, his mathematical development during the early forties and shortly thereafter was most influenced by the work of H. Weyl, in addition to that of Μ. H. Stone, J. v. Neumann, W. V. D. Hodge, A. Weil, and, last but not least, 0. Zariski, whose book "Algebraic Surfaces" provided a stimulating and fas­ cinating challenge that strongly influenced Kodaira's later work. At the University of Tokyo, Kodaira graduated first from the Department of Mathematics in 1938, from the Department of Physics in 1941, and from 1944 to 1951 was an associate professor there, where he obtained his Ph. D. in mathematics in 1949 by submitting a thesis on the subject of harmonic forms on Riemannian manifolds. In the fall of 1949, Kodaira became a mem­ ber of the Institute for Advanced Study in Princeton, N. J., U. S. A., at the invitation of H. Weyl. This turned out to be the start of an 18 year residence in the United States, during which time he collaborated with several mathe­ maticians, including W. L. Chow, F. Hirzebruch, L. Nirenberg, G. de Rham, and D. C. Spencer. With the latter, Kodaira collaborated on no less than twelve papers, a most fruitful partnership. Kodaira divided his time and serv­ ices between the Institute for Advanced Study and Princeton University until 1961 when he went to spend a year at Harvard University. From 1962 until 1965, he was Professor of Mathematics at the Johns Hopkins University, and from 1965, held the same position at Stanford University until 1967 when he returned to Japan to become a professor at the University of Tokyo. In 1954 Kodaira was co-recipient with J. P. Serre of the Fields Medals at the International Congress of Mathematicians held in Amsterdam that year. In 1957, he received the Japan Academy Prize and, in the same year, the Cultural Medal, the highest level of recognition in Japan for cultural achievement. Since 1965, he has been a member of the Japan Academy. Kodaira's earliest papers reflect the breadth of his interests spanning a wide range of subjects, all of which were to play a role in his later work. Several of these papers were co-authored with S. Iyanaga, M. Abe, and S. Kakutani, and one of them [10], on Haar measure was based on ideas of A.

Preface

Weil. Kodaira's subsequent work may be divided approximately, according to his interests and research, into several periods, though naturally there is some overlapping. Among the earliest works there are several, [13], [18], [19], concerned with the application of Hilbert space methods to differential equations, for example the method of orthogonal projection of H. Wey1, and with the use of elementary solutions to prove regularity theorems for strongly elliptic systems of partial differential equations. In Kodaira's devel­ opment of these ideas, which were destined to be central themes of Kodaira's later work throughout, he was influenced to some extent by the work of M. H. Stone, but above all, which was to become increasingly apparent, by the work of H. Weyl. The influence of the latter, as well as that of W. V. D. Hodge, is felt still more strongly in the; second period when Kodaira did his fundamental work on harmonic integrals. However, it should be kept in mind that due to the isolation of the war years, Kodaira was not able to keep up with everything that was going on in other countries and many of his early results were developed more or less independently of the work of others in the same area. The third period is characterized by the application of har­ monic integrals and of the theory of sheaves to algebraic geometry and to complex manifolds, and it was here that he received much encouragement and stimulation from A. Weil and that his collaboration with D. C. Spencer became fruitful. This collaboration enjoyed a fruitful culmination in the de­ velopment of deformation theory in the years that followed. A natural subdi­ vision of deformation theory is the theory of moduli of algebro-geometric structures, in particular the classification of special families of algebraic varieties and of complex manifolds depending on a continuously varying parameter in some manifold (real or complex). It was apparently at the beginning of the 1960's that Kodaira became absorbed in his next major project, the classification of compact, complex analytic surfaces. This topic was subsequently to become the inspiration for a large group of his students both in the United States and in Japan, many of whom have written highly interesting and original papers on this subject. It is this work and this project which still continue, both with Kodaira as with his students, because of which, at least, we are forced to dispense with the word "complete" in the title of these volumes. We wish now to review briefly some of the main ideas, themes, and me­ thods which play the dominant role here. We do this by referring in broad terms to some of the major works of Kodaira. In what follows, the numbers in brackets refer to the items in the bibliography. As indicated above, one of the dominant themes of Kodaira's work has been the use of orthogonal projection and the proof of a regularity theorem

Preface

for strongly elliptic systems of partial differential equations on manifolds, using the notion of local elementary solution. When the manifold is compact, all the characteristic spaces of solutions of the system have finite dimension. These ideas all appear in Kodaira's first major work [17], which was his the­ sis, though the main ideas were sketched in [12] which appeared some years earlier. In [17], the de Rham theory, giving the isomorphism between the real cohomology of a differentiable manifold and the cohomology of differential forms on it, is summarized, along with a statement of Hodge's theorem for Riemannian analytic manifolds which shows, with some restrictions, how to pick a harmonic differential form from each cohomology class. In this situa­ tion, one assumes that all manifolds are orientable. In principle, Hodge as­ sumed that the manifolds he dealt with were compact and proved his results starting from an idea he ascribes to H. Kneser. But there was a defect in his proof. The latter was repaired by H. Weyl, using the parametrix method sug­ gested by Kneser, and independently by Kodaira using Weyl's method of or­ thogonal projection. Kodaira developed his results for a countable, connected, but otherwise unrestricted orientable Riemannian manifold, using a modified elementary solution, and working within the context of measurable functions or differentiable forms (eventually, distributions or currents). Going beyond the improvement and extension of Hodge's theorem, Kodaira further es­ tablishes existence theorems for harmonic fields with singularities, and then proves a generalization of the classical Riemann-Roch theorem extended to harmonic fields with singularities, and applies this in particular to the case of compact Riemann surfaces to obtain the well-known form of the RiemannRoch theorem in that case. When the manifold is compact, of course the facts mentioned above about elliptic operators guarantee the finite-dimensionality of the spaces in question. The influence of H. Weyl is also quite noticeable in [18] and [19], where Kodaira applied Hilbert space methods to the eigenfunction expansions for ordinary differential equations, of order two [18], or more generally of any even order [19], with singularities, and gave an explicit formula for weight differentials. These papers, in fact, stem in large measure from Weyl's paper, in vol. 68 of the Mathematische Annalen, on ordinary differential equations with singularities. At this point, we should again stress that much of Kodaira's work during this period was actually independent of related work elsewhere because of a lack of communications. The generalized Riemann-Roch theorem for harmonic forms on Riemannian manifolds and its proof are clearly related to ideas in Weyl's book, "Die Idee der Riemannschen Flachen", both in connection with existence theorems for differential forms (of the first, second, and third kinds) on complex ma-

Preface

nifolds of dimension >1, and in connection with a generalization of Abel's theorem through the construction of multiplicative meromorphic functions corresponding to characters of the fundamental group of the manifold. The latter theme is prominent in [16], [23], and § 9 of [30], and is spelled out in careful detail in [21]. The theme of the Riemann-Roch theorem plays a major role in Kodaira's work. This and the related subject of arithmetic genera form the basis for most of the content of [28] and [30], as well as of a large part of the early joint work of Spencer and Kodaira. Severi has defined three prospectively different kinds of arithmetic genera for projective algebraic varieties, denot­ ing them by pa, Pa, and a, of which the first two are defined in connection with a polynomial formula for the dimensions of the linear spaces of multi­ ples of certain divisors, and of which the last is defined as the alternating sum of the integers gk(V), where Qk(V) denotes the dimension of the space of the &-ple holomorphic differentials on the algebraic variety V. Severi con­ jectured that Pa=Pa=a. In [28] and [30], by demonstrating certain functorial properties of a, Kodaira proved that Pa—a. Later, in [31],Spencer and Kodaira proved Pa=Pa using the theory of sheaves. Section 8 of [28] contains a re­ production of the proof of a result of Chow and Kodaira [25] to the effect that a compact Kahler surface with two algebraically independent meromorphic functions is an algebraic variety; this uses the Riemann-Roch theorem for Kahler surfaces, and Kodaira later [52] removed the restriction for the sur­ face to be Kahler by proving a version of the Riemann-Roch theorem for arbitrary compact, complex surfaces. The themes in [30] are similar, but it might be pointed out also that here Kodaira proves a conjecture of Weil to the effect that a finite, unramified covering of a non-singular, projective algebraic variety (over the complex numbers, of course)- is an algebraic va­ riety. One notices that in the course of his work, not only has Kodaira aimed to generalize his results from algebraic varieties to Kahler varieties and then to arbitrary compact complex manifolds, but has also attacked with notable success [37, 38] the related question of when a compact, complex manifold is algebraic. We shall discuss this result again later. The Riemann-Roch theorem also formed one of the early themes of the joint work of Spencer and Kodaira, where other concepts that proved fruitful were those of sheaf, of complex line bundle, and of differential form with coefficients in a complex line bundle. Here, in [32], relying on results about strongly elliptic systems of differential equations, Kodaira proved a Hodge­ like theorem for the Dolbeault (5-) cohomology; and Spencer and he toge­ ther proved results about divisor class groups on compact Kahler manifolds, leading up to new proofs of Igusa's duality theorems on Picard varieties and of the Lefschetz-Hodge results on the identification of the space of algebraic

Vlll

Preface

cycles with a certain subspace of the Hodge-Dolbeault decomposition of the cohomology space. At the same time, Kodaira took up an idea due earlier to Bochner to de­ monstrate sufficient conditions for the vanishing of certain Dolbeault coho­ mology groups. This, by way of showing the surjectivity of certain homomorphisms of spaces of holomorphic cross-sections of complex line bundles, proved the crucial factor in applying another idea frequently to make its ap­ pearance : that of projective imbeddings in terms of a basis of holomorphic cross-sections of a complex line bundle. This is the main point in [37, 38], where it is shown that a compact, complex Kahler variety with a Kahler metric in an integral cohomology class (briefly, a Hodge metric) is an alge­ braic variety. This most remarkable result effectively generalizes Lefschetz' theorem showing certain conditions (the Riemann period relations) on a pe­ riod matrix sufficient for a complex torus to be an Abelian variety. In fact, Kodaira's result also implies, for example, that if Γ is a discontinuous group acting freely on a bounded domain D in C" (complex w-space) such that the quotient space DjT is compact, then DITis a complex projective variety; and there are other applications as well which there is not space to mention here. The main point of the Kodaira vanishing theorem is, as Kodaira himself has said, that it is a disguised potential-theoretic proof for the existence of meromorphic functions. Perhaps the deepest subject of the joint collaboration of Spencer and Kodaira is that of the deformation of structures on manifolds. This is an ex­ tension of the notion of moduli for Riemann surfaces, but technically in many ways is far more difficult, raising questions that don't even occur in the theory of moduli of Riemann surfaces. This subject is a natural outgrowth of ideas and results from algebraic geometry. A foretaste of some of the con­ cepts and problems related to it may be found in [40] and [42]; in [40] one considers the family of characteristic divisors of a complete continuous sys­ tem on a variety and proves its completeness as a linear system. Later, in [47], Spencer and Kodaira proved a sharper form of this result in the con­ text of deformation theory. This proof reveals the true nature of the theorem of completeness of the characteristic systems of complete continuous sys­ tems. In [42] there are some initial developments concerning the technical matters at the heart of the main problem, notably the principle of upper semicontinuity and the fact that a Kahler almost complex structure is integrable. Spencer and Kodaira's first major work on deformation theory is [43], where the authors, taking as their starting point a paper of Frohlicher and Nijenhuis, approach in comprehensive generality the topic of variation of complex structure on differentiate manifolds. Although the general notion of a space of moduli is more or less intuitively clear, as a kind of parameter

Preface

space for some class of algebro-geometric structures, the question of what, precisely, shall be called moduli and the actual existence of a "reasonable" moduli space are technically difficult problems. In many areas of mathema­ tics, the description or classification of the types of superstructures that may be superimposed on a given underlying structure is given in terms of some first cohomology group or set. So, here, too, the problem of classifying com­ plex structures on a fixed C~-manifold can be formulated in terms of a certain first cohomology group. In the present instance, the first cohomology group one considers is that of the given manifold V, with a given complex structure, with coefficients in the sheaf of germs of holomorphic vector fields on V. Suppose we are given a differentiate fiber space (U, π, Μ), where π is a differentiate map of U onto M, such that all the fibers are complex manifolds, such that the map π has maximal rank everywhere, and such that Zr 1 (O) =V for some point 0 of M. Then there is a natural linear mapping p 0 : To-^H1CV1 Θ), where T0 is the tangent space to M at 0. The properties of this fiber space, viewed as a family of variations of complex structure on the underlying C°°-manifold V, may be characterized at 0 in terms of properties of the mapping p0. If the mapping p0 is injective, then the family is "effective", while if the mapping is surjective, then the family will be "complete", or, in some sense, maximal at 0. If it is both, then M may be viewed, locally, as the space of moduli of V. But this characterization is only approximate and many details must be verified to justify fully the designation of HX(V, Θ) as the tangent space to the space of moduli of V at the point 0. This paper has the quality of a pioneering work in the subject, and besides laying the theoreti­ cal groundwork for a variation of complex structures, or "moduli", contains many examples to illustrate its applicability and suggests many open ques­ tions for future investigation. Two questions that remain open at the close of this paper are those of the general existence of M and of the possibility of supplying the moduli space M, if it exists, with a suitable complex structure of its own, which one would of course anticipate in the case of algebraic varieties with moduli lying on some parameter variety. These problems are attacked in [44], [45], and [46], In the first of these, produced in collaboration with L. Nirenberg as well as with Spencer, a theorem for the existence of variation of complex structure on V is proved, in [45] it is proved that if p 0 is surjective, then the family is complete, and in [46], the question of the actual existence of a complex structure on M is treated. Another way in which one may supply a given differentiate manifold with an additional structure is to be found in the subject of pseudo-group structures which occur when we restrict the types of transformations be­ tween systems of local coordinates on the manifold. In [49] Kodaira extended the theory of deformations of complex structures to a class of complex pseu-

Preface

do-group structures. This extension delineated certain features of a general pattern in the theory whose outlines were much less discernible in the special case of complex structures. In [50], Spencer and Kodaira developed a theory of deformations of both the real and the complex multifoliate struc­ tures which fits into the general pattern. In the complex case, one may again apply the theory of strongly elliptic systems of partial differential equations and recover results analogous to those in the variation of complex structures; however, in the case of real multifoliate structures such tools are unavailable and the results in this case are less complete. Volume III of this set is primarily on surfaces, i. e., compact, complex analytic surfaces, their structure and classification. This is a natural sequel to the papers on variation of complex structure, as well as a further develop­ ment of some of the themes of the earlier papers on algebraic geometry. The concentration on the case of complex dimension »=2 makes possible the illumination, by specific examples, of some of the ideas on moduli as well as some highly interesting applications of general theorems, e. g., the RiemannRoch theorem for surfaces. And while far more complicated than the subject of Riemann surfaces (complex dimension w=l), the possibilities are still suf­ ficiently limited to achieve some meaningful classification and clarification of details. As pointed out by Hermann Weyl in his address awarding the Fields Medals in 1954, this restriction to lower dimension has proved a wise limitation; it affords the hope of mastering some outline of a theory that might hint at things to come in higher dimensions, at the same time giving us some grasp of a theory of beauty in its own right. Here we can give no more than the briefest hint of the nature of the contents of some of the papers in this field. Let S be a compact, complex analytic surface. The field M(S) of meromorphic functions on S is a finite algebraic extension of a field of transcendence degree r over the complex numbers. We have r=0, 1, or 2. If r=2, then by the results of Chow-Kodaira [25] and of Kodaira, S is an algebraic surface; the sharper results are proved in [60], using a general form of the Riemann-Roch theorem established by M. F. Atiyah and I. M. Singef. If r=l, then there exists a non-singular algebraic curve Δ and a reg­ ular analytic map Φ of S onto Δ such that for all but a finite number of points of Δ, the inverse image :' " ..v, .Wj J • Dann ist es kIar, dass diese Homomorphismen die Bedingung 1) im § 1 erfiiI1en. Es gilt nun auch die Bedingung 2). Denn aus ZjC Xj,

Grc L: Xi. sSr-l

und C;-1

C

L; X h

sSr-2

folgt

2J L:j Z,:;;I = - Cr k J

~ G>:-1 j

J

Grc

2J Xi"

sS r-l

~ j

C;-1

C •

L:

XI;.

s; r-2

4) P. Alexandroff: a. a. O. 2). 5) iff ist die abgesehlossene Hiille von xi, d. i. die Vereinigungsmenge aller Seite von :rr. Diese Bezeichnung ist weiterhin beibehalten; wir bezeichnen niimlich aas Urbild von xi' immer mit xi. 6) Wegen der Bezeichnung, siehe die vorhergehende Fussnote 5). C 2)

9

No.2.] tiber deli Allgemeinen Zellenbl·gritf und die Zellenzerspaltung der Komplexe. 51

Daraus folgert man leicht

'" in X;.;-l mod X;.;-I. Andererseits, wenn man die Homologieklasse von Zi tl mit ti'+! bezeichnet, ~ Zj;;1 C 2:; ti+1[xi+1: xj] [x;: X,;-I]. j

j

Also folgt ~

[xi+! : xi] [xi: xZ- 1] = 0 .

j

Daher konnen wir nach ~ 1 die Bettischen Gruppen Br(D) von D in bezug auf .0i und [xi: x;-t] definieren. Dann gilt der folgende Satz. 3. Beweis des Satzes. Wir brauchen folgende Bezeichnungen: Kr = ~ X1. Zr bedeutet * s:S.r immer einen Relativzyklus von Xi mod Xi. C1=CZ (Kr) bedeutet dass Ct - C2 c: Kr ist. Dann beweisen wir der Reihe nach folgende Tatsachen a}-e).

a) Fur r =\= s ist jeder Relativzyklus zr aus KS mod KH homolog Null in K" mod K8-I. 1st insbesondere r> a, so wt jeder Zyklus zr aus KS homolog Null in K8. Beweis. Zr lasst sich darstellen in der Form: zr=2:; Ci (Ks-l), ,. Ci c: Xi. Dann ist Ci ein Relativzyklus von Xi mod Xi, ist also -- 0 nach unserer Definition der Zelle. Also ist zr auch '" 0 mod Kg-I. b) Jeder Zyklus Zr ist einem Zyklus aus Kr homolog. Beweis. Nach a) ist ein Zyklus aus K8 einem Zyklus aus Ks- I homolog, wenn s =\= r ist. Daraus folgt die Behauptung. c) Jeder Zyklus zr aus Kr Uisat sich folgendermasaen darstellen:

zr

Zr mit tr bezeichnet,

Wenn man die Homologieklasse von

ein Zyklus von D. Beweis. Sei Z[::=::E Z[;-1 (Kr-2) ,

80

ist

Z[,;-1 C X;-I.

i

Dann ist "Z>:;-I= ~ J.) i

"0 ~

L...J~ k~j i

Z>:-1 tk

+ ()"; zr-zr). ~

1-

Da ::E Zr - zr C Kr-l ist, folgt hieraus dass ::E Z[j-l '" 0 in Xi ,. ; mod Xi-I. Namlich ist 2:; ti[xi: xi- I 1= 0, also ist ::E ti xi ein Zyklus. i

Zu jedem K von der Form: d)

Z]Jlclu.~

zr = ::E tr xi

1~n

D gibt

c.~

z;-c ti. ( 2)

10

1

einen ZyklnN Z r aus

K.

[Vol. 14.

KODAlRA.

Beweis. Sci Zi c ti, und Z[=2.:; Z[j-l (Kr-2). j ~ Dann ist, da zr ein Zyklus ist, 2.:; Zij-l -- 0 ill Xi- l mod Xj-l. Also gilt ~ Zi;l=Cj- (Kr-2) und Gj' c XI-I. SO ist (~Zi - ~ C.il=O i

Nach a) ist (2.:; Zi - ~ Cj)'''''' 0 in Kr-l, es gibt also einen Komplex Ct' c K,--l mit C~::; Zi- ~Cn·=C". Wir setzen zr=~ Zr~ Gj'-cr. So ist zr ein Zyklus und Z"=~ Zi (Kr-l). e) Es sei zr=~ Zi (Kr-l) und Z[ c fi. Zr ist .- 0 in K dann und nnr dann, wenn zr=~ t[xi '" 0 in D Beweis. i) Sei zr=Cr+l. Da cr+1 ein Relativzyklus mod Kr ist, erhiilt man durch Anwendung von a) einen Komplex 1'1'1-1 aus Kr+1 mit cr+1 = Q'-12+ /,,-+1. r1'+l lasst sich so darstellen: 1'1'+1=2.:; Z;;+l (K,--2).

wt.

(Kr).

Dann ist

Sei Z;;+l=2.:; Zki (Kr-l). So erhii.lt man ~ (Zr - ~ Zr,i)==(rr+l- ~ Zk+ 1). i

(K1'-l).

i

Daraus folgt

Zr - ~ Zi;i ...... 0 in

k

,.

Xi mod Xi.

Bezeichnet man also die" Homologieklasse von Zr,+l mit t;;+l, so bekommt man tt=~t;;+l[x;;+l:xi], namlich Z1'=(~tk+lx;;+l)·. ii)

" (~ t;;+lX;;+l)..

Es sei zr =

Man wahle einen Zyklus Z;;+1 aus t;;+1

und setze i;;+l=~ Zki (Kr-l), Z;;i c Xi. Dann ist ~ Zki ..... Zi in Xi ,. i . k mod Xi, d. h. es gibt Ci+ l c Xi mit Zi=Ci+l+ ~ Z;;i (Kr-l). Hierk

aus erhalt man Z1'=(~ Z;;+l+ ~ C[+l). (Kr-l). Nach a) ist aber der Zyklus zr - (~Z;;+l+:8 Ci+ 1) ...... O. So ist auch Zr '" O. Unser Satz folgt nun aus c), d) und e). Bemerkung I. Durch Anwendung des projektivcn Komplexes, konnen wir im Fall der kompakten Koeffizientenbereiche die obigen Uberlegungen wortlich auf die Kompakten iibertragen, genau wie bei P. Alexandroff a. a. O. 2). Bemerkung II. Man wahle als Koeffizientenbereich die additive Gruppe der reel1en Zahlen mod 1 und als Xi die Homologiesimplexe. Dann sind die stetigen Homomorphismen [xi: a:jl] durch ganzen Zahlen darstellbar. In diesem Spezialfall kann also die Tuckersche Theorie direkt angewandt werden.

C 2)

11

45.

Eine Bemerkung zur Dimensionstheorie. Von Kunihiko KODAIRA. Mathematical Institute, Tokyo Imperial University. (Comm. by T. TAKAGI, M.I.A., June 12, 1939.)

Man definiert die Dimensionszahl von einem Kompaktum F bekanntlich auf zweierlei Weise: Nach Lebesgue-Brouwer wird sie namlich als die kleinste nicht-negative ganze Zahl n definiert, sodass F beliebig feine Uberdeckung mit der Ordnung n+ 1 besitzt. Die Dimension von F in diesem Sinne wollen wir mit dim F bezeichnen. Andererseits heisst F nach Urysohn-Menger· Mehstens n-dimensional, wenn jeder Punkt p von F in einer beliebig kleinen Umgebung U(p) enthalten ist, deren Rand U(p)- U(p) Mehstens (n-l)-dimensional ist; die Dimension von F wird· hier somit rekursiv definiert. indem als Dimension der leereD Menge die Zahl - 1 zugesehrieben wird. Die beiden Definitionen sind bekanntlieh aquivalent falls dim FF2) ... F").

Da

p~j.)

(j =1, 2, .. , r) den Punkt p enthiilt, so gibt es fur jedes j mindestens

ein Flhi mit

1) Siehe z. B. Menger: Dimensionstheorie, S. 155. FI'cudenthal, Entwicklungen von Illiumen und ihren Gruppen. (Compositio Muth. 11 (1937), So 2282:H). 2) Vgl. Alexanuroff, Dimensionstheorie. (Math. Ann. Bu. 106, S. 161.)

( 3)

12

No.6.]

175

Eine Bemerkung zur Dimensionstheorie.

Andererseits gilt auch F,,0k) 3

P,

k=1,2, ... ,).J.

Diese r+).J Mengen Fp h ,), ... , Ff rhr ), F,,0!), "', Floy) sind aber offenbar aBe voneinander verschieden. Also muss r +).J kleiner als die Ordnung von Ul sein: d. h ).J < n-r+l. Da die Uberdeckung {Pkl} mit wachsendem l beliebig fein wird, fo\gt hieraus unsere Behauptung: dim F:')Fj,) ... ?:/ < n - r . Nun sei p ein beliebig gewahlter Punkt von F.

Um(p)=F-

~ p,t:)~p

Setzen wir

F/:t),

dann ist offensichtlich mithin

Hieraus folgt, nach dem Hilfssatz und dem bekannten Summensatzl> dim (Um(p) - Um(p»)=dim (~~ F,!)F,t:»

< n-l.

Die Umgebung Um(p) kann nun mit geeigneter Wahl von m beliebig klein gemacht werden. Damit ist also der folgende Satz bewiesen: Satz 1. Es sei dim F=n. Dann besitzt jeder Punkt p von F

cine belicbig klcine Umgebung U(p) mit der Eigcnschaft: dim (U(p) - U(p»)

< n-l.

Umgekehrt gilt nun: Satz 2. Es sei dim F < + 00.

Wenn jeder Punkt p von F cine beliebig kleine Umgebung U(p) mit dcr Eigenschaft: dim (U(p)- U(p»)

hesitzt, so ist dim F

< n-l

< n.

Zum Beweis setzen wir dim F=d und nehmen an, dass d> n sei. Da dim F bekanntlich mit der Homologie-dimension mod. 1 iibereinstimmt, gibt es einen (d-l)-dimensionalen wesentlich berandeten Zyklus Zd-l in F. Es gibt also ein Teilkompaktum Z von F, sodass Zd-l

C

Z.

in F.

aber 1) Vol. Alexandroff, a. a. O. S. 216. C 3)

13

176

K. KODAIRA.

[Vol. 15,

Es sei (/) (=> Z) ein minimalel' Homologie-tl'agel' von Z'l-J. Z,l-J soIl also co 0 in (/1, abel' 01J 0 in jedem echten Teilkompaktum von f/J sein. Es sei Cd ein Komplex von f/J, del' Zd-l bel'andet:

p sei nun ein Punkt aus (/1 - Z. Wil' wahlen eine Umgebung U = U(p) von p so, dass dim (U - U) ~ n -1 und [j. Z = 0 wil'd. Das oben eingefiihl'te Cd lasst sich dann offen bar folgendermassen darstellen :

Crcf/J-U.

Hieraus folgt

und daher, da Cgc Uf/J und Zd-l_Ctcf/J-U ist, C'l=Zd-l_Ctc Uf/J-Uc U-U.

C~ ist nun ein (d-1)-dimensionaler Zyklus. Da aber nach unserer Voraussetzung dim (U - U) < n -1 < d -1 , ist, folgt in Uf/J- U;

also muss es einen Komplex Dd aus Uf/J - U so geben, dass C~=Zd-I-Ct=iJd

gilt.

odeI' Z,I-l=(ct+Dd)'

Wegen

bekommt man hieraus Zd-l co 0

in f/J - U,

entgegen der Minimalitat von f/J. Also muss dim F' < n -1 sein, w. z. b. w. Die A.quivalenz d~; beiden Definitionen del' Dimension von F folgt nun aus unsern beiden Satzen 1, 2.

( 3)

14

52.

On some Fundamental Theorems in the Theory of Operators in Hilbert Space. By Kunihiko KODAIRA. Mathematical Institute, Tokyo Imperial University. (Comm. by T. TAKAGI,

M.I.A.,

July 12, 1939.)

The purpose of this note is to point out that some fundamental theorems in the theory of operators in Hilbert space, viz. I. the possibility of the canonical decomposition of closed linear operators with an everywhere dense domain;1) II. the possibility of the integral representation of normal and especially self adjoint operators2) are easily deducible from a certain lemma contained in the proofs of the lemmas 9. 1. 2, 9. 1. 3, 9. 1. 4 in RY We state this lemma in § 1, and then prove these fundamental theorems in the following two paragraphs. The sole knowledge presupposed to our demonstrations is that of operational calculus for bounded operators exposed for example in E,2) Anhang III. Our proofs need not be altered, if we have to consider the generalized complex Euclidean space instead of the Hilbert space, as we make no use of the separability or of the infinite dimensionality of the space. §§ 1, 2 hold good also for the generalized real Euclidean space, so that our proof for I has somewhat larger validity than von Neumann's given in A.3) 1. Lemma. Let III be a linear, everywhere dense set in Hilbert space (or in generalized Euclidean space) .~: [~n=,j), and Q(f, g) be a complex-valued (or real-valued, if one has to consider the real Euclidean space) function of f, gEm, having the properties of inner product in m. We suppose that m is Q-complete, i. e. complete with respect to the metric determined by Q(f,!). If Q(f,f) > (f,f) for all fE m, then there exists a unique operator B in .j) mapping ,\j in mso that (1)

Q(Bf, g)=(f, g) for fE.\j,

gEm.

B has the following properties: 1) B is a bounded Hermitian operator and 0 < (Bf, f) < (f, f) if f ~ 0 ; so that we can form operators as V B, -11- B in the sense of F. Riesz. 2) ~{ = Range 1/ B 3) Q(I/BJ, ''/Bg)=(f, g) for all f, gE.\;!. 1) J. v. Neumann: Uber adjungi8rte Funktionaloperatoren, Ann. of Math. 33 (quoted as A). See also F. J. Murray and J. v. Neumann: On rings of operators, Ann. of Math. 37 (quoted as R) especially p. 141-142. As to the notation and terminology we follow the usage in R 2) J. v. Neumann: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (quoted as E.) and Zur Algebra der Funktionaloperatoren, ibid. 3) We wish to remark, by the way, that we could verify that all results of R hold with 'slight modifications for complex or real genet'alized Euclidean space. We reserve it for later publications.

C 4)

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We have not to dwell upon the proof of this Lemma, as it is fully contained in the proofs of above quoted lemmas in R. 4) The following consequences are of importance for us. 2) and 3) imply that -IB maps Sj isometrically on &, (when the latter is metrized with Q). So we can form the inverse V B-1 of liB. Let us now consider the operator (2)

H=vl-B VB-I.

H has evidently the domain &.

We will show that it is self adjoint and semi-definite. Remark, to the purpose, that each f € ~ can be put in the form f=l/Bfl' If g*=Hg, we have therefore (Hf, g)=(VI-Bfbg)= (fh l/l-B vB 11 B-lg) = (fh -IBHg) = (vBfb g*)=(f, g*) for allf€~. Conversely, suppose (Hf, g)=(f, g*) for all fe &, and some g, g*. Put f=vBfh then we have (VI-Bfb g) = (vBfh g*), or (fh VI-Bg) =(ji,l/Bg*) for all fi€Sj. Therefore vI-Bg=l/Bg*, (l-B)g= vl-B vBg* or g=v1i(J/Bg+1/1-Bg*). So we have g€~ by 2), and Hg=vI-B (-vBg+-II-Bg*)=v:B-vl-Bg+(l-B) g*=g*. H is thus self adjoint. H is positive semi-definite, as (Hf, f) = (1/1 - 13fl vBfl)=(l/B VI-Bfb};) > 0 by 1). 2. Canonical decomposition. Let A be a linear closed operator with an everywhere dense domain &. Let us put, after K. Friedrichs5) (3)

Q(f, g) =(Af, Ag)+(f, g).

One verifies immediately that this Q(f, g) satisfies the conditions of the lemma. (In particular, ~ is Q-complete, because A is closed.) We may therefore write in virtue of the lemma (All B};, AvB!lI)+(YB./i, -I Bg1 )=(};, gl)

for all li, gl € Sj.

Now we have (fb gl) - ('1/}3ft. V Bg1) = (ft. gl) - (Bft. gl) = ((1- B)fr. gl) =(1/1- Bfb vI- Bgl), so that we obtain (A-IBft. AI Bg1)=(1/I- Bfl. -11- BgI ) and specially I Afll = I Hf II, if we use the notation (2) and put 1/ Bli =f. In putting (4)

A=WH,

we define therefore a partially isometric operator W with the initial set [Range H]=Sj-(f; Hf=O)=Sj-(f; Af= O}= [Range A*] and the final set [Range A]=Sj-(f; A*f=O). As H is self adjoint, one derives easily from (4) A*=HW*. 4) space, Cf. F. 5)

The proofs in R. are valid also for complex or real generalized Euclidean for the Riesz's theorem holds, as is well-known, also for the non-separable case. ReJlich: Spektraltheorie in nichtseparablen Riiumen, Math. Ann. 110. K. Friedirchs: Spektraltheorie halbbeschriinkter Operatoren, Math. Ann. 109. C 0;

E()..) being defined by (5).

If, moreover, A is positive semi-definite, we must have E- = 0, A=H. As H is uniquely determined by A by (3), (1), (2), we see

that the canonical decomposition of a linear closed operator is possible in a unique manner.

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[Vol. 16,

31.

On the Theory of Almost Periodic Functions in a Group.

By Sh6kichi IYANAGA and Kunihiko KODAIRA. Mathematical Institute, Tokyo Imperial University. (Comm. by T. TAKAGI, M.I.A., April 12, 1940.)

The theory of almost periodic functions (a. p. f.) in a group, due originally to J. von Neumann,]) has been simplified by W. Maak. 2) The last author starts from a modified definition of a. p. f., and obtains a shorter proof of the existence of the mean value. His proof necessitates, however, a certain combinatorial lemma, .which is indeed very interesting in itself, but somewhat alien to the theory of a. p. f. We propose here another way of founding this theory, which seems to us also simple and natural. 1. We begin with some general remarks on metric spaces. An abstract space ffi with" points" x, y, Z, ... is called a metric space, if there is defined a "metric," i. e. a real-valued function p(x, y) for x, y E ffi satisfying the following conditions: 1) p(x, y) > 0, p(x, x) = 0, 2) p(x, y} = p(y, x), 3) p(x, y) + p(y, z) > p(x, z}. The separation axiom: " p(x, y)=O implies x=y" will not be' postulated here. 3 ) Such spaces are topological spaces, i. e, they satisfy the first three Hausdorff axioms ; it is therefore clear what are to be understood under the terms such as: open or closed sets in ffi, the continuity of a mapping of ffi in another such space ffil etc. One can, moreover, speak of the equi-continuity of a family of mappings and also the uniform continuity of a mappingY Theorems such as the following are evidently true thereby: If f maps ffi continuously in )H', and f' maps ffi' in ffi" in the same way, then f" = f'f maps ffi also continuously in ffil!. If, moreover, f and l' are uniformly continuous,so is also f". (Transitivity of continuity and uniform continuity.) We can speak also of the diameter of a set & in ffi, e-covering, e-net, the boundedness and the totally-boundedness of &. If we have to do with several metrics of a fixed space ffi, we will say also that a metric p is bounded or totally bounded (t. b.), when the entire space ffi has this property with respect to p. For two metrics p, PI of ffi we will write p < Ph if p(x, y) < PtCx, y} for all x, Y E ffi. The following lemmas are all fairly obvious: 1) J. von Neumann: Almost periodic functions in a group I. Trans. Am. math. Soc. Vol. 36 (1934). 2) W. Maak: Eine neue Definition der fast periodischen Funktionen. Abh. math. Sem. d. Hans. Universitiit. 11. Bd. (1936). 3) Such p(x, y) is often called "quasi-metric" in opposition to the usual "metric" Eatisfying the separation axiom. We prefer to call p a "metric" in the general case, and "separated metric" when it satisfies the separation axiom. 4) In this sense m is a "uniform space"; cf. Andre Wei!; Espaces a structure uniforme. Act. sc. et indo 551 (1937). A. Weil postulates, however, the separation axiom. C 5)

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No.4.]

On the Theory of Almost Periodic Functions in a Group.

137

Lemma 1. Totally bounded me tries are bounded. If p < Ph and PI is t. b., then P is also t. b. Let fit be another space and pi, p~ metrics of fit. Suppose P < PI, P; < p'. If f is a uniformly continuous mapping of fi in fi/, when these spaces are metrized with p, p' resp. then f is also uniformly continuous, when they are metrized with Ph p~. Lemma 2. If Ph "', Pr are t. b. metrics of fi, so is also PI + .. , + Pr' There exist namely finite €-coverings of fi corresponding to Pi, i= 1, ... , r. The superposed covering of these coverings constitutes clearly an r.:-covering of fi for the metric PI + ... + Pr' Lemma 3. The uniform limit of t. b. metrics is also a t. b. metric; i. e. if t. b. metrics Pv(x, y) tend to p(x, y) uniformly in x, y, then p is also a t. b. metric. We will add here further the following remark: Let 10 be a set of elements a, b, .... Suppose there exist a mapping .f of 10 in a space fi with a metric p. Then 11 (a, b)=p(f(a), flb») is clearly a metric for 10: the "transferred metric from fi into 10 by means of.f." If p is hereby t. 1., so is also 11 (as a metric of 10). 2. Now let @ be a group, and p a metric of @. P is called leftinvariant (l-inv.) if p(ax, ay) = p(x, y) for all a E @; right-invariant (rinv.) if p(xb, yb)=p(x, y) for all bE@; invariant (inv.) if it is both rinv. and l-inv. From any metric P we can form a l-inv. metric pl, a r-inv. pr and an inv. pi=plr=prl in putting: pl(X, y)=L u. b. p(ax, ay),

pr(x, Y):::::l.

U.

b. p(xb, yb),

pi(X, y)=l. U. 1. p(axb, ayb) where a, b run over the elements in @. This process to obtain pi, pr, p' from p will be called l-, r-, and i-operation resp. Lemma 4. If one of the metrics pl, p" and pi is t. b., so are also the others. Proof. As pl < pi and pr < pi, pl, p" are t. b. in the same time with pi according to the lemma 1. Now let pr be t. b. and a], "', an an €-net for pro Put pr(OiX, aiy)=pi(x, y). pi, i=l, ... ,n are t. b. by the last remark in § 1. Let U be the superposed covering of €-coverings for pi, i = 1, ... , n. We will see that U is a 3€-covering for pi = prl and recognize thus pi as t. b. Indeed, let a be any element of @ and x, y two points belonging to an element of U. Then we have for a certain ai pr(ax, ay) < pr(ax, aix)+pr(aix, aiY) + pr(aiY, ay)=2pr(a, a.)+ piCx, y) < 3€, therefore prl(x, y) < 3 a-Ix von G auf sich ansieht. Das Weilsche Mass m* ist dann G-invariant und G ist messbar uber G im obigen Sinne. Satz 2. Wenn G messbar uber Q ist, so ist die Abbildung P x x --"> p. X-I auch messbar. Beweis. .f(P) sei messbar. Dann ist f(p· x) messbar in Q x G, also nach Satz 1 ist f(p'x-1y) messbar in QxGxG. f(p·x-I).y) ist also nach dem Satz von Fubini fUr fast aIle y messbar in P und x. Da p* G-invariant ist, ist daher f(p· X-I) messbar in Q x G, w. z. b. w. Auf den Fall Q=G angewandt, ergibt diesel' Satz den folgenden 1) en(x) bedeutet die charakteristische Funktion von D. 2) Vgl. E. Hopi: Ergodentheorie, S. 9, Definition 3.4.

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[Vol. 17,

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Satz 3. Die Abbildung x x y - yx von G x G in G ist nwssbar.D Es seien S';)~) der (nicht notwendig separable) Hilbertsche Raum aller p* -quadrat-summierbaren Funktionen auf Q, und U'" der dutch Uxf(P) = f(p· x) definierte unitii.re Operator von S';)~). Es gilt dann der folgende Satz 4. 1st G messbar uber Q, so ist fur jedes f e S';)~) die durch aUe Uxf aufgespannte abgeschlossene lineare Mannigfalt?'gkeit [Uxf; x e G] separabel. 2) Beweis. G liisst sich nach der Voraussetzung als die Summe hOchstens abziihlbar vieler Aj e (m) darstellen. In jedem Q x Aj gehOrt f(p· x) zu V. Man kann also gff) e S';)~) und hjr)(x) so wahlen, dass

gilt. Hieraus folgt nach dem Satz von Fubini, dass U.,j fiir jed~s x ausserhalb einer x-Nullmenge X in der durch rfJf!) aufgespannten separablen Mannigfaltigkeit W1 enthalten ist. G- X ist also in bezug auf die Iinks-invariante "Quasi-metrik" p(x, y) =II U.,j - Uyfll separabel. Man setze nun 8,= (x; p(x, 1) < eine p-uberall dichte Folge

Xj

E).

8, ist dann messbar, und da fUr

U=l, 2, ... )

aus G-X

G-X=~xj8 j

gilt, muss m(f),) >0 sein. yx $ X Daher gilt

Es gibt also fur jedes yeX ein xe8, mit

II UlI f- UlI.,jII=11 U.,j- fll < E, und Uy.,je W1. Also muss Uyf e W1 auch fUr ye X sein. 2. Definition 2. Man sagt, dass in Q eine Un?Jorm-topologie (die wir notigenfaUs mit :t bezewhnen) definiert ist, wenn in Q ein System der jedem Punkte P e Q zugeordneten Teilmengen (V,,(P); a e I) mit dem " Indizen-bereich" I als das "Umgebungssystem" so a1-tsgezeichnet wird, dass die folgenden Bedingungen erfullt sind: i) ViP) '3 P. Fur jede a, (3 e I gibt es ein reI mit Vr(P) C vacP) (\ Vi>(P), ii) Fur jedes a-e I gibt es ein reI, 80 dass aus P' e Vr(P), pI! e Vr(P) P' e ViP") folgt. iii) Fur jedes a e I gibt es Mchstens abzahlbar viele Punkte Ph P z, ••• € Q mit ~ V,,(P,) = Q. j In diesem FaU heisst !} ein uniform-topologischer (kurz: u-t.) Raum.3 ) In einem u-t. Raum werden die Stetigkeit, die Gleichmiissig-stetigkeit, die Total-beschriinktheit, u. s. w. wie ublich definiert. Ein u-t. 1) Vgl. B. M. T. Kap. IV. Satz 16. 2) Vgl. B. M. T. Kap. V. Satz 22. 3) Der u-t. Raum wurde von Herm A. Weil ausfiihrlich behandelt. V gl. A. Weil: Espaces a structure uniforme, Act. sc. et indo 551 (1937). Hier wird die Bedingung (Ut ) (in S. 7) von A.Weil- d. i. das Trennungsaxiom: II r- Va(P)=P - ausa geiassen, und die Bedingung iii) neu hinzugefiigt.

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No.1.]

Uber die Gruppe der messbaren Abbildungen.

21

Raum' heisst im Kleinen total beschrankt (kurz: im Kl. t. b.), wenn fiir ein (J. E I aIle Va(P) total beschrankt sind. Definition 3. Man sagt, dass in der Gruppe G eine Topologie definiert ist, wenn in G im obigen Sinne eine Uniform-topologie definiert ist, die nook der Bedingung: aVix) = V,.{ax) genugt, und wenn dabei die Abbild~tng x xy->y-1x stetig ist. G heisst dann eine topologische Gruppe. Jede topologische Gruppe im iiblichen Sinne ist es auch im Sinne der obigen Definition. wenn sie der zu iii) entsprechende Abzahlbarkeitsbedingung geniigt In dieser Nr. setzen wir stets voraus: Es seien Q ein im Kl. t. b. u-t. Raum und G eine im Kl. t. b. topologische Gruppe. Definition 4. Eine (komplex-wertige) Funktion in Q heisst Bairesch, wenn sie aus gleichmassig stetigen Funktionen durch sukzessive Anwendungen der elementaren Rechenregeln und Limes-bildungen erhalten wird. Eine Teilmenge aus Q heisst Borelsch, wenn ihre charakteristische Funktion Bairesch ist. Eine offene Menge ist in diesem Sinne im allgemeinen nicht notwendig Borelsch. Es lasst sich aber zeigen, dass Q ein Umgebungssystem besitzt, das lauter aus Borelschen Mengen besteht. Umgekehrt enthiilt jeder absolut additive Mengenkorper, der ein Umgebungssystem enthalt, auch aIle Borelschen Mengen.l) Es gilt also Satz 5. Die Familie aUer Borelschen Mengen ist der minimale absolut additive Mengenkorper, der uberhaupt ein Umgebungssystem enthiilt. Definition 5. Das Mass p* heisst zur Topologie von Q gehorig, wenn i) jede total besckriinkte Borelsche Teilmenge in (p) enthalten, und ii) es fur jede Teilmenge A ein Borelsches B von der Art gibt, dass B::::> A und p(B)=p*(A) gilt. 2 ) Aus Satz 5 folgt dann leicht der folgende Satz 6. Es seien Q}, Q 2 zwei im Kl. t. b. u-t. Riiume. Sind pi bzw. ,.,; zu den Topologien gehOrige Masse von Q1 bzw. Q2, so gehOrt das Produkt-mass PIp.; zur "Produkt-topologie" von Q1 X Q2.3) Definition 6. G heisst stetig uber Q, wenn die Abbildung Px x->P'x in jeder total beschriinkten Teilmenge von Q x G gleichmassig stetig ist. Z. B. ist G, als Abbildungsgruppe von sich selbst betrachtet, stetig iiber G.4) Satz 7. G sei stetig Uber Q, und p* bzw. m* gehare zur Topologie von Q bzw. G. Dann ist G messbar uber Q. Beweis. f(P) sei messbar. Dann gibt es ein Bairesches h(P), so dass f(P) = h(P) fUr fast aIle P gilt. Man wahle eine Borelsche NulImenge N, die die Nullmenge (p;f(P) ~h(P») enthiilt, und bezeichne 1) Fur eine ausfuhrliche Darstellung siehe B. M. T. Kap. I. Nr. 5, Nr. 6. Vgl. J. von Neumann, Recueil Math., I (43) (1936), S. 721 u. 722. Siehe auch B. M. T. Kap. IV. Nr. 1. 3) Vgl. B. M. T. Kap. II. Satz 5. 4) Denn jede im Kl. t. b. topologische Gruppe ist komplettierbar. Vgl. B. M. T. Kap. I. Satz 3. 2)

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die charakteristische Funktion von N mit eN(P)' eN(P'x) und k(P·x) sind nun Bairesch,l) also messbar in Q x G, da nach Satz 6 p.m zur Topologie von QxG gehOrt, und es gilt

f f eN(P·x)dPdx=O.

f(p·x)

!JxG

stimmt also fast iiberall mit h(P· x) iiberein. Daher ist f(p· x) messbar, w. z. b. w. In diesem Beweis wird der Weilsche Charakter von m* nicht benutzt. Auf den Fall Q = G angewandt, ergibt dieser Satz also den folgenden Satz 8. Jedes zur Topologie gehOrige Mass von G ist Weilsch. 2) Es gilt fUr G noch der folgende wichtige Satz 9. Ein in G erklartes Weilsches Mass m* gehOrt zu Topologie von G, wenn i) jede total beschrankte Borelsche Menge in (m) enthalten wird, und ii) es .fur jedes A E (m) ein Borelsches B mit meA - B) + m(B-A)=O gibt. 3 ) 3. Einen wichtigen Satz von Herrn A. Weil4) k6nnen wir nun folgendermassen formulieren: Satz 10 (A. Wei]). Jedem in einer Gruppe G erklarten Weilschen Mass m * wird eine Topologie :r so zugeordnet, dass G eine im Kl. t. b. topologische Gruppe und m * eben das zu :r gehOrige Mass wird. ;;r ist dabei durch m * eindeutig bestimmt. - Wir nennen dieses ;;r die zu m * gehOrige Topologie. m * ist dann umgekehrt durch die zugehOrige Topologie bis auf multiplikative Konstant eindeutig bestimmt. Die Zuordnung der Topologie zum Weilschen Mass ist also umkehrbar eindeutig.5 ) 1m folgenden nehmen wir also immer G (0 u. s. w.), als eine topologische Gruppe mit dem zugehOrigen Weilschen Mass an. Es gilt dann der Satz 11 (A. Wei!). Das Umgebungssystem von G ist durch (VA(x);

VA(x)=xAA-l,

Ae(m), m(A»O)

gegeben. m( G) ist dann und nur dann endlich, wenn G total beschrankt ist. 6) Satz 12. Eine homomorphe Abbildung G aUf G ist dann und nur dann messbar, wenn sie stetig ist. Beweis. Es ist nach Satz 11 klar, dass die Stetigkeit aus der Messbarkeit folgt. h: x ~ x =h(x) sei nun eine stetige homomorphe Abbildung. Man fasse Gals eine Abbildungsgruppe von G, indem man x.a = h(a-1)x (a e G) setzt. Dann wird G stetig iiber G. Gist also nach Satz 7 messbar fiber G,. daraus folgt die Messbarkeit von h. 1) Dies ersieht man leicht, indem man !J und G komplettiert. Vgl. B. M. T., Kap. I. Nr. 6. 2) VgJ. B. M. T. Kap. IV. Satz 16. 3) Fiir den Beweis vgl. man B. M. T. Kap. IV. Beweis von Satz 19. 4) VgJ. A. Wei!, C. R. t. 202 (1936), S. 1147-1149. 5) Eine topologische Gruppe besitzt aber im allgemeinen nicht notwendig ein zugehoriges Weilsches Mass. VgJ. B. M. T., Fussnote 29). 6) Einen ausfiihrlichen Beweis dieser Satze findet man in B. M. T. Kap. V.

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No.1.]

Uber die Gruppe der ;messbaren Abbildungen.

23

Aus diesem Satz folgt z. B., dass jede messbare Darstellung durch Matrizen von G stetig ist.l> Es gilt nun noch allgemeiner der Satz 13. Fur jedes x € G sei ein unitarer Operator U", von einem separablen Hilbertschen Raum .~ homomorph zugeordnet, so dass (f, U",g) sich fur jede f, g € s:? messbar verhalt. Dann ist die Abbildung x ~ U", stark-stetig. 2) Beweis. Man setze B.=(x; II U",j-fll < e) fiir ein festes f. B. ist dann messbar, und man kann zeigen, wie im Beweis des Satzes 4, dass m(e.) > 0 ist. Xe ee;1 ist also nach Satz 11 eine Umgebung von x. Fur jedes y € xeet)";1 gilt dabei I U.,j- Uyfll < 2E:. x - U'" ist daher stetig im Sinne der starken Topologie. Es seien nun s:? = s:?~) und U'" der durch U",f(P) = f(p· x) definierte unitare Operator. s:?~) kann zwar nicht separabel sein, aber, wenn G messbar uber fJ ist, kann der obige Satz auch auf diesen Fall angewandt werden, wie man aus Satz 4 ersehen kann. Es gilt also der Satz, 14. G sei messbar, dann ist die Abbildung X,---c> U'" starkstetig. 1) Vgl. A. Wei!, C. R. t. 202 (1936), S. 1147-1149. 2) Dies bildet eine Erweiterung von einem bekannten Satz, der die Stetigkeit jeder messbaren Stromung behauptet. Vgl. z. B. E. Hopf: ~rgodentheorie, S. 10.

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45

· Uber die Beziehung zwischen dej~ MltsSen und den Topologien in einer Gruppe.

Von Kunihiko KODAIRA. (Gelesen am 16. November 1940.)

In 1936 hat Hfrr Andre Weil in einer C.R.-Note (1 ) angezeigt, dass zwischen gewissen Massen und Topologien in .einer Gruppe eine enge Beziehung bosteht. Eine Gruppe G mit Elementen x, y, ... , nennt man bekanntlich eine topologische Gruppe, wenn sie zugleich ein topologischer Raum ist, und y-Ix eine stetige Funktion von x, y ist. Die hier in Frage komlfiende Topologie von Gist also nicht eine beliebige: sie muss mit der Gruppenoperation von G in einer gewissen Relation stehen. Als Topologie in einer Gruppe wollen wir im folgenden nur solche Topologie in Betracht ziehen. Dementsprechend ist es naturgemass, dem Mass in einfr Gruppe eine gewisse Bedingung aUfzuerlegen. Welche Bedingung soIl es sein? Eine Abbildung von einem mit einem Mass behafteten Raum in einen ebensolchen Raum heisst naeh J. von Neumann messbar, wenn das Urbild jeder messbaren Menge wifder mess bar istYJ Wenn also in einer Gruppe G ein Mass erklart ist, konnte man die Messbarkeit der Abbildung (x, Y)-4y-I x von G x G auf Gals das annehmen, was der Stetigkeit von y-Ix entspricht. Das ist in der Tat was Herr A. Weil tut, indem er die Masse betraehtet, die die Eigenschaft besitzen, dass mit j(x) auch j(y-1x) eine messbare Funktion sein sollY) Ein linksinvariantes Mass mit dieser Eigensehaft wollen wir ein Weilsehes Mass nennen. Herr A. Weil behauptet nun, dass sich zwischen solchen Massen und den Topologien in einer Gruppe G eine eineindeutige Zuordnung in der Weise herstellen lasst, dass das betreffende Mass mit dem Haarschen Mass derjenigen Gruppe iibereinstimmt, die aus G dUTch Komplettierung in bezug auf die entsprechende Topologie entsteht, und immer im Kleinen bikompakt ausfallt.-Der Beweis dieses wichtigen Resultates seheint jedoeh noeh nicht publiziert zu sein. 1m folgenden (1) Vgl. A. Wei! [1l Die Nr. bezil,hen sich auf dM Literaturverzeichnis am Ende der Einleitung. (2) Vgl. J. VOn Neumann [2l (3) Herr. A. Wei! nennt eine Gruppe mit einem solchen Mass "gronpe mcsure". VgL A. Weil [lJ Er stellt dort allerdings eine etwas stiirkere Bedlngung fUr das Mass nuf. In Wirklichkeit ist sic aber mit dieser iiquivalent. Siehe Fussnote (28).

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Kunihiko KODAIRA.

68

[Vo!. 23

solI der Beweis mitgeteilt werden, den ieh naeh kurzen Andeutungen von Herrn A. Weil durehgefiihrt habe. In I wird die Vorbereitung getroffen, die fur die genaue Formulierung des Weilsehen Ergebnisses notwendig ist. Am wesentliehsten ist dabei, dass wir einerseits den Begriff der Topologie in einer Gruppe etwas allgemeiner fassen als ublich (namlieh von dem Trennungsaxiom absehen), und andererseits ihr eine gewisse Abzahlbarkeitsbedingung (Bedingurig (B) I, Nr. 6) auferlegen. Solche "alIgemein-topologische" Gruppe, die im Kleineu total beschrankt sind, spielen im folgenden eine Hauptrolle. Als im Kleinen bikompakte Gruppen betrachten wir aber nur die topologischen Gruppen im gewohnlichen Sinne (mit getrennter Topologie). II bringt also eine leichte Verallgemeinorung der bekannten Theorie des Haarschen Masses auf den im Kleinen bikompakten Fall. In III wird das Mass, das durch ein Haarsches Mass induziert wird (Fur die Definition siebe III, Nr. 1) behandelt. Es wird insbesondere gezeigt, dass soIehes Mass ein Weilsches Mass ist (Satz 12). In IV fiihren wir den Begriff des Haarschen Masses in einer im Kleineu totalbeschrankten allgemein-topologischen Gruppe im genauen Analogen mit dem in II behandelten Fall ein. Jede solehe Gruppe G lasst sieh nach I, Nr. 6 in eine im Kleinen bikompakten Gruppe & homomorph abbildell, und es zeigt sich, dass das so definierte Mass eben das durch ein Haarsches Mass von & induzierte Mass, also ein Weilsehes Mass ist (Satz 14, 15). In V wird dann bewiesen, dass jedem WeiIschen Mass eine Topologie von G entspricht, sodass G mit dieser Topologie im Kleinen total besehrankt wird, und das gegebene Mass mit dem Haarsehen Mass von dieser Gruppe iibereinstimmt. So entsprechen sich die WeiIsehen Masse und gewisse Topologie von Gin eineindeutiger Weise. Der Hauptsatz in genauer FormuIierung Hest sirh in V, Nr.3. VI behandeIt die speziellen FaIle der separablen Topologie. Hierfiir gelten die Dichtigkeits- und Dberdeckungssatze, die auch Interesse beanspruchen diirften.

Literaturverzeichnis. P. Alexandroff und H. Hopf: [1) Topologie. I. G. Birkhoff: [1] Moore-Smith convergence in general topology. Annals of Math. 38 (1937), S. 34-56.

D. van Dantzig: [1] Zur topologische Algebra.!. Math. Ann. 101 (1933), S. 587-626. A. Haar: [1] Der Massbegriff in der TheOlie del' kontinuiel'lichen Gruppen. Annals of Math. 34 (1933), S. 147-169. S. Kakutani: (1) On the uniqueness of Haar's measnre. Proc. Imp. Acad. Japan, XIV (1938), S. 27-81.

J. von Neumann: [1] Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren.

C 10) 47

1941]

J.

Jfasse unci l'opologic

YOIl

~n e~nC1'

(fruppe.

69

)fath. Ann. 102 (1929), S. 49-131. [2] Uber messbare Abbildungen. Annals of Math. 33 (1932), S. 574-586. [3] Zur Algebra der Funktionaloperatoren. Math. Ann. 37 (1936), S. 116-229. [4] The uniqueness of Haar's measure. Reeueil Math. I, (43) (1936), S. 721-734. :N"eumann uncI F. J. Murray: [1] On rings of operators. Annals of Math. 37 (1936), S. 116-229.

S. Saks: [1] Theory of Integral. A. Tychonoff: [1] Uber die topologische Enveiterung von Ranmen. Math. Ann. 102 (1930), S. 543-561.

A. Weil: [1] Sur les groupes topologiques et lea groupes lnesures. C.R. 202 (1936), S. 1147-1149.

[2] Sur les espaces it structure uniforme et sur la topologie generale.

I. Vorbereitung. 1. Mass. In der vorliegenden Arbeit verstehen wir unter dem in cinem Raum R definie1·ten Mass m*=m*(A), AeR, immer das reguHire Oaratheodorysche aussere Mass(4) mit der folgenden Eigenschaft: R Hisst sich als die Sum me von hochstens abzahlbaren Teilmengen mit endlichen ausseren Massen darstellen: ~

(A)

R=2;Ah

mit

m*(AJ) < +00.

J~l

Eine Teilmenge A VOll R heisst m-messbar, falls A im Oaratheodoryschen Sinne messbar ist; fur messbares A schreiben wir meA) statt m*(A). Eine im R definierte komplexwertige F'unktionj(x) heisst m-messba·i', wenn das Urbild j-l(O) jeder offenen Menge 0 immer m-mesabar ist. Das "Lebesguesches Integral": f/(x)m(dx) einer m-messbaren F'unktion j(x) uber eine m-messbare Teilmenge A ist daun ein wohlzuverstehender Begriff. 1m Prodnktraum R x R ist das Produktmass mm* definiert: fur jede Teilmenge J'eR X R setzen wir namlich:

mm*(J') = inf:2m* (A,)m* (BJ) , J=!

wobei inf die untere Grenze fur aUe Wahl von A, und B, aus R bedeutet. Die F'nnktion j(x, y) zweier Veranderlicher x, y aus R heisst m-messbar, wenn sie als Funktion im R x R mm-messbar ist. ,\Vir bezeichnen ein Integral solcher Funktion mit (4) Vgl. S. Saks [1], S. 43-47. Wir schliessen dabei natiirlich den tri\'ialen Fall aus, dass totale Mass von R selbst gleich Null ist.

C 10)

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Kunihiko KODAIRA.

70

[Vol. 23

1f(x, y)rnrn(d(x, y)), oder als "Doppelintegral"

f 1f(:r,

y)m(d:r)rn(dy).

Der diesbezugliche, bekallnte Satz dungen wichtig.

VOll

FubiniC'a) ist Hir l111Srre Anwen·

2. Absolut additiver und absolut muItiplikativer l\[engenkurper. DEFINITION .• Eine Farnilie iJ von Teilrnengen A, B, .... , aus R heisst absol7Jt rnultiplikative?' JJfengenkMper, wenn sie den folgenden Bedingungen genugt :

i) ii)

aus A, B

€

iJ

jolgt

A.+B, A-B, AI\B

a1W Aij=1,2, ' ... ) €

iJ

folgt

II Aj

€

iJ,

iJ,

€

i-I ~

iii)

esgibt

A/j=1,2, .... )€iJ,

R=~AJ

sodass

j~l

gilt. DEFINITION. Ein absol~d rnultiplikative?' JYIengenko1'pe1' lieisst absol1a additiv, wenn er 170ch die Bedingung e1:fullt: ~

iv)

aus A/j = 1,2, .... ) €

iJ

folgt

L;AJ €

iJ.

j=l

Es sei iJ ein absolut multiplikativer Mellgenkorper. Dann bildet die aus allen Summen der hochsteJ)s abzahlbaren Mengen aus iJ bestehende Familie einen absolut additiven Mengenkorper. Wir bezeichnen dies mit @::

~=(x; .

X=i'A j, AjeiJ)(5J. J~l

~ enthaJt offen bar den ganzen Raum R . . Eine Funktion f(x) im_ R heisst iJ-rnessbm', wenn das Urbild FICO) jeder offenen Menge 0 in iJ enthaltrn wird. 1m Fall, dass im Rein Mass m* erkHirt ist, bilden aIle m-messbarcn Teilmengen mit endlichem m-MaEs einrn absolut muItiplikativen Mengenkorpcr. Bezeichnen wir dies mit (rn): (rn)=CA; m(A) < +00). Dann stimmt die m-Messbarkeit mit (rn)-Messbarkeit uberein. Nun sei fur aHe zu iJ gehorendell Mengen eine llichtnegative, abVgl. S. Saks [1], S. 76-88. (5) 1m folgenden bezeichnen wir nach J. von Neumann und F. J. Murray, die Menge, die aus allen Elementen x mit der Eigenschaft G: besteht, mit (x; (%(:r». (4a)

( 10 J 49

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JIasse und 'l'opologie in eine1' G1'uppe.

solut additiw, endlichwertige Mengenfunktion(6) ,.,. erklart. nieren dann .das Z1t ,.,. gehorige ~Mass ,..,* wie folgt:

71

\Vir defi-

00

,.,.*(X) =inf 1;,.,.(A,) , j~l

~AJ~X,

AJ e~.

J~l

Die absolut additivE: Mengenfunktion ,.,.=",,(A), die in ~ el'klart isi, heisst m-absolut stetig, wenn sie folgenden Bedingungen geniigt: i) iJc(m), ii) ans ?1l(A) = 0 folgt fJ.(A) = O. Es gilt der SATZ VON RADON-NIKODYM. Eine m-absolut stetige, absol1Lt additive Mengenfunktion ,.,.(A) , A e~, lasst sick stets als ein "indefinites" Integral da?'stellen:

",,(A)=i0 ist ep(x) eine Baircache Funktion, kann also als eine Funktion im separablen metrischen Raum mt aufgefasst werden. p besitzt in 91' ein in mt kompakte abziihlbare Umgebungssystem {Uj!. Es sei U; die ofi'ene Teihnellge ami 91, die der ofi',!lnen Menge UJ ontspricht. UJ ellthalt dann p, UJ ist

bikompakt und

II U;=p.

Daraua erkennt man, dass {Uj ! ein Umge-

J~l

bungssystem von p bildet. Betraehten wir nun ein Dmgebungssystem, odor eine Basis von 9'(. Dafiir gilt: Jeder absolut additiven Mengenko1'Per, dcr eine Basis von m enthdlt, enthiilt die Familie aller Borelschen Mengen. Zum Beweis dieser Tatsacho goniigt os zu zeigen, dass fUr jede stetige reelle Funktion f(~) in m die Menge (x, f(x)??=a) im gonannten Mengenkorper t'nthalten ist. Es sei 0 beliebig gegeben, Man wahle fiir jeden Punkt x f'inp Umgebung ll(x) aus der gegebenen Basis, so dass aus ye

( 10)

53

1941]

75

Masse nnd Topologie in einer Grnppe.

H(x) folgt If(y) -f(x) I < e/2. Die Gesamtheit aIler H(x) bildet offen bar ('me offene UberdeckuDg von m. Nach Voraussetzung uber m kann man also daraus eine abzahlbare Teiliiberdeckung !H(xJ)} wahlen. He sei die Snmme aller U(Xj), fur die !(xJ ?:a- e/2 ist: Ue = _L lUxj). f(TJ)~a-'/2

Daun ist ('S klar, dass fii.r xell e !(;r»a-E, unA,

und

p.*(A):==p.(B)

gilt. In diesel' Definition verlangen wir die Messbarkeit del' offenen Mengen nicht. Bei den separablen Gruppen ist offenbar jede offene Menge p.-messbar; im al1gemeinen Fall abel' ist dem Verfasser nicht gelungen, zu entscheiden, ob man die Messbarkeit der offene Menge aus obiger Definition folgern kann oder nicht(19). Die Existenz des Haarschen Masses ist unter Voraussetzung del' Separabilitat von & erst von A. Haar bewiesen worden(20). Die Haarsche Beweismethode kann man auf den allgemeinen Fall iibertragen, indem man die gewohnliche mathematische Induktion durch die transfinite Induktion ersetzt(21). Hier geben wir abel' einen Beweis, del' sich auf einem Satz von Tychonoff(22) stiitzt. Es sei & eine im Kleinen bikompakte Gruppe. Die bikompakte Teilmenge von & bezeichnen wir allgemein mit F (notigenfalls mit Akzent oder Indizes), die offene Teilmenge mit U oder V. IB} sei ein Umgebungssystem von 1. Zunachst zeigen wir, dass sich eine Mengenfunktion l1lo(F) fiir jedes F so definieren lasst, dass die foIgenden Bedingungen erfullt sind: i) Oa;mo(F) < +00; ii) moCv) > 0(23), (U*O); (1)

iii)

mo(F)=mo(aF),

ae&;

iv)

mo(F+F')a;mo(F)+mo(F');

v)

mo(F+F')=mo(F)+mo(F'),

falls F"F' =0 ist.

(19) Der Verfasser vermutet, dass die Antwort auf diese Frage vielleicht negativ sein wird. (20) Vg1. A. Haar [1). (21) Diese Bemerkung verdanke ich einer miindlichen Mitteilung von Herrn S. Kakutani. Neuerdings hat Herr Kakutani iibrigens noch einem neuen Bcweis fUr diese Tatsache gegeben, der auf eine Erweiternng yon einem Lemma von W. Maak beruht. (22) Vg1. A. Tychonoff [1). (23) fj bedeutet die abgeschlossene Riille von U.

( 10) 59

1941]

.:1Iassc

~md

81

Topologie in cine?' Gruppe.

Es seien Fund U(=t::O) belie big gegeben. n

endlich vielen aU iibcrdecken: Fc ~ ajU,

Dann lasst sich F mit

Die kleinst mogliche An-

J-t

zahl solcher aj sci mit n(FjU) bezeichnet. Eine offene, in pakte Menge Uo sci nun fest gewahlt. Wir setzen

@

bikom-

l(F; 8)=n(~/8) . n(Uo/8)

Dieses 1(F; 8) hat die folgenden Eigenschaftcn, wie leicht nachzupriifen ist: i) (2)

O;'2;l(F; 8);'2;n(F/Uo);

ii) iii)

1(U; 8) ?=.1/n (Uo/ U) ;

iv)

l(F+F'; fJ);'2;l(F; EJ)+l(F'; fJ);

v)

leaF; e)=l(F; fJ);

Falls F f\F' =0 und 80 geniigend klein ist, so gilt l(F+F'; fJ)=l(F; e)+l(F'; 8)

fur jedes 8ceo•

Nun fassen wir die Familie {F I aller F als einen "Raum" f (mit "Punkten" F) auf. Bezeichnen wir die reelle Zahlengerade mit £, so Hisst sich l(F; 8)=lr.(F) mit einem festen 8 als ein Element von dem Abbildungsraum £1 ansohen. L sei derjenige Teil vom £1, der aus solchon l(F)e£ besteht, die O;'2;l(F);'2;n(F/Uo) geniigen. Wegen (2) i) ist l&(F)eL fUr jedes e. Nach dem Satz von Tycbonoff ist ferner L bikompakt im Sinne der "schwachen" Topologie. Setzen wir also

so folgt II f\L(8)=t::0,

ee!"'}

da aus fJcEJ' folgt L(8)CL(8'), und mithin der Durchschnitt endlichBl" Anzahl VOll L(8) nie leer ist. Es sei nun mo(F) ein Element aus II 1\ r.0 und fur ein beliebiges System endlicher Anzahl von Elemonten von {FI: FJ, j=1,2, .... , h, gibt es dann offen bar ein ecfJo, so dass

j=1,2, .... , h gilt, da ja moe L(eo) ist. Aus (2) folgt dann leicht, dass mo(F) den Bedingungen (1) genugt. Von jetzt an solI F bzw. U oder V immer bikompakte bzw. offene

( 10 J 60

Kunihiko KODAIRA.

82

Borelsche Teilmenge von

(3)

@

bedeutell i21 ).

[Vol. 23

Man setz0 nun

'1nt(U)=sup moCP), Fc::{f

und

(4)

fur beliebiges Ac@.

Dann behaupten wir: m* ist ein Haarsches Mass. Es ist l1amlich leicht zu zeigen, daEs m* ein links-invariantes Caratheodorysches ausseres Mass ist, und dass m *(A) < + 00 fUr jedes total beschrankte A gilt. Ferner ist offen bar m*(U)=m1(U), also gibt ss nach (4) eine Borelsche Menge B fur jedes Ac@, so daFiFi m*(A)=m*(B) und B~A

gilt. Um die obige Behauptung zu bewoison, haben wir also nUl" noch zu zeigen, dass jedes U m-messbar ist, d.h. dass (*) m*(A)=m*(A"U)+m*(A-U) fUr jedcR AC(Sj gilt. Dazu geniigt es, dies nul' fUr d('ll Fall A= V (offene Borelsche Menge) nachzuweisen: anderufalls hat man namlich nUl' ein paEsendes V mit AcV zu betrachten. Seien also U, V und e>O beliebig gegeben. Nehmen wir ein F mit Fc U" V, mo(F) >?nl (U" V) - e, dann eine weitere bikompakte Borelsche Menge FI und offene Borelsche Menge V}, so dass U"V:::JFI~Vl-:;:)F gilt. Dann gilt wegen V-UCV-FI m*(V"U)+m*(V - U)~ml(V"U)+ml{V -FI) A,

und p.(B-A)O offene Borelsche Mengen

n, U~,

Naeh Satz und bikom-

pakte Borelsche Mengen l5 und l5f, so dass

U::JiB::Jl5, fL(U-\5)0 belie big gegeben. Man kann dann nach dem obigen KorolIar eine Umgebung 6 von EillS so klein wahlen, dass fUr jedes (J,efJ

II Uae .. -eAII < m(A) gilt. Daraus folgt nnn also gilt aeAA- 1 oder 8cAA-l. Damit i"t bewiesen, dass (AA- 1; Ae(m), m(A) >0) ein Umgebungssystem von 1 biJdet. KOROLLAR. Es seien G1, G2 zwei im Kleinen total befchrankte Gruppen, mf bzw. m~ die Haarsche Masse von G1 bzw. Ga. Dann ist }ede messbare(31b) homomorphe Abbildung r von Gs auf Gl stetig. BEWEIS. Es geniigt zu zeigon, dass es fUr jede Umgebung 8 1 von 1 in G1 eine Umgebung 8z·von 1 in Oz mit r(8z)CfA gibt. Zu diosem Zweck sei ~ eine Borelsche Umgebung von 1 in G1 mit B:~-lcfJl' Dann ist r-l(8;) ma-messbar, und m2(r- 1(8;)) >0. Wir wahlen eine mzmessbare Menge Az mit einem endlichen positiven Mass, die in r-l(fJ;) enthalten ist, und setzen 8z=AzA;I. Dann ist offen bar r;(82)cfJ1. (31) Vgl. A. W.·il [lJ. (31 a) Dies folgt aus der Bedingung (B). (81 b) Vgl. S. 67. Siehe auch J. yon Neumann [2].

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99

3. Verschiirfter Eindeutigkeitssatz. Wir wollen indieser Nr. den folgenden wichtigen Satz beweisen, der als eine Versharfung des Eindeutigkeits~atzes (Satz 15) anzusehen ist. 'SATZ 19. (VERSCHARFTER EINDEUTIGKEITSSATZ). Es seien G eine im Kleinen total beschriinkte allgemein-topologische Grnppe, nnd m* ein in G erkliirtes Weilsches ]rfas8. Damit m* das Haarsche .LWass von G .~ei, ist notwendig nnd hinreichend, da.~s m* den folgenden Bedingungen genugt: i) Jede total beschrankte B07'elsche Menge Be Gist ?n-messba1' nnd m(B) < +00,

ii)

Fur jedes m-rnes8bm'e A gibt es eine Borelsche 111enge BeG, so dass dm(A, B)=O

BEWEIS. Die Notwendigkeit der Bedingungen ist ohne Weiteres klar. Urn zu zeigen, dass die Bedingungen hinreichend sind, bezeichnen wir die Familie aller total be~chrankten Borelschen Mengen aus G mit (B), Man setze fur

mo(B)=m(B)

B e(B).

Dann ist offen bar mo eine total additive Mengenfunktion III (B). Das zu mo gehorige Mass mt ist nun das Haarsche Mass von G, wie man leicht ersehen kann. Fur unseren Zweck genugt es also zu zeigen, dass mt=m* ist, Wir zeigen zunachst, dass an8 m(X)=O mt(X)=O folgt. Zurn Beweis dieser Tateache, sei r die Teilmengc aus G x G, die aus allen Elernenten (x, y) mit der Eigenschaft: y-1xeX besteht. Also gilt (32) Hierbei ist es wesentlieh, dass m* ein WeiIsehes Mass ist. Es gibt niimlieh ein nicht-Weilsches. links-invariantes Mass m*, das denoben genannten Bedingungen i) nnd ji) geniigt, aber nieht ein Haarsehes Mass ist. Beispiel: Es sei. G eine Gruppe mit mehr als abziihlbaren Elementen, nnd man fasse Gals eine total besehriinkte allgemein-topologisehe Grnppe mit derjenigen Topologie auf, die dureh die homomorphe Abbildung v,on G auf die kompakteGrnppe mit einem einzigen Element 1 induziert wird. Es gibt also nur zwei offenen Mengen in G: namlieh die leere Menl,YB und die ganze Grnppe G I Das HaarseheMass p* von Gist dann offenbar dureh

o

,.*(A)= {1

erkliirt.

'NUll

fiir A=O, sonst

definieren wir m* = rn",(A) \Vie folgt: m*(A)= {O im Fall, dass A. hoehstens abzahlbar jst, 1 sonst,

Dann ist es klar, dass m* oen Bedingungen i) und ii) gentigt. Uno sieher ist m*=I=p*.

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Kunihiko KODAIRA.

100

fur die charakteristische Funktion er(x, y) von l' er(X, y)=ex(y-I x ). Nun ist ex(y-I x) m-messbar, da m* ein Weilsches Mass ist.

mm(l') =

DalH'r ist

JJex(y-Ix)m(dx)m(dy) = f m(dy) 1e.r(y-Ix)m(dx)=O. G

a_a

G

Nach Definition des Produktmasses gibt os also fur beliebiges e::>O m-messbare Mengen XJ, YjeG, so dass ~

~

rC~XjXYf'

.2;'m(Xj)m(Yj)O hinreichend gross gewiihlt, so gilt daher fi'lr jedes x€ Vj Ii UzC[1-c.c 1 W-"" m(UJ)

bz) 1st Ae(m),

80

kann man fur jedes £>0 (ine .Polge von Ele-

( 10)

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Kimihiko KODAIRA.

118

menten aN und einc Teiljolge U},,, m (A-.?; aN U}N) =0,

und

[Vol. 23 allS

U, so wahlen, dass

~m(~)0 eine Borelbche, sogar eine offene Menge U mit dm(A, U) 0 there exists an A E W(G) satisfying III B- A III < e. The purpose of this paper is to determine a general form of maximal ideals of ffi(G). It will be shown that there exists a one-toone correspondence between the family 9J1( G) of all maximal ideals M of ffi(G) and the familyx(G) of all algebraic (=not necessarily continuous) characters2) lea) defined on G. This correspondence is even 1) I. Gelfand, Normierte Ringe, Recueil Math., 9 (1941), 3-25. 2) Under a character of a locally compact abelian group G, we understand a continuous representation of G by the additive group of real numbers mod. 1. Some· times it is also necessary to consider representations of G which are not necessarily continuous. In order to distinguish these cases, we usually say continuous characters and algebraic characters of G.

( 11)

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No.7.]

Normed Ring of a Locally Compact Abelian Group.

361

a homeomorphism if we take the usual weak topology of W1(G) with respect to which W1(G) is a compact Hausdorff space, and if we consider leG) as the compact character group G(d)* of a discrete abelian group G(d) which 'is algebraically isomorphic with G. It will also be shown that meG) is isometrically isomorphic with the normed ring

C(W1(G») = C( l(G»)

of all complex-valued continuous functions defined on 9JC(G)=l(G). From these two facts follows immediately that the normed ring meG) is uniquely determined up to an isometric isomorphism by the algebraic structure of a locally compact abelian group G, and so is independent of the topology or the I:Iaar measure of G which we needed in defining L2(G). Thus it turns out that in order to investigate the normed ring meG) of a locally compact abelian group G, it suffices to discuss the case when G is a discrete abelian group. § 2. Let M be an arbitrary maximal ideal of m(G). Then there exists a continuous natural homomorphism B-- CfJM(B) or meG) onto the ring of complex numbers such that \ CfJM(B) I < III B III for any BE m(G) and M = {B \ CfJM(B) = O}. It is then clear that a-- Ua-- CfJM( Ua) is an algebraic representation 'of G by complex numbers. Further, -I

since ICfJM(Ua) I

90(fl) = 11 (Xo)

=:S~~lapexp (2rrixo(ap»)=~~..!lapexp (27ri(a p ,Xo»).

This shows that 9o(fl) is obtained first by extending each function fl(g*) on G* to a continuous function fl(X)=S~-lapexp (27riX(ap») =~~~lapexp (2rri(ap, X») on G*=l(G), and then by taking the value of Il(X) at a particular point Xo € G~ = l( G). Since G* is dense in G* =l(G), so we see (14)

=

1!Po(fl) 1 ¥?o(A) of %f(G) by complex numbers which is given by (12) can be uniquely extended to a continuous representation B -> fJo(B) of ffi( G) by complex numbers such that the maximal ideal M::;: {B If;o(B) = O} determined by f;o(B) gives an algebraic character XM(a) which satisfies XM(a) = Xo(a) for all aE G.

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[Vol. 20,

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42. tiber die Harmonischen Tensorfelder in Riemannschen Mannigfaltigkeiten, (I). Von Kunihiko KODAIRA. Physikalisches Institut der Kaiserlichen Universitiit, Tokyo. (Comm. by TAKAGI,

M.I.A.,

April 12, 1944.)

1. Existenz der Harmonischen Tensorfelder mit gegebenen Perioden.

Die Beziehung zwischen Topologie und Tensorfeld auf Mannigfaltigkeiten ist schon von mehreren Autoren untersucht wordenl). 1m folgenden geben wir davon eine kurze Zusammenfassung, wobei aber bemerkt werden sol1, dass man eine formal ubersichtlichere Theorie bekommt, wenn man dem Tensorbegriff den dazu dualen Begriff der Tensordichte gegenubersteIlt. Der" obere Randoperator r*" operiert auf Tensorfelder; der "untere Randoperator t" auf Tensordichte. Erst in Riemannschen Mannigfaltigkeiten, wo wegen der mittels Metrik bestimmten Zuordnung der Unterschied zwischen den beiden Begriffen sozusagen verschwindet, lassen sich die Operatoren r* und t beide auf Tensorfelder operieren. Harmonisch heisst dasjenige Tensorfeld e, wofUr te=t*e=O gilt2). In diesem Teil I beweisen wir die Existenz der uberall regularen harmonischen Tensorfelder mit gegebenen Perioden. Dabei bedienen wir uns der Weylschen "Methode der orthogonalen Projektion "3). Den Beweis eines dabei benotigten wichtigen Lemmas, das bei Weyl nur im Fall des Euklidischen Raumes aufgestellt ist, tragen wir im § 4 nacho Die Tensorfelder mit Singularitaten betrachten wir im Teil II. Es wird sich zeigen, dass mit unserer Methode auch die klassischen Theoreme der Existenz der Abelschen Integrale auf der Riemannschen Flache sich leicht beweisen lassen. § 1.

Kombinatorische Topologie der Homologie-Mannigfaltigkeiten.

Zunachst erinnern wir uns an einige fundamentalen Eigenschaften der Homologie-Mannigfaltigkeiten. Es sei K=Kn ein n-dim. endlicher simplizialer Komplex, t P ein p-dim. Simplex von K. Die p-dim. algebraischen Komplexe mit reellen Koeffizienten auf K bezeichnen wir mit AP, CP, r]JP, etc.; t bzw. t* sei unterer bzw. oberer Randoperator4); [t P : tp-J] Inzidenzzahlen; dann gilt (1.1)

rt P= 2[t P : tp-1]t p- 1 ,

(1.1)*

t* t P=

2.'[tP+1 : tP]tP+1

•

1) U. a. von G. de Rham, E. Cartan, W. V. D. Hodge. Siehe insbesondere G. de Rham: Dber mehrfache IntegraJe, Abh. Math. Sem. Hans. Univ. 12 (1938), 313-339, wo auch Literatur angegeben wird. 2) Hodge nennt das "harmonic integral ". W. V. D. Hodge: Proc. London Math. Soc. Ser. 2 Vol. 36, 257-303; Vol. 38, 72-95; Vol. 41, 483--496. 3) H. Weyl: Method of orthogonal projections in potential theory, Duke Math. Jour. Vol. 7 (1940), 411--444. 4) V g1. H. Freudenthal: Alexanderscher und Gordonscher Ring und ihre Isomorphie, Annals of Math. (2), Vol. 38 (1937), 647-655. C 12)

105

No.4.]

fiber die Harmonischen Tensorfelder.

187

t P> t solI bedeuten, dass t a eine Seite von t P ist. LP(K) sei der Iineare Raum der p-dim. algebraischen Komplexe auf K, ZP(K) der Raum der p-dim. unteren Zyklen, HP(K) = t£P+1(K), BP(K)=ZP(K)/HP(K) ist dann * die p-dim. untere Bettische Gruppe. Entsprechend sei ZP(K) der Raum * * * * p-dim. oberer Zyklen, HP(K)=t*£P-l(K), BP(K)=ZP(K)/HP(K) ist dann die p-dim. obere Bettische Gruppe. Das Produkt zweier Elemente + oo(j -> 00) und zugeh6rige Eigenfunktionen ej, die zu :tl gehoren und einen 'vollstandigen normiertenorthogonalen System aus r bilden. Fur beliebiges 7) € n gilt dabei

Aiej, 7) = (Hej,

7)

= (ej H7) = (ej, L(7)) .

Setzt man D = L - At = - t1 + q* "- Aj, dann gilt also (ej, D( 7) ) = O. Nach dem Hauptlemma muss also ei in we regular sein, und L(ej)-Ajej= D(ej) = 0 genugen. Damit ist bewiesen der Satz 2. Es gibt jur: L ein vollstandiges normiertes orthogonales System von Eigenjunktionen ej, welche in we regular sind und zu :tl gehOren. Die zugehOrigen Eigenwerte Aj sind alle positiv, und es gilt: lim Aj=+OO. Das Randwertproblem kann nun folgendermassen formuliert werden ;3) Randwertproblem. Es sei eine in we beschrankte, stetig difJerenzierbare Funktion f mit 1 ojl1 < + 00 gegeben. Gesucht ist dann eine in we zweimal stetig difJerenzierbare Funktion S2, ••• mit G-SC ~ Sk(::) •• Es muss also m(e,) > 0 sein. Hieraus folgt jedes s ein t E (::), mit st $ S. Daher gilt

II TJ-Tstfll=lif -Ttfll

/c

s(-I,

q: S, also gibt es fUr

< c,

und dabei T,d (. 9fl ist. Fur jedes s € S muss also TJ € I)J~ sein. Aus diesem Satz folgt der wichtige Satz 6. Es seien hOchstens abziihlbar viele Bairesche Funktionen it(t),fzCt), ... auf G gegeben. Dann gibt es e'inen bikompakten Normalteuer N mit der separablen metrischen lokal kompakten Restklassengruppe G::::::G/N, sodass jede fit) als eine Funktion auf G aufgefasst werden kann. Jede Aussage ilber die hochBtens abziihlbar vielen Baireschen Funktionen, die fur jede separable, metrische, lokal kompaJcte Gruppe1) richtig ist, gilt auch fur allgemeine lokal bikompakte Gruppe. Beweis. Es geniigt den Fall zu betrachten, wo aIle f/c(t) stetig sind. Es sei U eine so kleine Umgebung von I, dass UU-I relativ bikompakt ist, und fo(t) eine stetige Funktion mit

fo(t)={ . 01

t=l,

fUr fur

tE G-U.

Ferner sei ~.m der durch aIle TJ/c aufgespannte abgeschlossene lineare Teilraum von :p: \))(=[T.fk; 8 E G, k=O, 1, 2, ... J, und 9'~=:p-\))(. G wird durch die Abbildung 8 - , T8 in U topologisch isomorph eingebetett. \))(, und also W sind aber G-invariant. Bezeichnet man die unitiire Transformationsgruppe von \))( bzw. IJl mit UIJJI bzw. U% so lasst sich G also als eine Untergruppe von U~JI x U91 auffassen: man kann jedes 8 € G so schreiben: 8=8 x r,

Die aus alle s bestehende Untergruppe von U~JI sei mit G bezeichnet. Nach dem obigen Satz ist '.m separabel. Wahlt man ein vollstiindiges orthonormiertes System {hI> h2 , ••• } aus '.m, so ist G also eine separable metrische Gruppe mit der Metrik:

pes, t) =k-l ~ 2\ II T,h" - T.thd . Fur ojede Teilmenge .A C Gist offen bar

A={s; s€.A}=Gn(.A x U91 ).

A ist also dann und nur dann offen, wenn A offen ist. Die Abbildung 8-

s

is also homomorph, stetig und gebietstreu. Setzt man N = {s ;

1) Unter einer metrischcn Gruppe vel'stehen wir cine topo\ogischc GruJlJlc mit der Iinks-invarianten Metrik.

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140

No.7.]

Uber das Haarsche Mass in der lokal bikompakten Gruppe.

449

s=l}, so ist daner G=G/N. Fur genugend kleines e>O ist andererseits {s; p(s, 1) < €} relativ bikompakt. Denn: aus

J)

!o(S-lt) - fo(t) 12m(dt) S lifo li2

folgt s E UU-I. Hieraus folgt, dass N bikompaKt und G lokal-kompakt sind.

Ferner ist fur sEN fk(S-lt)=fk(t), da Llfk(S-lt)-fk(t) 12m(dt)=

II T.fk-fkll=O und fk(t) als stetig vorausgesetzt ist. sich also als eine Funktion auf G auffassen. Fur jede Teilmenge A c G setzen wir nun

Jedes fk(t) lasst

m*(A)=m*(A) ,

wobei A die zu A entsprechende Teilmenge von G bedeutet. Dann ist m" das Haarsche Mass von G. Denn: erstens ist es klar, dass m" ein links-invariantes Caratheodorysches ausseres Mass in Gist und jede Borelsche Menge in G m" -messbar ist; zweitens gilt m*(A)=infm"(V} ,

U: offen und :::>A,

da fUr jede offene Menge U:::> A die (J = {x; xN C U} :::> A offen ist und nt*(V} S m*(U) geniigt. Der iiber N aufgenommene Mittelwert von einer stetigen Funktion 9(U) auf N 'sei mit M9(U) bezeichnet. u

1st \p(t) stetig auf G, so ist offenbar M¥'Cut) eine stetige Funktion auf u

G. Normiert man nun das Haarsche Mass n* von N sodass n*(N)=l, so gilt fUr jede stetige Funktion \P(t) auf G nach dem Satz von Fubini

f .M9(ut)m(df) = JA f M9(ut)m(dt)=f m(dt)J 9(ut)n(du) u A N

JAU

=

t

n(du) LIP(ut)m(dt) = L9(t)m(dt) ,

wobei A eine m*-mess bare Teilmenge von G und A die zu A entsprechende Teilmenge von G sind. § 4. Einzigkeit des Haarschsen Masses. Wir beweisen nun, dass das links-invariante Lebesguesche Mass erster Art u'uf G mit dem zweiter Art, niimlich mit dem Haarschen Mass, ubereinstimmtl). Zu diesem Zweck genugt es den folgenden Satz zu beweisen: Satz 7. Jede offene Teilmenge von G ist m*-messbar. Beweis. Es sei U eine offene Teilmenge von G mit m"'(U) < + 00. Setzt man m*(U)=sup m(B), B d(G), (m*(U) ist das m*-innere Mass BP)k"'l = -

]yoi-/Y9>ik" 'l),

(t*9>P)iik ... Z= Oi9>ik .. ·l- Oj9>ik .. 'Z+Ok9>ij" ·l- +"', 1 rjk ... /)1) - --sgn 12 if) (rnp*)pq'''T r - , ,rjk· .. !,

(pq ...

p

.-v(/

· ........

n

where OJ=o/oxj, g=det(gjk), 9> jk"'l=gjpgk q... gIT9>pq"'r. The operations t, t*, * satisfy obviously

= tOt' = 0, = (-l)p(n- p)9>p, t(9)P*) = (- V-P(t*epp) *. tt

(1. 1) (1. 2)

9>P* *

A p-chain Cp in ID1 is a formal linear combination r 1 Tf+"'+rmn,of geometrical simplices n lying in ID1. Introducing the plane elements dx iTe "' l =

O(Xixk'''Xl ) 1 2 du du .. ·dup 0(U 1U2 .. ·UP)

on each TZ by means of a parameter system u 1 , u 2 , integral (epP, Cp) of 9>P on Cp= Lrk T~ by (epP,O)

"',

uP on TZ, we define the

-i-

= Lrk P.JTf r epjk ... l(x)dx jk " ·l.

Then, denoting the boundary of C by tC, the Green-Stokes' formula is represented as 1) sgn

(1%:::1) means the sign of the permutation (1Z:::!), if j, k, ''', I coinci:le with p, q, "', r

in a certain order; otherwise it means O.

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143

K.

KODAIRA

(r*S9, C) = (S9, rC).

We introduce furthermore the product

1

(S9 P·CP'k··/c(x) = (CP'·S9 P)j ... /c(x) = ,S9j ... /cpq ... rcppq"·r, a.

and the inner product (S9 P, CPp)G = LS9 P·CPP(x).fgdG,

(dG = dx 1dx 2 .. ·dx n),

where G is an arbitrary sub domain of m. Especially if G=m, we write (S9, CP) for (S9, CP)'J]/' Then we have S9*'CP* = S9'CP, (S9*, cP*) = (S9, CP),

and the Green-Stokes' formula (1.3)

(r*S9,cp)G-(S9,rcp)G=

l{S9,cp}j~gdoj,

where r is the boundary of G and do j = (-l)j-ldxl .. ·dxJ-ldxJ+l .. ·dxn. Now let G be an open subset of m (or m itself). A field S9 is said to be regular harmonic in G, if S9 admits continuous first derivatives and satisfies rS9 = r*S9 =

°

everywhere in G. By a harmonic field we shall mean a field e which is regular harmonic in m except at most a nowhere dense compact subset S of m ; then e is said to be regular in m-s and singular in S. Especially if e is regular everywhere in m (i. e. if S is empty), e is called a harmonic field of the 1st kind. For a fixed harmonic field e, the integral (e, C), considered as a linear functional of variable chains C, is called a harmonic integral. It is obvious by (1. 2) that, if S9 is regular harmonic in G, the dual field S9*, is also regular harmonic in G. In the simplest case n=2, every Riemannian manifold m is conformally fiat, i. e. in a suitable system of coordinates xl, X2, the metric ds 2 has the normal form ds 2 = ./g{(dxl)2+ (dX 2)2}.

m can be considered therefore as a Riemann surface with the local uniformization variable z=xI+.;=y x 2 , so that harmonic integrals (e, C) =

J

e1dx1+

e2dx2 coincide with the real (or imaginary) part of abelian integrals. In fact, since e= (el , e2) and its dual field e*= (e2, -el) satisfy (he2-a 2el = r*e = 0, a1e2*-a2el* = r*e* = -re = 0, the integral

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144

Relations between Harmonic Fields in Riemannian Manifolds

is locally univalent. Moreover, J(z) is an analytic function of z, since, by t*u=e, t*v= -e, u and v satiSfy Cauchy-Riemann's differential equation t*v = -(t*u)*.

Thus we see that the harmonic integral]ejdx j coincides with the real part of the abelian integral]df(z). The notion of abelian integrals can be extended to general Riemannian manifolds of even dimensions. In the case of Riemann surfaces, the integrand w

= e-.J-1 e*

of an abelian integral! wjdx j can be characterized as a vector field satisfying tw = tOw = 0, w* =.J-1w.

This suggests the following Definition 1. Let fin be a Riemannian manifold of even dimension n=2).1. By an abelian field on fin we shall mean a field w of the rank ).I satisfying tw=t*w=O, w* = rw, is a constant. The integrals (w, C) of an abelian field w will be called

where r abelian integrals. Since w**=(-l)"w, the constant r in this definition satisfies r2=(-1)"; hence we have r = ±.J-l, for odd).l, r = ±l, for even).l. In case ).I is odd, abelian integrals are therefore always complex integrals; while, in case ).I is even, abelian integrals lie in real domain. The existence of harmonic integrals of tne 1st kind was established by W. V. D. Hodge!); he treated also harmonic integrals with singularities in the case that fin is conformally fiat. In this note we shall investigate harmonic integrals with singularities in general Riemannian manifolds fin. In § 2, we state the existence theorems of harmonic fields. The proof of these theorems based on H. Weyl's method of orthogonal projections will be given elsewhere. 2) In § 3, we discuss the relations between various harmonic fields; 1) W. v. D. Hodge: Proc. London Math. Soc. Ser. 2 Voi. 36, 257-303; Vol. 38, 72-75; Vol. 41, 483-496. K. Kodaira: Proc. Imp. Acad. Tokyo Vol. 20, 186-198; 257-261 ; 353-358. 2) K. Kodaira : Harmonic tensor fields in Riemannian manifolds.

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145

K.

KODAIRA

especially we prove the reciprocity formulae and Riemann-Roch's theorem. In § 4 and § 5, we treat abelian fields. § 2.

Existence theorem. We begin with the following Theorem 1. For every point ~ E fin and every set of p indices A, fl, "', II, (p ~ n), there exists one and only one harmonic field

e = ejk ... lj Jp ....(x ; ~) of the rank p, which is regular in fin except for

~

and satisfies

(e, rC) = 0

for arbitrary chains C in fin, (2.1)

(e*, Z) = 0

for arbitrary cycles Z in fin, and pr2,

./iiYi> .... ", liiYI

·lily"

g,;, .... " ''', g'l

·liiy, , g,l, where r=r(x;

~)

......... , g,l

is the geodesic distance from

~

to x and

=

gpi gpi(~)' Ylc = glci(~)(XL~i),

1

(J}n = the surface area of the n-dim. unit sphere. The function ejk... ll ,p... ,(x ; ~) is holomorphic with respect to x, x=~ and satisfies

~

except for

(2.2)

Thus, as a function of ~, ejk... ll JP".,(x ; ~) constitutes a tensor field with suffixes A, fl, ...... , II.

In the case n=2. p=l,!ej I ,(x; ~)dxj. (A=l, 2) represents real and imaginary parts of the abelian integral!d!'(z) having the pole of 1st order at (=;1+ .; -1 e2 • As in the case of Riemann surfaces, various harmonic fields can be obtained from ejk"' ll 'p".,(x ; ~) by integrations and differentiations. First we introduce Definition 2. For arbitrary chains C in fin, we put e[C] = e[C]j/c ... 1(X) =

~ rejk"' 'p... ,(x ; ;)dew",. p .le ll

Then we have

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146

Relations between Harmonic Fields in Riemannian Manifolds

Theorem 2. e[C] is regular harmonic in Wl-rC. e[C] satisfies (e[C], r¢) = 0

(2.3)

for arbitrary fields ¢ and (e[C], cp) = (cp, C)

(2.4)

for fields cp satisfying r*cp=O, where the integrals (e[C] , r¢), (e[CJ, cp) converge absolutely. Conversely, if C is given, e[C] is uniquely determined by the conditions (2. 3) and (2. 4). The dual field e[C] * of e[C] satisfies (e[C]*, Z) = ifJ(Z, C)

(2.5)

for arbitrary cycles Z, where ifJ(Z, C) denotes the intersection number of Z and C. Theorem 3. For arbitrary chains C and D, we have (e[C], D)

(2.6)

= (e[D], C) =

(e[C], e[D]).

For arbitrary cycles Z, e[Z] is obviously a harmonic field of the 1st kind. The linear space consisting of all harmonic fields of the 1st kind will be denoted by ~f. Then a theorem of W. V. D. Hodge1) can be stated as follows: Theorem 4. The mapping Z --+ e[Z] maps the p-dim. homology group of Wl isomorphically on @;p. Corollary. If a harmonic field of the 1st kind e of rank p satisfies (e*, Zn- p) =0 for all cycles Zn- p , then e=O.

Proof. By the above theorem, e is represented as e=e[Z], so that we have, by (2. 5), (e*, Zn- p ) =ifJ(Z, Zn- p ) , from which follows immediately the corollary. Now we shall investigate point singularities of harmonic fields. Theorem 5 (Poisson's formula). Let C be a subdomain of Wl with the boundary rand cp be a harmonic field which is regular in CUr; assume, furthermore that there exist two fields f/J, 'Iff such that

cp = t*f/J = t'lff,

Then, for every x (2.7)

E

CPjk •.• I(X)

c, (jJj/C ••• l(X)

=

-1

in CUr.

can be represented as the surface integral

{e jk"' 11 (x;

)·f/J+ejk"'11 (x;

). 'Iff}a..j(jdo ••

Proof. Let C be an arbitrary chain lying in C. Then, considering that f/J and 1) VV. V. D. Hodge: loco cit.

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147

K. KODAIRA

'Iff are extended over whole an and applying the Green-Stokes' formula (1. 3), we get from (2.3) and (2.4)

0= (e[C], t'lff) = (e[C], t'lff)a+ (e[C], r'lff)'l1l_0 = (e[C], ~)a+ 1ce[C]. 'Iff)"./Ydoa,

(~, C) =

(r*$, C)

= (e[C], r*$) = (e[C], ~)a-1(e[C] .$)a.jYdoa ;

hence we have

(~, C) =

-1

{e[C] ·$+e[C]· 'Iff}a.jYdoa,

which yields immediately (2. 7). Theorem 6. Let G be a subdomain of an, ~o be a fixed point in G, and ~=~p be a harmonic field which is regular in G except for ~o; in case p=l or p= n-l we assume furthermore (cp*, r) =0 or (cp, F) =0, resp., where r is the surface of a sphere with the center ~o. Then, if cp satisfies r = rex, ~o),t)

cp(x) = O(r-n-l+l) ,

can be represented in a neighbourhood of ~o as the series !-1 1 1 '" '" 'p"" pq···rM M M e (e:. ) (2.8) CPjk ••• l( X) =~-I~-Ic ' VpVq"'Vrjk"'!I,p""X;O

cp(x)

m~om.

p.

~

+regular harmonic field, where Vq means the covariant differentiation with respect to ~q. Proof. Let r" r R be the surfaces of the concentric geodesic spheres of the radius e, R with the center ~o. For~, we can readily construct $, 'Iff such that cp = t*$ = t'lff,

$(x), 'Iff(x) = O(r- n- l +2 ).2)

Hence, applying the Poisson's formula to the ring domain between

r R , we get cp(x) =

(1. -1J

{e(x;

)·$+e(x;

). 'lffV.jYdoa,

Now we expand e(x, ~) in the first integral e(x' ~) = ,

!-1

1

am

m~O

m !

a~Pa~q"'a~r

1: -

Jr,r

(e1+>2··· ···+>8

be a divisor of the rank p in [n and A, B, b, d the constants defined as follows: A: the number of linearly independent harmonic fields ~ in [n satisfying ~

== 0 (d),

(~,

Zp) =

(~*,

Zn- p) = 0,

B: the number of linearly independent harmonic fields (J)

== 0 (d- i );

b: the p-Betti number of [n; d : the order of the divisor d. Then we have

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152

(J)

in [n satisfying

Relations between Harmonic Fields in Riemannian Manifolds

(3.13)

A = B-d-b.

Proof. Let el, ea, ...... , eb be the base of the space ~p. By theorem 10, 'P is represented as 'P= ~aie[pi], so that we have, using the reciprocity formulae (3. 8), (3. 9), ('P, Z)

= ~ai(e[piJ, Z) = ~ai(e[ZJ, Pi), ~ai(e[piJ, qj) = ~ai(e[qjJ, J;li),

(cp, qj) =

('P, C k ) = ~aie[1'iJ, C k ) = ~ai(e[CkJ,1'i)'

Hence A is equal to the number of linearly independent solutions aI, az, ...... , as of the linear equations ~ai(e" 1'i) = 0, (1 = 1,2, ...... , b), (3.14) ~ai(e[qjJ, J;li) = 0, (j = 1,2, ...... , t), ~ai(e[CkJ, Pi) = 0, (k = 1,2, ...... , 1). On the other hand, w is represented in the form

1

w

= ~h,e,+ ~bje[qjJ +~eke[CkJ ;

then we have (W,1'i) = ~h.(e" 1'i) + ~bj(e[qjJ, Pi) + ~ek(e[CkJ, J;li),

and thus B is equal to the number of linearly independent solutions hI, ...... , hb' b1, ...... , bt , e1, ...... , Cl of the linear equations (3.15) ~h,(e" +1t)+ ~bj(e[qjJ, l'i)+ ~ek(e[CkJ, 1'i) =

0,

(i

= 1,2, "', s).

The equation (3. 15) is obviously the transposed equation of (3. 14). Hence we have b+t+l-B=s-A, or A=.B-d-b, q. e. d. § 4. Abelian integrals, 1). The case: lJ=n/2 is odd. In this § we shall investigate abelian fields w such that

(4.1)

w* =..J-1w

(see definition 1); abelian fields w with w* = -..J -1 w will be obtained from w simply by conjugation: ..J-l-->-..J-l. Obviously an abelian field w (satisfying (4. 1» has the form (4.2)

w = cp-..J-l cp*,

where 'P=91ew is a real harmonic field, and, conversely, a field w with the form (4. 2) is an abelian field, if 'P is harmonic. First we shall investigate the structure of ~'. Since 'P** = -'P, we have, by the formula 'P*.¢*='P.¢, 'P·'P*='P*·'P**=-CP·'P*, or cp.cp* = 0,

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153

K. KODAIRA

and therefore (q;, q;*) =0.

Hence we can choose a system of harmonic fields of the 1st kind (4.3) so that (4.3) constitutes an orthogonal normal real basis of the space Introducing the cycles Z" Z, * such that

e,

= e[Z,J,

e,*

=

e[Z'*J,

(.< = 1,2,

~'.

······,P),

we get a real basis {Z, Zl *, ...... , Zp, Zp *} of the Ii-homology group of ~. This will be called a normal basis. The Ii-Betti number b' of ~ is therefore even: b'=2 p. Putting w,

we get a (complex) basis {Wl' the 1st kind. The system

{J

= e,-H e,*,

W2, ..••.• ,

wldx, ......

wp } of the space of abelian fields of

,J

wpdx} constitutes obviously a basis

of abelian integrals of the 1st kind. Thus we see Theorem 12. The Ii-Betti number b' is even: b'=2p. There exists on exact P linearly independent abelian integrals. Using (2. 6) , we get the formulae

~

(w" Z.) = 0,., (w" Z. *) = --/ -1o, •.

(4.4) (4.5)

Now we put

= e[1'J --/ -1 e* [1'J, = e[eJ --/ -1 e* [eJ, where l' means an arbitrary pole, e an arbitrary chain. Then we have w[1'J w[eJ

w[1'*] = --/ -1 w[1'] +w[Z(1')] ,

(4.6)

as is readily verified by (3.3) and (3.4). w[1'] will be called an abelian field of the 2nd kind, w[eJ an abelian field of the 3rd kind; corresponding integral

J

w[p]dx

orJ

w[eJdx will be called an abelian integral of the 2nd or the

3rd kind, resp. The meaning of divisor d

= qlq2 ...... q e l .. · .. ·ed1'l1'2 ...... 1'. t

is similar as in

§ 3, but we assume here that the abelian fields W[l'l], ...... , w[l'.J, W[ql], ...... , w[qt], w[elJ, ...... , w[elJ are linearly independent mod (Wl,

W2, •••..• ,

wp ) with resp. to complex coefficients. For arbitrary abelian fields

", we shall mean by T=:O (d) that T satisfies

(15)

154

Relations between Harmonic Fields in Riemannian Manifolds

T=:O mod(w!, W2, '" ... , lOp, w[l'!], ...... , w[l's])

and (r, qj)

= (r, C = 0 k)

(J' -- 1" 2 ...... " t· k

= 1" 2

...... /) "

by r=:O (d- I ) that r satisfies r =: 0 mod(w!, ...... , lOp, W[ql], ...... , w[qtJ, w[CI ],

.••..• ,

w[C t ])

and (i

= 1,2, ... "', s).

Theorem 13 (Reciprocity formulae). (4. 7) (w[~], q) = (w[q],~) +~-l(e[Z(~)], q), (4. 8) (w[~J, C) = (w[C], ~) +~ -l(e[Z(~)J, C).

Proof. (4. 7) follows immediately from (3.8) and (3.10); (4.10) follows from (3. 9) and (3. 11). Theorem 14. For arbitrary poles ~ and cycles Z, we have (4. 9) (w[~], Z) = ¢(Z, Z(~», (4.10) (w[Z],~) = ¢(Z,Z(~»+~-l¢(Z,Z(~*».

Proof. (4.9) foHows immediately from (3.5) and (3.6). As to (4.10), we have (w[Z],l') = (e[Z], 1') -~ -l(e* [ZJ, 1')

= (e[ZJ, 1') +~ -l(e[Z], 1'*),

whence we get (4. 10) , using (3. 12). Now we shaIl investigate abelian fields r satisfying Cr, Z) =0 for all cycles Z (of rank 1.1). Theorem 15. Let ~I, ~2' •••••• , ~8 be a system of poles. Then, for given complex numbers ai=ai+~-l ai, (i=1,2, ...... , s), there exists an abelian field r such that

r"" Laiw[l'i], if and only if

ai=ai+~ -1 a~

(T, Z) = 0 for all cycles Z,

satisfy the condition

(4.11) and, if (4.11) is satisfied, such T is determined uniquely and represented as (4.12) Proof. Since, by (4. 6) ~-1 W[~iJ- -W[~i*J, an abelian field r satisfying r - Laiw[~i] is represented as T

= L{aiw[l'i]-a~w[l'*J}+LXAWA'

( 15)

155

K.

KODAIRA

For such 't" we have, by (4.4), (4.5) and (4.9), ('t", Z,) = ¢(Z" L{aiZ(~\) -a~Z(pi*)J) +X', (r-, Z,*) = ¢(Z/, L{aiZ(Pi) -a/Z(pi*)}) -"f-l x" whence we conclude that T satisfies (T, Z) =0 for all Z=Z" Z,* if and only if Xl=O and L{aiZ(Pi) -aiZ(Pi*)}CIJ 0, since the intersection number ¢ is always real, q. e. d.

Theorem 16 (Riemann-Roch's theorem). Let d = qIq2······q t CI······Cz/PIj:J2······Ps

be a divisor in IDl of the order d and A, B the constants defined as follows: A : the number of linearly independent abelian fields T satisfying T

== 0 (d),

(r, Z) = 0 for all cycles Z,

B: the number of linearly independent abelian fields w satisfying w

== 0 (d- I ).

Then we have A=B-d-p,

(4.13)

where p means the half of the ].I-Betti number of IDe. Proof. By virtue of theorem 15, we may suppose that the form T

=

L

T

is represented in

{aiw[j:Ji] -aiw[j:Ji*]}

by means of the coefficients ai, a: satisfying (4. 11). Then we have for arbitrary poles q (4.14)

(T, q)

= La;(w[q] , j:Ji),

since, by (4. 7) (r, q)

= L{ai(w[q], Pi) -a~(w[q], j:Ji*)} = L{ai(W[q], j:Ji) +a:(w*[q], j:Ji)} = L(ai+"f -1 a:)(w[q] , j:Ji).

Using (4.10), we conclude similarly (4.15)

Again, the condition (4.11) is equivalent to (4.16) in fact, we have, by (4.10), Lai(W[Z], j:Ji) = ¢(Z,

L {aiZ(j:Ji) -a:Z(j:Ji*)} +J -IL {aiZ(j:Ji*) +a:Z(j:Ji)}) ,

(15)

156

Relations between Harmonic Fields in Riemannian Manifolds

while the identity ~{aie[ZO\*)] + a!e[Z(Pi)] } = (~{aie[Z(pi)] -a;e[Z(pi*)]}) *,

deduced from (3.7), shows that ~[aiZ(Pi*)+a~ZO)i)}ClJO if and only if ~{aiZ(pi) -a;Z(p;*)} co o. From (4.14), (4. 15) and (4. 16) we conclude that A is equal to the number of linearly independent solutions aI, a2, ...... , as of the linear equations ~ai(WI,.pi) = 0, (A = 1,2, ······,P), ~ai(W[qjJ, Pi) = 0, (j = 1,2, ...... , t), ~ai(w[CkJ, Pi) = 0, (k = 1,2, ...... , /). On the other hand, B is the number of linearly independent solutions Xl, ...... , "J.p, 131, ...... , f3t, 71, ...... , 7l of the transposed equations

1

1:XI(W" Pi) + ~f3iw[qjJ, Pi) + ~rk(W[CkJ, Pi) = 0,

(i =1,2, ...... , s),

since, w is represented in the form w = ~;VWI+ ~f3jw[qjJ + ~rkW[CkJ.

Hence we get s-A=p+t+l-B or A=B-d-p, q. e. d. § 5. Abelia.n integra.ls, 2). The case: l.I=n/2 is even. By definition, an abelian field W is a (real) harmonic field of rank l.I satisfying

W*=±W;

thus an abelian field is a special harmonic field. An abelian field (1) will be called a (+) -abelian field, if W* =W, and a (-) -abelian field, if W* = -w. Every harmonic field cP is decomposed uniquely into the sum of a (+) -abelian field cp+ and a (-) -abelian field cp- : cp = cp++cp-,

1 cp± = 2(cp±CP*).

The (+) - (or (-)-) components of harmonic fields of the 1st, 2nd, and the 3rd kind will be called (+) - (or (-) -) abelian fields of the 1st, 2nd, and the 3rd kind, resp. ; we write, for example, e+[~J instead of (e[pJ)+. The corresponding integrals will be called (+) -or ( -) -abelian integrals of the 1st, 2nd, and the 3rd kind. Since cp+·1r=cp+*·(Z, Z(\)+» , = , it is sufficient to consider the case that l:> lies on a single branch r/ of IDI. In N(\) , e[C] is represented as e[C] = 21

where 8 pq{x, .;) =

rt*8 q(x, ';)[d';Pd.;q] +regular form,

Jr'p'

p

~ 8 jk ;pq(x, ';)[dxfdx k ]

means the elementary solution of La-

place's equation .:18=0.10) Choose the local coordinates zl, Z2 with the origin \) so that the parametric representation of r/ takes the form (4). Then, using the explicit expression of e[C] mentioned above, we infer readily le[C]1 =

°(1: 1). 2

for

Zl

= 0,

while it is obvious that

I -lap

O(1Z2J' 1)

for

Zl

= O.

Zl

= O.

The difference 0= e*[C]-ap

satisfies therefore also the inequality (7)

101 = °(1: 1). 2

(16)

for

167

K.

KODAIRA

On the other hand, 0 is regular harmonic in N('p) -.j) and satisfies IloIIN(~)

< + 00.

o is therefore represented as o(x) = co·dE(O, x) + regular form,

where 8(x, ~) means the elementary scalar solution of Ll8=0. Now (7) shows that the coefficient Co in this expression must vanish, since 18(0, x)l- {lz l l2 +IZ212}-1. Consequently o=e*[C]-a~ is regular harmonic also in .j). Thus we conclude that, for every point .j) E an, e* [C] is represented as

(8)

e* [C] =

a~+regular

harmonic form

in a neighbourhood N(.j) of .j). Now we write 41r.J -1 e* [C] as 41r.J -1 e* [C]

= cp.dz·-ip.dz·

and put f/J = cp.dz·.

Then, since 41r.J -1 a~ = Lmk d log fk~- Lmk d log fk~' it follows from (8) that (9)

is regular (with respect to real coordinates xl, x 2, x 3, X4) in N(.j). This shows that t*f/J is regular everywhere in an. On the other hand, f/J satisfies, by Lemma 3, Llf/J=O in an-IDI, whence we get Llt*f/J = t* Llf/J = O.

Thus t*f/J satisfies Llt*f/J=O everywhere in an. Consequently, by Lemma 2, t*f/J is a harmonic form of the first kind. To prove r*f/J=O, it is sufficient therefore to show (10)

for every 2-cycle Z. Assuming that Z meets with IDI only in a finite number of simple points of IDI, we infer from the fact that the difference (9) is regular in N(.j) the equality

l

r*f/J = 21r.J -1 J(Z, D) ,

where J(Z, D) means the intersection number of Z and D. Now, since D;::::;O.

(16)

168

EXISTENCE OF ANAL YTlC FUNCTIONS

we have I(Z, D) =0, proving (10). Thus we get t* aj in the above introduced partial ordering.

Obviously

(1.4)

hence, the Alexander product is defined also for the elements of *B(K). the simplicial mapping 0 preserves the partial order>, we have 0*(.1

0

B) = 0*.1

0

Since

o*B.

This implies that the isomorphic mapping from *B(K) to *B(N) preserves the Alexander product. Hence, the Alexander product defined in *B(K) does not depend on the artificially introduced partial ordering in the set of vertices of K. Now let K be an n-dimensional closed orientable lwmology manifold. Then [tnlt n, there exists on K a fixed n-dimensional "fundamental cycle" zn = where [tl denotes the "orientation" of t" (so [tnl = ±1). Using [tl, we define the "dnal-cell-operator" b to be a linear operator mapping L(K) to L(N) such that DS P = [t][t n, tn-I, .. , ,tP+I, sPl(t n, ... , tP+l, sP).

Lt

Lt

Then the absolute complex 1 bs P 1 is always a (n - p)-dimensional homology simplex, and is called the dnal-cell of sp. N is divided into the cells 1Ds PI. The cell-complex D consisting of all cells 1 bs P1 is called the dnal complex of K. One can prove easily that rbP

(1.5)

= (-1) n-Pb r*q,p.

* BP(K) is therefore mapped by b isomorphically on Bn-p(D). On the other hand, Bn-p(D) "" Bn-p(N) and

0

maps Bn-p(N) isomorphically on Bn-p(K).

Hence,

ob maps *BP(K) isomorphicallyon Bn-p(K).

Finally, we define the bilinear functional bq,q X CP(u bs" X uP = [n P, tP-I, ... , t"+\ s"](n P, t"-\

~

p) of q," and CP by

Lt

.. , ,t+l, SO). P b ." >

l)

597

HARMONIC FIELDS IN RIEMANNIAN MANIFOLDS

will satisfy r*if;

= I{!, but the formula y = (x\ '" ,x"-I, x

(3.3) is useful only for such points x, that

P

,

•••

,xi - \ yi, 0, '" ,0), (yi

=

(Jx

i

,

°

~

(j

~ 1, p ~ i ~ n)

are all contained in G. Hence, for example, 1f G has the property (3.4)p For any x E G the set 1 P 1 t. = p, P {y I y = (x,"', x ,,-1 ,y,"', y n) ,y i = (jix,i 0 < = (ji .(x) be a function with continuous second derivative defined in the interval (0, 1) so that

°
,

(e*( 2),

1

+ vex, ~),

{ - 4'/1' log P(x, ~). U(x, ~)

(n = 2)

satisfying AZ(x,~) = 0 as a function of x, whereby w,. denotes the surface area of the n-dimensional unit sphere, n-2 T = -2-' U(x,

~)

and vex,

~)

are holomorphic matrix functions of x, U(~, ~) =

~,

and

1(= unit matrix). 13

Our next task is to prove the existence of an elementary solution. Let l(P) be any scalar function of P and W(x, ~) any matrix function of x and~. Then, W(x, ~) as a function of x, we have considering l{P)· W = UP(x,

m·

A·l(P)W Putting

= l(P)'AW + l'(P) {g"fJo"OfJP + A "P" + 2P"o,,} W + l"(P)P"p,,· W. 4N = 2n - g"fJ O"OfJP - A "'P"

and using (7.7), (7.8), we obtain from this

(7.13)

A·l(P)W

= l(P)·AW + l'(P) {2n - 4N +

Using the "normal variables" '/I'i and put

= '/I';(x,

4r :r} W

~) instead of x\ we write N

(7.14)

11

We shall follow the method of J. Hadamard. Cf. Hadamard [1].

C 17

J

+ 4l"(P)PW.

197

=

N(~, '/1')

613

HA.RMONIC FIELDS IN RIEMANNIAN MANIFOLDS

whereby 11'; = 11'\X, ~). To prove the convergence of the series (7.14), we shall use for a moment the notion of the norm 1 X 1 of matrices X defined by

1X 1 =

v'L: 1xrt::[ 12,

where L: means the sum for aIIj, k, ... , l, p, q, .. , , r such thatj < k < . " < l, p < q < ... < r. Since N(~, 11') = 0(11') by (7.10), (7.9) and (7.3), l/t N(~, t11') is holomorphic with respect to ~, 11' and t, and there exists a constant k such that for 0 where r

= rex, ~).

I

~

t

~

1,

Hence

II ··· I

N(~, t111') •.. N(~, tm11') td2 ... tm

dtl .. , dtm

1< ktnr m!

m ,

1 2, applying the above procedure to the row Uo , U1 , ••• , U.- 1 , Vo , VI, Vi, ... , we first obtain

14

-2.-1 ) , -2,-1 (1 - ~) ,

U.

«'l (1

V

r

p

-

« 'Y

P

-

~

(v = 0, 1, .. , , (v

Hadamard [l].

( 17 J 201

= T, T

T -

1),

+ 1, ... ),

Hence.

1+

fJ )-1

--(I'

p,

PUp for

617

HAR:IIONIC FIELDS IN RIEMANNIAN MANIFOLDS

= (T + 1/2)a'.

where'Y

r

Putting f32

V.

= 'Yg, we have therefore

« 'Y' e:r

~)

(1 _

-2.-2.-1 ~

where

"" (f30')2' ( 1 - 0'- )-21'-2.-1 « ( 1 - -0')-2T ( 1 L.-oP P P Hence the series

L , 1(; ,
-1 «:

L~-l ( f3


..

=

-p

a~ a~a a~

at;

-"

•••

a~ a~'Y

p

q

r

ax ax ax _afl··· 'Y axi axk ••• ax! ':!'pq ••• r

•

Then the quantity Z~~:::'i(x, ~) will appear as a "double skew-symmetric tensor", i.e. a quantity transformed by arbitrary transformations of coordinates ~ or x as a skew-symmetric tensor with suffixes >., p., •.. , v or j, k, ... , l, respectively. Thus we conclude THEOREM 10. For every point ~o on 911, the Laplace's equation double skew-symmetric tensor Z has an elementary solution

1 Iog P . U).I'''·V( - 41r ik .. ·! x, ~ ) , _"I""'() ,:!,ik·.·1 X, ~

1

+ V~I" .. p( x, '>I!) , ik·"!

p-r U).I'''·' ( ) ik··· I x, ~ ,

1

( 2w ')'

= 1n-

~ =

0 of

(n

=

2),

(n

=

3, 5, 7 ... ),

(n

= 4,6,8, ... )

n

1 p-r U~,,···p( ) • (n - 2)w n ik···1 X, ~

I,

+ log p. V~~:::i(x, ~),

defined for x, ~ lying in a certain neighborhood of ~o , where T = (n = P(x, ~), U(x, ~) and Vex, ~) are holomorphic with resp. to x, ~, and

2) /2,

p

U~~:::~(~, ~)

=

I~~:::'i

(l~~:::i denotes the component of the skew-symmetric unit tensor I, i.e. I~~:::~ = sgn (~r::m. Now let G be a small open subset of 911 with the regular closed boundary r such that the elementary solution Z(x, ~) is defined for arbitrary x, ~ E G + r. Suppose ~ E G as fixed, and let S(~) be a geodesic sphere with the radius ~ having ~ as its center, r(~) the surface of S(~). We shall apply the Green's formula (4.10) to Z and arbitrary field ({J having continuous second derivatives with respect to the domain

Ga = G -

S(~),

considering Z = 'i'1'''''( , ~) as a tensor function of x (co-rresponding to the abbreviation ({J for ({Jjk· •. I(X) we write ~... p( ,~) for Z~:: :'i(x, ~), 'omitting the variables x and suffixes j, k, ... ,l). Then, since ~ = 0, we lilive (7.23)

(d({J, 'Z)Oa

+ Jfr m«({J, Z)i v(j dOi = JrrQ) m«({J, Z)i v(j do j •

(17)

203

619

HARMONIC FIELDS IN RIEIliL

(n

= 2).

41r'(IPW)(~),

From these results we see immediately lim a-o

while, by U~~:::i

r WIP, z)i v'o dO i = Jr(6)

l~~:::I' (IPU>q.···,( ,

=

mm

=

-(IPU)W,

lip! lk ... lm u~r::~(~, ~)

= i!'· .. ·(~).

Hence, we obtain from (7.23) the identity (7.25)

-1P}.p.····W = (6.\0, Z).!'····( , ~»G

C

in

+ Ir lR(lP, Z).p.····(

204

,

m v'o do i

i•

2),

620

KUNIHIKO KODAIRA

=

If especially I{! satisfies the Laplace's equation AI{! I{!AI""'W

(7.26)

=-

l

m(l{!, 'ZAI' ...•(

0 in G, then we have

,m yg do i

i•

Since 'Z(x, ~) is holomorphic with respect to x, ~ except for x = ~, the integral in the right side of (7.26) represents a holomorphic function of ~ defined in G. Thus we conclude THEOREM 11. If afield I{! defined in an open subset G of'1R has second continuous derivatives and sati8fies AI{! = 0 everywhere in G, then I{! is holomorphic. §8. The proof of the fundamental theorem

In virtue of Theorem 11, to prove Theorem 8 we have only to show that f has continuous second derivatives. In fact, if f has continuous second derivatives and satisfies (f, A71kr = 0 for every 71 C D having continuous derivatives up to the order v, then Af = 0 by (t::.f, 71)0 = (f, t::.71)0 = 0, and hence, by the above theorem, f must be holomorphic. Assume now the closure 5 of the open set D in Theorem 8 to be compact and contained in a certain open subset 0 of '1R such that the elementary solution :E(x, ~) is defined for arbitrary x, ~ EO. Since, as was already remarked, Theorem 8 has local character, it will be sufficient for our purpose to prove Theorem 8 under such assumptions. Let G be an arbitrary open subset of D such that the closure G is also contained in D. We denote further II //p,o by II lip: III{!

lip = II I{! IIp,o.

Under these conditions, we choose a positive constant a SO small that for every ~ E G the "geodesic sphere" {x / rex, ~) ~ 3a} is contained in D, and construct functions n(p) and A(P) of P = r2 having continuous derivatives up to the order v 2 so that

+

n(p)

A(P)

=

=

r-' r

for

r

~

a,

0,

for

r

~

2a,

Ogp

for

r

~

a,

for

r

~

2a,

0,

' ,

(P

=

r2)

whereby the behaviors of n(p) and A(P) for a < r < 2a may be freely determined. Replacing the functions p-r and log P in:E by the functions n(p) and A(P), respectively, we define the "modified elementary solution" Y(x, ~) for x E 5, ~ E G by

Y(x, ~)

=

en _1 )wn n(p(x, 2

fl-

1

~». U(x, ~) + A(P(x, m· Vex, ~),

411' A(P(x, ~». U(x, ~)

+ vex, ~),

C 17)

205

(n

> 2),

(n

=

2).

621

HARMONIC FIELDS IN RIEMANNIAN MANIFOLDS

< iX, put

Again, we choose a positive constant {3

IIp(P) =

~ {nt J2(3"

-

en -

2)P}

,

lll(P) ,

=

A~ep)

and define Y~(x, ~) for x

Yfj(x, ~)

=f

2 log {3 - 1 { E

+ (3~'

A(P) ,

for

r ~ {3,

for

r

> {3,

for r

~ {3,

for

> {3,

r

5, ~ E () by

en _1 2)w" lliP(x, m· U(Xi ~) + A(l(P(x, m· Vex, ~),

l- 41r1 Afj(P(x, ~». U(x, ~) + vex, ~),

(n

> 2),

(n = 2).

(ll/l(P) and A~(P) are so defined that i) they are smooth functions, that ii) for ~ (3 they coincide with ll(P) and A(P) respectively, and that iii) they are linear for r ~ (3). Yex,~) admits continuous derivatives up to the order II 2 except for x = ~, and

r

+

Y(x, ~)

Z(X' ~), for r(x,~) ~ a,

={ 0,

r(x,~) ~

for

2a j

hence /lY(x, ~) has continuous derivatives everywhere up to the order /lY(:t, ~)

= 0,

a

except for

II,

and

< r(x,~) < 2a.

Y~(x, t) adroit continuous first derivatives everywhere, but their second derivatives present certain discontinuities for rex, ~) = (3. Y,,{x,~) differs from Y(x, ~) only for rex, ~) « f3j hence

(8.1)

AY~(x,

t) = /lY(x, t),

for

rex, ~)

> f3.

While, by definition, Yfj(x, ~) is holomorphic for rex, t) < f3, and therefore /lYfj(x, t) is also holomorphic for rex, ~) < f3. Moreover, as is easily verified by (7.13), if f3 tends to zero, then f3"/lY/l(x, ~) tends to -n/wn ·1 (1 denotes the "unit double tensor" sgn (~r::m uniformly for rex, ~) < (3:

(8.2)

lim f3"/lYfj(x, ~) (J.... O

= (-n/w,,) '1,

uniformly for

r(x,~)

< f3.

Now we shall prove that for any f3 < a there exists a sequence of double tensor fields Y.. (x, ~), 1'11, = 1, 2, ... , having continuous derivatives up to the order II + 2 everywhere, such that

Y..(x, ~)

= Y(J(x, ~),

for

r(x,~) ~ {3

and (8.3)

lim (ip, /lY m)O m-+oo

(17)

=

(ip; /lY(J)o

206

622

KUNIHIKO KODAIRA

for. every field", with finite nonn functions

II", lip defined in D.

w = l(P)· W,

or

l(P) = IIII(P)

AII(P) ,

To do this, we observe the W = U

or

V

separately. l(P) has continuous first derivatives everywhere; its higher derivatives exist and are continuous up to the order II 2 unless r = yIP = {3. Again, Now we construct a uniformly bounded for r = yIP < (3, we have l"(P) = O. sequence of functions l~(P) having derivatives up to the order II such that

+

=

t:.(P)

(8.4)

l"(P) ,

lim t:.(P) = 0 (= l"(P»,

for

r ~ (3,

for

r

< (3;

then we define l~ and lm by

1 l~(P) =1

=

'(P)

p

dP,

+'"

lm(P)

P '(P) dP +'" (remember that l~(P) = l"(P) = 0 for r = VP > 2a). defined admits continuous derivatives up to the order II

=

lm(P)

l(P),

for

Obviously lm(P) thus

+ 2 and

r ~ (3.

Again, by unifonn boundedness of the sequence {lm) and (8.4), we have 11'

lim m-+CIO

1I 0

t:.(P) I dP = 0,

II'

whence, by 1 '(P) - l(P) 1 ~ l i t : . 1 dP, we see

lim '(P) lim lm(P) m"''''

Using (7.13) and putting W' and w = l(P)W

1(""

=

l/(P),

=

l(P)

m .... '"

(8.5)

Aw",)o - ("', Aw)o I

+ i~1I1 '(P) while, by

=

{2n - 4N + 4r a/or) W, we have for wm

~ i~1I Ilm{P)

- l/(P)

(uniformly for r ~ (3).

- l(P)

= lm(P)W

II ",·AW I yg dG",

II ",W' I yg dG", + 4{32 i~1I Il~(P) II ",W I Vg dG""

II", lip < + 00, it is obvious that the three integrals i~1I I",· AW I ygdG.,

1~~ I",W' I yg dG.

and

i~~ I gW I yg

( 17)

dG", converge absolutely.

207

Hence, by

623

HARMONIC FIELDS IN RIEl'.IANNIAN MANIFOLDS

(8.4) and (8.5), we conclude lim (cp, Liwm),o = (cp, ~w),o . Now the desired sequence Ym will be easily constructed, by applying the above procedure to each term rr~· U, A~· V or A~U, V of Y~, according to n > 2 or n = 2. By hypothesis of Theorem 8, I satisfies (f, .117),0 = 0 for arbitrary 'YJ C 0 having continuous derivatives up to the order 'V. Since the field Ym(X, 0, considered as a function of x, admits continuous derivatives up to the order 'V + 2 and is CO, we see, by (8.3), (f, AY~),o

=

lim

(I,

AY m),o

= O.

Hence, putting (8.6)

we obtain the important relation (8.7)

As was already remarked, AY~Z:::~(x, ~) has continuous derivatives up to the order 'V everywhere in ~ f G, x f .0, and is equal to zero for rex, V ~ 2 a. Every derivative tXI""'(

of the order ;:;;;

'V

~ .. , ~ LlYxw "'( ~) , ~) = ~ a~a a~~ ap ,

is therefore C 0 as a function of x, and hence the integral

converges absolutely and represents a continuous function of ~ E G. This shows that, considered as a function of ~ E G, (f, AYxl""'( ,~»,o admits continuous derivatives up to the order 'V. Our Theorem 8 will therefore be proved, if one shows that (~

(8.8)

E

G)

holds except for a ~-set of measure zero, since G is an open subset chosen arbitrarily under the only restriction that G is contained in O. The relation (8.8) is proved in the following way. First we shall consider the general skew-symmetric double tensor field e~r::l(x, ~) with p suffixes X, /I., '" ,'V and II suffixesj, k, .,. ,l, defined for xED, ~ EG. The inner product (cp,exw ",(

,W ,0

--If!1 1,0f cpik".le~l'· "'( ,k···1

is obviously a tensor field with suffixes X,

eclPi'l'oO·'W =

1-', •• ,

(cp, eXI""'(

(17)

208

,~) ygg dG

,'V.

,~»,o,

We put (~E G)

624

KUNIHIKO KODAIRA

e

and consider as a linear transformation from the functional space on D to the functional space on G. Now we introduce the "invariant norm" of the double tensor eex, ~)

and put

r I e(x,~) 1v'U(x) dG""

kl =

k1ee)

=

sup ~,a Jo

ky. ...• ( (16.1)

,~), Z)

(e*;AI' ••.• (

for arbitrary (n- p)-cycles Z in fIn iii) in a neighborhood of ~, e;AI'" .p(

~, ,~)

~)

satisfies

=

0

~,

,~)

has the form

2

pr YA Y,. nyj giA gil' 1 w"r n+2

(16.2)

~~~ . .g.k~ . .~k~ nYI

glA

gil'

.,.

y. gi.

.. ,

gl.

....... ~k: + 0(r1-"),

where gil' = gil'W, Yi = gi"(X" - ~"), and r = rex, ~) denotes the geodesic distance from ~ to x. In fact, it is obvious that e;>y. ...• ( ,~) has the properties i) and ii), while, since e(x, ~)

=

r*r:Z(x,~)

+ holomorphic field,

(16.2) can be readily verified by actual calculation of r*r:Z(x, ~). Conversely, let e' be another field having the properties i), ii) and iii), and let us consider the difference

= e' - e;>y. ...• ( ,~). e"(x) = 0(r 1-"), and, in case elf

Then we have. by (16.2), p = n - 1 or p = 1, (elf, r) = 0 or elf X r = 0, respectively, for the surface r of a geodesic sphere

(17)

240

656

KUNIIiIKO KODAlRA

with the center ~, since e" has no singular point other than~. Hence, by virtue of Theorem 25, e" is everywhere regular in IDe, while by (16.1), we have (be", Z) = 0 for every (n - p)-cycle Z in IDe. Hence e" must be equal to 0, i.e. e' = e;>.,. ...• (

,~).

The field ejk ... l;>.,. ...• (x, ~) thus defined is holomorphic with respect to 2n variables (xl, ... , xn, ~\ ... , ~n) except for x = ~ (Theorem 22), and has the symmetric property (16.3) (Theorem 23, A».

Now we put, for arbitrary p-poles p

=

(0",

~),

m-l

[ )() ep x

( »,p = '( = (ex, p. m

1 >..,. ...• pq ... T~ (I!) -1),0" VpVq'" vre;>..p ...• x,,,,

.

for arbitrary p-chains C,

e[C](x)

=

(e(x, . ), C)

=~ p.

f e;>..p ...•(x, ~) d(w"', c

and e*[p]

e*[C]

= be[p], = be[C).

Then the fields e[p], e*[p] are "harmonic fields of the seeond kind," and e[C], e*[C] are "harmonic fields of the third kind," if rC ~ O. The fields e[C], e*[C] are regular in IDe - rC and singular in every point on tC. e[C] satisfies tl}e "integral equations"

= (t, C),

(16.4)

(e[C], t)

{l6.5)

(e[C] , tT/I) = 0,

(r*t = 0),

where r is an arbitrary exact upper field (having continuous first derivatives) and y; an arbitrary fields with continuous first derivatives (Theorem 16). The period of e*[C] on a (n - p)-cycle Z is given by the formula

(16.6)

(e*[C], Z)

=

I(Z, C),

which can be considered as the dual expression of (16.4) (Theorem 17). Furthermore we have (16.7)

(e[C], e[B])

= (e[B], C),

where B is an arbitrt,try p-chain such that rB C IDe - C (Theorem 18). For a p-cycle Z, e[Z] is a "harmonic field of the first kind." Moreover, the mapping Z ~ e[Z] maps the homology group B P of IDe isomorphically on the space cgP, consisting of all harmonic fields of the first kind (Theorem7). The mapping Z -> e*[Z] maps therefore B Pisomorphically on cgn- p • Denoting, for simplicity's sake, (n - p)-cycles by Z*, we have, by (16.4) and (16.6),

(16.8)

(e[Z*], e*[ZJ) = J(Z*, Z).

(17)

241

657

HARMONIC FIELDS IN RIEMANNIAN MANIFOLDS

The group B n - p is dual to BP with respect to the "product" I(Z*, Z), while es n - p is self-dual with respect to the inner product. The relation (16.8) shows that, under the isorrwrphic correspondence: Z* ~ e[Z*], Z ~ e*[Z] , the dualism between Bn- p and BP corresponds to the self-dualism of es n- p • Let {el' . .. ,eb} be a normalized orthogonal basis of the space es p • Then, putting

= bep,

e;

=

({3

1,2, ... ,b)

and introducing the cycles Zp , Z; such that

ep = e[Zp],

«(3

ep* = e[Zp], *

=

1,2, .. , ,b),

we obtain the normalized orthogonal basis {ej, e: , ... , e;} of the space es n - p , n p the basis {Zl, Z2, ... ,Zb} of B P and the basis {Zi, zi, .. ,', Zil of B - • The basis {Zi , Z: , ... ,Zil is dual to {Zl , Z2 , ... ,Zb} in the sense:

= Oap.

I(Z: , Zp)

(16.9)

§17. Reciprocity formulae ~)

Now we introduce the double tensor field e**(x, e**(x,~)

(17.1)

=

by

b~b~e(x, ~)

or

}.W eab"'D;ap...,/x, ~),

e**ik ... /;XI'····(X,~) = }.(x)

Uk ... 1ab ... c) or (Xp. ... II a{3 ... 'Y) being an even arrangement of the indices 1, 2, ... , n, and put, for arbitrary p-chains C and p-poles ).1,

e**(x,

~)

e**[C] (x)

(e**(x,

), C),

e**[p] (x)

(e**(x,

), p).

is obviously symmetric:

(17.2)

**... I;X" .... (x,~) eik

For fixed ~ and fixed X, p., ... , ~, and satisfies in

m-

II,

= ex"**...•;ik ... /(~, x).

the field e~:

..(

,~) is regular harmonic

e~:...• ( ,~),...., -e;x" ...• ( ,~),

(17.3) Indeed, since

e~:.... (x, ~)

+ e;x" ....(x, ~)

=

l

O(r -

n

),

(r

= rex, m,

as is readily deduced from (16.2), we infer, by Theorem 25, the relation (17.3). Now we put (17.4)

( 17)

242

658

KUNIHIKO KODAIRA

,~)

Then the field w;>.p ... ,( is also symmetric:

is regular harmonic everywhere in 9.n.

w(x,~)

(17.5)

Putting

= (w(x, w[p] (x) = (w(x,

w[CJ(x)

), C), ), p),

we get from (17.4) (17.6)

etC]

=

-e**[C]

(17.7)

e[p]

=

-e**[p]

+ w[C], + w[p];

again, from (17.5) follows (17.8)

(w[C], Z) = (w[Z], C),

(17.9)

(w(Cl, p)

= (w[pl, c),

etc. w[C] and w[p] are obviously harmonic fields of the first kind. The field etC] can be considered as the "harmonic representative" of the chain C; the formula (17.6) represents the decomposition of etC] into the "singular part" -e**[C] and the "regular part" w[C]. For arbitrary p-cyc1es Z, we have, by (16.1),

e**[Z]

(17.10)

0,

=

and therefore

e[Z] = w[Z].

07.11)

Again we have, assuming that Z and tC have no common point,

(e[C], Z) = (w[C], Z),

(17.12)

since, by (16.7), (17.11) and (17.8),

(e[G), Z)

= (erC), e[Z)) = (e[Z),

= (w[Z), C) = (w[G), Z).

G)

(17.12) shows that the period of etC] on Z depends only on the homology class of Z in 9.n. From (17.12) follows

(e**[C], Z) = O. Similarly, e[p] is the "harmonic representative" of the pole p; -e**[p] or w[p] is the singular or the regular part of e[p]. Again, we have (17.13)

(17.14)

(e[p], Z)

= (w[p], Z),

(17.15)

(e**[p], Z)

=

For an arbitrary p-po1e p

p* = bp = (0'*,

~),

0'

= (0',

~),

*",~ ... "/ pq ... r

o.

we define its dual pole p* 1

= p1Vg _r:: sgn

(In

243

= bp by

(afJ .. , 'YAp. .•• 12

")

............ n

pq ••••

O'AI'"··'·

HARMONIC FIF.1LDS IN RIEMANNIAN MANIFOLDS

659

Then we have (17.16) and, for arbitrary fields cp, (17.17) the formula (17.7) can be written therefore in the form

(17.18)

=

e[p]

_(_ly(n- p)e*[p*]

+ w[p].

Replace p by p*, this yields then

(17.19)

= bw[p].

w[p*]

THEOREM 26 (RECIPROCITY FORMULAE).

For an arbitrary p-chain C and

p-poles p, q, we have (17.20)

(e[C], q)

(17.21)

(e[p], q)

(17.22)

(e*[C] , q*)

(17.23)

(e*[p], q*)

= (e[q], C), = (e[q], p),

_(_ly(n- p){(e*[q*], C) - (bw[q*J, C)},

= =

_(_I)p(n-p){(e*[q*),p) - (bw[q*],p)}.

PROOF. (17.20) and (17.21) follows immediately from the symmetric property of e($, ~). As to (17.22), we have, by (17.17), (17.20), (17.18),

(e*[C), q*)

= (e[C],

p q) = (e[q), C) = - (-l)p(n- )(e*[q*], C)

+ (w[q), C).

Combined with (17.19), (17.16) this yields immediately (17.22). (17.23) is also proved similarly. Now, using above results, We shall prove the following theorem, which can be considered as a generalization of the Riemann-Roch's theorem in the theory of algebraic functions: THEOREM 27 (GENERALIZED RIEMANN-RoCH'S THEOREM). Let, in IDe,

p-poles PI , P2 , ... , pr , ql , q2 , ... , q. , p-chains Cl , C2 , ••• ,Ct , and (n - p)chains ci , C:, ... , C: be given, so that e[PIJ, ... , e[p,], e[ql], ... , e[q.], e[Cl], ... , e[C t ], e*[Ci], ... , e*[C!] are linearly independent mod~P, and A, B, b be the numbers defined as follows: A: the number of linearly independent harmonic fields e satisfying

e ==

°

mod (~P, e[Pl], ... ,e[p,]),

r(e, Z)

J (be,

= 0,

Z*) : 0,

1(e, Ck )

for all

for all (n - p)-cycles Z*,

0,

(k

I (be, C:) = 0,

(k

l (e, qk) = 0,

(Ie

-

p-cycles Z,

( 17)

= 1, 2, = 1, 2, = 1, 2,

244

.. , , t), ... , u), ... , s),

660

KUNIHIKO KODAIRA

B: the number of linearly independent harmonic fields e' satisfying

e', == 0 ~Od (Q:P, e~q~ ... , e[q.], e[Cl], '" , e[CI], e*[C~], ... , e*[C!J),

I

(J - 1, 2, .,. , r),

(e, Pi) - 0,

b: the p-Betti number of m. Then we have

= B - b + r - s - t - u. •.• ,ebl be a normalized orthogonal

(17.24)

A

PROOF. Let {el' C!, basis of Q:P, and {Zl, Z2, ... ,Zb I the corresponding basis of the p-homology group of m. Combined with the third condition: (be, Z*) = 0, the first condition

e

== 0 mod

(Q:P, e[l-h], ... ,e[PrJ)

means that e has the form (17.25) Indeed, since, by (14.12), (b(e -

L aie[PiJ), Z*)

= (be, Z*),

the relation (17.25) follows from e == L aie[Pi] (mod Q:P) and (be, Z*) = 0. We may assume therefore that the fields e have the form (17.25). Then we have, by the reciprocity formulae (17.20), (e, Zp) =

L aj(e[Pi], Z(1)

=

L a/e[ZIl], Pi)

=

L aj(e{3 , Pi),

while, by (17.14),

L Hence e satisfies (e, Z)

ai(e[Pi], ZIl)

=L

ai(W[Pi], ZIl).

= 0 for all Z if and only if

(17.26) and, if (17.26) is satisfied, then the identity (17.27) holds. Now, assuming (17.27), we have, by virtue of the reciprocity formulae,

L ai(e[Pi], C = L ai(e[C Pi), ~ (be, CZ) = L ai(e*[pi]' CZ) = - (-ly(n- )L ai(e*[CZ], Pi), lee, qk) = L ai(e[Pi], qk) = L ai(e[qk], Pi)'

r(e, C

k)

=

k)

k ],

p

Hence A is equal to the number of linearly independent solutions at , a2 , ... ,ar of the system of linear equations

(17 )245

661

HARMONIC FIELDS IN RIEMANNIAN MANIFOLDS

0,

({3

} L ai(e[Ck), Pi)

= 0, Pi) = 0,

(k

= 1, 2, ... = 1, 2, ...

1L ai(e*[C:], lL ai(e[qk], Pi)

(k

= 1,2,

= 0,

(k

=

L ai(e{J , Pi) (17.28)

=

, b), , t),

... , u),

1, 2, '"

,s).

On the other hand, the first condition for the fields e' means that e' have the form

e'

=

L }..{Je{J

+L

(3ke[qk)

+L

'Yke[Ck] + L 'Y:e*[C:].

B is therefore equal to the number of linearly independent solutions }..l, ••• ,,,! of the system of linear equations

,}..b,

{31, .. , ,(3. , 'Yl , ... ,'YI , 'Yi , . "

L }..{J(e{J, Pi)

+

L (3k(e[qk], Pi)

+L

'Yk(e[Ck], Pi)

+L

,,:(e*[C:], Pi)

=

(j

which is just the transposed system of (17.28). A - r

=

B - b - s - t - u,

or

- b

+r

q.e.d. The double tensor field w(x, ~) has a simple meaning. consider the integral transformation I(J~W[I(J],

defined for alll(J

E

S3 P•

W[I(J)(X)

=

(w(x,

w(l(J)

I(Jfi

I(J)

1, 2, .. , , r),

-

s - t - u,

To illustrate this, we ) '1(Jv(J dG

= J'1)/ w(x,

Then we have

(17.29) where

),

0,

Hence we have

=B

A

=

means the f3. P-component of

I(J.

= I(Jfi , In fact, since w(x,

)

E

f3.P

for fixed x,

we have w[I(J](x) - W[l(Jfi](X)

= w(l(J

- I(Jfi](X)

= (w(x,

),

I(J -

I(Jfi)

= 0.

On the other hand, we get from (17.5), (17.11) and (16.4) (W(IPfi], Z)

=

(w[Z],

I(Jfi)

= (e[Z], I(Jfi) = (1(J1f, Z),

f3. P,

which shows, combined with w[l(Jlfj E that W[I(JIf] coincides with 1(J1f. Hence we have w[l(J] = I(JIf - . Thus we conclude THEOREM 28. w(x,~) is the kernel of the integral operator which represents the projection to the space f3.p. COROLLARY. w(x,~) is represented in the form

(17.30) where ell, (fj

Wik ..• I;~ ••.• (x,

= 1, 2, '" ,

~)

=

L{J (e{J) ik ... I(X)· (e{J)~~ .... (~),

b) denote normalized orthogonal basis of the space f3.p.

§18. Riemann surfaces

m

2 As is well known, every 2-dim. Riemannian manifold is conformally fiat, i.e. in a suitable system of coordinates xI, X2, the metric di -takes the normal

( 17) 246

662

KUNIHIKO KODAIRA

form

M = wI (dXI)2

+ (dX2)2}.

m can be considered therefore as a Riemann surface with the local uniformization variable z = Xl + vi - 1 x2 , so that, in 2, the notion of analytic functions can be introduced. In this section, we shall discuss the relation between harmonic 2 2 fields and analytic functions in , assuming that m is closed. 2

m

For every field e

=

m

(el' e2), we associate the differential

vi -1~) dz.

dr = (el -

Then the field e is harmonic if and only if the differential dr is analytic, since the conditions: r*e = 0, re = 0 mean that the functions el, e2 satisfy the Cauchy-Riemann's differential equations

Thus there exists the one-to-one correspondence

e ~ dr = (el -

(18.1)

vi -1~) dz

between harmonic fields e and analytic differentials dr. dr[p]

=

dr[C]

= (dC]

(el[p] -

-

Put now

vi -le2[p]) dz, vi -le2[C]) dz,

where e[p] or e[C] means a harmonic field of the second or the third kind. the explicit form of the elementary solution

:e(z, 5") =

Using

2~ log Iz ~ 5" I'

we infer readily the structure of the singularities of e[C], so that we obtain, for a chain C with rC = L'Yp'p(P denote points in m2),

dr[C] = -2'YP

(18.2)

dz ( )

1rZ-ZP

+ regular differential,

where z(p) means the local coordinate of p.

Thus r[C] =

in p,

f dr[C]

represents

an abelian integral of the third kind. For a pole p = (u, p) of order 1, dr[p] is obtained from dr[C] by a simple limiting process. As the result, we get (18.3)

=

dr[p] I

2: C

+

d

_l (p») Z

+ regular diff.,

in p,

(p = (u, p»,

_ /- 2

where a = u V -lu. Now, consider a pole p = (u, p) of order m >1. For an arbitrary harmonic field e, the value m-l

1 ~ !I () ( e, p) = (m _ 1)1 11 Xik··.1 0iOk ... Vie). p,

C 17)

247

663

HARMONIC FIELDS IN RIEMANNIAN MANIFOLDS

of e at \l is represented as (18.4)

= (m _1

(e, \.1)

{rf"r}

1)! m a dzm (p) ,

where

= L( V -1) i+k+ .. -+I-m+l(lTlik ... 1 + V _li ik .•. / ).

a

Indeed, (e, \.1) is written as ( e, ) \l

=

rue

(m _1 I)! v~

while, since el -

(v'=1)i-l

alaz.

Thus, r[p]

=

+ v.

1 - 2Jk"'I)!l !l !l ( -lIT ViVk ••• VI

) el - v- I-I - e2 )( p,

v'=1 e2 is an analytic function, ai can be replaced by Comparing (18.4) with (18.3), we obtain

dr[pJ =

(18.5)

Ijk···l IT'

2;dC -\(p)T +

regulardiff., inp.

Jdr[pJ represents an abelian integral of the second kind.

As a

"representative" of singularities, a pole p is determined by a, p and m, so that p can be written as

= a·pm.

p

With respect to real coejficients, there remam therefore only two linearly independent poles of the order m: pm, pm* = -V _ lpm, for which We have

!

(e, pm)

(18.6)

1)!

mfz: (p), (dr = (el -

I(

l

(18.7)

= (m ~

V -1~) dz),

m*) _ 1 rf" r ( ) e,p - (m - 1)r 3 dz m p,

f ';"(p"( ~ !;d,(' -"((P»): + ,)e~Ular diff., l dr[pm*j

= 211'v'=1 d

pm* is dual to pm in the sense of §I7. For every e E ~\ we have (e, be) harmonic fields of the first kind (~

z _ z(p)

=

= 1, 2,

in p.

+ regular diff.,

O. Hence we can choose a system of ... , p, p

= genus of 9R),

so that {el' ei' , e2, '" ,ep , e;l constitutes a normalized orthogonal basis of the space ~l. The corresponding basis of the I-homology group of ID1 will be denoted by {ZI, ,Zp, Z;l (see §16); we have therefore

zi , ...

ell =

e(ZIl],

C 17)

* = e[Z/lJ, *

ell

248

664

KUNlHIKO KODAIRA

Putting ({3 = 1, 2, .. , , p),

we get a basis first kind.

{J dTl, . .. , JdT p}

of the space of all abelian integrals of the

Obviously we have

1

(18.8)

dT(3 = -

z~

V -1 Oa(3 •

Let d = p';lp';2 ... p";' Iq~lq;2 . . . q;' be a divisor of the order d = Lm; Lni , A the number of linearly independent one-valued analytic functions on l 9Jl2 such thatj SE (cr ) , and B the number of linearly independent differentials 2 dT on mc such that dT = Oed). The classical theorem of Riemann-Roch asserts that there exists the relation

°

.a

(18.9)

=

B - p

+ d + 1,

where p denotes the genus of iJ)1z. Now we shall show how to deduce the relation (18.9) from our Theorem 27 in §17. Every (many valued) analytic function f is determined up to an additive constant uniquely by the differential df. Hence we infer, by (18,7), that A is equal to the number of linearly independent (with respect to complex coefficients) differential df with the form

LiL:!.l (aim dTfpj]

=

df

(18.10)

- a;m dT[pj*])

+ regular diff.,

(aim, a;m denote arbitrary real coefficients) satisfying the conditions

fl ~f

Z

1

(18.11)

df

= 0,

df

= 0,

Ck

dnj

l dz"

\

(qk)

= 0,

for all cycles Z,

(n

= 1,2,

'"

,nk -

1, k

= 1,2,

.,. ,s),

where [qlqkl means a curve joining qk with ql. Transferred to the harmonic fields e corresponding to df in the sense that df = (el - v!=1ez) dz, (18.10) and (18.11) mean

e

=

LiLm(aime[pi'l - a;me[pj*])

(e, Z)

(18.12)

{ (e, Ck)

lee, q:)

= (be, Z) = 0,

= =

(be, Ck)

= 0,

(e, q:*) = 0,

+ regular field,

for all cycles Z, (k

= 2,3,

... ,s),

(n = 1, ... , nk - 1, k

( 17) 249

=

1,2, '" ,s),

.HARMONIC FIELDS IN RIlDMANNIAN MANIFOLDS

665

as one readily infers by (18.6). Thus 2A is equal to the number of linearly independent harmonic fields e satisfying (18.12). We infer similarly, by (18.2), (18.7) and (18.6), that 2B is equal to the number of linearly' independent harmonic fields e' such that e' = LkL:~ll(bkne[q:l - b~n[q:*])

I

+ L~_2(cke[Ckl + c~e*[Ck]) + regular field,

(e', pi) = (e', pi*) = 0, (m = 1, 2, . " ,mj, j = 1, 2, ... ,r).

Hence we have, by Theorem 27,

2A or

=

2B - 2p

A = B - p

+ 22:i_l mi

-

22::=1 (nk - 1) - 2(8 - 1),

+ d + 1, q.e.d.

UNIVERSITY OF TOKYO TOKYO, JAPAN. BIBLIOGRAPHY P. ALEXANDROFF AND H. HOPF: [I) Topologie, 1. Berlin, Springer, 1935. H. FREUDENTHAL: [I) Alexanderscher und Gordonsher Ring und ihre Isomorphie. Ann. of Math. 38 (1937), 647-655. J. HADAMARD: [I) Le Probhlme de Cauchy ct les equations aux d(irivees partieJIes lineaires hyperboliques. Paris, Hermann, 1932. W. V. D. HODGE: [I) A Dirichlet Problem/or Harmonic Functionals, with Applications to Analytic Varieties. Froc. London Math. Soc. 36 (1934),257-303. [2] Harmonic Functionals in a Riemannian Space. Proc. London Math. Soc. 38 (1935), 72-95. (3) The Existence Theorem for Harmonic Integrats. Proc. London Math. Soc. 41 (1936), 483-496. K. KODAIRA: [I] Uber die Harmonischen Tensorfelder in Riemannschen Mannigfaltigkeiten, Proc. Imp. Acad. Tokyo 20 (1944),186-198; 257-261; 353-358. [2] Relations between Harmonic Fields in Riemannian Manifolds, Math. Jap. 1 (1048), 6-23. G. DE RHAM: [1] Sur l'analysis situs des varieUs Ii n dimensions. J. Math. Pures App!. 10 (1931),115-200. [2] Uber mehrfache Integrale. Abh. Math. Sem. Hansischen Univ. 12 (1938), 313-339. H. WEYL: [I] Die Idee der Riemannschen Flache. Berlin, Teubner 1913. [2] Method of Orthogonal Projection in Potential Theory. Duke Math. J. 7 (1940), 411--444. [3] On Hodge's Theory of Harmonic Integrals. Ann. of Math. 44 (1943), 1-6. I am indebted to the referee and editors for the following references to material to which I did not have access during the preparation of the present paper: (1) S. LEFscHETz: Algebraic Topology. New York, Amer. Math. Soc., 1942. (2) W. V. D. HODGE: The Theory and Applications of Harmonic Integrals. Cambridge Univ. Press, 1941. (3) G. DE RHAM: Sur la theorie des formes differentielles harrnoniques. Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.)., 22, 135-152 (1946): Math. Rev. 8, p. 603 (1947). (4) BIDAL AND DE RHAM: Lea for-mes differentielles harmoniques. Comment. Math. Helv. 19,1-49 (1946): Math. Rev. 8, p. 93 (1947). (5) A. WElL: Sur la theorie des formes differentielles attachees Ii une variete analytique complexe. Comment. Math. Helv. 20 (1947), 110-116.

THE EIGENVALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HEISENBERG'S THEORY OF S·MATRICES.*

By

KUNIHIKO KODAIRA..

Recently E. C. Titchmarsh has treated the theory of expansion of arbitrary functions in terms of the eigenfunctions of a differential operator of the second order by a new method and obtained results of importance for applications. 1 The method of Titchmarsh is based solely on the calculus of residues. In the present paper we shall first give another proof of Titchmarsh's results based on the general theory of linear operators in Hilbert space and, secondly, applying them to differential equations of Schrodinger type, show that a theorem of Heisenberg 2 concerning the S-matrix can be founded on these results. 1.

Spectral theorem.

Let us consider the differential expression:

L[u] ==- (djdx) {p(x)djdx}u

(a

+ q(x) . u,

< x < b,

-

00 0 for a < x < b; for x ~ a or x ~ b, p(x), q(x) may behave arbitrarily, e. g. increase infinitely, oscillate infinitely many times, etc. Classification.

Consider the differential equation

(L 1)

L[u]

=

l· u,

where 1 means a complex parameter. * Received May 12, 1949. Titchmarsh [13]. The results of the present paper were in August, 1948 independently of Titchmarsh's work and other was inaccessible in Japan. The paper was revised following the H. Weyl, who kindly informed the author of the literature. express here his best thanks to Professor Weyl. • Heisenberg [7]. See also Ma [9], Jost [8]. 1

obtained by the author recent literature which suggestion of Professor The author wishes to

921

( 18) 251

922

KUNIIIIKO KODAIRA.

TIIEOREM 1. 1. Choose a fixed point c, a < c < b, arbitrarily. If every solution u of L [u J = lo . u is square summable in (a, cJ (or [c, b» for some lo, then, for arbitrary l, every solution u of L [u J = l . u is also square iummable in (a, cJ (or [c, b) ).3

Since the solution u is square summable in (a, cJ (or [c, b» or not independently of the choice of c, we can classify, by virtue of Theorem 1. 1, the differential expressions L in two types with respect to a (or b) : if every solution u of L[uJ = l· u is square summable in (a, cJ (or [c, b», L is said to be of the l. c. type (limit circle type )at a (or b); otherwise L is said to be of the l. p.type (limit point type) at a (or b). Thus there exist the following four cases: I.

L is of the 1. p. type at both a and b.

II.

L is of the 1. c. type at a and of the 1. p. type at b.

II'.

L is of the 1. p. type at a and of the 1. c. type at b.

III.

L is of the 1. c. type at both a and b. 4

In what follows the case II' will be omitted, since this case can be reduced to the case II by the transformation: x ~ - x. Bracket.

For arbitrary functions u, v, we introduce the bracket: [uv](x)

=

p(x)[u(x)v'(x) - v(x)u'(x) J, (u' = du/dx, v' = dv/dx).

In case u and v satisfy one and the same equation L [u J = l . u, we write [uv] for [uv] (x), since, in this case, [uv J (x) does not depend on x. Fundamental solutions. By a system of jundamental solutions we shall mean the system of two solutions S1 (x, l), S2 (x, l) of the equation L [u J = l . u having the following properties:

( i) (1. 2)

~lm ~~~)

[S2S1J = 1, Sk(X,1) = Sk(X, l), as functions of l, Sk(X, l) and (d/dx)sk(x, l)

(k = 1,2) are regular analytic in the whole l-plane. 3 Weyl [14], Theorem 5, p. 238. • Weyl [14], [15]. • The bar means the conjugate complex number.

( 18

J

252

(k=1,2)/

923

EIGENVALUE PROBLEMS AND S-MATRICES.

Such a system of solutions Sl, S2 is obtained, for example, by solving the equation L[u] = l· u under the boundary conditions: (a")

lim lim 1/'7l'

=

6-++0 ...

6

As a function of >.., the matrix function P(>") = (Pi"(>"») is continuous on the right and monotone non-decreasing in the sense that) for p. < >.., the symmetric matrix P(>") -P(p.) is positive semi-definite. Put) for every finite interval /l. = (p., >"], E(/l.)

=

E(>..) -E(p,).

Then) for every ue.\5, E(/l.)u(x) can be represented as follows: (1. 16)

where

and the integral

i

b

dy in (1. 16) converges absolutely. u(x) itself is repre-

sented as follows: (1. 17)

u(x)

lim

=

A-++"

f

bu(y)dy a

fA~ Sj(x, >..)s,,(y, >..) dpi,,(>") , "

j,k

p.-+- "

where the limit converges in the mean)' especially if u belongs to the domain of H, we have (1.18)

Hu(x)

=

lim A-+-t"

fb u(y)dy fA~ Sj(x, >..)s,,(y, >..)>..dpj,,(>"). a

p. 1,k

p.-+- " 13

Cf. Titchmarsh [13], Chap. III.

( 18) 256

927

EIGENVALUE PROBLEMS AND S-MATRICES.

For every v > 0, the residual ter.ms (1.19)

Rik(v) (l)

=

Mik(l) -

f:

(A -l)-ldpik (A)

are regular in the I-plane except for real 1 such that the singularities of Mik(l) in the finite I-plane are by Pik(A). Especially every isolated singular point of order 1. Incidentally, if we take as Sl, S2 the special solutions S1 0, S20 satisfying

1< - v or 1> v; thus completely determined Mik(l) is a pole of the system of fundamental

Mik(l) can be represented as follows: (1. 21) where 10 means an arbitrary fixed point with ~lo~O and the integral converges absolutely. The existence of the matrix Pik(A) so that (1.16), (1.17) hold was first proved by H. Weyl14; another proof based on the modern theory of Hilbert space was given by M. H. Stone. 1S The formula (1. 15) was obtained by E. C. Titchmarsh. 16 (1. 15) will be called the spectral formula; it can be represented also in the following form: (1.22)

Pik(A) =-lim lim (21r'i)-1 /l-+A+O

f-+O

r

Mik(l)dl,

J O(,..,..,f)

where G(JL, IX, €) means the contour consisting of two oriented polygonal lines whose vertices, in order, are JL k, JL i..-l)dE(A)U,

G(l)v(ll.).

Hence we have

~)dy = (v(~), G(x,

(A-1)dE(A)U,G(X,

,1»

= (u,

,1»

Sa (>..-l)dE(>..)G(x, ,l».

Thus we obtain the formulae 11 18

We follow the method of H. Weyl. Of. Weyl [14], pp .. 238-25l. Weyl [14], pp. 224-231. Of. also Stone [11], Thl;lorem 10.19, 10.20 and 10.21.

( 18)

259

930

KUNIHIKO KODAIRA.

(2.3) (2.4)

u(x,~)

(u,

i

It.

(A-l)dE(i\)G",(x, ,1»,

=

du(x,~)/dx=(u,

(A-l)dE(A)G(x, ,1», (Gz=dG/dx).

From these formulae follows that the functions u(x, ~), (d/dx)u(x,~) are continuous with respect to :c and of bounded variation with respect to ,\ in every finite interval A]. Considering u(~) as an element of ~, we have the relation

eft,

L [u (~) J = H u (~)

ft.

=

AdE (,\) u (~),

showing that the function u(x, A) satisfies the integro-differential equation:

(2.5)

L[u(x, A)]

=

f\dU(X, A);

u (x, i\) satisfies furthermore the boundary conditions: u(x, 0)

(d/dx)u(x, 0)

=

=

o.

Under these circumstances, the solution u(x, A) of the equation (2.5) is given by

(2.7)

u l ( A)

=

(u' = du/dx).19

P( c) u' ( c, A) ,

In what follows we assume that the additive functions Uk(~), {k(~)' etc. of the interval ~ are always combined with the corresponding functions Uk(A), ~k('\), Pik('\), etc. by the relations as follows: Pfk(~)'

(~=

Now, putting Yl(l) =p(c)GIJJ(c, 1), (2. 8)

~k(~) =

It.

(}-" AJ).

yz(l)=G(c,l) and

(A-l)dE('\)yk(l),

(k=1,2),

we get from (2.3), (2.4) and (2.7) (2. 9)

Uk(~) =

(U'~k(~»'

(k

= 1, 2).

For fixed~, Uk(~) can be considered as linear functionals of u, not depending ,. Weyl [14], Hilfssatz 1 and its proof, pp. 240-241.

( 18) 260

931

EIGENVaLUE PROBLEMS AND S-MATRICES.

on l. Hence we infer, by (2. 9), that the functions ~k (A) do not depend on 1. Insert now u=~j(A), (A= (O,A]) in (2.6). Then we have (2. 10) where

Pik(A) = (MA), ~k(A».

(2.11) Inserting (2.8)

III

(2.11), we get

whence we conclude

(2.12) where the integral converges absolutely. From (2.11) follows that the matrix P (A) = (Pjk (A» is positive semi-definite and independent of 1. Using the explicit expression of G (x, y, l), we can readily calculate the right hand side of (2.12). As a result, we obtain

Z51· (Yj(l),Yk(l»

=

Z5Mjk(l) ,

and therefore the formula

(2. 13) which yields immediately the spectral formula (1. 15). we choose real numbers Ao, A1,' . " An so that }J-

put 8 = max I Am - Am-l

= Ao

< A1 < ... < An =

I , and consider the

To prove (1. 16),

A,

sum

m

8B(Y)

=

~

I. 8i(X, Am)~j(y, Am),

m i

Then we have, by virtue of (2.10), lim 86(Y) 6->0

while, since

I. II MAm}ll

2

=f

t.

=

I. Sj(x, A)Sk(y,A)dpik(A), i.k

II H,A) 11

2

,

there exists lim 86 in the sense of

m

II and thus lim 86 e Sj.

i"ay I i

This proves the inequality

~Sj(X'A)Sk(Y'A)dpjk(A)12 < + 00.

( 18) 261

932

KUNIHIKO KODAIRA.

Again we have, by (2.6) and (2.9), E(A)u(x)

=

lim

~ 8j(X,

'\",)u(x, A",)

0""0 J

= (u,limSo) = (u, 0-+0

f

~8j(x,A)8k(

J.k

lI.

,A')dpik('\»,

proving (1.16). (1.17) and (1.18) follow immediately from (1.16). Again, (1. 21) follows from (2. 13) . Thus the spectral theorem has been proved for the special case mentioned above. To prove the spectral theorem in general cases, we denote the special fundamental solutions 81, 82 and the corresponding M jk , Piic mentioned above by 81°, 82°, M;kO, Piko, respectively. Then general 8 1, 82 are related to S10, 82° by a unimodular transformation: (2.14)

det (fJj1o (l) ) = 1,

where fJJ1o(l) are holomorphic functions of land /3/10(1) = /3j1o(l). By the transformation (2. 14), the characteristic matrix is transformed according to the rule:

"'... This shows, combined with the relation:

that (2.15)

are regular except for real 1 such that 1< - v or 1> v; hence the functions Pik (,\) defined by (1. 15) are given by

(2.16)

PJk('\) =

("" ~ {3m;(>..){3"1o(>..)d pO",,,(A). Jo m,n

Inserting this in (2.15), we see immediately that RJ1o(v}(l) are regular except for real 1 such that 1< - V O l ' 1>v. Again, from (2.14) and (2.16) follows that the relations (1.16), (1.17) and (1.18) are preserved by the transformation 810° -+ 810, Pik° -+ Pik' Thus the spectral theorem is completely proved.

3. Hill's equation.

As an example, we consider (q(x

+ 1) =

( 18) 262

q(x), -

00

< x < + 00),

933

EIGENVALUE PROBLEMS AND S-MATRICES.

where q(x) is a continuous periodic function with the period 1. 20 Choose the fundamental solutions 81, S2 of Hill's equation L[u] = 1· u so that S1(0) =8'2(0) =0,82(0) =8'1(0) =1. Then we have 8i(X

+ 1,1) =

2

~ hj1. (1) Sk (x, 1),

det (hik(l»

=

1,

1c=1

where hjkCl) are holomorphic functions of 1 and hJk(l) be the roots of the secular equation

=

hik(l).

Let z.(l)

such that I z+(l) I >1, I z-(l) I 1, I z_(l) I < 1, we infer, by (3.2) and (3.1), that our L belongs to the case I and f.,.,(l) = f.(l). Hence we get M(l)

(3.3)

= 4-1{T2(1) -l}-§ (271,21(1)

h22(1) - hu(1)). h22(l)-hu(1) -2h12(1)

°

+

i· 0) is therefore :1= if and only if '7'2(>..) -1 < 0, sinne hik (>..) and T(A) are real for real A. Now, it is known 21 that the equation '7'2 (A) - 1 = has infinitely many roots Ao, >"1, >"2,' . • such that

~Mjk(>"

°

and that

'0 Many examples of applications to the spectral theorem are to be found in Titch. marsh [18]. Here we shall consider Hill's equ'ation, which has been treated by Wintner [20], [18]. Wintner has determined the spectrum of this equation; above, the matrix P (X) determining the spectral resolution and, therefore, the spectrum, will be obtained. 21 Cf. Strutt [12].

C 18)

263

934

KUNIHIKO KODAIRA.

1"2(A) -1

5 > 0, 1 < 0,

for A < Ao and A2m-1 for A2m < A < A2m+l'

< A < A2m,

Using the spectral formula (1. 15), we get therefore, from (3.3)

dP(A) _ -

{

(-I)m(4,,-)_l{l_ r2(A) }-~ ( - 2h21 (A), hII (A) - h2i A») dA hu(A) - h22(A), 2hI2 (A) , (for A2m < A - 00 and a is the regular singular point of the differential equation L[u] = O. Then, fa(l) is a meromorphic function, if (x-a)2jp(x) ~O (x~a).

Combined with Theorem 4. 3, this yields immediately the following

+

THEOREM 4. 5. Assume that L is analytic, 00 < a, b < 00 and a, b are both regular singular points of the differential equation L[u] = O. Then H has only the discrete point spectrum, if

(x-a)2/ p (x)

~

0

(x~

a),

5. Schrodinger equation.

(X-b)2jp(X)

~O

(x~b).

Let us consider the differential operator of

Schrodinger type: (Ot, the system of fundamental solutions 81, S2 dJined above is normal (i. e. fa (1) ""'" 00), since

In case v < t, we take, as the boundary condition at 0,

(5.1) then the system 81, S2 is also normal. Thus Theorem 4. 1 and Theorem 4. 2 can be applied here. In order to investigate the asymptotic behavior of the solution of L[u] = l·u at 00, we put

k=

l~,

Then we have 5. 1. If ~k :> 0, 7c =1= 0, the equation L[ u] and only one solution u (x, le) 8uch that THEOREM:

(5.2)

u (x, 7c) '-' exp [i7cx -

=

7c 2u has one for

ti O. Furthermore we have

(5.3)

u' (x, le) '-' (d/dx) exp [ilex -

(ii O. Thus Theorem 5. 1 is proved. 23 From (5. 2) follows

1: i

( 5. 4)

u (x, k) I 2 dx


0;

We need therefore no

Put now

u(x, k)

=

A (k)S2(X, l) -B(k)Sl(X, l),

Then we have, by (1.10b) and (5.4), (5.5)

f",(l) =-B(k)/A(k),

A(k) and B(k) are given by

(5.6) Hence, by Theorem 5. 1, A (k), B (k) are regular in

~k

> 0 and continuous

•• After the variation of constants employed in. the proof above, the theorem 5. 1 is contained in a result of BOcher [2]; cf. also Wintner [17], [19], [21].

( 18) 269

940, inr~k

KUNIHIKO KODAIRA.

> 0, k =rf= O. Again, since u (x, k) is not identically 0, A (k) and B (k)

have no common zero point. We conclude therefore, by virtue of (5. 5), that I", (l) is meromorphic except for 1 >0 and every pole of I", (1) which is not > 0 corresponds one-to-one to the zero point of A (k), since, by the relation 1= k2, k with ~k > 0 corresponds one-to-one to 1 which is not > O. On the other hand, by Theorem 4. 1, every pole of I", (1) lies on the real axis and is of the order 1. Hence, in ~k > 0, all zero points of A (k) lie on the imaginary axis and are of the order 1. Denote these zero points by km=ilkml, (m=1,2,3,· .); then the poles of 1",(1) which are not > 0 are given by Am = k 2 m =-1 km 12, (m = 1, 2, 3,' .. ). Now, for real k=;'= 0, we conclude from (5. 2) and (5. 3) the formulae (5.7)

u(x,-k)=u(x,k), (k>O);

[u(-k)u(k) ]

=

2ik, (k

> 0);

which yields (5.8)

A(-k) =A(k),

B(- k)

=

B(k),

(k

> 0),

(5.9)

A(k)B(- k) - A ( - k)B(k)

=

2ik,

(k

> 0).

Hence A (k) and B (k) do not vanish for real k =;'= O. Now, using these results, we can readily calculate the function p(A). First, for A > 0, we have p(A)

=; iX~I"'(A+iO)dA=-~ iX~{B(A~)/A(A~)}d'\,

while, by (5.8) and (5.9), ~{B(k)/A(k)}

=-k/I A(k)12,

(k

> 0).

Hence we obtain dp(A)

=

(k = Ali, A > 0).

(2/'lT) (k 2 /1 A(k) 12)dk,

Secondly, for A < 0, p(-O) -p(A) is, by virtue of (4.5), equal to the sum of the residues pm of - '00 (1) at the poles Am, A < Am < 0, i. e., (5.10)

Obviously

p(- 0) -p('\)

pm

is positive.

=

~

pm,

X 0).

Putting

(5.13)

A(- k)

=

A(k)

=

I A(k) I ei~(k),

(k> 0),

2< Cf. Titchmarsh [12], Chap. V. The location of the continuous spectrum can be read off from more general theorems; see Wintner [18]; Hartman and Wintner [4], [5], [6]; Hartman [3].

( 18) 271

942

KUNIHIKO KODAIRA.

we have therefore

This yields, combined with (5. 2), the asymptotic formula: (5.14)

(x~

00).

6. Heisenberg's S-function. In this section we shall investigate the Schrodinger operator treated in 5 under the following assumption: ASSUMPTION I. The functions u(x, k) and 1I(x, k) introdu.ced in Theorem 5. 1 can be extended analytically over the whole lower half k-plan6 across the negative part of the real axis.

Under this assumption, the functions A (k) and B (k) are also extended analytically over the whole lower half plane across the negative part of the real axis, and the relations (5.8), (5.9), (5.12) proved for k> 0 are extended for ~k > 0, i. e., we have (6.1)

A(- k)

(6.2)

A(k)B(- k) - A ( - k)B(k) = 2ik,

(6.3)

Sl

> 0), (~7c > 0), 7c)}, (~7c > 0).

= A (k),

(x, k 2 )

(~k

= (2i7c )-l{A ( - k )u(x, 7c) - A (7c)u(x, -

Now we make furthermore ASSUMPTION

II. u (x, 7c) and u' (x, 7c) are regular in the whole k-plane

except for k > O.

Then the functions A (k) and B (7c) are also regular in the whole k-plane except for 7c > O. Furthermore, using (6. 1), (6. 2) and (6. 3), we can eliminate the function B (k) from the formula (5. 10). In fact, inserting k=7c m in (6.2), we get A(-7c".)B(k".)=2\k".\, which shows that A(-7cm ) is real, since B(7c".) =B(7cm ) by (6.1). Putting (6.4)

S(k)

(~k

= A(-7c)/A(lc),

we obtain therefore p".=

(2j-lT) I k"./A(-k m )

(18)

I 1.

272

2

Jk..S(k)dk.

> 0, lc =1= 0),

943

EIGENVALUE PROBLEMS AND S-MATRICES.

Again, from (6. 3) follows

Hence the normalized eigenfunction

Urn (x)

is represented as follows:

S (k) will be called the Heisenberg's S-function, since it is a diagonal element of the Heisenberg S-matrix!5 By virtue of (5. 13), we have S(k) = e2t6 (kl, (k > 0). Hence S(k) is obtained, under the Assumption I, from e2i6 (k) by analytic continuation, if one knows the "phase shift" 8 (k) from the asymptotic behavior of the" normalized eigenfunction" Uk(X). Under the Assumptions I and II, S(k) is meromorphic in ~k > 0, has poles k m = i 1 k m 1 (m = 1, 2,' .. ) of the order 1, and except for these poles, S (k) is regular in ~k > 0, since A (k) and A ( - k) are regular in ~k > 0 and have no common zero point, as one sees from (6. 3). Thus the following theorem of Heisenberg has been strictly founded: 6.1 (HEISENBERG).26 the S chrodinger equation: THEOREM

Let Uk(X), k> 0, be the solution of

(0 satisfying the

U

< x < 00)

boundary condition JJ (x~O)

and having the asymptotic form

Uk(X) '-' (2/lT)i sin [kx - talk log x

+ 8(k)],

(x~

00).

Then, under the Assumptions I and II, the function S(k) = e2i6 (k) can be extended analytically over the whole upper half k-plane . . The extended S (k) has, on the imaginary axis, the poles km = i 1 k m 1 (m = 1, 2,' .. ) of the order 1, and, except for these poles, S (k) is regular in ~k > O. The negative eigenvalues of the Schrodinger equation are given by Am = k m2 = -I k m 12, (m = 1, 2,' .. ) and the corresponding normalized eigenfunctions um(x) have the asymptotic expressions: 25 Heisenberg [7]; see also Pauli [10], Ma [9], Bargman [1] . •• Heisenberg [7J, Part III and IV. Cf.. also Ma [9J, Pauli nOJ.

( 18) 273

944

KUNIHIKO KODAIRA. (x~ct)),

where the constants cm are given by cm 2

=

1. s (k) dk.

)k..

It is an open question whether the Assumption I is valid for every Schrodinger equation or not. If Assumption I were not valid, it would be impossible to define S (k) for ';Jk > O. Incase the Assumption II is not fulfilled (while the Assumption I is valid), S(k) has, in general, singular points in ';Jk > 0 other than k m (m = 1, 2,' .. ), arising from the numerator A ( - k) of S (k). Thus, in this case, it might be impossible to determine the negative eigenvalues of the Schrodinger equation from the singular points of S (k). 27 The necessary and sufficient condition for V (x) in order that the Assumptions I and II are valid is not known. But it can be proved that the assumptions I and II are valid if the potential V (x) is analytic in a neighborhood of ct). This asserts the validity of Heisenberg's theorem for Schrodinger equation appearing usually in application.

BIBLIOGRAPHY. [1] Bargman, V., "Remarks on the determination of a central field of force from the

elastic scattering phase shifts," Physical Review, vol. 75 (1949), pp. 301-303. [2] Bocher, M., "On regular singular points of linear differential equations of the

second orde. whose coefficients are not necessarily analytic," Transactions of the American Mathematical Society, vol. 1 (1900), pp. 40-52. [3] Hartman, P., "On the spectra of slightly disturbed linear oscillators," American Journal of Mathematics, vol. 71 (1949), pp. 71-79. [4] - - - and Wintner, A., "An oscillation theorem for continuous spectra," Proceedings of the National Academy of Sciences, vol. 33 (1947), pp. 376-379. [5] - - - and Wintner, A., "On the location of spectra of wave equations," American Journal of Mathematics, vol. 71 (1949), pp. 214-217. [6] - - and Wintner, A., "A criterion for the non-degeneracy of the wave equation," American J01Jrnai of Mathematics, vol. 71 (1949), pp. 206-213. [7] Heisenberg, W., "Die beobachtbaren Grossen in der Theorie der Elementarteilchen," I, Zeitschrift fur Physik, vol. 120 (1943), p. 513; II, ibid., p. 673. III, IV, unpublished. [8] Jost, R., Helvetica Physica Acta, vol. 22 (1947), p. 256. 27

S. T. Ma has pointed out this fact by an example.

( 18) 274

Cf. Ma [9].

EIGENVALUE PROBLEMS AND S-MATRICES.

945

[9] Ma, S. T., "On a general condition of Heisenberg for the S matrix," Physioal Review, vol. 71 (1947), pp. 195-200. [l0] Pauli, W., Meson theory of nuclear forces, New York (1946). [ll] Stone, M. H., Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15 (1932).

[l2] Strutt, M. J. 0., Lamesche, Mathieusche und verwandte Funktionen in Physik und Technik, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 1, no. 3 (1932) . [13] Titchmarsh, E. C., Eigenfunction expansions associated with second-order differential equations, Oxford (1946). [l4J Weyl, II., "uber gewohnliche Differentialgleichungen mit Singularitaten und die zugehorigen Entwicklungen willkurlicher Funktionen," Mathematische Annalen, vol. 68 (1910), pp. 220-269. [15] - - - , "uber gewohnliche Differentialgleichungen mit singularen Stell en und ihre Eigenfunktionen," Gottinger N achrichten (1910), pp. 442-467. [l6] - - - , "uber das Pick-Nevanlinnasche Interpolationsprob1em und sein infinitesimales Analogen," Annals of Mathematics, vol. 36 (1935), pp. 230-254. [17] Wintner, A., "Asymptotic integrations of the adiabatic oscillator," American Journal of Mathematics, vol. 69 (1947), pp. 251-272. [18] - - - , "On the location of continuous spectra," American Journal of Mathematics, vol. 70 (1948), pp. 22-30. [19] - - - , "On the normalization of characteristic differentials in continuous spectra," Physiaal Review, vol. 72 (1947), pp. 516-517. [20] - - - , "Stability and spectrum in the wave mechanics of lattices," Physical Review, vol. 72 (1947), pp. 81-82. [21] - - - . "Asymptotic integration of the adiabatic oscillator in its hyperbolic range," Duke Mathematical Journal, vol. 15 (1948), (IV), pp. 55-67.

( 18) 275

ON ORDINARY DIFFERENTIAL EQUATIONS OF ANY EVEN ORDER AND THE CORRESPONDING EIGENFUNCTION EXPANSIONS.* By

KUNIHIKO KODAIRA.

The general theory of expanding an arbitrary function in terms of the eigenfunctions of a singular differential operator of the second order was first given by H. Weyl,1 An alternative method, based on the general theory of linear transformations in the Hilbert space, is to be found in a treatise by M. H. Stone. 2 Recently E. C. Titchmarsh has treated the same theory by still another method and obtained some new results of importance for applications. 3 The purpose of the present paper is to generalize the Weyl-StoneTitchmarsh theory of eigenfunction expansions to the case of formally selfadjoint ordinary differential operators with real coefficients of any even order. Our method is also based on the general theory of linear transformations in the Hilbert space and goes along the lines of H. Weyl,4 In Section 1 we shall give preliminary remarks on the formally self-adjoint differential operator of real coefficients, which will be denoted by L. In Section 2 we shall analyse the singularities of the operator L. For that purpose, we introduce a complex projective space ~, each point of which is associated with a general solution of the differential equation L[u] = l· u, and construct the subsets raU), fb(l) of qs corresponding to the" limit circles" in the theory of H. Weyl,5 Furthermore it will be shown that the formally self-adjoint differential operators of the even order n can be classified into (i·n 1) 2 different types according to the nature of their singularities. In Section 3 we shall introduce Green's function and determine the closure and the adjoint of the operator L. The boundary conditions will be discussed in Section 4. It will be shown that, under suitable boundary conditions, L becomes a self-adjoint operator. The next section, 5, will be devoted to the proof of the spectral theorem, which can be regarded as a generalization of the Weyl-Stone-Titchmarsh results. In

+

* Received September 22, 1949. Weyl, [10], [11], [12]. Numbers in brackets refer to the bibliography at the end of the paper. 2 Stone, [7], Chap. X, §3. 3 Titchmarsh, [9]. Another proof of Titchmarsh's results is given in Kodaira [3]. • Weyl, [11], [12]. 5 Weyl, [11], pp. 221·231. 1

502

( 19) 276

503

EIGEN:FUNCTION EXPANSIONS.

Section 6, the expansion theorem will be deduced from the spectral theorem. Finally it will be shown in Section 7 that the spectral and the expansion theorems can be readily extended to the case of simultaneous differential equations. The author wishes to express his grateful thanks to Prof. H. Weyl for his encouragement and his aid in acquainting the author with the recent results of Prof. E. C. Titchmarsh. 1.

Introduction.

L

Po(x) (d/dx)n

=

Consider a formal differential operator

+ Pl(X) (d/dx)n- l + ... + Pn(X),

of the even order n = 2v defined in a (finite or infinite) open interval (a, b), where each coefficient Pm (x) (m = 0, 1,' .. , n) is a real valued continuous function defined in (a, b) having continuous derivatives up to the order n-m andpo(x) > [or po(x) < 0]; for x--.?a or x--.?b, Pm(x) may behave arbitrarily, e. g. increase infinitely, oscillate infinitely many times. 6 Furthermore we assume that L is f01·ma.lly self-adjoint, i. e. L coincides with its Lagrange adjoint or

°

"m-l

(1. 1)m 1'",(x)

=

~

(_1)kC,n_k n - kpk(m-k) (x)

+ (-1)"'pm(x),

k=O

(m='1,2,·· ',n), where Cr" mean binomial coefficients. For odd m, (1. 1) m is rewritten as

( m = 1, 3, 5,' . ., n - 1), while, for even m, (1.1)", follows from (1. Ih, (1. Ih,' .. , (1.1)m-1' Hence, in general, L is formally self adjoint ,if the coefficients po (x) ,. . ., pn-l(X) satisfy the conrliiion8 (1.2)",> m=1,3,·· ·,n-l. Thus, in our operator L, the coefficients po (x), P2 (x), P4 (x),' .. , pn (x) may be chosen arbitrarily> while pdX),P3(X),· " must be determined successively by (1. 2h, (1. 2)3,' .. , (1. 2)n+ Put now

for arbitrary functionsu(x) having continuous derivatives up to the order n. Then we have Green's formula • These conditions on the coefficients may be replttccd by weaker ones, but ill the present paper we are not intercsted with snch generalizations. Cf. Stone, [7], pp. 448453, Halperin, [2].

( 19

J

277

504

KUNllUKO KODAIRA.

(1. 3)

i"'(L[U] . v - U· L[v] )dx = [uv](x) -

[uv](y) ,

where [uv] (x) means the bilinear form ~ Bjk(X)U(j) (X)V(k) (x) j+k;iiin-1 of two "vectors" (u,u',u",· .. ,U(n-l», (v, v', V",' .. ,v(n-l» coefficients

[uv](x)

=

with the

The bilinear form [uv] (x) is obviously skew-symmetric, i. e. [uv](x)

(1. 5) For j

+k =

=-

[VU](X).7

n --1, we have

U+ k =n-1).

(1. 6)

The bilinear form [uv] (x) is therefore non-degenerate, since by hypothesis po (x) =F O. Now we introduce the " WrOllskian " [UIU2' .. un](x)

=

{ - Fo(X)}"'

U, k =

det( u/k-1»,

1, 2,' . " n).

Then we have the identity (1. 7)

(1/2"1'!) ~ ± [u1uv+d [U2UV+2]' .. [uvun]

=

[U 1U2 '

••

Un],

where ~ ± means the alternating sum extending to all n! permutations of n functions U I , U2,' .. , Un. (1. 7) is proved as follows: 8 As one readily infers, the left hand side of (1. 'I) is equal to

[U 1U2 '

••

un] . (-l)v(v+I)/2Fo-V(1/2vJ!!)~ ± B 01B 23 •

••

Bn- 2 "-1,

~ ± being the alternating sum extending to all n! permutations of the suffixes 0,1,2,' . " n -1; whlIe, since B jk = 0 for j k > n, we get, using (1.6),

+

(1/2"1' !) ~ ± B Ol B 23 •

••

B n_2 ,,-1 = Bo n-IB I n-2'

••

BV_l V = (- 1) v (v+l) /2pO '.

Hence we have (1. 7).

2. Classification. (2.1)

We shall consider the differential equation

L[u]=l·u,

where l means a complex parameter.

From (1. 3) follows that, if u and v

7 This identity can be deduced readily from Green's formula. 8Weyl, [14], pp. 166·167.

( 19) 278

EIGEN~'UNCTION

505

EXPANSIONS.

satisfy one and the sa.me eq1wtion (2. 1), [uv] (x) does not depend on x. Combined with (1. 7), this shows further that the Wronskian [1J,lUZ' • . un](x) of n solutions U 1 (x), uz (x),' . " Un (x) of the equation (2. 1) is a constant. In these cases, we write therefore [uv] or [U1U2' . • uon] for [uv] (x) or [Ul' . . un] (x), respectively. DEFINITION 2. 1. By a regular system of fundamental solutions of the differential equation L [u] = l . u we shall mean a system of n solutions sj(x,l) (j=1,2," ·,n) such that

(2.3)

(2.2)

and that the functions dmslx, 0/dxm (j = 1, 2,' . " n; m = 0, 1,' .. , n -1) are entire functions of l. A regnlar system of fundamental soltdions will be called a canonical system, if the skew symmetric matrix ([SjSk]) has the " canonical form" in the sense that (2.4)

€jk being the quantities defined by €jk k = j - v), = 0 (otherwise).

=

1 (for k = j

+ v),

= - 1 (for

It should be noted here that (2.4) implies (2.2)' as (1. 7) shows. A canonical system of fundamental solutions is obtained, for example, as follows: Choose a fixed point c, a < c < b, arbitrarily (in what follows c b). Then there exist n real means always a fixed point such that a c vectors (j = 1, 2,' . " n)

<
O}, ~O(x,

1)

=

{(f); h(f; x, 1)

=

~-(x,

1)

=

{(f); h(f; x, 1)

< O}.

O},

Evidently ~+(x, 1) and ~-(x, 1) are open subsets of ~; ~O(x, 1) is a quadratic hypersurface, constituting the common boundary of ~+(x, 1) and ~-(x, 1). Denote in general the closure of a subset £l in ~ by [£l]. Since h(f; x, 1) is a monotone increasing function of x, ~-(x, 1) is monotone decreasing in the sense that [~-(x, 1)] C ~-(y, 1) for x> y; similarly we have [~+(y, 1)] c ~+(x, 1) for x> y. Putting fb(l) = ~(x, 1), fa(l) = ~+(x, 1), we

n

n

III

III

infer therefore that fa(l) and fb(l) are compact, not empty, and have no common points. 9 Furthermore we have (2.9) Indeed, if fdb(l), then f lies in all ~-(x,l), ti v-I, dimmb(l) >v-l. Furthermore dimma(l) and dimmb(l) do not depend on the parameter 1, provided ~l ¥= 0, as will be proved in 3 below (see Theorem 3. 2) . Putting (2.13)

Ta = dim ma(l) - v + 1,

we obtain therefore two non-negative integers Ta, To which are characteristic for L. DEFINITION 2. 3. L will be called a differential operator of the type (Ta, To), Ta, Tb being the non-negative integers defined by (2.13). The sum T = Ta + To will be cg,lled the excess index of L.

Possible values of Ta, To are obviously 0,1,2,' . " v; thus formally selfadjoint differential operators L of the order n = 2v can be classified into (v 1) 2 different types. This classification is a generalization of R. Weyl's one 10 concerning the differential operators of the second order. For a later purpose we introduce the space m(l) = ma(l) mo(l). Since fa(l) and fb(l) have no common point, Pa and Po have also no common point, and therefore ma(l) + mb(l)~ Pa + ~lb = jJ5,u Hence we get

+

n

(2. 14)

dim m(l)

=

T-l.

The space m (1) is clearly the set of all points (f) such that w (x, 1, f) is square summable in (a, b). The excess index T is therefore equal to the number of linearly independent solutions of the differential equation L[u] =l·u which are square summab1e in (a,b), provided ~1¥=0.

3. Green's function.

Let S) be the Hilbert space consisting of all

1·Weyl, [11], pp. 221-231; [12], p_ 443; [13]. means the" join" in the sense of the projective geometry.

11 "

+"

C 19)

283

510

KUNIHIKO KODAIRA.

square summable functions in the interval (a, b) ; the inner product of the functions u, v in .\S will be denoted by (u, v), the norm of u by II u II. DEFINITION 3. 1. We denote by Sl) the linear subspace of .\S constituted of all functions u (x) satisfying the following four conditions:

u(x) is square summable in (a, b), in the open interval (a, b), u (x) admits continuous deriva.tives up to the order n-1, (n - 1) -th derivative u (n-1) (x) is absolutely continuous in every closed interval [Xl> X2], a < Xl < X2 < b (so that L [u] can be defined for almost all x, a < x < b), L [u] is square summable in (a; b).

i) ii) iii)

iv)

Considered as a linear operator having Sl) as its domain, the differential operator L will be denoted by T, i. e. we put (3.1)

for u e Sl).

Tu=L[u],

Obviously the bilinear form [uv] (x) can be defined for two arbitrary functions u, v in Sl) so far as a < x < b. Furthermore we infer from Green's formula that, for arbitrary u, v e Sl), the limits [uv]( a) = lim [uv]( x),

[uv](b)

=

lim [uv](x) exist. 0:""0

Now we shall introduce Green"s function. Let 1 be a fixed complex number with ~l # 0 and .po., .po be two (v - 1) -dimensional null planes with respect to 1 contained respectively in fa (l), fa (l). Then, choosing n linearly independent vectors f1' f2" . " fv, fV+1" . " fn such that fa. e.pa (for IX = 1, 2," ',v), ffJe.po (for .B=v+1,·· ·,n) arbitrarily, we put (3.2)

G(x, y; l, .po., .po)

= v

G(y, x; l, .po., .po) n

(x > y), FafJ(l)wfJ(x)wa(y), 0.=1 fJ=v+1 where w'Y(x) = w(x, 1, f'Y) (y = 1,' . " v, v 1,' . " n) and (FafJ(l» means the inverse matrix of ([faffJ]/). The function G(x,y;l,.pa,.po) thus defined will be called Green's function. The matrix ([faf.a] I) is skew-symmetric and, since f1,' . ,fv or fV+1,' . , fn lie on the null plane Po. or Po, respectively, we have (3.3) [!af.a] 1 = 0, for IX < v, .B < v, and for IX > v,.B > v. =

~

~

+

Consequently we see

(3.4)

FafJ(l) =-FfJa(l),

( 19) 284

511

EIGENFUNCTION EXPANSIONS.

(3.5)

for a -< v, f3 -< v, and for a

Fa.p(l) === 0,

> v, f3 > v.

Introduce now the important matrix (3,6)

Then we obtain, using (3.3) and (3. 5) , ~~ k m

M",i (1, -Pa, .j:)b) [s",(1)s,,(l)] . f,,/ =

j 1-/,

!

0,

(1- _

18

( 19) 292

519

EIGENFUNCTION EXPANSIONS.

(~l ~

(3.33)

0).

Proof. It is sufficient to show that every element u e SD is decomposed uniquely in the form

(3.34)

+

By Theorem 3. 4, u is represent!Jd as u = Gv Ws (ws e ~ (l) ), where G means G(l, lJa, lJb)' Decompose now v into two parts: v = Vo

+

+ w, + + +

(vo e H -@(l), W e ft(l» and put U o = Gvo, W 1 = Gw w/2i~l W a, W 2 = w/2i~1. Then we have the desired decomposition u = ' U o Wl wz, (Wl' W2 e ~ (l». To prove the uniqueness or the decomposition, it is sufficient to show that the relation (3.35) implies Uo = W1 = W z = O. Assuming (3.35), we have 2i~l' (1Vl' w1 ) = (Wl' Touo) - (Tw 1, uo) = 0, proving that W 1 is equal to 0. Similarly we get U'z = 0, and thererore u = 0, q. e. d. COROLLARY.

We have

(3.36) T

dim (SD/SD o) =2T,

being the excess index of L.

4. Boundary conditions.

First we shall study the properties or the skew-symmetric bilinear functionals [uv]( a), [uv J( b) defined by (4. 1)

[ uvJ ( a)

=

lim [uv J (x) ,

[uv ] (h)

=

lim [uv] ( x) , ( u, v e SD). .2" .........&

We start with the important formula: (4.2)

(Tu,v) -

(u,Tv) = [uv] (b) -

[uu](a),

deduced immediately from Green's rormula. Equation (4. 2) shows that the necessary and sufficient condition for u to be contained in SD o is

(4.3)

[uv] (b) -

[uv](a)

=

0,

ror all v e SD.

On the other hand, an arbitrary function v e * consists of all v 8::D satisfying

(4.16)

[uv] (b) -

[uv] (a)

=

0,

for all n 8 ::D1>.

Since the bilinear form [uv] (a) is non-singular on ::D/~a and dim (~/~a) 'I

C 19)

295

522

KUN1HIKO KOllAIRA.

= 2Ta, the condition (4. 13) a determines a linear subspace of SO which is Ta-dimensional over SOa, while, since the system of conditions (4. 13) a is assumed to be self-adjoint, Ta functions ¢aj (j = 1, 2,' . " To) which are linearly independent mod SOa satisfy the condition (4. 13) a. Hence the condition (4. 13) a is satisfied if and only if u can be represented as

u=~yi.paj

(modSD a ).

j

Thus we see that an arbitrary element u € ~ satisfies the boundary conditions ( 4. 13) nand (4. 13 h if and only if u can be represented as (4. 17)

U t== ~

yicpai (mod ~a),

/

=== ~ 8/CPbi (mod ::Db) .

y" /, ..

i

Moreover, for arbitrary constants ',8" 82 , ' • " 10e can readily construct a function ueSO sati.sfying (4.17). Inserting (4.17) in (4.16) and using' (4. 11), we infer therefore that v belongs to S01>* if and only jf v satisfies the condition

~ 8i [cpbjV](b) - ~ r i [ dimma(l)-Ta=v-1. On the other hand, ra(l) has no common point with feel) and, as Theorem 2.1 shows, fe(l) contains (at least) one (v-I)-dimensional null plane lJ with respect to l. Hence lJa(l) has no common point with ,p and therefore dim lJa (l) '< dim

\l3 -

dim lJ -1

=

v-

1.

Thus we see that dim ,pa(l) is equal to v··-1. THEOREM

5. 2. As functions of the pararneter l, the characteristic planes

lJa(I), lJb(l) are holornorphic, provided ;;S1#0, in the sense that, for every 10 with ;;Slo# 0, the suitably norrnalized 20 Plucker's coordinates lJaii ... k(I), ,pbii ...k(l) of ,pa(l), ,pb(l) are holomorphic with respect to 1 in some neighborhoods of 10, Proof. We shall prove the theorem with respect to ,pa (l). For that purpose, we observe the operator L in the interval (a, c], a < c < b, as above. Then we have dim Q:(l) = Ta v. Now let D be an arbitrary compact domain in the half-plane ;;Sl > (or ;;Sl < 0). Then, by virtue of Lemma 3. 1, the base ro- (x, I) «1" = 1, 2, ... , Ta v) of the space Q:(l) can be chosen so that ro-(x, 1) are holomorphic in D with respect to I in the sense of the norm: I Ii and that d"'1'0-(x, l)/dx nt (rn = 0,' .. , 11, - 1) are holomorphic in D with respect to I. Putting 1'0-(X,l) =w(x,l,fo-(l», we obtain the points fo-(/) «1"=1,·· ',Ta+V) generating the space ma(l). Then an arbitrary point f in ma(Z) can be represented as f=~'fJufo-(l), so that have w(x,l,f)=~'fJuro-(x,I). Using v) as coordinates of points f in ma(l), we these 'fJu «1" = 1, 2,· .. , Ta infer from (5. 3)a that the subspace lJa(l) of ma(l) is determined by the linear equations

+

°

+

+

(j = 1, 2,· . " Ta).

(5.7) 20 ~('P

footnntr. 14.

( 19)

~98

525

ElGE.lIfFUNCTION EXPANSIONS.

These Ta equations are linearly independent, since dim Pa(l) = dim ma(l) - Ta; while the coefficients [ajr,,(l)](a) are holomorphic in D with respect to 1, as one readily infers from

[ of class 0"" with I eI> I C~. Then, a p-current in SD is, by definition, a linear functional T[ eI>] defined on {eI>} which is continuous in the following sense; For an arbitrary sequence {eI>(1), eI>(2), ... , eI>(m), ... } of forms eI>(m) e {eI>} such that all I eI>(m) I are contained in one and the same compact subset ~ C SD covered by a single system of local coordinates xl, x 2, • • • , x 2n, we have T [eI>(m)] ~ 0 (m ~oo) if each partial derivative 8seI>(m)jk ... 1/8xP8x Q • • ·8xr converges uniformly to zero for m ~ 00 • We consider every continuous p-form >11 defined in SD as a current by identifying w with the linear functional

(1. 2) similarly, we consider every differentiable (2n - p) -chain r as a p-current defined by r[eI>]

Ir

=

eI>.

Thus the notion of currents is a generalization

of both notions of differential forms and chains. We say that the p-current T vanishes at a point p e SD and write T = 0 at p if there exists a neighborhood Up of p such that T [eI>] = 0 for all eI> with ! eI> I C Up. By the carrier I T I of T will be meant the set consisting of all points p e SD such that T does not vanish at p. We get from (1. 2) the identities dw[ cp]

=

(-

1 )P+lw[d1t)[ cp]

(-1 )P>p[ *eI>]'

=

In Vlew of these identities, we define the exterior derivative dT and the adjoint *T of an arbitrary p-current T by

(1. 3)

dT[eI>]

=

(-l)P+IT[d],

(*T)[eI>]

=

(-l)pT[*eI>].

Then we have ddT = 0 and **T = (-l)PT. In case T is a (2n- p)-chai.n r, dT coincides with the boundary ~r of r up to the sign ( - 1 )p+l; dr = ( - 1 )p+l~r. We introduce further the operators il and A defined ild. Again, the exterior product T· wof a respectively as 0 = - *d*, A = dil p-current T and a q-form w of class Coo is defined as (T· w) [eI>] = T[w· cP]. For an arbitrary p-current T and p-form eI> of class 0"" defined on the whole manifold WC 2n, we define the inner product of T and eI> as (T, cp) = (eI>, T) = T[*eI>J. In case T = w is a p-form of class 0"", we have

+

(w, cp)

=

filJl W· *eI> = filJleI>. *w;

especially (cp, 4» > 0 if eI> does not vanish identically, where ip denotes the conjugate form (lip!) ~ ]. A p-current T is said to be regular at a point ,lJ e SD, if T comcides with a p-form of class 0'" in some

C 22)

342

817

THE THEOREM OF RIEMANN-ROCH.

neighborhood Uil of .p; otherwise T is called singular at.p. By the singular set of T will be meant the set of all singular points of T. The singular set of T will be denoted by ) T )8' Obviously) T )8 is a subset of the carrier) T ). In case T is defined in the whole manifold W(2n, the singular set I T 18 of T is compact and, for an arbitrary open set U::J I T I., we can find a p-form \[I' of class 080

=

i-pO (Ad -

dA)

=

0,

in

., Weil [24], p. 111, Eckma.nn a.nd Guggenheimer [7]. 2. See Hodge (12], pp. 188-192, Eckmann and Guggenheimer (7]. 23 Eckmann and Guggenheimer [7].

( 22) 345

~.

820

KUNIHIKO KODAIRA.·

thus an arbitrary holomorphic differential q, satisfies 8q, = 0. A holomorphic p-ple differential defined in the whole of WC 2 " is called a p-ple differential of the first kind. The above result shows that every p-ple differential of the first kind is a harmonic p-form of the first kind.

°

THEOREM 1. 3. Let T be a pure p-current of type in a domain SD C WC 2 ". If T satisfies dT = in SD, then T is a holomorphic p-ple differential in SD.

°

Proof· Since T satisfies CT = iPT and AT = 0, we get 8T = i- p 8CT = i-PC (Ad - dA) T = 0, in SD. Hence, by virtue of Theorem 1.2, T is a p-form of class Coo; however, since T is of type 0, T is represented as T = (lip!) ~ q,,,,a2"'''pdz'''' .. dz"p. N ow the relation dT = implies that 8q,"'''''''''pI8zf3 = «(3 = 1, 2,' . " n), proving that each q,"'''2''."p is a holomorphic function of Z2,' • " z", q. e. d.

°

The integral F (z) =

°

5

Z

z"

q, of an arbitrary holomorphic simple differential

q, defined in SD is a many valued holomorphic function in SD and q, coincides with its differential dF: q,

=

dF. The integralS· q, of an arbitrary simple

differential q, defined in the whole of

we

is called a simple Picard integral.

The simple Picard integralS" q, is said to be of the first kind if q, is a differential of the first kind.

In case q, has any singularities, the simple

Picard integralS" q, is called of the second or the third kind according as it is locally single valued or it is not!4 Let Ai, A 2 , ' • " Aq be a base of the space of all simple differentials of the first kind. Then Au A 2 , ' • " A q , Ai, A 2 , ' • " Aq are linearly independent and constitute a base of the space of all harmonic 1-forms of the first kind!5 Thus the number q of linearly independent simple differentials of the first kind is equal to a half of the I-Betti number bl of we 2 ,,: q = ib'. This number q will be called the irregularity of the manifold IDF".

+

THEOREM 1. 4. Let T be a pure (p 1) -current of type 1 defined in the whole of 9)(2" satisfying dT = and HT = 0, p being a positive integer p"(Tp), Z2=1>p2(Tp), where /, 1>p2 are holomorphic functions of Tp defined in Up satisfying 1>/(0) = 1>p2(0) = O. On the other hand, r is represented at each point .p 2 We by a minimal local equation Rp (z", Z2) = 0, where RlJ (z", Z2) is a holomorphic function of z", Z2 defined in a neighborhood Up of.p. In case .p =

'" for 1>p'" (0;=1,2). Now we choose for each P 2 r a sufficiently small neighborhood Up of p so that the closure [1>(Up)] of1>(Up) satisfies the following two conditions: i) [1>(Up)] -1>(p) contains no singular point of r (1)(p) may be a singular point of r) and ii) the local equation Rp(z", Z2) = of r is useful at every Uq is point q on [1>(Up)]. Consider two points p, q on r such that Up not empty. Then the ratio Tpq(z) = Rp(z)jRq(z) is a non-vanishing holomorphic function of Z= (z", Z2) defined in some neighborhood of [1>(UpnUq)]. Putting tpq(T) = Tpq(1)"(T),1>2{T)), we get, for each ordered pair {p,q} with Up Uq =1= 0, a non-vanishing holomorphic function tpq = tpq ( T) defined in Up Uq. It is clear that these functions tpq satisfy the following relations:

we

°

n

n

n

(2.1)

III

Up

n Uq,

(2.2)

III

Up

n Uq n U

r•

THEOREM 2. 1 (A. Weil) . We can find a system {hpIP 2 r} of meromorphic functions hp defined respectively -in Up such that

(2.3) Proof. It is sufficient to consider one irreducible component, say r 1, of r. r 1 is a compact Riemann surface; hence it can be decomposed into the sum of a finite number of 2-simplices so that r , becomes a finite siYlplicial complex~. We denote the vertices (O-simplices) of ~ by Pl, P2,' " , Ph' . " the oriented I-simplex with the vertices Pj, Pit (in order) by Sjk, and the oriented 2-simplex with the vertices Ph Pk, PI by Sjkl. The simplical decom-

C 22)

348

THE THEOREM OF RIEMANN-RoeH.

823

position ~ of I\ determines its dual cell decomposition ~* of Fl' The 2-cell of ~* which is dual to Pi will be denoted by E j . We assume that the simplicial decomposition ~ is chosen so that i) each E j is contained in one of. Up, say UP1 and ii) each E j is covered by the domain of the local uniformization parameter Tj with the center Pj' N ow, choosing for each E j a simply connected neighborhood U j such that i) E j C U j C UPi' that ii) U j does not intersect with any simplex SkIm not containing the vertex pj and that iii) each intersection U i n Uk is simply connected (if it is not empty), we put (2.4) Then we have

tjk = tp1Pk>

III

Uj

n Uk' n Uk,

(2.5)

tj1f,tkJ

= 1,

in U j

(2.6)

tjktkltlj

= 1,

III

u,n Ukn Ul.

In order to prove our theorem, it is sufficient to show the existence of a system {hj} of meromorphic functions hj defined respectively in Uj such that hj/h k = tjk' in U j n Uk; indeed, the desired system {hp} can be obtained from {hi} by putting hp = tpp,hJ, in Up n U i . Now, let us forget the definition (2.4) of tik and consider an arbitrary system {tik} of holomorphic functions tik defined respectively in some neighborhood of [Uj n Uk] on r 1 satisfying the conditions (2.5) and (2.6). We shall say, for a moment, that two such systems {tJd and {t'Jk} are equivalent if tjk/t'jk can be represented as tjk/t'jk = hj/h k , in U j n Uk, by means of a system {hj} of meromorphic functions hj defined respectively in Ui . Then, it is sufficient for our purpose to show that {tjk} is equivalent to {l}. Since U j n Uk is simply connected, the logarithmic function Pjk = -

PM =

is determined uniquely in Uj Putting Cjkl = Pjk

n Uk

(1/2?ri) log tjk

+ Cjk

up to an arbitrary rational integer Cjk.

+ Pkl + PI;'

we get, by virtue of (2.6), rational integers on the choice of Cjk; but the sum J

=

~

Cjkl.

These integers CP,I depend

(sgn Sjkl)Cjkl

i A p., r consists in Uk of a finite number of branches r(l), r(2),' .. corresponding to points Pk(l), Pk(2),' • " e cf>-l(.\.lk)' Obviously we have r t = ~ rt(r) in Uk and therefore

+ +

+

+

r

lk(r, r t )

~

=

lk(r, rt(r).

r

Let ®k be the (spherical) boundary of Uk' Then the intersection Zt = ®k' rt(r) of ®k with rt(r) is a I-cycle bounding the closure [Uk' rt(r)] of the intersection Uk·rt(r). Hence, using (2.14), we get her, r/r» where R(z)

(1/2".i)

=

r

Jz.

d log R(z),

Rp.(z). Let qt e Zt be an arbitrary point on Zt. Then we have

=

and therefore

R(qt) = t

~

DaR(q) . u"(q)

" R (qt) =

where R (z (qt) and (2.15) for v"(q), we get

+ oCt) = K

is a

t ~ DaR(q) . v"(q)

" constant

=

{

--

K'

tB

not depending on q.

+ KtB + oCt), t· N(q)/hp(q) + KtB + oCt), t. N(q)hp(q)/fk2

R(qt)

+

+ oCt), Inserting

(lp"', h p, etc. have the same meaning as in Section 2. First we shall show that the meromorphic function 1/R)) can be considered as (L O-current defined in p, where U\1' is a sufficiently small neighborhood of p. For this purpose, we take an arbitrary 4-form cI> of class C'" with I cI> I CUp and consider the integral f(l/Rp) . cI>. In case p is a simple point of 1', the integral f (l/Rp) . cI> converges absolutely; then we put

u.

(l/Rp)[cI>]

(3.1)0

=

f (l/Rp)

. cI>.

In case p is a singular point of 1', we define the" principal value" (11 Rp) [cI>] of the (divergent) integral f(lIRp) . cI> as follows: Choosing the system of local coordinates (w, z) with the origin p (we write w, z instead of zt,ZZ) so that Rp (w, 0) does not vanish identically, we put (l/R)!) [cI>]

(3.1)

=

lim (
with I X leu» - p, then we have, by (3.9),

~[Xw]

=

~

,p(P)=p

r

J u.

X(]

=

Q,[d4>]

=

j'r ~(014)2{3. -

-

024>1{3.

{J

=

~

-

+ op.4>12)fhp pd,pp{3. fJ

r, we get

Considering 4>lz(cpp)fhpap as a I-form on

d (4)12 (cpp )fhp(Tp)

Then we have

op*4>lz!hp(Tpd,pp{3

(3

and consequently

f. ~op*4>lz!hp(Tpd,pp{3=-j' r

r

{3

~

where the sum

d(4)12fhpap)

=~ ~

lim

P ¢(p):p 12fhpap,

J l'I"pl=
lz/hppl)k(rj>p2) IhpUp, there exists at least one differential ~:E

2:

~

on r satisfying

¢(p)=P

(rj>/)k(rj>p2)lhp pl)k(rj>p2)lhpup p k. I ¢(p)=P

~

satisfying

(-h-),

apkl being arbitrary constants, we infer readily that the space f* (I', h) = f(I', b) f (0) is the subspace of f (h+) consisting of all f £ f( h+) satisfying (~f) = 0, for all ~ e S; thus f* (1', b) is characterized as the space which is orthogonal to S in the dualism between fW)lf*(b - c) and S(bc)/S*(c - b) with respect to the product (~f). Obviously S contains S (0) but does not necessarily contain S (c - b). The space which is orthogonal to f* (I', h) is therefore S S ( c - b). Hence we get

+

+

+

(4.3)

dim f*(I', h) - dim f*(b - c) = dim S(b-

+ c) -

dimeS

+ S(c- b)].

Obviously we have dim f* (I', b) = dim f(l', b) +}1, dim f* (b - c) = dim f (b - c) dim S(hand, since S

n S (c -

b)

dim [S

+ S (c -

+ c) =

deg c

+ 1",

+ dim S(h-)

S (1', - b) ,

=

b)] = dim S

+ dim S (c -

b) - i,

while, applying the theorem of Riemann-Roch to each r., we get dim feb - c)

=

deg h -deg c- ~ 'lrv

+ J1. + dim S(c-b).

Inserting these formulae in (4.3) and using dim S - dim S(b-) = obtain dim fer, b)

=

deg b - ~ 7l'v

~

lp, we

\l

+ J1.- ~p lp + i.

Thus (4. 2) is proved in the case that b is subject to the conditions: b- v

C 22

J

375

> 0,

B50

KUNIHIKO KODAIRA.

j bin I c I is empty. N ow, consider an arbitrary characteristic divisor .h' not satisfying the above conditions. b' is determined by a system {h'p} which is obtained from {h p } by the transformation hp ~ h'p = g' hp, g being a meromorphic function on r. Then, as one readily infers, Hr, b') or S(r,-b') is obtained respectively from f(r,b) or S(r,-b) by the transformation f ~ f' = g-' . f or ~ ~ = g .~, so that we have

e

dim fer, b')

dim fer, b),

=

dim S(r, - b')

=

dim S(r, - b).

Hence (4. 2) is valid also for b', q. e. d.

5. The theorem of Riemann-Roch on compact analytic surfaces. Let D = ~ mvrv be the representation of an arbitrary divisor D on m in terms of the irreducible components rv of D. Then, at each point .\J e m, D can be represented by the meromorphic function

Rp (D; ~" Z2) where each Rvp(z" Z2)

=

=

II {Rvp (Z', Z2) }mv,

•

r.

0 is the minimal local equation of

hp(D; Tp)

=

at.p.

Put

RI C '11"-1 (U A), we get, putting { = ~\ and TJ = IRA, d·dlogC[ot>] =dlog{[dot>] =

f

= -

=2'11"i{

f

r

J 111=.

(d'R) .ot>-

r

J 1111=.

(dTJ/TJ) .ot>}

r ot>- Jm", r ot>} =2'11"i(WC -un",)[ot>], o

Jrrn"

proving the above formula. (5.3)

(dU') . dot>

d{ (dU') . ot>} = -lim d{ (d,/,) . ot>} ....+0·p (T» belonging to f(r, D) and the induced function f(T) vanishes identically if and only if F belongs to ~(D - r). Hence we have dim fer, D) > dim {D} - dim {D Now we define the deficiency def(D/I')

=

46

n.

def(D/r) of the divisor D on the curve r aB

dim f(r, D) -

[dim {D} - dim {D - r}].

As one readily infers, if D' is linearly eqwivalent to D, then def (D' /I') = def(D/r). In case D and r have some components in common, we define de:f(D/r) as def(D/r) = def(D'/r) by means of a divisor D':::::; D having no component in common with r, assuming the existence of such a divisor D' (this is the case if ill( is an algebraic surface). It is to be noted here that def(r/r) coincides with the characteristic deficiency chd(r) of r, whenever def (r/r) can be defined. In fact, if there exists a divisor D ;:::: r such that D and r have no component in common, then we have a meromorphic function F with the divisor (F) = D - r, and, putting (5.16) we get a meromorphic function RIl(D) representing Din Ut!. It follows from (5.16) that the functions hp(D) =R(p)(D;cf>p) satisfy

hp(D)/hq(D)

=

R (p)/R (q),

in Up

n Uq,

proving that {hp(D)} can be used as a system {hpJ defining the characteristic divisor b. Thus we see that D· r is the characteristic divisor b so that fer, D) coincides with fer, b). Hence we get def(D/r) proving that def(r/r) The difference sup {r}

=

=

dim {r}

=

dim f(r, b)- dim {r}

+ 1,

chd(r).

+ dim {K -

r} - I (r, I')

+ + q7lT

g- 2

is called the superabundance of the divisor class {n. From our deduction of Riemann-Roch's inequality it follows immediately that sup {r}

=

i - [g- dim {K - n]

+ q- chd(r).

Now, assuming the existence of a canonical divisor K such that K and r have •• Cf. Goldman [9], § 6. In the terminology of Italian geometers, def (D In is the deficiency of the linear series on r cut out by the complete linear system

,D ,.

( 22

J

385

860

KUNIHIKO KODAIRA.

no component III common, we shall show that i - [g - dim {K - r}] is equal to def(K/r). Let W = W12 dz 1 dz 2 be the double differential with (W) =K and consider the one-to-one correspondence f~~=fW12(CPp)hpcrp between meromorphic functions f and differentials ~ on r. Since the divisor (W12(cpp)h pu,,) of W12hpe-p is K·r+b-c, ~ satisfies (~)-b+c>O if and only if (f) + K· r > O. Again, since W12 (P) = hp(K), ~ satisfies the condition ~ Resp[(cppl)k(cp/)!hp-1eJ =0 ¢(P)=p

if and only if

f

satisfies ~

Resp[(cpp1)k(cpp2)!fhp(K)up] =0.

¢(p)=p

Thus ~ belongs to S(r, - b) if and only if f e f(r, K). Hence we have i=dimS(r,-b)=dimf(r,K), while, as (5.14) shows, g=dim}ffi(O) = dim {K}. We get thl:!refore i - [g-dim {K -r}] = dim f(r, K)- [dim {K} - dim {K -r}], proving that i the following

[g - dim {K - r} ] is equal to def (K /r) . Thus we obtain

THEOREM 5.4 (G. Castelnuovo 47). The superabundance of the divisor class of an arbitrary curve r is represented as sup {r} = def(K/r) + q - chd(r).

(5.17)

6. Double differentials. In this Section we shall determine the dimension of the linear space m! (r) consisting of all double differentials on IDl which are multiples of - r in the sense that (W) + r > O. As was shown in Section 3, the residue = 91 (W) = Rp W 12 (p) Up of an arbitrary double differential We (r) is a well defined differential on r. This differential = !R (W) satisfies obviously

e

m

e

again, as one readily infers by Theorem 3. 2, (~)o

~ Resp[(cpl)k(p2)!~J=0,

~

= 91(W) satisfies

(k,l=0,1,2,···)

¢(P)=p

for each singular .):J of r. We denote by S (r) the space consisting of all differentials ~ on r satisfying ('I) 0 and (8) o. The mapping tv ~ ~ = !R (W) .7 This theorem was proved by G. Castelnuovo in the case that !lII is an algebraic surface. Cf.. Castelnuovo [4]; see also Zariski [30], p. 71.

( 22)

386

THE THEOREM OF RIEMANN-ROCH.

861

is a linear transformation which maps IDl(r) into aer)o Clearly ~ = !neW) vanishes identically if and only if W is a differential of the first kind. Now, suppose an arbitrary differential ~ e a(r) as given. Then, does there exist a double differential We !mer) such that !neW) =~? In order to solve this question, we associate with ~ the 3-current T~ defined as (6.1) where X is a "variable" I-form of class C~ and the value of the integral in the right-hand side is to be interpreted as the Cauchy principal value, in case it does not converge absolutely (see (3.15». T~ is obviously a pure 3-current of type 1. Again we have (6.2) In fact, for an arbitrary function w(z) of class C'Z>, we have dTE[W]

while

w(cpp H is

=

Tddw]

= J~ ~. dw(fi,)

a I-form of class C'Z> in dTE[W]

=

~ ~ )J 1>(p)=)J

where

~ p

=

-

r - \. c \.

lim ( +0

J~ d(w(cpp)~), Hence we get

w(cpp)~,

J 17.!=
f), (6.2) is a consequence of (6.1). Conversely, for arbitrary constants c(g) satisfying the simultaneous equations

Ls (g. Aj)c(g) = 0,

(6.4)

(j = 1,2, ... ,f),

there exists at least one differential ~ E l~l(-) having the form (6.1). In fact, it is easy to construct 24 a differential ~ with ~ _·c satisfying (6.1) for given c(g)

m

such that 24

Ref ~ is a single-valued function on ,3.. For this ~, the differential

Weyl 129), pp. 91-100.

(24)

434

310

~ + J~

KUNIHIKO KODAIRA

is everywhere regular and Re

J(~ + J~)

is a single-valued function.

Hence we get ~ + J~ = 0, while (5.1) is a consequence of (6.1). Consequently ~ belongs to W(-). We denote by t the number of triple points 9 of S and by e the number of linearly independent conditions involved in (6.4). Again, let l~lJ-) be the subspace of W(-) consisting of all ~ E W(-) of the first kind. Then, since each ~ E l~l (-) is determined by the constants c(g) uniquely modulo WJ->' it follows from the above results that dim (~l (-)

(6.5)

= t - e + dim

wt).

We decompose the differential al> on ;i induced by an arbitrary simple differential ex of the first kind on S as follows: Then, for any ~

E

W(-J, the condition (5.3) is equivalent to the following one:

hhi~-) = °

(6.6)

for all simple differentials a of the first kind on S. Since a~-J E W~-), the dimension of the subspace Z(-) of (~l (-) determined by the above condition (6.6) is given by dim ;g j, we denote by Cj the number of cuspoidal points lying on Ilj. Then, since ;ij is a two-fold covering manifold of Ilj having Cj branch points of order 1, the genus 1r j of ;ij is given by 1rj

=

271'j -

1

+t

Cj ,

being the genus of Ilj. Denoting by WOj the space of all differentials the first kind on ;ij, we have therefore

71' j

dim WOj = 271'j - 1

C 24)

435

+ t Cj.

~

of

311

THEOREM OF RIEMANN-ROCH

On the other hand, the subspace W~t) of WOj consisting of all ~ ~ WOj satisfying J~ = ~ is isomorphic to the space of all differentials of the first kind on f)"j , and consequently dim Wat> = 7r j • Hence we get dim W~i) = dim {~}Oj - dim {~}at> = 7rj - 1

+! Cj,

(j

>

f).

Combining this with (6.8), we obtain (6.9)

dim W~-) = Li=d7rj - 1)

+ f + ! c,

where C = Ll=l+l Cj • Since every cuspoidal point of S lies on one of the irreducible components f)"j , f < j ;;;:; s, C represents the total number of the cuspoidal points of S. Denote by {a} the space of all simple differentials of the first kind on Sand put q = dim {a}. Then we have dim {a~-)} = q - dim {a I Ja!!. = a!!.}. Every simple differential A of the first kind on 9')1 induces on S a simple differential A s belonging to the space {a I J a!!. = a!!.}. Denoting by {As} the space of all such differentials As, we see therefore that the difference (6.10)

o = dim {a I Ja!!.

=

a!!.} - dim {As}

is always non-negative. This difference 0 will be called the deficiency of S with respect to simple differentials. Let k be the number of linearly independent simple differentials A of the first kind such that As = O. Then, denoting by rl the total number of linearly independent simple differentials of the first kind on 9R, we have dim {As} = rl - k and therefore (6.11)

dim {a~-)} = q - rl

+ k - o.

In order to evaluate the constant e appearing in (6.5), we associate with each irreducible component f)"j of the first kind an indeterminate X j and consider the simultaneous linear equations for all triple points g.

(6.12)

Denote by 1]* the number of linearly independent solutions X j of (6.12). Then, since (6.12) is the transposed system of (6.4), we get e = f - 1]*. Now we shall show that 1]* is not smaller than the difference JL - m between the number JL of irreducible components of S and the number m of connected components of S. Let S = L~=1 S. be the decomposition of S into the irreducible components SI , S2, ... , SI' and S = L~=l S., S. = cp-l(S.), be the corresponding decomposition of S. Clearly each A}m), 1 ~ j ;;;:; f, lies on one and only one component S•. ~ ow we associate with each S. an indeterminate Y. and set (6.13)

if

Then these quantities Z j satisfy the equation (6.14)

C 24)

436

312

KUNIHIKO KODAIRA

for each triple point g of 8. This can be verified as follows: First we consider the

case where there are three different components of 8, say 8 1 , 82 , 8a , passing through g. In this case there are three different components of .l, say .ll, 1l2, .la, passing through g. Obviously we may assume that g~m) f 8k for k == A + rn (mod 3) and that gA f .lA . Let (i,j, k) be an arbitrary even permutation of (1, 2,3). Then we have g)1) f 8k , g)2) E8i and therefore (g . .lj)

=

r-1 for il~ C

1

LA.m (-1) mo(g~m), ilj)

= i l+1 for

°

8k , 8i •

-1-

Il j C

Again it is obvious that (t· .l/) = for l ;?; 4. Hence we get Lh (g . .lh)Zh = 0, proving (6.14). In case there are just two different components of 8, say 81 , 82 , passing through g, we may assume without loss of generality that g~l), g~2) E 81 and gil), gi2), g~l), gi 2) f 82 , For the irreducible component of .l, say .ll containing gl , both il~ and ilr lie on 82 and therefore the corresponding quantity Zl vanishes' identically. If g2 and ga lie on the same irreducible component, say .l2, then we have (g . .l2) = 0, while it is obvious that (g . .lj) = for j;?; 3; hence we get (6.14). Otherwise we may assume that g2 f .l2, \Ja E .la. Then we have

°

(g . .l2)

=

(2)

-I

(1)

-I

0(g2 , .l2) - 0(g2 , .l2) =

1+

il~

C

-1, if .l2

C

1, if

81 ,

-1-

82 ,

while Z2 equals Y l - Y 2 or Y 2 - Y l according as il~ C 81 or 82 ; consequently we get (g . .l2)Z2 = Y l - Y 2 • Similarly we obtain (g . .la)Za = Y 2 - Y 1 , while (g . .lj) = for j;?; 4. Hence we get (6.14). Finally, (6.14) is valid trivially in case there is only one irreducible component of 8 passing through g. Thus the quantities Z; defined by (6.13) satisfy (6.14). These quantities Zl, Z2, ... , Zj contain J.I. independent variables Y l , Y z , •.. , Y J.I., while, if Zl = Z2 = ... = Z, = 0, then YA must be equal to Y. as long as 8 A and 8. belong to the same connected component of 8, as one readily infers from (6.13). This shows that at least J.I. - m quantities, say Zh , Zh , ... , Z;._m are linearly independent; thus we see that 'Y/* ;?; J.I. - m. Putting

°

(6.15)

'Y/

=

'Y/* -

J.I.

+ m,

we introduce the non-negative integer 'Y/ which will be calleu the deficiency of the singularities of 8. Clearly the constant e = f - 'Y/* has the value e =

(6.16)

f -

'Y/ -

J.I.

+ m.

N ow, inserting (6.9), (6.11), (6.16) into (6.7), we get dim Z(-) = t

+ tc+

L; (11';

-

1)

+

q + rl - k + 0 +

j.I -

71 -

m,

and finally, combining this with (5.5), we obtain the following formula (6.17)

dim

5ffi(8)

=

t+

!C+

L; (11'; +

( 24)

1)

+

ra - r2

437

g - q

+

rl

+ J.I. + l - k + 0+

'Y/ -

m.

THEOREM OF RIE:\IANN-ROCH

313

§7. Fundamental theorem tn this section we shall summarize the results obtained above and state our fundamental theorem. Let S be a surface on SJJl with only ordinary singularities, i.e. a double curve d on which there is a finite number of ordinary cuspoidal points and of triple points of S (see §1, §2). Again, let S' = L~=l S. or d = Lf=l IJ. j be the decomposition of S or d into its irreducible components S. or dj, respectively. Each component S. of the non-singular model S = L~=l S. of S is a Kahlerian surface (see §1). The double curve d = LJ=I IJ.} has the image A = Lf=1 Aj on S which is a two-fold covering manifold of IJ. (see §2). We denote by J the covering transformation of A with respect to d which interchanges two sheets of A (see §2). The difference

o=

(7.1)

dim {a I Ja~ = a~l

-

dim {As}

is called the deficiency of S with respect to simple differentials, where {a I J a~ = a~} is the space of all simple differentials a of the first kind on S such that J a~ = a~, a~ being the differential on A induced by (x, and where {As} is the space of all differentials As on S induced by simple differentials A of the first kind on 9Jl (see (6.10». {As} is a subspace of {a I Ja~ = a~} and therefore 0 is always non-negative. The irreducible component IJ.; of d is said to be of the first or of the second kind according as Kj is reducible or irreducible; in case A; is reducible, Aj consists of two irreducible components A~, A; = JA: (see §6). For each triple point g of S, there exist three places (i.e. branches) gl , g2, ga of d passing through g and each place g>. corresponds to two places g~l), g~2) on A, while there exist three points tl , t2 , ta on S corresponding to g. We arrange six places g~m)(A = 1,2,3, m = 1,2) in such a way that g~m) passes through tk if A m == k (mod 3) (see (2.1». For each irreducible component d of the first kind, the index (g. IJ. j) of IJ.j at the triple point g is defined to be

+

(7.2)

(g'dj)

=

L>..m (_l)mo(g~m), A\

where o(gim>, A~) is 1 or 0 according as g~m) E A~' or 1/ A~ (see (6.3». We associate with each IJ. j of the first kind an indeterminate X j and consider the simultaneous linear equations (7.3)

for all triple points g, where the sum Lj is to be extended over all irreducible components IJ. j of the first kind. Then the number '1/* of linearly independent solutions Xi of (7.3) is not smaller than the difference }J. - m between the number !J. of irreducible components of S and the number m of connected components of S. The non-negative integer (7.4)

'1/ = '1/* -!J.

+m

is called the deficiency of the singUlarities of S (see (6.15». It is to be noted here that this deficiency '1/ is determined uniquely by the topological configuration of double curves and triple points of S. Now we introduce the following

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KUNIHIKO KODAIRA

'DEFINITION I (FIRST DEFINITION OF VIRTUAL ARITHMETIC GENERA). By the virtual arithmetic genus of S will be meant the integer a(S) defined by

(7.5)

a(S) = t

+ ! c + Lj

(7rj -

1)

+ L:. (g.

- q.

+ 1)

- 1,

where t is the number of triple points of S, c is the number of cuspoidal points of S, is the genus of fl j , g. or q. is respectively the number of linearly independent double or simple differentials of the first kind on the non-singular model S. of s. , and the sum Lj or L. is to be extended over all irreducible components flj of fl or S. of S, respectively. In case S = S1 is an irreducible algebraic surface free from singularities, the virtual arithmetic genus a(S1) = g1 - q1 coincides with the arithmetic genus of S1 defined as the virtual dimension of the canonical linear system 25 on S1. Incidentally, a simple differential A (or a double differential B) on we is said to vanish on S if the differential As (or Bs) induced on S by A (or B) vanishes identically. Now, we set 7r j

(7.6)

where r A is the number of linearly independent X-pIe differentials of the first q. + 1) appearing in (7.5) is kind on we. Then, since the expression equal to g - q + f.L, we obtain from (6.17) the following THEOREM II (FUNDAMENTAL THEOREM). The dimension of the linear space 5ill(S) consisting of all triple differentials on we which are multiples of - S is given by

L.(9. -

(7.7)

dim 5ill(S) = a(S)

+ a(we) + l -

k

+ 0 + '1/

-

m

+ 1,

where a(S) is the virtual arithmetic genus of S, a(we) is the constant defined by (7.6) above, l or k is respectively the number of double or simple differentials of the first kind on we which vanish on S, 0 is the deficiency of S with respect to simple differentials, '1/ is the deficiency of the singularities of S and m is the number of connected components oj S.

§8. Adjoint systems on 3-dimensional algebraic varieties

In what follows we assume we to be a 3-dimensional algebraic variety without singularities imbedded in a projective space. A divisor on we is, by definition, a 4-cycle D = m.S. with integral coefficients rn. composed of a finite number of irreducible surfaces S. on we. Each S. associated with rn' -,6 0 is called a component of D. The set of all divisors on we constitute an additive group. Every meromorphic function F on we which is not identically zero determines its divisor in a well known manner. The divisor of F will be denoted by (F). We say that a divisor D' is linearly equivalent to D and write D' = D if there exists on we a meromorphic function F with (F) = D' - D. Obviously the linear equivalence thus defined is an equivalence relation and therefore the set of all divisors on we can be decomposed into mutually disjoint equivalence classes. Each equiv-

L

25 See Zariski [30), pp. 122-128. For a proof of this fact based on the theory of harmonic integrals, see Kodaira [12], pp. 873-874.

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THEOREM OF RIEMANN-ROCH

alence class is called a divisor class. We say that D = L m.S. is effective and write D ~ 0 if all coefficients m. are ~ O. A meromorphic function F on 'lR is called a multiple of D if (F) - D ~ 0 or F is identically zero. We denote by 'i3(D) the set of all meromorphic functions on 'lR which are multiples of -D. 'i3CD) is a finite dimensional linear space over the field K of all complex numbers. Denote by e>h an h-dimensional projective space and by C~o , ~l , .• , , ~h) the homogenuous coordinates of a point ~ f e>h . A linear system26 on 'lR is, by definition, a set {lh.l = {Dx I A f e>h l of effective divisors lh. on 'lR depending on A in such a way that lh. = (FA) + D, FA = AoFo + AIFI + '" + AhFh , where Fo , Fl , .,. , Fh are linearly independent meromorphic functions on 'lR and D is a divisor such that F; f 'i3(D) (j = 0, 1, ... , h). As one readily infers, the correspondence between DA and A is one-to-one; thus DA depends on h independent parameters. This number h is called the dimension of the linear system {lh.}. We say that the linear system {D A ) is composed of a pencil27 if the degree of transcendency of the field K(FdFo , FdFo , .. , ,Fh/Fo) over K is equal to 1. An irreducible surface S is called a fixed component of IDA) if DA ~ S for all A. Again, a point p on 'lR is called a base point of ID,,} if every D" has a component 28 passing through lJ. We have the following theorems of Bertini: 1) The general member D" of a linear system ID,,} without fixed components is irreducible,29 provided that ID,,} is not composed of a pencil. 2) The general member D" of a linear system ID,,) has no singular points outside the base points of {D,,). In case the functions Fo , Fl , .. , , Fh generate the linear space 'i3(D) , the linear system IDA I D" = ("2)\jF j ) + D} is said to be complete and is denoted by I D I . Apparently the complete linear system I D I consists of all effective divisors which are linearly equivalent to D and dim I D I = dim 'i3(D) - 1. A canonical divisor K on 'lR is, by definition, the divisor (W) of an arbitrary triple differential W on 'lR: K = (W). The set {K} of all canonical divisors on 'lR constitutes a divisor class which will be called the canonical class on 'lR. The corresponding linear system I K I is called the canonical system. Again, for an arbitrary divisor D on 'lR, the complete linear system I K + D I is called the adjoint system of D. Let E be a hyperplane section of IDl and D be an arbitrary divisor on IDl. Then, as one readily infers, the complete linear system I D + nE I has neither fixed components nor base points and is not composed of a pencil if n is sufficiently large. Hence, in virtue of the theorems of Bertini, the general member of I D + nE I £s an £rreducible non-singular surface £f n is sufficiently large. THEOREM III. For an arbitrary surface Son 9)1 with ordinary singularities only, the dimension of its adjoint system i K + S I is given by (8.1)

dim I K

+ S I = a(S) + a(5JJl) + l

where aCS), a(9)1), l, k, 0,

'1],

- k

+ 0 + 'l/ -

m,

m have the same meaning as in Theorem II.

O. For the theory of linear systems, see, for example, van der Waerden [26], pp. 179-205. 27 See Zariski [31). pp. 67-68. 28 See Zariski [31J, [32J; van der Waerden [26), pp. 200-205; Akizuki [lJ. 2t More precisely, there exists a subset lB consisting of a finite number of proper subvarieties of ~h such that D~ is irreducible for every A E @5h - lB.

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KUNIHIKO KODAIRA

PROOF. Take an arbitrary but fixed triple differential Wo and set (Wo) = K. Then the linear mapping F -> W = FWo maps lj(K + S) one-to-one onto \ffi(S) and consequently dim I K + S I + 1 = dim \ffi(S). Hence the formula (8.1) follows from Theorem II. Now, denoting by E a general hyperplane section of IDe, we shall show that I = k = 0 = Tf = 0 and m = 1 for any surface of the type S = S' + E, S' being a surface with ordinary singularities only (the case in which S = E is included). In virtue of the theorems of Bertini, E is irreducible and is free from singularities, and, moreover, S = S' + E has only ordinary singularities. First, it is obvious that S is connected; hence m = 1. Second, it is well known30 that every 2-cycle (or I-cycle) on IDe is homologous to a 2-cycle (or I-cycle) on E. This implies that a double (or simple) differential of the first kind on IDe must vanish identically if it vanishes on E and, therefore, I = k = O. Third, if a I-cycle 'Yon E is homologous to zero on IDe, then 'Y is homologous to zero on E, as has been shown by S. Lefschetz. 31 This implies that, for every simple differential a(E) of the first kind on E, there exists on illC a simple differential A of the first kind such that AE = a(E) , AE being the simple differential induced on E by A. In fact, if aiEl were not contained in the space IAEl consisting of all induced differentials

AE ,

that

then there would exist a I-cycle 'Y on E such that

i =i A

'r

'r

AE

=

0 for all A, so 'Y ",-,0 on E and

i

rv

a(E) .,t.

0 and such

0 on illc, leading to a

S'

contradiction. Now, let = L~:~S. be the decomposition of S' into irreducible components S• . Since the intersection SI n E, ... , S"-1 n E are irreducible components of the first kind of the double curve .:1 of S = S' + E, we may assume that .:11 = SI n E, ... ,.:1"-1 = S"-1 n E and that .:1", .:1,,+1, , ••. .:1J are irreducible components of the first kind of the double curve of S'. Let a be an arbitrary = L~:~S. + E satisfying J at:. = at:.. simple differential of the first kind on Obviously this a has the decomposition ex = L':~ a. + aiEl , each a. being a simple differential on S• . Suppose that Li: c S, and Li~ C E. Then the simple differential a. is determined uniquely by the differential (a.)t:.: induced on Li: . This can be proved as follows: The curve Li: is the inverse image 4>-I(S. n E) of the hyperplane section S. n E of S. on S•. Now, if a. vanishes on Li: = 4>-I(S. n E), then, as one readily infers,32 a. must vanish on q,-rcS. n E') for every hyperplane section E' and therefore a. must be identically zero, proving that a. is determined uniquely by (ap)JI:. Combined with the relation (a.)JI: = J(a(E»)t:.:' which follows from J at:. = at:., this shows that a. is determined uniquely by a( E) • Thus every a = L~:~ a. + a(E) satisfying J at:. = at:. is determined uniquely by its last component a(E) . As was shown above, there exists on illc a simple differential A of the first kind such that AE = a(E) , which induces a simple

S

30 Lefschetz [13], pp. 88-91; see also Hodge [9], pp. 182-185. Foc another proof of this fact, see Eckmann [5]. 31 Lefschetz [13], pp. 88-91; see also Hodge [9], pp. 182-185. 32 Kodaira [12], pp. 864-865.

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THEOREM OF RIEMANN-RoeH

atE) , we differential a' = As on S satisfying Ja~ = a~. Now, since a~E) have (X = (X' = As. Thus every simple differential (X of the first kind on S satisfying J aa = aa can be represented as a = A s by means of a simple differential A of the first kind on we, proving that D = O. Finally, it is easy to see that /1 j , II- ;;;;; j;;;;; f, meets with E at one triple point g of S (at least) for which we have (g·/1k) = ±Dkj, II- ;;;;; k ;;;;; f. Hence the simultaneous equations (7.3) can be solved with respect to X~, X~+I' ... , XI' This shows that TJ* ;;;;; II- - 1, while TJ = TJ* - II- + 1 is always non-negative. Hence TJ is equal to zero. Thus we see that I = k = D = TJ = 0 and m = 1 for the surface S = S' E. Consequently we derive from Theorem III the following THEOREM IV. Let E be a general hyperplane section of we. In case the surface

+

S is of the form S = S' + E, S' being a surface with ordinary singularities only (the case in which S = E is included), we have

dim I K

(8.2)

+ S I = a(S) + a(we)

- 1.

§9. The second definition of virtual arithmetic genera The 4-cycle - K represents the third basic characteristic class of we, K being a

canonical divisor on we. 33 This can be verified as follows: Let W = W123 di dl di be a triple differential having the divisor (W) = K. Choose a simplicial decomposition .\'f of we whose I-dimensional skeleton .\'f 1 does not meet the carrier of K such that each simplex of sr is covered by a single system of local coordinates. and attach to each 2-simplex T of st the integer K(T)

= 2-~' d log '7I'dr aT

W123 ,

where aT denotes the boundary of T and W123 is the coefficient of TV with respect to the system of local coordinates which covers T. Then, considered as a fUIlctional of a "variable" 2-simplex T, K(T) defines a 2-cocycle on we. The third basic characteristic class 34 of we is, by definition, the cohomology class of the 2-cocycle - K(T). N ow, as one readily infers, K(T) equals the intersection number I(K, T) of T and K: K(T) = I(K, T). Hence the 4-cycle -K represents the third basic characteristic class·5 of we, q.e.d. Now, we denote by C a 2-cycle expressing the second basic characteristic class of we. It is to be noted here that the second basic characteristic class is represented by the canonical (virtual) curve C, on we introduced by Segre: 6 Thus we can choose as C the algebraic cycle C. , but we do not use this fact in what follows. 33 Recently Hodge has shown that Chern's basic characteristic classes on an irreducible non-singular algebraic variety are represented by canonical systems introduced by Ege[' and Todd. See Hodge [10], Eger [7], Todd [24]. 34 Chern [3], pp. 99-103. 3. From the point of view of the theory of currents, each homology class can be identified with its dual cohomology class. See de Rham and Kodaira [17], pp. 22-42. 36 Hodge [10]. See also Segre [19].

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KUNIHIKO KODAIRA

N ow we shall show that the virtual arithmetic genus a(S) of S can be represented by means of K and C as (9.1)

a(S) = i-S

3

+ tKS2 + n(K2 + C)S -1,

where S3, KS 2, ... denote the topological intersection numbers J(S, S, S), I(K, S, S), , respectively. For this purpose, we set, for a moment,

(9.2)

iJ'(D)

=

i-D3

+ tKD2 + n(K2 + C)D

D being an arbitrary divisor on (9.3)

'Ir(Dl , D2)

=

- 1,

m. Again, we introduce the binary functional

t(D~D2

Dl , D2 being arbitrary divisors on

+ DlD~ + KD lD2) + 1,

m.

Then we have

(9.4) Obviously iJ'(D) or 'Ir(Dl , D2) depends only on the homology classes of D or

Dl , D2 , respectively. In case S is an irreducible surface free from singularities

on m, 'Ir(D, S) represents the virtual genus 'lrDS of the divisor DS on S cut out by D, D being an arbitrary divisor not containing S as one of its components. To prove this, we first remark that, for an arbitrary pair of divisors D, D' on the topological intersection number I(D, D' , S) is equal to the topological intersection number I s(DS, D'S) of the divisors DS, D'Son S cut out by D, D', respectively.37 Now, the canonical divisor class {Ks} on S is determined by the adjunction formula 38

m,

(9.5)

~

Ks

(S

+ K)S,

on S,

where (S + K)S denotes the divisor on S cut out by a general39 divisor which is linearly equivalent to S + K. Hence we get 39a

+ J(D, S + K, S) = Is(DS, DS) + l s(DS, S2 + DS) = 1 s(DS, DS) + J s(DS, Ks) = 2'1rDS - 2,

2'1r(D, S) - 2 = J(D, D, S)

37 This is obvious in the special case where D, D' are both non-singular surfaces cutting out on S non-singular curves DS, D'S such that DS meets D'S at simple intersection points only. The general case can easily be reduced to this special case by making use of the following facts: Let D,. or En be the general member of I D + nE I or I nE I, respectively, E being a hyperplane section of lIn. Then D = D. - En and, if n is sufficiently large, D n , En are both non-singular and cut out on S non-singular curves DnS, EnS, respectively. 38 Hodge (IOJ, p. 146. 38 Let Dn or E,. be the general member of I S + K + nE I or I nE \, respectively, where E is a hyperplane section of lIn and n is a sufficiently large positive integer. Then Dn - En is a general divisor which is linearly equivalent to S + K. 3'. The virtual genus 7(c of a curve C on the surface S is given by the formula 27(c - 2 = 18 (C, C) + Is(Ks' C). See Kodaira [12J, p. 854, the formula (5.7).

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THEOREM OF RIEMANN-ROCH

proving that 1T"(D, S) = 1T"DS. We remark here that the adjunction formula (9.5) follows immediately from (3.2). In fact, let W = W 123 di di di be a triple differential on ffi1 having S as one of its polar surface of mUltiplicity 1. Then, as (3.2) shows, w = Rq,(p) W123ITp is a double differential on S, and its divisor Ks = (w) is equal to (K + S)S. Hence we have (9.5). In order to prove that a(S) = CP(S), we first consider the case in which Sis an irreducible surface free from singularities. In this case a(S) coincides with the arithmetic genus of S so that a(S) is related with the Zeuthen-Segre invariant I of S and the relative invariant p(1) = I s(K 8 , K s) + 1 by the relation48 I

while I (9.6)

+

p(l)

= 12a(S)

+ 9,

+ 4 equals Euler's characteristic Xs of S.41 Hence we get 12(a(S) + 1) = xs + K~ ,

where K~ denotes the intersection number Is(Ks, Ks). Now, Euler's characteristic Xs of S is given by the adjunction formula 42 (9.7)

Xs

= C·S

+ KS·S2,

where S2 denotes the divisor on S cut out by a general divisor which is linearly equivalent to S. This formula can be deduced as a consequence of the so-called duality theorem on complex sphere bundles in the following manner:43 We define on S three complex sphere bundles CB, CB t CB n as follows: CB is the bundle consisting of all unit tangential vectors of ffi1 at all points on S, CB t is the bundle consisting of all unit tangential vectors of S, and CB n is the bundle consisting of all unit normal vectors of S. Then it is obvious that the first or the second basic characteristic class of CB is represented by C . S or - K . S, respectively. Again, the first or the second basic characteristic class of CB t is represented by Xs or - K s , respectively. To prove that the basic characteristic class of CB n is represented by S2, take a meromorphic function F on ffi1 having S as one of its zero surfaces of multiplicity 1 and set D = S - (F). With the help of this function F, we define on S - C£(DS) , C£(DS) being the carrier of DS, the continuous field of normal unit vectors

n" =

L:Il g"P' dIJFI(L: g'YIJ* d'YFdIJF)\ '"(,Il

(d" = d/dZ").

For each point lJ on S, we choose the system of local coordinates (z! , z: , z!) with the center lJ such that S coincides with the coordinate plane z! = O. Then, setting F = z:M~ , we get a merom orphic function M~ defined in a neighborhood of lJ having the divisor (M~) = - D, and ncr has the representation (9.8)

n"

I = (M~/IM~)

" ·n~,

n~""s'/( =g ,g3S')! .

See Zariski [30], p. 62. See Zariski [30], p. 113. 42 Hodge [10], p. 146. 43 The following proof of the adjunction formula is due to Chern. 40

41

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444

320

KUNIHIKO KODAIRA

n;

Apparently is a continuous field of normal unit vectors of S defined in a neighborhood of lJ. Now, the field n" determines a 2-cocycle on S expressing the characteristic class of (En in the following manner:44 Choose a simplicial decomposition of S whose I-dimensional skeleton 1 does not meet with C£(DS) so that each simplex of ~ is covered by a single system of local coordinates. Then, for each 2-simplex T of we can define the continuous mapping 1): Z -7 Mp(z)/I Mp(z) I of the boundary aT of T into the unit circleu in the plane of a complex variable by means of the function Mp/IMp I expressed by (9.8) in terms of the system of local coordinates which covers T. The image f)(aT) of the I-cycle aT is homologous to an integral multiple p(T)·u of the 1-cycle u. Attach to each 2-simplex T the corresponding integer peT). Then the cochain so defined is a 2-cocycle expressing the characteristic class of (En . Now it is obvious that

sr

sr

sr,

peT)

= 21 . 7rt

i

aT

d log

Mp =

lCD, T).

This proves that the characteristic class of (En is expressed by the 2-cycle DS on S cut out by D, while, since D = S, we may consider that S2 = DS. Hence S2 represents the characteristic class of (En . Now, the bundle (E is the product of the bundles (Et and (En . Hence, by virtue of the duality theorem,45 the first basic characteristic class C· S of (E is represented by

=

C·S

Xs -

KS·S2,

and this proves (9.7). Now, inserting (9.7) into (9.6) and using (9.5), we obtain 12(a(S)

+ 1)

+ KS, 2S + KS) + C· S = 1(8 + K, 28 + K, 8) + C· 8 = 28 + 3K82 + = I S(S2

2

3

(K2

+ C)8,

proving that a(8) = CS'(S) for the surface 8 free from singularities. Now, consider the general case in which S is a surface with ordinary singularities only. Take a general hyperplane section E of W1. Then E is free from singularities and S + E is also a surface with ordinary singularities only. 8 therefore cuts out on E a curve 8E with only ordinary singularities (i.e. ordinary double points). Obviously the double curve of 8 + E is composed of SE and the double curve A of S. Moreover, the set of all triple points of S + E consists of all triple points of 8 and of all double points of SE, while S + E has no cuspoidal points other than those of S. Consequently, it follows from Definition I that (9.9)

a(S

+ E) = a(8) + t' + L:' (7rj -

1)

+ aCE) + 1,

'4 See Steenrod [23], pp. 177-181. 45 A proof of the duality theorem for real sphere bundles is to be found in Chern [4], pp. 101-102. The duality theorem for complex sphere bundles can be proved in a similar manner as in the real case. See also Steenrod [23], pp. 198-199.

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THEOREM OF RIEMANN-ROCH

where t' is the number of double points of SE and L:' is the sum extended over all irreducible components Aj of SE, 'lrj being the genus of Aj • Now, t' + L:'('lrj - 1) + 1 is equal to the virtual genus46 'Ir(S, E) of the curve SE on E, while, as was proved above, we have aCE) = (P(E). Hence we get, from (9.9), (9.10)

a(S

+ E)

= a(S)

+ (P(E) + 'Ir(S, E).

By Theorem IV, we have a(S

+ E) = dim

I

K

+S+E

1 -

a(9JC)

+ 1,

and this proves that a(S + E) is determined uniquely by the complete linear system I S I, while 'Ir(S, E) depends only on the homology class of S. Consequently, by (9.10), a(S) is also determined uniquely by the complete linear system 1S I· Furthermore, comparing (9.10) with (9.4), we obtain a(S) - (p(S)

=

a(S

+ E)

- (p(S

+ E),

- (PCS

+ L:kn_l

and this yields immediately (9.11)

a(S) - (P(S) = a(S

+ L::=l E(k»

E(k),

where E(l) , E(Z), •• , ,E(n) are general hyperplane sections such that S + L::-l is also a surface with ordinary singularities only. Now, choose n so large that the general member Sn of the complete linear system I S + L:k':..l E(k) I is an irreducible surface free from singularities. Then, since a(Sn) = (P(Sn) as was proved above, we have E(k)

a(S

+ L:k':..l E(k)

= a(Sn) = (P(Sn) = (p(S

+ L::-l

E(k),

and therefore, using (9.11), we get a(S) = (P(S). This proves (9.1). In view of the relation (9.1) thus proved, we introduce the following DEFINITION II (THE SECOND DEFINITION OF VIRTUAL ARITHMETIC GENERA). By the virtual arithmetic genus of an arbitrary divisor D on 9JC we shall mean the integer a(D) defined by (9.12)

a(D)

= iDa + tKD2 + n(K2 + C)D

- 1,

where K is the canonical divisor on 9JC, C is the 2-cycle expressing the second basic characteristic class of 9JC, and DB. KD2, ... denote the topological intersection numbers J(D, D, D), J(K, D, D), .... Obviously a(D) is determined uniquely by the homology class of D. Furthermore we have

(9.13)47 as (9.4) shows. The fact that a(D) is always an integer can be verified as follows: Choose a positive integer n so Jarge thf1,t 1 D + nE I contains a surface Sn free 46 47

See Kodaira. [12J, p. 850. See Severi [21], p. 41.

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from singularities. Then, denoting by En a general member of I nE I (En is also a surface free from singularities), we get, by (9.13),

a(D)

(9.14)

=

a(8n) - aCE,,) - 7r(D, En),

where the arithmetic generaa(8 n), a(En) of 8 n , En and the virtual genus 7r(D, En) of the divisor DEn on E" are all integers. Hence a(D) must also be an integer. §10. The theorem of Riemann-Roch for adjoint systems 48

For an arbitrary divisor Don WC, we call7r(D, D) the virtual curve genus of D and denote it by 7r(D2). In case 8 is a non-singular surface, 7r(82) represents the virtual genus of the curve 8 2 on 8 cut out by a general member of the divisor class {8). It follows from the definition (9.3) that (10.1)

Now, consider an arbitrary surface 8 on WC with only ordinary singularities and set I D I = I K + 8 I Then, using (9.12), we get easily

a(8) = D3 - 7r(D2)

+

a(D)

+

1-

fi KG.

Hence, putting (10.2)

Pa(WC) =

-fi KG - a(WC)

+ 2,

we obtain from Theorem III the following THEOREM V (THE THEOREM OF RIEMANN-RoCH FOR ADJOINT SYSTEMS). The dimension of the adjoint system I D I = I K + 8! of an arbitrary surface 8 on WC with ordinary singularities only is given by

(10.3)

dim I D I = D3 - 7r(D2)

+ a(D) -

pa(WC)

+3 -

m+l- k

+0+

7],

where 7rCD2) is the virtual curve genus of D, a(D) is the virtual arithmetic genus of D, pa(WC) is the constant defined by (10.2) and l, k, 0, 7], m have the same meaning as in Theorem II. Severi has proved the theorem of Riemann-Roch for the adjoint system of an arbitrary "general" surface in the form of an inequality.49 Now we shall deduce the result of Severi from the above Theorem V. Denoting by n the fundamental exterior form i L gafJ.dzad~/ associated with the Kahlerian metric on WC, we consider the integral (lOA)

(A, A)s = i

1

nAsAs = i

1

nAA,

where A is an arbitrary simple differential of the first kind on 9R. Then, since inA sA s is a positive definite density on 8, A s vanishes identically if and only if (A, A)s = 0, while it follows from the identity d(nAA) = 0 that (A, A)s depends only on the homology class of 8. Consequently, if As vanishes identically, 48

4~

Severi [21], p. 63. Severi [21}, pp. 63-64. See also footnote 4.

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THEOREM OF RIEMANN-ROCH

323

then A T vanishes identically for an arbitrary surface T which is homologous to S. N ow, assume that S is irreducible and that dim I S I ~ 1. Then there exists at least one meromorphic function F on IDe with (F) ~ - S. Setting St = (F - t) + S, t being a complex parameter, we get a linear system {Stl containing S( = S",). Obviously a general member St of {Stl is irreducible; more precisly, there exist a finite number of "exceptional values" t1 , t2 , ••• , tn such that St is irreducible for t ~ tl , t2 , ... , tn . In order to show that k = 0 for S, consider a simple differential A of the first kind on IDe satisfying As = O. Then, since each St is homologous to S, it follows from the above result that As, = 0 for each t ~ tl , t2, ... ,tn' This shows that A has the representation A = 1> dF, where 1> is a meromorphic function on IDe. Moreover, since d1>·dF = dA = 0,1> must be a constant on each St , t ~ tl , t 2 , ... , tn . Consequently 1> is a meromorphic function of F: 1> = 1>(F). Now, since A = .p(F) dF is a differential of the first kind, .p(t) dt is a differential of the first kind on the space of a complex variable t and therefore .p(t) dt must vanish identically, proving that A vanishes identically on IDe. Thus we see that As = 0 implies A = O. Hence the number k is equal to zero. Now, combining this with Theorem V, we obtain the following THEOREM VI (Severi). If S is irreducible and if dim I S I ~ 1, then the dimension of the adjoint system I D I = I K + S I of S satisfies the inequality dim I D

(10.5)

I~

D3 - '/r(D2)

+ a(D)

- pa(IDe)

+ 2.

In case S contains a general hyperplane section E as one of its irreducible components. we have l = k = 0 = 1) = 0 and m = 1, as was proved in Section 8 (see also Theorem IV). Hence we get from (10.3) the following THEOREM VII.50 In case a complete linear system I D Ion IDe is sufficiently ample in the sense that I D - K - E I contains a surface with ordinary singularities only (the case in which I D I = I K + E I isincluded), the dimension of I D I is given by dim I D

(10.6)

I=

D3 - '/rCD2)

+ a(D)

- paCIDe)

+2.

It is easy to see that

+ (;) ('/r(E2) - 1) + n(a(E) + 1) = n 2(n - 1) E3 + n 2 ('/r(E2) - 1) + 1.

aCnE) = (;) E3 '/r«(nE)2)

- 1,

Inserting these expressions into (8.2), (10.6), we get the postulation formulae (10.7) dim I K

+ nE I =

(;)

e + (;) ('/r - 1)

+ n(a + 1) + a(IDe)

- 2,

(for n (10.8) dim I nE

1=

(n

t 2) e - (n t 1)

('/r - 1)

+ n(a +1)

~

1),

- Pa(IDe) , (for large n),

50

See Severi [21), p. 63.

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KUNIHIKO KODAIRA

where e E3 is the order of 'iJJl, 7r = 7r(E2) and a = aCE). The formula (10.8) shows that pa('iJJl) is the arithmetic genus of the variety 'iJJl (see the formula (12.4) below). We note that D3 - 7r(D2) + a(D) = -a( - D) - 3, so that (10.6) can be written in the form s1 dim I D

1= -a(-D) - Pa('iJJl) - 1. I D I of I D I is, by definition, the dimension of the - D I increased by 1: i I D I = dim I K - D 1 + 1.

(10.6)'

The index of speciality i complete linear system I K In the simplest case in which 'iJJl is a (3-dimensional) projective space, we have the identity dim I D

I=

+ a(D)

D3 - 7r(D2)

- Pa(lJR)

+ 2 + i I D I.

This suggests that we define the superabundance of an arbitrary arbitrary variety 'iJJl) as (10.9) sup I D I = dim I D

I-

D3

+ 7r(D2) -

a(D)

+ Pa('iJJl)

I D I (on

- 2 - i I D I.

The formula (10.3) shows that the superabundance of the adjoint system 1 K of an arbitrary surface 8 with only ordinary singularities is given by (10.10)

+8

sup I K

1

1- k

=

+ 0 + 'T/

-

m

an

+8

I

+ 1.

sup I D I is not necessarily non-negative; moreover there exists a variety 'iJJl on which one can find, for any positive number N, a surface 8 with sup I K + 8 1 < - N. As an example, let us consider the direct product 'iJJl = ~ X @5 of a curve ~ and a surface @5. Take m points 1Jl, 1J2, ... , 1Jm on ~ and construct the surface 8 m = L:'=I1Jv X @5 on 'iJJl. Denoting by ~ a differential of the first kind on ~ and by Wk a k-ple differential of the first kind on @5, we infer readily that any simple, double or triple differential of the first kind on 'iJJl may be written in the forms ~ + WI , ~'Wl Wz or ~'W2 , respectively. Using this fact, we can readily compute the numbers 1, k, 0, 'T/ for 8 m • As result, we get

L

l

+

L

= 7rfJi' q@5 ,

where 7rfJi is the genus of

k ~

sup) K

=

o=

7rfJi,

(m -

l)q@5,

'T/ =

0,

and q@5 is the irregularity of @5. Hence we obtain

+8

m

1 =

(q@5 - l)(7rfJi

+m

- 1).

In case @5 is a regular surface (i.e. q@5 = 0), we have therefore

+ 8 I = - 7rfli - m + I, and this shows that sup I K + 8 I can take an arbitrary large negative value. sup I K

m

§11. Deficiencies Let S be an irreducible surface on 'iJJl free from singularities, D be a divisor on 'iJJl not containing 8 as one of its components and 'i5(D) be the space of all meromorphic functions F on 'iJJl which are multiples of -D. Again, denote by f(DS) 51

Cf. Zariski [34].

(20

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THEOREM OF RIEMANN-ROCH

the space consisting of all meromorphic functions f on S which are multiples of -DS, DS being the divisor cut out by Don S. Then each FE. 'J(D) induces on S a function P s belonging to f( D S) and F s vanishes identically if and only if F lies in 'J(D - S). Thus F ---+ F s is a linear mapping of 'J(D) into f(DS) whose kernel is 'J(D - S). Setting (ILl)

I D IS =

!(Fs)

+ DS I F

E

'J(D) - 'J(D - S)},

we define a linear subsystem I D IS of the complete linear system I DS I on S. I D IS is called the linear system cut out by I D 1on S, since I D IS consists of all divisors D'· S, D' E I D I. It follows from (11.1) that the dimension of the linear system I D IS is given by (11.2)

dim(1 D 18) = dim I D

I-

dim I D - 8

I-

1.

N ow we define the deficiency of the divisor Dover S to be (11.3)

def(DIS)

=

dims I DS

I-

dim(1 D 18),

where dims! DS I denotes the dimension of the complete linear system 1DS I on S. The deficiency def(DI S) is also called the deficiency of the linear system I DIS, provided that I D IS is not an empty set. Obviously def(DI S) vanishes if and only if the linear system I D 18 is complete. Furthermore we have (11.4)

def(DI S)

=

dims I DS

I-

dim I D

I + dim I D

- 8

I + 1.

It is clear that def(D IS) depends only on the divisor class of D. In case D contains S as one of its components, we define the deficiency of Dover S to be def(DI S) = def(D'1 S) where D' is a general divisor D' = D. By the characteristic deficiency of 8 will be meant the deficiency def(SI8) of S over itself. First we shall deduce a formula expressing the characteristic deficiency of S in terms of the simple differentials of the first kind on m. Take a divisor D = S not containing S as one of its components and put r = D8. Then we have (11.5)

def(SI S) = dims I r

I-

dim I S

I + 1.

The divisor class of r on S is called the characteristic divisor class of S. Now, let Fo be the meropmorphic function on m with the divisor (Fo) = D - S. Then, for each P E 'J(S), the ratio FIFo is a meromorphic function belonging to 'J(D), so that FIFo induces a meromorphic function f = (F/Fo)s belonging to fer). Thus we get a linear mapping F ---+ f = (FIPo)s of 'J(S) into fer). By means of the concept of currents, the relation between P and f = (FIFo) s can be described as follows: Since the integral F[ X2) be a system of local coordinates with the center p. We consider the direct product Up X S of a neighbor-

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PROC. N. A. S.

hood Up of p and a projective line 5 and, by means of the homogeneous coordinates (to, tl) on 5, construct the subvariety

W p = {(Xl, X2; to, tl ) [tOX2 - tlXl

=

O}

(1)

of Up X 5. Obviously Wp is an analytic surface without singularities and contains the projective line p X 5 which will be denoted by the same symbol 5: 5 = P X 5 c Wp • The mapping

Pp': (Xl> X2; to, tl ) -- (Xl, X2) is a regular mapping of W p onto Up which maps S onto p and W p - S bi-regularlyonto Up - p. Hence, replacing Up by Wp, we get from Va new analytic surface VI = (V - Up) U W p having the following properties: There exists a regular mapping Pp of VI onto V such that S = Pp ~l(p) is a projecti~e line and that Pp is bi-regular between VI - S and V - P (Pp is a "natural" extension of PP'). The inverse mapping Ql' = Pp ~l of Pp will be caned the quadratic transformation with the center p. It is easy to see that the quadratic transform VI = Qp( V) of V is determined uniquely by p (and V) and is independent of the choice of the local coordinates (Xl> X2) appearing in the above construction. Every meromorphic function F = F(x) on V induces a meromorphic function F(Pp) on Qp(V). Conversely every meromorphic function Fl on Qp( V) induces a meromorphic function F on V such that F(Pp) = Fl. In fact, it is obvious that Fl induces on V - P a meromorphic function F such that F(Pp) = Fl. To prove that F is meromorphic also at p, consider the meromorphic function Xl ~m Fl in a neighborhood of 5, m being the multiplicity of the component 5 in the divisor of Fl. Then, on the curve 5, Xl ~m Fl equals a rational function R(t) in t = tI/to, so that R(X2/Xl) ~lXl ~m Fl is constant on S. This shows that R(X2/Xl)~JXJ~mFl induces a holomorphic function H(xl' X2) in a neighborhood of P. proving that F = xlmR(X2/Xl)H(xl> X2) is meromorphic at p. The mapping F -- F(Pp) gives therefore an isomorphism between \)'(V) and \)'(Qp(V)). Thus the quadratic transformation Qp does not affect thefield of aU meromorphicfunctions on V: \)'(Qp(V)) = \)'(V). LEMMA 1. If V is algebraic, then Qp( V) is also algebraic. Proof is well known. 3 LEMMA 2. If V is a Kahlerian surface, then Qp( V) is also a Kahlerian surface. Proof: Using the system of local coordinates (Xl, X2) with the center p, the Kahlerian metric on V can be expressed in a neighborhood Up of p as

ds 2 =

2

L

(o2K/oxi)xk)dxj dXk'

j, k=l

K being a function of class C" defined in Up.

unit sphere {(Xl> X2) [

[XI[2 +

[X2[2


= - = 1, 2) instead of 9)1, S, we infer that, for an arbitrary element WI + W2 E llJo, the exterior derivative ~ = dWl = -dw2 pelongs to @m-z(r) and satisfies as a current on S}. the relation

(>. = 1, 2)

(7)

and that, if an element ~ of @m-z(r) satisfies (7), thereexistw}. E $fi(r; S}.) (>. = 1, 2) satisfying dWl = -dw2 = ~ and w... [,9}.l =

(8)

0,

(see the remark at the end of Section 4). Obviously (8) implies (6). W2 - ~ = dWl = -dw2 maps llJo onto the Hence the linear mapping WI subspace .p(r) of @m-2(r) consisting of all ~ E @m-2(r) satisfying (7). The kernel f of this mapping is' given by

+

f = {WI

+ w21 w}.

E

@m-l(S},), wdBs,l

+ w2[Bs,l

= 0 (B

E

@m-l(9)1» J.

Setting l = dim IBI BE @m-l(9)1), B s, = B s, = 0), we get dim f (Sl) + gm-l(S2) - gm-1(9)1) + I and therefore dim llJo = gm-l(Sl)

+ gm-l(S2)

- gm-l(9)1)

+ I + dim .p(r).

= gm-1 (9)

It is obvious that the condition (7) is equivalent to

(>Hence, setting gm-2(S) = dim {a1

+ 0!21 a}. E@m-2(Sh),

=

1,2).

(10)

air == a2rJ, we get

Combining this with (9) and inserting it in (5), we obtain the following THEOREM 2. In the second case (b), we have dim m3(S; Wl) = gm(Wl) - grn-l(IDl) + gm-l(SI) - gm-2(Sl) gm-l(S2) - gm-2(S2) + gm-z(r) + gm-2(S)

+

+ l,

(11)

where I is the number of linearly independent differentials B E@m_l(IDe) satisfying B St = B s, = 0 and gm-2(S)is the dimension of the linear space {al + a2\ a}. E @m-2(S}.) (>- = 1, 2), air = a2r l· 6. In this section we shall consider the case in which 9)1 is an (irreducible non-singular) algebraic variety imbedded in a projective space. Let E be an arbitrary hyperplane section of WI which is irreducible and non-singular. Then a theorem of Lefschet:;8 asserts that, in case p ~ m - 1 (m = dim IDe), every p-cyc1e Z on 9)1 is homologous to a p-cyc1e ZE lying on E and that, in case p ~ m - 2, a p-cyc1e Z E lying on E is homologous to zero on E if Z E is homologous to zero on Wl. It follows from this theorem that

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526

(1) in case p ~ m - 1, a p-ple d~tJerential P f @p(WC) vanishes identically if P E vanishes identically on E and that (II) in case p ~ m - 2, the mapping P - P E maps @p(WC) isomorphically onto@p(E). The second proposition (II) implies obviously that gp(E) = gp(WC) for p ~ m - 2. Incidentally we denote by K the canonical divisor on WC. Now, for an arbitrary irreducible non-singular algebraic variety V of dimension n, we set

Then we have THEOREM 3. is given by

+ E I on WC

The dimension of the complete linear system IK dim IK

Proof: Since dim IK

+ EI

+ EI

=

a(9.n)

= dim

+ aCE)

(13)

- 1.

)ID(E; WC) - 1, we get, by Theorem

1,

I + EI

dim K

=

gm(Wlj - gm-l(WC)

+ gm-l(E) + Z -

1.

(14)

Now, it follows from the above proposition (I) that I = 0, while gp(E) = gp(WC) for p ~ m - 2. Hence we obtain from (14) the formula (13), q. e. d. THEOREM 4. Let S be an arbitrary irreducible non-singular subvariety of WC of dimension m - 1 and E be an irreducible non-singular hyperplane section of WC which cuts out on S an irreducible non-singular subvariety ES with the intersection multiplicity 1. Then the dimension of the complete linear system K + S + E is given by

I I dim IK + S + EI = a(WC) + a(S) + aCE) + aCES)

- 1. (15)

Proof: By virtue of Theorem 2, we get dim IK

+ S + EI = gm(WC) - gm-l(WC) + gm-l(S) - gm-2(S) + gm-l(E) - gm-2(E) + gm-2(ES) + gm-2(S + E) + I -

1.

Now, we have gp(E) = gp(WC) for p ~ m - 2 and, since ES is a hyperplane section of S, gp(ES) = gp(S) for p ~ m - 3, while it follows from the proposition (I) that I = O. Hence we obtain dim IK

+ S + EI

=

a(WC)

+ a(S) + aCE) + aCES) - 1 + gm-2(S + E) - gm-2(WC).

It is sufficient therefore to show that gm-2(S gm-2(S + E) is the dimension of the linear space @ =

where

r

=

I al + a21 al f ES.

@m-2(S), a2 f @m-2(E), air

+ E)

=

=

gm-2(WC).

a2r},

For an arbitrary element A f@m-2(WC), we have

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VOL. 38, 1952

527

MATHEMATICS: K. KODAIRA

+

As ~ @m-2(S), AE ~ @m-2(E) and (As)r = (AE)r; thus A - As AE is a linear mapping of @m-2(9'J1) into @. It follows from the proposition (II) that, for every element ar + a2 of @, there exists one and only one element A of @m-2(9'J1) with AE = a2. This element A satisfies (ar - Ash = a2r - (AE)r= 0, while r = ES is a hyperplane section of S. Hence, by the proposition (I)! we get As = ar. Thus we see that A - As + AE is a one-to-one mapping of @m-2(Wl) onto @. Consequently we obtain gm-2(Wl) = gm-2(S + E), q. e. d. 1 In case there exists on IJJl at least one non-trivial meromorphic m-ple differential W o, the canonical divisor K on 9)/ may be defined by K = (Wo). Then the dimension of !ffi(S; 1JJl) is equal to the dimension of the adjoint system \K S\ of S increased by 1. 2 For the special case in which IJJl is a Kiihlerian surface, see K. Kodaira, "The Theorem of Riemann-Roch on Compact Analytic Surfaces," Am. J. Math., 73, 813-874 (1951), §6. 3 See Kodaira, lac. cit .. §1, where the reader will find a summary of the results in the theory of harmonic integrals which are necessary for the present note. 4 de Rham, G., and Kodaira, K.; "Harmonic Integrals" (mimeographed notes), Princeton (1950), pp. 63-68; see also Kodaira, lac. cit., p. 817. , Kodaira, lac. cit., Theorem 1.4. t Kodaira, lac. cit., Theorem 1.3. 7 Hartogs, F., Acta Math., 32, 57-79 (1909). 8 Lefschetz, S., L'Analysis situs et /.a g,'ometrie alg:bri'lue, Paris, 1924, pp. 89-91. • It can be shown that a( V) is equal to the arithmetic genus of V. See K' Kodaira. "Arithmetic Genera of Algebraic Varieties," these PROCEEDINGS, 38, 527-533 (1952).

+

(26)

480

ARITHMETIC GENERA OF ALGEBRAIC VARIETIES By

KUNIHIKO KODAIRA

INSTITUTE FOR ADVANCED STUDY, PRINCETON,

N. ].

Communicated by Hetmann Weyl, March 10, 1952

Let us consider an arbitrary irreducible non-singular algebraic variety Vn of dimension n imbedded in a projective space, and denote by gk(Vn) (1 ;;;; k ;;;; n) the number of linearly independent k-ple differentials of the first kind attached to V n' In the present short note, we shall prove a conjecture of F. Severi 1 to the effect that the arithmetic genus Pa(Vn) of Vn is represented in the form

Paevn)

=

gnevn) - gn-l(Vn )

+ gn-2(Vn )

-

+ ..... + (_l)n-lg1 (V n). (1)

It follows from (1) that the arithmetic genus Pa(V n) is a birational invariant

of V n , since each gk(V n ) is known to be a birational invariant 2 of V n • 1. We set

C 27

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MATHEMATICS: K. KODAIRA

528

PROC. N. A. S.

+ (_l)n-'g,(V n). (2)

Then we have to prove the identity (3)

First we consider an arbitrary but fixed irreducible non-singular variety = V m of dimension m ~ 2, A linear system {A of effective divisors A on 9.n without 'fixed components corresponds to a rational mappingS of 9.n into a projective space @3, and this correspondence is one-to-one, We shall say that the linear system {A} is sufficiently ample if the corresponding rational mapping induces a biregular imbedding of 9.n into @3. By virtue of theorems of Bertini, a general member A of a sufficiently ample linear system {A is an irreducible non-singular subvariety of 9.n. It is obvious that, if {A 1 is sufficiently ample, the complete linear system IA I is also sufficiently ample. Again, if A} is sufficiently ample and if {B has neither fixed components nor base points, then the complete linear system IA + B I is also sufficiently ample. Assuming that 9.n is imbedded in a (fixed) projective space @3d of dimension d, we denote by Eh the section of 9.n cut out by a general hypersurface (!h of order h. Then it follows from the latter proposition that, for a given divisor D, the complete linear system I D + E h \ is sufficiently ample if h is sufficiently large. In what follows we denote by S, T, U irreducible non-singular subvarieties of 9.n of dimension m - 1 and by K the canonical divisor on 9Jt Incidentally, for an arbitrary divisor D on an irreducible non-singular variety V, we denote by D v the complete linear system on V consisting of all effective divisors which are linearly equivalent to D. THEOREM 1. 4 If S belongs to a sufficiently ample linear system on 9Jl, then we have

I

9.n

I

I

I

I I

dim I K

+ SI9)/ =

a(9JO

+ a(S)

- 1.

(4)

THEOREM 2.5 Assume that m = dim m ~ 3. If S belongs to a sufficiently ample linear system on m and if the intersection S· T is an irreducible non-singular variety with the intersection-multiplicity 1, then we have dim IK

+ S + T\ m =

a(9.n)

+ a(S) + aCT) + a(S, T)

- 1.

(5)

The above two theorems can be proved with the help of the theory of harmonic integrals. s THEOREM 3. If = TIIJ.);' then a(S) = aCT). Proof: In case m( = dim 9Jl) = 2, the genus a(S) of the curve S on the surface m is determined by the relation 7 2a(S) - 2 = I(S, K S), where I(S, K + S) denotes the intersection number of Sand K + S. Hence a(S) = aCT) if I sim = I Now, assume that our theorem is

Isim I

+

Tim.

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MATHEMATICS: K. KODAIRA

.529

true for m < n and consider the case in which m = n > 2. Let E = El be a general hyperplane section of IDe. Then S or T cuts out on E an irreducible non-singular subvariety S· E or T· E, respectively. Since IK + S + ElilJI = IK + T + ElilJI' we get, using (5), a(S) + a(S·E) = aCT) + a(T·E). On the other hand, it follows from IslilJI = TlilJI that Is· E = IT. E IE. Hence, by the induction hypothesis, we get a(S· E) = a(T·E). Consequently we obtain a(S) = aCT), q. e. d. THEOREM 4. Let V be an irreducible non-singular subvariety oj IDe oj dimension m - 1. IJ sl WI is sufficiently ample, then K + S + WI cuts out on V a complete linear system. Proof: Denote by IK + S + WI' V the linear system cut out by IK + S + WI on V and by K u the canonical divisor on V. Clearly, a general member S of Isl ilJI cuts out on V an irreducible non-s;ngular variety S· V with the intersection-multiplicity 1, and Is· u is sufficiently ample. Now, we infer readily that IK + S + :m' V is contained in the complete linear system K u + S· u, while

I

IE

I

vi

I

vi

vi

vi vi = dim IK + S + vi dim {I K + S + vi I

WI'

WI -

vi

vi

I + sl WI -

dim K

1.

By virtue of (4) and (5), we get dim Ku + S· dim K + sl ilJI dim K + S +

vi u vi

+ a(S· V) - 1, + a(S) - 1, WI = a(9)() + a(S) + a(V) + a(S· V) - 1. Consequently, we see that dim 11 K + S + vi V I = dim IK u + s· vi u. This shows that IK + S + vi ilJI' V coincides with the complete linear system IKu + S· vi u, q. e. d. =

a(V)

= a(IDe)

9)1'

COROLLARY. Let V be an irreducible non-singular subvariety of IDe OJ dimension m - 1. Then, Jor an arbitrary divisor D on IDe, D + Ehl WI cuts out on Va complete linear system iJ h is sufficiently large. 2. We denote by Ek the intersection of k general hyperplane sections of 1JJl. For each k ~ m - 1, Ek is an irreducible non-singular subvariety of IDe of dimension m - k and, as one readily infers from Theorem 3, a(Ek) is independent of the choice of Ek, i.e., a(Ek) is determined uniquely by IDe, IEI WI and k. Em is a set consisting of a finite number of points on IDe. We denote by a(Em) the number oj points oj Em decreased by 1. Clearlya(Em) + 1 is equal to the order of IDe. LEMMA 1. Assume that the intersection S· T is an irreducible non-singular WI = Is + TI WI and iJ variety with the intersection-multiplicity 1. IJ sl WI and WI are both sufficiently ample, then we have

I

I

I vi

I vi

a(V)

=

a(S)

+ aCT) + a(S·T).

ProoJ: The lemma is an immediate consequence of (4) and (5). 5. We have

THEOREM

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483

(6)

MATHEMATICS: K. KODAIRA

530

aCE,,)

+ (_l)m-1 =

PROC.

N. A. S.

'f (h)k [a(Ek) + (_l)m-k].

(7)

k=1

Proof: We prove the relation (7) by induction on m and h. In case = 2, (7) is an immediate consequence of the relation 2a(E h ) - 2 = I(E", E" + K). Now, assume that (7) is true for m ~ n - 1 and consider the case in which m = n > 2. For h = 1, (7) is valid trivially. Assume therefore that (7) is true for E"_I' Since E"I m = E"_I EI WI> we get, using (6), m

I

I

+

(R)

while, by the induction hypothesis, a(E"_I) a(Eh_I.E)

+ (_l)n-1 = kti + (_l)n-2

==

e 1) ~

[a(Ek)

+ (_l)n-k],

"f (h -k 1) [a(Ek+l) + (_l)n-k-I]. k=l

Inserting these two identities into (8), we obtain (7), q. e. d. It follows from Theorem 5 that there exist a polynomial v(l; m) tn of degree m and an integer 10 (depending on m) such that

I

dim Ell m = vel; m) - 1,

for I ~ 10 •

(9)

In fact: choose a positive integer 10 such that IElo - K[ m is sufficiently ample and set I = h + 10 • Then, denoting by S an irreducible nonsingular variety belonging to IElo m, we get, using (5),

KI

I

I + E" + sl m =

dim Ell m = dim K

aCID?)

+ aCE,,) + a(S) + a (E"S)

- 1

while, as (7) shows, aCE,,) or a(EhS) is a polynomial in h of degree m or m - 1, respectively. This shows the existence of a polynomial vel; m) satisfying (9). This polynomial v(l; m) is called the postulation of 9J1. The arithmetic genus8 Pa(m) of m is defined by (10) LEMMA

2.

For sufficiently large I, we have

vel; S)

=

I

dim Ell m - dim lEI -

sl m.

(11)

Proof: For large I, we have v(/; S) = dim [EIS[ s + 1, while, since, as the Corollary to Theorem 4 shows, Ell m cuts out on S a complete linear system, dim IEsl s is equal to dim IEll m - dim lEI - SI m - 1 Hence we get (11). From (11) we get immediately the following

I

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484

VOL.

531

MATHEMATICS: K. KODAIRA

38, 1952

If / S/ m

6.

THEOREM

= / T/ roll then v(l; S) = vel; T) and Pa(S) =

PaCT). THEOREM 7. Assume that the intersection S· T is an irreducibl~ nonsingular variety with the intersection-multiplicity 1. If I m = IS + TI m, then we have (12) vel; U) = v(l; S) + vel; T) - v(l; S·T),

ul

Pa(U) = Pa(S)

+ PaCT) + Pa(S, T)

(13)

Proof: Considering S· T as a divisor on T, we get, by (11), vel; S· T) = dim EITI T - dim EIT - sTI T for large I, while, by virtue of the Corollary to Theorem 4,

I

I

dim IElTIT

=

dim IEllm - dim IE 1 - Tim - 1,

I

dim EIT - STI T = dim lEI for sufficiently large I.

v(l;S·T)

=

sl m -

TI IDI -

dim lEI - S -

1

Hence we obtain

+

dim IE1jm- dim jEz- sim -dim jE z - Tjrol dim IEz -

ul m·

Using (11), we obtain (12) from this immediately. The formula (13) is an immediate consequence of (12). 3. Let a be a projective space of dimension d and Va-II Va_2' "', V a- k , '" be irreducible non-singular subvarieties of a such that each V a _ k is the section of V a- k+1 cut out by a hypersurface ~' nk of order nk (in particular V a_1 = ~' n,)' Denoting by ~I a general hypersurface of order 1, we infer readily that the canonical divisor on V a- k is given by K d _ k = ~Sk' V d _ k , where Sk = nl + n2 + ..... + nk - d - 1. Since ~l' V a- k+1 - Vd-kj V a-k+l = \ ~I-nk' V a- k +1 \·Va-k+l and v(l; V d- k +I ) =

e

e

I

dim I~l' Va-HIj Vd-k+l

+ 1 for large I, we get, by (11),

vel; V d- k )

=

vel; V d -

H1 )

vel -

-

It follows from the well-known formula vel;

v(-l- d - 1; f6w=m(W) is a linear mapping of jill(S) into m(l'). LEMMA

1. 3.1.

me W) =0 if and only if W is of the first kind.

m( W) =0 if and only if Rp Wp=O on S. But Rp Wp=O on S if and only if (Rp Wp)jRp is holomorphic, q. e. d. PROOF.

Let mdenote the mapping

m:

W ---> w

= m( W) ,

and let jill(O) = gt-l(O)

be its kernel (space of all differentials of the first kind), let (1. 3. 9)

\U = m(m3(S»,

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\Us;;;; m(l').

492

APPLICATIONS TO ALGEBRAIC GEOMETRY

Then

m~ 5m(S) /5m(0) ,

(1. 3.10)

and therefore dim 5m(S) = !/m(an) +dim m

(1. 3.11)

where (1. 3.12)

!/m(an) = dim 5m(O).

To find dim 5m(S) , it is thus sufficient to find dim m. 1. 4. The'dimension of 5m(S) when S is irreducible and non-singular. We shall find a formula for 5m(S) under the assumption that S is irreducible and non-singular. Then

Moreover, in the neighborhood of any point of S we can choose local coordinates z; of an such that S is defined by z~=O and such that u1=z:, , .. , u m-1 =z';' on S. A minimal local equation therefore is Rp(z) =0 where R~(z) =z~. In the equation (1. 3. 3) take a=l. Since a«R~=azl/azl=l, Z2=¥J2=ul, , .. , zm =¥Jm=um -\ we then have ap

= du 1du

2

...

du m -1,

I

= -Cap) = O.

It follows that ru(.!') = ru(O) = {wi w: (m-I)-ple differential on S of the first kind},

LEMMA

1. 4.1. Wand w=9't( W) are currents on

an of types (m, 0)

and (m,

1) respectively. PROOF,

Consider

(1. 4.1)

WUPJ

= LWotP

where tP is an m-form, tP E Coo. Since R~ W = V where V is a local holomorphic function, we have

V

W=-.

z!

The integral

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493

K.

KODAIRA

converges absolutely since

It follows that the integral (1. 4. 1) converges absolutely; hence W is a current, obviously of type (m, 0). Next, consider (1. 4. 2)

w[WJ =

Is

w' Ws

where Ws denotes the restriction of the form Wto the subvariety S, defined as follows. Let W = 'J:, W"""'P,P, ... (z)dz· ' ... dz P1 dz P, ...

be an (m-1)-form of class Ws

C~

on ftn. Then

= 'J:,W"""'P,P, ... (cp(u))dcp·'dcp·' ...dql'dql' ... = W(cp(u)).

Since w[WJ =0 if Ws contains no terms of type (0, m-1), w is a current of type (m, 1). LEMMA PROOF.

1. 4. 2. The residue of W is given by the formula d W =2rriiR( W). By definition dW[WJ

= (_1)m+l W[dWJ = (_1)m+lj W·dW

where Wis an (m-1)-form of class

C~.

Let

J:> E S;

it is sufficient to show that

d W[ WJ = 2rri!R( W) [WJ

for any Wsuch that We have

~(W)

cU(J:».

(_1)m+l

rWdW = (_1)m+lJfIR-S r W.dW.

JfIR

In ftn-S, dW=O since aw=o automatically and dW=O because W is holomorphic. Therefore d(W.W) = (-1)mWdW,

and hence dW[WJ = -

r

J'4Il-S

d(W. W) = -lim .--0()

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r

J I .1 I ;,.

494

d(W· W) = lim .-.0

r

J I .' I =.

W· W

APPLICATIONS TO ALGEBRAIC GEOMETRY

· = I,-0 1m

J

\ Z1 I

=,

P' dW 12"'m ~1Tr 2 d Z md':l' pI",Pm-1 d Z 1dZ... Z... Z Pm - ' • 0 "'"'

Since z1z1=e2 on IZ11 ==e, we see that dz 1dz 1=0; therefore

dW[W]

= limJrI .-0

Z1

I =,

W 12 •.. mo"1:.W•... mdz1 ... dzmdz2 ... dzm

= 27ri Lm( W) oW. LEMMA

1. 4. 3. The residue of W satisfies d9t( W) =0, H9t( W) =0.

In fact, 1 d9t(W) == 2---;d odW 7rt

= 0,

1 H9t(W) = -2.HdW = 0. rrt

LEMMA 1. 4. 4. If w is an (m-l)-ple differential of the first kind on S which satisfies dw=O, Hw=O, then there exists aWE 5ill(S) with ffi( W) =W. PROOF. Let

8 = (dA +Ho)Gw.

By Theorem 1.1.1, 8 is of type (m, 0) and d8=.J-1w; in particular, 8 is a holomorphic differential in IDC-S, ~(w) =S. It is thus sufficient to show that 8 is meromorphic and that R~8 is holomorphic. To prove this, let PES and consider a neighborhood U(p). In this neighborhood Let

then m( W~) = (Z1 W~)du1 ... dum-l = w.

We see that W~ is a current in U(j:J) of type (m, 0) which satisfies dW~=2rriw there. Let T~=2rr8- W~. Then T~ is a current of type (m, 0) in U(p) and dT~ =2rri(w-w) =0. By Lemma 1.1. 3, T~ is holomorphic in U(p) and hence 27r8 = T~+ W~ is meromorphic. If we take W=27r8, the conditions of the lemma are satisfied. 4. 5. If WE w(O) and if w is interpreted as a current, then dw=O. We have

LEMMA 1.

PROOF.

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495

K.

dw[qJ]

KODAIRA

= (-I)mw[dqJ] = (-I)ml w·(dqJ)s = (-I)m lWdqJs

since (dqJ)s=dqJs. Here to 5,

qJ

is an (m-2)-form. Since dw=O if w is restricted

1

dw[qJ] = (-I)m wdqJs = - ld(wqJs) +ldw./Ps = O.

Now we consider Hw. Let {B l , B z, ... , B h _,} be an orthonormal·basis for the (m-I)-ple differentials on Wl of the first kind. A basis for the harmonic (m-I)-form is then given by {B l , " ' , B um -" qJ, "', Bl , " ' , Bum _,} where the Bi are of type (m-I, 0), the qJ of type (r, s), (r, s) *- (m-I, 0) or (0, m-I). Then Hw =

~(w,

*X)*x

= (-I)m-l~w[x]*x

where the x's run through the harmonic basis differentials. Since w is of type (m, 1), W[E] =0 unless x is one of the Bi and hence gm.-l

Hw

_

= (-l)m-l~w[Bi].Bi. i~l

This proves the following LEMMA 1. 4. 6. If WE m(G) and if w is regarded as a current, Hw=O if and only if weB] =0 for all (m-I)-ple differentials of the first kind B on Wl. Let (1. 4.4) l=dim{BIB: an (m-I)-ple differentials of

the first kind on Wl with Bs=O}. Then the number of non-trivial conditions involved in Lemma 1. 4. 6 is Ym-l -I, so (1. 4. 5)

dim iB

= dim 11)(0) -Ym-l +1.

Hence we obtain finally from (1. 3.11): THEOREM 1. 4.1. If the subvariety 5 is irreducible and non-singular, the dimension of the space 5m(S) is given by the formula

dim 5m(S) = gmOln) -gm-l(Wl) +gm-l(S) +1. CoROLLARY.

If K= (Wo) exists, then

dimlK+SI

= Ym(Wl) -gm-l(Wl) +Ym-l(S) -1+1.

1. 5. The dimension of5m(S) when S is the union of two subvarieties which

are irreducible and non-singular together with their intersection. Let S be an

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APPLICATIONS TO ALGEBRAIC GEOMETRY

analytic subvariety, scan, and assume that S=Sl +S2 where Sl, S2 and Sl S2 are irreducible and non-singular. Then 5=51 +5 2 where 5 is mapped into an by a mapping As=As.+As, and (As.}r= (As,)r=A r. Therefore {As} ~ {ala Ir=a2r}.

Let (1. 5.15)

Since dim{As} = dim{A}-k =

gm-2(~)-k

where (1. 5.16)

k = dim{AIAs=O},

we have (1. 5. 17)

dim {(al-a2)r} = gm-2(Sl) +gm-2(S2) -O-gm-2(~) +k

and therefore, by (1. 5. 14), (1. 5.18)

dim E = gm-2(r) -gm-2(Sl) -gm-2(S2) +O+gm-2(~) -k.

Finally, by (1. 3. 11), (1. 5. 5) and (1. 5. 10) : (1. 5.19) dim m3(S) = gm(~) -gm-l(~) +gm-2(~) +gm-l(Sl) -gm-2(Sl) +gm-l(S2) -Ym-2(S2) +gm-2(F) +l-k+o.

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502

APPLICATIONS TO ALGEBRAIC GEOMETRY THEOREM 1. 5.1. If S=SI +S2 where SI, S2 and r=SI,S2 are irreducible and non-singular, then dim ~(S) = (7m(9.n) -(7m-l(9.n) + (7m-2(9.n) + (7m-l(SI) -(7".-2(SI) + (7m-l(S2) -(7m-2(S2) + (7m-2(F) +l-k+o where 1= dim{BIB E @"'-1(9.n), Bs! = B s, = O}, k = dim{AIA E @m-2(9.n), As! = As, = O}, i3 = dim{al+a2Ia, E @m-2(S,), alr = azr} -dim{As!+As,IA E @m-2(9.n)}. The above method of computing the dimension of ~(S) may be applied to more general types of S, provided that the structure of singularities of S is explicitly known. For example, we may consider the case S=SI +S2+S3 where SI, S2, S3, r,,=s,'S, and r 123 =SI,S2,Sa are each irreducible and nonsingular. The dimension formula then takes the form

dim ~(S)

3

3

z

= L( -1)kgm-k(9.n) + L L( -l)k(7m-l-k(SJ k=O ,=lk=O 1

+ L L( -1)kgm _2_ir1,) +Ym-a(F123) +additional terms. J. is a l-form of class Coo on 9n, dTp[q>]

= Tp[dq>]

= isf3dq>s

=

J

d(f3q>s)

= O.

Now (5.4.6)

dTw,[q>] = Tw,[dq>] = isw,odq>s = w,[dq>s] = 2;ril'ift(w,) oq>J

= dw.[q>s]

= 2;ril~oq>J

for every l-form q> of class Coo on 9n. If \) E A and if \) is not a cuspidal or triple point, then SO-l(\) = W, .f/). Hence J is a 2-fold covering space of A. Let J be the covering transformation of J with respect to A which interchanges the two sheets of J ; that is,

J.f = p",

]j;" =

.f.

We observe that J is an involution: P=1. J induces in a natural wayan involution of the space of differential forms on J. Let q> be a I-form on 9n. Then and hence (5.4.7) if and only if

JJ~oq>J = O. Therefore (5.4. 7) is true if and only if

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543

K.

KODAIRA

(5.4.8) Since r[J is an arbitrary I-form, we conclude from (5.4.5) and (5.4.6) that dTw=O if and only if (5.4.9) Let (5.4.10) We remark that w=w,+j3 is uniquely determined by w. In fact, if w=w,+ fi=w/ + fit, then w,-w/ =fi'- fi where

lsw

-fi) -pr = 0

for an arbitrary double differential fil of the first kind on vanishes identically. It follows that (5.4.11)

5. Hence

fir - fi

dim roo = dim BC-) +dim{fi I fi satisfying (5.4. 4)}.

Let

= dim{fi I fi: double differential of the first kind on 5} be the geometric genus of 5, and let (5.4.12) g

= g2(5)

(5.4.13) 1 = dim{B I B: double differential of the first kind on an, Bs :::: OJ. Since dim{Bs}

= dim{B} -dim{BIBs = OJ,

we have (5.4.14) and therefore by (5. 4. 11) (5.4.15) The problem of determining dim QB(S) is thus reduced to that of determining dim BC-) (a I-dimensional problem). 5.5. Dimension of the space BC-). Consider a triple point t, and let (iI, h 1a) The points 11 ,12 , la are double points of J, and they correspond to six points i.e m ) (m=l, 2,; l=l, 2, 3) on the non-singular model of J. We suppose that l.cm) lies over 1k if k=l+m (mod 3). By (5.3.6) =~-I(t).

(5.5.1)

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544

APPLICATIONS TO

ALGEB~AIC

GEOMETRY

by (5.4.10) (5.5.2) EC-)

= W~~ -c,

(5.3.13) and (5.3.14) satisfied,

R= -~}.

The differential ~ has a :pole at each place l,em) and is otherwise regular. Therefore, at each place l,em) , (5.5.3)

~ =

e,Cm) (t)dlog v [i,em)] +holomorphic function

where vet,em)] is a local uniformization variable at (,em). The condition R=-~ implies that (5.5.4)

e,,(t) = -e,n(t)

since ]L,=l,,,. In the notation of Section 5. 3, let pI and pI! be the two points of J which correspond to the double point ~ E £1. The situation is then given by the following scheme : pI pI!

fl fll fa' f2 fa" l/ fa fI" fl. The condition (5. 3. 13) implies that

hence by (5.5.4)

e/I

(5.5.5)

= -ea' = ea" = -e/ = e/' = -e/ = e.

Thus (5.3.13) and the condition (5.5.4) imply that (5.5.6)

~ =

(-l)me(t)dlog v [i,em)] + holomorphic function

e

at each t,cm). On the other hand, if satisfies (5. 5. 6), then (5. 3.13) is satisfied and ~+ R is a differential of the first kind on J. Finally, (5.3.14) states that

1~ .

(5. 5. 7)

al =

0

for all simple differentials a of the first kind on S. Let

~~~

£1=±~ j=1

where the £1j are irreducible components. Then (5.5.9)

J = ±Jj, j=1

Jj = lp-l(£1j ).

We say that £1j is of the first kind if Jj is reducible, Jj=J/ +J/, and that £1 j is of the second kind if it is irreducible. Assume that £11, £12, ... , £1f are of the

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545

K.

KODAIRA

first kind, J,+1, J'+2, ... , J 8 of the second. Then J = J/+J/'+···+J/+J/'+J'+1+···+J8'

(5.5.10)

Given j, l::O;j::O;s, let j ::0;/,

(5.5.11)

j>1

and consider the condition LRes[PJ(~) =

(5. 5.12)

o.

pEr

If i) f'=J j

U> I),

then ll' E Jj implies that l,,, E J j • Hence Res[L,J(~) +Res[l,,,J(~) =

O.

Since Res(l,4, and let Zo, Zl, ... , Zd be homogeneous coordinates in @Sd. We may assume without loss of generality that l' is the point of V with the homogeneous coordinates zo=I, ZI=O, ... , Zd=O and that V does not pass LEMMA

through any of the coordinates points (0, ... ,0, Ii, 0, ... , 0) (I~i~d). Further we assume that each coordinate line meets with V at simple intersection points only. Let e=(d!I)_I, and let Pik, i, k=O, 1, ... , d, if/J/(ZI) =f/JQ-pl(ZI) of V' onto V* which corresponds to the linear system ICll of curves C,'=QvCC,) on V'. This new system IC,'I might have a base point 1:1' on S= Qi+» , butthe multiplicity m(1:1 /) of +>' is strictly smaller than me+»~ since

Ip,(Cl, C/)

= Ip(C"

Cp) -m,m p

where m, and mp are the multiplicities of the curves c" Cp at +>, respectively. This shows that, by applying a finite number of suitable quadratic trans-

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576

APPLICATIONS TO ALGEBRAIC GEOMETRY

formations Qv, Qv', ... successively, we obtain an analytic surface

V=

... Qv"Qv,Qv( V)

such that the mapping iP=(/JQplQ-;lQ;;f ... of V onto its algebraic equivalent V* is a regular mapping. By Lemma 8. 2. 4, V is a Kahler surface. The inverse mapping iP- l has possibly a finite number of fundamental points and branch curves and, except for these singular points, iP- l is locally bi-regular. Hence, for any P* on V* which is not a singular point, the set iP-l(p*) consists of a finite number n of points on V, the number n being independent of P*. We now show that n=l, and the proof is based on the Riemann-Roch theorem for Kahler surfaces. Let E: be the section of V* cut out by a general hypersurface of order h containing no fundamental point of iP- l and set Eh=iP-l(E~). Since V* is the algebraic equivalent of V, the complete linear systems IEhl on V and IE: Ion V* have the same dimension: dimlEhl =dimIE:j. By (4.2.7) (8.3.1)

dimlE:1 =

~ I(E:, E:) - ~ I(E:, K) +a(V*) +1

if h is sufficiently large (in which case IE: I is sufficiently ample). On the other hand, by the Riemann-Roch theorem for Kahler surfaces (K. Kodaira, American Journal of Math., vol. 73 (1951), 813-875) (8.3.2)

1-1-dimlEhl 2 ZI(Eh, E h) -ZI(Eh, K) +a(V) +l-g

where g denotes the geometric genus of I(E:, E:) = h2I(Ei, En,

V.

Since IE:I=lhEiI, we have

I(Et, K*) = hJ(Ei, K*).

On the other hand, I(Eh, E h) =n·I(Et, Et) =nh2J(E,,{, En. Combined with dimlEhl =dimlEt I, the above two formulas show (by taking' h sufficiently large) that n=l. It follows from n=l that iP- l has no branch curve; thus iP- l is regular and single-valued except for a finite number of fundamental points each of which corresponds to a fundamental curve on V. We show finally that these fundamental curves can be eliminated by applying quadratic transformations to V*. Let P* be a fundamental point on V*. Then L=iP-l(p*) is a (possibly reducible) fundamental curve on V. Let L = Ll + L2 + ... + Ln be the decomposition of L into a sum of irreducible components L j . Now let F l , F2 be two meromorphic functions in {'Y={'Y(V) ={'Y(V*) such that (Fl, F 2 ) constitutes a system of local coordinates on V* with the center P*. Then the curves B, of zeros of the meromorphic functions ).lFl + ).2F2 in V constitute a linear pencil {B,} on V. The image B!=iP(B,) is the curve of zeros' of ).lFl+).2F2 on V* and {Bn is a linear pencil on V, having P' as one of its base points. By a suitable choice

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J

577

K.

KODAIRA

of F l , F 2, we may assume that {Bn has no fixed component and that no base point of {Bn other than P* is a fundamental point of $-1. Then {B,} has no fixed component outside L. Let Bo="LkjLj be the fixed component of {B,}. Then we have B,

=

k j :::::: 1.

"LkjLj+C"

The same argument as in the case of fundamental curves on algebraic surfaces* shows that a general C, meets one irreducible component of L, say L 1 , in one variable point with intersection multiplicity 1 (that is, I(C" L 1 ) =1) but does not meet other components L 2, La, ... of L. In fact, take a divisor D* ~B! on V* such that no component of D* passes through any fundamental point of $-1 and set D=$-l(D*). Then D~Bo+C" while lCD, L j ) =0 for each L j . Hence we get (8.3.3)

lCD, Bo) = O.

Now, take another general Cpo Clearly Ip*(Br, B;) =1 and, for each intersection point q* of Br, B;, we have q = $-l(q*),

I.(C" Cp) = Iq*(B";., B;),

since, by hypothesis, q* is not a fundamental point of $-1 and $ is therefore bi-regular in a neighborhood of q. Hence we get I(C" Cp )

::::::

"Llq(G" Cp) = "Llq*(B"I., B;) q

q*

= I(B";., B;) -1 = I(D*, D*) -1 = lCD, D)-1.

Consequently, we obtain, by (8. 3. 3) , lCD, D) = lCD, Bo+G,) = lCD, C,) = leBo, C,) +1(Cp , C,) :::::: leBo, C,) +1(D, D) -1, and this shows that leBo, C,)::.;;1. On the other hand, since $(G,) =$(B,) =B! passes through P*=$(L), C, must meet L, and therefore leBo, C,)::::::1. Hence "Lkjl(Lj , C,) =1(Bo, C,) =1, where k j ::::::1. Since a general G, does not contain j

any Ll as its component, we have I(LJ, C,) ::::::0. This shows that I(LJ, C,) =1 for one j, say j=l, and that I(L j , C,) =0 for all other j*l. Moreover, since C,~Cp, I(Lj, G,) =1(Lj, Cp ) . Hence every general G, meets Ll in one point with intersection multiplicity 1 but does not meet other components L 2 , La, ... of L. It is easy to see that C,·Ll*Cp·L1 if ).*/1. In fact, if C, and Cp intersect with Ll at the same point, then the images B;=$(Cp) and B";.=$(C,) must have the intersection number Ip*(Bt B;) ~2 at j:J*=oo. For example, we can consider an arbitrary divisor D= 'L,m,S, as a 2-current by identifying D with the linear functional D [(li] = 'L,m,

r

(li. The exterior dejs, rivative dT and the adjoint * T of a p-current T are defined by dT[qj] = (-l)p+l T[dqj] , (* T)[qj] = (-l)pT[ *qj], respectively. Furthermore oT and JT are defined by oT= -.d.T, JT= (do+od)T. We say that a p-current T vanishes at a point,» if there exists a neighborhood U(j;l) of p such that T[qj] =0 for all qj vanishing outside U(,»). By the carrier of T will be meant the set consisting of all points p on jIRn such that T does not vanish at p. For example, 1) The theorem of Riemann-Roch on algebraic surfaces is w~l1-known. See Zariski [16J, pp. 63-76. Recently the author has proved the corresponding theorem for complete linear systems IDI on an arbitrary Kiihlerian surface lIn. in the case where D has no multiple components. See Kodaira [3J, p. 852. 1a) This paper has been presented to the conference as a preliminary report of a more detailed exposition: Kodaira[4J. 2) de Rham and Kodalra [9J, Chap. II. For the theory of currents on Kiihlerian varieties, see Spencer [11]. Cf. also Kodaira [3J, § 1, where the reader will find a summary of the results in the theory of harmonic integrals which are necessary for the present paper.

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584

THE THEOREM OF RIEMANN-ROCH

the carrier of a divisor D=,£m,S, is the sum '£ S, of all components S, of ml.l'*O

D. Every p-current T can be written in the form 1 T -- r+s=p L L Ta1"··"p""P. dz·'···dza'dzP'···dz~· , r!s! where the coefficients T., ...•.Pt .•• a. are distributions on the space of local analytic coordinates z\ Z2, ···zn. By using this expression, we define the operators II and A as follows: r,B

1

P1 P IIT -- -"'T , ,"-' ., ...••p, ... p.dz"···dz·'dz- ···dz- •, r,s r.s.

AT =

~ _,1, ~(-I)'ig'PT""""ppl... p.dzl .. ·dzrdi l ... dzs. r. s.

r+s=p-2

A p-current T is said to be of the type (r, s) if T=II T. Let {el, e2, "', eb} be r.B

a base of the linear space consisting of all harmonic p-forms of the first kind on IDC n which is normalized in the sense that

r

ej' *ek=Ojk. With the help j!Uln of this base, we associate with any p-current T the harmonic p- form HT = '£j T[ *ej]ej. HT is called the harmonic part of T. As was shown by de Rham 3), the Laplace equation LlX= T -HT has one and only one solution X satisfying HX=O (where the solution X is also a p-current). Denoting this unique solution X by GT, we introduce Green's operator4) G. Then we have LlGT=GLlT= T -HT and GHT=HGT=O for an arbitrary p-current T. Incidentally, a p-form f/)= (I/P!)Lf/)"""'.pdz· 'dz·'···dz· p of type (P, 0) is said to be holomorphic [or meromorphic] in a domain ~ s;;: IDC n , if, at each point p in ;tI, the coefficients f/)., ...• are holomorphic [or meromorphic] functions of the local coordinates z1, Z2, "', zn. By a p-ple differential we shall mean a meromorphic p-form f/) defined in the whole of IDC n satisfying df/)=O except for the singular points of f/). A p-ple differential f/) is said to be of the first kind if f/) is holomorphic everywhere in IDC n. P

THEOREM 1. (Existence Theorem) .5) Let T be a current of type (P, 1) on IDC n, p being a positive integer ~n. If T satisfies dT=O and HT=O, then 8= (dA +io)GT is a current of type (P, 0) on IDC n and satisfies d8=iT; moreover 8 is a holomorphic p-form outside the carrier of T. This Theorem 1 can be regarded as a generalization of the classical existence theorem on compact Riemann surfaces.6 ) We remark here that the 3) 4) 5) 6)

de Rham and Kodaira [9J, p. 65, Theorem 1. de Rham and Kodaira [9J, pp. 63-66. Kodaira [3J, Theorem 1. 4. Weyl [15J.

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K. KODAIRA

singularities of the "analytic" differential e is determined by the relation de=iT. As an example of applications of the above existence theorem, we shall prove the existence of a multiplicative meromorphic function with given divisor.7) Let D be a given divisor on mn which is homologous to zero. Then, considering D as a 2-current, we have dD=HD=O. Moreover D is a 2-current of type (1, 1). Hence, by the above Theorem 1, e= (dA+io)GD is an "analytic" simple differential which is regular outside the carrier of D. Furthermore we infer from the relation de=iD that each component S, of the divisor D = "Lm,S, is a logarithmic polar variety of the simple Picard integral

! 2;re

with the residue m,. Consequently exp

(!2;re)

is a multiplicative

meromorphic function with the divisor D. § 3.

Adjoint systems on n-dimensional varieties

For an arbitrary divisor D on wen, we denote by m3(D) the linear space consisting of all n-ple differentials which are multiples of -D. Then, assuming that there exists on wen at least one n-ple differential Woe *0) and setting K = ( W o), we infer readily that the correspondence F ---> W = F Wo between FE 'iJeK + D) and WE m3(D) is one-to-one. Hence we get dimlK+DI = dim m3(D)-1.

(3.1)

This relation (3. 1) is valid also in the case where wen has no n-ple differentials other than 0, since, in this case, the dimension of the empty set IK+ D I is equal to -1. Incidentally we denote for an arbitrary compact Ktihlerian variety wen the number of linearly independent A-pIe differentials of the first kind on wen by r.(we n) and set Now, let S be an irreducible non-singular subvariety of wen of complex dimension n-1. In order to deduce the theorem of Riemann-Roch for the adjoint system IK +SI of S, it is sufficient to compute the dimension of the linear space m3(S) , as (3. 1) shows. Denote by tv the linear space consisting of all (n-1) -pIe differentials w of the first kind on S and associate with each WEtv the (n+1)-current T(w) of type (n, 1) on wen defined by T(w)[fl>J= lW.(fJs,

where

(fJ

is a "variable" (n-1)-form of class C~ on

wen and (fJs de-

notes the (n-1) -form induced by (fJ on S. Then, as one readily infers, T(w) satisfies dT(w) =0. Each n-ple differential WE m3(S) can be considered as an 7) Wei! [14J, de Rham and Kodaira [9J, Part II.

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THE THEOREM OF RIEMANN-ROCH

n-current of type (n,O) on ann, since the integral W[sbJ =

r W°sb converges

JilII.

absolutely for an arbitrary n-form sb of class Coo on ann. Now, it can be shown8) that the exterior derivative d W of this n-current W is represented in the form dW=21!'iT(w), where WE hl. The (n-I)-ple differential w determined uniquely by the relation d W = 21!'iT(w) is called the residue of Won its polar surface S. The residue w of W will be denoted by m( W). Obviously the relation dW=21!'iT(w) implies that HT(w) =0. Moreover we can prove, with the help of Theorem 1, that, if conversely, for given WE hl, the (n+ 1)current T(w) satisfies HT(w) =0, the n-current W=21!'(dA+io)GT(w) is an n-ple differential belonging to 5m(S) and satisfies dW=21!'iT(w). Thus, for given WE hl, there exists WE 5m(S) with m( W) =W if and only if T(w) satisfies HT(w) =0. It is easy to see that the condition HT(w) =0 is equivalent to the following one:

lwo13s =

(3.3)

°

for all (n-I) -pIe differentials B of the first kind on ann. Denoting by hlo the subspace of hl consisting of all WE hl satisfying (3. 2), we infer therefore that w-->w=m( W) is a linear mapping which maps 5m(S) onto hlo. The kernel of this mapping w-->w=m( W) is the space 5m(0) consisting of all n-ple differentials of the first kind on ann. Hence we obtain (3.4)

Denote by I the number of linearly independent (n-I) -pIe differentials B of the first kind on ann such that Bs=O. Then the number of linearly independent conditions involved in (3.3) is equal to r n-l(ann) -I and consequently dim hlO=rn-l(S) -rn-l(ann) +1. Inserting this into (3.4) and combining it with (3. 1), we get the following THEOREM 2.9) Let S be an irreducible non-singular subvariety 01 ann 01 dimension n-l. Then the dimension of the adjoint system IK+SI of S is given by

(3.5)

dimlK+SI = rn(an n) -rn-l(ann) +rn-l(S) +1-1,

where I is the number of linearly independent (n-I) -pie differentials B of the first kind on ann which vanish on S in the sense that Bs=O, Bs being the (n-I) -pie differential on S induced by B. This Theorem 2 can be regarded as the theorem of Riemann-Roch for the adjoint system IK+SI of S. Now we consider the case in which ann is an irreducible non-singular alge8) Kodaira [5J, § 3. 9) Kodaira [5J, Theorem 1.

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K.

K:ODAIRA

braic variety imbedded in a projective space. Let E be a general hyperplane section10) of WCn. Then, applying formula (3. 5) to IK+EI, we get dimlK+EI =rn(WCn) -rn-l(WCn) +rn-1(5) -1, since, in this case, 1=0, while it follows from a theorem of Lefschetz ll ) that r,(E) =r,(WCn) for A::::::n-2. Hence we obtain the following THEOREM 3.12) The dimension of the adjoint system IK+EI of a general hyperplane section E of WCn is given by

(3.6)

dimlK+EI = a(WC n) +a(E)-1.

It is not difficult to generalize the formula (3. 5) to the case in which 5 consists of two irreducible non-singular components 51, 52 such that the intersection 51 n 52 is also an irreducible non-singular variety with the intersection-multiplicity 1. In particular we can prove the following THEOREM

4. 13) Let 5 be an irreducible non-singular subvariety of WCn of di-

mension n-I and E be a general hyperplane section of WCn cutting out on 5 an irreducible non-singular variety En 5 with the intersection multiplicity 1. Then we have (3.7)

dimIK+5+EI = a(WC n) +a(5) +a(E) +a(En5)-1.

The arithmetic genus of an arbitrary irreducible non-singular algebraic variety WCn is defined as follows 14) : There exists a polynomial v(h ; WCn) in h such that (3.8)

dimlhEI = v(h; WCn)

for sufficiently large positive integer h, where E is a general hyperplane section of WCn. Then the arithmetic genus Pa(WC n) of ID'ln is defined by Pa(ID'ln) = (-I)nv(O ; WC n). With the help of the above formulae (3.6), (3. 7), we can prove1S ) that the numerical characteristic a(ID'ln) defined by (3. 2) coincides with the arithmetic genus Pa(WC n) : (3.9) Incidentally, it follows from (3.9) that the arithmetic genus Pa(WC n) is a birational invariant of the algebraic variety WCn, since each r,(WC n ) is known to 10) It is well known that a general hyperplane section is an irreducible non-singular subvariety. 11) Lefschetz [7J, pp. 88-91. 12) Kodaira [5J, Theorem 3. 13) Kodaira [5J, Theorem 4. 14) Muhley and Zariski [8J, p. 82. 15) Kodaira and Spencer [6].

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588

THE THEOREM OF RIEMANN-ROCH

be a birational invariant16) of 9n n . In view of the above identity (3. 9), we define the arithmetic genus of an arbitrary Kiihlerian variety 9n n to be the numerical characteristic a(9n n ). § 4.

Adjoint I!!ystems on 3-dimensional varieties

In this section we consider a 3-dimensional Kahlerian variety 9n3. The system of local coordinates on 9n 3 with the center .\J will be denoted by Z~= (z~\ Z~2, Z~3). By a surface S on 9n 3 will be meant a compact complex analytic (reducible or irreducible) subvariety of 9n 3 of complex dimension 2. For each point .\J on 9n 3, the surface S is represented in a neighborhood U(.\J) of .\J by a minimal local equation Rp(z~) =0, where R~(z~) is a holomorphic function of Z~ defined in U(.\J). A singular point .\J of S is said to be ordinary, if, by a suitable choice of the system of local coordinates (z/, Z~2, Zp3) , the minimal local equation R~=O of S at .\J takes one of the following three forms: i) z/· Z~2=0, ii) Z/·Zp2.Z~3=0, iii) (Z~2)2_ (Z~1)2.Z~3=0 ; .\J is called a double, a triple or an (ordinary) cuspidal point of S according as the equation R~=O has the form i), ii) or iii), respectively. It is to be noted here that, in a neighborhood U(q) of a cusp ida I point q, S has the parametric representation Zq1=U, Zq2= U·V, Z q3=V 2, where u, v are local uniformization variables on S with the center q. In what follows we assume that the surface S has ordinary singular points only.17) Now we construct a non-singular model S of S in an obvious manner and consider S as the image cp(S) of the compact non-singular (possibly reducible) analytic surface S by a holomorphic mapping cp of S into 9n 3 • It can be shown that S has a Kahlerian metric. 1S) The set LI consisting of all singular points of the surface S constitutes a (possibly reducible) compact analytic curve on 9n 3, which will be called the double curve of S. Clearly each triple point t of S is also a triple point of LI and LI has no singular point other than these triple points. The inverse image J =cp-1(LI) is a compact analytic curve on S. Incidentally we denote by {tl (or {q}) the set consisting of all triple points t (or of all cuspidal points q) of S. Take a triple point t E {tl and denote by S(1), S(2), S(3) three sheets of S passing through t. The inverse image cp-1(t) consists of three points 1'1, 1'2, 1'3 on S corresponding respectively to S(1), S(2), Sm. On the other hand, there exist three branches ±1 =S(2) n sm, i 2 =S(3) n S(1), t3 = S(1) n S(2) of LI passing through t and, for each t" the inverse image cp-1(t,) consists of two branches V, V' of J. We arrange these six branches 16) van der Waerden [13J, § 6. 17) We assume this in view of the following well-known fact: Every algebraic surface can be birationally transformed into a surface in a 3-dimensional projective space with ordinary singularities only. 18) Kodaira [4J, § 1.

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K.

KODAIRA

!:i(m)(j/=l, 2, 3, m=l, 2) in such a way that tim) passes through 7:k if j/+m::::: k (3). Clearly each 7:k E ~-l(t), t E {t} is a double point of J and J has no singular points other than these double points. A place on a compact analytic curve is, by definition, a pair [P, 0] of a point p on the curve and a branch 0 of the curve passing through p. We denote the places [t, t,], [7:k, t,rm)] simply by t" t,+rl on 6 increased by 1. BIBLIOGRAPHY [1] Chern, S., Characteristic classes of Hermitian manifolds, Annals of Math., vol. 47 (1946), pp.85-121. [2] Hodge, W. V. D., The characteristic classes on algebraic varieties, Proc. London Math. Soc.,

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KODAIRA

Ser. 3, vol. I (1951), pp. 138-15I. [ 3 J Kodaira, K., The theorem of Riemann-Roch on compact analytic surfaces, Amer. Jour. of Math. vol. 73 (1951), PI>. 813-875. [ 4 J --, The Theorem of Riemann-Roch for adjoint systems on 3-dimensional algebraic varieties, Annals of Math., vol. 56 (1952), pp. 298-342. [ 5 J - - , On the theorem of Riemann-Roch for adjoint systems on Kahlerian varieties, Proc. Nat. Acad. Sciences, vol. 38 (1952), pp. 522-527. [ 6 J Kodaira, K., and Spencer, D. C., On arithmetic genera of algebraic varieties, Proc. Nat. Acad. Sci. U. S. A. 39, (1953) pp. 641-649. [ 7 ] Lefschetz, S., L'analysis situs et la geometrie algebrique, Paris (1924). [ 8 J Muhley, H. T., Zariski, 0., Hilbert's characteristic function and the arithmetic genus of an algebraic variety, Trans. Amer. Math. Soc., vol. 69 (1950), pp. 78-88. [9] de Rham, G., Kodaira, K., Harmonic integrals, (mimeographed notes), Institute for Advanced Study, Princeton (1950). [10J Severi, F., Fondamenti per la geometria sulle varieta algebriche, Rend. Circ. Math. Palermo, vol, 28 (1909), pp. 33-87. [l1J Spencer, D. C., Green's operators on manifolds, these Proceedings. [12] Todd, J. A., The arithmetic invariants of algebraic loci, Proc. London Math. Soc. 2nd Series, vol. 43 (1937), pp. 190-225. [13J van der Waerden, B. L., Birational invariants of algebraic manifolds, Acta Salmanticensia, vol. II (1947). [14J Weil, A., Sur la theorie des formes differentielles attachees a une variete analytique complexe, Cemm. Math. Helv., vol. 20 (1947), pp. 110-116. [15J Weyl, H., Die Idee der Riemannschen FHiche, Berlin (1913). [16J Zariski, 0., Algebraic surfaces, Ergeb. Math. Grenz., vol. 3, no. 5 (1935).

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SOME RESULTS IN THE TRANSCENDENTAL THEORY OF ALGEBRAIC VARIETIES* By

KUNIHIKO KODAIRA

(Received April 15, 1953)

The object of the present paper is to prove some results in the theory of algebraic varieties, e.g. the theorem of Riemann-Roch for adjoint systems of ample linear systems, a conjecture of Severi concerning arithmetic genera of algebraic varieties, the completeness of the characteristic systems of sufficiently ample complete continuous systems, etc., by means of the theory of harmonic integrals. First, in Section 1, we summarize the necessary preliminary materials concerning linear systems on non-singular algebraic varieties. In Section 2 we derive from a theorem of Lefschetz several necessary results concerning differentials of the first kind on non-singular algebraic varieties. In the following two Sections, 3 and 4, we prove the theorem of Riemann-Roch for adjoint systems on Kahlerian varieties in two simple cases. The result in Section 3 leads immediately to the theorem of Riemann-Roch for adjoint systems of ample linear systems on nonsingular algebraic varieties, which will be given in Section 5. The following Section, 6, is concerned with arithmetic genera of' algebraic varieties. In particular we prove a conjecture of Severi~ to the effect that the arithmetic genus2 P a of an n-dimensional non-singular algebraic variety V n is expressed as an alternating sum: P a = (jn - (jn-l + (jn-2 - + ... + (-1) n-~ (jl of the numbers (jk of linearly independent k-ple differentials of the first kind on V n • In Section 7, we show the existence of Picard integrals of the second kind on non-singular algebraic varieties with given polar varieties and with given residue functions, provided that the polar varieties are non-singular prime divisors. Then we derive a necessary and sufficient condition for the residue functions in order that the corresponding Picard integrals should be single valued merom orphic functions. In Section 8 we consider an arbitrary but fixed non-singular prime divisor S and an arbitrary complete linear system I D I on a non-singular algebraic variety V .. such that I S - D I contains a connected variety (where the case in which I S I = I D I is included) and deduce from the results in Section 7 a formula expressing the deficiency of the linear system \ D \ S on S cut out by \ D \ in tierms of simple differentials of the first kind on Vn . The formula shows, in particular, that the deficiency of the characteristic linear system \ S \ S of a complete linear system I S \ on its general member S is always not greater than the number q( = (jl) of linearly independent simple differentials of the first kind on V .. . In Section 9 we give a brief summary of some known results concerning theta functions and * This work was carried out under Office of Ordnance Research, U. S. Army Contract No. DA-36-034-0RD-639RD. Severi [22], p. 87. The arithmetic genus Po of a non-singular algebraic variety V .. is defined to be the virtual dimension of the canonical linear system on V. increased by 1 - (-1)". 86 1

2

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J

599

ALGEBRAIC VARIETIES

87

Picard varieties. Using these materials we prove, in the following Section 10, that the deficiency of the characteristic linear system I SIS of a complete linear system I S I on its general member S is equal to q if I S I is sufficiently ample. Thus the number q of linearly independent simple differentials of the first kind on V n can be characterized geometrically as the maximum value of the deficiencies of the characteristic linear systems of complete linear systems on F n • In Section 11, we prove a classical conjecture to the effect that the characteristic linear system of a complete continuous system e on a non-singular algebraic variety V" of any dimension n is complete3 in th~ case where e contains a sufficiently ample linear system. In the final Section 12, we prove a conjecture of Weil' to the effect that every finite unramified covering manifold of a nonsingular algebraic variety is also a non-singular algebraic variety: §1. Linear systems In the present paper we shall mean by a non-singular algebraic variety of dimension n a compact complex analytic variety V" of complex dimension n such that 7 there exists a bi-regular mapping }L of V" int06 a complex projective space C0. By virtue of a theorem of Chow,s the image }L(Vn ) is then an irreducible nonsingular algebraic subvariety of C0 of dimension n. Since, from the abstract point of view, two bi-regularly equivalent analytic varieties can be considered as not essentially distinct, we may identify V" with }L(V,,) and regard V" as a subvariety of C0. Then we say that V" is imbedded in C0 and that C0 is an ambient space of V" . By a prime divisor on Vn we shall mean an irreducible (n - I)-dimensional subvarietl S of V" . A prime divisor S will be called non-singular if S is a nonsingular subvariety of Vn . From the topological point of view, the prime divisor S is a (2n - 2)-cycle on V" . A divisor on V" is, by definition, a (2n - 2)cycle D = LjmjSj with integral coefficientsmj composed of a finite number of prime divisors S; on V n • Each S; associated with mj ;t. 0 is called a component of D. The set of all divisors on V" constitutes an additive group. Every merolO morphic function F on V n which does not vanish identically determines its 3 Severi [22], p. 57. In the case where n = 2, the completeness of the characteristic linear systems of complete continuous systems has been established by the combined efforts of Poincare, Severi, Lefschetz, andB. Segre. Cf. Zariski [30], pp. 82-88. 4 Weil [28J, p. 876. 6 The results in Sections 3, 4, 5 and a part of Section 6 have been announced by the author with some outlines of the proofs in Proc. Nat. Acad. Sciences, Vol. 38 (1952), pp. 522-533. 6 By a bi-regular mapping into @5 will be meant a bi-regular mapping onto a subvariety of @5. 7 We introduce this definition of non-singular algebraic varieties simply in order to make clear that we are mainly concerned with the properties of algebraic varieties not depending on the choice of the projective space in which the varieties are imbedded. 8 Chow [3], Theorem V. 9 By a subvariety of Vn we shall mean a compact complex analytic subvariety of V". In view of Chow's theorem every subvariety of V n imbedded in a projective space @5 is a.n algebra.ic variety in @5. 10 In case Vn is imbedded in its ambient space @5, every meromorphic function on Vn is a rational function. Chow [3J, p. 914.

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KUNIHIKO KODAIRA

divisor in a well known manner. The divisor of F will be denoted by (F). We say that a divisor D' is linearly equivalent to D and write D' ~ D if there exists on Vn a meromorphic function F with (F) = D' - D. Obviously the linear equivalence thus defined is an equivalence relation and therefore the set of all divisors on Vn can be decomposed into mutually disjoint equivalence classes. Each equivalence class is called a divisor class. We say that D = LjmjSj is effective and write D ~ 0 if all coefficients mj are non-negative. A component Sj of an effective divisor D = LjmjSj is called a simple component if mj = 1. An arbitrary (possibly reducible) subvariety of V" of dimension n - 1 is obviously an effective divisor with simple components only. A meromorphic function F on Vn is called a multiple of D if (F) - D ~ 0 or F is identically zero. We denote by \J(D) the set of all meromorphic functions on Vn which are multiples of - D. \J(D) is a finite dimensional linear space over the field C of all complex num-

bers.ll Denote. by ];d a d-dimensional complex projective space and by (AO , Al , . .. , Ad) the homogenous coordinates of a point A E ];d . A linear system on V n is, by definition, a set ID~ I = ID~ I A E ];d I of effective divisors D~ on V n depending on Ain such a way that D~ = (F~) + D, FA = AoFo + AIFI + ... + AdFd , where FQ , F 1 , .•• , Fa are linearly independent meromorphic functions on V" and D is a divisor on V,,·such that (F j ) + D ~ 0 for j = 0,1, '" , d. Clearly the correspondence between Dx and A is one-to-one; thus Dx depends on d independent parameters. This number d is called the dimension of the linear system IDx I and is denoted by dim IDxl. The linear system IDx I is said to be composed of a pencil12 if the degree of transcendency of the field C(Ft/Fo , F2/Fo , .. , , Fa/Fo)

over C is equal to 1. A prime divisor S on V n is called a fixed component of {Dxl if Dx ~ S for all A. Again, a point II on Vn is called a base point of {D~I if every Dx has a component pa;sing through ll. We have the following theorems of Bertini:13 (I) The general member14 Dx of a linear system {Dx} without fixed component is a prime divisor provided that {Dx} is not composed of a pencil. (II) The general member Dx of a linear system IDx} without fixed component on

V n is a (possibly reducible) subvariety of dimension n - 1 having no singular point outside the base points of {Dx I. In case the functions Fo , Fl , '" , Fa generate the linear space

= IF I (F) + D ~ 0 or F = O}, the linear system {Dx I DA = (LAjF j) + D I is said to be complete and is denoted \J(D)

Wei! (28J, pp. 885-886. See Zariski [31J. 13 See Zeriski [31J, [32J; van der Waerden [25], pp. 200-225; Akizuki [lJ. 14 More precisely, DA is irreducible unless A belongs to a subset consisting of a finite number of proper subvarieties of 2:d. 11

11

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601

ALGEBRAIC VARIETIES

89

by \ D \ . Clearly the complete linear system I D I consists of all effective divisors which are linearly equivalent to D and dim I D I = dim !jeD) - 1. Introducing another d-dimensional projective space @Sa (which may be regarded as the "dual" space of ~a), we associate with the linear system

the meromorphic mapping p: IJ --- p(lJ) = (Fo(IJ), Fl(IJ), ... , Fa(IJ)) of V .. into @Sa , where IJ denotes a "variable" point on V n . The image p(Vn) of V .. is then an algebraic subvarietyl. of @Sa . Now, we shall say that the linear system {D,,} is ample if {DA} has no fixed component and if the corresponding mapping p is a bi-regular mapping of V n into @Sa . By identifying V .. with p(V n), the ample linear system {DA} may be therefore regarded as the system consisting of all hyperplane sections of V .. = p(V n ) in its ambient space @Sa . By virtue of theorems of Bertini, the general member DA of an ample linear system {D A } on V .. , n ~ 2, is a non-singular prime divisor. It can be easily verified that, if a linear system {D" I D" = (FA) + D} is ample, the complete linear system I D I is also ample (we note that I D I is the only complete linear system containing {D A }). In fact, let {F o , Fl , ... ,Fa, Gl , ... ,Gh } be a base of the linear space !jeD). Then the meromorphic mapping

associated with I D I maps V n into @Sd+h :::> @Sh and, denoting by 7r the projection: (to, ... , td , td+l , ... ,td+h) --- (to, ... , td) of @5d+h onto@Sh , the mapping p(lJ) associated with {D A } is written in the form p(lJ) = 7ru(IJ). Now, I D I contains the system {D A } having neither fixed component nor base point. Therefore IJ --- u(\3) is a regular mappingl6 and the image u(V .. ) does not intersect with the linear subvariety @Sd of @5d+h defined by to = ... = td = 0, while, by hypothesis, IJ --- p(lJ) = 7ru(lJ) is bi-regular. Hence the mapping \l --- u(\l) must be also bi-regular and consequently I D I is an ample system. In what follows we shall often make use of the following theorem: If {D A I DA = (FA) + D} has neither fixed component nor base point and if {E~ I E~ = (G~) + E} is ample, then the complete linear system I D + E I is also ample. PROOF. By hypothesis, the mapping IJ --- (Fo(lJ) , ... ,Fd(lJ» associated with {D A } is regular and the mapping IJ --- (Go(IJ), ... ,G.(IJ)) associated with {Ep} is bi-regular. Consequently the mapping

is a bi-regular mapping of V n into a projective space @Sd.+d+< of dimension de + + e, where FjGk(lJ) denotes F j(IJ)Gk(\3). The set {(H.) + D + El consisting of all divisors (H.) + D + E, H. = L LlljkFjGk , constitutes therefore an ample

d

Chow [3], Theorem V. It is easy to verify that the mapping p associated with a linear system {D,,} without fixed component is regular if and only if {DAI has no base points. 16

18

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602

90

.KUNlHIKO KODAlRA

linear system on V n • Hence, by the above result, the complete linear system I D + E I is also an ample linear system on V .. , q.e.d. We add here several remarks on the notion of intersections. Let U be an irreducible non-singular subvariety of V nand D be an arbitrary divisor on V n • Then the divisor DU on U cut out by D can be easily defined, provided that no component of D contains U. In fact, let F~ be a meromorphic function defined in a neighborhood U(p) of p E U such that (F~) = Din U(p). This function F~ induces a meromorphic function Fw on Un U(p). The divisor DU is then defined by DU = (F~u) in U n U(p) for each point p E U. If U = 8, D = Tare both non-singular prime divisors on Vn and if T8 is a non-singular prime divisor on 8, then 8T is also a non-singular prime divisor on T, and T8, 8T both coincide with the set theoretical intersection S n T: TS = ST = S n T. In this case, the divisor D8T = D· 8T on the variety ST cut out by D can de defined for an arbitrary divisor D on V n , provided that no component of D contains 8T. If, moreover, D = U is also a non-singular prime divisor on V .. cutting out a non-singular prime divisor U· 8T on 8T and if SU, TU are both non-singular prime divisors on U, then we have U·ST = T·SU = S·TU = Un S n T. Assume that Vn is imbedded in its ambient space @Sa and denote by IE} the linear system consisting of all hyper-plane sections E of V n in @Sa • Let @Sd-n+l be a general linear subspace of @Sa of dimension ,d - n + 1. Then the intersection L = @Sd-n-tl n V n is a non-singular irreducible curve on V n which will be called a general I-dimensional linear section of V n • Given an effective divisor D on V n, the degree of the divisor D· L cut out by D on L is equal to the intersection number lCD, L) of D and L. This number lCD, L) is called the order of the effective divisor D. Now we shall show that the complete linear system I hE-D I has neither fixed component nor base points if h ~ lCD, L). For this purpose it is sufficient to consider the case in which D = S is a prime divisor. Take a general linear subspace @Sd--n-l of @Sd of dimension d - n - 1 and let (:l be the cone joining @Sd-n-I to S i.e. the variety consisting of all (d - n)-dimensionallinear varieties @Sd-n joining @Sd-n-l to points on 8. Obviously (:l is a hypersurface of order i = 1(8, L) containing 8 and therefore (:l cuts out on Vn an effective divisor Ei E I iE I containing 8 as one of its components; thus Ei - 8 is an effective divisor belonging to I iE - 8 I. Moreover, for any given point q on V n , we can choose @5d-n-1 in such a way that Ei - 8 contains no component passing through q. This shows that the complete linear system I iE - 8 I has neither fixed component nor base points. Hence I hE - 8 I has neither fixed component nor base point if h ~ 1(8, L). Combined with the above theorem, this result proves that the complete linear system I hE - D I is ample if h > lCD, L) where D is an effective divisor. From this we infer readily the following theorem: Given an arbitrary divisor Do on V n , there exists an integer ho depending on Do such that I hE + Do - D I is ample for h > I (D, L) + ko, D being an effective divisor on V n' In fact, letting Do = DI - D 2 , DI ~ 0, D2 ~ 0 and setting hI = ICDI , L) + 1, we see that I hIE - DI I is ample. Let D~ be a general member of I hIE - DI I . Then D~ is a prime divisor and

( 30)

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91

ALGEBRAIC VARIETIES

I hE + Do

I = I (h + hl)E - D~ - D2 - D I , while the above result shows that I (h + hl)E - D~ - D2 - D I is ample if h + hI > I(D~ + D2 + D, L). Setting ho = I(D~ + D2 , L) - hI, we conclude therefore that I hE + Do - D I is ample if h > lCD, L) + ho, q.e.d. -- D

§2. Application of a theorem of Lefschetz

By the rational homology we shall mean the homology over the field of rational numbers. Now, let V" be an n-dimensional non-singular algebraic variety, let {E} be a,n ample linear system on V" , i.e. the system of all hyperplane sections of V" in an ambient space ®d ~ V", and let E be a member of lEI which is a non-singular prime divisor on V" . Then we have THEOREM 2.1. (Lefschetz 17 ). In case 1 ~ k ~ n .:.... 1, an arbitrary k-cycle Z with rational coefficients on V n is rationally homologous to a k-cycle Z E with rational coefficients lying on E.

The original proof of thi,s theorem due to Lefschetz is based on a purely topological consideration. For a later purpose, we shall reproduce here a proof of l8 Eckma,nn b3,sed on the theory of harmonic integrals in somewhat different version. Let (to, tl , .. , , td) be the homogenous coordinates in ®d of a point z = (i, l, .,. , zn) on V" , where (z\ .. , , z") denotes the local coordinates of z. a Then the "standard" Kiihlerian metric di = 2 ga{J.(dz di) on Vn is given by

L

2

gafJ'

=

aN

N

!l a !l-fJ' uZ uz

1

d

= 2-7r 1og L k~O

I

tk2

-;: I ' ~

= Aoto + Altl + ... + Adta is an arbitrary but fixed linear form of to , ... ,td' It is easy to see that ga~' are independent of the choice of this linear form t. We choose t in such a way that the hyperplane t = cuts out

where t

°

on V" the hyperplane section E mentioned in our theorem. Then N is a function of claBs C'" on Vn - E, and, by using K

the fundamental 2-form n

=

L uz ~N . dz a

a

,

iL: ga~' dz a d-l is written in the form n = -idK,

(2.1)

in Vn - E.

Take a point p on E and, assuming that p does not lie on the hyperplane to = 0, set to = 1 in a neighborhood U(p) of p. Then the inhomogenous coordinates tl, ... , td of a point z = (z\ l, "', z") are holomorphic functions of z\ Z2, , z" in U(p) and therefore (2.2)

K

=-

1 dt -2 7r

17

18

t

+ a form of class C"',

Lefschetz [16], pp. 88-91. Eckmann [7J.

( 30)

604

in U(p).

92

KUNIHIKO KODAIRA

=

This shows that the integral K[iI]

1

K·iI converges absolutely for an arbi-

Vn

trary (2n - I)-form iI of class C"'; thus K can be regarded as a I-current of type (1, 0). Moreover, since the current drIr satisfies d(drIr) = 27riE in U(\», we get from (2.1) and (2.2) the following relation: (2.3)

12

=

E - idK.

For an arbitrary k-form iI on V n , we denote by iIa the k-form on E induced by iI. Clearly d(il a) = (dil)B and therefore dila = 0 if dil = O. Now, the above Theorem 2.1 can be easily derived from the following THEOR,EM 2.2. For 1 ;;; k ;;; n - 1, a harrnonic k-form tI> on V" vo,nishes identically if the induced f()rrn tl>a is a derived forrn on E. In fa.ct, denoting by HIIPB the brrnonic part of tI>.E on the K1thlerian va,riety E, we infer from Theorem 2.2 that the linear mapping tI> - t HIiP~ ffill,DS the space {tI>} of all harmonic k-forms tI> on Vn isomorphically onto a subsp9,ce of the space {] = ZB[tI>] for all harmonic k-forms tI> on V" ; thus we see tht there exists a k-cycle ZB with real coefficients on E which is homologous to Z with respect to real coefficients. This proves the existence of a k-cycle ZIiJ with rational coefficients on E which is ration9,lly homologous to Z. 9 Now, it is easy to prove the above Theorem 2.2. As wa,s shown by Hodge/ every harmonic k-form tI>, k ~ n, is decomposed uniquely in the form ~

':l'"

=

",(k/2] nP. ~ LJP>=O U ':I:"p,

where ea,ch tl>p is a harmonic (k - 2p)-form satisfying AtI>p of ea,ch 12PtI>p is given by

= 0, while the adjoint

= ap · 12,,-k+PCtI>p ,

*12Pp

ap being a real constant depending only on n, k, p. Consequently, if k

~

n - 1,

the adjoint *' 'Ct» denote the homogeneous coordinates of the point /k)(t) in ~d. Hence EU) depends holomorphically on t. Obviously S is a member of 1 E - Aa 1 and therefore S has the form S =

( 30) 634

122

KUNIHIKO KODAIRA

E a - aa, a E Z( a). Let ~d-r be a general linear subspace of ~d of dimension d - r passing through a. Assuming that a = (1, a\ ... , ad) and expressing the intersection point cr(t) = Z(t)· ~d-r in the form cr(t) = (1, crl(t), ... , crd(t», we infer that each component (}'i(t) is a holomorphic function of t E 'U(a) and that cri(a) = a i . Clearly St = E,,(t) - at is an effective divisor belonging to IE - at I and Sa = S. Now, let p be an arbitrary point on Ea. Then, in a neighborhood U( p) of p on V n , we may assume that the coordinates r i = r i(Z) of a point z E U(p) are holomorphic functions of z and that at least one of the ri(Z) does not vanish in U(p). Under these assumptions,

cr(t)· r(z)

= ro(z) + /(t)Mz) + ... + crd(t)riz)

is a holomorphic function in (t, z) for t E 'U(a), z E U(p). For each fixed t, we have (cr(t)· r(z» = E,,(t) and «(J(P(z) + = at in U(p), while St = E,,(t) - at is effective. Hence the quotient

t»

R~(z, t)

u(t). r(z)

= (J(P(z) + t)

is a holomorphic function of (t, z) for t E 'U(a), z E U(p), and, for each fixed t, this function has the divisor (R~(z, = St in U(p). Denoting by a, the partial differentiation a/at, with respect to t, we get from R~(z, t)·(J(P(z) + t) =u(t)·r(z) the relation

t»

(10.5)

a, R~(z, a) R~(z, a)

+ a.(J(p(z) + a) (J(P(z) + a)

_ -

~ a.cri(a)· ri(Z)

L airi(Z) i

This equality holds in U(p) for each point p on Ea . Now, setting F ( ) •z

+ a) + a) ,

= a,(J(p(z) (J(P(z)

we infer readily that F,(z) is an additive meromorphic function on V n • In fact, it follows from (9.8) and (9.10) that d. (J(t

(J(t

+ Wi)

+ Wi)

_ d. (J(t) - (J(t)

+ .. v'J ,

hence, by the analytic continuation along a closed curve 7 which is rationally homologous to 7i on V n , F.(z) is increased by V.i:F.(Z) - t F,(z) + V.i. Clearly F.(z) is multiple of -.ia . On the other hand, G,(z)

= L a.cri(~) . r;(z)

L

aJr;(z)

is a single-valued meromorphic function on Vn with the divisor (G,) !i;; -Ea = - Sa - aa. Consequently the difference F, - G. is an additive merom orphic function with (F. - G.) !i;; - Sa -.ia, while

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635

123

ALGEBRAIC VARIETIES

F ( ) _ G()

.z

.Z

= _

a.Rp(z, a) R p (z, a) '

in U(Il),

for each point Il on E" = Sa + Doa , as (10.5) shows, and (Rp(z, a» = Sa in U(Il). The additive meromorphic function F. - G. is therefore a multiple of

- Sa . Moreover, the period v.j

=

Joy; deFy - G.) of F. - G. over

'Y j

is given

by Vvj = LJo,hvAwxj, as (9.10) shows. Hence the additive meromorphic functions Fl - G1 , ••• , Fq - Gq , P 1 , ••• , Pq are independent. Thus we obtain 2q independent additive merom orphic functions which are multiples of - Sa = - S and therefore, by Theorem 8.5, def(S/ S) must be equal to q. This proves the following THEoREM 10.2. If a non-singular prime divisor S belongs to a complete linear system I S I which is sufficiently ample in the sense that I S - K and I S Doa - Dot I are both ample for all t E 'U(a), then we have

+

(10.6)

def(S/ S)

+

Doa

-

Dot

I

= q.

Given an ample linear system I E I on V n, the general member S of I hE I satisfies the hypothesis of the above theorem if h is sufficiently large, as one readily infers from a theorem in Section 1. Hence we obtain THEOREM 10.3. Let Eh be the section of V n cut out by a general hypersurface of order h in its ambient space. Then the equality def(Eh / E h ) ciently large.

=

q holds if h is suffi-

Combined with Theorem 8.5, this result proves the following THEOREM lOA. There exists on V" exactly 2q independent simple Picard integrals of the second kind. By the irregularity of a non-singular algebraic variety V we shall mean the number q = gl(Vn ) of the linearly independent simple differentials of the first kind on V". Then, combined with (8.8), the above Theorem 10.2 shows that the irregularity q of an algebraic variety V n is equal to the maximum of the characteristic deficiencies def(S/S) of non-singular prime divisors S on V n • This gives a geometric characterization of the irregularity q. fl

§11. Characteristic systems of complete continuous systems

Let I Do I be a complete linear system on V" which is sufficiently ample in the sense that I Do - K + Doo - Dot I and I Do + D.o - Dot I are both ample for all t, where Doo , At are non-singular prime divisors introduced in Section 9. Then it can be shown in the same manner as in Section 10 that I E I = I Do + Llo I is also an ample linear system and (1Ll)

dim I E - Dot

I=

r,

for all t,

where r is a constant not depending on t. Now, denote bye the set of all effective divisors on V" which are homologous to Do :

e = {D I D '" Do , D > 0 I·

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636

124

Clearly I E - At

KUNIHIKO KODAIRA

Ice and,

for each member

DEI E -

At

I , we have

[7)(D - Do)] = [7)(Ao - At)],

while, as (9.12) shows, the mapping [t] -7 -[h't] = [7)(Ao - At)] maps a onto CP, where h' is the transposed matrix of h = (h~.). Hence D -7 [7)(D - Do)] is a mapping of e onto CP. Let a be an arbitrary point on CP and 'U(a} be a small neighborhood of a. Then there exists a point a E luI and a neighborhood 'U(a) such that t -7 r = - [h't] is a bi-regular mapping of 'U(a) onto 'U(a) which m m maps a intoa. Choosing the theta function O(a)(u) = LmCm 01(U + U )01(U - u ) such that Aa = (f/a\p(z) + a» is a non-singular prime divisor on V n , we set A~a) = «(J(a)(p(z) + t», where r = -[h't] E'U(a), t E'U(a). Then A~") is a nonsingular prime divisor on V n for T E 'U(a), provided that 'U(a) is sufficiently small, and A~a) depends on T E 'U(a) holomorphically. Moreover, we have T = [7)(Ao - A;a»]. Consider the subset e a = {D I [7)(D - Do)] E 'U(a) I of e. Then for any D E ea , we have [71(D- Do)] = [71(Ao - A~a»], where r = [71(D - Do)] E'U(a). This shows that D - Do.~ Ao - A;a) and therefore DEI E - A;a) I . Thus we see that ea consists of all linear systems I E - A;") I , r E 'U(a). The system I E I may be regarded as the system I E. I consisting of all sections E. of Vn cut out by hyperplanes 0",\ = 2:1=0 O'i.\j = 0 in its ambient space @3d and, as was shown in Section 10, the complete linear system I E - A;a) I consists of all divisors E. - A~a) such that E. ~ A;a). Letting ;&;a(r) be the linear subspace of the projective space ~d = {ul defined by ;&;,,( T) = {u I O'. A;a) = 0 I, we infer that (11.2)

and that dim Za(r) = r. Let ~(k)(k = 0, 1, '" , r + 1) be a general linear subspaces of ~d of dimension d - r and set 0' "k( r) = ;&;a( r)' ~(k) • Then r + 2 points O'aO(r) , .•• , Uak(T), ... , u" r+l(T) determine a system of homogeneous coordinates on Za(T) in the following manner: We normalize the homogeneous ordinates 0' a'( T) of each point 0' ak( r) by the conditions ",+1

i() T

"'-'1:=0 0' ak

= 0;

then the homogeneous coordinates (t~, . " ,~:) of ~n arbitrary point is determined uni~uely up to a factor p by O'i =

0'

on ;&;a( r)

L~-o Pt..Ua~(r).

We may assume here that each Ua~(r) is a holomorphic function of r E 'U(a), since, as was shown in Section 10, the point O'"k(r) depends holomorphically on r. Denote by;&; the projective space consisting of all points t = ~\ Then, setting

aD, ... ,n.

D(~a, r)

=:;

E~ - A~a) , O'i

=

Lk-oPt~Ua~(r),

we infer therefore that ta X r -+ D = D(ta , r) gives a one-to-one correspondence between Z X 'U(a) and ea. Now, consider two points a, (3 on CP such that 'U(a) n 'U({3) is not empty. Then each member D E e a n e~ has two representations:

( 30) 637

125

ALGEBRAIC VARIETIES

(11.3) where T = ['leD - Do)] and ~a X T E:=: X '\l(a) , points ~a , ~P appearing in (11.3) are related by p~~

(11.4)

=

~P

X T E:=: X '\l(fj). The two

L~ M~(T)~:,

(p

~ 0),

where (Mi(T» is a non-singular matrix depending holomorphically on T E '\lea) n '\l(fj). In fact, the relation (11.3) implies that, as meromorphic functions on V .. >

(J(P)(P(z) e(a)(p(z)

+ t)

and

+ t)

:E ~~ U"k(T)·t(Z) :E ~~Uak(T)·t(Z)

have the same divisor, where T = - [h't]. Hence, normalizing the homogeneous coordinates (~~, ... , ~~) by mUltiplying a suitable factor p, we have the equality (11.5)

:E ~~ Upk(T)·t(Z)

L ~~ Uak(r) 'r(Z)

= (J("l(p(Z) e(a)(p(Z)

+ t) + t)'

T

E '\lea) n '\l(m.

This shows that, for each fixed T, the one-to-one correspondence linear. Thus we infer that ~J can be expressed in the form

~a

-7

~"

is

~~ = L~Mt(1')'~~' where the matrix (M~(r) is non-singular. For each point p on V .. , ('Yl() U'Yk(T)' r(z) R'k z, T = (J('y)(P(z) + t)'

('Y = aorfj)

is a holomorphic function of z and T for z. E U(p), T E '\l('Y), as was. shown in Section 10. By using these functions, (11.5) can be written in the form :E~ ~ R~~l (z, T)

=

:E~..o ta R~kl (z, T)

and therefore R~kl(Z, T)

=

L~-oM~(T)RW(z, T),

while, for each fixed 1', r + 1 functions R~~J(z, T) (k = 0, 1, ... , r) of z are linearly independent. Hence Mi(T) are holomorphic functions in T. For simplicity's sake, we write (11.4) as follows: ~" = M"a(T)·~a. Consider the system fax '\l(a)} of the product spaces a X '\lea), a ( A' is analytic in the sense that the graph of the mapping A ---> X' is an analytic subvariety of the product space A X A'. Hence A' must be an irreducible variety. Thus we see that e is an irreducible algebraic system, i.e. a continuous system in the sense of Italian geometers. This continuous system e is complete,·l since e consists of all effective divisors D '" Do on V" . Let S = Dxo be a member of e which is a non-singular prime divisor on V n • Then the characteristic linear system52 of the continuous system e on S may be defined as follows: Let (A\ A2 , ••• , A,+q) be a system of local coordinates in a neighborhood 'U(Ao) of Ao on A, IJ be a point on S, and R~(z, A) be the holomorphic function defining the divisor X in a neighborhood U(IJ) X 'U(AO) of P X Ao on V" X A. Then each partial derivative ojRp(z, Ao) = [oRp(z, A)/OAi).=).o induces on S n UCIJ) a holomorphic function ojRp(z, AO)8. Take another point q on S such that S n U(p) n U(q) is not empty. Then, since Rq(z, A) = U(z, A). Rp(z, A), U(z, A) being a holomorphic function not vanishing on [U(p) n U(q») X 'U(AO), we have OJ Rq(z, Xo)8

=

U(z, AO)8'Oj Rp(z, AO)8 . q

This shows that, for arbitrary constants p.\ '" , p.i, ... , p.r+ , 'L,jp.;ojRp(z, XO)8 and 'L,j)/OiRq(Z, Xo)8 have the same divisor on S n U(IJ) n U(q). Consequently, setting (11.8) Weil [281, p. 881. Wei! [28], pp. 887-888. 60 Wei! [281, p. 885. 61 For the various definitions of complete continuous systems, see Zariski [30], pp. 79-82. 62 For the notion of the characteristic linear systems of continuous systems, see Zariski 1301, p. 82. 43

43

( 30)

639

127

ALGEBRAIC VARIETIES

in U(ll) for each point II on S, we get a well defined effective divisor Dp on S, provided that the function Lj jJ.iajRp(z, Xo)s does not vanish identically on S. Obviously Dp depends "linearly" on the parameter jJ. and therefore the set IDp} of all such divisors Dp constitutes a linear system on S. This linear system {Dp} is called the characteristic linear system of eon S. As one readily ipfers, the characteristic linear system {Dpi is determined uniquely by e and S = Rx.. and is independent of the choice of the system of local coordinates (X \ ... , Xi, •.. , X +q ). T

Now we shall show that the characteristic linear system {Dp ) of the complete continuous system e on S = Dxo is complete. We may assume without loss of generality that Xo = 0" X a, 0" = (1, 0, ... , 0). Now as local coordinates of X = ~a X T, ~a = (1, ~\ ... , f), T = - [h'tl, we choose Xl = ~\ ... , xT = X +l tl , ••. , XT +q = tq , where (tl, ... , tq ) = t. Then, using (11.6), we get

r,

T

(11.9) where Rpj(z, T) = R~i)(z, T) and a.RpO(z, a) = (aRpO(z, T)/dt,]r=,,' With the help of this formula we prove that L/ajRp(z, Xo)s does not vanish identically on Sunless jJ.l = / = ... = jJ.T+q = O. Since S = (RpO(z, a», we infer readily that. T

L/

uak(a) ·t(z)

k=l

M(z)

is a single-valued meromorphic function on V n with (M) ~ - S, while, as was shown in Section 10, each M. = -a,RpO/RpO is an additive meromorphic function on V" with (M.) ~ -So Now, if LjJ.iajRp(z, Xo)s vanishes identically on S, it follows from (11.9) that M - L~=ll+·M. is everywhere holomorphic on V n , so that (11.10) where {PI, ... , P ql is the base of the simple Picard integrals of the first kind on V n • But this is possible only if p,T+l = ... = jJ.T+q = Cl = ... = Cq = 0, since 2q integrals M l , . . . , M q , P l , ••• , P q are independent, as was shown in Section 10. Moreover we infer from (11.10) that M = Co, so that Co UaO(a)·

vanishes identically on V n above results shows that (11.11)

•

t-

Lr=l /

Uak(a)' t

This implies that

dim

{Dpl =

Co

=

T

jJ.l

+ q - 1. L:1=l (akR p(z,

jJ.

O. The

r

It follows from (11.9) and (11.6) that Xo)s = Rp(z, ~ X a)s. while the complete linear system I S I consists of all divisors D~xa = (Rp(z, ~ X a», ~ E Z. Hence {D,,} contains the linear system I SIS cut out by I S I on S. Denoting by r a member of I SIS, we get therefore {Dp} C I r Is. On the other hand, we have dim

I r I s = dim I S I -

( 30)

1

640

+ def (S/ S),

128

KUNIHIKO KODAIRA

while'dim 18 I = rand def (8/8) ~ q by (8.8). Using (ILll), we get therefore dim I r I 8 ~ dim {D".}. Hence {Dp.} must coincide with the complete linear system I r I 8. Thus we obtain the following THEOREM 11.1. Let I Do I be a complete linear system on Vn which is sufficiently ample in the sense that I Do - K + Ao - At I and I Do + Ao - At I are both ample for all t, e be the complete continuous system consisting oiall effective divisors D "" Do on V n, and 8 be a non-singular prime divisor belonging to e. Then the characteristic linear system of e on 8 is complete. We note that, given a divisor Dl and an ample linear system {E} on V n , [ Do [ = I Dl + hE I satisfies the hypothesis of the above theorem if h is sufficiently large. §12. Finite covering manifolds of algebraic varieties Let V be a non-singular algebraic variety, U be the universal covering manifold of V and 9 be the covering transformation group of U with respect to V. As is well known, every finite unr.amified covering manifold Vof V corresponds one-to-one to a subgroup X of 9 with finite index [g:X] in such a way that V is obtained from U by identifying each point p of U with IT' P for all (T EX; then we write V = U Ix. Denote by ;reV), ;reV), . , . the fields of all meromorphic functions on V, V, ... , respectively. Every meromorphic function FE ;reV) or E ;reV) induces a meromorphic function Fu on U in an obvious manner. Identifying F with Fu , we may suppose that ;reV) C ;reV) c ;r(U). For an arbitrary meromorphic function F = F(p) E ;r(U), we define ITF by (ITF)(p) = F(IT -lp) , where IT is an element of g. The mapping IT:F ---+ ITF is obviously an automorphism of the field ;r(U). Clearly ;reV) or ;reV) consists of all meromorphic functions F E ;r(U) satisfying ITF = F for all ITE 9 or E X, respectively. Now, for an arbitrary sub field ;r of ;r(U), we set

9'(;r)

= {IT lIT E g, ITF = F for all F E ;r}.

Then we have THEOREM 12.1. For any finite unramified covering manifold ihe equality

V

=

U IX of V,

(12.1)

holds. PROOF. We denote by 7(' the natural mapping of V onto V. Let E be a nonsingular prime divisor belonging to an ample linear system {E} on V. Then .ihe inverse image E = 7('-\E) of E is a non-singular prime divisor on V. To prove this it suffices to show that an arbitrary pair of points ji', ji" on E such that 7('(ji') = 7('(ji") can be combined by a continuous curve t lying on E. Let.yo be a continuous curve on V combining ji" with ji'. Then 'Yo = 7('(.yo) is a closed continuous Curve on V passing through 13 = 7('(ji') = 7('(p"). Hence, by a theorem of Chow,63 'Yo is homotopic relative to 13 to a closed continuous curve 'Y lying 53

Chow [4], p. 731.

( 30)

641

129

ALGEBRAIC VARIETIES

on E. Corresponding to this, 'Yo is homotopic relative to ji', ji" to a continuous curve 'Y on E combining ji" with ji'. Now we show that Theorems 2.1, 2.2, 2.3 and 2.4 concerning the divisor E on V hold also for the divisor E = 7r-I(E) on V. For this purpose, consider the ample linear system {E I as the system of hyperplane sections of V in its ambient space ~ and let ds 2 = 2Ly,,~.(dz" dzfJ) be the "standard" Kahlerian metric on V c ~. Then ds2 induces a Ktihlerian metric d'i = 2 LY",fJ*(dz" dfi) on V in an obvious manner. Now, the proof of Theorem 2.2 in Section 2 is based solely on the relation n = E - i dK between n = iLY"'fJ*dz"'di and E, while the corresponding relation n = E - i di( holds between n = iLY"fJ* dz" dzii and E, where K is the I-current on V induced by K. Hence, Theorem 2.2 hOlds jor E on V. It follows that Theorems 2.1 and 2.3 hold also jor E on V, since they are immediate consequences of Theorem 2.2. The proof of Theorem 2.4 in Section 2 is based on Theorems 2.1, 2.2, 2.3 and on the fact that, for each point II on V, we can find a system of local coordinates (z~ , ... , z; , ... , z:) with the center II such that each z; is a meromorphic function defined on the whole of V and that each coordinate plane: = const. coincides with a non-singular prime divisor belong to {E}. Letting {E} = 1I"-I{E} be the inverse image of {E} and z; the meromorphic function on V induced by z; , we infer that, for each point Ii E 7r-1(ll) on V, (z~ , .,. , z; , ... , zn is a system of local coordinates with the center ji and that each coordinate plane: z; = const. coincides with a nonsingular prime divisor belonging to {E}. Hence Theorem 2.4 holds also for the divisor E on V. Clearly the inverse image K = 7r-I(K) of a canonical divisor K on V is a canonical divisor on V. Furthermore, it follows from the above results that Theorem 5.1 concerning the adjoint system I K + E Iv of E on V holds also for the adjoint system I K + E I-v of E = 7r-\E) on V, since the proof of Theorem 5.1 in Section 5 is based on Theorem 3.1 which is valid for arbitrary Kahlerian varieties and on Theorems 2.3,2.4. Namely, we have

z:

(12.2)

dim I K

+ E Iv = a(V) + aCE)

- 1.

Similarly, letting 8 be a non-singular prime divisor on V such that S = 11"-1(8) is also a non-singular prime divisor and assuming that E cuts out on 8 a nonsingular prime divisor E8, we have (12.3) dim I K + E + S Iv = a(V) + aCE) + a(S) + aCeS) - 1 for E = 11"-I(E), S = 7r-1(8), since the proof of Theorem 5.2 in Section 5 is based on Theorems 2.3, 2.4 and 4.1. Now, let Eh be the section of V cut out by a general hypersurface of order h in its ambient space ~, Ek be the intersection of k general hyperplane sections of V in ~, and set Eh = 7r-I (Eh ) , Ek = 7r-l(~). Then it follows from the above results that each E\ k ~ n - 1J is an irreducible non-singular SUbvariety of V of dimension n - k, where n is the dimension of V. Furthermore, with the help of (12.2) and (12.3), we can prove the formula (12.4)

a(Eh )

= ~ (~) [a(~)

( 30)

+ (- I)"-kl + (- 1)"

642

130

KUNIHIKO KODAIRA

in the same manner as in the proof of Theorem 6.1, where a(En) + 1 denotes the number of points of the finite set En. Now, using the above results, we prove the equality (12.1). It is obvious that 8'(5{V» ::> X. Setting 8' = 8'(;r(V», we infer therefore that V is a covering manifold of V* = U/8', while V* is obviously a covering manifold of V. Since every function F E ;r(V) satisfies uF = F for all u E 8', we get ;r(V) C ;r(V*) and consequently 5'CV) = ;r(V*). Let Eh or Et be the inverse image of E" on the covering manifold V or V* of V, and K or K* be the inverse image of K on V or V*, respectively. Then it follows from ;r(V) = ;r(V*) that dim

I K + E" Iv =

dim I K*

+ Et Iv •.

Using (12.2), we obtain therefore

a(V)

+ aCE,,)

= a(V*)

+ a(Et),

while (12.4) shows that

aCE,,) =

~~ {a(~) + 1 + 0 (D},

a(Et) =

~~ {a(E· + 1 + 0 (D}. n

)

Hence we get the equality a(En) = a(E'n). On the other hand, denoting by s the number of sheets of the covering manifold V over V*, we have a(En) + 1 = s(a(E· n) + 1). Consequently, the number s must be equal to 1 and therefore V = V*. This proves that X = 8', q.e.d. THEoREM 12.2. Let V be a non-singular algebraic variety imbedded in a projective space. Then every finite unramified covering manifold Vof V is also a nonsingular algebraic variety imbedded in a projective space:4 PROOF. The field ;r(V) is a finite algebraic extension of ;reV). In fact, letting V = UIx and 9 = X + u 2X + ... , + UIX, we infer that for an arbitrary member F of ;r(V) the polynomial