Kunihiko Kodaira, Volume III: Collected Works 9781400869879

Table of contents :
Cover
Contents
[51] On Compact Analytic Surfaces, Analytic Functions, Princeton Univ. Press, 1960, Pp. 121-135.
[52] On Compact Complex Analytic Surfaces, I, Ann. Of Math., 71(1960), 111-152.
[53] A Theorem of Completeness for Analytic Systems of Surfaces, with Ordinary Singularities, Ann. Of Math., 74(1961), 591-627.
[54] A Theorem of Completeness of Characteristic Systems for Analytic Families of Compact Submanifolds of Complex Manifolds, Ann. Of Math., 75(1962), 146-162.
[55] On Stability of Compact Submanifolds of Complex Manifolds, Amer. J. Math., 85(1963), 79-94.
[56] On Compact Analytic Surfaces, II-III, Ann. Of Math., 77 (1963), 563-626, 78(1963), 1-40.
[57] On the Structure of Compact Complex Analytic Surfaces, Proc. Nat. Acad. Sci. U. S. A., 50(1963), 218-221.
[58] On the Structure of Compact Complex Analytic Surfaces, II, Proc. Nat. Acad. Sci. U. S. A., 51(1964), 1100-1104.
[59] On the Structure of Compact Complex Analytic Surfaces, Lecture Notes Prepared in Connection with the A. M. S. Summer Institute on Algebraic Geometry Held at the Whitney Estate, Woods Hole, Mass. July 6-July 31,1964.
[60] On the Structure of Compact Complex Analytic Surfaces, I, Amer. J. Math., 86(1964), 751-798.
[61] On Characteristic Systems of Families of Surfaces with Ordinary Singularities in a Projective Space, Amer. J. Math., 87(1965), 227-256.
[62] Complex Structures on S¹ x S³, Proc. Nat. Acad. Sci. U. S. A., 55(1966), 240-243.
[63] On the Structure of Compact Complex Analytic Surfaces, II, Amer. J. Math., 88(1966), 682-721.
[64] A Certain Type of Irregular Algebraic Surfaces, J. Analyse Math., 19(1967), 207-215.
[65] Pluricanonical Systems on Algebraic Surfaces of General Type, Proc. Nat. Acad. Sci. U. S. A., 58(1967), 911-915.
[66] On the Structure of Compact Complex Analytic Surfaces, III, Amer. J. Math., 90(1968), 55-83.
[67] Pluricanonical Systems on Algebraic Surfaces of General Type, J. Math. Soc. Japan, 20(1968), 170-192.
[68] On the Structure of Complex Analytic Surfaces, IV, Amer. J. Math., 90(1968), 1048-1066.
[69] On Homotopy K3 Surfaces, Essays on Topology and Related Topics, Mémoires Dédiés A, Georges De Rham, Springer, 1970, Pp. 58-69.
[70] Holomorphic Mappings of Polydiscs into Compact Complex Manifolds, J. Differential Geometry, 6(1971), 33-46.

Citation preview

Kunihiko

KODAIRA:

Collected Works Vol. EI

Kuniliiko KODAIRA: Collected Works Vol.in

Iwanami Shoten, Publishers and Princeton University Press 1975

Copyright (C) 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

CONTENTS YoLI PREFACE

by W. L. Baily, Jr.

Uber die Struktur des endlichen, vollstandig primaren Ringes mit verschwindendem Radikalquadrat

1

C 2 D Uber den allgemeinen Zellenbegriff und die Zellenzerspaltungen der Eomplexe

8

Japan. J. Math., 14(1937), 15-21.

Proc. Imp. Acad. Tokyo, 14(1938), 49-52.

C3

Eine Bemerkung zur Dimensionstheorie Proc. Imp. Acad. Tokyo, 15(1939), 174-176.

12

C O On some fundamental theorems in the theory of operators in Hilbert space

Proc. Imp. Acad. Tokyo, 15(1939), 207-210.

C 5 D On the theory of almost periodic functions in a group (collaborated with S. Iyanaga) Proc. Imp. Acad. Tokyo, 16(1940), 136-140.

C6

19

Uber die Differenzierbarkeit der einparametrigen Untergruppe Liescher Gruppen

24

Uber zusammenhangende kompakte abelsche Gruppen (collaborated with M. Abe)

26

Proc. Imp. Acad. Tokyo, 16(1940), 165-166.

C7

15

Proc. Imp. Acad. Tokyo, 16(1940), 167-172.

C 8 D Die Kuratowskische Abbildung und der Hopfsche Erweiterungssatz

32

C9D Uber die Gruppe der messbaren Abbildungen

40

ClOD Uber die Beziehung zwischen den Massen und den Topologien in einer Gruppe

46

Compositio Math., 7 (1940), 177-184.

Proc. Imp. Acad. Tokyo, 17(1941), 18-23.

Proc. Phys.-Math. Soc. Japan(3), 23(1941), 67-119.

ν

Contents

CllD Normed ring of a locally compact abelian group (collaborated with S. Kakutani)

99

Proc. Imp. Acad. Tokyo, 19(1943), 360-365.

(12} tJber die Harmonischen Tensorfelder in Riemannschen Mannigfaltigkeiten, (I), (II), (III)

105

Proc. Imp. Acad. Tokyo, 20(1944), 186-198, 257-261, 353-358.

(13} tJber die Rand- und Eigenwertprobleme der linearen elliptischen Differentialgleichungen zweiter Ordnung · · · · 129 Proc. Imp. Acad. Tokyo, 20(1944), 262-268.

(14} Uber das Haarsche Mass in der lokal bikompakten Gruppe (collaborated with S. Kakutani)

136

Proc. Imp. Acad. Tokyo, 20(1944), 444-450.

(15} Relations between harmonic fields in Riemannian manifolds

143

Math. Japonicae, 1(1948), 6-23.

(16} On the existence of analytic functions on closed analytic surfaces

161

Kodai Math. Sem. Reports, 1 (1949), 21-26.

(17} Harmonic fields in Riemannian manifolds (generalized potential theory)

172

Ann. of Math., 50(1949), 587-665.

(18} The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices · 251 Amer. J. Math., 71(1949), 921-945.

(19} On ordinary differential equations of any even order and the corresponding eigenfunction expansions

276

Amer. J. Math., 72(1950), 502-544.

(20} A non-separable translation invariant extension of the Lebesgue measure space (collaborated with S. Kakutani) · · · 319 Ann. of Math., 52(1950), 574-579.

(21} Harmonic integrals, Part II

325

Lectures delivered in a seminar conducted by Professors H. Weyl and C. L. Siegel at the Institute for Advanced Study (1950).

(22} The theorem of Riemann-Roch on compact analytic surfaces

339

Amer. J. Math., 73(1951), 813-875.

(23} Green's forms and meromorphic functions on compact analytic varieties Canad. J. Math., 3(1951), 108-128.

402

Contents

(24) The theorem of Riemann-Roch for adjoint systems on 3-dimensional algebraic varieties

423

Ann. of Math., 56(1952), 298-342.

( 2 5 ) On analytic surfaces with two independent meromorphic

functions (collaborated with W.-L. Chow)

468

Proc. Nat. Acad. Sci. U. S. A., 38(1952), 319-325.

(26) On the theorem of Riemann-Roch for adjoint systems on Kahlerian varieties

475

Proc. Nat. Acad. Sci. U. S. A., 38(1952), 522-527.

(27) Arithmetic genera of algebraic varieties

481

Proc. Nat. Acad. Sci. U. S. A., 38(1952), 527-533.

(28) The theory of harmonic integrals and their applications to algebraic geometry

488

Work done at Princeton University, 1952.

(29) The theorem of Riemann-Roch for adjoint systems on Kahlerian varieties

583

Contributions to the Theory of Riemann Surfaces, Annals of Math. Studies, No. 30,1953, 247-264.

(30) Some results in the transcendental theory of algebraic varieties

599

Ann. of Math., 59(1954), 86-134.

Vol. II (31) On arithmetic genera of algebraic varieties (collaborated with D. C. Spencer)

648

Proc. Nat. Acad. Sci. U. S. A., 39(1953), 641-649.

(32) On cohomology groups of compact analytic varieties with coefficients in some analytic faisceaux

657

Proc. Nat. Acad. Sci. U. S. A., 39(1953), 865-868.

(33) Groups of complex line bundles over compact Kahler varieties (collaborated with D. C. Spencer)

661

Proc. Nat. Acad. Sci. U. S. A., 39(1953), 868-872.

(34) Divisor class groups on algebraic varieties (collaborated with D. C. Spencer)

665

Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 872-877.

(35) On a differential-geometric method in the theory of analytic stacks Proc. Nat. Acad. Sci. U. S. A., 39(1953), 1268-1273.

671

Contents

(36] On a theorem of Lefschetz and the lemma of EnriquesSeveri-Zariski (collaborated with D. C. Spencer)

677

Proc. Nat. Acad. Sci. U. S. A., 39(1953), 1273-1278.

(37] On Kahler varieties of restricted type

683

Proc. Nat. Acad. Sci. U. S. A., 40 (1954), 313-316.

(38] On Kahler varieties of restricted type (an intrinsic characterization of algebraic varieties)

687

Ann. of Math., 60(1954), 28-48.

(39] Some results in the transcendental theory of algebraic varieties

708

Proc. Intern. Congress of Mathematicians, 1954, Vol. Ill, 474-480.

(40] Characteristic linear systems of complete continuous systems

715

Amer. J. Math., 78(1956), 716-744.

(41] On the complex projective spaces (collaborated with F. Hirzebruch)

744

J. Math. Pures Appl., 36(1957), 201-216.

(42] On the variation of almost-complex structure (collaborated with D. C. Spencer)

760

Algebraic Geometry and Topology, Princeton Univ. Press, 1957, pp. 139-150.

(43] On deformations of complex analytic structures, I-II (collaborated with D. C. Spencer)

772

Ann. of Math., 67(1958), 328-466.

(44] On the existence of deformations of complex analytic structures (collaborated with L. Nirenberg and D. C. Spencer)

910

Ann. of Math., 68(1958), 450-459.

(45] A theorem of completeness for complex analytic fibre spaces (collaborated with D. C. Spencer)

920

Acta Math., 100(1958), 281-294.

(46] Existence of complex structure on a dififerentiable family of deformations of compact complex manifolds (collaborated with D. C. Spencer)

934

Ann. of Math., 70(1959), 145-166.

(47] A theorem of completeness of characteristic systems of complete continuous systems (collaborated with D. C. Spencer) Amer. J. Math., 81(1959), 477-500.

956

Contents

(48^ On deformations of complex analytic structures, III, Stability theorems for complex structures (collaborated with D. C. Spencer)

980

Ann. of Math., 71(1960), 43-76.

(49] On deformations, of some complex pseudo-group structures

1014

Ann. of Math., 71 (1960), 224-302.

(50] Multifoliate structures (collaborated with D. C. Spencer) · · · 1093 Ann. of Math., 74(1961), 52-100.

Vol. Ill (51] On compact analytic surfaces

1142

Analytic Functions, Princeton Univ. Press, 1960, pp. 121-135.

(52] On compact complex analytic surfaces, I

1157

Ann. of Math., 71 (1960), 111-152.

(53] A theorem of completeness for analytic systems of surfaces, with ordinary singularities

1199

Ann. of Math., 74(1961), 591-627.

(54] A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds

1236

Ann. of Math., 75(1962), 146-162.

(55] On stability of compact submanifolds of complex manifolds

1253

Amer. J. Math., 85(1963), 79-94.

(56] On compact analytic surfaces, II-III

1269

Ann. of Math., 77 (1963), 563-626, 78(1963), 1-40.

(57] On the structure of compact complex analytic surfaces

1373

Proc. Nat. Acad. Sci. U. S. A., 50(1963), 218-221.

(58] On the structure of compact complex analytic surfaces, II

1377

Proc. Nat. Acad. Sci. U. S. A., 51(1964), 1100-1104.

(59] On the structure of compact complex analytic surfaces Lecture Notes prepared in connection with the A. M. S. Summer Institute on Algebraic Geometry held at the Whitney Estate, Woods Hole, Mass. July 6-July 31,1964.

1382

Contents

£60} On the structure of compact complex analytic surfaces, I

1389

Amer. J. Math., 56(1964), 751-798.

(61] On characteristic systems of families of surfaces with ordinary singularities in a projective space

1437

Amer. J. Math., 87(1965), 227-256.

(62] ComplexstructuresoniS1XS3

1467

Proc. Nat. Acad. Sci. U. S. A., 55(1966), 240-243.

(63] On the structure of compact complex analytic surfaces, II

1471

Amer. J. Math., 88(1966), 682-721.

(64} A certain type of irregular algebraic surfaces

1511

J. Analyse Math., 19(1967), 207-215.

(65} Pluricanonical systems on algebraic surfaces of general type

1520

Proc. Nat. Acad. Sci. U. S. A., 58(1967), 911-915.

(66] On the structure of compact complex analytic surfaces, III

1525

Amer. J. Math., 90(1968), 55-83.

(67] Pluricanonical systems on algebraic surfaces of general type

1554

J. Math. Soc. Japan, 20(1968), 170-192.

(68] On the structure of complex analytic surfaces, IY

1577

Amer. J. Math., 90(1968), 1048-1066.

(69] On homotopy KZ surfaces

1596

Essays on Topology and Related Topics, Memoires dedies £l Georges de Rham, Springer, 1970, pp. 58-69.

(70] Holomorphic mappings of polydiscs into compact complex manifolds J. Differential Geometry, 6(1971), 33-46.

χ

1608

vol. in

On Compact Analytic Surfaces Kunihiko Kodaira PRINCETON UNIVERSITY AND INSTITUTE FOR ADVANCED STUDY

THE present note is a preliminary report on a study of structures of compact analytic surfaces. 1. Let Kbe a compact analytic surface, i.e. a compact complex manifold of complex dimension 2. Let (V) be the field of all meromorphic functions on V and denote by dim .Jt (V) the degree of transcendency of (V) over the field C of all complex numbers. By a result due to Chow [2] and Siegel [9], we have dim J((V) • Δ induces an isomorphism Φ* : Ji(V) 2, where JK(IS) denotes the field of meromorphic functions on Δ. Moreover V contains no irreducible curve other than the components of the fibres of V. 121

( 51 )

1142

KUNIHIKO KODAIRA We denote by (CD) the intersection multiplicity of two curves C and D on Vand write (C2) for (CC). A curve C on V will be called an exceptional curve (of the the first kind) if C is a non-singular rational curve with (C2) = — 1 (compare Zariski [10], pp. 36-41. By an exceptional curve we mean always an exceptional curve of the first kind). For an arbitrary point pe V we denote by Qp the quadratic transformation with the center ρ (see Hopf [5]). Moreover we call any surface W — ... QilsQviQlll(V) obtained from V by applying a finite number of quadratic transformations a quadratic transform of V. We recall that 5 = Q Jp) is an exceptional curve on the quadratic transform V = Qv(V) and that Qfl 1 is a holomorphic map of V onto V which is biregular between V-S and V — p. Moreover Q~x induces an isomorphism: Jt(Qv(V)) ^ Jt(V). THEOREM 3. Assume that dim -Jf(V) > 1. If V contains an exceptional curve S, then there exists a compact analytic surface W and a point pe W such that V — Qv(W) and S = Qv(p). This theorem has been established for algebraic surfaces by Castelnuovo and Enriques [1] (compare also Kodaira [7]). In view of Theorem 1, it suffices therefore to consider the case in which dim Jf(V) = 1. Now, if dim Jf(V) — 1, Fis an analytic fibre space of elliptic curves over a curve Δ and the canonical projection Φ maps S onto a single point u on Δ. Let U be a small neighborhood of u on Δ. A detailed analysis of the structure of the fibre space V shows that the neighborhood Φ_1(ί/) of S can be imbedded in an algebraic surface. Hence the theorem is reduced to the case of algebraic surfaces. As to compact analytic surfaces with no meromorphic functions except constants, we have the following THEOREM 4. (Kodaira [6].) Let V be a compact Kahler surface with dim Jf(V) = 0. The geometric genus pg(V) of V is equal to 1 and the first Betti number b}( V) of V is equal to either 4 or 0. Ifb1(V) — 4, then V is a quadratic transform of a complex torus (of complex dimension 2). If 6X(V ) = 0 and if V contains no exceptional curve, then the canonical bundle K of V is trivial and the second Betti number b2(V) of V is equal to 22. It can be shown that c'f( V) 2. It is clear that Cap is a multiple singular fibre if and only if Cap is of type mlb, m > 2. Suppose that Cap' 1 < p < /, are of types "'/bp' mp > 2, respectively, and that Cap' P > 1+ 1, are simple. Let mo be the I.c.m. of m1 , •.. , mp' ... , mZ, and let d = m1m2 ..• mz' Moreover let ao be an arbitrary point on ~ - {aJ Then there exists ad-fold abelian covering surface Zi of ~ which is unramified over ~ - {a o, aI' ... , az} and has dlmp branch points b pk , k = 1,2, ... , dlmp of order mp - lover each point a p ' P = 0, 1, 2, ... , I. Letting 1lT be the canonical projection of Zi onto ~, we define the analytic fibre space V over Zi inducedfrom V by the map 1lT to be the minimal non-singular model of the subvariety of V X Zi consisting of all points (z, u) E V X Zi satisfying (z) = 1lT(u). The canonical projection d>: V -+ Zi is induced from the projection V X Zi -+ Zi. V is also an analytic fibre space of elliptic curves. It is clear that, for U =1= b pk' the fibre C:u of V over u has the same type as the fibre Cu of V over u = 1lT(u) and, in particular, C:u is regular if 1lT(u) 1= {a p }, while the fibre Cb pk' 1 < P < t, is of type 11m pb p and Cb Ok is regular. Thus the induced fibre space V is free from multiple singular fibres. On the other hand V is a d-fo1d abelian covering manifold of V which is unramified over V - Ca and has dlmo branch curves Cb ,k = 1,2, ... , dlm o, of o u order mo - lover _Ca.0 The covering map II : V -+ V is induced from the projection V X ~ -+ V. Thus we obtain the following THEOREM 6. An analytiC fibre space V of elliptic curves over a curve ~ having no exceptional curve induces over a suitably chosen finite ramified mIb

:

+

124

( 51 J 1145

ON COMPACT ANALYTIC SURFACES

abelian covering surface Δ of Δ an analytic fibre space V of elliptic curves free from multiple singular fibres. V is a finite abelian covering of V having branch curves over a regular fibre C„o of V and contains no exceptional curve. V is represented as the factor space: V = Vj(5, where (5 is the covering transformation group of V over V. 3. Now let V be an analytic fibre space of elliptic curves over Δ having neither exceptional curve nor multiple singular fibre and let Φ : V —>- Δ be FIGURE 1 IV

II*

11

IV*

in*

Each line represents Θρ8; the integers attached to the line gives /Jps.

125

[ 51 ] 1146

KUNIHIKO KODAIRA

the canonical projection. Moreover let {a p } be a finite set of points a1, ... , a p' ... , Gr on L\ such that the fibre C" = -1(U) is regular for u =/0 a p and let L\' = L\ - {a p }, where, for convenience, we do not assume that each fibre Co is singular. Co is therefore either a regular fibre or a singular P P fibre of one of the types Ill' I:, II, Il*, III, IIl*, IV, IV*, where we write Ib for 1 lb' The restriction V I L\' = -1( L\') = U "E~' C" of V to L\' is a differentiable fibre bundle of tori over L\' and therefore the first homology groups H1 (C u ' Z)(:::::: Z ffi Z) of the fibres C" with coefficients in the integers Z forms a locally constant sheaf G'

=

U"E~' H 1(Cu ' Z)

over L\' in a canonical manner. Let Ep be a circular neighborhood of a p on L\ and let E; = Ep - a p' Then the group reG' I E;) of sections of G' over is independent of the size of We extend G' to a sheaf Gover L\ by defining reG' I E;) to be the stalk Ga p of Gover a p :

E;

E;.

G=

U pGaUG', p

Gap= r( G' I Ep')

and we call G the homological invariant of the fibre space V over L\. We denote by J(w) the elliptic modular fu.nction defined on the upper half plane C+ = {w I ~w > o}. As is well known, w ~J(w) is a holomorphic map of C+ onto the whole plane C and the equality J(w) = J(w') holds if and only if w' is obtained from w by a modular transformation

,

aw+b cw+d

S:w~w = - - - ,

ad- bc = I,

where a, b, c, d are rational integers. For each point u E L\', we represent the elliptic curve C u as a complex torus with periods (w(u), 1), ~w(u) > 0, and set feu) = J(w(u)).

w(u) is determined by C u uniquely up to a modular transformation and depends holomorphically on u E L\'. Hence feu) is a single-valued holomorphic function of u defined on L\' = Ll - {ap}. Moreover it can be shown that each point a p is either a pole or a removable singular point of feu). Thus feu) is a meromorphic function on L\. We call feu) the functional invariant of the fibre space V of elliptic curves over L\. By a suitable choice of the finite set {a p }, we may assume that ;feu) i= 0, 1, 00 for U E L\' = L\ - {a.}. Suppose conversely that an algebraic curve L\ of genus p and a meromorphic function feu) on Ll are given. Take a finite set {ap} of points on Ll such that feu) i= 0, 1, 00 for U E L\' = L\ - {a.}. Then there exists one and only one multi-valued holomorphic function ro(u) with

126

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1147

ON

COMPACT

ANALYTIC

SURFACES

on A' satisfying Take a p o i n t o on A' and suppose that each element of the fundamental group of A' is represented by a closed arc ft on A' starting and ending at o. By the analytic continuation along the arc fi, co(u) is transformed into Sj»(U), where is a modular transformation depending only on the homotopy class of ft relative to o. We indicate this fact by the formula

(2) The correspondence f l - t - S p gives a representation of 771(A'), provided that we take a fixed branch of co(u) at the starting point o of fi. In the particular case in which a>(u) = cu0 is a constant, the condition (2), which states simply that w 0 is a fixed point of Sf!, may not be sufficient to determine the modular transformation Then we choose arbitrarily, provided that S p satisfies (2) and that gives a representation of TT^A'). For each modular transformation , we choose

in such a way that /S —^(Sf,) gives a representation: This is possible in ^ different manners, provided that r > 1, since is generated by 2p + r generators with the single relation:

The representation (3 —>- (Sp) defines a locally constant sheaf G' over A' whose stalks are isomorphic to Z © Z and G' can be extended uniquely to the sheaf over A, where Under these circumstances we say that the sheaf G belongs to the meromorphic function V2 such that 2/ = j, where 3>1; (1»2 are the canonical projections of Vlt V2 onto A, respectively. In the above definition of the family -^(J^, G), analytically equivalent fibre spaces may be considered

127

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]

1148

KUNIHIKO

KODAIRA

as the same fibre space. More precise classifications of analytic fibre spaces of elliptic curves will be introduced later. Let a p be a small circle with positive orientation around a p . The restriction G | Ep of G to a circular neighborhood £ p of ap is determined by the representation of a p . THEOREM

7.

The type of the fibre Ca of any fibre space

is determined uniquely by (S a ) (see Table I). By a holomorphic section of the fibre space V over an open subset TABLE I Normal form of (S,)

°fCap

1

regular

t; (i?) ti =;) r:) r:)

Type

c?dp

0*PO

torus

torus

r* Jo

C x Z2 x Z2

C

h

C* x Z„

c*

Behavior of Jf{u) at a p

regular

pole of order b

r* h

C x Z2 x Z2

c

II

c

c S(af) = 0

II*

c

c

III

CxZ,

c /(Op) = 1

III*

CxZj

c

IV

CxZ,

c /(fl p ) = 0

IV*

CxZ,

Zj, denotes the cyclic group of order b.

128

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]

1149

c

ON

COMPACT

ANALYTIC

SURFACES

E 1 is a deformation of an algebraic surface unless V is a quadratic transform of an analytic quasi-bundle of elliptic curves over a curve. In case W = B" is an analytic principal bundle of elliptic curves, it can be shown that, if V = ... Qp2Qp.(B"/(f») has a Kahler metric, V is a deformation of an algebraic surface. Combining this with Theorem 21, we obtain THEOREM 22. Every compact Kahler surface having at least one nonconstant meromorphicfunction is a deformation of an algebraic surface. Now we consider a compact Kahler surface V with dim Jt(V) = If the first Betti number b1(V) of V is positive, then, by Theorem 4, Vis a quadratic transform of a complex torus, and therefore V is a deformation of an algebraic surface. Combining this with the above Theorem 22, we obtain THEOREM 23. Every compact Kahler surface V with the first Betti number b1(V) > 0 is a deformation of an algebraic surface.

o.

REFERENCES [I) G. CASTELNUOVO and F. ENRIQUES, Sopra alcune questioni fondamentaIi nella teo ria delle superficie algebriche, Annali di Math., Ser. Ill, 6 (1901),

pp. 165-225. [2] W. L. CHOW, On complex analytic varieties, to appear in the Amer. J. Math. [3] W. L. CHOW and K. KODAIRA, On analytic surfaces with two independent meromorphic functions, Proc. Nat. Acad. Sciences 38 (1952), pp. 319-325. [4] A. GROTHENDIECK, A general theory of fibre spaces with structure sheaf, University of Kansas, 1955. [5] H. HOPF, SchIichte Abbildungen und lokale Modificationen 4-dimensionaler komplexer Mannigfaltigkeiten, Comm. Math. Helv. 29 (1955), pp. 132-156. [6] K. KODAIRA, On compact complex analytic surfaces I, to appear in Ann. of Math. [7] - - - , On Kahler varieties of restricted type, Ann. of Math. 60 (1954), pp.28-48. [8] K. KODAIRA and D. C. SPENCER, On deformations of complex analytic structures, I-II, Ann. of Math. 67 (1958) pp. 328-466. [9] C. L. SIEGEL, Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, Nachr. Akad. Wiss. Gottingen (1955), pp. 71-77. [10] O. ZARISKI, Algebraic surfaces, Ergeb. Math. Berlin (1935).

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ON COMPACT COMPLEX ANALYTIC SURFACES, 1* By

K. KODAIRA

(Received June 30, 1959)

The present paper is the first part of a study of structures of compact complex analytic surfaces. The main results of this study were announced in a short note ' which will serve as an introduction to this paper. Naturally that note, written two years ago, does not cover some recent results among which the following would be worth mentioning here: Every compact Kahler surface is a deformation of an algebraic surface. This result was conjectured earlier by Sir William Hodge. 2 1. Exact sequences First we fix our notations. We denote by Va compact complex analytic surface. By a complex line bundle F over V we mean an analytic fibre bundle over V whose fibre is a complex line C and whose structure group is the multiplicative group C* of complex numbers acting on C. The bundle F can be described as follows: Let {UJ } be a sufficiently fine finite covering of V and let 'tiJ' be the canonical projection of F onto V. Then the inverse image 'tiJ'-l(UJ ) has a product structure: 'tiJ'-l(UJ ) = UJ X C, and (z, ~ J) EO UJ X C is identical with (z, ~~) E Ule X C if and only if ~ J = fJIe(z) • SIe, where fJ.(z) is a non-vanishing holomorphic function defined on UJ n Ule. Under these circumstances, we say that the bundle F is defined by the system {fJk} of the transition functions fJIe = fJIe(z), and we call ~J the fibre coordinate of the point (z, SJ) on F over the neighborhood U j • We identify two complex line bundles which are analytically equivalent. For any pair of complex line bundles F, F' determined respectively by {fJIe}, {f~Ie}' we define the sum F" = F + F' to be the complex line bundle determined by the system {f~D of the product f~~ = fJk' jj". Then the set {F} of all complex line bundles over V forms an additive group. For any divisor D on V defined in each U J by a local meromorphic function RiD) = Rj(z; D), we denote by [D] the complex line bundle over V defined by the syst~m {RJiD)} of the functions Rjk(D) = RiD)/Rk(D). It is clear that [D] coincides with [D'] if and only if D is linearly equivalent

* This work was supported by a research project at Princeton University sponsored by the Office of Ordnance Research, U. S. Army. 1 "On compact analytic surfaces," in Analytic Functions, Princeton University Press, 1959. 2 A lecture delivered at Princeton University, 1954. 111

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to D' in the sense that D - D' is the divisor of a (global) meromorphic

function on V. Given a complex line bundle F, the set of all eff,ective divisors D satisfying [D] = F forms therefore a complete linear system on V. We denote this complete linear system by the symbol/ F /. Clearly 1[D] 1 coincides with 1D I. A holomorphic section cp of F over an open set U ~ V is, by definition, a holomorphic mapping z -+ cp(z) of U into F satisfying 1liCP(z) = z. Letting cpiz) be the fibre coordinate of cp(z) over Uj, we have the relation cp;(z) = fjk(Z)CPk(Z) for z € Uj n Uk n U. Thus each holomorphic section cp over U may be identified with a collection {cp;(z)} of holomorphic functions cp/z) defined respectively on Uj n U such that cp;{z) = fjk(Z)CPk(Z) for z € Uj n Uk n U. By a meromorphic section '0/ of F over U we mean a collection {'o/iz)} of meromorphic functions 'o/;(z) on Uj n U such that 'o/;(z) = fjk(Z)'o/k(Z) for Z € Uj n Uk n U. Since fjk(Z) for Z € U j n Uk' the divisor ('o/j) of 'o/;(z) coincides with the divisor ('o/k) of 'o/k(Z) in Uj n Uk n U. We may therefore define the divisor (0/) of the meromorphic section '0/ by setting ('0/)= ('o/j) on each Ujn U, provided that '0/*0. Given a divisor D on V defined in each U j by a meromorphic function Rj(z; D), we say that a meromorphic section '0/ of F over U ~ V is a multiple of D if 'o/lz)/Rlz; D) is holomorphic on Uj n U for each Uj. We denote by il(F) the sheaf over Vof germs of holomorphic sections of F and by il(F - D) the sheaf over V of germs of meromorphic sections of F which are multiples of D. (This symbol il(F - D) is to be considered as an abbreviation of il(F, - D), since the difference F - D has no meaning). We note that il(F - 0) coincides with il(F). Moreover, we have the canonical isomorphism

*°

il(F - D)

~

il(F - [D]) .

In fact, in terms of the fibre coordinates of germs of sections, the isomorphism is given by viz) - - + cp;(z) = 'o/;(z)/Rlz; D) • For an arbitrary sheaf S over V we denote by Hq( V, S) the q'h cohomology group of V with coefficients in S. We have, a basic theorem 3 to the effect that Hq(V, il(F - D» is a finite C-module. We write r(F - D) for HO( V, il(F - D». Obviously r(F - D) consists of all meromorphic sections of F over V which are multiples of D. As a generalization of the notation 1F I, we denote by 1F - D 1 the complete linear system of all divisors X ~ D satisfying [X] = F. The relation [X] = F holds if and 3 K. Kodaira, On cohomology groups of compact analytic varieties with coefficients in some analytic faiscea'Ua!, Proc. Nat. Acad. Sci. U.S.A., 39 (1953), 865-868.

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113

only if there exists a meromorphic section.jr of F over V with ("") = X and moreover"" is determined by its divisor ("") uniquely up to a multiplicative constant. Hence we have dim I F - D I = dim r(F - D) - 1 . Now let V be a compact complex analytic surface and let C be a (possibly reducible) curve on V. Letting F be a complex line bundle over V, we consider the sheaf f2(F) over V of germs of holomorphic sections of F and the subsheaf f2(F - C) of f2(F) consisting of germs of holomorphic sections which vanish on C. Letting Fa be the restriction of F to C, we define the sheaf f2(Fo) to be the factor sheaf f2(F)/f2(F - C) and denote by ro the canonical homomorphism of f2(F) onto f2(Fo) = f2(F)/f2(F-C). The sheaf f2(Fo) may be regarded as a sheaf over C, since the stalk f2(Fo)p over a point p on V vanishes unless p € C. By a germ of holomorphic section of Fa we shall mean a member of f2(Fo). Let Cbe the non-singular model of C and let p. be the canonical mapping of Conto C. Obviously p. can be regarded as a regular mapping of C into V. Let p be a point on C and let (zt, Z2) be a system of local coordinates on V with the center p. The inverse image p.-'(lJ) consists of a finite number of points p" P2' ••• on C:

p.-'(lJ) = {PH·'·, PA' ' •• , Pr} . We denote by tl>. the local uniformization variable on Cwith the center PI>.' In a neighborhood of PA the mapping p. is written in the form

z'

= p.WI>.) ,

Z2

= P.WA) ,

where each P.~(tA) is a convergent power series in tl\ satisfying p.~(O) = O. In case lJ is a simple point of C, p.-'(lJ) consists of a single point PI and z', Z2 can be chosen in such a way that p.i(t,) = t, and !!:(t,) = O. Letting Rp(z\ Z2) = 0

be the minimal local equation of Cat lJ, we set (1.1)

(] I>. -

dp.~ _ - dp.l f),Rp(pL pO - fJ 2R p(pl, pO

where fJ«Rp = fJR>e/fJz« and p~ = P~(tA)' The differential (h thus defined is written in the form (1.2)

where the non-negative integer CA is positive if and only if lJ is a singular point of C (see Appendix I).

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K. KODAIRA

Let Fo be the induced bundle fl-1F over Cand let U(Fo) be the sheaf over C of germs of holomorphic sections of Fo. Clearly fl induces a homomorphism fl* : U(F) --'; U(Fo)

of U(F) into U(Fo) in a canonical manner. The kernel of this homomorphism fl* is equal to the sub sheaf U(F - C) of U(F). Hence we have the exact sequence (1.3)

Clearly fl* maps the stalk U(Foh into the direct sum EA U(FO)PA of the stalks U(Fe)p. In order to describe the induced bundle Fe = /rl F A ~ explicitly, let {UJ} be a (finite) covering of V and, for each point p on C, let Up be a neighborhood of p on C such that fl(Up) is contained in one of the open sets U j , say UJ(p). Moreover, let {fJk(Z)} be the system of transition functions defining the bundle F with respect to the covering {U j } . Then the bundle Fo is defined with respect to the covering {Up} by the system {jpq{t)} of the induced functions jpq(t)

= fj(p)J(q)(fl(t»

.

Now let Dp be the ring of convergent power series in (z\ Z2) and let OA be the ring of convergent power series in tAo The mapping /1: tA ~ (z\ Z2) = (/LHt A), flWA» induces a homomorphism fl*: Dp - - - 7

,LA 0,,-

in a canonical manner. Denoting by , we define an isomorphism

'P(z)

j: U(Fh ~

Dp

by j : If'(z)

---7

(jlf')(z)

= 'P/z)

.

Moreover, we define an isomorphism jA: U(FiJ)p,,- ~

0,,-

in a similar manner as j by means of the fibre coordinates of Fe corresponding to the system {jpq{t)} introduced above. Then we have the commutati ve diagram

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115

(1.4)

Denoting by o/o.(m) the ideal t,!:oA of

0,1.,

we set

0; = I:A oA(c A) , . where C/o. is the integer defined by (1.2). Moreover, we set Cp

=

I:/o. C

o~

is a subring of !l*D~ and

A •

Then we have THEOREM 1.1. (Gorenstein)4.

dim [U*Dp /o;J =

(1.5)

t

Cp •

A proof of this theorem will be given in Appendix I below. Letting D~=fl*-'(o;) and D!(F)p=j-l(D;), we observe the exact sequence (1.6)

0 ----+ n'(F)p~

n(F)p~ Mp-->

0,

where Mp = O(F)pjD,'(F)p. By the above theorem,

Mp ~ £VO~ ~ fl*O)io~ is a finite C-module of dimension iCp. We note that O'(F)p = j-l(D;) is independent of the choice of the system of fibre coordinates. In fact, 0; = fl*-l(O;) is an ideal of Dp' since o~ is an ideal of EA 0,1., while the fibre coordinate on V is positive and write 'Y > 0 if the Hermitian matrix ('Y",~) is positive definite at each point on V. Moreover, we say that a cohomology class c E H2(V, Z) is positive and write c > 0 if CR contains a closed real (1, 1)form ry > O. --j.

THEOREM

2.49 • If the characteristic class c(F - K) of F - K is positive,

then we have s(F) = i(F) = 0 . We note that H2(V, Z) contains a positive cohomology class if and only if V is an algebraic surface (see Theorem 3.4 below). Hence the above theorem is useful only if V is an algebraic surface. We can deriveo from Theorem 2.3 the following 2.5. Let C be a connected curve on a Kahler surface V with 1. Then we have s(K + C) = i(K + C) = o.

THEOREM

dim I CI

~

3. Algebraic surfaces

In this section we derive several criteria for compact analytic surfaces 8 K. Kodaira, The theorem of Riemann-Roch on compact analytic surfaces, Amer. Jour. Math., 73 (1951), p. 866. 9 K. Kodaira, On a d~tferential-geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. U.S.A., 39 (1953), p. 1272. 10 Kodaira, loco cit. in ref. 8, p. 865.

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125

to be algebraic. For an arbitrary compact complex manifold W of complex dimension n, we denote by .5Ii( W) the field consisting of all meromorphic functions on W. According to a result of Siegel't, the field .5Ii( W) is a finite algebraic function field whose degree of transcendency is not greater than the complex dimension n of W. We denote the degree of transcendency of .5Ii( W) by dim .5Ii( W) and call it the algebraic dimension of W. THEOREM 3.1. Any compact analytic surface V with two algebraically independent meromorphic functions is (analytically homeomorphic to) a non-singular algebraic surface imbedded in a projective space. This theorem was proved by Chow and the authorl ' under the assumption that V carries a Kahler metric. The assumption was necessary only in order to be able to apply the theorem of Riemann-Roch on compact Kahler surfaces to V. Now that the theorem of Riemann-Roch on compact analytic surfaces has been established, their proof is valid for any compact analytic surface V. We give here an outline of the proof. For an arbitrary point p on a compact analytic surface V, we denote by Q~ the quadratic transformation l3 with the center p. We note that 8 = Q~(p) is a non-singular rational curve on V = QiV) with (8 2 ) = -1 and that Q;;I is a regular map of iT onto V which is bi-regular between V - 8 and V - p. It is easy to prove the following propositions.1 4 (i) Q;;I induces an isomorphism: .5Ii(Q~(V» ~ .5ti(V). (ii) If V is a non-singular algebraic surface in a projective space, then QiV) is also a non-singular algebraic surface in a proJ'ective space.

Moreover we have

Let Vbe a non-singular algebraic surface in a projective space and let 8 be a non-singular rational curve on V with (8 2) = -1. Then there exists a non-singular algebraic surface V in a projective space and a point p on V such that Qp( V) = V and CASTELNuovo-ENRIQUES' THEOREM I5 •

11 C. L. Siegel, Merorrwrphe Funktio'lllJn auf leompa1cten analytishen Mannigfaltigkeiten, Nachr. Akad. Wiss. Gottingen, 1955, 71-77. 12 W. L. Chow and K. Kodaira, On analytic surfaces with two independent meromorphic functions, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 319-325. 13 For a systematic theory of quadratic transformations, see H. Hopf, Schlichte Abbildungen und loleale ModifiJcationen 4-dimensionaler komplexer Mannigfaltiglceiten, Comm. Math. Helv. 29 (1955), 132-156. 14 See Chow and Kodaira, loco cit. 15 G. Castelnuovo and F. Enriques, Sopra alcune questioni fondamentali nella teoria delle superjicie algebriche, Annali di Math., Ser. III, 6 (1901), pp. 165-225, Chap. III. See also K. Kodaira, On Kahler varieties of restricted type, Ann. of Math., 60 (1954), 28-48.

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K. KODAIRA

Qp(p)

= S.

The following proposition 16 is a corollary to this theorem: (iii) If Qp( V) is a non-singular algebraic surface in a projective space, then V is also a non-singular algebraic surface in a projective space. Now let V be a compact analytic surface with two algebraically independent meromorphic functions x, y. Then, by the result of Siegel mentioned above, 31(V) is a finite algebraic extension of the field C(x, y) of rational functions in x, y. Hence there exists a non-singular algebraic surface V* in a projective space such that 3Ji(V) ~ 31(V*). Letting (1, X[, ••• , xi) be the generic point of V* and letting x" = xk(z) be the meromorphic function on V corresponding to x; by the isomorphism 31(V) ~ 31(V*), we define a meromorphic mapping of V onto V* by Z~

(z) = (1, x1(z), ••• , xa(z» •

This mapping is regular' on V except for a finite number of its fundamental points. These fundamental points can be eliminated by means of quadratic transformations. Namely, by applying a finite number of suitable quadratic transformations Qp, Qpl, Ql,JII, ••• to V successively, we obtain a compact analytic surface

V=

... QpllQplQp( V)

such that the mapping q; = Qp1Qp/Qpl~ ••• of V onto V* is everywhere regular17 • Since

-1 is regular and single-valued except for a finite number of fundamental points. ¢ is therefore bi-regular on if except for a finite number of fundamental curves. Now these fundamental curves can be eliminated by quadratic transformations. Namely, by applying a finite number of suitable quadratic transformations Q.,,*, QlJ*" ••• to V* we get

Vi = ...

Qp*"Qp*'Qp*( V*)

such that the mapping ;PI = •. • Qp*"Qp*'Qp*~ of V onto Vl* is bi-regular!". By the proposition (ii), V!* is a non-singular algebraic surface in a projeclR

Chow and Kodaira, loco cit., p. 324.

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K. KODAIRA

tive space. Hence V is a non-singular algebraic surface in a projective space and therefore, by the proposition (iii), V is also a non-singular algebraic surface in a projective space, q.e.d. THEOREM 3.2'9. Let C be an irreducible curve on V with (C2) ~ 1 and let mF = F + [mC], where F is an arbitrary but fixed complex line bundle over V. Then there exists an integer m o such that

(3.1)

s(mF) =

So

for m

= const.,

~

mo

,

and that the exact sequence

holds for m ~ m o + 1, where rc denotes the restriction map to C. PROOF. Since the canonical bundle f over C is given by (2.2), we have f - (mFo - [cD = Ko

+ [C]o

- mFo .

By the duality theorem, we obtain therefore H'(C,O(mFo - c)) ~ l'(Ko

+ [C]o

- ",Fo) ,

while we have (mFo - Ko - [C]o) = (m - 1)(C2)

+ (FC)

- (KC) .

Letting m, be an integer such that (m, - 1)(C2)

+ (FC) -

(KC)

>0 ,

we infer therefore that H'(C,O(",Fo - c)) = 0,

Combined with (1.11), this shows that H'(C,O(",Fc)) = 0,

form ~ m,.

Hence we obtain from (1.13) the exact sequence (3.3)

o-l'(m-,F) _

r{mF) HO(C, O(mFc)) H'(V, O(m_,F)) H'(V, O(mF)) 0,

form

~

m,.

It follows that

Thus s(mF) = dim H'( V, n(mF» is a non-increasing function of m and consequently there exists an integer m o such that

~

m,-l

'9 In case V is an algebraic surface, this theorem is reduced to a result of G. Castelnuovo. Cf., O. Zariski, Algebraic Surfaces, Berlin 1935, p. 71.

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ON ANALYTIC SURFACES, I

for m

s(mF) = const.,

~

mo.

This implies that for m

~

mo

+ 1.

Combining this with (3.3), we obtain immediately the exact sequence (3.2), q.e.d. THEOREM 3.3. Let V be a compact analytic surface. If there exists on Van irreducible curve C with (C2) ~ 1, then V is an algebraic surface. PROOF. Letting mF = [mC], we infer from Theorem 3.2 that the exact sequence (3.4)

holds for m

~

mo. It follows from (1.11) that dim HO(C, D(mFa»

~

dim r(mFC - [e]) ,

while we have (mFC - [e]) = m(C") - degc .

Hence we obtain dim HO(C, D(mFa»

-+

+

for m

00,

-+

+00

,

form-++ oo

•

and therefore, by (3.4), dim r{mF) We fix an integer m

~

-+

+

00 ,

mo such that

Then it follows from (3.4) that there exists a holomorphic section rp E r(mF) with rorp O. Let D = (rp) be the divisor of rp. Obviously D is an effective divisor belonging to \",F\ = \mC\, and moreover, since rarp 0, D does not contain C as one of its components. Denoting by 5.?(D) the linear space consisting of all meromorphic functions on V which are multiples of -D, we see therefore that the restriction ro5.?(D) of 5.?(D) to C is well defined and that the exact sequence

'*

(3.5)

'*

0 ----> 5.?(D - C)

---->

5.?(D)

---->

ro5!(D)

---->

0

holds, where 5.?(D - C) is the subspace of 5.?(D) of functions which vanish on C. In view of the canonical isomorphisms 5.?(D)

~

l'(",F),

we infer from (3.4) and (3.5) that

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K. KODAIRA

ro2(D)

~

HO(C, n("Fo)) .

Since dim HO(C, n(mFO)) ;:;: 2, we have therefore dim ro2(D) ;:;: 2 . Thus there exists a meromorphic function x E 2(D) such that rox is not equal to a constant on C. Moreover we have dim 2(D - C)

= dim r(m-1F)

;:;: 1 .

Hence there exists a meromorphic function y E £(D - C) which is not identically equal to zero. This function y is not equal to a constant, since y vanishes on C. Now it is clear that these two functions x, yare algebraically independent. In fact, there is no functional relation between x and y, since, on the curve C, y vanishes identically and x is not equal to a constant. Thus there exist on V two algebraically independent functions and therefore, by Theorem '3.1, V is an algebraic surface, q.e.d. Now we consider the case where V carries a Kahler metric ds 2 = 2Eg",~(dz"'dzll). The Kahler metric ds 2 is called a Hodge metric if the associated exterior form Q) = iEg",~dz'" /\ dz ll belongs to the cohomology class CR E H2(V, R) induced by an integral class C E H2(V, Z), 3.4. 20 A compact analytic surface V is (analytically homeomorphic to) a non-singular algebraic surface (imbedded tn a projective space) if and only if V carries a Hodge metric. It is easy to derive from this theorem the following THEOREM

3.5. A compact analytic surface V with a Kahler metric is (analytically homeomorphic to) a non-singular algebraic surface if there exists no holomorphic 2-form on V. PROOF. We write the second cohomology group of V in the form THEOREM

H2(V, Z) = ZC1

+ ZC o + ... + ZC + torsion group, b

where ZC1 + ZC 2 + ... + ZC b is the free abelian group generated by the basis {c u c., "', cb } . For any c E H2(V, Z), we denote by Hc the harmonic 2-form belonging to CR defined with respect to the given Kahler metric on V. Any harmonic form of type (2, 0) is a holomorphic 2-form and any harmonic form of type (0, 2) is the conjugate of a holomorphic 2-form, while, by hypothesis, there is no holomorphic 2-form on V. Hence, for each ck , HC k is a closed real 2-form of type (1, 1). Now let Q) = iEg",~dz'" /\ dz~ be the 2-form associated with the Kahler metric on V. Obviously Q) is represented in the form 20

Kodaira, loe. cit. in ref. 15.

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ON ANALYTIC SURFACES, I

where the coefficients Pk are real numbers. Letting c be a sufficiently small positive number, we take a positive integer n and integers nu •.. , n", ••• , nb such that

(k=1,2,···,b), and set n k H ck w~ = "'" '-'/C~l-'

n

•

Clearly w is a closed real (1, I)-form and, since (V is positive, co is also positive. Hence nw is positive and moreover nw belongs to (L: n"c")R' where E n"c" € H2(V, Z). Setting nw =

iE {j",r;dz'" 1\ d-Z{3 zE g",r;(dz"'dz f3 ) is

we infer therefore that ds 2 = a Hodge metric on V. Consequently V is a non-singular algebraic surface imbedded in a projective space. 4. Analytic surfaces of algebraic dimension 1

Let V be a compact analytic surface of algebraic dimension 1. In view of a result of Siegel quoted in Section 3, the field ~1i( V) of all meromorphic functions on V is isomorphic to the field 5li(D.) of all meromorphic functions on a non-singular algebraic curve D. in a projective space en. Obviously the curve D. is determined uniquely up to a bi-regular transformation. We call D. the algebraic equivalent of V. THEOREM 4.1. There exists a unique regular analytic map cP of V onto D. which induces the isomorphism: .5'Yt(D.) ~ c.'M(V). PROOF. Let (Xo, Xl' "', Xn) be the generic point of Ll and let Xv = xv(z) be the meromorphic function on V which corresponds to X"/Xo by the isomorphism 3i( V) ~ .5'Yt( D.). Take a sufficiently fine covering {UJ } of V. Then, on each U j , xAz) can be written in tho form

xv(z) = 'Pv;(z)/'Poiz) ,

where rpo;(z), rp1J(z), ••• , rpnAz) are holomorphic functions on U j • Moreover we may assume that, at each point \.) € U j , 'Polz), ••• ,'P"iz) have no common divisor as mem bers of the ring Op. It follows that ej 1«z) = 'Po;(z)!'P01«z) = ... = 'Pv;(z)/rp,,,(z) = ...

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K. KODAIRA

is a non-vanishing holomorphic function on U j n Uk. Thus, considering ('Po/z), 'Pl/Z), ••• , 'Pn/z)) as the homogeneous coordinate of a point in 10m we have

Hence

(4.1) is a well defined "meromorphic" map of V into I0 n • A point Z E Uj is called a fundamental point of the map if 'POj(z)='PlJ(z) = ... ='Pnlz) =0. We denote by {o} the set of all fundamental points of . Since 'POj, 'Pl}' ••• , 'PnJ have no common divisor, {v} consists of a finite number of points. Obviously the map: Z -+ (z) is regular analytic on V - {o}. The curve Ll is defined' by a finite number of homogeneous equations (8=1,2,··.). Since the isomorphism 5J1(A) ~ 5J1(V) induces the correspondence Xv/Xo-+ Xv = 'Pv)'POj, we have (8 = 1,2, ••• ) .

This shows that is a regular map of V ~ {o} onto A. The image (o) of a fundamental point 0 is "indefinite". We set therefore (0) = A for each point 0 E {o}. Letting 0,. = ojoz"', a = 1,2, we have 'PVjf),,'P7j - 'P7j o,,'PvJ = (e jk )2{'PVkO,.'PTk - 'PTkf)",'PVk} •

Hence the simultaneous equations (4.2)

'Pv/Z)O",'PT/Z) - 'Pd(Z)a",'Pv/z) = 0 ,

«(t

= 1,2; Ii, r

= 0, 1, .•• , n)

determines a (possibly reducible) subvariety S of V containing {o}. Obviously S consists of a curve and a finite set of points. Moreover it follows from (4.2) that o",('PVj!'PT}) = 0 at each point on S outside {o}. Hence (z) is constant on each connected component of S - {o} and therefore the image (S - {o}) consists of a finite number of points au ••• , ap , • • • , as: (S - {o})

= {a

p}

•

For an arbitrary point u =I=- a p , 1 ;;;;; P ;;;;; 0, on A, we denote by C(u) the curve on V defined by the simultaneous equations (Ii,

(4.3)

where (u o, Uu in en. Since

••• ,

T

= 0, 1, ••• , n) ,

un) is the homogeneous coordinate of the point u

'

(a = 1,2) ,

where N(z)

=" g"'~(z) aT(Z) ~0l,1> az'"

• aTiP(Z) GEl>

We note that the right hand sides of (4.6) are independent of the choice of the local uniformization variable r on~. In order to indicate explicitly the dependence of the arc z(s) on Zo and /3, we write z(s; zo, (3) for z(s). Considering Zo as a variable point on C(u o), we infer from (4.6) that, for fixed (3, z"'(s; zo, (3) are functions of class C>-l in (zo, s). Moreover it is clear that, for each point z on C(u), u = u(l), there exists one and only one solution z(s) of (4.6) satisfying z(l) = z. Thus we see that (zo' s) ~ z(s; zo, (3) is a differentiable map of C(u o) x [0,1] into V which maps C(uo) x 0 and C(u o) x 1 topologically onto C(u o) and C(u), respectively. This proves that C(u) is homeomorphic to C(uo) and also homotopic to C(u o) on V. Now, letting To be the local uniformization variable with the center un' we consider for each e, 0 ;;? e ;;? 2rr, the arc /3e with the parametric representation u(s; e) defined by To(U(S;

where

$

e»

= e • sel9

,

is a small positive constant. Then, denoting by T the circular

( 52

J

1181

136

K. KODAIRA

disk I To I < c with the center u o, we infer that (zo, c.seO) -+ z(s; zo, (30) is a topological map of C(uo) x Tonto the "cylindrical" neighborhood cp-l(T) of C(u o). Thus we see that CP-l(Ll - {a p } ) is topologically a fibre bundle over Ll - {a p } . Let C(u) = E~=l C,(u) be the decomposition of C(u) into a sum of its irreducible components C,(u), where u is a point on Ll- {a p } . Since C(u) is a non-singular curve, the components C,(u) are also non-singular and do not intersect each other. Hence, in view of the bundle structure of V - Up cp-l(ap ), the number n of the irreducible components is independent of u, and moreover we may assume that on V. It follows that the genus 'Jr,

= n(C,(u»

is independent of u. For our purpose it is sufficient to show that n = 1 and n1 = 1. Let {u1 , u., ••• , u j , • • • } be an infinite sequence of points Uj on Ll- {a p } such that u j of=. U i for j of=. i. Moreover, letting C'j = C,(u j), we set

We have (4.7) In fact, this is obvious if i of=. j, and therefore, by using CAj ,......, CAi, we get (C'jCAj) = (C,jC Ai ) = 0 .

Obviously (4.7) implies that

Moreover, since 2;r'(D) - 2 = (D2) n'(mD,)

+ (KD),

= mnv -

we get

+1,

m

n'(mD) = m E~=l nv - mn

+ 1.

Using the theorem of Riemann-Roch (2.10), we obtain therefore (4.8)

dim I mD, I = m(1 - n,)

+P

g -

q - i(mD,)

Similarly we get, by using (2.16),

( 52)

1182

+ s(mD,)

.

137

ON ANALYTIC SURFACES, I

dim I K

(4.9)

+ mD 1= m E~~l 7r, + P

q

g -

+k -

1,

where 0 :;:;; k :;:;; q. Given a complex line bundle F over V, there exists a constant f such that dim I F

(4.10)

+

m

D

I :;:;;

m

+f ,

for all m

~ 0 .

In fact, if \ F + mD \ is empty for all m, this is obvious. Otherwise there is an integer l such that I F + zD I contains an effective divisor D' and consequently F is written in the form F = [D"] , where D" = D' - zD. Now it is clear that for m

= a(D'). Hence we get, with the help of Lemma 4.1, dim IF + ",D I = dim I tI' + E7-1+1 u 1= m - l + d' -

~

l ,

where il'

j

P

for sufficiently large integer m, where d' is the degree of i)' and P is the genus of A. Thus we obtain (4.10). Now, comparing (4.9) with (4.10), we infer immediately that m • E~~l lry

:;:;;

m

+ const.

for all m

,

~

0 .

This proves the inequality (4.11) By Lemma 4.2, the index of specialty i(",D,) vanishes for sufficiently large m. Hence we get from (4.8) the inequality (4.12)

dim I mD, I ;;;; m(l - 7r,)

+P

g -

q

for sufficiently large m. Now we prove that n = 1. If n would have A(mD,) = 0 and therefore, by Lemma 4.1,

~

2, then we

dim I mD, I = 0 . Combined with (4.12), this would imply that 7r, ",n 1....J'~1

> 2 7r, > = n =

~

1 and consequently

.

This contradicts (4.11). Thus we see that n = 1. The inequality (4.11) shows that 7rl :;:;; 1. In order to prove that 7rl is strictly pol:litive, we consider the complete linear system I - K + mD \. Since, by Lemma 4.2, i( - K + mD) = 0 for sufficiently large m, we get from (2.9) the inequality

( 52)

1183

138

K. KODAIRA

dim l- K + ,,,D l;:;; - 11:'(mD) - (KmD) + xCV, - K) for sufficiently large m. Clearly we have (K ",D)

= 2m(11:, - 1) .

We obtain therefore dim I - K

+ mD I ;:;; 3m(1

- 11:1 )

-

1

+ x( V,

- K)

for sufficiently large m. Comparing this with (4.10), we infer that 11:\;:;; 2/3. Consequently the integer 11:\ is equal to 1, q.e.d. Clearly we may assume that, for each point a p , the divisor C(a p ) is not a non-singular elliptic curve. We set (4.13) where

@p,

are irreducible curves and nps are positive integers.

THEOREM

and C(u), u

4.3. There exists on V no irreducible curve other than (8 p ,

"* ap.

Let C be an irreducible curve on V. Then the image (w, z)

--*

(/1[cI»(t A ) = cI>(p;..(t A ), t'";:>.) ,

where cI> = cI>(w, z) is an arbitrary element of ,0 form the direct sum o = 01

Now we

+ ... + OA + ... + 0,.

and define the homomorphism /1* of D into /1* : cI> --* /1*cI> =

= C{w, z}.

/1~cI>

0

by

+ ... + /1J:cI> + ... + p;cI> .

Moreover, we set 0'

where

t~ AOA

=

t '01 + ... + t/o), + ... + t;ror ,

denotes the ideal of OA generated by dim [0/0'] = EA CA

t~A.

Obviously we have

•

To prove Gorenstein's theorem, it is sufficient therefore to show that /1*D ::J 0' and that dim [/1*D/o'] = dim [0//1*5)] . Letting fA = C({ t;..} ) be the quotient field of 0A' we form the direct sum

f = f1 + ... + fA + ... + fr . Moreover, let ~ be the subring of the quotient field of D = C{w, z} consisting of all elements of the form cI>/W, /W) = EAP:'(cI»/pnW) .

The homomorphism /1* maps ~ onto f. Moreover, for any element ~ of f, there exists an element F = F(w, z) of mof the f'Orm (A.2)

satisfying

p*F =

( 52)

~,

1192

ON ANALYTIC SURFACES, I

where m

=E

147

m" is the degree of the polynomial R(w, z) in w. This

can be verified "as follows: Let wt = ptw = P,,(tA), zt = ptz = tr;:" and let ft be the subfield C( {zt}) of h. Clearly h = ft(t,,) is a finite algebraic extension of ft of degree m, while the element w(: of h satisfies in ff the irreducible equation R,,(w, zt) = 0 of order m". Hence h coincides with U'(wt) and therefore an arbitrary element Yj" of fr. is written in the form

Expressing each coefficient in zt. we set

F~

as a meromorphic function F0c = F"k(Zt)

Then we have Now, letting

we set s"

= ptS" =

S,,(wt, zt) .

Clearly s" is not equal to zero. Writing [; € f as a sum [; = E" [;'" [;" € fA, we set Yj" = s;:'[;" and express Yj" in the form Yj" = pt F". Then, letting F(w, z) =

E" S,,(w, z)F,,(w. z) •

we obtain and therefore p*F=[;. It is obvious that this function F(w. z) has the form (A.2). The explicit expression of the function F(w. z) can be obtained with the

help of Lagrange's interpolation formula. Since, by (A. 1), R(w, z) =

It, II;;, (w -

we get (A.3)

Letting

( 52)

1193

t"k) ,

148

K. KODAIRA

we set B,,(w, z)

= w" + A1(z)W"-1 + ... + A,,(z)

•

Then we have R(w, z) -'--'--'- = Wm - 1 + B1(s j..k' Z)Wm - 2 + W -

SAle

... + Bm-1(s j..k, Z)

•

Moreover, the relation F(Pj..(tj..), trj..) = (fl* F)(tj..) =

~j..(tA)

implies that 1

F(Sj..k, z) = ~A(€~Z-;;j..) •

ConsequentJy we infer from (A.3) that the coefficients of the polynomial F(w, z) = Fo(z)w m - 1 + F 1(z)W m - 2

+ ... + Fm-b)

are given by F,,(z) =

E j.. E k B,,(s j..k' z) •

~j..(€~zmj..) --=-'-"----"--

(JwR(sj..l" z)

Now, for an arbitrary element r; = Er;j.., 'f)j.. " residue" p(r;) of r; by

€

h, of f, we define the

(AA) where

f denotes the integral over a small circle I

tj.. I = const. Then, ex-

panding F,.(z) in the form F,,(z) =

E

n

F"". zn ,

we infer readily that each coefficient F',n is given by (A.5)

B"

In fact, letting

we have while

( 52)

1194

= B,,(w, z) .

ON ANALYTIC SURFACES,

f ~A(t/.)(t'::A)-n-lB,,(w:, = f ~A(tA)p:(z-"-lB,,)

149

I

Zt) • (h(t/.)

=

• (J /.(t/.) .

Hence we obtain (A.5).

f belongs to the subring p*D of f if and

THEOREM. An element ~ of

only if for aU

(A.6)

q> €

p*D .

PROOF. Let F= F(w, z) be the function satisfying p*F =!; introduced above. Now, if g satisfies (A.6), we get, by (A.5), F"n = 0 for n ~ - 1. Thus F belongs to 0 and therefore I; = p* F belongs to p*O. Conversely, if g = p*iJ>, cJ> € D, then F belongs to D. To see this, suppose F were not contained in D. Then we can find a positive integer b such that (A.7)

G(w, z) = zbF(w, z)

0,

E

G(w, 0)"", 0 .

Obviously G -- zbiJ> vanishes on C and therefore G - Zb is divisible by R(w, z): G(w, z) - Zb(W, z) = Q(w, z)R(w, z),

Q(w, z) ED.

Setting z = 0, we get from this G(w, 0) = Q(w, 0) • w m

,

while G(w, 0) is a polynomial in w of degree;;;; m - 1. Hence G(w, 0) must be identically equal to zero, but this contradicts (A.7). Thus we see that F belongs to D. We have therefore, F"n = 0 for n ~ - 1 and consequently, by (A.5), p(1; • p*(z"B,,» = 0,

for n = 0, 1, 2, •••.

In particular, since Bo(w, z) = 1, we get p(l;)

=0 .

Thus the "residue" p(l;) of an arbitrary element !; E fl*D is equal to zero. Now, if I; E p*D, then, for an arbitrary element q> € p*O,!; • q> belongs to p*D and therefore q.e.d. Now we derive Gorenstein's theorem from the above Theorem. For an arbitrary submodule tn of f, we denote by tn' the submodule of f defined by tn':::: {!;lp(I;r;) = 0 for all

( 52)

1195

r;

E tn} •

150

K. KODAlRA

We infer from (AA) that and that 0"

If 0'

~ tn ~ 0,

we have therefore

(A.S)

=

0 .

0' ~ tn' ~ 0

and

dim [tn'lo'] = dim [olm]

The above Theorem asserts that (p*O)' f/*D C 0 that p*O = (p*O)' Moreover, applying (A.S) to

m =

= p*D.

Hence we get from

=:J 0' •

p*O, we obtain

dim [f/*O/o']

= dim [0111*£)],

q.e.d.

Let ® be the ring of germs of functions in (w, z) of class c~ defined at the origin p = (0,0) on W, 8/\ the ring of germs of functions in tA of class c~ at 0 and let () be the direct sum () = (),

+ ()2 + ... + ()/\ + ... + ()r •

It is obvious that Dc ® and 0 c (). Each germ g E ® is represented by a function g(w, z) of class C= defined in a neighborhood of p. g(w, z) induces on each branch C/\ the function (Ptg)(t>J = g(p/\(t/\), t'J::>.) of class C~. We denote by p;g the germ determined by the function (p;g)(h) and set p*g = p;g + ... + p;g + ... + p;g. Clearly the mapping g ~ p*g is a homomorphism of ® into e.

We have

PROPOSITION.

(A.9)

0

n p.*® =

p*O .

PROOF. It is sufficient to show that 0 n p*® ~ p*O. By virtue of the above Theorem it suffices for this purpose to verify that every element cp EOn p*® satisfies

p(cp'fr) = 0 ,

for all 'fr

E

p*D .

Let g be an element of ® such that p*g = cp and let g(w, z) be the function of class C~ defining the germ g. Moreover, take an integer h > CA , A, = 1, "', r, and express g(w, z) in the form g(w, z) =j(w, z, iV, z)

where f(w, z, iV, we have

+ O({lwl + Izl}k) ,

z) is a polynomial in w, z, iV, z of degree ~ h -

( 52)

1196

1. Then

151

ON ANALYTIC SURFACES, I

2nip(ep'o/)

==

E /.

f

(f.L:g)(t/..)'o//..(t/..)a/..(t/..)

== E limo~o f

JIt/..I =8 g(wt, zt)'o//..(t/..)a;.(t;..}

/.

== E /..lim8~o \

J

where wt == PA(t)..), zt ==

tr

2nip(ep'o/) =

A•

E /.

f(wt, z't, :wt, zt)'o//..(t/..)a/..(t;..) ,

It/..I=8

We get from this the equality

f

f(w:', z:', 0, O)'o/A(t).)a/..(t;.,) ,

since for k == 0, ± 1, ± 2, "', n = 1,2,3, •••. Thus, setting

A

~ 5 ------+ 0

rol _rol ----'>

Ao~ So ------+ 0 .

where ro denotes the canonical homomorphism. Moreover the horizontal sequences in (A.10) are both exact. In fact, the exactness of the first sequence 0--> n

-->

A

~5

------+

°is well known.

To prove the ex-

actness of the second sequence 0------+ no - - 7 Ao ~ So ------+ 0, it is sufficient to show that, if a € Ap satisfies area = 0, there exists ep € np

( 52)

1197

152

K. KODAIRA

with reT = rca, where 1.J is an arbitrary point on C. Letting Irl(~l) = {PH···' p", .•• }, we observe the ring Eo" introduced in Section 1. It follows from area = 0 that af1-*a = f1-*aa = 0 .

This implies that f1-*a € Eo". Hence, by (A.9), there exists rp € Dp such that f1-*T = p*a, while the equality f1-*T = p*a is equivalent to reT = rca. Thus we see the exactness of the second sequence. We get from (A.l0) the commutative diagram a (3* ---> l'(A) ---> 1'(S) ---> Hl( V, D) ---> 0

(A.ll)

r~1

fj

r~1

*

r~1

---> l'(Ae) ---> 1'(Se) ~ Hl(C, Dc) ---> 0 ,

where the horizontal sequences are both exact. On the other hand, we have 1'(S)

= a1'(A) EB l'(Dl) ,

where qDl) is the linear space of holomorphic i-forms on V. Denoting by YJ an arbitrary element of r(Dl), we infer therefore that the mapping ' i j - 8*fj gives an isomorphism23 between r(Dl) and Hl( V, D). Consequently the dimension k of the kernel of the homomorphism r~: Hl(V, D)---> Hl(C, Dc) is equal to the number of linearly independent holomorphic 1forms YJ satisfying r~8*fj = O. It is therefore sufficient for our purpose to show that r~8*ij = 0 if and only if f1-*YJ = O. We infer from (A.ll) that r~8*Yj = 0 if and only if r"%Yj € ar(Ae). On the other hand, r~ maps 1'(A) onto r(Ae), since the sheaf A( - C) is fine, and therefore the inclusion r~ij € ar(Ae) holds if and only if there exists ry € r(A) such that r~f; = ar"%ry. Obviously the equality r"%f; = ar-;ry is equivalent to p*YJ = 8f1-*7 while, since p*YJ is a holomorphic i-form on C, f1-*YJ = 8p*7 implies that f1-*YJ = O. Hence we infer that r"J8*f; = 0 if and only if f1-*YJ = 0, q.e.d. PRINCETON UNIVERSITY AND INSTITUTE FOR ADVANCED STUDY

23 P. Dolbeault. Sur la cohomologie des varietes analytiques complexes. C. R. Acad. Sci. Paris. 236 (1953), 175-177.

( 52)

1198

A THEOREM OF COMPLETENESS FOR ANALYTIC SYSTEMS OF SURFACES WITH ORDINARY SINGULARITIES By K. KODAIRA* (Received April 6, 1961)

Recently D.C. Spencer and the author proved a theorem of completeness of characteristic systems of complete continuous systems of nonsingular submanifolds of co-dimension 1 of ambient spaces, under the assumption that the submanifolds are semi-regular (see Kodaira and Spencer [2]). The purpose of the present paper is to derive an analogous result for continuous systems of surfaces with ordinary singularities in ambient spaces of dimension 3. Our main theorem may be stated as follows: Let V be a surface, with ordinary singularities only, imbedded in a 3-dimensional compact complex manifold W, and let 'l' be the sheaf of infinitesimal displacements of germs of the surface V (see Definition 3). Assume moreover that the surface V is semi-regular in W (see Definition 7). Then there exists a system of surfaces V t with ordinary singularities in W depending holomorphicaUy on several parameters ttl t 2 , " ' , t"", I tl I + ... + I t", I < 1, such that Vo = V, and such that the infinitesimal displacements (8 Vt/8t")t=o, ).J =1, 2, •. " m, span the linear space HO(V, 'l') (see §3). Thus the formulation of our main theorem is an exact analogue of that of [2]; however, an examination of examples would seem to indicate that the requirement of semi-regularity imposes a strong restriction on surfaces with ordinary singularities. Applications of our result to some examples of surfaces will be discussed elsewhere. 1. Surfaces with ordinary singularities

Let W be a compact complex manifold of complex dimension 3 and let V be an irreducible analytic subvariety of Wof complex dimension 2, briefly: a surface in W. A point p on V is called an ordinary singular point of the surface V if and only if there exists on Wa local (complex) coordinate (x, y, z) with the center p such that, in a neighborhood of p, the surface V is defined by one of the following three equations:

( 1 )d ( 1 ),

yz = 0, xyz

= 0,

* The author was supported during a portion of the period of preparation of this paper by a research project at the Institute for Advanced Study sponsored by the National Science Foundation; during another portion, by a research project at Princeton University sponsored by the Office of Ordnance Research, U.S. Army. 591

( 53)

1199

592

K. KODAIRA

We assume that the surface V has ordinary singular points only and denote by A the singular locus, i.e., the set of all singular points of V. A is a (possibly reducible) algebraic curve in W. Let N p be a "polycylindrical" neighborhood of p consisting of all points (x, y, z), I x I < e, Iy 1< e, I z I < e, where $ is a small positive number, and let

Up = Vn N p . ( i) In case V is defined in N p by yz = 0, Up is composed of two planes and U~ in N p defined respectively by the equations z = and y = 0, i.e.,

°

U~

and A n N p = U~ n U~ coincides with the segment of line defined by I x I < e, Y = z = 0. p'is called a double point of the surface V. (ii) In case V is defined in Np by xyz = 0, Up is composed of three planes U~, U~, U~' in N p defined respectively by the equations z = 0, x = 0, y = 0, i.e.,

Up =

U~

U U~ U U~' ,

and A n N p is composed of three segments of lines U~' n U~", U~' n U~, U~ n U',;. p is called a triple point of the surface V. (iii) In case V is defined in N p by xy' - 4z' = 0, Up admits a parametric representation: (3)

X =

u2

,

1

Y = v, z = -uv , 2

1

u

1

< e'12 ,

I

v

1

(x, y, z) = yt;(x, y, z)

+ z1J(x, y, z) ,

where t;(x, y, z) and r;(x, y, z) are arbitrary convergent power series. The restriction rg.> of g.> = g.>(x, y, z) to V n N p = U~ U U~' is composed of a pair of convergent power series g.>' g.>"

= g.>'(x, y) = yt;(x, y, 0) =

g.>"(z, x)

=

,

zr;(x, 0, z) .

Thus rrp may be written in the form rg.> = g.>'

+ g.>" •

(ii) In case p is a triple point of V, the stalk Hp( -~) consists of all convergent power series in x, y, z of the form rp(x, y, z) = xYC;(x, y, z)

+ yzg(x, y, z) + zxr;(x, y, z)

,

where t;(x, y, z), t;(x, y, z) and r;(x, y, z) are arbitrary convergent power series. The restriction rg.> of rp=rp(x, y, z) to VnNp= U~n U~n U~' may be written in the form ( 6),

rrp = g.>'

+ rp" + rpm

,

where rp' = rp'(x, y) = xyt;(x, y, 0) ,

= yzt;(O, y, z) , = g.>"'(z, x) = zx r;(x, 0, z) •

rp" = g.>"(y, z) g.>'"

(iii) In case p is a cuspidal point of V, the stalk np( -~) consists of all convergent power series in x, y, z of the form g.>(x, y, z) = yt;(x, y, z)

( 53)

+ zr;(x, y, z)

1201

•

594

K. KODAIRA

Consider the restriction of rq; = q> = q;(x, y, z) to Up = V n N p as a power series in the local uniformization parameters u, v defined by (3). Then we have rq;(u, v)

=

v·r(u, v) ,

where (7)

r(u,v)=S(U\V,+uv)++U~{U2,v, ~uv).

It is clear that an arbitrary convergent power series r(u, v) in u and v can be written in the form (7). We infer therefore that the stalk !1 p ( -A) I V consists of all convergent power series "ll'(u, v) in the local uiformization parameters u, v such that 'fr(u, 0) = 0 . Now we fix our notation. Let {Wi} be a finite covering of W by coordinate neighborhoods Wi and let (Xi' Y;, Zi) be a local coordinate defined on Wi' For any point WE Wi' we denote by (xi(w), Yi(W), Zi(W» the value of the local coordinate (Xi' Yi> Zi) at W. We assume that the holomorphic functions Xi' Yi' Zi can be extended to holomorphic functions on a larger domain containing the closure of Wi> and that Wi is the polycylinder consisting of all points (Xi' Yi' Zi)' I Xi I < 1, IYi I < 1, I Zi I < 1. Moreover we assume that the surface V is defined in each neighborhood Wi by a holomorphic equation (8)

and that the holomorphic function ei following five forms: ( 9 )0 ei = 1 ,

=

ei(x i , Yi' Zi) in (8) has one of the

(9 )s ( 9 )d

( 9), ( 9 )0

(Compare (l)d' (1)" (1)e; we indicate by setting ei = 1 that V n Wi is empty.) We denote by I the set of the indices i, and by 10' Is> I d , I" Ie, respectively, the subsets of I consisting of those indices i for which (9)0' (9)s> (9)d, (9)" (9)c hold. Since the triple and the cuspidal points of Vare isolated, we may assume further that the covering {Wi} of W satisfies the following two conditions: (a) If i and k belong to I, U Ie and if i *- k, then Wi n W" is empty; if, moreover, Wi n W j is not empty, then W j n WI; is empty.

( 53)

1202

595

SYSTEMS OF SURFACES

(b) If i E I d , k E Ie> and if Wi n W k is not empty, then the function 11 Xl: is holomorphic, single-valued and bounded away from zero on the domain W;nWk • For an arbitrary sheaf S over W[or V], we denote the space of all sections of S over an open subset N of W [or U of V] by the symbol r(SIN)[or r(S I U)]. In particular, in case S is a subsheaf of D, we denote the linear space of all sections of S I V over an open subset U of V by r(S I U). Now we set U;= vn Wi and consider the linear space l'(D( -A) I U;) of all sections of D( -11) I V over U;. It is pretty obvious that every section tl E r(D( -11) I UI) can be extended to a section 1>; E r(D( -11) I Wi)' We fix a "canonical extension" 1>.; = Exti [til of ti as follows: ( i) In case i E Is, U; is a piece of plane consisting of all points (Xi' y" 0), I Xi I < 1, I Yi I < 1, and the space r(a( -11) I Ut) consists of all holomorphic functions t, = 'h(xl> y,) defined on U,. We define (10).

(ii) In case i E I d , Vi is composed of two pieces of planes U;, VI', where U: consists of all points (x" Yi, 0), I Xi I < 1, IYi I < 1, V;' consists of all points (xu 0, Zi), 'Xi' < 1, 'Z;' < 1, and 11 n Wi = V: n U;'. We infer from (6)d that any section ti E r(D( -11) I Ut ) may be written as a formal sum

of holomorphic functions

t: : :. : ,/r:(x

p

y,) = Yi{;i(X" Yi) ,

t:' = t:'(Zi' x,)

= ZiY;!i(X i , Zi) ,

where {;,(x i , Y;) and Y/;(x i , Zt) are holomorphic functions defined respectivelyon U: and U;' . We define (10)d

Ext, [t,](x;, Y;, Zi)

= y;f;i(X;, Yi) + Zi'li(X"

Zi) •

(iii) In case i E I" Ui is composed of three pieces of planes U;, UI', Vt, where VI consists of all points (Xi' Yt, 0), I X; I < 1, I Y; 1t, (21)0 respectively, the formulae (24)d

~S;(w, t) = Zi Y: + YiZ: ,

at

C 53)

1209

602

K. KODAIRA

These formulae show that aSi(w, t)/at vanishes on Ll t + Ec; and therefore 'o/j(a/at) = 'o/,(w, a/at) is a section of the sheaf D( -Ll t - Ec;) I V t over Uti' Combining this with (23) we infer that the collection 'o/(a/at) = {'o/i(a/at)} of 'o/i(a/at) forms a global section of the sheaf We over Ve' DEFINITION 6. We call 'o/(a/at) E HO(Vt, 'l"t) the infinitesimal displacement of Vt along the tangent vector a/at and denote it by (Je(a/at). The infinitesimal displacement of Ve along a/at may also be denoted by the symbol aVe/at. Let Te be the tangent space of Mat t. It is clear that

(Jt: :t ~ (Jt ( :t) is a linear map of T t into HO( VI' 'l"t). The range (Jt(Tt) of (Jt may be called the characteristic system on V t of the family cV (compare Zariski [5], p. 99). The characteristic system (Jt(Tt) is said to be complete if and only if (Jt(Tt) coincides with HO( Ve, We). 3. Main theorem

Let V be a surface in W with ordinary singularities only, and let Ll, Cj , F, ... have the same meaning as in § 1. Consider the sheaf 'I" = D(F - Ll I V (see (18». Since the sheaf D(F - V) is isomorphic

Ec')

to D, we obtain from (16) the exact sequence (25)

0 --'> D

-->

D(F - Ll -

Ec')

r --'>

'I" - - ' > 0 .

This yields the corresponding exact cohomology sequence 0--> C --> HO(W, D(F - Ll (26)

--> Hl(W, D(F - Ll -->

H2(W, D)

Ec'» ~ HO(V, '1") --> 1J (W, D) Ec'» ~ Hl(V, '1") 1

--> •.. ,

where C denotes the complex numbers. DEFINITION

7. The surface V is said to be semi-regular in W if and

only if (27)

r* Hl(W, D(F - Ll -

Ec'»

= 0,

(compare Kodaira and Spencer [2], p. 481; Severi [4]). Now we shall formulate our main theorem. In what follows we denote by M. a spherical domain in the space of several complex variables of radius c > 0 and of center O. MAIN THEOREM.

Let V be a surface in W with ordinary singularities

( 53)

1210

603

SYSTEMS OF SURF ACES

only. If V is semi-regular in W, then there exists an analytic family CV ~ M. of surfaces V t = -ur- 1(t), t € M" with ordinary singularities in W such that V o = V and such that, for each point t € M" the linear map at: B/Bt --; at(B/Bt) maps the tangent space T t of M. at t isomorphically onto HO( Vt' 'V t), where at(B/Bt) denotes the infinitesimal displacement of V t along a/Bt. Thus the characteristic system of CV on Vt' t € M.. is complete. Moreover the family CV ~ M. is maximal at t = O. 4. Construction of an analytic family of surfaces with ordinary singularities

Let V be a surface in W with ordinary singularities only, and let 6. be the double curve of V. Moreover let {WI}' (XI' YI' ZI)' ei,J~k' CI , 1,10 , Is,' .• have the same meaning as in § 1. Consider the sheaf 'V = n(F- 6. -

.Ee') I V,

F = [V] .

It is clear that the linear space HO( V, 'V) is finite dimensional. Let m = dim HO( V, 'l') and let {,8H "', ,8» "', ,8",} be a base of HO( V, 'If). We represent each element ,8, as a collection {,8'I} of sections ,8,i of n( -6.- .Ee'} IV over Ui = Vn Wi such that

on uln Uk' Let M be the space of m complex variables tH " ' , t"" denote a point t, and define

(tlf "', t",) in M by

I t I = -VI t, I' + ... + I t", 12

.

Moreover, we denote by M. the spherical domain in M of center 0 and of radius c> 0; i.e., M. = {t I I t I < c}. THEOREM 1. If V is semi-regular in W, then there exists an analytic family qJ ~ M. of surfaces with ordinary singularities in W such that -ur- 1(0) = V and such that ao maps the tangent space To of M. at 0 isomorphically onto HO( V, 'If). Let a be a positive number. In order to prove Theorem 1, it suffices to construct holomorphic functions S;(w, t), Xi(w, t), Yi(w, t), Zi(W, t), i € I, defined respectively on Wi >< Mal, and non-vanishing holomorphic functions flk(w, t), i, k € I, defined respectively on (Wi n W.t) >< Mal, such that

(28) (29) (30)

SI(W, t) = elXI(w, t), Yi(w, t), Zi(W, t)) ,

= fik(w, t)Siw, t) , Xj(w,O) = xj(w), Yj(w, 0) = y;(w), Si(W, t)

( 53 J 1211

Z;(w,O)

= Zt(w)

,

604

K. KODAIRA

for we Vn Wt

(31)

•

In fact, letting c be a sufficiently small positive number < a, we infer readily that the family CV ~ M. is given as the subvariety CVe Wx M. defined by the system of holomorphic equations St(w, t) = 0, i e I (compare Kodaira and Spencer [2], p. 485). We remark that, combined with (28) and (29), the boundary condition (30) implies fik(W,O) = f~k(W) .

(32)

In what follows we write the power series expansion of an arbitrary holomorphic function P(t) in t" ••• , t", defined on a neighborhood of the origin (0, ... , 0) in the form P(t) = P(O)

+ E :=1 Pp.(t)

,

where each term Pp.(t) denotes a homogeneous polynomial of degree p in t" ... , t"" and we set

Moreover we write [P(t)]p. for Pp.(t) when we substitute a complicated expression for P(t). If Q(t) is another holomorphic function in t" ••• , t"" we indicate by writing P(t) =p.Q(t) that Pp.(t) = Qp.(t). For the sake of simplicity we write Xt = xj(w), Yi = Yt(w), •.• , Si(t) = Si(W, t), X,(t) = Xi(w, t),·. ·,fik(t) = fik(W, t). Assuming the equalities (30) and (32), we expand Xi(t), Y!(t), Zi(t), ftk(t) into power series in

t" ... , t,,,,. Then we have Xi(t) = Yt(t) = Zi(t) =

+ E:=lXtIAt) , y, + E:=l YilP.(t) , Zt + E:=lZi!P.(t) ,

Xi

fik(t) = f~k

+

E:=,fiklP.(t),

+ E~=lXW,(t) , Y;,(t)= Yt + E~=l Yw(t) , Z,:(t) = Zt + E~=lZW..(t) , P:k(t) = f~k + E~=,ftklh(t) . X~(t) =

Xt

We set (33)p.

S~P.)

(t) = e,(Xt(t), Y;(t), Zt(t» .

Then we have (34)

Therefore the equation (29) is equivalent to the system of congruences (35)p.

S!P.) (t) =p.fMt)S!:) (t) ,

Define

( 53)

1212

f1

=

1,2,3, ••..

605

SYSTEMS OF SURF ACES

(3,(t) = (3i(W, t) = E:~lt>(3),(W) .

Then the boundary condition (31) can be rewritten in the following form:

i¢ 10

(36)

,

where r denotes the restriction map to the surface V. We remark that S;I'-I (0) = S~ = ei(x i , Vi' Z;} •

(37)

We shall construct, by induction on fl, the polynomials X~(t), Y~(t), S!li-I (t)'!;k(t) which satisfy (33)1-" (35)1'- and (36). In the sequel we writeff., fi~II-" Xr, XI11-'" .• forfMt), fikll-'(t), X;'(t), Xill-'(t), • •• when we need not indicate the variable t explicitly. ( i) First we define Z~(t),

(38)

Xil1l-

=

XliII-

=

XiiII-

= 0,

l

= Zi/II- = YIII-' = 0 , 1';/11-

for i € 10 , fl = 1,2,3, ... , for i € I., It = 1, 2, 3, ... , for i e I d , fl = 1, 2, 3, .•. ,

0,

(ii) It is clear that (33)1 implies

for i e Is,

Zi/1' [Sill (t)]l =

+ Yi Z '/l , YiZ:XiI1 + ZiXI Y![l + XiYiZil1 Zi 1';/1

1

Y,XiI1 + 2x i y, Y,ll - 8z,Zm ,

,

for i € I d , for i e It , for i € Ie .

On the other hand, the canonical extension of (3i(t) has the form i e Is , i € Id ,

(39)

i e It,

i

€

Ie,

where ~ill1 1)1111 till are linear forms in tl1 " ' , t", whose coefficients are holomorphic functions defined on Wi (see (10)8, (10)d' (10)t and (15». We define for i e It UIe , (40) for i e Id U It U Ie , for i ¢ /0 and let Xi

= Xi + XiII

,

+ 1';11 , Zi + ZiI1 ,

Yl = y, Z~

=

S;1) (t) = ei(Xl, Y:, ZD .

(53)

1213

606

K. KODAmA

Then we have

i¢ L,.

(41)

Obviously this implies (36). (iii) The congruence (35)1 is equivalent to the equation [SF 1(t)]1 =f~k,[Sk11(t)]1

+ SZ·fikI1(t).

In view of this we define

fi~il

(42)

Since (3,(t) =

= fiki1(t) = ~k [S;1I (t)

f~k' (3k(t)

- f~~,Sk1l (t)]1 •

on Ui n Uk' we infer from (36) that

[S;1I (t) -

f~k,S21

(t)]1

vanishes on Vn Win W k, while Vis defined in W., by the equation S2=0. Consequently the coefficients of the linear form f!ki1(t) are holomorphic functions on Wi n Wk' It is clear that f;k(t) = f~k + fiki1(t) satisfies (35)1(iv) Suppose that, for an integer p. ~ 1, the polynomials X~(t), Yr(t), Z~(t), S~I'I (t), fMt) satisfying (33)1' and (35)1' are already determined. Clearly (35)1' implies that (43)1' We determine homogeneous polynomials E iki l'+1(t) in tH"" t", of degree p. + 1 by E iki l'+1(t)

(44)

=1'+1f~k(t)SI:I

(t) - S;I'I (t) ,

on Wi n W k ,

and define "f!kil'+I(t) to be the restriction of Eikil'+1(t) to the surface V: (45) Obviously the coefficients of "i"!kil'+1(t) are holomorphic functions on U! n Uk' LEMMA

1. We have

(46) PROOF.

EiJll'+1(t)

We nave

+ f~j·Ejk'I'+I(t)

+ f~JfYt,lSI:) f~j)St) + f~jffkSkl') -

=1'+1f~JS(r - S;I'I =1'+I(f~j

-

while

( 53)

1214

f~jSjl')

Sil') ,

607

SYSTEMS OF SURF ACES

Hence we obtain

and therefore

"/t'iJIp.+1(t)

+ f~J·"/t'Jklp.+l(t)

Since SkP.) =oS%, we have rSjt)

=p.+1 r(ffJjkS't) - S;P.) .

=0 O.

Consequently, we obtain, using

(43)",

This proves (46), q.e.d. Now we shall prove that the coefficients of "/t'iklp.+1(t) = rS ik lP.+1(t) are sections of the sheaf D( -A- Ee') I Vover Ui n Uk' We have, by (46),

"/t'kilp.+l(t) = - f2i·"/t'iklP.+1(t) , "/t'!![P.+1(t) = 0 ,

*

while Ui n Uk' i k, contains neither triple point nor cuspidal point of V. Therefore it will suffice for our purpose to show that, for i e Id , k € Id U It U Ie, the coefficients of SiklP.+1(t) vanish on A. LEMMA 2. Let $ be a domain in the space of three complex variables x, y, z and let r be the space of all holomorphic functions of x, y, z defined on $. Moreover let Y(t), Z(t), P(t), Q(t) be formal power series in tu •• " t", with coefficients in r which satisfy the conditions

(47)

Y(O) = P(O)

=y

,

Z(O) = Q(O)

=z.

Let Jl be a positive integer. If (48)p.

P(t)Q(t) - Y(t)Z(t)

then there exists a polynomial get) in t l , such that (49)p.

P(t)

(50)p.

P(t)Q(t) - Y(t)Z(t)

=p.

g(t) Y(t) ,

=,,0 , ••• ,

tm with coefficients in r

Z(t) =p.g(t)Q(t) , =p.+l

(P(t) - get) Y(t»z - (Z(t) - g(t)Q(t»y .

PROOF. By induction on p.. For the sake of simplicity we write P, Q, Y, Z for pet), Q(t), Yet), Z(t). Suppose that we have already determined a polynomial g*(t) in tu ••• , t", with coefficients in r such that

( 53 J 1215

608

K. KODAIRA

P

=1'-1

PQ -

g*(t) Y ,

Z

g*(t)Q ,

=1'-1

YZ =I'(P - g*(t) Y)Z - (Z - g*(t)Q)y .

Then, by (48)", we have (P - g*(t) y)z

=I' (Z -

g*(t)Q)y .

It follows that there exists a homogeneous polynomial gl'(t) of degree f1 in tH ,••• , tm with coefficients in r such that

(51)

gl'(t)

=1'

~(P - g*(t) Y) =I'~(Z - g*(t)Q) .

z

y

Setting

=

g(t)

+ gl'(t)

g*(t)

,

we obtain from (51) the congruences (49)1'. Now, since by (47), Q(t)=oz, we get, using (49)"" PQ

=1'+1

P(Q - z)

+ pz =1'+1 g(t) Y(Q

- z)

+ pz .

Similarly we have YZ

=1'+1

(Y - y)g(t)Q

+ yZ.

Consequently we obtain PQ - YZ = (P - g(t) Y)z - (Z - g(t)Q)y .

This proves (50)1" q.e.d. Now, with the aid of Lemma 2, we prove that the coefficients of Sikll'+1(t), i € I d , k € Id U It U Ie, vanish on~. Setting Y(t) = Y;:(t), Z (t) = Z;:(t) , we have (52)

Y(t)Z(t) =

S;"I (t)

,

Y(O)

=

Yi' Z(O)

=

Zi •

For our purpose, it suffices therefore to show the existence of formal power series P(t) and Q(t), of which the coefficients are holomorphic functions on Wi n W k , such that (53)

P(t)Q(t) =

irkS iii (t)

P(O) = Yu Q(O) = Zl .

,

In fact, combining (52) and (53) with (35)1" we obtain P(t)Q(t) -

Y(t)Z(t)

=1'

0 ,

while Sikll'fl(t) =I'fl P(t)Q(t) -

Y(t)Z(t) .

Consequently, by Lemma 2, there exists a polynomial g(t) whose coefficients are holomorphic functions on Wi n W k such that Slkll'+1(t) =1'+1(P(t) - g(t) Y(t»z; - (Z(t) - g(t)Q(t»y; .

( 53)

1216

609

SYSTEMS OF SURF ACES

This proves that the coefficients of 8'~II'+l(t) vanish on ~. Now, in case k € I d , we have f~kYkZk = yjZj •

There exist therefore two alternatives, namely, either Zi = h~kZk or Zt = h~1GYk' where h~1G is a non-vanishing holomorphic function defined on W, n Wk' Assume, for instance, the first alternative. Then, setting we obtain (53). In case k

€

I" we have f~kXkYkZk

= YiZi

.

Therefore there are three alternatives, namely, Zi = h~kZk' Z, = h~kYk' Zj = h~kXk' where h~k is a non-vanishing holomorphic function on Wi n Wk' Assume the first alternative. Then, setting we obtain (53). In case k e Ie, we may assume that, on, Wi n W k , V X k is holomorphic, single-valued and bounded away from zero, as was mentioned in § 1. We define

=VX(1 +"~ (_1)n-l(2n)! (X~ - Xk)n) . k L....Jn-cl (2"n!)2(2n - 1)x;: Then we have SkI') (t) =

(Y%v x~ -

2Z~)( Y%v:Xt

+ 2Zr) .

We have f~k' (Yk V Xk - 2z k)(Yk -V Xk

+ 2zk) =

YiZi .

There exist two alternatives, namely, either Zi = h~k'(YkVXk

+ 2z

Z, = h~k' (Yk vX;

- 2zk) ,

k)

,

or where h~k is a non-vanishing holomorphic function on Wi the first alternative and ~et P(t) = (f'Mh~k) (Yrv Xr Q(t) = hZk'(Y~VXr

- 2Zr) ,

+ 2Z~) .

( 53 J 1217

n Wk'

Assume

610

K. KODAIRA

Then we obtain (53). Thus the coefficients of +iki,u.+I(t) are sections of the sheaf O( - 6.- Ee')1 V over Ui n Uk' while, by Lemma 1,

"I"ikilL+I(t)

= +!JilL+l(t) + f~j'+jkllL+I(t) ,

on Ui n Uj n Uk .

Hence the collection {+UIIL+l(t)} of 'tlkllL+I(t) may be considered as a homogeneous polynomial of degree f-1 + 1 whose coefficients are 1-cocycles on the nerve of the covering n = {Ui } of V with coefficients in the sheaf '1'. Consequently, by means of the canonical homomorphism: (54)

the collection {+ikllL+l(t)} determines a homogeneous polynomial +IL+I(t) of degree f-1+1 in t" "', t", with coefficients in Hl(V, '1'). We call +IL+I(t) the flth obstruction. As will be shown below, the obstruction +IL+I(t) vanishes if V is semi-regular in W. We remark that the canonical homomorphism (54) is actually an isomorphism. In fact, in view of the exact sequence (25), 'I' may be considered as a coherent analytic sheaf over W, while the neighborhoods Wi are polycylinders. We have therefore the canonical isomorphism: or (55) (see Cartan [lJ, Leray (31). However we do not make use of this fact in our proof of Theorem 1. (v) Assume that the obstruction +1L+I(t) vanishes. Then, in view of an elementary fact that the canonical homomorphism (54) concerning the first cohomology group is injective, there exists a collection {lPillL+l(t)} of homogeneous polynomials lPillL+I(t) of degree fl + 1 in t" "', tm with coefficients in I'(O( - 6. - E c') 1 Ui ) such that (56)

We write the canonical extension of lPiIILH(t) in the form ie Is, i e Id , ie It, i e Ie, where

~illL+I'

iJlIIILH, SillLH are homogeneous polynomials of degree f-1

( 53)

1218

+ 1 in

611

SYSTEMS OF SURFACES

t" ... , t", whose coefficients are holomorphic functions defined on Wi' We define

I

Xil"'+1

(57)

= ~II"'H ,

Yi:"'+l =

for i E It U Ie , for i e Id U It U Ie , for i ¢ 10

7}il",+1 ,

ZiJ"'+l = SiJ",+l ,

(see (38», and let

+ X'IIJ.Cl , Y;' + Y"",+1 , Z;- + Z'I",H ,

X;-+l = X;y;-+l =

Z;-+l = S ;fI.+lI (t) = e,(X;'+l,

Y~+t, Z~+l)

.

Then, setting (58)

we obtain for i ¢ 10

(59)

(compare (24)d, (24)" (24)c)' Note that S ;"'111(t) = 1 for i € 10 , Now we define homogeneous polynomials f'~I",rl = f'~I"'+l(t) of degree f1- + 1 in tl! •. " tm by (60)

Since [8;",+11 (t) - !':k(t)Sr+l l (t)]"'+l = [S.]"'+1(t) - f~k' (Sk]fl.H(t) - S'k:fI.+1(t) ,

we infer from (56) and (59) that the coefficients of the polynomial [S;fI.+ 1) (t) - !fk(t)Skf'+l) (t)]fI.+1 vanish on vn W, n W k, while Vis defined in W k by the equation S% = O. Consequently the coefficients of fiklfl.+l(t) are holomorphic functions defined on W, n Wk' It is obvious that the polynomial !~tl(t) = Rk(t) + !,klfl.H(t) satisfies the congruence (35)"," 1Thus we obtain Xf+\ Y;,+\ Z;'+l, 8;"'+1 1 (t) andf;-/l which satisfy (33)1'-+1 and (35)",+" provided that the obstructiono/fl.H(t) vanishes. (vi) Now we prove that, if V is semi-regular in W, the obstruction 'tf'+l(t) vanishes. For this purpose it suffices to construct a collection {Wik(t)} of homogeneous polynomials Wik(t) of degree f-l + 1 in t" •• " tm with coefficients in A - I)') I W, n W,,) satisfying

rene -

W,k(t) = wi;(t)

+ f~;·w;it)

,

such that (61)

( 53 J 1219

on Wi n W; n W k

,

612

K. KODAIRA

In fact, the collection {W'k(t)} represents a homogeneous polynomial w(t) with coefficients in Hl(W, fl(F - ~ - Ee'» such that r*w(t) = "hJ.+l(t). Therefore, if V is semi-regular, we obtain 'tp.H(t) = O. Denoting by log X the formal power series 1 )n-l

(

log X = E:=l we define polynomials glk(t), i

€

I, k

n €

(X - l)n ,

I, of degree p in tl/ "', tm by

glk(t) =- I" log (f':k(t)/f~k) •

We have fMt) _ f':/t) f'Mt) - f o =p.-fo '-fo ' tIc

ij

:Jk

It follows that

Letting we obtain therefore (62)

Obviously the coefficients of the power series Jlk(t) in t morphic functions on WI n Wk' We have

j ,

"',

tm are holo-

(63)

Hence we obtain from (35)1" the congruence (64)

Jlk(t)S !:)(t) - S ~p.)(t) =-1"0 ,

Now we determine homogeneous polynomials wolt) of degree p tu "', tm by wlit) =-p.HJ,it)S!:)(t) - S;p.)(t) •

+ 1 in

It is clear that the coefficients of Wlk(t) are holomorphic functions on Win Wk' We have (65) Wik(t) = wiit) + f~J'WJk(t) , on Wi n WJ n W k •

In fact, using (62) and (64), we obtain Wi;(t)

+ f~J'WJk(t) =-p.HJijS'j)

- S;p.)

+ f~JJJkSkp.) - f~JSjp.) + f~jJjkS 1:) - S ~p.) k + fOd fAjk k -

== p.H (JiJ - f~j)S jp.)

-= p.H (fAij - fOij )fAjk SII')

= fA fAJk SIP.) -1L+1 k ij

-

SIp.) t

( 53 J 1220

SIp.)

= fA SIP.) -J.L+l tk k

SIP.) i

-

SIp.) t

= (t) • -1L+l{J)i1c

613

SYSTEMS OF SURF ACES

This proves (65). On the other hand, with the aid of Lemma 2, we infer in the same manner as in (iv) above that, for i e I d , k e Id U It U Ie, the coefficients of m,it) vanish on ~. Combining this with (65), we conclude that the coefficients of m,,,(t) are sections of the sheaf D( -~- Ee') over Win Wk' Since Sk"')(O) = SZ vanishes on Vn W k , we get from (63)

j,,,(t)S)t) (t) =1-'+JMt)Sl"l(t) ,

on Vn Win WIG .

Consequently we obtain rm,k(t)

=

",+1

r(jik(t)S kl-'I(t) - S V· I (t»

=1-'+1 r(j;k(t)Sl"I(t) - SlP.)(t» =P.+1Vikl"'+l(t) .

This proves (61). Thus, in case V is semi-regular in W, we can construct the polynomials X;(t), yt(t), Zt(t), S ;P.I(t) , fMt) satisfying (33)1-' and (35)", by induction on p., and therefore we obtain Xi(w, t), Yi(w, t), Z,(w, t), S,eW, t).!i"(W, t) satisfying (28), (29), (30), (31) which are formal power series in t l1 • •• , t",. 5. Proof of convergence

A l-cochain on the nerve of the covering q; = {Ui } of V with coefficients in the sheaf 'l' may be defined to be a collection V = {vilol of sece') I V over Ui n U" such that 'tlei = - f2i •'tile' tions V,,, of D( - ~ Since each section 'ti/' is a holomorphic function defined on Ui n Uk' we may define the norm II V II of by

r:

-+

II V II

= maXi,,. supw IVi,.(W)

I,

where 't'k(W) denotes the value of the fUnction 'tile at a point W € Ui n Uk' Similarly, a O-cochain on the nerve of the covering {U,} of V with coefficients in 'l' may be defined to be a collection

2 ••• t;;.'" , and let

= {1{r,kl",-H"1'2' ...",} 'P1'+1"l'2'"'''' = {'Pi/I'+I"I'2"""'} • """'+1"1'2""'"

Obviously the relation (56) is equivalent to the system of equations

+

l.Il

l.I2

+ ... + l.I", = P. + 1 .

Hence, by Lemma 3, we can choose, for each p. = 1,2,3, ..• , the polynomials 'Pil"-f1(t) satisfying (56) in such a way that (67) II 'P",+l"I'2'"'''' II ~ CO II """,+1"1'2'"'''' II . Now, assuming (67) and letting a be a sufficiently small positive number, we shall prove that the power series X,(w, t), ~(w, t), Zi(W, t), S,(w, t) andfi1.(w, t) in tH "', t"" converge for I t 1< a. Consider a formal power series f(t)

= f(w, t) = Ef'I'2"'V",(w)tN~2 •. ·t;;,'"

whose coefficients fv 12 • ... v",(w) are holomorphic functions of w defined on a domain in W [or V] and a power series We indicate by writing f(w, t) «a(t) that

I fV 1V2"""'( w) I ~ aVIv... "", . Moreover we write f(t) «a(t) if and only if f(w, t)

« a(t) for each point

w in the domain. Let A (t)

c'" bL,,= = -16c J 1-(t +t +···+t",)I', "'- p.2 l

2

where band c are positive constants. Then we have (68)

A(t)2« !.A(t) . C

In what follows we denote by C1> cz•••• positive constants. For our purpose it suffices to prove the inequalities

« c1A(t) , Xi « A(t) ,

fik(t) - f~k

Xi(t) -

Y;(t) - Yi

« A(t) ,

Z,(t) -

«A(t) •

Zi

( 53)

1222

615

SYSTEMS OF SURF ACES

provided that we choose the constants b, c and C1 properly. For an arbitrary holomorphic function CfJ; =CfJi( w) defined on Ui, we set

II 8

6

for

,

1z 1= 1 .

we have

I abc I = I S~(p) I ;2;

'(} •

Assuming that I a I ;2; / b / ;2; / c /, we set (3='1=0, a=(3=o, '1=0, a=(3='Y=o,

if if if

lal~02 06

for I z 1= 1 ,

,

q.e.d. We may assume that (70)

/fMw)/;2;c 2

for w e Wi n Wk

,

•

In fact, we obtain (70) by replacing, if necessary, each neighborhood Wi by a smaller one. Similarly we may asSUme that the coefficients of the linear forms Xi\l(t), Yi\lt). Zi\/t) are bounded. We obtain therefore the inequalities X!(t) -

Xi

« A(t) ,

Y!(t) - Yi «A(t) , Z!(t) -

Zi

«A(t) ,

provided that the constant b is sufficiently large. Now we prove, by induction on fl, the inequalities (71)1'-1 (72)1'

« clA(t) , X:(t) = Xi « A(t) ,

frk-l(t) - f~"

l

Y. (t)

Z~(t)

Yi

-

Zi

« A(t) ,

«A(t) .

We have the recurrent formulae: (73)1'

( 53)

1225

618

K. KODAIRA

- fl'-IS(I')(t)] · (t) =_l_rS\I'J(t) f 'kll' S2 L , ,k k I' ,

(75)1'

'tikll'+1(t) = r [IrkS kl') (t) - S ;1') (t)]I'+l ,

(76)1'

'Pill'+1(t) - f~7/l)kll'+l(t) = 'tikll'+1(t) ,

i € Is, i € Id , i € It, i € Ie, (compare (33)1" (42), (44), (45), (59) and (60». Suppose that (71)1'-1 and (72)1' are already proved for an integer p"ii:;.l, we estimate I,kll'(t), X il l'+l(t), Y'II'+l(t) and Zill'+l(t) by means of the above recurrent formula. Assume that c > b. Then we obtain from (72)1' and (73)1' the inequalities (78) S ;1') (t) - S~ 19A(t) ,

«

(79)

[8;I'l(t)JfJ.t1

« sbc A(t) .

In fact, if i e I d , we have S;I')(t) - S~ = (Y;- - Yi)(Zt - Zi)

+ Zi(Y;' -

Yi)

+ Y,(Zt

- Zi)

and therefore, using (68), we get S;I'J (t) -

S~«

A(t)2

+ 2A(t) « (~ + 2)A(t)« 3A(t)

,

[S;I'J(t)]l'+l« A(t)2« l.A(t) •

c

If i e It> we get

+ 3A(t)2 + 3A(t) «7A(t) [S;I'J(t)]I'+1« A(t)3 + 3A(t)2 « 4b A(t) .

SlI'J(t) -

S~«

A(t)3

,

c

If i e Ie, we have 8;I'J(t) - S~ = (Xr - Xi) (yr - Yi)'

+ 2Yi(Xr + YHXf -

+x

i(

yr - Yi)'

xi)(Yr - Yi) - 4(Zt - Zi)2 Xi)

+ 2X iYi(Y;- -

Yi) - 8z i (Zt - Zi) •

Hence we get

+ 7A(t)2 + llA(t)« [8 ;iJ-) (t)JI'+1 « A(t)B + 7A(t)'« sbA(t) .

Sll'l(t) - S~ «A(t)S

c

Thus we obtain (78) and (79). We have

( 53)

1226

19A(t) ,

619

SYSTEMS OF SURF ACES

f':k-1S1/L) (t) -

S~/L) (t) =

(ffk-1 - f:,,) (Sl/L) - sZ)

+

+ f~,,(S!t)

- SZ) 1 SHftk- - f~k) - Sj/L)

+ S~

.

Hence, using (68), (70), (71)P.-1 and (78), we obtain (80)

where

ca = 19bc1 /c

+ 19c + 19 . 2

Now we estimate fiklP.(W, t). Let {Wi} be a covering of W such that the closure of each neighborhood W7 is contained in Wi and let 8 be a small positive number. Take a point p € Wi n Wk' By Lemma 5, there exists a disk D with R(D) < 28 such that p € Dc W" and such that

I S~(w) I S

(81)

86

for W € aD.

,

It is clear that D is contained in W; n W k , provided that the positive constant 8 is sufficiently small. We have fiklP.(W, t) = SZ;W) [SjlLl(W, t) - f':k-1S!t)(w, t)lp. ,

(see (74)1')' Hence, by (80) and (81), we get fil'Ip.(W, t) «8- 6caA(t) ,

for w € aD,

while the coefficients of the polynomial fil'liw, t) are holomorphic on the disk D. Therefore, by the maximal principle, we get Thus we obtain for

(82)

W

e W; n Wk

We have It follows that fiklP. - ndijlP. -

f~djklfL =

[(ftj-l -

Hence, by (71)P.-H we obtain (83)

This implies that

( 53 J 1227

f~j)(f~;;l

-

f~k)]P.

•

620

K. KODAIRA

Combining this with (82), we obtain (84)

fik,,.(W, t)

«(Ci~s + C2:C~)A(t) ,

for we Win W: .

Let w be an arbitrary point in Wi n Wk' w is contained in one of the neighborhoods, say Wi. Combining (83) with (82) and (84), we infer therefore that for we Win Wk' Now we set Assume that Then we have Cs

< 19(c. + 2)

and therefore (c;

+ 1)(C C + 2 S

86

Consequently we obtain fikl,.{t)

bcr )

!!:... ds

= dt,(s)...!.. ds at,

+ ... +

dt",(s) ...!.. ds at""

+ _0_ . at",,+!

Then we have at(s.t.)(d/ds)

=0 .

This proves that the fibre Vt(S,t') = 11f-'(t(s, t*» is independent of s. Clearly, for each point t IS Mil) there exists one and only one curve C(t*) passing through t. Consequently, writing hl(t) =t* if and only if t E C(t*), we obtain a holomorphic map h! of M~ll into Mi l - ll with h!(O) = 0 which obviously satisfies 'tlJ"-'(t) = 'tlJ"-'(h!(t», q.e.d. Now we prove our main theorem. Let V be a surface in W with ordinary singularities only. If V is semi-regular in W, then, by Theorem 1, there exists an analytic family q; ~ M. of surfaces with ordinary singularities in W such that 11f-'(O) = V and such that a o: To ----> HO( V o, '\)10) is bijective. By Theorem 3, the family q;~M. is maximal at O. Moreover, as was shown in the proof of Theorem 3, the infinitesimal displacements at(ajat,), I.i = 1, 2, "', m, of V t = 'tlJ"-'(t) form a base of HO( VI' '\)It), provided that I t I is sufficiently small. It follows that at: T t ----> HO( V" '\)It) is bijective if I t I is sufficiently small. This completes our proof of the main theorem. PRINCETON UNIVERSITY AND INSTITUTE FOR ADVANCED STUDY REFERENCES

1. H. CART AN , Varietiis analytiques complexes et cohomologie, Colloque sur les fonctions des plusieurs variables tenu it Bruxelles, 1953, pp. 41-55. 2. K. KODAIRA and D. C. SPENCER, A theorem of completeness of characteristic systems of complete continuous systems, Amer. J. Math., 81 (1959), 477-500. 3. J. LERAY, L'anneau spectral et l'anneau fiUre d'homologie d'un espace localement compact et d'une application continue, J. Math. Pures et Appl., 29 (1950), 1-139. 4. F. SEVERl, Sui teorema jondamentale dei sistemi continui di curve, Annali di Mate· matica, s. IV, vol. XXIII (1944), 149-181. 5. O. ZARISKI, Algebraic surfaces, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Bd. 5, No.5, 1935.

( 53) 1235

A THEOREM OF COMPLETENESS OF CHARACTERISTIC SYSTEMS FOR ANALYTIC FAMILIES OF COMPACT SUBMANIFOLDS OF COMPLEX MANIFOLDS By K. KODAlRA

(Received April 26, 1961)

Recently D. C. Spencer and the author proved a theorem of completeness of characteristic systems of complete continuous systems of compact submanifolds of co-dimension 1 of ambient spaces. ' The purpose of the present note is to prove a similar theorem of completeness2 for analytic families of compact sub manifolds of an arbitrary co-dimension of ambient spaces. A formal aspect of the theorem has been studied earlier by A. Haefiiger. 3 1. Analytic families of compact submanifolds of complex manifolds

In what follows, we assume that all manifolds under consideration are paracompact and connected. Let Wbe a complex manifold of (complex) dimension d + r. We denote a point in W by wand a local (complex) coordinate of w by (wI, w 2, "', WTH). Letting M be a complex manifold, we form the product space W x M and denote by 1C the canonical projection of W x M onto M. DEFINITION 1. By an analytic family of compact submanifolds of dimension d of W we shall mean a pair (CV, M) of a complex manifold M and a complex analytic submanifold c:v of W x M of co-dimension r which satisfies the following two conditions: (i) for each point t E M, the intersection n W x t is a connected, compact submanifold of W x t of dimension d; (ii) for each point P E C:V, there exist r holomorphic functions

cv

fl = flew, t), ···,iT = f.(w, t)

defined on a neighborhood CUp of p in W x M such that 1 See K. Kodaira and D. C. Spencer, A theorem of completeness of characteristic systems of complete continuous systems, Amer. J. Math., 81 (1959), 477-500.

2 When applied to systems of submanifolds of co-dimension 1, this theorem does not cover the result of Kodaira and Spencer, loco cit. 3 A. Haefiiger, Structures feuiUetees et cohomologie a valeur dans un faisceau de groupoides, Comment. Math. Helv., 32 (1958), 248-329, Chap. IV. 146

( 54

J

1236

147

THEOREM OF COMPLETENESS

rank

a(fl' f2' "', fr) =r a(w l , w 2, "', w r +d )

and, in cUp, the submanifold cV is defined by the simultaneous equations (1)

fr(w, t) = f2(W, t) = ... = fr(w, t)

=0 .

We call M the parameter manifold or the base space of the family (c(l, M). We denote the family (C(l, M) simply by c(l when we need not indicate the base space M. For each point t E M, we set (2)

The submanifold V t of W thus defined will be called the fibre of cV over t. We may identify Vt x t with Vt and consider C(l as a family consisting of compact submanifolds Vt , t E M, of W. 2. An analytic family (C(l, M) of compact submanifolds Ve, t E M, of W will be said to be maximal at a point to of M if and only if, for any analytic family (CV', M') of compact submanifolds V;, s EM', of W such that V:o = Vto for a point 8 0 of M', there exists a neighborhood N' of So in M' and a holomorphic map h: s t = h(s) of N' into M sending So into to such that V~= VIolS) for SEN', where we indicate by writing V~ = V t that V~ and V t are the same submanifold of W. Now we shall introduce the concept of infinitesimal displacements. Let C! be the space of several complex variables t l , t 2 , • • • , t!, denote any point (tl1 t 2, "', t!) in C! by t, and define DEFINITION

-7

I t I = max I t I . p

p

In what follows we denote by M. the polycylindrical domain in 0 of radius e> 0 and of center 0; i.e., M. = {t II t I < e}. Consider an analytic family (C(l, Mr) of compact submanifolds V" t E Ml1 of Wand let V = Yo. Obviously V is covered by a finite number of coordinate neighborhoods Wi in W. We choose a local coordinate on each neighborhood W, such that V n Wi coincides with the subspace of Wi determined by wl = ... = w~ = 0, where we write z;, "', z~ for w~+1, "', W~+d, and we define

I w, I = max>.. I w~ I ,

I Zi I = max", I z~ I .

Moreover we assume that the local coordinate (Wi' Zi) can be extended to a domain containing the closure of Wi and that Wi is a polycylinder: Wi

= {(Wi' Zi)

II Wi I < 1, I Zi I < I}

( 54)

1237

.

148

K. KODAIRA

On the intersection Wi n Wk , the coordinates w!, ... , wi, holomorphic functions of W/c and Zk:

A. = 1,2, ... , r ,

W: : f:(w/c, z/c) , { Z; - g,k(W/C, Zk) ,

(3)

z;, ... , z~ are-

a = 1,2, ... , d.

Setting, for instance, fil,(W k, Zk) = (fA(w", Zk), •• ',fA(w k, z,,), •• ',fMw", z,,» ,

we write the above formulae (3) in the form Wi: fik(W k, Zk) , { Zi - gi.l:(W , Z/c) . k

( 3 )'

Obviously we have (4)

We set Ui = V

n Wi =

{(O, Zi)

II Zi I < 1}

.

We denote a point on V by Z and, if Z = (0, Zi) E Ui , we consider Zi = (zl, "', zt) as the coordinate of Z on Ui • Thus, for instance, we indicate by writing Z = (0, Zk) E U" n Ui that Z is a point in Uk n Ui whose coordinate on U,. is Z/c' We define

and let F..I:(Z)

= (fA.(Z)h..=1.2"".r .

Obviously the matrix-valued functions Fi/c(z) thus defined satisfy (5)

for Z E U.

Fil,(z) = F;iz)Fjiz) ,

n Uj n U" •

DEFINITION 3. By the normal bundle of V in W we shall mean the complex vector bundle F over V defined by the system {FiT,(z)} of transition matrices Fik(Z) = (fi~'(Z», We denote by 'l" the sheaf over Vof germs· of holomorphic sections of the normal bundle F. Let e be a sufficiently small positive number. It is clear that, for I t I < e, the submanifold V, of W is defined in each neighborhood Wi by simultaneous equations of the form

A. = 1,2, "',

(6)

r ,

where the g1(Zi, t) are holomorphic functions of Zi' I Zi I < 1, depending holomorphically on t, I t I < e, and satisfying the boundary conditions

( 54)

1238

149

THEOREM OF COMPLETENESS gJ~(Zi' 0)

= 0,

J\,

= 1,2, "',

r .

Setting gJ;(Zi, t) = (gJl(zi' t), "', gJ~(Zi' t), "', t) ,

E

Uk n Ui •

We have the equality (17)

'o/ik(Z, t) = 'o/tJ(z, t)

+ F,iz)''o/Jk(Z, t)

for Z E Ui n UJn Uk •

,

To prove this let

SJ = gJk('Pk'(t» , where we write gJk(W k) for gJk(W k, Zk). From the congruence we get 'P';(t;J' t)

=".+1

!Jk('Pk'(t»

+ 'o/Jk(Z, t)

,

while we have the identities !ik(W k) = !i/fJk(W k), gJk(Wk» , g,k(W k) = gij(fJk(W k), gJk(W k

».

Hence we obtain !iirp~(tJ' t), !;j)

(18)

="'+1 !iJ(fJk(rp'k(t» + 'o/Jk(Z, t), !;J) ="'+1 f,ifJk('Pk'(t», gJk('Pk'(t» + Fij(Z)''o/Jk(Z, t) ="'+1 fik(rp;:(t» + Fij(Z)''o/Jk(Z, t) ,

and similarly (19)

( 54)

1242

153

THEOREM OF COMPLETENESS

Obviously we have (20)

V'iiZj, t)

=",+1

IP':'(g,ig:>';'(Zj' t), Zj), t) - f,ig:>';'(zj' t), Zj) •

Now we take a point Z E Ui n U j n u" and let ZJ and Zk be the local coordinates of Z on Uj and Uk' respectively. Since ~j = gJk(IPk'(t), Zk)

=0 gjk(O, Zk) = Zj ,

we have

V'iis;, t) =m+11h,(z;, t) = 1{ri'(Z, t) . Hence, substituting Sj for Zj, we obtain from (20) the congruence 1{r,;(Z, t) =",+1 IP;:'(gi;(g:>';'(S;, t), Sj), t) - fu(g:>';(Sj, t), Sj) .

Combining this with (18) and (19), we get 1{rij(Z, t) =",+1 IP;:'(g,k(IPk'(t», t) - fik(IPk'(t» - F ii (Z)'1{r;k(Z, t) ,

or V'ik(Z, t)

=",+1

V'i;(Z, t)

+ Fij(Z)'V'ik(Z, t)

.

Thus we obtain the equality (17). The equality (17) shows that the collection {V'ik(Z, t)} of 1{rik(Z, t) may be considered as a homogeneous polynomial of degree m + 1 in tIl "', tz whose coefficients are 1-cocycles on the nerve of the covering {U,} of V with coefficients in the sheaf 'l'. Thus the collection {'fr,k(Z, t)} represents a homogeneous polynomial 'fr",+1(t) of degree m + 1 in tIl "', t z with coefficients in H1( V, 'l'). We call 'fr"'+l(t) the mth obstruction. By hypothesis, the cohomology group H1( V, '1') vanishes. There exists therefore a collection {g:>ilm+1(Z, t)} of homogeneous polynomials IPil",+1(Z, t) of degree m + 1 in t 1, •• " tl whose coefficients are vector-valued holomorphic functions of Z defined on U i such that (21) Considering the coefficients of IPilm+1(Z, t) as fUnctions of the local coordinate Zi of z, we write IPilm+1(Zi, t) for g:>ilm+1(z, t). Then the formula (21) is written in the form (21)'

V'ik(Zk, t)

= F,iz)g:>klm+1(Zk' t)

- g:>ilm+1(g,k(0, Zk), t) .

Now we define . I ~>'I . Consider a power series

...

~(z, u) = E~'m "(z)u;u;' .•. u~

in Uu U 2 , •• " U q whose coefficients Z and a power series We indicate by writing ~(z, u)

~!1" .•. "(z)

are vector-valued functions of

< a(u) that

I ~Im .•. n(z) I ~

aim···,. •

Let

where a and b are positive constants. We have for

(22)

).i

= 2,3, ...•

For our purpose it suffices to prove the inequalities cp;(Z;, t)

< A(t)

In what follows we denote by Co, C" c2 ,

( 54

J

iEI.

, •••

1244

positive constants which are

155

THEOREM OF COMPLETENESS

greater than 1. We may assume that

Iftkiz) I < Co, It follows that the coefficients bounded. Hence we have

/3~,(z)

Co>

1.

of the linear form 'PiIJ(Zi, t) are

(23)

provided that the constant a is sufficiently large. Now, assuming the inequalities 'P';(z" t)

~

iEI,

A(t) ,

for an integer m ~ 1, we shall estimate the coefficients of the homogeneous polynomials 'ti.(Z, t) defined by (15). We expand fi.(W k) = fi.(Wk> Zk) and g,k(W k) = g,k(W k, Zk) into power series in wl, "', W'k whose coefficients are vector-valued holomorphic functions of Z = (0, Zk) defined on Uk n Ui • Obviously we may assume the inequalities

+ w% + ... + wk)" , ~ E;=oc~(wl + w% + '" + w;;)n .

fik(Wk) ~ E;=,c~(wl

(25) (26)

gik(Wk)

First we estimate [f,k('Pk{t))J"'H' The terms which are linear in 'Pk'(t) contribute nothing to [fik('Pk'(t))]m+1. Hence, using (22), we get from (24)", and (25) the inequality [fik('Pk'(t))]",+1

~ 1::=2c~rnA(t)" ~ c,rA(t) 1::=1 C~1;ar

Assuming that (27)

we obtain therefore (28)

Denote by

U~

the sub domain of Ui consisting of all points

Z

= (0, Zi).

I Zi I < 1 - o. We fix a positive number 0 such that {U~ liE J} forms a covering of V. Take a point Z E Uk n U~ and let Zk and z, be the local coordinates of z on Uk and U" respectively. Obviously we have

I Zk I < 1, I Zi I < 1 - 0 •

(29)

Now we estimate ['P:,,(gik('Pk'(t», t)]m+1. Letting y = (y" "', YO" "', Ya), we expand the coefficients of the polynomial 'P':(Zi + Y, t) into power

( 54 J 1245

156

K. KODAIRA

series in YH "', Y/}.. Since the coefficients are holomorphic functions of Yl> "', Yd' I Yl I < 0, "', I Ya I < 0, we obtain from (24)", the inequality (30) Let g,k(g>'k(t» = Zi

+ YJ •

Then we get from (24)... , (26) and (27) the inequality YJ ~ E:=lc;r"A(t)" ~ 2c1rA(t) .

Combining this with (30) and using (22), we obtain [g>:"(Zi

+ YJ, t)]m+l = ~

[g>7(Zi A(t)

+ YJ, t) -

{(I -

r

g>:"(z;, t)],.+l d

2c1r:(t)

-

I}

~ E:=l(d + ; - 1)(2~lrrA(t)tHl ~ A(t) E=

,,=1

(d + nn -

1)(2C1ra)" bO'

Assuming that M> 4c1ra,

(31)

w.e have therefore Thus we get [g>i"(gik(g>Z'(t», t)]",+l

~

2d+2c,rab-'o-lA(t) .

Combining this with (28), we obtain for

(32)

ZE

Uk n U~,

where (33)

C2 = 4c 1rab- I(2 d o-1

Now we take an arbitrary point which contains z. We have

ZE

+ rCI)

Uk n U. and choose a domain U;

while, by (32),

( 54)

.

1246

157

THEOREM OF COMPLETENESS

Consequently we obtain (34)

where (35)

The following lemma can be proved easily by an elementary consideration.· LEMMA.

We can choose the homogeneous polynomials CJ>i\m+1(Z, t), i E I.

satisfying Fik(Z)CJ>k\m+l(Z, t) - CJ>i\m+l(Z, t) = "friiz, t) in such a way that

(36) where the constant

C4

> 1 is independent of m.

By (33) and (35) we have C4C3 = 8cOclc4r2ab-l(2ao-l

+ rCI)

+

rCI) ,

.

Therefore, setting (37)

b = 8cOcIc4r2a(2do-I

we obtain from (36) the inequalities (38)

CJ>ilm+l(Zi, t) ~ A(t) ,

i

E

I.

Note that (37) implies (27) and (31). Combining (38) with (24) .. , we obtain (24),,+1

CJ>f+ 1(Zi, t) ~ A(t) • poly~

Thus we conclude that we can choose for each m the homogeneous nomials CJ>ilm+r(Zi, t), i E I, in such a way that (39)

CJ>JZi. t) ~ A(t) ,

i

E I.

It follows that the power series CJ>lZi, t) converges for I t I < co, where l/lb. If we consider CJ>i(Zi, t) and CJ>k(Zk, t) as formal power series in tl> ••• , t!, we have

Co.. =

We denote by W i5 the sub domain of Wi consisting of all points (Wi' z;), I Wi I < 1, Iz, I < 1 - O. Obviously the domains wt, i e I, cover the manifold V. Now we fix a sufficiently small positive number c and consider CJ>,(Zi, t) and CJ>k(Zk. t) as holomorphic functions of t, I t I < c. We infer 5

See Kodaira and Spencer, loe. cit., § 6.

( 54)

1247

158

K. KODAlRA

from (39) that, if I t

I < c and if ( t = h(s) of a neighborhood N' of 0 in M' into M" sending 0 into 0, such that V~ = V"'IS). Let Wi' Ui , Zi' Wi' fik(W k, Zk), gik(W k, Zk), Fik(Z), t, B, p) forms a complex analytic family of compact complex manifolds. For any sub domain N of B we call the complex fibre manifold (p-l(N),N,p) the J'estriction of 'Jt to N and denote it by the symbol IN. Let 'lJ be a complex submanifold of IN such that p('V) =N. We call 'V a fibre submanifold of the complex fibre manifold IN if and only if the triple

'*

'*

'*

* llecei \Teu N ovembel' I, 1962. This research was partially supported by the United States Air Force through the Air Foree Office of Scientific Research of the Air Research and Development Command, muler contract No. AF49 (638) -253. Reproduction in whole or in part is permitted for any purpose of the United States Government. 1

'Y9

( 55 J 1253

80

K. KODAIRA.

('V,N,p) forms a complex fibre manifold. If, moreover, each fibre V .. 'V n w.., u EN, of 'V is compact, we call 'V a fibre submanifold with compact fibres of the complex fibre manifold IN. Consider a compact complex submanifold V of a complex manifold W.

'*

=

Definition 1. We call Va stable submanifold of W if and only if, for any complex fibre manifold (,*, B, p) such that pol (0) = W for a point 0 E B, there exist a neighborhood N of 0 in B and a fibre submanifold 'V with compact fibres of the complex fibre manifold IN such that 'V n W = V. In this section we shall prove the following theorem:

'*

THEOREM 1. Let '11 be the sheaf over V of germs of holomorphic sections of the normal b1tndle of V in W. If the first cohomology group IF (V, '11) vanishes, then V is a stable s1tbmanifold of W.

Let (,*,B,p) be. a complex fibre manifold such that pol(O) = W for a point 0 E B and let (u1 , 1£2,' . " u q ) denote a loeal coordinate on B with the center o. Considering V eWe as a submallifold of ,*, we cover V by fl finite number of coordinate neighborhoods '1Lt in and choosc a local coordinate (1)

'*

'*

on each neighborhood '1Lt such that the map p takes the form

and such that the simultaneous equations

Wi 1 = '

.. =

define the submanifold V.

wi' = u 1

=' .. =

uq = 0

For brevity we write

Zi = (z/,' . " Zi d ) , Wi = (w/,' . " wt), u = (ut,· . " uq ), and define

1Zi 1=

max 1z{" a

I, 1Wi 1=

1w/"I, 1u I =

max X

max' uP ,. p

We assume that each neighborhood 'l4 is a polycylinder consisting of all points (Zi' Wi, u), I Zi I < 1, IWi I < 1, I u , < 1. On the intersection 'U; n 'U" the local coordinates z;''', Wi X are holomorphic functions of Zk, Wk, u:

(2)

j z,"=gi1""(Zl,,Wk,U),

I wix=fikX(zk,wk,u),

Note that

[ 55)

1254

i(Zi, t)

=

(,fJ/(z., t),' .. ,(g(Zi, t), tn-q+l,' .. , tn)

satisfying the boundary conditions

i(Zi, 0) = 0, [Dc/>t (z" t) 100 p]too = f3pi (z),

v=1,2,' .. ,n,

such that (11 ) We write tbe power series expansion of

for p=I,2,· . . ,q. Thus the coefficients of the homogeneous polynomial {ifik (z, i)} are l-cocycles with coefficients in the subsheaf.y of tf>. By hypothesis the cohomology group Hi (V, .y) vanishes. Hence the obstruction 1/tm+1 (i) vanishes. Moreover the l-cocycle {1{Iik(Z, t)} is a coboundary of a O-cochain {Xi(Z, t)} composed of homogeneous polynomials Xi(Z, t) of degree m 1 in t 1 ,· • • , in whose coefficients are sections of the sheaf .y over Ui . Thus

+

and ifikA(Z, i)

We define

=

r

~ ai/,A/L (Z)X"IL (z, t)

/L=1

-xhz, t),

and set r+l(Zi, t)

=

rpim(Zi, t)

for A= 1,· .. , 'r.

+ rpi[m+i(Zi, t).

The polynomials .1n+i (z., t) thus obtained satis£y the congruence (12) m+1 and the condition (13)m+i. We remark that, by (9), the linear terms n

i1(Zi' t) =~f3.i(z)t. 1>=1

satisfy the condition (13 h. This completes our inductive proof of the vanishing of ifm+l (t), q. e. d. Remark. In view of (10) the family ~ is defined on each neighborhood 'l4 X ME by the simultaneous equations

(14)

{

1, 2,· .. ,1", p= 1,2,· .. , q.

w/' = e,A(Zi, t), uP =

.\ =

tn-q+p,

Proof of Theorem 1. Let NE denote the neighborhood o£ 0 in B consisting of all points u = (ul,· .. , uq ), Iu I < £, where £ is a small positive number. Considering the family fJ of compact complex submanifolc1s Vt. iE M., of defined by (14), we define a linear map: tf--'j.t(U) of X. into Me by setting (15) feu) = (0,0,· . ·,0,u\u 2, • • ·,uq ).

'*

( 55)

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85

STABILITY OF COMPACT SUBMANIFOLDS.

U VI(u) of the submanifolds

Then the union 'lJ =

Vt(u), U

EN

f,

of '}t forms

u

a fibre submanifold with compact fibres of the complex fibre manifold '}t IN. and 'lJ n W = V. Thus we conclude that V is a stable submanifold of W, q.e. d. Let (')ft, B, p) be a complex fibre manifold and let W = p-l (0) be the fibre of '}t over a point 0 E B. Moreover let V be a compact complex submanifold of Wand denote by 'It the sheaf of germs of holomorphic sections of the normal bundle of V in W. The above Theorem 1 asserts that, if H' (V, 'It) vanishes, then, for a sufficiently small neighborhood N of 0 in B, there exists a fibre submanifold 'lJ with compact fibres of the complex fibre manifold ')ft IN such that 'IJ n W = V. The fibre sub manifold 'IJ is, in general, not unique. In fact, in the above construction of 'lJ = U V t (",); we

"

may replace (15) by

(15)'

t (u)

=

(t' (u),' . " p. (u),- . "

tr (u), u\' . " uq )

where the {)'(u) are holomorphic functions in u such that th(O) =0, < E for \ u ! < £.

I th(U) I

THEOREM 3. If the cohomology groups HI(V, 'It) and HO(V, 'It) both 'l:anish, then, for a sufficiently small neighborhood N of a in B, there exists one and only one fibre submanifold 'IJ with compact fibres of the complex fibre manifold '}t IN such that 'IJ n W = V.

pj·oof. It suffices to prove the uniqueness of 'lJ. In terms of the local coordinate (1) the submanifold 'lJ is defined on each neighborhood 'U; by simultaneous equations of the form

wi' =

A= 1, 2,' .. ,1'.

(J/'(Zi' u),

Writing t for u we consider 'V as an analytic family consisting of compact submanifolds Vt='Vnp-l(t), tEN of '}to Obviously the submanifold V t is defined on each neighborhood 'l4 by the simultaneous equations

wih=8ih(z;,t), { uP = tp,

A= 1, 2,' p= 1, 2,'

. ,1', ., q.

We compute the infinitesimal displacement [8Vt/8tl t=o and find that (16) Since, by hypothesis, HO (V, 'It) vanishes, we infer from (5) that the map K: HG(V, -1. Hence we conclude that, if i( C, C) > -1, C is a stable submanifold of S. A simple example of surface containing a non-singular rational curve C with i( C, C) < is given by a rational ruled surface. Let P be a projective line and denote by ~ a non-homogeneous coordinate on P. Moreover let U 1 and U z be two copies of the space C of a complex variable z. For each integer n > 0 we form a rational ruled surface

°

Sn=PXU1UPXlJ z

by identifying (~l' Zl) E P X U 1 with ({z, zz) E P X U z if and only if

and we denote by C the curve on S", defined by the equations ~1 = ~2 = O. Clearly C is a non-singular rational curve with i (C, C) = - n. Moreover, for any irreducible curve l' on Sm we have the inequality: i(1',1') > - n. Now we show that, in case n > 2, C is an unstable submanifold of Sn. For each complex number t we define a surface

S",t = P X U1 UP X U z by identifying (~l' Zl) E P X U1 with (~z, zz) E P X U z if and only if (17)

5~l = 1Zl =

zzncz l/z z,

+ tzz\

where 7c is a fixed positive integer < in. The surface Sn.t depends holomorphically on t. Thus the set of all surfaces S",t form an analytic family.

( 55)

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87

STABILITY OF COMl.'ACT SUBMANIFOLDS.

It is obvious that 8".0 coincides with 8n , while, for t =1= 0, 8 n•t is complex analytically homeomorphic to the ruled surface Bn-2k' In fact, introducing new variables ,/ =

U=

(Z1 k'1 - t)/t'1' '2/ (iZ.n- k'2 + t 2),

we infer readily that the equalities (17) are equivalent to I"

~1

{

_ -

Zl=

Z n-2kl' , 2

':.2,

liz •.

It follow8 that there exists on Bn,t, t =1= 0, no irreducible curve Ct with i ( Cbet) = - n. Thus we see that the curve C on Bn is unstable.

2. Stability of fibre structures. Let W be a complex manifold. By a fibre siruct1tJ'e on W we shall mean a pair (B, p) of a complex manifold B and a holomorphic map p of W onto B such that the triple (W, B, p) forms a complex fibre manifold. A (complex) analytic family of compact complex manifolds is, by definition, a complex fibre manifold with compact fibres. Let ']f = (']f, s, w) be an analytic family of compact complex manifolds, By an a1talytic family of fibre structures on the analytic family 'X' we shall mean a pair (18, 'P) of an analytic family 18 = (18, B, 7r) of compact complex manifolds and a holomorphic map 'P of ']f onto 18 such that 7r'P = wand such that the triple ('X', 18, 'P) forms a complex fibre manifold. For each point 8 of B we set W8= w- 1 (s), B.=7r- 1 (s) and denote by p. the restriction of the map 'P to the submanifold Ws of 9f, Obviously the pair (BS) Ps) "', ®" ... such that (®.®t) ~ 1 (Le., ®. and 0 t have at most one simple intersection point) for s < t and ®r n 0. n ®t is empty for r < s < t. These types are therefore described completely by showing all pairs ®" ®t with (®,0 .

Thus, for any integral unimodular matrix (8) = (~ ~), we denote by 8 the corresponding modular transformation of the upper half plane C+. The holomorphic 1-form qJ(u) on Cu is obviously unique up to a, multiplicative constant factor. Hence, combining (7.1) with (7.3), we obtain THEOREM 7.2. By the analytic continuation along a closed arc f3 on D.', the holomorphic function w{u) is transformed into 8f3w(u). We define

!leu) = J(w(u») . Clearly !leu) is a single-valued holomofphic function of u defined on .:1' = .:1 - {a p }. In order to determine the singularity of !leu) at a point ap , we describe a small circle a p of center a p on .:1 with positive orientation and set Ap = 8",p .

THEOREM 7.3. The point ap is a pole or a removable singula1- point of !leu) according as the order of Ap is infinite or finite. Thus g(u) is a merornorphic function on.:1. If ap is a pole of g(u) of order bp , then Ap is equivalent to the modular transformation:

w ~ U)

+b

p •

PROOF. We denote by r the local uniformization variable on.:1 with the center ap and let w(r), r = r(u), be a branch of the multi-valued holomorphic function w(u) such that the analytic continuation along Q'p induces the transformation: w(r) ~ Apw(r). In case the order of Ap is finite we infer readily tha't a p is a removable singular point of g(u). In fact, if A;' = 1 for a positive integer m, then w(a m ) is a single-valued holomorphic function of a defined on the domain: 0< I a I < $, where $ is a small positive number, and moreover Sw(a m ) > o.

C 56

J

1281

576

K. KODAIRA

Hence w(a m ) is regular at a = 0 and therefore Sw(O) > O. It follows immediately that a p is a removable singular point of g(u) = J(w(r"». Now we consider the case in which the order of Ap is infinite and let A w = aw

+b

ad - be

ew + d '

p

=1.

First we shall show that Ap is equivalent to a modular transformation of the form: w ---+ W + bp • For this purpose it suffices to prove the equality a + d = 2 under the assumption that e 0 and a + d ~ O. In fact, if a + d = 2, we have

'*

be = (1 - a)(a - 1) .

Hence we can find integers p, q, r, s such that ps - qr = 1 .

bq = (1 - a)p ,

ep = (a - l)q,

We obtain

where

bp = (a :- d)rs + bs 2

-

er 2

•

This proves that Ap is equivalent to the modular transformation: w ~ w + bp • To prove that a + d = 2 we consider the fixed points Wi- and w_ of Ap which are given by the formula w± =

_l_(a - d ± V(a 2C

+ d)2 - 4) .

+ d = 0 or 1, then SWi- > 0 and therefore the order of Ap would be equal to 2 or 3. This is contrary to our hypothesis. If a + d ~ 3, then the transformation Ap: w ~ w' would be represented in the form If a

where k is a positive constant g(l)

=

'* 1.

Hence, setting

w(!') -

Wi- ,

w(r) -

w_

l = _I_log!,

2ni

•

we would obtain a single-valued holomorphic function g(l) of 1 with 3g(1) > 0 defined on the domain: 31 > &, K being a large positive number,

( 56)

1282

577

ON ANALYTIC SURFACES; II

such that (7.4)

g(l

+ 1) =

kg(l) .

For our purpose it suffices therefore to show that, if a single~valued holo~ morphic function g(l) of I, Sl > /C, with Sg(l) > 0 satisfies (7.4), the positive constant k must be equal to 1. Take a point lo, Slo > /C, and let 1-10 l - ld '

y _ \> -

where It =

Io+ 2i/C •

The quotient h(t) = g(l) - g(lo)

g(l) - g(lo)

is a holomorphic function of t defined on the unit disk: 1 t 1 < 1 and satisfies 1 h(t) 1 < 1, h(O) = O. Hence, by Schwarz's lemma, we have

I g(l) 1

- g(lo) I < g(l) - g(lo) I =

II - 10 I II - It I

•

Setting l = lo + n, n = 1,2,3, "', and using (7.4), we obtain from this the inequality 1

kng(lo) - g(lo) 1 ~ - g(lo) 1

1 kng(lo)

n

In + lo

-It

1

or kn

+ k- n -

2

~

21lo - It 1-2(1 - cos O)n2 ,

for n = 1, 2, 3, ... ,

where cos 0 = m[g(lo)jg(lo)] . This proves that k must be equal to l. Thus Ap is equivalent to a modular transformation of the form: w ~ w + bp • We may assume therefore that Ap itself has the form (7.5)

b

* o.

Now we prove that the integer b is positive and that a p is a pole of g(u) of order b. It follows from (7.5) that the difference f(T)

(7.6)

• l = -1. logT,

bl ,

27ft

single~valued

is a

o
K , 2dw where ~(e2~iw) is a convergent power series in e , provided that K is sufficiently large. Combining this with (7.7) we see immediately that ap is a pole of g(u) = J(w(r» of order b, q.e.d. We call g(u) the functional invariant of the fibre space Vof elliptic curves over~. Bya suitable choice of the finite set {a p }, we may assume that g(u) 0, 1, = for u E ~' = ~ - {ap }.

*'

8. Construction of an analytic fibre space of elliptic curves

Suppose that a non-singular algebraic curve ~ of genus p and a meromorphic function g(u) on ~ are given. We take a finite set {ap } of points a p , p = 1, 2, "', r, on ~ such that g(u) 0, 1, for u E ~' = ~ - {a p }. Tllen there exists on~' one and only one multi-valued holomorphic function w(u) with Sw(u) > 0 satisfying J(w(u» = g(u). We fix a point 0 on ~' and suppose that each element of the fundamental group IT1(A') of ~' is represented by a closed arc fJ on A' starting and ending at o. By the analytic continuation along the arc fJ, w(u) is transformed into S{3w(u) , where Sfl is a modular transformation of the upper half plane C+. We

*'

( 56)

1284

=

579

ON ANALYTIC SURFACES; II

indicate this by the formula (8.1)

W(u) ~ 8 tlw(u) = a/lw(u) + btl . cllw(u) + d tl

The correspondence: fJ ~ 8/l gives a representation of niLl') (see the formula (8.2) below). In the particular case in which w(u) = Wo is a eonstant, the condition (8.1) may not be sufficient to determine the modular transformation 8/3' Then we choose 8/3 arbitrarily, provided that 8/3wo = Wo and that fJ -+ 8/3 gives a representation of rr,(Ll'). The integral unimodular matrix

is determined by 8/3 uniquely up to a factor ±1. For each modular transformation 8/3 we choose (8/3) in such a way that the correspondence fJ ~ (8/3) forms a representation of the fundamental group rr/Ll'). This is possible in 2p + r -1 different manners, provided that r ~ 1, since rr/Ll') is generated by 2p + r generators fJH ... , f32P' a H "', a r with the single relation:

fJlfJi31W;lfJJ34 ••• fJ;;,l__1f3;}a1a 2 ••• a r

=

1.

The representation: fJ ~ (8/3) defines a locally constant sheaf G' over Ll' whose stalks are isomorphic to Z E9 Z in an obvious manner. We extend G' to the sheaf G = U P G",P UG' over Ll, where G", P = reG' IE;), and we say that the sheaf G belongs to the meromorphic function g(u,). It is clear by Theorem 7.2 that, given an analytic fibre space Vof elliptic curves over Ll free from mUltiple singular fibres, the homological invariant G of V belongs to the functional invariant g(u) of V. DEFINITION 8.1. Given a non-singular algebraic curve Ll, a meromo'rphic function g = g(u) on Ll and a sheaf Gover Ll belonging to g, we denote by gCg, G) the family of aU analytic fibre spaces Vof elliptic curves over Ll free from multiple singular fibres whose functional and homological invariants are g and G. We say that analytic fibre spaces VI and V 2 of elliptic curves over Ll are analyticaUy equivalent if and only if there exists a biholomorphic map J1. ()f Vlonto V2 such that 1' Thus M+m is a non-singular model of N+m and the curve 6>1 corresponds to the singular point .p of N+ m. (,8) N_m- Take m - 1 projective lines Lb k = 1, 2, "', m - 1, with

'*

'*

( 56)

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584

K. KODAIRA

= Uk/V k, tk = sk' =

respective homogeneous coordinates (Uk' Vk) and let Sk Vk/U k' We consider the rational transformation

T: (a,

n

->

(x, y, SI> .

of P into X x L, x

0

0

0

0,

sm-')

= (am, sm, a/sm.-',

000,

s/a m-')

x L m - 1 which is invariant under Z-m and define

0

M_ m

=

T(P).

We infer readily that M_ m is the subvariety of X x L, x defined by the simultaneous equations V;;'Xk - u;;,ym-k = U;V{yk-J - v;u{ = U';-kV;;'-JXk- J - v';-ku'k- J = v~-Ju;-"v{-" -

ut-Jv~-"ur-"

=

ut-Ju;-kVZ-J - vZ-JV;-kUZ-J =

°, °, °, °, °,

000

x L m- 1

l~k

[esa, cv(a)-lt] ,

gS: [a, t]

->

[-a, -t] .

The automorphism g has one fixed point P1 = [0,0]. g2 and gS have respectively two and three fixed points besides P1 which are congruent to P2 = [0, (1/3)r; + 2/3] and Ps = [0, 1/2] with respect to C. In terms of the local coordinates (a, tv), ).I = 1,2,3, defined by (8.16), the automorphisms g, g2 and gS are represented as follows:

( 56)

1295

590

K. KODAIRA

g: (a,

S,)

---t

(esa, ea' s,)

,

g': (a, s,) ---t (ega, ea's,) , ga: (a, Sa)

---t

(-IJ, -Ss) .

Hence the quotient space FIChas three singular points ).\, p" 1Ja corresponding to p" p., ps and having neighborhoods in Fie which are analytically homeomorphic to N_ 6 , N_ a, N_" respectively. We define Bp to be the nonsingular model of F/ Cobtained by a canonical reduction of its singularities. Bp is the union of B:, the proper transform ® of the rational curve Fo/C and eight non-singular rational curves ®" ®., "', ®8 corresponding to the singular points P" p" ps of F/ C. For the first five curves ®" "', ®b "', ®5 which correspond to p, we have six coordinate neighborhoods W" "', Wk , " ' , W6 with respective local coordinates (y" 8,), •. " (t k - l , 8k ), " ' , (x" t 5) such that Wk and Wk+' cover ®k. The neighborhood W, consists of all points (y" 8,), \y,\ < 0, \y~s1\ < 6, W k of all points (t k -" 8 k ), \t~_,8t\ < 0, \ tt=-M-IH' \ < 6, and W6 of all points (x" t 5 ), \ X, \ < 6, \ xitg \ < o. The curve ® is defined in W, by 8 , = 0 and does not meet W" "', W6 • The curves ®" "', ®k, "', ®5 are defined respectively by the equations Y, = 8, = 0, •• " t k-, = Sk+l = 0, "', t4 = X, = O. For the curves ®6 and ®7 which correspond to p" we have three coordinate neighborhoods W,,, W", l¥;s with respective local coordinates (y" 8 6), (t6' 8 7), (X2' t 7) and ®6 is covered by W21 and W", ®7 is covered by W'2 and W'S< The neighborhood W" consists of an points (y., S6), \ y,l < 0, I y~sg \ < 6, W" of all points (t6' 8 7), I t~s71 < 0, I t~s~ I < 6, and W'S of all points (X2' t7), I x~ I < IS, I xit~ I < 0. The curve (8} is defined in w" by 8 6 = 0 and does not meet W" and W2S • The curves ®6 and ®7 are defined respectively by the equations Y2 = 8 7= and t6 = X2 = o. The curve ®s corresponding to Ps is covered by two coordinate neighborhoods Ws, and Ws, \lith respective local coordinates (Ys,8a) and (xs , t s), where WS1 consists of all points (Ys, 8 a), I Y3 \ < 0, \ y~8: \ < 6, and WS2 of all points (xs, t a), I x~ I < 6, I xat~ ( < o. The curve ® is defined in TVa, by 8a = and does not meet W3,. Consequently ® is non-singular. The curve ®a is defined by the equations Xs = Ys = o. By (8.11) we have

°

°

= x, = Yi8~ =-- t;8g = t~s~ = tis! = t48~ , Sf = Y, = xft~ = tis, = t;s; = tis! = t1si ,

as

(8.21)

as = s~

a

2

x~

= y;s~ = t6S~ ,

= y, = x~t~ = t~87

= Xs = Yss;

,

,

s; = Ys = xst~ . Moreover we may use (IJ, n as local coordinates at any point on Be -

( 56)

1296

U®k'

591

ON ANALYTIC SURFACES; II

Hence T = a 6 is a holomorphic function on Bp and therefore the canonical 6 -> T = a of B; onto E; can be extended to a holoprojection 'l': «a, morphic map of Bp onto Ep. Thus Bp is an analytic fibre space of elliptic curves over Ep which is an extension of B;. We infer from (8.21) that the fibre of Bp over a p is

m

(8.22)

Cap = 6®

+ 5®1 + 4®2 + 3®a + 2®4 + ®5 + 4®6 + 2®7 + 3®8 .

Since (C ap ®) = (Cap®k) = 0, we obtain

=

(®2)

=-

(®D

2.

Thus we see that Cap is a singular fibre of type II*. (23 ) The case: (Ap) = -~). The order of (Ap) is 3. Hence we 3 set 7: = a • We have

(-i

"'(a) =

W'

r;1-

lt

h -= 1(3)

r/aI t '

-a

.

The cyclic group C of analytic automorphisms of F of order 3 is generated by

g: [a, S] ---> rep, co(a)-lS] . The automorphism

g

has three fixed points

Po = [0, 0], P1 = [0,

~ r; + ~

1

Pa = [0,

~ r; +

+J.

Let

(8.23)

So

= (1 - a")s ,

Sl

= (1 - a")

S2

= (1 -

(s - +co(a) - :), an) (t - : coCa) - +) .

Then the pairs (a, sv), )) = 0,1,2, form local coordinates with respective centers Pv and g: (a, tv)

---->

s

(e 3 a, e 1tv) .

Hence the quotient space FIC has three singular points p" v = 0,1,2, corresponding to Pv and each singular point Pv has a neighborhood in FIC which is analytically homeomorphic to N_ a• We define Bp to be the nonsingular model of Fie obtained by a canonical reduction of its singularities. Bp is the union of B;, the proper transform ® of the rational curve FolC

( 56)

1297

592

K. KODAIRA

and six non-singular rational curves 0 vk1 ).J = 0, 1, 2, k = 1, 2, corresponding to ).\, ).J = 0, 1, 2. For each pair of curves ®VH ®vz corresponding to ).1" we have three coordinate neighborhoods WVI ' Wvz , WV3 with respective local coordinates (Yv, 8V1 ), (tv1' 8V2 ), (x", t v2 ) such that W,k and W,k+l cover ®Vk for k = 1,2. The curve

[e.a, -w(a)-It] •

The automorphism g has two fixed points Po = [0,0] and PI = [0, (1/2)i + (1/2)]. g' has two fixed points besides Po and PI which are congruent to Ps = [0,1/2]. We set

So (8.29)

= (1 - a 2h )s ,

SI = (1 -

a

2h

)(s - +w(a) - +) ,

Clearly the pairs (a, sv), ).I = 0,1,2, form local coordinates on F with respective centers Pv and, in terms of these local coordinates, g and g' are represented in the form g: (a, sv)

-->

(e.a, e.sv)

g2: (a, Sa) -> (-a,

,

-s.) .

( 56)

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J..I

= 0,1,

595

ON ANALYTIC SURFACES; II

Hence the quotient space FIC has three singular points +\, 1) = 0, 1, 2, corresponding to p,. Moreover +'0 and.p1 have neighborhoods in FIC which are analytically homeomorphic to NH and +'2 has a neighborhood which is analytically homeomorphic to N+2' Let M denote the non-singular model of FIC obtained by a canonical reduction of its singularities. M is the union of B:, the proper transform ® of the rational curve FolC and three rational curves ®" 1) = 0, 1, 2, corresponding to +". Each curve ®, is covered by two coordinate neighborhoods W. 1 and W'2 with respective local coordinates (y" s.) and (Xv. tv). The rational curve ® is defined in W. 1 by S, = and does not meet W'2' Hence ® is non-singular. The curve ®, is defined by the equations x, = y, = 0. We have

°

(8.30)

a4

yvs; ,

= Xv =

{a 2 =

X 2 = Y2S; ,

tt = t; =

Y. = xvtt ,

for

1)

= 0,1 ,

Y2 = X2t; •

Thus M is an analytic fibre space of elliptic curves over Ep which is an extension of and the fibre of Mover a p is

B:

Co

= 4® + ®o + ®1 + 2®2 .

We infer in the same manner as in (21) above that there exist a nonsingular surface Mil and a holomorphic map p of M onto Mil which maps ® and ®2 onto a point q" and M - ® - ®2 biholomorphically onto Mil - q". The curves ®~ = p(®.), 1) = 0, 1, on Mil are non-singular, rational and ®~' . ®~'

= 2q" ,

We define (8.31)

Clearly Bp is an analytic fibre space of elliptic curves over Ep which is an €xtension of B: and the fibre of Bp over a p is (8.32) Thus Cap is a singular fibre of type III. The map p maps W.2> 1) = 0, 1, biholomorphically onto W:' = p( WV2 ) ' Hence we may consider (x" tv) as a local coordinate on the neighborhood W:' in Bp. We note that W:' consists of all points (Xv. tv), I x. I < e, I xJ~ I < 0, and that ®;' - q" is the subset of W:' consisting of all points (0, tv), t. E C. (3 2) The case: (Ap) = r4. We have

(~

-5).

w(a) = i

The order of (Ap) is 4. We set a =

+ ia

1_ a

( 56) 1301

2h 2h

596

K. KODAIRA

The cyclic group C is generated by g: [a,

s] ---> [e 4a, w(a)-'s] .

g has two fixed points Po = [0,0] and P, = [0, (1/2)i + (1/2)]. g2 has two fixed points besides Po and p, which are congruent to P2 = [0,1/2] with respect to C. In terms of the local coordinates (a, q, )..I = 0,1,2, defined by (8.29), the automorphisms g and g2 are represented in the form

v = 0, 1,

g: (a, sv) --> (e 4a, e;'sv) , g2:

(a, S2)--> (-a, -S2)'

Hence the quotient space FjChas three singular points Po, P" P2 corresponding to Po, P" P2 and having neighborhoods in FIC which are analytically homeomorphic to N-4' N-4' N-2' respectively. We define Bp to be the nonsingular model of FI C obtained by a canonical reduction of its singularities. Bp is the union of B;, the proper transform ® of the rational curve FolC and seven non-singular rational curves ®Ob ®'k' k = 1,2,3, and ®2 which correspond to Po, p, and P2' For the curves ®Vk, k = 1, 2, 3, we have coordinate neighborhoods Wv" WV2 ' Wva , WV4 with respective local coordinates (Yv, 8 v,), (tv" 8 V2 ), (tv2' 8 V3 ), (xv> t v3 ) such that W'k and W'k+1 cover ®Vk' The curve ® is defined in W" by the equation 8" = and does not meet W'k' k ~ 2. The curves ®'H ®'2> ®va are defined respectively by the equations y, = 8'2 = 0, tv, = 8'3 = 0, Xv = tV2 = O. The curve ®2 is covered by two coordinate neighborhoods W2" W22 with respective local coordinates (Y2' 8 2), (x 2 , t 2 ). The curve ® is defined in W 21 by the equation 8 2 = 0 and does not meet W22 . Hence ® is non-singular. The curve ®2 is defined by the equations X 2 = Y2 = Q. We have

°

(8.33)

=

0'4

= Xv = y~s~, = t~,S~2

s~

==

y""

==

x;t;a := t;l S )/2 == t~28;3 ,

0'2

=

X2

=

Y28 ; ,

1s; = Y2 =

t v2 8;3

,

for for

)..I

=

)..I

=

0, 1 , 0,1 ,

X2t; •

Bp is an analytic fibre space of elliptic curves over Ep which is an extension of B;. The fibre of Bp over a p is (8.34)

where (®2) =

(®~k) =

(®D = -2 .

Thus Ca p is a singular fibre of type III*. (v) Now we consider the case in which the order of (Ap) is infinite. We have, by (7.7),

( 56)

1302

597

ON ANALYTIC SURFACES; II

w(l) = bl

+ f( r)

,

where f(r) is a holomorphic function of -r, I r I < e. Hence, by an appropriate choice of the local uniformization variable -r, we may assume that w(l) = bl .

(8.35)

We have

±(~ ~).

(Ap) = (1) The case: (Ap) =

(6 f).

We have

B; = Up >< Cjgp , and, in this case, the discontinuous group gp consists of analytic automorphisms where k, n n 2 are arbitrary integers. Let 'J2 denote the normal subgroup of gp consisting of g(k, 0, n 2), k, n 2 = 0, ±1, ±2, .... Then, denoting by C* the multiplicative group of complex numbers, we have j ,

Up >< C/'J2

=

E;

x C* ,

where the canonical projection of Up >< C onto (l, t)

->

E;

>< C* is given by

(r, w) ,

Hence, considering the factor group fl, = gp/'J2 as a group of analytic automorphisms of E; >< C* in an obvious manner, we obtain

B;

=

E;

>< C*{fl, .

It is clear that Z is the infinite cyclic group generated by the auto-

morphism (8.36)

E; x C*. We insert here a general consideration. Let M denote a complex manifold and let S be a submanifold of M. Moreover let fD be a properly discontinuous group of analytic automorphisms without fixed point of M' = M - S satisfying the condition that, for each point Z E S, there exist open subsets 0 ::J Sand N:7 Z of M such that g(N n M') n 0 is empty for any element g 1 of fD. Then there exist a complex manifold Wand a holomorphic map 'tit of M onto W which is locally biholomorphic such that

of

*'

( 56 J 1303

598

K. KODAIRA

Ui(Zl) = Ui(Z2) if and only if Zl and Z2 belong to M' and Z2 = gZl for an element g of [f). Obviously the image Ui(S) of S is an analytic submanifold of W, Ui maps S biholomorphically onto Ui(S) and W - Ui(S) coincides with the quotient space M'I[f). Hence, identifying Ui(S) with S, we may consider Was a union of M'j[f) and S. We denote Wby the symbol MIg). Applying the above consideration to M = Ep x C*, S = ap x C* and g) = C, we define

(8.37)

W= Ep x C*jC= B; UC*,

and, for any point (r, w) E Ep x C*, we denote the corresponding point Ui(r, w) on W by the symbol «r, w». It is clear that : «r, w»

--->

r

is a holomorphic map of Wonto Ep which is an extension of the canonical projection 'l" of B: onto E:. By a compactification of W relative to Ep we shall mean an analytic surface W containing W as its open subset together with a proper holomorphic map ci> of W onto Ep which is an extension of . We shall define subsequently the extension Bp of B: to be a compactification Wof W relative to Ep. (11) First we consider the case in which b = 1. We define (8.38)

(8.39)

-21::=1 (1 - rn)-2rn + 1::=-= (1 y(r, w) = 1::=_= (1 - wr n)-3(1 + wrn)wrn .

x(r, w) =

wrn)-2wrn .

We remark that x(r, w) = - -

1 1 - -if-l(n , 12 4n2

~if-l'(S) a

y(r, w) =

8n

,

where

We have y(r, wr) = y(r, w) .

x(r, wr) = x(r, w) ,

Hence we may consider x(r, w) and y(r, w) as merom orphic functions of «r, w» defined on W = Ep x C*jC. The functions x(r, w) and Y(T, w) satisfy the equation y2 _ 4x 3 - X2

+ gir)x + ga(r) =

where

( 56)

1304

0 ,

599

ON ANALYTIC SURFACES; II

glr) = 20 E;=l (1 - 'tn)-lnS'tn , gl't) = ~ ~""'\~~ (1 - 'tn)~1(7n5 3 L..Jn~l

+ 5nS)'tn .

Denoting by P z a projective plane with non-homogeneous coordinates (x, y), we define Bp to be the surface in Ep x Pz determined by the

equation (8.40)

The surface Bp is non-singular and the canonical projection: Ep X Pz ---> Ep induces a holomorphic map 'I' of Bp onto Ep. Moreover the inverse image 'I'~l(ap) of a p is the plane curve defined by the equation yz _ 4 xs

-

X2 =

0

which has an ordinary double point at q: x = y = O. We infer readily that J.1:

«'t, w»

---+

't X

(x('t, w), Y('t, w»)

is a biholomorphic map of Wonto Bp - q. We have (8.41)

J.1«0, w» = 0 x (w(l - w)~z, w(l

+ w)(l -

w)-S) .

This shows that J.1 maps C* c W biholomotphically onto 'I'-l(ap) - q. Obviously the composite map 'l'p, coincides with . Hence, identifying W with p,( W) = Bp - q, we may consider Bp as a compactification of W relative to Ep. Thus Bp is an analytic fibre space of elliptic curves over Ep which is an extension of B;. The fibre Gap = 'I'-l(ap) of Bp over a pis a rational curve with one ordinary double point. Thus Gap is a singular fibre of type 11 , (1 2) Now we consider the general case in which b is an arbitrary positive integer. We form

W = Ep x C*/C = W' U C* , We then take b copies

Wk

=

W' = E; x C*/C .

W~ U C: ,

k = 0, 1, 2, "', b - 1 ,

of W = W' U C* and identify «'t, w»j E W; with «'t, w'tk-J)h E the union

W~.

Then

(8.42)

forms a non-singular analytic surface containing subset and (8.43)

where

qp -

B: =

Cd u c,* u ...

C: n C: is empty for J *- k.

( 56)

We note that

1305

B;

W;

=

U C:- 1

,

as an open

600

K. KODAIRA

CU, = Eo

X

C*/C' ,

where C' is the infinite cyclic group generated by the automorphism (r, w) -+ (r, wr) of Ep x C*. We denote by «r, w))' the point on q]' corresponding to (r, w). In what follows we consider the index k of WkJ Ct, ... as an element of the cyclic group Zb = Z/(b) of order b. Thus, for instance, «r, W))-l will denote (Cr, w»b-l. It is clear that CfP is a b-fold un ramified covering of CU,. In fact the covering map U1 of CUb onto CU' is defined on each open subset W k of CUb b~ the formula (8.44)

U1:

«r, w)h

-+

«r, w))'

and the covering transformation group of qp over CU' is the cyclic group of order b generated by the analytic automorphism

For

r * 0,

h:

the point «r, h:

«r, w»H1 «r, W»k . -+

W»k is identical with «r, wr»k+,'

«r, w»o «r, wr»o ,

Hence we have on

-->

B;.

Let Bp' denote the compactification of CU' relative to Ep defined by the equation (8.40). B; is the union of V' and a point q, and q has a neighborhood N q in B; such that N q - q is simply connected. Hence the b-fold unramified covering CfP of Cf]' can be extended uniquely to a b-fold unramified covering Bp of Bp'. Clearly Bp is the union of CUb and b points qo, q" "', qb-I which cover the point q. The holomorphic map «r, W»k -+ T of CUb onto Ep can be extended to a holomorphic map 'l' of Bp onto Ep which is an extension of the canonical projection of B; onto E;. Thus Bp is an analytic fibre space of elliptic curves over Ep which is an extension of B;. Let ®k denote the closure of Ck* on Bp. We infer readily that each ®k is a non-singular rational curve on Bp and that

provided that the points qk are numbered properly. Moreover the point Wk = on ®k and W k- 1 = 00 on ®k-" where W k denotes the inhomogeneous coordinate on the projective line ®k = Ck U 00. Clearly the fibre of Bp over a p is qk has the coordinate

(8.45)

°

Cap = ®o

+ ®, + ... + ®b

Thus Cap is a singular fibre of type lb. (2) The case: (Ap) = (-~ We have

=f).

B: = Up x Clgp ,

( 56)

1306

.

601

ON ANALYTIC SURFACES; II

where gp is the group consisting of analytic automorphisms k, n H n 2 E

Z,

of Up x C. Let J: denote the subgroup of gp consisting of g(k, nlf n 2), k == 0(2), nlf n 2 E Z, and let

w' =

Up x CjJ: .

= a 2, a = er.'!. Moreover let D denote the circular disk: 1a 12 < $

We set 'r and let D' = D - O. Obviously we have

W' = D' x C*jC,

where C is the infinite cyclic group of analytic automorphisms of D' x C* generated by g:

(a, w) ----> (a, wa 2b )

•

We construct a compactification F of W' relative to D in the manner described in (1) above. Namely we take 2b copies Wk = W£ u Ct, k = 0, 1, "', 2b - 1, of

W = W' U C* = D x C*/C , and, identifying «a, w»j Eo WI with «a, wa"-J)h E

W~,

we form the union

(S.46)

Then we define F to be the union (S.47)

F = qpb U qo U ql U .•• U q2b-l

C(j2b and 2b points qk' k = 0, 1, "', 2b - 1. We note that ®k = C: U qlt U qk+l is a non-singular rational curve on F and that qk and qk+l have the coordinates W k = 0 and W k = 00 on ®k' respectively.

of

We have (8.48)

B; = W;/{t'} ,

where {t'} denotes the cyclic group of order 2 generated by the analytic automorphism

t': «a, w»o -> « -a, w-1»o of W;. The automorphism t' can be extended uniquely to an analytic automorphism t of F. In fact, since

«a, W»k = «a, wa-k»o ,

for a

=1= 0 ,

t' can be represented in the form t': «a, W»k -> «-a, (-1)kW- 1»2b-k •

( 56)

1307

for a*,O •

602

K. KODAIRA

Hence an extension t' of t' to

q}2b

t#: «a,

W»k

-->

is given by

« -a, (-1)k W -I»2b_k

•

Obviously t# maps Wk E C: onto (_I)kw;1 E C';-k' Hence t~ can be extended to an analytic automorphism t of F which maps ®k onto ®2b-k and qk onto q2b+1-k' The automorphism t has four fixed points on F. They are POV = (-1),,).) = 0, 1, on C~ and PI> = (-l»ib,).) = 0, 1, on Ct. We set (8.49)

t >..>

=

~ log (-I»i- Ab w ,

for

27n

11" ).)

= 0, 1 ,

and consider s>..> as a holomorphic function of «a, W»);...b' Then we obtain a local coordinate (a, t>..v) on F with the center P>..v' In terms of the local coordinate (a,X>..» the automorphism t is represented in the form t: (a, ~>..V) ~ (-a,

-s>..»

•

Hence the quotient space FI{t} has four singular points P>..> corresponding to P>..> and having neighborhoods which are analytically homeomorphic to N+ 2 •

We define Bp to be the non-singular model of F/{t} obtained by a canonical reduction of its singularities. For brevity we write Ek in place of ®k and let ®k denote the proper transform of 8 k on Bp. Obviously the curve ®k coincides with ®2b-k' The surface Bp is the union of B;, the nonsingular rational curves ®O, ®11 "', ®b and four non-singular rational curves ®),.v corresponding to the singular points Pl.> of FI{t}. Each curve ®AV is covered by two coordinate neighborhoods WAVI and WAV2 with respective local coordinates (Y>..vr S>..V) and (x>..vr t>..», t>..> = sl:;, where W>"VI consists of all points (Y>..>r s>..», IY>..v I < 0, I Y>..vs~> I < $, and W AV2 of all points (x Avr t>..V) , I XAv I < $, I XAvt~v I < o. The curves ®o and ®b are defined in WO>I and WlVl by the equations SOV = 0 and Sn = 0, respectively. The curve ®AV is defined by YAv = XAv = O. Hence the curves ®o> and ®1> intersect ®o and ®b transversally at (-l)V and (-l)"ib, respectively. The curves ®k-l and ®k have one simple intersection point which corresponds to qk = 8 k - l • 8 k• We have

{r = a

(8.50)

2

sL =

= XAv = YAv8~> ,

YAV = XAvt~v •

Consequently Bp is an analytic fibre space of elliptic curves over Ep which is an extension of B; over E; and the fibre of Bp over a p is (8.51)

Cap = ®oo

+ ®01 + ®1O + ®u + 2®0 + 2®1 + ... + 2®b •

Thus Cap is a singular fibre of type It.

( 56

J

1308

ON ANALYTIC SURFACES; II

603

This completes the construction of the analytic fibre space B of elliptic curves over Jl. It is clear that B belongs to the family 9'(g, G). We call B the basic member of 91 (g, G). 9. Sheaves of structure groups

In what follows we shall mean by an analytic group a complex Lie group. We shall denote by Z", C and C* respectively the cyclic group of order n, the additive and the multiplicative groups of complex numbers. DEFINITION 9.1. Let @ and M be complex manifolds and let 'l' be a holomorphic map of @ onto M such that, at each point of @, the rank of the jacobian of'l' is equa~ to the dimension of M and such that the inverse image @u = '¥--'(u) of each point u E M is an ana~ytic group whose complex structure is that of the complex submaniJold CS:I u of @. Let ~ be the complex submanifold of @ x @ consisting of all points (g" g2) E @ x @ satisfying 'l'(g,) = 'l'(g2)' We call @ an analytic fibre system of groups over M if and only if the map: (g" g2) ---> g,g~' is a holomorphic map of SD onto @. Let B be the analytic fibre space of elliptic curves over Jl constructed in the preceding section of which the canonical projection is denoted by 'l'. Consider the open subset

B' = W-'(Jl') = U x Clf}

of B. It is clear that the formula: (9.1)

defines on B' a structure of analytic fibre system of abelian groups over Jl'. We define B~ to be the open subset of B consisting of all points z satisfying

I8ra ('l'(z»)/8z, I + I8ra ('l'(z»)/8z2 > 0 1

(compare (6.1». We write the fibre of B over ap in the form

and let

where U"ps=' indicates the union extended over all s for which nps = 1. Obviously we have

B# THEOREM

= Up C!p u B' .

9.1. There exists on B# a unique st1'ucture of analytic fibre

( 56)

1309

604

K. KODAIRA

system of abelian groups over ~ which is an extension of the structure on B' of analytic fibre system of abelian groups over ~' defined by the formula (9.1). The analytic group structures of the fibres Cfp of B# are shown in Table 1. TABLE Normal form of

C~p

I

(Ap)

C# Gp

®#

regular

1

torus

torus

1*0

(-1o -10)

C X Z2 X Z2

C

C* X Zb

C*

Type of

h

-

(~

p

regular

~)

pole of order

1*b

(-1o -1 -b)

C X Z2 X Z2 or C X Z.

C

II

C~ ~)

C

C

II*

(~

C

C

III

C~ ~)

C X Z2

C

IIl*

(~

-~)

C X Z2

C

IV

(-~ -~)

C X Za

C

IV*

(-~ -~)

C X Za

C

PROOF.

Behavior of g(u) at a p

-~)

b

0

g(a p)

=

g(a p )

=1

g(a p )

=

0

We employ the notation of the preceding section. Let

B#p = B#

n B = C# p

ap

U B'p

,

B;

has a structure of analytic fibre system of abelian groups over E; which is the restriction to B: of that of B'. Consider an analytic group structure @ on Cf. We shall say that @ is an analytic continuation of p the analytic group structures of the fibres of B: if the union Cfp U B; of Cfp with the analytic group structure @ and B; forms an analytic fibre system of abelian groups over Ep. The uniqueness of the analytic continuation of the analytic group structures follows from the uniqueness of analytic continuation of holomorphic functions. For our purpose it suffices therefore to define an analytic group structure on Cfp which is an analytic continuation of the analytic group structures of the fibres of B;.

( 56)

1310

605

ON ANALYTIC SURF ACES; II

(Ib) In case Cap is of type I b, we have, by (8.42) and (8.43),

B; = LUb =

Wo U WI U ••• U WH

and

C!p =

C~ U C: U ••• U C:- I

•

We define the analytic group structure on C!p by the formula: «0, w)h - «0, v»J

(9.2)

= «0, wv-l)h_j .

Note that the indices J, k are elements of Zb' The analytic group structures of the fibres of B; = W~ are defined by

«T, w»o - «T, v»o = «T, wv-l»o .

(9.3) Since

«T, w)h = «T, wrk»o ,

for

T

* °,

the formula (9.3) is equivalent to

«T, w)h - ({T, v»j = «T, wv-l)h_j .

(9.4)

Hence the analytic group structure of C!p defined by (9.2) is an analytic continuation of that of the fibres of B;. It is clear that C!p ~ C* x Zb' (II) In case Cap is of type II, we have, by (8.19) and (8.20), B; = M'" - q'" , C!p = @'" - q'" . Moreover there is a coordinate neighborhood Will C M'II consisting of all points (Xl' t l ), I Xl I < c, I xltf I < 0, and C!p is the subset of W'" consisting ef all points (0, t l ), tl E C. We identify C!p with the space C by means of the biholomorphic map: (0, t l ) -> tl of C!p onto C and define the analytic group structure of C!p to be that of C. From (8.16), (8.17) and (8.18) we obtain (9.5)

r; = (1 - a 2h )-latl {T = a = Xl •

,

6

*

«a,

Thus, for Xl 0, (Xli t,) represents the point 1;) in B; whose local coordinates a and r; are determined by (9.5). It follows that the analytic group structure of Ca#p is an analytic continuation of the analytic group structures of the fibres of B;. (III) In case Cap is of type III, C!p is composed of two curves @~' - q", ).I = 0, 1 (see (8.32». Each curve @~ q" is covered by the coordinate neighborhood W~' consisting of all points (Xv. t,), \ x, \ < c, \ x,tt \ < 0, and @~' - q" consists of all points (0, t,), t, E C. We identify C!p with C x Z2 by means of the biholomorphic map: (0, tv) --> t, x ).I, )..I = 0, 1, of C!p onto

( 56)

1311

606

K. KODAIRA

C X Z2 and define the analytic group structure of C!p to be that of C x Z2' From (8.29) and (8.30) we get (9.6)

i t

= (1 - a2h)-lato = (1 - a 2h )-lat,

+ ~(lJ(a) + ~ , 2

T =

a 4 = Xo

=

2

X, •

Hence, denoting by (x, t), the point (x" t,) in W:' with x, = x, t, = t, we obtain

t 'at + ~ w(a) + ~)),

a4 = x ,

For any pair of points (x, t'h and (x, t)" x

* 0, we

(x, t), = ((a, (1 - a

provided that x have therefore (x,

* 0.

t'h -

2h

(x, t), = (x,

t' - t)". ,

f1-

:=0

A, -

}.i

(mod 2) .

Hence we infer that the analytic group structure of Cfp is an analytic continuation of that of the fibres of (IV) In case Cap is of type IV, we are concerned with a similar situation q", }.i = 0,1,2, as in (III) above. Cfp is composed of three curves and each curve q" consists of all points (0, t,), t, E C (see (8.28». We identify C!P with C x Z3 by means of the biholomorphic map: (0, t,)---> t, x }.i, }.i = 0, 1, 2, of C!p onto C x Z3 and define the analytic group structure of Ca~p to be that of C x Z3' From (8.23) and (8.26) we get

B:.

®:: -

®:: -

t = (1 -

ah)-lato

=

(1 - ah)-'at,

+ ~w(a) + 2 3

3

(9.7)

Therefore, writing (x, t), in place of (x" t,), we obtain (x, t),

= ( (a, (1

- a h)-lat

+ ~

w(a)

+

2:)),

a3 = x .

Hence we infer that the analytic group structure of Cfp is an analytic continuation of that of the fibres of B:. (In In case Cap is of type I:, we have, by (8.14) and (8.51), Each curve ®L is covered by the coordinate neighborhood WA'2 consisting of all points (XA" t A,), I XA, I < e, I XA,t~, I < 0, and ®!, consists of all points (0, t A,), t A, E C. We identify C!p with C x Z2 X Z2 or C x Z4 by means of

( 56)

1312

607

ON ANALYTIC SURFACES; II

the biholomorphic map: (0, h,) ----+ t/,.v x A, x J.i or hv x (A, + 2J.i) and define the analytic group structure of C!p to be that of C x Z2 X Z. or C x Z. according as b is even or odd. In case b = 0, we have, by (8.12), (8.13) and (8.15),

I

A,

J.i

2

2

S = ahv + -w(a) + - ,

(9.8)

'Z'

~

In case b

= a' =

x/,.v •

1, we have, by (8.48),

B;

= W~I{t'} .

Thus each point on B; corresponds to a pair of points «a, «-a, w-'»o on W~. Moreover, by (8.49) and (8.50), we have w =

(9.9)

{

s

'Z'

=

(_1)Vi/"ba/,.be2~iO"tAv

a'

=

w»o

and

,

X/,.V

Setting = (1/2ni) log w, we denote by «a, S» the point on B; correand w-l»o on W;. Obviously (9.9) is equivalent sponding to to (9.10)

«a, w»o

s=

j

'Z'

atAv

«-a,

+ ~w(a) + A,(~4 - [~J) + ~2 , 2 4

= a 2 = XAV

w(a) =

~ log a nt

,

,

where [b/4J denotes the maximum integer which is not greater that b/4 From (9.8) and (9.10) we obtain

'*

provided that X AV 0. Hence we conclude that the analytic group structure of C!p is an analytic continuation of that of B;. (II*) In case Cap is of type II*, we have, by (8.22),

The curve 8~ is covered by the coordinate neighborhood Wr6 consisting of all points (x" t~), I x,I < $, I xftg I < 0, and consists of all points (0, to), to E C. We identify with C by means of the biholomorphic map: (0, t o)to of onto C and define the analytic group structure of to be that of C. From (8.16) and (8.18) we obtain

8:

(9.11)

8:

8:

8:

s = (1 - a {t' = a = 6

( 56)

2h

)-'a5t. ,

XI •

1313

608

K. KODAIRA

It follows that (J6 =

Xl ,

'*

provided that Xl O. Hence the analytic group structure of C!p is an analytic continuation of that of the fibres of B;. (UP) In case Gap is of type UP, we have, by (8.34), Cap#

--

~# ~03

+ ~#

\:?JI3 ,

Each curve ®~3 is covered by the coordinate neighborhood W N consisting of all points (x" t"3)' I x" I < c, I x~tt31 < 0, and ®~3 consists of all points (0, t"3)' tva E C. We define the analytic group structure of C!p by means of the biholomorphic map: (0, t v3 ) -> t"3 x J) of C!p onto C x Z2' From (8.29) and (8.33) we obtain (9.12)

I t;

= (1 - (J2h)-1(J 3t03 = (1 - (J2htl(J3t13

T

= (J4 =

+ ~w«J) + ~, 2

Xo

=

2

Xl •

This shows that (x"' t"3) = (((J, (1 - (J2h)-la 3tv3

+ ~ w«J) + ~)),

'*

provided that Xv O. Hence we infer that the analytic group structure of Cfp is an analytic continuation of that of the fibres of B:. (IV*) In case Cfp is of type IV*, we are concerned with a similar situation as in (III*) above. We have, by (8.25),

Each curve ®t2 consists of all points (0, t v2 ), t"2 e C. We define the analytic group structure of C!. by means of the biholomorphic map: (0, t"2)-> tV2 x !.I of C!p onto C x Z3' From (8.23) and (8.24) we obtain t; = (1 - (Jk)-1(J2t02 = (1 - a h)-la 2t12

+ ~w«J) + 2 3

3

(9.13)

Hence we conclude that the analytic group structure of Clp is an analytic continuation of that of the fibres of B;. This completes the proof of Theorem 9.1. Let E denote an open subset of~. By a holomorphic section of B# over E we shall mean a holomorphic map rp: u -> rp(u) of E into B# such that

( 56)

1314

ON ANALYTIC SURFACES; II

609

'IT(t~

=Xm-ktk,

t m-,

---->

t'm,

=

for k

t m-, + cp(O'm) •

( 56)

1316

=

1, 2, ... , m - 2 ,

611

ON ANALYTIC SURFACES; II

We have

x=

1

= 1

+ y".-2Sf'-lcp(y".-18f') + t;:~lk-18;:-kcp(t;:~lkS;:-k+I)

.

Therefore the function X is holomorphic on the neighborhoods W10 W2 , •• " W".-l and is equal to 1 on the curves @10 @2' •• " @".-2' It follows from (9.14) that, on the neighborhoods Wu "', W k , " ' , W".-u the auto~ morphism L* is represented in the form (y,

81)

->

(y',

8D

(x, t m - I )

--->

(x',

t:._

= (X"'y,

I)

t-

= (x, t m - 1

m

81 )

,

+ cp(x»)

.

Hence we conclude that L* can be extended to an analytic automorphism L of M_".. Note that L leaves invariant all points on the curves @u ®2' " ' , @".-2'

(iv) Now we consider the case in which Gap is of type II*. The fibre space Bp is the non-singular model of FIC obtained by a canonical re~ duction of its singularities. Let cp denote a holomorphic section of over Ep. In terms of the local coordinate (xu t 6) we write

B:

cp: T -> (XI' t 5) = (T, cp(T») ,

where cp('r) is a holomorphic function of have V(cp): «a, !;)

---->

«a,

t + (1 -

7:,17: I < e. Then, by (9.11), we

a 2h )-Wcp(a 6») ,

for a

=1=

0.

Hence V(cp) induces an analytic automorphism L F : [a, t]

->

[a,

t

+ (1 -

a 2h )-la 5cp(a 6 )]

of F which commutes with the elements of C. In terms of the local coordi~ nates (a, tv) on F defined by (8.16), the automorphism LF is represented in the form L F : (a, tv)

--->

(a,

tv + a 5cp(a B})

•

Therefore, by the above lemma, LF induces an analytic automorphism L(cp} of the non-singular model Bp of FIC which is obviously an extension of V(cp). We remark that L(cp) leaves invariant all points on the curves ®, ®u @2' @3' ®4' ®6' @7 and ®8 (compare (8.22}). (v) The case in which Cap is of type III*. The fibre space Bp is the non-singular model of FIC obtained by a canonical reduction of its singu~

(56)

1317

612

K. KODAIRA

larities. Consider the analytic automorphism T F : [a,

n

->

[a,

s + 1- w(a) + 1-]

of F of order 2. It is clear that TF commutes with the generator of C and that TF maps the fixed points Po and p, of g onto p, and Po, respectively, and the fixed point P2 of g2 onto gP2' Moreover, in terms of the local coordinates defined by (8.29), TF maps a point (a, so) in a neighborhood of Po onto the point (a, s,), S, = so, in a neighborhood of p, and g-' TF maps a point (a, S2) in a neighborhood of P2 onto (a', s~) = (-ia, it2)' Since it follows that the map g-1TF induces the automorphisms (Y2'

(-Y2'

8 2 ) ->

- 82 )

and (X2' t 2) ----> (-X., -t2)

of the neighborhoods W21 and W22 in Bp (see (8.33». Thus we infer that TF induces an analytic automorphism T of the non-singular model Bp of FICand that T maps the curves ®, ®Ok' ®'b ®2 biholomorphically onto ®, @1k' ®Ok' ®2 respectively (see (8.34». Consider a holomorphic section rp of B:. The fibre G!p of Bp# over a p is composed of two components ®~., J.i = 0, 1. Suppose that rp meets ®~2 and represent rp in the form rp: r

---->

(xv> t v3 ) = (r, rp(r») ,

where rp(r) is a holomorphic function of r, I r have

I < c. Then, by

(9.12), we

for a*-O . Hence we infer in the same manner as in (iv) above that T-V£#(rp) can be extended to an analytic automorphism of the non-singular model Bp of FIC. It follows that V(rp) can be extended to an analytic automorphism L(rp) of Bp. (vi) In case Gap is of type IV* or It, we obtain an extension L(rp) of V(rp) in the same manner as in (v) above. (vii) In case Gap is of type It, b ~ 1, the fibre space Bp is the nonsingular model of FI{t} obtained by a canonical reduction of singularities, where F is the compactification of CU 2! relative to D (see (8.46) and (8.47». For each pair (;\" J.i) of integers ;\', J.i = 0, 1, we define an analytic automorphism T>-'VF of F to be the extension of the analytic automorphism T:vF:

«a, w)h «a, (-l)Vi>-.b w )hHb ---->

of QJ2b. The existence and the uniqueness of the extension T>-'VF of TfvF

( 56)

1318

613

ON ANALYTIC SURFACES; II

are obvious. T;"VF commutes with t and maps the fixed point Poo of tonto P;"V' Moreover, in terms of the local coordinates (0", t;..v) defined by (8.49), T;"VF maps a point (0", too) in a neighborhood of Poo onto the point (0", tAV)' t;..v = too, in a neighborhood of P;"V' Hence T;"VF induces an analytic automorphism T;..v of the non-singular model Bp of F/{t} which maps the curve ®oo biholomorphically onto ®;..v. Consider a holomorphic section ep of B: over Ep. Suppose that ep meets ®L, and represent ep in the form ep: r

---->

(XAV1 t AV ) =

(r, ep(r») ,

where ep(r) is a holomorphic function of r, I r we obtain T;;l V( ep): «0",

t»

«0",

I < c. Then, using (9.10),

t + O"ep(0"2»)

,

*

for 0" 0 . Hence we infer in the same manner as in (iv) and (v) above that Veep) can be extended to an analytic automorphism L(ep) of B p , q.e.d. --7

In view of the above theorem we may consider the sheaf n(B~) of germs of holomorphic sections of B~ as a sheaf of structure groups6 acting on the fibre space B. We consider the first cohomology group H1(A, n(B~» of A with coefficients in the sheaf n(B~). Given an element TJ of Hl(A, n(B#», we choose a finite covering {EJ } of A by sufficiently small circular disks E J and let {TJJk}, TJjk E r(B~ I E J n E k), be a 1-cocycle on the nerve of the covering {E;} which represents the cohomology class TJ. DEFINITION 9.2. We define B~ to be the fibre space of elliptic curves over A obtained from the collection {B I E j } of the pieces B I E j of the fibre space B by identifying z; E B I E J with Zk E B I Ek if and only if Zj = L(TJjk)Zk' It is clear that the fibre space B~ is determined uniquely up to an analytic equivalence by the cohomology class TJ of the 1-cocycle {TJik}.

10. Families of analytic fibre spaces of elliptic curves

In this section we shall show that the family '3'(g, G) consists of all fibre spaces B~, TJ E Hl(A, n(B~», where B denotes the basic member of '3'(g, G). First we prove functicn theoretic lemmas. LEMMA

s,

10.1. Let w(s) be a holomorphic ftmction of a complex variable

lsi < 1, with ,s'w(s) > 0 and let t(s) be a holomorphic function of s,

6 See A. Grothendieck, A general theory of fibre spaces with structure sheaf, University of Kansas, 1955.

( 56)

1319

614

K. KODAIRA

o < I s I < 1, such that the analytic continuation along a circle: I s I = constant with positive orientation induces the transformation: t(s) ----> t(s)

+ mlw(s) + m

2 ,

where m 1 and m 2 are (rational) integers. If for n H n 2 E Z, 0

(10.1)

< Is I < 1 ,

then the integers m 1 and m 2 vanish and t(s) is (extended to) a holomorphic function of s defined on the disk: I s I < 1. PROOF. First we show that the inequalities (10.1) lead to a contradiction unless m 1 = m 2 = O. Suppose that at least one of m H m 2 does not vanish and let 1

l = -.logs. 27r'/,

Obviously f(l) is a holomorphic function of l defined on the upper half plane: Sl > 0 and satisfies f(l

(10.2)

Hence

e2~if(!)

o < I s I < 1.

+ 1) =

f(l)

+1.

is a non-vanishing single-valued holomorphic function of Setting

if m 1 = 0 , otherwise,

W(S) ,

w1(s) = { (mlw(s)

s,

+ m 2)-1 ,

we obtain from (10.1) the inequality (10.3)

It follows that therefore that

f(l) e2~if(Z)

*- mw1(s) + n ,

for m, nE Z.

*- 1. With the aid of Picard's theorem we infer

or f(l) = kl

+ g(s)

,

where k is an integer and g(s) is a holomorphic function of s defined on the disk: I s I < 1. Combining this with (10.2) we obtain f(Z) = l

+ g(s)

,

Hence the range of the function f(l) of l, Sl > 0, covers the domain: Sf> c, where c is a large positive number. This contradicts the inequality (10.3). In case m1 = m 2 = 0, t(s) is a single-valued holomorphic function of s,

( 56)

1320

ON ANALYTIC SURFACES;

615

II

o < I s I < 1.

Hence, with the aid of Picard's theorem, we infer that t(s) satisfies the inequalities (10.1) only if t(s) is holomorphic at s = 0, q.e.d. LEMMA 10.2. Let w(s) be a holomorphic function of s, 0 < lsi < 1, such that the analytic continuation along a circle: I s I ;= constant with positive orientation induces the transformation w(s)

-->

smw(s) ,

where m is an integer. If w(s) *- 0, *-1 ,

for 0

< Is I < 1 ,

then the integer m vanishes and w(s) is (extended to) a meromorphic or holornorphic function of s defined on the disk: I s I < 1. PROOF. By virtue of Picard's theorem it suffices to show that w(s) *-

0,_ *- 1 for 0

< I s I < 1 only if m = O.

Let 1

l = -.logs.

f(l) = w(s) ,

211:~

Obviously f(l) is a single-valued holomorphic function of l, ;Jl f(l

(10.4)

+ 1) =

> 0, and

e 0.

Let D denote the domain obtained from the space of the complex variable fby deleting 0 and 1. The universal covering of D is the upper half plane H: S'z > 0 in the space of a complex variable z and the covering map of H onto D is given by the modular function 7 z -----> f = 1 - ,.;-2(Z). We denote the inverse function by p.: f -----> z = p.(f). We note that, on the domain: If I > 2, there is a branch of p.( f) of the form (10.5)

where

p.(f) = ~(llf)

.i log f + ~(1/f) 11:

,

denotes a convergent power series of Ilf. We set h(l) = p.(j(l») .

Obviously h(l) is a holomorphic function of l defined on the upper half plane: ;Jl > O. Hence h(l) is single-valued. Now, assuming that the integer m were positive, we shall derive a contradiction. (In case m is negative, we replace w(s) by 1/w(s». Setting f(l)

= e~iml!L!)g(s)

,

7 See A. Hurwitz and R. Courant, "Allgemeine Funktionentheorie und elliptische Funktionen," pp. 400~405.

( 56)

1321

616

le. KODAlRA

we infer from (lOA) that g(s) is a single-valued holomorphic function of s,O < 1 s 1 < 1. We substitute -x + i for l, where x denotes a real variable, and let r(x)

>0,

where r(x) and O(x) are real-valued continuous functions of x. We have

+ 1) = + 1) =

r(x { O(x

(10.6)

r(x) , O(x)

+k,

where k is an integer. Setting r;(x) = m(x'

+

x - 1)

+

28(x) ,

we get f( -x

(10.7)

+ i) = r(x)e~m(2X+ll+~i~(x)

.

Hence, assuming (10.5), we obtain h( -x

+ i) =

-r;(x)

+ im(2x + 1) + ~ log r(x) + p(x)

,

11:

where p(x) =

?l3(l/f( -x + i») .

For brevity we set (10.8)

h(-x

+ i) =

- mx'

+ x,8(x).

Let c denote a large positive constant. The formula (10.7) shows that f( -x + i) -> co for x -> + co. It follows that p(x) is bounded for x > c. Moreover, by (10.6), the functions log r(x) and O(x)/x are bounded for x > c. Hence we conclude that the function ,8(x) is bounded for x > c. Since h(l) is a single-valued holomorphic function of l with :Jh(L) > 0 defined on the upper half plane Sl > 0, we have, by Schwarz's lemma, the inequality h(l)-al:s; I~~-I l +i '

I

h(l) - a

where a

=

h(i). Combining this with (10.8), we obtain

11+

a-a + a I;;:::- 11 -2£1 x

mx' - x,8(x)

or x 3 1 a,8(x) 1 ~ 1mx' - x,8(x)

+ a I' ,

for x> c .

This contradicts the above result that the function ,8(x) is bounded for

( 56)

1322

ON ANALYTIC SURFACES; II

617

x> e, q.e.d. Let V be an analytic fibre space of elliptic curves over 4 belonging to the family ::f(/J, G) and let denote the canonical projection of V onto ~. Moreover let B denote the basic member of g(g, G). For any open subset E ~ 4 we shall mean by a biholomorphie fibre map of B I E onto V I E a biholomorphic map A of B I E onto V I E satisfying the condition that A = 'P, where 'l' denotes the canonical projection of B onto 4. We fix a locally finite covering {E;} of 4' = 4 - {a p } by sufficiently small circular disks E;. We take a base {"YJu), "Yjz(u)} of i-cycles on the fibre C" of V over each point u E Ef such that "Yjl(U) and "Yjlu) depend continuously on u and let (wiu), 1) be the periods of the complex torus C" which corresponds to the base {"Yjl(U), "Yj2(U)}. For U E E/ n E£ we have

+ bjk"Yk2(U) , "Yj.(u) ~ Cjk"Ykl(U) + djk"Yk.(U) , "Yjl(U)

~

ajakl(U)

where ajb bjk , ejk, d jk are integers satisfying ajkd jk - bjkcjk

=

1.

We set jk jk (SJk) = (a b ). Cjk d jk

Obviously G' is the sheaf of germs of "locally constant" sections of the fibre bundle over 4' with the fibre Z EB Z defined by the system of transition matrices (SJk) operating on the right. Let G; be the discontinuous group consisting of analytic automorphisms

of Ej x C. We consider the quotient space Ej x CjGj as an analytic fibre space of elliptic curves over Ef in an obvious manner and, for any point (u, n on Ef x C, we denote the corresponding point on Ef x CjGj by the symbol [u, Clearly the restriction V I E/ of V to Ef is analytically equivalent to the fibre space Ef x CjGj. Thus there is a biholomorphic fibre map

n.

(10.9)

of E/ x C/G; onto Vi

E/ which satisfies A/u, S-) = u .

Setting

( 56)

1323

618

K. KODAIRA

we infer readily that the equality

= Ak(u, tk)

A,(u, t J) holds if and only if

(10.10) where n 1 and n 2 are integers and 'lj;k(U) denotes a holomorphic function of u defined on E} n E£. The fibre space B' = Bib..' is obtained from the collection of the fibre spaces E; x C/G'; by identifYing [u, tiL with [u, tkh if and only if (10.11)

tf = fik(u)(tk

+ n1{J)k(u) + n

2) ,

where n 1 and n 2 denote integers. This follows immediately from the definition (8.5) of B'. Consequently we may identify the fibre space E} x C/G~ with the piece B I E; of B and consider Aj as a biholomorphic fibre map of BI E; onto VI E;. Let 7};k denote the holomorphic section u

--->

[u, 7}ik(u)lj

of B over Ej nEt. The fibre preserving analytic automorphism L(7};k) of B I E; n E~ associated with 7}ik is given by the formula L(7};k): [u, tL

---+

[u, t

+ 7}jk(U»);

.

Hence, writing zJ = [u, tjlj and Zk = [u, tkh and comparing (10.10) with (10.11), we infer that the equality A;z; = Akzk holds if and only if Z; = L(7}jk)zk. Thus we conclude that (10.12)

Aj1A k = L(7}Jk) ,

on B IEj

n E£ .

For each point u E,i we denote by o(u) the unit of the fibre C! of the analytic fibre system B# of abelian groups. By (9.1) we have for uEEj .

o(U) = [u,O); ,

Note that 0:

u

o(u)

--->

is a holomorphic section of B over ,i. 10.3. Let E' be an open subset of ,i'. If there exists a holomorphic section ojr: U -> ojr(u) of V over E f , then there exists a biholomorphic fibre map A' of B I E' onto V I E' such that LEMMA

A'o(u) = ojr(u) ,

for UE E' .

We may assume that E' is covered by those disks Ej which are contained in E'. For each disk E; c E' we choose the map Aj in such a PROOF.

( 56)

1324

619

ON ANALYTIC SURFACES; II

way that for

UE

E; .

Then we have for

U E

E; n E~,

E; c E',

E~

c E' .

This implies that the holomorphic section r;;k coincides with 0 and therefore the automorphism L(r;Jk) is reduced to an identity map. Hence the equality AJz; = Akzk is equivalent to ZJ = Zk, provided that E; c E' and E; c E'. Thus we conclude that A':

Zj ---> Ajzj

defines a biholomorphic fibre map of B I E' onto VI E' and A' o(u) = 1fF(u) for U E E', q.e.d. It follows from the above lemma that, in case the fibre space V possesses a global holomorphic section 'VI': '/,~ ---; 1fF(u) over A, there exists a biholomorphic fibre map fl' of V' = VI A' onto B' = B I A' such that fl'1fF(u) = o(u) for U E A'. LEMMA lOA. Assume the existence of a holomorphic section 1fF: U ---; t(u) of V over A. If a biholomorphic fibre map fl' of V' = VI A' onto B' satisfies the condition that fl't(u) = o(u) for U E A', then fl' can be extended to a biholomorphic fibre map fl of V onto B. PROOF. In what follows we denote by v a "variable" point on V. Let fl' be a biholomorphic fibre map of V' onto B' such that fl't(u) = o(u) for u E A'. First we shall show that, for any merom orphic function ~ = ~(z) defined on B, the meromorphic function Hfl') = ~(fl'(v» induced on V' can be extended to a meromorphic function = g*( v) defined on V. Let Va be the open subset of V obtained by deleting all points t(ap) and all singular points of the curves -I(a p), a pE {a p}. By Levi-Hartogs' theorem any meromorphic function on Va can be extended to a meromorphic function on V. For our purpose it suffices therefore to prove that the meromorphic function /;(fl') on V' can be extended to a meromorphic function go defined on Va. Take a point Va E Va - V' and a small neighborhood N of Va in VO. Obviously Va is a simple point of a curve - I(ap) and fl' maps N n V' biholomorphically into B: = Bp n B'. Moreover the image fl'(N n V') does not meet the holomorphic section 0: u --> o(u) of B, provided that N is sufficiently small, since Va *' t(ap) andfl't(u) = o(u)foruEA'. Thefibre space Bp is, by our definition, a non-singular model of the quotient space FIC, where F is a fibre space of elliptic curves over a disk D: I a 1m < c

e

( 56)

1325

620

K. KODAIRA

and C is a cyclic group of analytic automorphisms of F of order m """ 1, 2, 3, 4 or 6. Moreover the fibre Fo of F over the center: a = 0 of D is either regular or of type Ib (see § 8. In case the fibre C",,, of Bp is regular or of type lb' we set F = Bp and C = 1. In case Cap is of type I:, we set C = {tn. We have F'

B; = F'/C,

=

F- Fo.

We recall that, = am is a local coordinate on Ep. We choose a local coordinate (x, y) on the neighborhood N of Vo such that ,

=

xn ,

where n is a positive integer, and suppose that N is the cylinder consisting of all points (x, y), I x I < 0, I y I < 0, where is a positive number. We set N' = N n V'. Obviously N' consists of all points (x, y), 0 < I x I < 0, I y I < 0. We denote by M' the m-fold unramified covering surface of N' which consists of all points (8, y), 0 < 181 m < 0, I y I < 0, where 8 = xllm. Let t; denote a meromorphic function defined on B and let

°

f'(x, y)

= t;(,Lt'(x, y») ,

for 0

< I x I < 0, I y I


p(v)

= (1, ti(v),

"', !;t(v»)

defines a "meromorphic" fibre map of V onto B which is an extension of the biholomorphic fibre map p' of V' onto B'. Since, by hypothesis, the fibres of V and B contain no exceptional curve, the meromorphic map p is biholomorphic, q.e.d. THEOREM 10.1. The family q(g, G) consists of all fibre spaces

B~,

r; E H'(il, O(B#». PROOF. Let V be a member of q(g, G). For each point a p E {a p } we choose a holomorphic section "yp of V defined over a neighborhood N p of a p in il. This is possible since V is free from multiple singular fibres. We take a small circular disk Ep c N pof center a pand fix a locally finite covering {E;} of il' = il - {a p} such that the disks E; c Npcover E; = Ep - a p. Let Aj denote a biholomorphic fibre map of B I EJ onto V I EJ of the form (10.9). Obviously we may assume that for

U E

E; c N p

•

Then we have Ajz

=

Akz ,

for Z E B I E;

n E~,

E; c N p ,

E~

c N

p

(see the proof of Lemma 10.3). Hence there exists a biholomorphic fibre map A~ of B I E; onto V I E; such that for

ZE

B I E;

n E;

I

E; c N p

•

Now we show that A; can be extended to a biholomorphic fibre map Ap of B I Ep onto V I Ep. We form the union W=UpVIEpUB'

by identifying each point Z E B I E; c B' with its image A;Z E V I Ep. Clearly W is an analytic fibre space of elliptic curves over ~ and belongs to the family q(g, G). Moreover W possesses a global holomorphic section over ~ which is an extension of the holomorphic section: U -> o(u) of B' c Wover A', since A;o(u) = "yp(u) for u E E; and "yp: u -> "yp(u) is a

( 56)

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624

K. KODAIRA

holomorphic section of VI Ep over Ep. Hence, by Lemma 10.4, the identity map of B' c B onto B' c W can be extended to a biholomorphic fibre map of B onto W. This proves that the biholomorphic fibre map A; of B IE; onto V \ E; can be extended to a biholomorphic fibre map Ap of B \ Ep onto

VIEp. We have (10.14)

on BI E;

n E{,

where f)jk denotes a holomorphic section of B' over Ej n E{ (see (10. 12». Hence we infer the existence of a holomorphic section f)jp of B' over Ej n Ep such that (10.15)

on BI Ej

n Ep.

It is clear that the collection of the holomorphic sections f)jk and f)JP of B' c B# defined respectively on E; n E£ and Ej n Ep form a 1-cocycle which determines a cohomology class f) E H'(6., D(B#». Moreover the formulas (10.14) and (10.15) show that V = B~. THEOREM 10.2. Let V denote an analytic fibre space of elliptic curves over 6. belonging to the family 9{!/, G). If V possesses a global holomorphic section over 6., then V is (analytically equivalent to) the basic member B of 2'(!/, G). PROOF. This theorem follows immediately from Lemmas 10.3 and 10.4.

Appendix

A proof of Theorem 6.1. Let V denote an analytic surface which contains an exceptional curve ®. Our purpose is to construct an analytic surface W such that Qp( W) = V and Qp(p) = ® for a point pEW. We take a point a on ® and denote by U a coordinate neighborhood of a in V. We choose a local coordinate (z" z.) of the center a defined on U such that z. vanishes on the curve ®. Moreover we assume that U is the cylinder consisting of all points (z" z.), I z, I < 1, I z. \ < 1, and let A denote the curve on U defined by the equation: z, = o. Obviously the curve A intersects ® transversally at a. Let N denote a sufficiently small neighborhood of ® in V. LEMMA. There exists a holomorphic function x defined on N whose d,ivisor (x) is equal to ® + AN' where AN = AnN. PROOF. Let U* denote the cylinder consisting of points (w" w.), I w, I < 1, I w2 < 1. Consider the holomorphic map 1

h: (ZlJ Z2)

->

(w" w 2)

( 56)

= (z"

1330

Z,Z2)

ON ANALYTIC SURFACES; II

625

of U into U*. Obviously h maps the curve A onto the center (0,0) of U* ar..d U - A biholomorphically into U*. Let U1 denote the subdomain of U consisting of all points (ZlJ Z2) with [Zl [ < 1/2. We take a small neighborhood M of ® in V and choose an open subset Ml of M - U1 such that the union Ml U U covers M. We then form a union M*

= Ml U U*

of Ml and U* by identifying each point Z in Ml n U with the point h(z) of U*. The union M* is a non-singular surface, provided that M is sufficiently small, since h is biholomorphic on Ml n U. Let PI denote the biholomorphic map: Z --> z of Ml c M onto Ml c M*. We extend PI to a holomorphic map /1 of Minto M* by defining for zEMn U.

p(z) = h(z) ,

Clearly the image C = p(®) of ® is a non-singular rational curve on M* and p maps ® biholomorphically onto C. Let Fa denote the restriction to the curve C of the complex line bundle F = [CJ over M* and let 'P' be the sheaf over C of germs of holomorphic sections of Fa. By hypothesis the intersection multiplicity (®2) is equal to -1. Hence we infer that (C 2) vanishes, while C is a non-singular rational curve. Consequently the complex line bundle Fa is trivial and therefore the sheaf 'P' is isomorphic to the sheaf of germs of holomorphic functions. It follows that the cohomology group Hl(C, \(1') vanishes and dim Ho(C, "IJ') = 1 . Hence, using a theorem of completeness of characteristic systems,R we conclude the existence of a family {C t } of curves Ct on M* depending holomorphically on a complex parameter t, [t [ < 1, such that Co = C, of which the characteristic system on C is complete. Letting w denote a variable point on M*, and setting x*(w) = t for WE Ct , we obtain a holomorphic function x* = x*(w) defined on a neighborhood of C in M* with (x*) = C. We verify this as follows. We cover C by a finite number of coordinate neighborhoods U,*, i = 1, 2, ... , in M* with respective local coordinates (U i , Vi) such that Vi vanish on C. The partial derivatives [8v;!8v k Jo k =o form a system of transition functions defining the trivial complex line bundle Fa. Hence we may assume that

(A.1) In each neighborhood U,* the curve Ct is defined by an equation of the B See K. Kodaira and D. C. Spencer, A theorem of completeness of characteristic sY8tems of complete continuous systems, Amer. J. Math., 81 (1959), 477-500.

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626

K. KODAIRA

form (A.2)

where fi(u i , t) denotes a holomorphic function of u, and t. We infer from (A.l) that the partial derivative

o = [8fi(u i, t)f8t]t~o is a constant which is independent of i. Note that 0 represents the infinitesimal displacement of Ct along the tangent vector dfdt at t = O. The completeness of the characteristic systems of {C t } on C implies that o*- O. Hence the equation (A.2) is equivalent to an equation of the form t = g;('U'i' v,) ,

where g,(Ui! v,) denotes a holomorphic function of Ui and v,. It is clear that g,(U i , 0) = 0 ,

while for w = (u" v,) .

x*(w) = g,(u i , v,) ,

Thus we see that x* = x*(w) is a holomorphic function defined on a neighborhood of C in M* with (x*) = C. Now, setting x(z)

=

x*(,u(z») ,

for zEN,

we obtain a holomorphic function x = x(z) defined on N with (x) = ® + AN' We take a point fJ on ®, fJ *- a, and a non-singular curve BN on N which intersects ® transversally at fJ. Then, by the above lemma, there exists a holomorphic function y = y(z) on N with (y) = ® + B N • The quotient s = s(z) = y(z)/x(z) is a meromorphic function on N with (s) = BN - AN' This implies that the map: z -> s(z) maps @ biholomorphically onto a projective line Pl' Let C' denote the space of the complex variables x and y. The quadric transform QoCC') of C' with respect to the center 0, = (0,0) is realized as the surface in C' x PI defined by the equation: y = sx and Qo(O) = 0 x Pl' We infer from the above result that :\.: Z

-->

(x, y, s) = (x(z), y(z), s(z»)

is a biholomorphic map of N onto an open subset of the surface Qo(C') c C' X P, and that :\.(®) = QoCO). Hence we conclude the existence of a surface W such that Q~(W) = V and Q/p) = ® for a point p on W. PRINCETON UNIVERSITY AND HARVARD UNIVERSITY

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ON COMPACT ANALYTIC SURFACES, III* By K. KODAIRAt

(Received April 19, 1962)

11. An exact sequence

In the preceding section we have shown that the family '3(g, G) of analytic fibre spaces of elliptic curves over A consists of all analytic fibre spaces B~, r; E Hl(A, n(B#». We denote by ®~o the component of the fibre C!p of B# over ap containing the unit o(ap) and let

m=

Up®!oUB' .

B~

has a structure of analytic fibre system of abelian groups over A which is the restriction of that of B#. Let n(m) denote the sheaf over A of germs of holomorphic sections of B~. Obviously n(B~) is a subsheaf of n(B#) and the stalk of the quotient sheaf n(B#)jn(BD over each point U E A' vanishes. Hence we obtain the exact sequence (11.1)

For each element r; E H 1(11, n(Bm we define B~ in the same manner as in Definition 9.2. In view of the exact sequence (11.1) we obtain from Theorem 10.1 the following THEOREM 11.1. The family q(g, G) consists of all analytic fibre spaces B"fI, r; E Hl(l1, n(BD). The notion of fibre spaces and their equivalences depends on the sheaf of structure groups.l By a @-fibre space we mean a fibre space with a sheaf @of structure groups, and we say that two fibre spaces are @-equivalent if they are equivalent as ®-fibre spaces. The fibre space B~ may be considered as an analytic fibre space, as an n(B#)-fibre space, or as an n(m)-fibre space. The cohomology class r; E H 1(11, n(m») represents the n(Bn-equivalence class of B~. Let cg" denote the fibre of B~ over U E 11. It is clear that

*

On compact analytic surfaces, I, Ann. of Math., 71 (1960), 111-152; II, Ann. of Math., 77 (1963), 563-626. t The author was supported during the final phases of the preparation of this paper by a research project at Harvard University sponsored by the National Science Foundation. I A. Grothendieck, A general theory of fibre spaces with structure sheaf, University of Kansas, 1955.

1

( 56)

1333

2

K. KODAIRA

for

U E~'

,

for

u =ap

•

It follows that

(11.2)

Cgu

=

a complex torus, C*,

if

l

Cu is regular ,

if C" is of type Ib , otherwise.

C

Let f" denote the tangent space of cgu at o(u). The union

f = U UEA f" of the tangent spaces f" forms a complex line bundle over ~ in an obvious manner. Since the fibre f" of f over u can be regarded as the infinitesimal group of the complex Lie group cgu which is isomorphic to a complex torus, C or C*, we have a canonical homomorphism h,,: f" ~ Ggu •

m

We define h to be the map of f onto which coincides with hI' on each fibre f" of f. Let O(f) denote the sheaf over ~ of germs of holomorphic sections of f. We infer readily that the map h is locally biholomorphic. Hence h induces a homomorphism of the sheaf onto the sheaf D(Bg). We denote this homomorphism by the same symbol h. The stalk of the sheaf Gover a p is, by definition,

Om

Ga p = r(G'IE') p

,

where Ep denotes a circular neighborhood of a pin~. Hence Gap is (isomorphic to) the submodule of Z EB Z which consists of all pairs (11,11 n~) of integers 11,11 11,2 satisfying the linear equation (11", n2)(Ap)

=

(11",

11,2) •

Using the normal form of Ap (see Table I in § 9) we infer, therefore, that (11.3)

~

Ga p =

lZZ,EB Z,

if Ca p is regular, if Cap is of type 16 , otherwise.

o, THEOREM 11.2. (11.4)

We have the exact sequence

0 ---+ G ---+ Dm ~ D(Bg) ~ 0 .

PROOF. Let K denote the kernel of the homomorphism h: O(f)---+O(Bg). Our purpose is to prove the isomorphism K~G.

( 56)

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ON ANALYTIC SURFACES, III

3

First we consider the restriction K' = K I A' of K to A'. Let f' = f I A' be the restriction of f to A'. We have, by (8.5), B~IA'=B'=

U'xClfl.

Hence, letting flo denote the subgroup of fl consisting of all analytic .automorphisms gOC/3)

= g(f3, 0, 0): (u, S) --> (f3u, fj3(u)r) ,

of U' x C, we infer that (11.5)

f'

= U'xCfflo •

We denote by {u, S} the point on f' corresponding to (u, S) E U' X C. We have h{u, S} = «u, S) . Consequently the kernel K' of the homomorphism h: n(f') -> nCB') consists of germs of holomorphic sections ·of f'. Thus the stalk K~ of K' over each point U E A' is the free abelian group with two generators U -> {u, w(u)} and u -> {u, 1} . Since w«(3u) = ff\(u)(aj3w(u) 1 = fj3(u)(cj3w(u)

+ bj3) , + dj3) ,

we have (f3u, w(f3u») = aj3gO(f3)(u, w(u») (f3u, 1) = Cj3gO(f3)(u, w(u»)

+ bj3gO(f3)(u, 1) , +

df3gO(f3)(u, 1) ,

while the sheaf G' is defined by the representation: bf3 ) df3

of niA').

Hence we conclude the isomorphism K':::::;G'.

We have the equality Ga p =

reG' I E~)

.

To prove the isomorphism K:::::; G, it suffices therefore to show that the eorresponding equality

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1335

4

K. KODAIRA

holds for each point a p E {a p }. For any section cp of Kover E p , we denote by cp' the restriction of cp to E~. The section cp is uniquely determined by cp'. Hence we may identify each section cp E r(K I Ep) with its restriction cp' E r(K' I E~) and consider r(K I Ep) as a subgroup of r(K' I E;), while, since K' is a locally constant sheaf, we may identify r(K I Ep) with the stalk Ka p' Thus we obtain Ka p = r(K I Ep) S; r(K' IE;) .

For our purpose it suffices therefore to verify that the equality (11.6) holds for each point a p E {a p }. (i) In case Cap is a regular fibre, the sheaf K I Ep is locally constant. Hence we obtain (11.6). (ii) In case Ca p is a singular fibre of type I b , we have, by (8.42), B~ I Ep =

Wo

= EpxC*jC,

where C denotes the infinite cyclic group generated by the analytic automorphism: Cr, w)-{r, W1: b) of E~ x C*, 1: being a local uniformization variable of the center a p on A. Hence we infer that

flEp

= EpxC

and that h: (1:,

n

«1:, w))o , Consider a germ ,,;r: 1: -> (1:, ,,;r(t")) of holomorphic section of f over U E E p • In case u=Fap , the germ ,,;r belongs to K,. if and only if exp [2ni,,;r(1:)] = 1:nb for an integer n. In case U = ap, the germ ,,;r belongs to Ka o if and only if exp [2ni,,;r(1:)] = 1. Assume that Ep is a disk: 11: I < s, take a section 6 E r(K' I E~) and write it in the form ->

6: 1: -> (1:, 6(1:)) ,

The fibre coordinate 6(1:) is a single-valued holomorphic function of 1:,0< 11: I

H'(A, O(f)) ~ H'(A, n(Bg)) ~ H2(A, G) - - > 0 .

We define for ~ E H'(A, O(BD) . c(~) = o*(~) , We call c(m E H2(A, G) the characteristic class of the n(BD-fibre space B~ over A. The cohomology group H'(A, n(f)) has a canonical structure of complex vector space. Given a cohomology class c E H2(A, G), we choose an element ~ of H'(A, O(Bm such that c(~) = c and define V t = B"*ltI+~ , for t E H'(A, n(f») . 11.3. The collection of the compact analytic surfaces H'(A, om), forms a complex analytic family.' PROOF. Let {f" "', fv. "', fd} denote a base of the linear space H'(A, O(f)) and let t,E C . THEOREM

Vt, t

E

We fix a finite covering {Ej } of A by sufficiently small circular disks Ej, and represent each cohomology class f, by a l-cocycle U,jk} composed of holomorphic sections f,jk of f over E j nEk such that, if E j n Ek contains a point a p E {a p }, the holomorphic section f,jk vanishes. Moreover, we represent ~ by a l-cocycle {~jk}, ~jk E r(B~ I E; n E k ). The analytic fibre space V t = B,,·(tlH is the union of the pieces B I E, obtained by identifying Z; E BI E; with Zk E BI Ek if and only if Zj

=

L(h(E,tJ,;k)

+ ~jk)Zk

.

It follows immediately that the complex structure of V t depends holo-

morphically on t, q.e.d. We denote by qlle) the complex analytic family of the surfaces V t = B"'ltI+~, t E H'(A, n(f)). In view of (11.7), the family qJle) is composed of all analytic fibre spaces BO, c(O)=c, with repetitions. Hence we obtain THEOREM 11.4. If c(8) = mation of B~.

c(~),

then the analytic surface BO is a defor-

2 Compare K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I-II, Ann. of Math., 67 (1958), 328-466, § 18.

( 56)

1337

6

K. KODAIRA

THEOREM 11.5. B~ is an algebraic surface if and only if r; is an element of finite order of HV1, PROOF. (i) We set V = B~. Let {Ei} denote a finite covering of A by sufficiently small circular disks Ei and let Bi = B I E;, Vi = V IE)' Moreover, let {r;jk}, r;jk E r(Bg I E j n E k), be a 1-cocycle representing the cohomology class r;. Then there exist biholomorphic fibre maps f-lj of V j onto B j such that

n(m».

(11.8)

(ii) Consider a fibre C! of B#. An effective divisor b of degree n on C! is a formal sum of n points Zil) , Z( 2), " " Zin ) on ct. We define s(b) =

Zll)

+

Z(2)

+ '" +

zln) ,

where the symbol + denotes the group operation on the abelian group C!. Let E be an open subset of ~, and let S denote an analytic curve on B I E containing no components of fibres of B I E. Assume that S is contained in the open subset B# I E of B I E. ThEm S cuts out on each fibre C! of B# I E an effective divisor S· C! whose degree is independent of u, and the map u --> s(S· CD of E into B# I E is holomorphic. Hence, setting ~(u) =

s(S·C!) ,

we obtain a holomorphic section ~: u --> Hu) of B# I E over E. (iii) Assume that V is an algebraic surface imbedded in a projective space. Let X denote a general hyperplane section of Vand let (11.9)

Clearly Sj is a non-singular curve on B j • We define a holomorphic section u --> ~~(u) of B' over E j n ~' by setting

~;:

~;(u)

=

s(Sj'Cu )

for u E E j n

,

~'

.

The holomorphic section~; can be extended to a holomorphic section ~j of B# over E j • To prove this, we consider the case in which B; contains a singular fibre Cap of B. Obviously we may assume that B j is the open subset Bp of B defined in § 8. In case the singular fibre Cap is of type I b , II, III, or IV, the curve Si is contained in B# I E i • Hence the extension ~j is obtained simply by setting ~;(u) = s(S;·C!) for u E E j • In case the singular fibre Cap is of type I:, II*, III* or IV*, B j = Bp is a non-singular model of the quotient space FIC obtained by a canonical reduction of singularities, where F is an analytic fibre space of elliptic curves over a disk D: I (J I < c, of which the fibre Fo over the center 0 of

C 56)

1338

7

ON ANALYTIC SURFACES, III

D is either of type 12b or regular, and where C is a cyclic group of analytic automorphisms of F of order 2,3,4, or 6 (see § 8). Thus there is defined a meromorphic correspondence between B j and F composed of holomorphic maps cp of B J onto FICand y of Fonto FIC:

Bi~FIC~F. Let Ej = E j n A'. To prove that the holomorphic section ~~ of B' over Ei can be extended to a holomorphic section f;j of B~ over Ej, it suffices to show that the closure in B j of ~;(Ej) is an analytic curve. In fact, assuming that the closure in B j of f;;(Ej) is an analytic curve A, we infer readily that the intersection multiplicity (AC ap ) is equal to 1. It follows that A is a non-singular curve on B# I E j intersecting simply with each fibre C!, U E E J • Hence we obtain an extension 1;j of ~j by setting ~;(u)= A·C! for U E E j • The curve B i on B j determines the corresponding analytic curve Y = t-1cp(B j ) on F. The curve Y is contained in F#. Hence, denoting by F~ the fibre of F# over U E D and setting y(U) = s( y. F~) ,

for

U

ED,

we obtain a holomorphic section y: U --> y(u) of F# over D. The analytic curve y(D) on F determines the corresponding analytic curve cp-lyy(D) on B j • Moreover, since Y = y-lcp(Bj), the analytic curve cp-lyy(D) contains f;~(E~). Consequently the closure in B j of ~;(E~) is an analytic curve. Thus we obtain a holomorphic section f;j of B# over E J such that (11.10)

for

U E

Ej

n A'

.

Denoting by m the least common multiple of the orders of the finite abelian groups C~ pICg a p , ap E {a p }, we note that m~j: U

-->

m1;/u)

is a holomorphic section of Bg over E j • Let n denote the intersection multiplicity of X and the general fibre of V. From (11.8) and (11.9) we obtain B j = L(1Jjk)B k .

Combining this with (11.10) we get ~j(u) = n7jjk(U)

+ f;k(U)

,

for U E E j

n Ek .

for U E E j

n Ek .

Hence we obtain

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1339

8

K. KODAIRA

This proves that mnr; vanishes. Thus we conclude that r; is an element of finite order of Hl(6., n(Bm. (iv) Assume conversely that nr; vanishes for a positive integer n. Then there exists a O-cochain {~;}, ~; E r(Bg I E;), such that

nr;;k(U)

= ~k(U)

~;(u)

-

,

for U E E;

n Ek .

By an elementary consideration we infer that H!(E;, G) vanishes. Therefore, we obtain from (11.4) the exact sequence

... ~ HO(E;, n(f)) ~ HO(E;, n(Bm ~ 0 . Consequently, we can find for each section f over E; such that

~j

a holomorphic section fj of

We define Then we have ~; .

nr;; = Hence, setting r;Mu) = r;3k(U) - r;k(U)

+ r;;(u)

,

we obtain

nr;Mu)

= O.

Thus the cohomology class r; is represented by the 1-cocycle {r;!k} satisfying nr;!k = O. We may assume therefore that the 1-cocycle {r;i~} satisfies

nr;;k

= 0 .

Let Z! denote the subgroup of the abelian group C~ consisting of those elements z of C! which satisfy nz = O. Z# = UuZ! is an analytic curve on B#. We define Z to be the closure of Z# in B. We infer readily that Z is a (possibly reducible) algebraic curve on B. Since, by hypothesis, r;;k(U) belongs to the subgroup Z! of C!, we have Hence we get

Z n B; n Bk

=

L(r;;k)(Z n B; n B k )

,

while (li(l;! = L(r;ik)' Consequently, we obtain

(ljl(Z)

n Vk =

(l;!(Z)

n Vi

.

This proves the existence of an algebraic curve X on V which coincides

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ON ANALYTIC SURFACES, III

9

with p;;1(Z) in each open subset V j of V. The intersection multiplicity of X and the general fibre of V is equal to n 2• Hence we obtain the inequality dim 12X + E~~1«l>-1(Ut.) I ~ 2vn2

+ constant,

where «l> denotes the canonical projection of V onto~. The inequality proves the existence of two algebraically independent meromorphic functions on V. Consequently V is an algebraic surface (compare Theorem 4.3). THEOREM 11.6. If c(1}) is an element of finite order of H2(~, G), then the analytic surface B~ is a deformation of an algebraic surface. PROOF. Assume that nC(1}) vanishes for a positive integer n. Then, in view of the exact sequence

... ~ H1(~, n(f)) ~ H1(b., n(Bn) ~ H2(b., G)

---0>

0

(see (11.7», there exists an elementfe H1(b., n(f») such that

n1} = h*(f) . Setting

we obtain n{)

c(O) = c(1}) .

= 0 ,

Hence we conclude with the aid of Theorems 11.5 and 11.4 that BO is an algebraic surface and that B~ is a deformation of BO, q.e.d. THEOREM 11.7. If the sheaf G is non-trivial, the cohomology group G) is finite. PROOF. The cohomology group H2(~, G) is isomorphic to the homology group Ho(~, G). Hence, letting M denote the submodule of Z EB Z generated by the elements H2(~,

(nu n 2)(Sfl) - (nu n 2)

(nu n 2)

,

E

Z EB Z, (3 E nib.') ,

we obtain the isomorphism H2(~,

G)

~

Z EB Z/M.

After a suitable change of the base of Z EB Z, we may assume that M consists of all elements (n1e1, n 2e2 ), n 1 E Z, n 2 E Z, where e1 and e2 are nonnegative integers. Suppose that H2(~, G) is an infinite group. Then either e1 or e2 must vanish. If e1 vanishes, we have

( 56 J 1341

10

K. KODAIRA

btJ ) 1 '

and therefore w«(3u) = w(u)

+ b/3 ,

for

(3 E triA') ,

while ,s'w(u) > O. Hence exp [2niw(u)] is a single-valued holomorphic function on A' with I exp [2niw(u)] I < 1 and therefore exp [2niw(u)] is reduced to a constant. It follows that w(u) is a constant, and consequently (8/3) = 1 for (3 E n1(A'). Thus the sheaf G is trivial. If e. vanishes, then we have

and therefore l(w(f3u) = l(w(u)

+ C/3 •

Hence, considering exp [-2ni/w(u)] instead of exp [2niw(u)], we conclude that the sheaf G is trivial, q.e.d. The following theorem is an immediate consequence of Theorems 11.6 and 11.7. 11.8. In case the sheaf G is non-trivial, every member Vof the family 9(g, G) is a deformation of an algebraic surface. THEOREM

Now we consider the case in which the sheaf G is trivial. The triviality of G implies that the meromorphic function g(u) has no pole. Hence g(u) is reduced to a constant, and therefore w(u) is independent of u. Consequently the basic member B of 9(g, G) is reduced to an analytically trivial fibre space; i.e., B=A x Co in the complex analytic sense, where Co denotes an elliptic curve (see (8.5»). Any member B~, r; E Hl(A, n(B», of '3(g, G) is therefore an analytic fibre bundle over A whose fibre is the elliptic curve Co and whose structure group is the translation group acting on Co. Let p denote the genus of A. 11.9. In case the sheaf G is trivial, the first Betti number b1 of B~ E g(g, G) is given by the formula THEOREM

2P b1 = { 2p

(11.11) PROOF •

+ 2, + 1,

We write

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1342

if c(r;) = 0 , otherwise.

11

ON ANALYTIC SURFACES, III

where Go denotes the discontinuous subgroup of C generated by the periods 1 and wo, ,swo > 0, of Co. For any point!: E C we denote by [1:] the corresponding point on Co = C/G o• Let {Ej } denote a finite covering of Ll by small circular disks E j • The restriction B~ I E j of B~ to E j is analytically trivial, i.e., B" I E j = E j x Co , and (u, [Si]) E E j x Co is identical with (u, [!:k])

[si1

= [Sk

+ !jk(U)]

E

Ek x Co if and only if

,

where fik(U) is a holomorphic function of U defined on E) n E k• We have (fik(U)

+ fkJU) + fi)(U)]

for

= 0 ,

U E

Ei

n E n Ek j

.

It follows that Cijk = f,;(u)

+ j"ik(U) + fk>(u)

is a constant belonging to Go. We infer readily that the collection {C,jk} of the constants Cijk forms a 2-cocycle on the nerve of the covering {E j } of a which represents the characteristic class c(1J) E H2(a, Go) of B~. We calculate b1 with the aid of de Rham's theorem. We associate with each element [1:] of Co the analytic automorphism (u, [!:j)) ~ (u, [!:j

of

+ 1:])

B~.

Thus we consider Co as an analytic group of automorphisms of Let R denote the field of real numbers. Since Co is compact, any 1dimensional cohomology class of B~ with coefficients in R is represented by a d-closed realI-form cp which is invariant under Co. It is clear that, on each open subset B~ I E j of B~, the I-form cp can be written in the form B~.

(11.12) where A is a constant which is independent of} and Xj(u) denotes a realvalued differentiable function of u defined on E j • Since H2(Ll, Go) is a subgroup of H2(a, C), we may consider the characteristic class c(1)) as an element of H2(a, C)~C. The constant A satisfies the linear equation (11.13)

AC(1J)

+ A c(1))

= 0 .

In fact, since (11.14)

on

it follows from (11.12) that

( 56

J 1343

B~

I E; n Ek ,

12

K. KODAIRA

Adf'k(u)

+ Adf'k(u) =

dXk(U) - dXj(u) ,

C,k = Afjk(u)

+ Af'k(u) -

for

U E

Ej

n Ek

.

Hence, setting Xk(U)

+ X,(u)

,

we obtain constants Cjk satisfying the equations (11.15) Thus we get (11.13). Conversely, for any constant A satisfying (11.13), we can find a 1form cp of the form (11.12). In fact the equation (11.13) asserts the existence of real constants Cjk satisfying (11.15). It follows from (11.15) that the differences gjk(U) = Af'k(u)

+ Af'k(u) -

C,k

satisfy the conditions

Hence we can find real-valued differentiable functions X,(u) such that Af'k(u)

+ Af;k(u) -

C,k = Xk(U) - X;(u) .

Combining this with (11.14) we obtain Adsj + AdC; + dX,(u) = Adsk + AdCk + dXk(U) . Thus we conclude the existence of cp of the form (11.12). Let n denote the number of solutions of the linear equation (11.13) which are linearly independent over R. We infer from the above results that the first Betti number bi of B~ is equal to 2p + n. Since c(1J) is an element of HVl, C) ~ C, the number n is equal to 2 or 1, according as c(1J) = 0 or O. Consequently, we obtain (11.11), q.e.d.

"*

We conclude from Theorems 11.6 and 11.9 the following THEOREM 11.10. In the case in which the sheaf G is trivial, a member B~ of the family '3(g, G) is a deformation of an algebraic surface if and only if the characteristic class c(1J) vanishes.

12. Numerical invariants

In this section we shall calculate several numerical invariants, e.g., the arithmetic genus Pa and the geometric genus Pg , of the algebraic surface

B. It is a matter of triviality to verify the following lemma for each type of singular fibre listed in Theorem 6.2.

( 56 J 1344

ON ANALYTIC SURFACES, LEMMA. Let Cap

13

III

=" £....J nps®p, be a simple singular fibre of an analytic s

fibre space of elliptic curves over a curve and let D = E m pS ®p8 be a divisor composed of the compononts ®P8 of Cap. If the inters~ction multiplicity (D®p,) vanishes for all components ®p" then D is a multiple of Cap' Let p denote the genus of A and let f be the canonical bundle of A.

THEOREM 12.1. The canonical bundle K of B is induced from the complex line bundle f -

f over A by the canonical proJ'cction 'It of B onto

(12.1)

K

A:

= 'It*(f - f) .

PROOF. We take a sequence of distinct points and consider the divisor C(n) = "n C 4-1';=1

U 1, U 2,

"',

U>, •••

on A'

'IL'l.I

on B. The virtual genus rr'(c(n») of C(n) is equal to 1. Therefore, using the formula (2.16), we obtain (12.2)

dim I K

+ C(n) I =

Pa+ k

+n -

1,

k

~

0.

This implies that the complete linear system I K + C (n) I contains an effective divisor D, provided that n> I-Pa. Tp.e intersection multiplicity (DC,,) of D and any fibre C" of B vanishes. Hence D can be written in the form D = EpE,mps®p, + E"m"C" ,

where the ®ps denote irreducible components of the singular fibres Cap of B and the coefficients mu vanish except for a finite number of points u on A'. Since we have (®ptEsmps®ps) = 0 .

Hence, by the above lemma, the divisor Esmp,®p, is a multiple of Cap. The canonical divisor D - C (n) can be written therefore in the form Ek"C" ,

where the coefficients k" vanish except for a finite number of points u on A. Thus we see that (12.3) K = [Ek"Cu ]. We consider the holomorphic section 0: u -> o(u) of B which maps each point u E A onto the unit o(u) of the fibre C! of B#. For the sake of brevity we identify A with the curve O(A) on B by means of the biholomorphic map: u --> o(u) of A onto 0(A). Denoting by K~ the restriction of

( 56 J 1345

14

K. KODAIRA

K to the curve

~ = o(~),

we infer from (12.3) that

(12.4)

K = 'l'*(KJ .

The complex line bundle f over ~ coincides with the normal bundle of ~ in B, i.e., the restriction [~]o of the complex line bundle [~] over B to ~. The canonical bundle f of ~ is given by the adjunction formula

+ [~L

f = Ko

(see (2.2». Hence we get K~ =

f - f.

Combining this with (12.4) we obtain (12.1), q.e.d. Let cz denote the Euler number (or the second Chern class) of the surface B. The formula (12.3) implies that (KZ) vanishes. Hence, by Noether's formula (5.8), we have (12.5)

We denote by v(T) the number of the singular fibres of B of type T and by J' the order of the meromorphic function g, i.e., the total multiplicity of the poles of g. THEOREM 12.2. The arithmetic genus Pa of the surface B is given by the formula (12.6)

12(Pa

+ 1) =

j

+ Eb6J.i(I:) + 2J.i(II) + 10J.i(II*) + 3J.i(III) + 9J.i(III*) + 4J.i(IV) + 8v(IV*)

.

PROOF. By the Euler number c(Cap ) of a singular fibre Cap = E np.®ps we shall mean the Euler number of the polyhedron Us®p.. The values of c(Cap ) are listed in Table II. Since the Euler number of any general fibre 8

Table II type

b

It

I

II

b+6

I

2

I I

II* 10

I I

III

\

III*

\

IV

I

IV*

3

I

9

I

4

I

8

Cv, of B vanishes, we have

Combining this with (12.5) we get 12(Pa

+ 1) =

Epc(Cap ) ,

while each fibre of type Ib or It corresponds to a pole of order b of g. Consequently we obtain (12.6), q.e.d.

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1346

ON ANALYTIC SURFACES,

III

15

The formula (12.6) implies that P. + 1 is non-negative and that P. + 1 vanishes if and only if g is reduced to a constant and B has no singular fibres. Let c(f) denote the characteristic class of the complex line bundle f over ~. In view of the canonical isomorphism: H2(~, Z)~Z, we may consider c(f) to be an integer. THEOREM 12.3. We have c(f) = - Pa - 1 .

(12.7)

PROOF. It follows from (12.1) that dim I K + GIn) I is equal to the diI on the curve ~. mension of the complete linear system I r - f + Hence, for all sufficiently large values of n, we have

r::=lU,

dim I K

+ G(n) 1=

p - c(f)

+n

- 2 .

Combining this with (12.2), we get p - c(f) = P.

+k +1,

while, by Theorem 2.3, the integer k is equal to p for all sufficiently large values of n. Consequently we obtain (12.7), q.e.d. It follows from (12.1) that the geometric genus Pv of B is equal to the dimension of the complete linear system It - f I increased by 1. Hence, using Riemann-Roch's theorem, we get Po = P

+ P. + 1 + dim I fl.

The formula (12.6) shows that c(f) = - p. - 1 ~ O. Hence dim) f I is equal to 0 or -1 according as f = 0 or =1= O. Consequently we obtain (12.8)

p g -

if f = 0 , if f =1= 0 .

P, { p + P.,

By the duality theorem we have dim Hl(~, D(f)) = dim I f - f I + 1 . Hence we obtain (12.9)

Consider the complex analytic family

CV(C)

consisting of the surfaces Hl(~, D(Bg» with

VI = B"'(t)+" t E Hl(~, D(f)), where r; is an element of c(r;) = c.

THEOREM 12.4. If the (lrder of c is finite and if the geometric genus Po of B is positive, then the general member of the family qJ(C) is non-

algebraic, or, more precisely, the member B"'(t)+>I of qJ(C) is an algebraic

( 56)

1347

16

K. KODAIRA

surface if and only if t belongs to a countable subset of HVl., n(f)). PROOF. Let F denote the subgroup of Hl(b., n(f) consisting of those elements t which satisfy the condition that h *(t) are of finite order. Since the group Hl(b., G) is finitely generated, we infer from the exact sequence (11.7) that F is countable. Letting n denote the order of c, we have c(m}) = 0 .

Hence, by (11. 7), there is an element f of Hl(b., n(f)) such that h*(f) = nr; .

Substituting s - n-1f for t, we obtain h*(t)

+ r; =

h*(s)

+0 ,

o=

r; - h*(n-1f) .

Since nO vanishes, we infer from Theorem 11.5 that B h *(sH8 is an algebraic surface if and only if s belongs to F. Consequently, the surface B"*(tl+~ is algebraic if and only if t belongs to the countable subset -n-lj + F of HV~", n(f)), q.e.d. 13. A spectral sequence

In this section we denote by b. a compact differentiable manifold. Let S be a sheaf over b. and let 'It denote the canonical projection of S onto b.. By an automorphism of S we shall mean a pair (gfl, gs) of a bidifferentiable map gfl of b. onto b. and an isomorphism gs of S onto S such that 'Itg s = gfl'It .

We assume that a finite group

@

and a homomorphism

g -> (gA, gs)

of @ into the group of automorphisms of S are given. For brevity we set gu = gflU, for u E b. • By a @-invariant covering of b. we shall mean a covering {E} of b. by open subsets E satisfying the following two conditions: (i) if E E {E}, then gE E {E} for all g E @, (ii) if E E {E} and if g E @, then either gE n E is empty or gE = E. Let {E} be a finite G-invariant convering of b., and let N denote the nerve of {E}. A q-cochain on N with coefficients in S is, by definition, an anti-symmetric function 8 of q + 1 elements Eo, E l , " ' , Eq of {E} whose value s(Eo, Ell "', Eq) is a section of S over Eo n El n ... n Eq. We denote by ds the coboundary of s. Moreover, for any element g E@, we denote by gs the q-cochain defined by the formula

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1348

17

ON ANALYTIC SURFACES, III

(gs)(Eo, E1> "', Eq) = gsS(g-1 Eo, g-1 El> •. " g-lEq) .

Let Cq(N) denote the module of q-cochains on N with coefficients in S. A p-cochain on @ with coefficients in Cq(N) is, by definition, a function s = S(gl. "', gp) of p elements gl •.• " gp of (» with values in Cq(N). We denote by AM(N) the module of p-cochains on @ with coefficients in Cq(N). The coboundary os of a p-cochain s = s(g1> .. " gp) is defined by the formula (OS)(gl' g2' .. " gp+l) = glS(g2, "', gP+l) (13.1) +2:~=1( -1)'s(glt "', g,g>+1> ... , gP+l) +(-I)P+ls(g1> "', gp). We form the bigraded module A"(N) = '-..Jp+q=n " AP,q(N)

,

of which the bigradation is defined by the indices p and q, and the total gradation by n. We set Ds = ds + (- Was, for s E Ap.q(N) . It is clear that D2 = 0 and that DA n(N) ~ An+l(N). Thus D is a coboundary of total degree 1 acting on A(N). We define H(A(N») = 2:"Hn(A(N»)

to be the graded D-cohomology group of the graded module A(N). Consider a finite @-invariant covering {F} of d which is a refinement of {E}, and denote by M the nerve of {F}. We choose a map n of the set {F} onto the set {E} which maps each element F of {F} onto an element E::J F of {E} such that ng = gn. This is possible since {E} and {F} are both @-invariant. For any eochain s = 8(gl' "', gp, Eo, "', Eq), we then define ns by the formula (ns)(glt .. " gp, Fo, "', Fq)

= r p s(gl, ., ., gPt nFo, ... , 1CFq )

,

where rp denotes the restriction map to F o n Fl n '" n Fq. We infer readily that the homomorphism n of A(N) into A(M) commutes with D. Hence n induces a homomorphism, 1C~: H(A(N»)

--->

H(A(M») .

Moreover, the homomorphism 1C~ is independent of the choice of the map n. It depends only on the coverings {E} and {F}. Consequently, we can define in a usual manner the direct limit limNH(A(N» of the modules H(A(N» with respect to the pro}ection8 n~. DEFINITION 13.1.

The graded cohomology group

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1349

18

K. KODAIRA

H(@x.:1, S) =

E

n

Hn(@x.:1, S)

of @x.:1 with coefficients in the sheaf S is the direct limit limNH(A(N» of the graded D-cohomology groups H(A(N». We set BM(N) = AM(N)

+ AP+l,q-l(N) + AP+2,q-2(N) + ....

The filtration A"(N) = BO'''(N)

~

BI,H(N) ~ ...

~

B",O(N)

~

0

of A"(N) induces a filtration (13.2)

H"(@x.:1, S) = KO,,,

~

K 1 ,n-l

~

•••

~

Kn,O

~

0

of the cohomology group Hn(@ x .:1, S) in a natural way. We define (13.3)

We note that each element g of @ induces an automorphism of the cohomology group Hq(.:1, S) in a natural manner. Hence the cohomology group HP(@, Hq(.:1, S» of @ with coefficients in Hq(.:1, S) is defined. THEOREM

13.1. There exists a spectral sequence E:,q in which

S») ,

(13.4)

Ef,q

~

HP(@, HQ(.:1,

(13.5)

E":.;q

~

HM(CM x.:1, S) .

PROOF. Let E:,q(N) denote the spectral sequence derived from the bigraded module A(N) with the coboundary D. The map 7r of {F} onto {E} induces a homomorphism

7rt;:

E~,q(M) --->

E:,q(N)

which depends only on the coverings {E} and {F}. Forming the direct limit E:,q = limNE:,q(N) with respect to the projections nt;, we obtain a spectral sequence E:,q in which (13.4) and (13.5) hold. 14. Automorphisms

Let V be an analytic fibre space of elliptic c"Q,rves over .:1 belonging to the family '3(g, G). By an automorphism of the analytic fibre space V over A we shall mean a biholomorphic map g of Vonto V such that g-l is a biholomorphic map of A onto A, where denotes the canonical projection of V onto.:1. In this section we shall examine finite groups of automorphisms of V.

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19

ON ANALYTIC SURFACES, III

First we consider the basic member B of 9!(g, G) and assume that there is given a homomorphism of a finite group @ into the group of automorphisms of B which leave invariant the curve O(A) on B. For each element g E @ we write the corresponding automorphism of B in the form zE B .

z --> gz ,

For brevity, we identify A with 0(A) by means of the biholomorphic map: ----> o(u) of A onto o(A). Then we have

u

(14.1)

'l" g = g'l" ,

where 'l" denotes the canonical projection of B onto A. The abelian group structure of the fibre C" of B over u E A' is determined uniquely by the position of the unit u = o(u). Hence g maps the abelian group C" isomorphically onto the abelian group Cg". It follows that g induces an automorphism of the sheaf n(B~) over A. We denote the induced automorphism of D(B#) by the same symbol g. Thus we define a homomorphism of @ into the group of automorphisms of D(B#). Applying Definition 13.1 to the sheaf D(B#) over A we form the cohomology groups

n = 0,1,2, .... We shall show that each element (J of Hl(@xll, D(B#» determines an analytic fibre space V E 9!(g, G) and a finite group @v of automorphisms of V which is homomorphic to @. We represent the cohomology class (J by a I-cycle s E Al(N), where N denotes the nerve of a finite @-invariant covering {Ej } of 11 by small circular disks E j • We write and set Note that y)jk and A,lg) are respectively holomorphic sections u ----> y)jk(U) and u -> A,j(g, u) of B# over E j n Ek and E j . For each element g of @ we define a permutation: j ----> gj of the indices j by putting g-lEj = E gj .

The equation: Ds = 0 is then written in the form JY)i3(U)

(14.2)

+ Y)Jk(U) + Y)ki(U) =

0,

for

Ig1)qjgk(g-~:) - 1)jk(U) = A,k(g, u) - A,~, u) , ~ gA,gj(h, g u) -

A,j(gh, u)

+ A,lg, u) -

0,

E; n E j n Ek for U E E j nEk for UE E j

U E

, , •

The first line of (14.2) implies that the collection {Y)ik} of y)jk forms a 1cocycle on N with coefficients in the sheaf n(B#) and determines an

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20

K. KODAIRA

element 1) of H'(fl, f2(E#». We set V = E~. We have a collection of biholomorphic fibre maps fl, of V j = V IE, onto E j = B I E j such that (14.3) The second line of (14.2) implies that L(\,,(g»)gL(1)yjYk)g-l = L(1)jk)L(\'k(g») .

Combining this with (14.3) we get fl;'L(\,/g»)gflYJ

=

fl,,-lL('A-k(g»)gflYk .

Hence, setting on each V gj ,

(14.4)

we obtain a biholomorphic map gv of Vonto V. It is clear that (14.5) Thus gv is an automorphism of the analytic fibre space V over A. Now the third line of (14.2) implies that L(\'j(gh») = L(\,/g»)gL(\,gj(h»)g-l .

Hence we obtain (gh)v = fl;'L(\'j(gh»)ghPhyj

=

flj'L(\,j(g»)gflYJ· fl;,'L(\,y/ h »)hflhgj

=

gvhv .

Thus we conclude that the automorphisms gv, g EO @, form a finite group @v which is homomorphic to @. DEFINITION 14.1. We denote by the symbol (E, @)"" the pair (V, @v) of the analytic fibre space V = E~ and the finite group @v of automorphisms gv, g E @, of V.

THEOREM 14.1. For any pair (V, @) of an analytic fibre space V E '3(!f, G) and a finite group @ of automorphisms of V, we can find a homomorphism of @ into the group of automorphisms of E leaving invariant the curve A and an element (J of H 1(@x fl, !1(B#» such that

(14.6)

(V, @) = (E, @)"" •

PROOF. Each element g of @ determines an automorphism g; of A such that O

.

We derive from this the exact cohomology sequence ... _ $(l(n(f)) _ $l(n(Bn) _ $(2(G) _

....

Combining this with (14.16) we obtain the exact sequence

Hl(~, n(f)) ~ $(l(!1(Bn) ~ .9{2(G) ~ 0 . Note that HO(@, Hl(~, n(f)) consists of those elements of Hl(~, !1(f)) which

(14.17)

...

---->

HO(@,

are invariant under @. We define C(O'O) =

0*0'0 ,

We consider the quotient sheaf Q = n(BI)/!1(BD . Each element g of @ induces an automorphism of Q in an obvious manner. Hence the cohomology groups .9{"(Q), n = 0, 1, 2, ... , are defined and the exact sequence

( 56)

1355

24

K. KODAIRA

(14.18) holds. The sheaf Q has only a finite number of non-vanishing stalks each of which is a finite module. It follows that the cohomology groups $("(Q) are finite. In what follows we assume that @ operates effectively on the curve A in the sense that the homomorphism of @ into the group of automorphisms of A is an isomorphism. Letting (J be an element of $(l(n(B#» we set ( V, @) = (B, @)(gu, [tD for all g E @, (u, [m E A x Co. THEOREM 14.6. .9{2(G) is a finite group unless B is analytically trivial and @ acts trivially on the fibres on B. PROOF. Consider the spectral sequence E:,q associated with the cohomology group H(@xD., G). We have Er q = HP(@, Hq(D., G») .

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27

ON ANALYTIC SURFACES, III

The cohomology group Hq(A, G) is finitely generated, while, for p ~ 1, the order of each element of HP(@, Hq(A, G» divides the order of @. Hence the terms E: q, p ~ 1, are finite groups. It follows that, if Eg·2 is finite, then $(2(G) is finite. If the sheaf G is non-trivial, then, by Theorem 11.7, H2(A, G) is finite and therefore Eg·· is finite. Consequently, $(2(G) is a finite group. If G is trivial, then B is analytically trivial, i.e., B = A x Co. Hence, each automorphism g E@ of B has the form g: (u, [t])

~

where e4 or e6 = 1 .

(gu, reS]) ,

The automorphism g induces an automorphism (A) of H2(tl, G) ~ Z EB Z which depends only on the value of the constant e. If e=1, then (A)=1. If e =f= 1, then (A) is represented by one of the following seven matrices

(-~ -~) , ± (-~ ~),

±

(-~

5), ± (~ -D

(compare Table I). The term Eg·2 consists of those elements of H 2(A, G) which are invariant under ®. We infer therefore that E~·2 vanishes unless e = 1 for all g E ®. Consequently, $(2(G) is a finite group unless ® acts trivially on the fibres of B, q.e.d. Now we consider the case in which B is analytically trivial, i.e., B = A x Co, Co = CIG o, and @ acts trivially on the fibres of B. Note that, in this case, Bg and B# both coincide with B, the complex line bundle f is trivial, and G = AxG o• Hence the exact sequence (14.17) is reduced to (14.21)

...

---->

HO(@, Hl(A,

k*

0*

0») ~ $(l(O(B») ----> $(2(GO)

---->

0.

By a constant section of B over E j we shall mean a map of E j into B of the form u --+ (u, [dJ), where dEC is independent of u. Let (J be an arbitrary element of $(l(O(B». We can choose a cocycle s = So + S1 representing the cohomology class (J such that the values Sl(g, E j) of S1 are constant sections of B over the disks E j. We verify this as follows. Let s' = s~ + si be any cocycle representing (J and write si(g, E 3 ) in the form si(g, E j): u

-->

(u, [fj(g, u)]) ,

where f;(g, u) is a holomorphic function of u defined on E j • Since ® acts trivially on the fibres of B, the vanishing of osi implies that [fgj(h, g-lu) - f3(gh, u)

+ f;(g, u)] =

0,

for g, hE @

It follows that c;(g, h)

= fgj(h, g-lU) -

fj(gh, u)

+ fAg,

u)

is a constant belonging to Go. Denoting by l the order of @, we set

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1359

•

28

K. KODAIRA 1

d;(g) =

T I:hr;(lJc;(g,

r,(u) =

1- I:yE(!')f,(g, u) .

h) ,

Then we have d,(g) = fAg, u)

+ ry,(g-'u)

- rAu) .

Hence, letting r denote the cochain defined by r(E j ): u

->

(u, (r;(u)]) ,

and setting 8

= So

+ 8, =

8~

+ 8; + Dr

,

we conclude that 8,(g, E,) is the constant section: u -> (u, [d,(g)]). We set Tj,k = 80(Ej, E k ), A,j(g) = 8,(E;) and write Tjjk in the form y)jk: u -->

(u, (fjk(U)]) ,

where fjk(U) is a holomorphic function defined on E j A,)(g); u

n E k•

Note that

(u, [dig)]) .

->

We write (V, @y)

=

(B, @)" .

The surface V = B~ is the union of the product spaces E j x Co where (u, [SjD E E j x Co is identical with (u, [sd) E Ek x Co if and only if [Sj] = [Sk

+ fjk(U)]

.

The automorphism gy of V corresponding to g E @ is given by the formula (14.22)

*"

Consider a fixed point a e A of an automorphism g E @, g 1, and denote by @a the subgroup of @ consisting of those automorphisms which leave invariant the point a. Since, by hypothesis, @ acts effectively on A, @a is a finite cyclic group. Let n denote the order of @a. We choose a local coordinate 1: on A of the center a and a generator ga of @a such that ga transforms 1: into e,,1:, e,. = exp (2rcijn). Suppose that a E E j and that Ei is the circular disk: /1: / < 1. Then we have (14.23)

(g.)y: (1:, [Si])

->

(e,,1:, [Si

We infer readily that

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1360

+ dj(ga)])

.

ON ANALYTIC SURFACES,

29

III

Thus the order m of [dj(ga)] divides n. The formula (14.23) shows that the quotient surface Vj/(@a)v, V j = V I Ej, is an analytic fibre space of elliptic curves over the disk Ej/@a whose fibre over the center of E j /@. is of type ",lo. Consequently, the quotient surface V/@v is an analytic fibre space of elliptic curves over Aj@ of which each singular fibre is of type .,10' m depending on the singular fibre. Let p denote the genus of the curve !1/@. THEOREM 14.7. In case B is analytically trivial and @ acts trivially on the fibres of B, the first Betti number bI of the quotient surface v/@v, (V, @v) = (B, @)", is given by the formula (14.24)

b

I

=

{2P + 2 , 2p

if the order of c(O") is finite,

+ 1,

otherwise.

PROOF. We infer from (14.22) that, for any element [S] of Co, the automorphism

of V induces an automorphism of the quotient surface V/@v. Thus Co acts on V/@v as an analytic group of autorilorphisms. Hence, any I-dimensional cohomology class of V/@v with coefficients in R is represented by ad-closed real1-form oft which is invariant under Co. We denote by f the l-form on V induced from 1fr' in an obvious manner. In view of (11.12) we have

where A is a constant. As (11.13) shows, the constant A satisfies the equation (14.25)

AC(1J)

+ AC(1J) =

0.

Conversely, for any constant A satisfying (14.25), we can find ad-closed reall-form oft on V/@v such that "'" = Ad!;j

+ AdCj + dWi(U)

.

In fact, we have seen in the proof of Theorem 11.9 that there exists a d-closed real l-form cp on V of the form

cp = Ad!;i

+ AdCj + dXi(U)

.

*

Assuming that each fixed point a E!1 of g E @, g 1, is covered by (one and) only one disk E j E {Ei }, we choose the l-form cp such that the functions Xj(u) vanish in a neighborhood of each fixed point a E A of g E @,

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30 g

*' 1.

K. KODAIRA

We have, by (14.22), gvcp = Adtj

+

Ad~j

+

dXg,(g-lU) .

Hence, setting

t = Adtj + Ad~j + dw,(u) ,

w,(u)

=

+

Ege~Xg,(g-lu) ,

we obtain a d-closed real I-form ton V which is invariant under @v. Moreover, the functions w;(u) vanish in a neighborhood of each fixed point a E A of g E @, g *- 1. Consequently, the I-form t is induced from a d-closed realI-form '1fr on the quotient surface V/@v. It is clear that '1fr is (induced from) a d-closed real I-form on A/@ if and only if the constant A vanishes. Hence, we infer that h, is equal to 2p + 2 or 2p + 1 according as c(J7) = 0 or *- O. For our purpose, it suffices therefore to verify that the order of c(a) is finite if and only if c(r;) vanishes. Let E:,q denote the spectral sequence associated with the cohomology group H(@x A, Go). The term E'/;,2 can be considered as a subgroup of E~,2 in a canonical manner. Hence, we have the exact sequence 0----> K',' ----> ,g{2(GO) ----> Eg,2 ,

while, since

@

acts trivially on the fibres of B, we have Eg,2

=

HO(@, HO(A, Go»)

=

H2(A, Go) .

Consequently we obtain the exact sequence 0---->

Kl,l

----> ,g{2(GO) ~ H2(A, Go) .

We infer readily that (3c(a) = c(J7) .

As was shown in the proof of Theorem 14.6, the terms Ef,q, p ~ 1, are finite groups. It follows that the group K',' is finite, while H2(A, Go) is isomorphic to the free abelian group Go. Hence we conclude that the order of c(a) is finite if and only if c(r;) vanishes, q.e.d. 15. Elliptic surfaces

By an elliptic surface we shall mean an analytic fibre space of elliptic curves over a non-singular algebraic curve. In view of Theorem 6.1, any elliptic surface is obtained from an elliptic surface whose fibres contain no exceptional curves by applying a finite number of quadric transformations.

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ON ANALYTIC SURFACES, III

Let V be an elliptic surface, Le., an analytic fibre space of elliptic eurves over an algebraic curve A, of which the fibres contain no exceptional curves, and let denote the canonical projection of V onto A. We denote the singular fibres of V by Cap' P = 1,2,3, ... , and assume that the fibres Cap' l~p~l, are of types "'plop' mp ~ 2, respectively, and the fibres Cap, P ~ l + 1, are simple. Let mu denote the least common multiple of the integers mlJ m 2, ••• , m z, and let d = m 1m 2 • • • mz. We fix an arbitrary point au on A - {a p } and construct a d-fold abelian covering Li of A which is unramified over A - faa, au ... , az}, and has d/mp branch points of order mp - 1 over each point ap , P = 0, 1, "', l. We then define V to be the analytic fibre space of elliptic curves Li induced from V and denote by the canonical projection of V onto Li. V is an abelian covering surface of V which is unramified over V - Cao and has djmo branch curves over Cao (see Theorem 6.3). Let @ be the covering transformation group of Li with respect to A. Each covering transformation g E @ induces a covering transformation of V with respect to V in an obvious manner. We denote the induced covering transformation of V by the same symbol g. We have

g

=

giP •

Hence g is an automorphism of the analytic fibre space Vover E. Thus @ acts on V as a group of automorphisms. It is clear that V = V/@. V is free from multiple singular fibres. Let!/ and G denote respectively the functional and the homological invariants of V, and let B be the basic member of the family g(!/, G). Note that B is an analytic fibre space of elliptic curves over the curve Li. We infer from Theorem 14.1 that @ acts on B as a group of automorphisms which leave invariant the curve o(E) c B and that CV, @) = (B, @Y, where (J E Hl(Li, O(B#» . Thus we conclude the following THEOREM

15.1. Every elliptic surface is represented in the form (V, @)

Q",Q"'-l '" QzQlV/@) ,

= (B, @Y ,

where (J is an element of Hl(@ x Li, !2(B#» and the symbols QlJ Q2' .•• , indicate quadric transformations. We infer from Theorems 14.5, 14.6, and 14.7 that the quotient surface

Vj@ is a deformation of an algebraic surface if and only if the first Betti number of iT /@ is even, while the first Betti number of a surface is invariant under quadric transformations. Consequently, we obtain the following

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32

K. KODAIRA

15.2. An elliptic surface is a deformation of an algebraic surface if and only if its first Betti number is even. THEOREM

16. Kahler surfaces

In this section we shall prove the following theorem. THEOREM 16.1. Every compact Kahler surface is a deformation of an algebraic surface. Let V be a compact Kahler surface. If V possesses two algebraically independent meromorphic functions, then V is an algebraic surface (see Theorem 3.1). If V possesses one and only one algebraically independent meromorphic function, then V is an elliptic surface (see § 4) and, since V carries a Kahler metric, the first Betti number of V is even. Hence, by Theorem 15.2, V is a deformation of an algebraic surface. In case V has no meromorphic function except constants, the geometric genus Prt of V is equal to 1, and the irregularity q of V is equal to either 2 or 0 (see Theorem 5.2). Moreover, if q = 2, the surface V is obtained from its Albanese variety by applying a finite number of quadric transformations (see Theorem 5.3). The Albanese variety is obviously a deformation of an algebraic surface. Hence, in case q = 2, the surface V is a deformation of an algebraic surface. In what follows we consider the case in which V has no meromorphic function except constants and the irregularity q of V vanishes. In view of Theorem 6.1 the surface V is obtained from a compact analytic surface W containing no exceptional curves by applying a finite number of quadric transformations Ql> Q2' .. " Qm, i.e.,

Let p" denote the center of the quadric transformation Q". Note that p" is a point on the surface QHQ"-2 ... QI( W). We obtain a small deformation of V by small displacements of the centers PI' P2' "', Pm, while any small deformation of a compact Kahler surface is a Kahler surface." Hence we may assume without loss of generality that for 1

~ A,


'" be the irreducible curves on W and let C; denote the proper transform of Cn on the surface V. Since the irregularity and the geometric genus of Vare 0 and 1, respectively, we infer from the inequality (5.3) that n'(C;)

=

for n = 1, 2, 3, ....

0,

Moreover, applying (5.4) to the curve Kv (KvC;)

~

+ C;,

we obtain the inequality

1.

Hence we conclude that Cn is a non-singular rational curve. We have, by (5.5), the inequality (C;,)

~

- 1,

while the surface W contains no exceptional curve. Hence, we obtain (C;,)

~

- 2.

It is obvious that the holomorphic 2-form f on V is induced from a holomorphic 2-form h on W. We infer in the same manner as in the proof of Theorem 5.4 that the divisor (h) of h vanishes. Consequently, the divisor (f) of f is the sum of the exceptional curves K 1 , K" "', K", introduced by the quadric transformations Qu Q2' ... , Qm, q.e.d.

We set (16.1)

K = (f)

= Kl + K2 + ... + Km •

Thus we denote by K a canonical curve instead of the canonical bundle. Let ® denote the sheaf over Vof germs of holomorphic vector fields. LEMMA 16.2. The second cohomology group H2( V, ®) vanishes. PROOF. By the duality theorem we have

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34

K. KODAIRA

Take an arbitrary element rp of HO( V, n'@an(K». Note that rp is a meromorphic 1-form on V which is holomorphic outside the canonical curve K. The map P maps V - K biholomorphically onto W - P(K), and P(K) is a finite set of points. Hence rp induces a holomorphic 1-form 1/r' on W - P(K), and 1/r' can be extended to a holomorphic 1-form 1/r on W. 1/r induces a holomorphic I-form on V which obviously coincides with rp. Thus rp is holomorphic on V. We conclude therefore that q.e.d. We define 0.'( - K) to be the subsheaf of il' consisting of germs of those holomorphic I-forms whose coefficients vanish on K, and denote by c the inclusion map of il'( - K) into 0.' . The inclusion map c induces a homomorphism c*: H'(V, n'( -K» -> H'(V, il') . In terms of the local coordinates

on V we write

Z" Z2

j = -; E:.f3~J"'f3dz", /\ dZf3 ,

jfJ"'= -j"'f3'

For any germ of holomorphic vector field 8

= £....J ,",'

CII =l

8"'f)/f)z

(»

we set F(8) = E",.fJj"'f38"'dzf3 .

The map F: 8 -> F(8) maps the sheaf ® isomorphically onto ill( - K). Thus we obtain (16.2)

We denote by C' the vector space of LEMMA

(16.3)

J.)

complex variables.

16.3. We have the exact sequence

0 ---> C2m

--->

Hl(V, ill( -K»)

,*

--->

Hl(V, 0.' ) ---> cm

--->

0.

PROOF. Let e be the restriction of the anti-canonical bundle [-K] to the canonical curve K and let il(e) denote the sheaf over K of germs of holomorphic sections of e. Moreover, let ni- denote the sheaf over K of germs of holomorphic I-forms. Note that, on each component K, of K, e coincides with the complex line bundle determined by a divisor of degree 1. We have the exact sequences'

• See K. Kodaira and D. C. Spencer, On a theorem of Lefschetz and the lemma of EnriquesSeveri-Zariski, Proc. Nat- Acad. ScL, U. S. A .• 39 (1953). 1273-1278.

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35

ON ANALYTIC SURFACES, III

o ----> [2,"

--'> 0 ,

-------> [2,1 -------> [2,k

0--'> [2,1( - K)

o,(e)

---> [2," ---)

---->

0.

Combined with (16.2) the above Lemma 16.2 implies the vanishing of H2( V, [2,1( - K», while, by hypothesis, HO( V, 0,1) vanishes. Moreover, it is obvious that HO(K, [1~J and Hl(K, o,(e» vanish. Combining these results with exact cohomology sequences derived from the above exact sequences, we obtain the exact sequences

o- , HI( V, 0,") -------> HI( V, 0,1) -------> H"(K, 0,1) ---> 0 , 0---) HO(V, [1(e»

--->

Hl(V, o,l(-K» __ H1(V, [1") ---) 0,

while we have HO(K, [1(e» ~ C2m HI(K,

[2,k) ~

c

m

,

•

Consequently, we obtain (16.3), q.e.d. Let b, denote the second Betti number of V. Noether's formula (5.8) asserts that while, by (16.1), (K2) = -m.

Hence, we obtain b, = 22

(16.4)

+ m.

Since dim H'(V, D) = dim HO(V, [12) = Pu = 1 , it follows from (16.4) that (16.5)

dim HI(V, [11) = 20

+m

.

Combining this with (16.3) and (16.2) we obtain (16.6)

dim HI( V, ®) = dim HI( V, 0,1(_ K»

= 20 + 2m

.

Let M be a small spherical neighborhood in C'0+2m of the center 0, and let To(M) denote the tangent space of Mat O. Since, by Lemma 16.2, the second cohomology group H2( V, ®) vanishes, there exists5 a complex analytic family eVof compact analytic surfaces Vt, tEO M, such that Vo= V and such that the map Po: fJlat ----> Po(fJlfJt) maps the tangent space To(M) isomorphic ally onto HI( V, ®), where the symbol Po(fJlfJt) denotes the in5 See K. Kodaira, L. Nirenberg, and D. C. Spencer, On the existence of deformations of complex analytic structures, Ann. of Math., 68 (1958), 450-459.

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36

K. KODAIRA

finitesimal deformation of Vt along 81at. We may suppose that the analytic surfaces Vt, t E M, have one and the same underlying differentiable manifold X. We identify X with V, and consider the local complex coordinates Z1, Z2 of Vas local differentiable coordinates on X. We choose6 local complex coordinates of the analytic surface Vt such that = r1(Z, t) and r2=l;iz, t) are differentiable functions of ZlI Z2 depending holomorphically on t. By hypothesis, the surface V carries a Kahler form

rll r2

W

r1

V -1 E

=

gr»/idzr» 1\ dZj3 .

Hence we can define on each surface Vt, t E M, a Kahler form W t which depends differentiably on t and satisfies the condition that Wo=W, provided that M is sufficiently smalV The existence of a Kahler metric on Vt implies that the geometric genus of V t is independent of t. Hence there exists on each surface V t one and only one linearly independent holomorphic 2-formft. We may assumes thatf, depends holomorphically on t. This means that, when ft is written in the form (16.7)

it =

(1f2)

r::: ,/l=1 i "'/l(\;', t)d\;'..(z, t) 1\ dl; j3(z, t) ,

the coefficients i"'/l(r, t) are holomorphic in rlo 1;2 and t. It follows that the canonical curve K t = (ft) of V t is composed of m exceptional curves K tl , K t2 , • • • , K tm depending holomorphically on t ; thus Kt

=

(ft)

=

Ktl

+ Kt2 + ... + K tm •

Consider the integral

t = )!t 1\10.

c

We infer from (16.7) that Ct depends holomorphically on t. Hence, writing ft in place of C01/2Ct-1ft, we obtain (16.8) By (16.4) the second Betti number b2 of X is equal to 22 + m. We choose a Betti base {ZlI Z2' ••• , Z22+m} of 2-cycles on X such that Zlo Z2' ••• , Z22 Z22+Y = K y

C

X - K ,

for

,

J)

= 1,

2, ... , m.

See Kodaira, Nirenberg, and Spencer, loe. cit. Kodaira and Spencer, loco cit. in ref. 3. S See K. Kodaira and D. C. Spencer, Existence of complex structure on a differentiable family of deformations of compact complex manifolds, Ann. of Math. 70 (1959), 145-166, §4. 6

7

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ON ANALYTIC SURFACES, III

Let hj(t), j = 1, 2, .. " b2, be real harmonic 2-forms on Vt with respect to the Kahler form W t such that for j, k = 1,2, ., " b2

•

We set

~ x hlt)

ajk =

/\ hk(t) .

Denoting by (Z,Zj) the intersection multiplicity of Z; and Zj, we have

L:

(16.9)

j

(ZiZj)aJk = O,k .

Hence, the constants a Jlc are rational numbers. We set (16.10) Since Z22h A,22fV(t), ).)

=

=

Kv is homologous to the analytic cycle K tv on Vo the periods 1, 2, ... , m, vanish. Hence we obtain ft = E~~l t-J{t)h/t) \

We infer from (16.7) that the periods t-it) are holomorphic functions of

t. We associate with each Kahler surface V t the point t-(t) = (t-l(t), t-it), "', t- 22 (t»)

in a projective space P21 of dimension 21. Let Q denote the field of rational numbers. LEMMA 16.4. If the point t-(t) is rational over the field Q(-v=T), then V t is an algebraic surface. PROOF. Assume that t-(t) is rational over Q(V -1). Let

wt

=

EM ukh,,(t) .

TaKe a small positive number c. Since EN; aj"t-j(t)Uk = Lft /\

Wt

= 0,

we can find a positive integer n and integers n k satisfying the linear equation and the inequalities

I

Uk -

::

1< c,

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k = 1,2, "', b2

1369

•

38

K. KODAIRA

We define Then we have

~!tl\w=O. Hence, the real harmonic 2-form won Vt is of type (1, 1). Moreover, positive, since (/)t is positive, and the periods

\~Zkw= n

wis

k

are integers. Thus ill is the Kahler form associated with a Hodge metric. Consequently, Vt is an algebraic surface,9 q.e.d. We have

E;~"=l a'kr..j(t)r..k(t) = LIt 1\ It = 0 . Thus we see that r..(t) is a point on the hypersurface S in P2l defined by the equation lO (16.11)

Since (ZiZj) vanishes for i ~ 22 < j, we infer from (16.9) that the determinant Ia,k I of the coefficients ajk, j, k = 1, 2, "', 22, of the equation (16.11) does not vanish. Our purpose is to prove that some members Vt of the family CV are algebraic surfaces. The hypersurface S contains an everywhere dense subset of pointsl l which are rational over Q(v -1). Therefore, in view of Lemma 16.4, it suffices for our purpose to show that the subset r..(M) of S consisting of all points r..(t), t E M, contains a neighborhood of r..(0) in S. Let W,l denote the space of harmonic forms of type (1, 1) on the Kahler surface V. Identifying the cohomology group Hl( V, iV) with HI! by means of the Dolbeault isomorphism, we write the exact sequence (16.3) in the form 1* (16.12) 0 - . C'm ~ H!(V, D,l(-K») _ Hl,l-. Cm- . 0 . We have the isomorphism See K. Kodaira, On Kahler varieties of restricted type, Ann. of Math., 60 (1954), 28-48. This result is due to A. Andreotti. Deformations of compact regular Kahler surfaces with trivial canonical bundles have been studied by A. Weil, A. Andreotti, and H. Grauert. We apply their method to the case in which the canonical bundles are not necessarily trivial. 11 See Th. Skolem, Diophantische Gleichungen, Berlin, 1938, Kap. III, § 1. 9

10

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ON ANALYTIC SURFACES, III

39

F*: HI(V,®)~ HI(V,UI(-K»). Hence we obtain a homomorphism

c* F* Po: To(M)

--->

HU

of the tangent space To(M) into the linear space LEMMA

HI,I •

16.5. For any tangent vector a/at E To(M) the lollowing equal-

ity holds: (16.13) PROOF. We denote by H the orthogonal projection operator of the linear space of differential forms on X onto the subspace of harmonic forms with respect to the Kahler form (j). Moreover, we denote by athe exterior derivative with respect to the variables Zh Z2' The cohomology group Hl(V, ®) is canonically isomorphic to the a-cohomology group of vector forms of type (0, 1) and the infinitesimal deformation po(a/at) is represented by the a-closed vector form

1::=1 al;~ (z)(ajaZ.,)

,

Hence we obtain (16.14) For brevity we write I; for [alt!atJt=o. Obviously I; is ad-closed 2-form. From (16.8) we get

while, since It 1\ It vanishes identically, we have

H(>nce we conclude that the harmonic part HI; of I; is of type (1, 1). On the other hand we infer from (16.7) that

I;

= E/"'/lal;~(z) 1\

dZ/l

+ a form of type (2, 0).

Hence we get

HI; = HE/"'/l al;~(z) 1\ dz~ . Combining this with (16.14) we obtain

(16.15)

t* F* Po(8/8t) =

Writing ;\.;(0) for [a;\.lt)/atlt=o, we have

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1371

HI; .

40

K. KODAIRA

r fri. Jz}

\,'(0) = I

Hence we get

Hft

=

r::

2 =1

A.'lO)h;(O) .

Combining this with (16.15), we obtain (16.13),

q.e.d.

Let T~dO)(S) denote the tangent space of S at \'(0). The holomorphic map \': t -> \'(t) of Minto S induces a linear map A of To(M) into Tx(O)(S). To prove that \'(M) contains a neighborhood of \'(0) in S, it suffices to show that A maps To(M) onto Tx(O)(S). It follows from (16.8) that the coordinates \'j(t) of \,(t) are normalized by the condition: ",22

--

"-'J.k=lUjk\,it)A.k(O) = 1 .

Hence we have A(fJ/fJt) = E~:l \'~(O)fJ/fJ\'j ,

Since Po maps To(M) onto Hl( V, ®), we infer from (16.12) and (16.5) that dim t* F*Po(To(M»

= 20 .

Combined with (16.13) this implies that dim A To(M) = 20 , while it is obvious that dim T;..,(o)(S) = 20 . Hence we conclude that A maps To(M) onto Tx(o)(S). PRINCETON UNIVERSITY AND HARVARD UNIVERSITY

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ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES

By

K. KODAIRA

DEPAR'l'MEN'r OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY

Communicated by D. C. Spencer, June 10,1963

The purpose of this note is to outline our recent results on the structure of compact complex analytic surfaces. Details will be published elsewhere. Our proof of the results is based on the Riemann-Roch theorem of which the complete form has been established recently by M. F. Atiyah and 1. M. Singer.! 1. We denote by Z the ring of rational integers and by C the field of complex numbers. By a surface we shall mean a compact complex analytic surface free from singularities. We consider a surface S and denote by bv the v-th Betti number of S and by Cv the v-th Chern class of 8. In view of the isomorphism H 4(8, Z) "-' Z, we may consider C1 2 and C2 as rational integers. Letting ir!, ... , r j, ... , rb,} denote a Betti base of 2-cycles on 8, we define b+ and b - to be, respectively, the number of positive and negative eigenvalues of the symmetric matrix (rjrk ), where the symbol (rjrk ) denotes the intersection multiplicity of r, and r k • Then we have 2 b+ - b-

=

-

2 3

C2

+3 -1 '

(1)

C!2

We denote by and by nr , respectively, the sheaves over 8 of germs of holomorphic functions and of holomorphic r-forms. We define hr , v

= dim HV(8,nr ),

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= 0.

VOL. 50, 1963

MATHEMATICS:

K. KODAlRA

21$1

We set Pu = hO,2 h2 ,0, q = hO, \ and call Pu and q, respectively, the geometric genus and the irregularity of S. We denote by K the canonical bundle on S. Note that the Chern class of K is equal to -CI. The surface S is said to be regular if and only if the irregularity q vanishes. We have the exact sequence Ii

O-C-e-de-O

and the corresponding exact cohomology sequence

0- HoeS, de) -

HI(S, C) -

HI(S, 0) -

(2)

... ,

where dl'l is the sheaf over S of germs of d-closed holomorphic I-forms.

1. Every holomorphic I-form defined on Sis d-closed. 2. If d-closed holomorphic I-forms 'PI, 'P2, ... , 'Ph defined on S are linearly independent, then the d-closed I-forms 'PI, 0, then S is an algebraic surjace. In particular we have THEOREM 2. If CI 2 > 0, then S is an algebraic surface. If Pu ~ 2, then there exist on S nonconstant meromorphic functions. Hence, we get 3. If Pu Moreover, we have

THEOREM

~ 2, then S

is either an algebraic surface or an elliptic surface.

~

3, then S is either an algebraic surface or an elliptic surface. 4 Now we consider the case in which bi is even and Pu = 0. Since, by Theorem 1, b+ = 1, there exists a cohomology class c E H2(S, Z) with c2 > 0, and the vanishing of Pu implies the existence of a complex line bundle F over S with c(F) = c. THEOREM

4.

If hi.

0

Consequently, we obtain THEOREM 5. If bi is even and if Po = 0, then S is an algebraic surface. 3. Any surface is obtained from a surface containing no exceptional curve (of the first kind) by means of several quadric transformations. s Note that the first Betti number, the irregularity, and the geometric genus of a surface are invariant under any quadric transformation. In this section we assume that the surface S has no meromorphic function (except constants) and contains no exceptional curve. (A) The case in which bi is even. It follows from Theorems 1, 3, 4, and 5 that b] = 2q, q = hi, 0 ~ 2, and Pu = 1. Hence, we conclude in the same manner as ill the case of Kahler surfaces6 that q is equal to either 2 or 0, that, if q = 2, the surface S is a complex torus, and that, if q = 0, the canonical bundle K on S is trivial. Following A. Weil, we call a regular surface with trivial canonical bundle a Ka surface. In case q = 0, the surface S is, then, a Ka surface. We note that the second Betti number of any Kg surface is equal to 22. (B) The case in which bl is odd. It follows from Theorems 1, 2, 3, and 4 that bl = 2q - 1, q = hI, 0 + 1, b+ = 2pu, hi, 0 ~ 2, Pu ~ 1, and Cl 2 ~ 0. We have 3. Ifpo = l,thencl2 = 0. (i) If hi, 0 = 2, then q = 3, bl = 5, and there exist two linearly independent holomorphic I-forms '1'1, '1'2 E HO(S, de). Moreover, the exterior product '1'1 1\ '1'2 does not vanish identically. Hence, Pu = 1 and therefore, by Lemma 3, Cl 2 = 0. The formula (6) then proves that b2 = -4. This is a contradiction. (ii) If hi, 0 = 1, then q = 2 and b] = 3. We take a holomorphic I-form 'I' E HO(S, de), 'I' :;t. 0, and find a I-form IT of type (1, 0) defined on S such that dlT = 'I' 1\ iP and such that IT + 0", If' and iP generate the d-cohomology group of I-forms on S. We then obtain multivalued holomorphic functions wand z on S satisfying the differential equations LEMMA

dw =

IT

+ Z'I',

dz

=

'1'.

We infer that the exterior product dw 1\ dz never vanishes. Hence, the space C2 of the complex variables wand z forms the universal covering surface of S. The covering transformation group of C2 over S is generated by the linear transformations

g.:w - w

+ a.,z + (3., z ( 57)

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z

+

lX., V

=

1,2,3,4

VOL. 50, 1963

MATHEMATICS:

K. KODAIRA

221

such that CR,83 rf 0,

where m is a positive integer. It follows that S is an elliptic surface. This is contrary to the assumption that S has no meromorphic function (except constants). (iii) If hI, 0 = 0, then q = 1, and bl = 1. Therefore, the Picard variety (p = HI(S, 0)jHl(S, Z) is isomorphic to the Lie group ejz. We shall show that Pu = O. Suppose that Pu = 1. Then, for any complex line bundle P E (P, the inequality dim HO(S, 0(P»

+ dim HO(S,

0(K - P»

~

1

holds. It follows that there exist infinitely many irreducible curves on S. This contradicts the assumption that S has no meromorphic function 7 (except constants). Thus, we obtain THEOREM 6. Assume that S has no merom01'phic function except constants and contains no exceptional curve. If b1 is even, then S is either a complex torus or a K3 surface. If b1 is odd, then b1 = q = 1, and Pu = 0. 4. We infer from the above results that, if b1 is even, then S is an algebraic surface or an elliptic sUljace, or S is obtained from a complex torus or a K3 sUljace by means of several quadric transformations. If b1 is odd, then either S is an elliptic surface, or b1 = q = 1, and Pu = 0. Complex tori and K3 surfaces are deformations of algebraic surfaces,8 while an elliptic surface is a deformation of an algebraic surface if and only if its first Betti number is even. 9 Hence, we conclude: THEOREM 7. A compact complex analytic surface S is a deformation of an algebraic surface if and only if the first Betti number b1 of S is even. THEOREM 8. If the first Betti number b1 of S is odd, then S is an elliptic sUljace, except the case in which b1 = q = 1, and Pu = 0. 1 Atiyah, M. F., and I. M. Singer, "The index of elliptic operators on compact manifolds," Bull. Amer. Math. Soc., 69, 422-433 (1963). 2 Hirzebruch, F., Neue (opologische Methoden in der algebraischen Geometrie (Berlin: Springer, 1956). • Kodaira, K., "On compact complex analytic surfaces, I," Annals of Maih., 71,111-152 (1960). • Ibid., section 5. 6 Grauert, H., "Uber Modifikationen und exzeptionelle analytische Mengen," Math. Ann., 146, 331-368 (1962). 6 Kodaira, K., op. Ctt., section 5. 7 Kodaira, K., 0p. cit., Theorem 5.1. 8 The structure of K. surfaces has been studied by A. Weil, A. Andreotti, and H. Grauert. 9 Kodaira, K., "On compact analytic surfaces, III," to appear in the Annals of Jlath.

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ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES, II* By K.

KODAIRA

DEPAR'l'MENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY

Communicated by D. C. Spencer, April 13, 1964

This note is a continuation of our previous report! on the structure of compact complex analytic surfaces. 5. We shall employ the notation of our previous report. Thus we denote by S a surface and by bPI C" pg, q, ... , respectively, the I'th Betti number, the I'th Chern class, the geometric genus, the irregularity,. .. of S. Any complex line bundle over a regular surface is determined uniquely by its Chern class. Hence, a regular surface is a K~ surface if and only if its first Chern class vanishes. It follows that any deformation of a Ka surface is a Ka surface. In this section we shall outline a proof of the following theorem which has been conjectured earlier by A. Weil and independently by A. Andreotti. 2 THEOREM 9. Every Ka surface is a deformation of a nonsingular quartic surface in a projective 3-space. (i) Assume that S is a Ka surface. Then, using Theorem 1 and the formula (6), we get C2

= 24, b2 = 22, b+ = 3, b- = 19.

(8)

Let Ir 1, . . . , r J , ••• , rn} be a Betti base of 2-cycles on S and, for any cohomology class c E H2(S, Z), let c(r i ) denote the value of c on r j • We choose eJ E H2(S, Z) such that eJ(rk) = OJk for j, k = 1, 2, ... , 22 and define There exists on S a nonvanishing holomorphic 2-form y.,. Andreotti we associate with S the point

x=

XeS)

=

Following Weil and

0\1, .. "X" ... ,X 22),

in a projective space p21 of dimension 21. We have A(X, X) = Js y., 1\ y.,

=

O.

Thus, X is on the hypersurface M in p21 defined by the quadric equation: A(x, x) = O. This result is due to Weil and Andreotti. 3 LEMMA 4. A cohomology class c E H2(S, Z) is the Chern class of a complex line bundle over S if and only if the point m = (mI' ... , mJ, ..• , m22), mj = c(rJ), satisfies the linear equation: A(X, m) = 0, X = XeS). LEMMA 5. If the linear equation: A(X, m) = 0, X = XeS), admits one and only one rational solution m E p21 and if, moreover, that solution m satisfies the quadric equation: A(m, m) = 0, then S is an analytic fiber space of elliptic curves over a projective line 11 of which the singular fibers 4 are either of type II or of type II. Proof: We choose homogeneous coordinates mj of m such that ml, ... , mJ> ... , m22 are rational integers having no common divisor and let e = L mjej. Then, using Lemma 4, we infer from the uniqueness of the rational solution m, that, for any 1100

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1101

complex line bundle F over S, there exists an integer n such that c(F)-ne is an element of a finite order. Hence, we get (9)

This proves that S is nonalgebraic. By Lemma 4 there exists a complex line bundle E over S with c(E) = e. Using the Riemann-Roch inequality (7) we obtain

while E is nontrivial. Hence, we infer the existence of nonconstant meromorphlc functions on S. Consequently, S is an analytic fiber space of elliptic curves over a nonsingular algebraic curve.l. Since hl,o vanishes, .l is ratjonal. Since the canonical bundle of S is trivial, S has no multiple singular fiber, Moreover, by (9), S contains no curve G with (G2) oz!; 0, Therefore the singular fibers of S are either of type II or of type II, q.e.d. (ii) In this subsection we assume that S is an e11iptic K3 surface of which the singular fibers are either of type II or of type II. The base curve .l of S is a projective line. We indicate a point on .l by its nonhomogeneous coordinate u. We denote by G(u) the fiber of S over u and by ,0 = ,0(u) the functional invariant5 of S. Suppose that S has j singular fibers G(al), ... , G(aJ ) of type II and r singular fibers G(Tl), ... ,G(TT) of type II. Then we have6 j

+ 2r =

C2

= 24.

relations7

There are certain between the type of C(u) and the behavior of,0 at u. We infer from those relations that r is not greater than 8, that ai, ... , aJ are simple poles of ,0 and that 8

r

,01(,0 - 1) = a II (u - T.) p=1

12

II

(u - Tp)3j

p=7+1

II

(10)

",=r+l

where a oz!; 0, Tp oz!; (flo" and Tp oz!; T),. for JI < A ;;; r. S belongs to the family8 :reg, G), where G denotes the homological invariant of S. Let B be the basic member of :r(,0, G). The cohomology group H2(.l, G) vanishes. Hence, :rCg, G) consists of the elliptic surfaces 9 B h*(8), 8 E HI(.l, fl(f». Consequently, S is a deformation of B. (iii) Let p2 denote a projective plane on which a system of homogeneous coordinates (x,y,z) is fixed. Take two copies p2 X Co and p2 X Cl of p2 X C and form their union

W

=

p2

X Co U p2 X C1

by identifying (x,y,z,u) E p2 X Co with (XI, Yl, Zl, Ul) E p2 X C1 if and only if UUl = 1, U4Xl = X, U6YI = y, Zl = z. Note that W is a complex analytic fiber bundle of projective planes over the pro.jective line .l = Co U Cl • For any point T = (TO, TI, ... , TS, (fl, . . . , (f12) in the space C21 of 21 complex variables, we set 8

12

g(u) = ToII(u - Tp), h(u) =II(u - (fp), ,0, = g(u}3j[g(U)3 - 27h(u ~=1

v=l

and define a subvariety B, of W by the equations

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2J,

1102

MATHEMATICS: K. KODAIRA

y2Z - 4x 3 + g(U)xz 2 + h(u)z3

l

yl2z1 -

4Xl3

=

PROC.

N. A. S.

0,

+ UI g(1/UI)XIZI 2 + UI 12h(1/UI) ZI 8

3

= O.

BT is an analytic fiber space of elliptic curves over l:1 of which the functional invariant is gjT and the singular fibers are either of type II or of type II, provided that T satisfies the following conditions: (a) TO ~ 0; (b) if ITA = TI" J.I ~ 0, then ITp ~ ITA for v ~ A; (c) gjT has no multiple pole. Moreover, BT has a global holomorphic section over l:1 defined by the equations: x = Z = Xl = Zl = O. Clearly, BT is a deformation of B 8, 0 = (1,1, ... , 1,0, ... , 0). Now we let T = (3aY', TI, ••• , Tg, TJ, • . . , TTl ITT+I, ••• , ITI2). Then gjT coincides with gj, and therefore BT coincides with the basic memberlO B of g:(gj, G). Consequently, B is a deformation of Bo. (iv) Let S be an arbitrary K3 surface. Denoting by e the sheaf over S of germs of holomorphic vector fields, we have dim HI(S, e) = hl,1 = 20, dim H2(S, e) = h l,2 = O. Hence, we infer the existence l l of an effectively parametrized complex analytic family of small deformations St, t E U, of S, where U denotes a domain in the space C20 of 20 complex variables. Assuming that the surfaces St, t E U, have one and the same underlying differentiable manifold X, we define A(St) by means of a Betti base Ir j I of 2-cycles fixed on X. It has been'shown by A. Weil and independently by A. Andreotti that t - A(St) is a biholomorphic map of U onto an open subset of M. Moreover, by (8), the signature of the quadric form A(x, x) is (3, 19). Hence, we can find a point t E U such that A = lI(St) satisfies the hypotheses of Lemma 5. The corresponding surface St is then an elliptic K3 surface whose singular fibers are either of type II or of type II. Therefore, by the results of (ii) and (iii), St is a deformation of Bo. Hence, S is a deformation of B 8• This result implies that any nonsingular quartic surface in a projective 3-space is a deformation of Ba. Consequently, every K3 surface is a deformation of a nonsingular quartic surface in a projective 3-space. 6. Let C2 denote the space of two complex variables Zl and Z2. THEOREM 10. If the canonical bundle of S is trivial, then S is a K3 surface, a complex torus, or an elliptic surface of the form C2/G, where G is a properly discontinuous nonabelian group of affine transformations without fixed points of C2 which leave invariant the 2-form dZ I /\ dz 2 • The first Betti number of C2/G is equal to 3. We shall outline a proof of this theorem. By hypothesis the first Chern class CI of S vanishes. Hence, by a result of Atiyah and Hirzebruch,12 the Todd genus of S is even, while the geometric genus pg of S equals 1. Consequently, by (6), the irregularity q of S is even and 12q - 2bl

+ b+ + b-

= 22.

+

In case bl is even, we have, by Theorem 1, b+ = 3, b1 = 2q, and therefore 8q = 0, then S is a K3 surface. If

°

b- = 19. Hence, q is equal to either or 2. If q q = 2, then S is a complex torus.

In case b1 is odd, we have b+ = 2, 2q = b1 + 1 ~ 1, and 8q + b- = 18. Hence, q equals 2, and bl equals 3. In 'view of Theorem 6, S is therefore an elliptic surface. Moreover, we infer in the same manner as in section 3 (ii) that the universal cover-

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1103

ing surface of S is C2, and the covering transformation group G of C2 is generated by the affine transformations (ZI, Z2) -

(ZI

+ a pZ2 + {3p, Z2 + aJ,

"=

1,2,3,4,

of which the coefficients satisfy the condition that aa

=

a4 = 0, ll't{3a ~ 0, ala2 -

a2al

=

m{34 ~ 0,

where m is a rational integer. 7. Let S be a surface free from exceptional curve, and let bl , Pu, and Cl denote, respectively, the first Betti number, the geometric genus, and the first Chern class of S. LEMMA 6. If Pu is positive, then Cl 2 is nonnegative. LEMMA 7. If b1 is even, Pu > 0, and Cl = 0, then the canonical bundle of S is trivial. LEMMA 8. If Pu is positive, C1 2 = 0, and Cl ~ 0, then S is an elliptic surface. Proof: If Pu > 0 and if Cl ~ 0, then, in view of Theorem 6, S is either algebraic or elliptic. On the other hand, in case 8 is algebraic, this lemma is reduced to a classical result of Italian geometers.Ja Clas.

bl

Po

I II III IV V VI VII

Even 0 4 Even Even Odd 1

0

CI

+ + + + +

=0 =0 ,.,0

CI'

0 0 0

+ 0

0

Structure

Algebraic Ka surfaces Complex tori Elliptic Algebraic Elliptic ?

With the aid of these lemmas we derive from Theorems 2, 5, 6, 8, and 10 the following theorem (see above table). Note that an elliptic surface is free from exceptional curve if and only if its Chern number C1 2 vanishes. THEOREM 11. Surfaces free from exceptional curves can be classified into the following seven classes: I. the class of algebraic surfaces free from exceptional curves with Pu = 0; II. the class of Ka surfaces; III. the class of complex tori (of complex dimension 2); IV. the class of elliptic surfaces with b1 == 0(2), Pu > 0, and Cl 2 = 0; V. the class of algebraic surfaces free from exceptional curves with Pu > 0 and 2 C1 > 0; VI. the class of elliptic surfaces with bl == 1(2), Pu > 0, and C1 2 = 0; and VII. the class of surfaces free from exceptional curves with b1 = 1 and Pu = O.

* This work was partly supported by the National Science Foundation grant G7030. 1 Kodaira,

K., "On the structure of compact complex analytic surfaces," these

PROCEEDINGS,

50,218-221 (1963). 2 Compare Grauert, H., "On the number of moduli of complex structures," in Contribution8 to the Function Theory (Tata Institute of Fundamental Research, 1960). 3 Ibid. • Kodaira, K., "On compact analytic surfaces II-III," Ann. of Math., 77, S63-626; 78, 1-40 (1963), Theorem 6.2. 5 Ibid., section 7. • Ibid., Theorem 12.2. 7 Ibid., section 9, Table 1.

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PRoe. N. A. S.

Ibid., Definition 8.1. Ibid., section 11. 10 Ibid., Theorem 10.2. 11 Kodaira, K., L. Nirenberg, and D. C. Spencer, "On the existence of deformations of complex analytic structures," Ann. of Math., 68, 450-459 (1958). 12 Atiyah, M. F., and F. Hirzebruch, "Riemann-Roch theorems for differentiable manifolds," Bull. Amer. Math. Soc., 65,276-281 (1959). 13 Enriques, F., Le Superficie Algebriche (Bologna, 1949). S

9

( 58) 1381

ON THE STRUCTURE OF COMPACT COMPLEX ANALYTIC SURFACES by K.

KODAIRA

By a surface we shall mean a compact complex manifold of complex dimension 2. We fix our notation as follows. S : a surface, b,: the Ii-th Betti number of S, c, : the Ii-th Chern class of S, eJ : the sheaf over S of germs of holomorphic functions, q = dim H1(S, eJ) : the irregularity of S, pg = dim H2(S, eJ) : the geometric genus of S. Note that C1 2 and C2 are (rational) integers. By a theorem of Grauert [2J, any surface is obtained from a surface containing no exceptional curve (of the first kind) by means of a finite number of quadric transformations. Hence, in order to study the structure of surfaces, it suffices to consider surfaces containing no exceptional curves. In what follows we assume that all surfaces under consideration contain no exceptional curves. DEFINITION 1. By an elliptic sur face we shall mean a surface S with a holomorphic map 1[1' of S onto a non-singular algebraic curve d such that the inverse image 1[I'-1(U) of any general point u Ed is an elliptic curve. We call L1 the base curve of the elliptic surface S. DEFINITION 2. (A. Weil). We call a surface S a K3 surface if S is a deformation of a non-singular quartic surface in a projective 3-space. MAIN THEOREM. Surfaces (containing no exeptional curves) can be classified into the following seven classes: I) the class of algebraic surfaces with pg=O; II) the class of K3 surfaces; III) the class of complex tori (of complex dimension 2); IV) the class of elliptic surfaces with b1== 0 (2), pg~l,CdO; V) the class of algebraic surfaces with pg~l, C1 2 >0; VI) the class of elliptic sur faces with b1== 1 (2), pg~ 1 ; VII) the class of surfaces with b1=q=1, pg=O.

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STRUCTURE OF SURFACES 2 class bl Pu Cl C1 structure I even 0 algebraic II 0 1 =0 0 K3 surfaces III 4 1 = 0 0 complex tori IV even + =1= 0 0 elliptic V even + + algebraic VI odd + 0 elliptic VII 1 0 ? An elliptic surface is a deformation of an algebraic surface if and only if its first Betti number is even (see [4J). Therefore the following theorem follows from the main theorem.

THEOREM. A surface is a deformation of an algebraic surface if and only if its first Betti number is even. REMARK: The class VII contains many elliptic surfaces. In fact, for any preassigned finite abelian group A, we find an elliptic surface of the class VII whose first torsion group is isomorphic to A. We obtain examples of non-elliptic surfaces of the class VII as follows: Let C2 denote the space of two complex variables (Zl, Z2) and let U=C2- (0, 0). Choose a properly discontinuous group § of analytic automorphisms without fixed points of U in an appropriate manner. Then the quotient surface S= Vj!} is a non-elliptic surface of the class VII. Note that S= Vjg is a deformation of an elliptic surface. As far as we know there is no example of a surface which cannot be deformed into surfaces with non-constant merom orphic functions. We shall outline a proof of the main theorem. Let ()* be the multiplicative sheaf over S of germs of non-vanishing holomorphic functions and let Z denote the ring of rational integers. We have the exact sequence (1)

a* ··· _____ Hl(S, () _____ Hl(S, ()*) _____ H2(S, Z) ->H2(S, () _____ ....

Each element F of Hl(S, ()*) represents a complex line bundle over Sand c(F) =0* F is the Chern class of F. Let {)(F) denote the sheaf over S of germs of holomorphic sections of F. In the case of complex line bundles over surfaces, the Riemann-Roch-Hirzebruch theorem can be formulated as follows: (2)

2 1 1 H'(S, ()(F» = 2(C2+C1C) + 12(c12 +c 2)'

~(-l)'dim

c = c(F)

(see Atiyah and Singer [lJ). This theorem implies the Noether formula (3)

and the Riemann-Roch inequality

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1383

K.

(4)

KODAIRA

~ ~ (C 2+ClC) +Pu-q+I,

dim HO(S, 6(F» +dim HO(S, 6(K-F»

where K denotes the canonical bundle of S. 1. Every holomorphic I-form on a sur face is d-closed. 2. Let !Pl, !P2, ... , !pn be holomorphic I-forms on S. If !Pl, ... , !pn are linearly independent, then the d-closed I-forms !Pl, ... , !pn, iPt, ... , ipn are dcohomologically independent. Letting {Tl' ... , T j , .. " T b2 } be a Betti base of 2-cycles on S and denoting by I(rj , T k ) the intersection mUltiplicity of T j and T k , we define b+ and b- to be respectively the number of positive and negative eigenvalues of the nonsingular symmetric matrix (I(Tj, Tk»' Moreover we denote by h the number of linearly independent holomorphic I-forms on S. With the aid of Theorems 1 and 2, we obtain from the Hirzebruch index theorem and the Noether formula (3) the equality THEOREM THEOREM

2q-b l +b+-2pu = 1,

while we have the inequalities

1

q 2: 2bl 2: h 2: bl-q,

b+ 2: 2pu'

Hence we obtain the following 3. If bl is even, then bl =2q, b+=2Pu+I and h=q. If bl is odd, then bl =2q-I, b+=2pg and h=q-1. THEOREM

We have the formula (5) c 2+8q+b- _ {lOPg+9, if bl is even, 1 lOPu+ 8, if bl is odd. Let us consider the case in which bl is even. By the above results there exist q linearly independent d-closed holomorphic I-forms !Pl, !P2, ... ,!pq on S. Let {rl, ... , rj, ... , r2q} be a Betti base of I-cycles on S and let COROLLARY.

W'J

= l!p" TJ

Then, by Theorem 2, the vectors Wj

= (WlJ, ... , W,j, ... , Wqj),

j

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

are linearly independent with respect to real coefficients and generate a discontinuous subgroup ~ of the vector group Cq of dimension q. We call A =CWlJ the Albanese variety attached to S and define a holomorphic map