Complex Manifolds and Deformation of Complex Structures (Grundlehren der mathematischen Wissenschaften)

Classics in Mathematics Kunihiko Kodaira

Complex Manifolds and Deformation of Complex Structures

Kunihiko Kodaira

Complex Manifolds and Deformation of Complex Structures Reprint ofthe 1986 Edition

^

Spri]nser 'g'

Originally published as Vol. 283 in the series Grundlehren der mathematischen Wissenschaften

Library of Congress Control Number: 2004113281 Mathematics Subject Classification (2000): 32-01,23C10,58C10 ISSN 1431-0821 ISBN 3-540-22614-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronlinexom © Springer Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-fi-ee paper

41/3142YL-5 4 3210

Dedicated to my esteemed colleague and friend D. C. Spencer

Preface

This book is an introduction to the theory of complex manifolds and their deformations. Deformation of the complex structure of Riemann surfaces is an idea which goes back to Riemann who, in his famous memoir on Abelian functions published in 1857, calculated the number of effective parameters on which the deformation depends. Since the publication of Riemann's memoir, questions concerning the deformation of the complex structure of Riemann surfaces have never lost their interest. The deformation of algebraic surfaces seems to have been considered first by Max Noether in 1888 (M. Noether: Anzahl der Modulen einer Classe algebraischer Fldchen, Sitz. Koniglich. Preuss. Akad. der Wiss. zu Berlin, erster Halbband, 1888, pp. 123-127). However, the deformation of higher dimensional complex manifolds had been curiously neglected for 100 years. In 1957, exactly 100 years after Riemann's memoir, Frolicher and Nijenhuis published a paper in which they studied deformation of higher dimensional complex manifolds by a differential geometric method and obtained an important result. (A. Frolicher and A. Nijenhuis: A theorem on stability of complex structures, Proc. Nat. Acad. Set, U.S.A., 43 (1957), 239-241). Inspired by their result, D. C. Spencer and I conceived a theory of deformation of compact complex manifolds which is based on the primitive idea that, since a compact complex manifold M is composed of a finite number of coordinate neighbourhoods patched together, its deformation would be a shift in the patches. Quite naturally it follows from this idea that an infinitesimal deformation of M should be represented by an element of the cohomology group H^{M,&) of M with coefficients in the sheaf 0 of germs of holomorphic vector fields. However, there seemed to be no reason that any given element of H^(M, S) represents an infinitesimal deformation of M. In spite of this, examination of familiar examples of compact complex manifolds M revealed a mysterious phenomenon that dim H^(M, S) coincides with the number of effective parameters involved in the definition of M. In order to clarify this mystery, Spencer and I developed the theory of deformation of compact complex manifolds. The process of the development was the most interesting experience in my whole mathematical life. It was similar to an experimental science developed by

Vlll

Preface

the interaction between experiments (examination of examples) and theory. In this book I have tried to reproduce this interesting experience; however I could not fully convey it. Such an experience may be a passing phenomenon which cannot be reproduced. The theory of deformation of compact complex manifolds is based on the theory of elliptic partial differential operators expounded in the Appendix. I would like to express my deep appreciation to Professor D. Fujiwara who kindly wrote the Appendix and also to Professor K. Akao who spent the time and effort translating this book into English. Tokyo, Japan January, 1985

KUNIHIKO KODAIRA

Contents

CHAPTER 1

Holomorphic Functions §1.1. Holomorphic Functions §1.2. Holomorphic Map

1 1 23

CHAPTER 2

Complex Manifolds §2.1. Complex Manifolds §2.2. Compact Complex Manifolds §2.3. Complex Analytic Family

28 28 39 59

CHAPTER 3

Differential Fornis, Vector Bundles, Sheaves §3.1. Differential Forms §3.2. Vector Bundles §3.3. Sheaves and Cohomology §3.4. de Rham's Theorem and Dolbeault's Theorem §3.5. Harmonic Differential Forms §3.6. Complex Line Bundles

76 76 94 109 134 144 165

CHAPTER 4

Infinitesimal Deformation §4.1. Differentiable Family §4.2. Infinitesimal Deformation

182 182 188

CHAPTER 5 Theorem of Existence §5.1. Obstructions §5.2. Number of Moduli §5.3. Theorem of Existence

209 209 215 248

CHAPTER 6 Theorem of Completebess §6.1. Theorem of Completeness §6.2. Number of ModuH §6.3. Later Developments

284 284 305 314

T CHAPTER 7

Theorem of Stability §7.1. Differentiable Family of Strongly Elliptic Differential Operators §7.2. Differentiable Family of Compact Complex Manifolds

320 320 345

APPENDIX

Elliptic Partial Differential Operators on a Manifold by Daisuke Fujiwara §1. Distributions on a Torus §2. Elliptic Partial Differential Operators on a Torus §3. Function Space of Sections of a Vector Bundle §4. Elliptic Linear Partial Differential Operators §5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation §6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations §7. Elliptic Operators in the Hilbert Space L^{X, B) §8. C°° Differentiability oi (p{t)

363 363 391 419 430 438 443 445 452

Bibliography

459

Index

461

Chapter 1

Holomorphic Functions

§1.1. H o l o m o r p h i c Functions (a) Holomorphic Functions We begin by defining holomorphic functions of n complex variables. The n-dimensional complex number space is the set of all n-tuples ( z i , . . . , z„) of complex numbers z„ / = 1 , . . . , n, denoted by C". C" is the Cartesian product of n copies of the complex plane: C " = C x - • - x C . Denoting ( z i , . . . , z„) by z, we call z = ( z j , . . . , z„) a point of C", and Z i , . . . , z„ the complex coordinates of z. Letting Zj = X2j-i-^ix2j by decomposing z, into its real and imaginary parts (where / = V H [ ) , we can express z as Z = {Xi, X2, . . . , X2n-l9

X2n)'

(l-l)

Thus C" is considered as the 2n-dimensional real Euclidean space R^" equipped with the complex coordinates. Xi, X2,..., X2„_i, X2„ are called the real coordinates of z. Let z = ( z i , . . . , z„) and w = ( W j , . . . , w„) be points in C". We define the linear combination Az + fiw of z and w, viewed as vectors, by Az + /iw = (Azi + fjLWi,..., Az„ + /xw„), where A and /JL are complex numbers. This makes C" a complex linear space. The length of z = ( z i , . . . , z„) is defined by |z|=V|zi|^+--- + |z„p.

(1.2)

N = |A||z|,

(1.3)

| z + w | ^ | z | + |w|.

(L4)

Clearly we have

The distance of the two points z, w e C" is given by \z — w = V | z i - W i P + . . - + | z , - w J ^

(1.5)

2

1. Holomorphic Functions

We introduce a topology on C" by the identification with R^" with the usual topology. Thus, for example, a subset D c : C" is a domain in C" if D is a domain considered as a subset of R^". Again, a complex-valued function / ( z ) = / ( z i , . . . , z„) defined on a subset D in C" is continuous if/(z) is so as a function of the real coordinates Xi, X2,..., X2„. Now we consider a complex-valued function / ( z ) = / ( z i , . . . , z„) of n complex variables Z j , . . . , z„ defined on a domain D c : C". Definition 1.1. If/(z) = / ( z i , . . . , z„) is continuous in Dc= C", and holomorphic in each variable z^, /c = 1 , . . . , n, separately, / ( z ^ , . . . , z„) is said to be holomorphic in D. We also call / ( z ) = / ( z i , . . . , z„) a holomorphic function ofn variables Z i , . . . , z„. Here, by saying t h a t / ( z i , . . . , z^,..., z„) is holomorphic in Z/, separately, we mean that / ( z i , . . . , z„) is a holomorphic function in z^ when the other variables Z i , . . . , Zfc_i, z^+i,..., z„ are fixed. The fundamental Cauchy integral formula with respect to a circle for holomorphic functions of one variable is extended to the case of holomorphic functions of n variables as follows. Given a point c = ( c i , . . . , c„) e C" and positive real numbers r j , . . . , r„, we put Uric)

= {z\z

= ( Z i , . . . , Z J I |Zfc - Cfcl < Tfc, /C = 1, . . . , w } ,

(1.6)

where r denotes ( r i , . . . , r„). Let Ur^{Ck) be the disk with centre Cj, and radius r^ on the z^-plane. Then we have UM=Ur,ic,)X'"XUrSCn)-

(1.7)

Thus we call Uric) the polydisk with centre c. We denote by C^ the boundary of Ur^icj^), that is, the circle of radius r^ with centre c^ on the z^-plane. Of course C^ is represented by the usual parametrization Oj^-^ yiO^) = c^-^ r^ e'^^ where O^OJ^^ITT. The product of Ci, C 2 , . . . , C„ C" = C,X''XCn

(1.8)

is called the determining set of the polydisk Uric). C" is an n-dimensional torus. Given a continuous function i/^(^) = i/^(fi,..., ^„), with ^1G C i , . . . , ^„ G C„, we define its integral over C" by

f

ii^iaj^i•

Jo

• • d^„ = I • • • I

Jo

Hjiie,),...,

Had^i'"dCn

yn{e„))y[{0,)...

y'„{0„) dO,... dd„.

(1.9)

§1.1. Holomorphic Functions

3

Theorem 1.1. Let f=f(zi,..,, !„) be a holomorphic function in a domain DciC". Take a polydisk UXc) with [Ur(c)]c^ D. Then for ze Ur{c)J{z) is represented as

\27ri) Jc" •^«='^i I (^^^^^"tf^*--«-

v„|). Proof For simplicity we consider the case n = 2. The general case is proved similarly. Since, by hypothesis, P(z) is convergent, there exists a constant M such that |a^^^2wr^tv^2|^M; = - : ( z - z ) . li

Here z and z are not independent variables, but considering them as if they are independent, we define the partial derivatives of/(z) with respect to z and z by

J-i'l-i^, 2\ax

by)

^=i(^+,-^. dz 2\dx dy)

(1.18)

8

1. Holomorphic Functions

In terms of u and v, we have — = 2{Ux-^Vy)-\--{-Uy -r2{ux-^Vy)-\--(

dz

2

+ Vj,

(1.19) dz

2

Therefore by use of (1.18), the Cauchy-Riemann equation: u^ = Vy, Uy = -v^, is written as — = 0. dz

(1.20)

Thus a continuously differentiable function/(z) is a holomorphic function of z in a domain D if and only if df/dz = 0 identically in D. I f / ( z ) is holomorphic, dfI dz = u^ + iv^= f'{z) by (1.19), namely, for a holomorphic function/(z), the partial derivative {df/dz){z) is identical to the complex derivative df{z)/dz. Next consider a function/(z) = / ( z i , . . . , z„) of n complex variables. Put f{z) = u-\-iv as above, / ( z ) is said to be continuously differentiable, C\ C°°, etc. if u and v are continuously differentiable, C\ C°°, etc. in the real coordinates X i , . . . , X2„. L e t / ( z ) be a continuously differentiable function in a domain W yXO) = (z, re'^ + C2, Z 3 , . . . , z„), 0 ^ ^ ^ 277. By considering the integral fiiz)=z— 27n J

—

: i-Z2 l

rf^,

\Z2-C2\v"~^ + - • • + a,(z) is called a distinguished polynomial. Theorem 1.10 (Weierstrass Preparation Theorem). For any sufficiently small e > 0 , there exists 5 = 5 ( e ) > 0 such that in the poly disk L^g,s(0) = {(w, z)| I w| < e, IZ2I < 5 , . . . , |z„| < 5}, /(w, z) can he represented uniquely as the product of a distinguished polynomial P(w, z), and a non-vanishing holomorphic function M(W, z): (1.28)

/ ( W , Z ) = M(M;,Z)P(W,Z).

Proof. By the assumption,/(w, 0) = w*(fc5 + b5+iW + - • •) with b^T^O, where 5 is a natural number. Hence, e being taken sufficiently small, /(w, 0)^0 for 0 < I w| ^ e. Let JJL be the minimum of |/( w, 0)| on the circle | w| = e. Then |/(w, 0)1 ^ /x > 0 if I w| = e. Therefore taking 8 = 8(e) sufficiently small, we have |/(w, z ) | ^ / x / 2 > 0

if \w\ = e, |z2| < 5 , . . . , |z„| < 6.

Hence c7fc(z)=-—

—

-w dw,

/c = 0 , l , 2 , . . .

27TI J | w | = H / ( W , Z )

are holomorphic functions of z = ( z 2 , . . . , z„) in |z2| < 5 , . . . , |z„| < 5, where J|w|=e denotes the integration along the circle 6^w = s e'^, 0 ^ O^ITT. For each z, o-o(z) is equal to the number of the zeros of/(w, z) in the disk

14

1. Holomorphic Functions

|w| < e. Therefore (TO(Z) is a natural number independent of z. Since o-o(O) = s, ao{z) = s.LQtoji{z),..., cu^(z) be the zeros of/(w, z) in the disk |w| < £ . Then

Put P(w,z) = w^ + ai(z)w^-' + . . . + a,(z)= {[ (w-a),(z)). The elementary symmetric functions a i ( z ) , . . . , a^Cz) of a>i(z),..., ct>5(z) are polynomials of (ri(z),..., o-^Cz): For example ai{z) = cr^{z), a2(z)=j^icT,(zr-0-2(1)), a,iz)=-{a,(z)'-3(T,{z)a2{z)-^2cr,iz))

• - -.

Therefore a i ( z ) , . . . , a^iz) are holomorphic functions of z = ( z 2 , . . . , z„) in | z 2 | < 5 , . . . , |z„| 0 and 6 > 0 sufficiently small, we may assume that /(w, z) is defined on |>v|i(z) within Us(0)-^. is among cOj{z), j = 1 , . . . , 5. Conversely, cOj{z), 7 = 1 , . . . , 5, are all analytic continuations of coi{z). For suppose that co2(z),..., o)t(z) are analytic continuations ofwi(z), and that (Ot+i(z),..., (o^iz) are not. Put i'i(w,z)= n (w-(Oj(z)) = w'-\-b,(z)w'~'-^'

• • + b,(z),

7=1

and ^2(w,z)= n

(w-a>,(z)) = w^-^ + c,(z)w^-^-^ + --- + c,_,(z).

Then 5 i ( z ) , . . . , ^r(^), Ci(z),..., Cs-tiz) are one-valued holomorphic functions in ^ 4 ( 0 ) - A , which are bounded, hence by Theorem 1.15 these are extended to holomorphic functions in Us(0), which are denoted again by 6 i ( z ) , . . . , bt{z), Ci(z),..., Ct-s(z)' Then Pi(w, z) and P2(w, z) are distinguished polynomials in P[w], and P(w, z) = Pi(w, z)P2(w, z) which contradicts the irreducibility of P(w,z). Thus if 5 is irreducible at 0, Wi(z),..., Ws{z) are all branches in Us{0) — A of one and the same holomorphic analytic function. Theorem 1.16. Let S be irreducible at 0, andfo{zi,..., z„) = 0 an irreducible equation of S at 0. If g{z^,..., z„)e€o vanishes identically on S, g = g ( z i , . . . , z j is divisible by fo = / o ( z i , . . . , z j in OQI fo\gProof We may assume that /o, and g are both regular with respect to Zi. We write w = Zj and z = ( z 2 , . . . , z„). Since /o is irreducible, /o and g are relatively prime if g is not divisible by /Q. This being the case, by Theorem

22

1. Holomorphic Functions

1.13 there exist a(w, z), /3(w, z) e OQ such that «(w, z)/o(w, z) + i8(w, z)g(w, z) = r(z) ?^ 0,

riz) e R.

Denoting the roots of /o(>v, z) = 0 by coi{z), co2(z),..., we obtain g(cui(z), z) == 0 since (coi(z), z) e 5 for any z 6 ^4(0). Hence r(z) = 0 identically, which is a contradiction. I Next we consider the general case in which the analytic hypersurface S given by the e q u a t i o n / ( z i , . . . , z„) = 0 is not necessarily irreducible at 0 G 5. In this case by Theorem 1.12, the power series expansion/o(z) of/(z) = / ( z j , . . . , z„) with centre 0 is factored uniquely up to units into irreducible factors in ^o- Let pi{z),... ,priz) be distinct irreducible factors of/o(z). Then we have

/o(z)=u(z) n pAzr\ A= l

where w(z) is a unit in ^oPut

foiz)= n PAZ). A= l

Then/o(z) and /o(z) vanish simultaneously, hence /o(z) = 0 also gives an equation of S in a neighbourhood of 0, which is called a minimal equation of 5. Theorem 1.17. Ifg{z) e €Q vanishes identically on S, g{z) is divisible byfoiz): /o(z)|g(z). Proof. By Theorem 1.16 above, g = g(z) is divisible by each px=Px{^)^ Since /?i(z),... ,p;^(z) are mutually coprime, g is divisible by /o = n l = i JPA- I Corollary. Let f{z) and g(z) G ^o- Ifbothf(z) = 0 and g{z) = 0 are minimal equations of S at OG 5, / ( z ) and g{z) are associates. Theorem 1.18. /o(z) = 0 is a minimal equation of S at 0 if and only if fo{z) and at least one of its partial derivatives fo^^(z) = dfo(z)/dzj^ are relatively prime in €Q. Proof Suppose /o(z) is not minimal. Then /o(z) has at least one multiple factor /7i(z):/o(z)=/7i(z)^/i(z). Therefore fo,,{z)=p^{zfK^{z)-^ 2px^^{z)pi{z)h{z) has a common factor P\{z) with/o(z). Suppose in turn that /o(z) = 0 is minimal. Choose coordinates Zi,...,z„ such that

§1.2. Holomorphic Map

23

/ o ( z i , . . . , z„) is regular with respect to Zi, and write w = Zi, and z = (z2, . . . , z „ ) . Then by the Weierstrass preparation theorem, we have the factorization /o(w, Z) = M(W, z)P(w, z), where w is a unit in OQ, and P(w, z) is a distinguished polynomial. Let P(w, z) ==nr=i ^fc(^» ^) t>e the factorization of P{w, z) into irreducible distinguished polynomials in R[w]. Then m

/o(w, z) = w n P/c(w, z), with a unit

w = M(W, Z)

is an irreducible factorization of/o(z) in OQ. Since/O(W, Z) = 0 is minimal, the irreducible polynomials Pfc(w, z), /c = 1 , . . . , m are mutually coprime. Consequently

k

j=\

k9^j

is not divisible by any Pfc(w, z). Hence /o(w, z) and /ow(>^? ^) are relatively prime. I Corollary. Let f{z) he a holomorphic function in a domain of C", and S the analytic hypersurface defined by the equation f{z) = 0. We assume OeS. Supppose fo{z) = 0 is a minimal equation of S at 0. Then if e>0 is small enough, fiz) = 0 is a minimal equation ofS at cfor every ce S with \c\ < s. This follows immediately from Theorem 1.14. Let U he a. domain where / ( z ) is defined, / ( z ) = 0 is called a minimal equation of 5 in U if fiz) = 0 is a minimal equation of S at every ce SnU.By the Corollary to Theorem 1.17, if fiz) = 0 and g{z) = 0 are both minimal equations of S in U, u{z) = f(z)/g{z) is a non-vanishing holomorphic function in U.

§1.2. Holomorphic Map In this section we consider a map O: z^w = , respectively. Definition 1.3.0 is said to be holomorphic in D if Wj =
w^

Vi,..., w j

(1.40)

a(zi,...,zj

is called the Jacohian of . Introduce real coordinates by putting z^ = X2k-\ + i^ik-, and w, = M2J_I + i*W2j and represent ^ in these terms as , X2n) -> (Wi, . . . , M2n) = ^ ( ^ 1 , • • • , X 2 J .

Then w^e have det

(9(Mi,...,M2n) a ( X i , . . . , X2n)

\nz)\\

(1.41)

where the left-hand side of this equality represents the Jacobian of O with respect to ( x i , . . . , X2„) and ( w i , . . . , M2n)Plroo/ We give proof for n = 3. Since by Theorem 1.16, dwj/dzi^ = (awy/az^) == 0, using an elementary calculation, we obtain a(Ml,...,M6)

, , ^ ( H ^ I , H ' S , W 3 , W i , W2, ^ 3 )

det' (— 9 ( X i , . . . , X 6 ) = det = det

a ( z i , Z2, Z3, z i , Z2, Z3) a ( W l , W2, W3)

(9(Wi, M>2, W3)

a ( z i , Z2, Z3)

(9(zi, Z2, Z3)

= |/(z)P.

§1.2. Holomorphic Map

25

Let ^ : z-» w = :

a(zi,...,zj

(9(wi,..., w^)

(9(zi,...,z„) *

(1.42)

In particular if i^ = m = n,

a(zi,..., z j

d{wu . . . , w j

(9(zi,..., z j *

^: w^ z, ^ a(zi,...,zj

det

^a(wi,..., w j

(9(wi,..., w j

det

a(zi,...,zj

= 1.

Thus if a holomorphic map : z ^ w has an inverse 0~^: w->z which is holomorphic, then ^a(wi,..., w j ^^ det — # 0. a(zi,...,zj

Theorem 1.19. Let (^: z^ w = ^{w} be a holomorphic map ofDc: C" into C", andJ{z) its Jacobian. IfJ{z^) T^ 0 at a point ZQ e D, there exist a neighbourhood U ^ D of z^, and a neighbourhood W of (z^) = w^, such that maps U bijectively on W. Moreover the inverse 0~^ o/ restricted to U is holomorphic on W. Proof Consider 0 as a C°° map. Then by (1.41) the Jacobian of O as a C°° map is equal to |/(z)|^. Since |/(z°)|^>0 by the assumption, the inverse mapping theorem for continuously differentiable maps shows that there exist a neighbourhood U of z^ in D, and a neighbourhood W of w^ = (z^) such that maps U bijectively on W, and that the inverse ~^ of O restricted to U is continuously differentiable. Put ^-': w -> ( z i , . . . , z j = (il^iiw),...,

(A„(w)).

Then z^ = ij/hicpiiz),..., (^„(z)). Since dipj/dz^ = 0, taking the partial derivative with respect to z^, we obtain

j ^ \ oZk

dWj

26

1. Holomorphic Functions

Consequently, since det(aw//azfc)^fc=i „ = /(z)7^0 on U, we have d{jjh{w)/dwj = 0,7 = 1 , . . . , w, hence ^h{^) are holomorphic in w^,..., w„. I Corollary 1. Let ^ be a holomorphic map of a domain D c i C " into C^. If J{z) does not vanish in D, 0 ( D ) is a domain in C". I Corollary 2. Let ^be a one-to-one holomorphic map of a domain D ci C" into C". IfJ(z) does not vanish in D, the inverse ~^ of O is a holomorphic map of the domain E = ^(D) onto D. I If maps a domain D c= C" bijectively onto a domain E ci C" and O"^ is also holomorphic, O is called a biholomorphic map. Two domains D and £ are said to be biholomorphic if there exists a biholomorphic map O of D onto £. Theorem 1.20. Letfiz),... that

,fmiz) be holomorphic in a domain ofC.

Suppose

..(3(/i(z),...,/^(z)) rank ; ; =P a(zi, . . . , z j is independent of z. If z^ is a point of this domain such that ^ j(f{z),...JAz)) det ;

;

(9(zi,..., z j

^^ 7^0

at

0 z = z,.

then there exists a neightourhood U{z^) ofz^ such thatfj,+i{z),... holomorphic functions offi{z),... ,/^(z) in U{z^).

,/„(z) are

Proof Put Wi = / i ( z ) , . . . , w^ =f^{z). On a sufficiently small neighbourhood (7(z') of z\ det — = det — a ( z i , . . . , z^, z^+i,..., z„) (9(zi,..., z^)

T^

0.

Therefore by Theorem 1.19, 0 : ( z j , . . . , z^, z^+i,..., z„) -> (vvi,..., w^, z^+i,..., z„) is a biholomorphic map of U{z^) onto a neighbourhood (7( w^) of w'=:Vi,..., w^, z^+i,..., z„) are holomorphic functions Wi,..., w^, and do not depend on the variables z^+i,..., z„: g,+^(wi,..., w^, z , + i , . . . , z j = ;i^(wi,..., w j . Consequently /,+,(z) = /i,(wi,..., w^) = hjifiiz),...

,/,(z)).

I

of

Chapter 2

Complex Manifolds

§2.1. Complex Manifolds (a) Definition of Complex Manifolds Recall that a Riemann surface ^ is a connected Hausdorff space S endowed with a system of local complex coordinates {zi, 2 2 , . . . , Zj,...}. Each local complex coordinate Zj is a homeomorphism Zji p^ Zj(p) of a domain Uj in 2 onto a domain % c: C such that U j ^ = ^? ^i^ Zj{p), pe Ujn L4, is a biholomorphic map from the open set ^^j ^ ^k onto ^ ^ c: ^^.. The concept of a complex manifold is a natural generalization of the concept of a Riemann surface. In case of a Riemann surface, the local complex coordinate of a point p 6 S is a complex number. Using an n-tuple of complex numbers Zj(p) = {zi(p),..., z„(p)) instead, we obtain the concept of an ndimensional complex manifold. More precisely, let 2 be a connected Hausdorff space, and { ( 7 i , . . . , L^,...} an open covering of 1 consisting of at most countably many domains. Suppose that on each ^ ^ 2, a homeomorphism Zj' P -^ Zj(p) = iz](p), • . . , zj{p)),

p e Uj,

is defined, which maps Uj onto a domain ^^ciC". Then for each pair 7, k with Ujr\Uk^ 0 , the map Tjki Zk(p)^Zj{p),

peUjnUk,

(2.1)

is a homeomorphism of the open set ^uj = {zkip)\p ^ ^ ^ ^/d ^ ^k in C ^ onto the open set %k = {Zj(p)\p^^j^ ^ / d ^ %- If O ' fc ^^ biholomorphic for any 7,/c such that Ujn Uk9^0, each Zj:p^Zj{p) is called local complex coordinates defined on Uj, and the collection { z i , . . . , z^,...} is called a system of local complex coordinates on 2. Definition 2.1. If a system of local complex coordinates { z i , . . . , z^,...} is defined on a connected Hausdorff space S, we say that a complex structure

§2.1. Complex Manifolds

29

is defined on 2. A connected Hausdorff space is called a complex manifold if a complex structure is defined on it. We denote a complex manifold by the letters M, N, etc. The system of local complex coordinates {z^,..., z^,...} which defines the complex structure of a complex manifold M is called the system of local complex coordinates of M. The dimension or complex dimension of M is defined to be n. We often denote M " if we want to make explicit the dimension of M. Thus the concept of complex manifolds is an obvious generalization of that of Riemann surfaces, and, in fact, a Riemann surface is nothing but a 1-dimensional complex manifold. If M is a complex manifold, and { z i , . . . , z,,...} is the system of local complex coordinates of M, each domain Uj is called a coordinate neighbourhood. We call local complex coordinates simply local coordinates or a local coordinate system. The point Zj{p) = (z](;?),..., Zj{p)) of C" is called the local (complex) coordinates of p. Let pe M. Then if we choose a coordinate neighbourhood Uj with pe Uj, p is determined uniquely by its local coordinates Zj = {z],..., Zj) = Zj(p). For pe Ujn (7fc, the coordinate transformation Tj,,: zj^^zj = [z],...,

zj) = T,-fc(zfc),

(2.2)

which transforms the local coordinates z^ = ( z ^ , . . . , z^) = z^ip) into the local coordinates z, = ( z j , . . . , zj) = z^(;?) is, by definition, a biholomorphic map. Since peUj is determined uniquely by its local coordinates Zj = Zj{p), identifying Uj with % via Zj, we can consider that a complex manifold M is obtained by glueing the domains ^ j , . . . , ^ ^ , . . . in C" via the isomorphisms Tjj^: %L^j -> %k: M = Uj %. Then Zj e % and z^ G ^^ are the same point on M if and only if z, = ^^^(^k). Example 2.1. Any domain ^ c i C " is a complex manifold. M = ^ has a system of local coordinates {z} consisting of the single local coordinates

z^z =

(z\.,.,zn.

Example 2.2. For a point (^o, • • •, ^n) e C""^' - ( 0 , . . . , 0), ^ = {(A^O,...,A^J|AGC}

is a complex line through 0 = ( 0 , . . . , 0). The collection of all complex lines through 0 is called the n-dimensional complex projective space, and denoted by P". The Riemann sphere S is the 1-dimensional complex projective space P \ A point ^ of P" represents a complex line ^ = {(A^o,...,A^J}. (^0,''', in) is called the homogeneous coordinates of ^G P", and denoted by f = (^0, . . . , ^ J . The equality Uo, • - -, in) = Uo, ---, D means that

30

2. Complex Manifolds

Wo, • • • 5 Cn) and (^0? • • • J fn) ai"^ the homogeneous coordinates of the same point f, that is, fo = A^o, • • •, ^n = AZ;„ for some A T^ 0. Put U, = {^ G P" I Cj 5^ 0}. f G (7o is represented as ^ = (1, z \ . . . , z") where ^"^ = ^u/io' {^\ •' •, ^") is called the non-homogeneous coordinates of f. The map

gives local coordinates on UQ, where ^o = Zo(Uo) = C". Similarly on Uj we define local coordinates zj:^^zj(a = izl...,zr\zr\...,z^\

zJ=U(r

Then % = Zj{ Uj) = C". Of course ZQ = z". On Uj n L/^, we have zf=l/zi

zj = zl/zi,

p^j,^k.

(2.3)

Hence the coordinate transformations TJ^: Z^^ ZJ are biholomorphic. P" is considered as the complex manifold obtained by glueing the {n + 1)-copies of C" via the isomorphisms (2.3). (b) Holomorphic Functions and Holomorphic Maps Let M be an /t-dimensional complex manifold, { z j , . . . , z,,...} the system of local complex coordinates, Uj the dommn of z^, and % = Zj(Uj), L e t / be a real- or complex-valued function defined on a domain Da M. For pe Dn Uj, using the local coordinates Zj = Zj{p),wQ define a functionyj(Zj) by fip)=fji^j),

(2.4)

then fjizj) is a function of n complex variables ( z i , . . . , z „ ) defined in Q)j = Zj{DnUj)c:%. By (2.1), if z,-= z,.,(z,), we have/(z,) = A(zfc). Since p^ Zj = Zj(p) is 3. homeomorphism, / is continuous in D if and only if each fj(Zj) is continuous in ^j with respect to Zj. Definition 2.2. / is said to be a continuously differentiable function, a C^ function, C°° function in D c: M if each fj is a continuously differentiable function, a C function, a C°° function in 2j with respect to Zj respectively. If we consider a complex manifold M as obtained by glueing the domains ^ 1 , . . . , % . . . in C " : M = U ; % identifying peM with the point Zj = (z],..., Zj) = Zj{p)E %, the function f{p) is written as f{zj). Since in this notation Zj e % and z^ e ^^ are the same point of M if Zj = Tjj,(zk), we have

§2.1. Complex Manifolds

31

fj{Zj)=fk{zj,) if Zj = Tjk{zk). Note that here/(zfc) does not denote the function obtained from f{Zj) by substituting z^ for Zj. Also in this notation a function f{p)=f{zj) defined in a domain D of M is of class C\ (holomorphic) if and only if each/(z,) is of class C^ (holomorphic) with respect t o Zj.

Similarly we define a holomorphic map from a complex manifold M to another complex manifold N. Let { w i , . . . , w^,...} be the system of local complex coordinates of iV, W), the domain of w^, and W), = W)^{Wx)^C^ where m = dim N. Let 0:/7-^g = ^ ( / 7 ) b e a continuous map from a domain D w^ip) from W^, into C" is defined, then {vvi,..., W;^,...} makes a system of local complex coordinates of M". Hence there are infinitely many choices of systems of local complex coordinates for one and the same complex manifold M". In view of this fact we may define a complex manifold as follows: first let two systems of local complex coordinates {zj} = { z i , . . . , z^,...} and {W),} = { w , , . . . , W;^,...} be given on a connected Hausdorff space 2 , Uj the domain of z^, and W^ the domain of WA. For 7, A such that L^ n W^ ^ 0 , W;,z7^z,(/7)->W;,(/?),

pe UjnW^,

is a homeomorphism from the open set %), = {Zj{p)\p^ Ujn W^} onto the open set Wj^j = {Wj(p) \peW^n Uj}. Let M be a complex manifold defined by the system {Zj}. Then W;^: p^w^ip) is biholomorphic if and only if W;,zJ^ is biholomorphic for any j such that UjOW^,?^ 0 . Therefore if we say that

32

2. Complex Manifolds

{zj} and {w^} are holomorphically equivalent when W;^zJ^ is biholomorphic for any pair 7, A such that Uj-n W^T^ 0 , then {Zj} and {w;^) are two systems of local complex coordinates of the same complex manifold if and only if they are holomorphically equivalent. Thus instead of the definition previously given, we define a complex manifold as follows Definition 2.3. Let 2 be a connected Hausdorff space. A complex structure on S is defined as a holomorphic equivalence class of systems of local complex coordinates on 2. A connected Hausdorff space endowed with a complex structure M is called a complex manifold and denoted by the same M. The complex structure M is called the complex structure of M, and a system of local complex coordinates belonging to M is called a system of local complex coordinates of the complex manifold M. Two complex manifolds M and N are called complex analytically homeomorphic or biholomorphically equivalent if there is a biholomorphic map O from M onto N. In this case we consider M and N as the same complex manifold by identifying pe M and q = ^{p)e N. In fact, since O is homeomorphic, M and N can be considered as the same Hausdorff space E. Next, let {Zj} be a system of local complex coordinates of M and {w^} a system of local complex coordinates of N. Then since O is biholomorphic, ^^j defined in (2.5) is biholomorphic, hence {zj} and {w^} are holomorphically equivalent systems of local complex coordinates on 1. If, as stated above, we consider M = Uj % and iV = UA ^ A , a map O from a domain Dfq(p)^ ^ = ^ ( ^ ) , defined in a neighbourhood U{q) of q such that SnU{q) = {peU(q)\fl{p) = '"=r,(p) = 0}.

(2.6)

Thus 5 is a subset of M " which is defined in a neighbourhood of each point qe Shy a system of analytic equations {flip) = • • • =fq{p) = 0}. This system of equations is called a local equation of S at q. There are infinitely many choices of local equations for a given S. S is said to be smooth at q if a local equation fl(p) = • • • =fq(p) = 0 of S at q can be so chosen that

.

d(fl(p\...j;ip)) d{Zq{p),...,Zq{p))

This being the case, we call m = n — v the dimension of S at q. Otherwise we call qe S Si singular point of S. If S is smooth at q, taking a sufficiently small neighbourhood U{q) and an appropriate renumbering of z^(/?)'s, we may assume that

, , diKip),...,n{p)) det

:, d(Zq

,.

^T-TT^'O, {P),---,Zq{P)}

m = n-v,

34

2. Complex Manifolds

for pe U{q). Thus for a sufficiently small U(q), the map

z,(p)=(ziip),..., z-(p),fi,(p),... ,r,{p)) is biholomorphic by Theorem 1.19. Therefore we may use p -> (zlip),...,

z'^{p),f\ip),...

,r,ip)),

m = n~v,

as local coordinates with centre q. In terms of these, we have SnU{q)

= {peU{q)\z-^\p)

= ''' = z''^{p)==0}.

(2.7)

Definition 2.5. A connected analytic subset S of M" without singular points is called a complex submanifold of M. For each qe M, choose a coordinate polydisk UR/(p) as a map of the Riemann surface M^ to the Riemann sphere P^ = C u {00}, and letting (^0, ^1) be the homogeneous coordinates of P \ we see that the map

is holomorphic. In case n ^ 2 , even if hq and gq are relatively prime, we may have Ki^) = Sqi^l) = 0. This being the case, the value f{q) o f / ( p ) = hq/gq at q cannot be determined. For example, p u t / ( z j , Z2) = Z2/Z1, which is a meromorphic function in C^. Then /(O, 0) cannot be determined. (f) Differentiable Manifolds A connected Hausdorff space S is called a topological manifold if there is an open covering of 2 consisting of at most countably many domains L/i,..., Uj,..., such that each Uj is homeomorphic to a domain % in R'". In this case the homeomorphism of Uj onto %:

is called local coordinates or a local coordinate system defined on Uj. The collection of local coordinates {Xj} = { x i , . . . , x^,...} is called a system of local coordinates on the topological manifold 2. Forj, k such that Ujn U^?^

0, Tjk'-Xk(p)^Xj{p),

peUjn

t/fc,

is a homeomorphism of the open set ^/g = {Xk(p) \pe U^n Uj} a ^ ^ onto the open set %k = {Xj(p)\pe Ujn U^}. We call {Xj} a system of local C°° coordinates if these TJ^ are all C°°, which means that x]{p),..., xj'ip) are C°° functions of xl(p),..., x1^{p). Suppose given two systems of local C°° coordinates {xj} and {u;,} on 1, and let Uj be the domain of x, and W^, the domain of U;,. If for any pair 7, A with L^ n W;^ 7^ 0 , the maps Xj{p)-^U;,(p)

and

M;,(/7)-»X^(;7)

are both C°° for pe Ujn W;,, {Xj} and {M;^} are said to be C°° equivalent. Definition 2.6. A C°° differentiable structure on a topological manifold 1 is defined to be an equivalence class of systems of local C°° coordinates on li. A topological manifold S endowed with a differentiable structure is called a differentiable manifold, whose differentiable structure is called the

38

2. Complex Manifolds

differentiable structure of the difierentiable manifold 2. A system of local C°° coordinates belonging to the differentiable structure of 2 is called a system of local C°° coordinates on the differentiable manifold 2. Let 2 be a differentiable manifold, {Xj} a system of local C°° coordinates on 2, and Uj the domain of Xj. A real- or complex-valued function f(p) defined in a domain D of 1 is represented on each D nUj 9^0 by SL function of local coordinates ( x j , . . . , xJ") = Xj = Xj{p) as f{p) =fj(Xj). We call f(p) continuously differentiable, C, C^ in D if each fj{Xj) is continuously differentiable, C\ C°°, respectively. The differentiability of a map of a domain Da^ to another differentiable manifold is defined similarly as follows. Let T be a differentiable manifold of dimension n, {u^} a system of local C°° coordinates on T, W^ the domain of MA, and O: /? -> g = ^{p) a continuous map of D to T. For kj with ^~\ W),)n Uj 9^ 0 , the map

^,y.xj{p)^u^mp)),

pe^-\W,)nUj,

is a continuous map of the open set %), = Xj{~^ are C°°, then 4> is called a diffeomorphism, and D is said to be diffeomorphic to E. We may identify two mutually diffeomorphic differentiable manifolds. Suppose that a complex structure M is defined on a connected Hausdorff space S. For a system of local complex coordinates {Zj} belonging to M, the domain Uj of Zj is homeomorphic to the image % = Zj{ Uj) in C" = R^". Consequently 2 is a topological manifold, which is called the underlying topological manifold of M. For local complex coordinates

z, = z,(p) = (zj(p),...,z;(p)), putting zJ(p) = xj''~\p)-\-ix]''(p), coordinates

p=l,2,...

,n, we introduce local real

Xjl p -> Xjip) = (Xj(p), . . . , X f (/?)).

Then {Xj} forms a system of local C°° coordinates on E. Consequently {Xj} defines a C°° structure on J., which makes 2 a differentiable manifold. This is called the underlying differentiable manifold of the complex manifold M Also we call M a complex structure on the differentiable manifold E. Conversely, let 2 be a Hausdorff space on which a differentiable structure is given, which makes 2 a differentiable manifold. Then a system of local complex coordinates {Zj} on the Hausdorff space ]S is a complex structure on the differentiable manifold 2 if and only if each Zj maps its domain Uj diffeomorphically onto a domain % = Zy( U,) cz C" = IR^".

§2.2. Compact Complex Manifolds

39

§2.2. Compact Complex Manifolds A complex manifold M is said to be compact if its underlying topological manifold S is compact. In this book we mainly treat compact complex manifolds. Let M be a compact complex manifold. Then since M is covered by a finite number of coordinate neighbourhoods, we may choose a system of local cotnplex coordinates on M consisting of a finite number of local coordinates { z i , . . . , z^v}. Let Uj be the domain of Zj: p-^ Zj{p), and put Zj{Uj) = %c: C". We have M = [J. Uj. Identifying Uj with % as usual, we may consider M = U^ %- Thus a compact complex manifold M is obtained by glueing a finite number of domains ^ i , . . . , ^^^ inC" via the identification o f Zfc G %L^j CZ OU. w i t h Zj = T,fc(Zfc) E %k C ^fc.

A holomorphic function defined on a compact complex manifold M is a constant, Proof Suppose that/(;7) is holomorphic on all of M. Since M is compact, the continuous function \fip)\ attains its maximum at some point qeM. Let q e Uj, and put f(p) =fj{Zj) on Uj where Zj is a local coordinate system on Uj. ThQnfj{Zj) =fj{z],. . . , z") is a holomorphic function on % = Zj{ Uj). We may assume that % is a poly disk with centre c, = Zj{q). Put

g(w)=y;(^j + w(zj-c]),...,c;+w(z;-c;)). Then for ( z ] , . . . , z") e Uj, g{w) is a holomorphic function of w on | w| < 1 + e if s is sufficiently small, and |g(w)| attains its maximum at w = 0. Consequently, by the maximum principle, g{w) is a constant. Thus fip) is a constant on Uj, and, by the analytic continuation, one sees that/(/?) is a constant on all of M. I In this section we give several examples of compact complex manifolds. A compact complex manifold is, theoretically, determined if a finite number of domains Uj and biholomorphic mappings TJ^ which glue them. But except for a few special cases as that of P" (Example 2.2), this method of construction is very complicated, hence is not practical. In the following we shall explain various methods of construction of a new compact complex manifold from given ones. (a) Submanifolds First we take P", and investigate submanifolds of P". In the following we denote a point ^ of P" by its homogeneous coordinates (^o, • • •, fn)- Let P(^o,''' •> Sn) be a homogeneous polynomial of degree m, which we often

40

2. Complex Manifolds

denote simply by PU)- Since P(A^o, • • •, A^J = A'^Pl^o, • • •, ^n), the equation P(^o, • • •, ^n) = 0 gives a well-defined subset of P". An algebraic subset S of P" is, by definition, a subset defined by a system of algebraic equations Pi(^) = • • - = P^{O = 0, where P i ( ^ ) , . . . , PAO are homogeneous polynomials. As stated in Example 2.2, P" is obtained by glueing (n + 1) copies ^^ of C", 7 = 0 , 1 , . . . , n: P " = UJ=o %- The local coordinates on % are given by ( z j , . . . , z j ~ \ . . . , z") where zj" = it/^j- Therefore Sn % is defined by the system of algebraic equations: i ^ , ( z ^ , . . . , z / - \ z f \ . . . , z ; ) = 0,

v = h2,...,K

(2.14)

on C" = %. Thus an algebraic subset of P" is an analytic subset of P". An algebraic subset M which is a complex submanifold of P" is called a projective algebraic manifold. In this case, M n ^j is a complex submanifold of C" = % defined by (2.14). In general a complex submanifold of C" which is defined by a system of algebraic equations is called an affine algebraic manifold. Thus a projective algebraic manifold M is obtained by glueing a finite number of affine algebraic manifolds Mn% = My. M = UJ^i ^jIn this book, by an algebraic manifold we always mean a projective algebraic manifold unless otherwise mentioned. A (projective) algebraic manifold is obviously compact. Let P(^) = P(^05..., ^„) be a homogeneous polynomial of degree m. The algebraic subset S of P" defined by a single equation P(^) = 0 is called a hypersurface of degree m. Put P^^{^) = dP{0/dCk- Then we have CoPdO + ^iP,U) + " ' + ^nPcSC) = rnP{0Consequently, if for any ^G P " , at least one of P^^iC) (^0, ^1, ^2, ^3) = (^0, tltu totl t',)

maps P^ biholomorphically onto C Therefore C is analytically isomorphic to p^

Example 2.5. The equation ^^ + ' • • + ^ r = 0 defines an algebraic surface in P ' since at every point ^eP\ at least one of dUo+CT-^C2-^^T)/dijc = ^^JT \ /c = 0, 2, 3, does not vanish. We only give a calculation of the Euler number of 5, putting aside various interesting properties of 5'. Let : ( ^ 0 , ^ 1 , ^ 2 , ^3) ^ ( ^ 0 , ^ 1 , ^ 2 , 0 )

be the projection. Then the restriction ^s of O to E is a holomorphic map of S onto the plane P^ defined by ^3 = 0. 0 , 0 < | a | < l . Therefore G* is properly discontinuous and fixed point free on C*. Clearly C = C / G = C*/G*. Let F * be the closed annulus {w||a| ^ |w| ^ 1, wG C*}. Then C is obtained from F * by identifying the points w and aw on the boundary of F * where |>v| = 1.

Figure 2

In general suppose given a group of automorphisms G of a complex manifold W, which is properly discontinuous and fixed point free. By a fundamental domain of G, we mean a closed domain F^W satisfying the following conditions: (i) (ii)

F = [{F)\ where (F) denotes the interior of F If/?G(F), GpnF = {p}.

48

2. Complex Manifolds

If F is a fundamental domain of G, the map p-^p maps (F) bijectively onto {p\pG (F)}, and F onto W= W/G. Therefore if F is compact, W is also compact. In Example 2.7 above F and F * are fundamental domains of G and G* respectively. A complex manifold W is called a complex Lie group if \ ^ is a group and the map of WxW to W defined by the group multiplication iq,p)-^ qp~^ is holomorphic. If \ ^ is a complex Lie group, identifying qe W with the automorphism p-^ qp of W, we see that T^ is a group of automorphisms of W itself. A subgroup G of W^ is called a discrete subgroup of W if it is a discrete subset of W. A discrete subgroup G acts on \\^ in a properly discontinuous manner without fixed point. In fact, for a compact subsets Ki, K2 of W, the set {ge G|gXi 0 X 2 7 ^ 0 } is contained in the compact subset K2K]^^ = {qp~^\qe K2,pe Ki}. Since clearly G is fixed point free, W = W/ G is a complex manifold and has a group structure as the quotient group of W by G. Thus W is SL complex Lie group. Example 2.8. A complex vector space C" is a complex Lie group with respect to the usual addition. Take 2n vectors co^ = ( c o j , . . . , w " ) e C " for j = l , . . . , 2 n , such that these coj are linearly independent over IR. Then coj generate a discrete subgroup

f"

I

1

of C". Since a fundamental domain F=

\l^tj(Oj\0^tj^l,j=l,...,2n\

of G is compact, T" = C"/ G is a compact commutative complex Lie group, which we call a complex torus. If a meromorphic function / ( z ) on C" satisfies the condition f{z + coj)=fiz) for 1 ^ 7 ^ 2 n and for any ZGC", / ( Z ) is called a periodic meromorphic function with the periods coi,..., a)2„. Such / ( z ) gives a meromorphic function on T", which we denote also by / ( z ) . w i , . . . , (02n is called the periods of T", and the matrix /

a-

1 1 (02

\^2n

is called the period matrix of T".

§2.2. Compact Complex Manifolds

49

For n = 1, as is stated in Example 2.7 above, T^ = C = C/G is always an algebraic curve. In case n ^ 2, a complex torus T" = C " / G is not necessarily an algebraic manifold. Let / be an invertible alternating real 2nx2n matrix. Then V - 1 ^n/~^n is a Hermitian matrix. By v - 1 ' i l / ~ ^ n > 0 , we mean that this matrix is positive definite. H is called a Riemann matrix if there exists a 2n x2n integral alternating matrix / satisfying the following conditions:

(i) ' a r ' a = o. (ii) V-rn/"'n>o. T" is an algebraic manifold if and only if its period matrix Ct is a Riemann matrix {see [28]). In this case we call T" an Abelian variety. In general the period matrix of a complex torus T" with n ^ 2 is not a Riemann matrix. Moreover it is known that there exist no non-constant meromorphic functions on most general complex tori T". For example, the complex torus T^ with the period matrix

a=

1 0

0 1 ,

has no non-constant meromorphic functions ([28], p. 104). Example 2.9. T" is an obvious generalization of an elliptic curve C = C/G given in Example 2.7 to the n-dimensional case. Here we give another generalization of C, considering C as C*/G*. Let ]¥ = £" -{0}, and G the infinite cycHc group generated by the automorphism g:z = ( z i , . . . , z j ^ g(z) = (a^Zu .. •, a „ z j of W, where a i , . . . , a„ are constants with |ai| > 1 , . . . , |a„| > 1. G acts on W^ in a properly discontinuous manner without fixed point. The quotient space M = W/G is called a Hopf manifold ([11]). M is diffeomorphic to the product S^ x 5 ^ " ~ \ We give a proof of this only for n = 2 below, but the generalization is straightforward. Let 5' = {(^„^2)eC^||^.P+|f2r=l}. Putting «! = £''• and oj = e^^, consider a C°° map * of IR x 5^ to W defined by

2. Complex Manifolds

50

Then O is bijective. In fact, since | a i | > 1 and \a2\> 1, ri = R e / 3 i > 0 and r2 = Rei82>0. Put N{t) = \z,e-'^f^Z2e-'^f

= \z,\^e-^'^' + \z2\^e-^'^\

Then iV(r) is a monotonously decreasing function of t. Moreover N(t)-^0 if r^oo, and N(t)-^+00 if ^->-oo. Hence there is a unique r such that N{t)=l. Putting ^1 = Zi e~^^\ and ^2 = ^2 ^"^^' for this r, we get the unique solution (r, ^1, ^2) of the system of equations fi e^^' = Zi, ^2 e'^2= ^2 with lfil' +1^2!'=1. Clearly ^-^(zi,Z2)->(ai,^2)eff«x5^ is C°°. Hence W is diffeomorphic to U x5^. The automorphism g"", m e / , of W corresponds via ^ to that of R x 5^ given by (^,^i,^2)^(^+m,z:i,^2). Therefore we obtain a desired diffeomorphism W / G - - R / Z x5^ = 5^ x 5 ^ Thus a Hopf manifold M - W/G is diffeomorphic to 5^ x 5 ^ " " \ hence, M is compact, and its first Betti number is equal to 1. From the theory of harmonic differential forms we know that the first Betti number of an algebraic manifold is even (see [14], p. 346). Consequently a Hopf manifold is not an algebraic manifold. A complex torus is, in general, not an algebraic manifold, but it has the same topological structure as algebraic complex tori, while in the case of Hopf manifolds, even their topological structures are different from those of algebraic manifolds. In the sequel we denote by z the point of M = W/ G corresponding to ze W. For a meromorphic function/(z) on M, putting/(z) =f(z), we obtain a G-invariant meromorphic function/(z) on W:f{gz) —f{z) for g e G. By Levi's theorem ([27], Band II, p. 220), any meromorphic function on W extends to a meromorphic function on all of C". Thus / ( z ) extends to a meromorphic function on C", which we denote also b y / ( z ) . From (e) of the preceding section, there are a neighbourhood (7^(0) of 0 and relatively prime holomorphic functions (p{z) and (/^(z) defined in L^e(O) such that / ( z ) = (p{z)/il/{z) in L^e(O). Using this fact we can determine all G-invariant meromorphic functions on W. For simplicity, we only treat the case n=2 below. Since / ( z i , Z2) =f(z) is G-invariant, / ( z ) =/(g~'"(z)) for any integer m. Since |a;i|> 1, |«2|> 1 and g~'"(z) = (a^""'^!, cti'^^^i), for any given z, if we take m sufficiently large, we have g~'^(z)e 11^(0). Hence f{z)=f{g-"{z))=

lim / ( g - ' " ( z ) ) = lim 7 7 ^ 3 ^ .

(2.16)

§2.2. Compact Complex Manifolds

51

Let + 00

M, we have

/ ( z ) = lim h,k

where the summation X^^^ is taken over all pairs (/i, /c) with |Q!ia2l = /^h,k

Thus laf 0:2/^1 = 1 for all a\ot2/^Ji appeared in the right-hand side of this equality. Let e^ = e'^>^ with 0^6^ (M-W)uW.

53

§2.2. Compact Complex Manifolds

Identifying peW-S with p = ^{p)e W-S, we may identify W-S and M-S with W-S and M-S, respectively. Then M is considered to be a complex manifold obtained from M by replacing S by 5.

M=

{M-S)uS.

Figure 3

This method of ^construction is called a surgery or a modification of M If M is compact, M is also compact.

54

2. Complex Manifolds

Example 2.10. Let ^ = ^i/^o be the inhomogeneous coordinates on P^ Putting ^ = 00 for ^0 = 0, we consider P ^ = C u o o . Let M = P^xP^ = {(z,0\zeP\^eP'}, S = OxP'c:M, and W=U,^xp' where U,= { z | | z | < e } . Further let !¥ = L^^ x p \ and 5 = 0xP^ c: W. We denote a point of W by (z, f) where ze U^, and ^ e P ^ Define a biholomorphic map $ of W - 5 onto T ¥ - 5 b y

^:izj)-^(z,n

=

{zj/zn,

where m is a natural number. Put Wi = U^/2XP^ and W^ = L^e/2xP^ Then since 0 ( Wi - 5 ) = ^ i - 5 , replacing 1^ by H^ by surgery, we obtain from M a new manifold M^ = ( M - W ) u H ^ - ( M - 5 ) u 5 . S and 5 are both P\ but M^^ has different complex structure from that of M. Moreover the complex structures of M^ and M„ are different if m T^ n. We shall prove this later. Here we only show that Mi is not homeomorphic to M. Since P^ is homeomorphic to 5^, M = S^ X S^. Hence H2(M, Z) is generated by 2-cycles S = OxS^ and T = S^xO, Therefore any 2-cycle Z on M is homologous to hS + kT with h,keZ. We denote the intersection multiplicity of two cycles Zj and Z2 by / ( Z j , Z2). Since 5 ~ Si = 1 xS^, and S does not intersect with S^ / ( S , 5) = /(S, ^i) = 0. Similarly we have I( T, T) = 0. Since S and T intersect transversally at the unique point 0 x 0 , / ( 5 , T) = / ( r , 5) = 1. Hence I{Z,Z) = 2hk is always even. On M = (M-S)uW, {z,^)eM-S with 0 < | z | < e is identified with (z, f) = (z, z^) G W: Consequently for any teC,

Z, =

{(z,t)eM-S}u{(z,zt)eW}

is a complex submanifold of dimension 1. Since Z^ depends continuously on t, Zt ~ ZQ. On the other hand, Z^ and ZQ intersect transversally at the unique point (0, 0) e W, Hence /(Zo,Zo) = /(Z„Zo) = l, which proves that M and M are not homeomorphic. Example 2.11. Let C = C*/G* be an elliptic curve given in Example 2.7 where G* = {g* | m € Z } . We denote a point C^w on C corresponding to w G C* by [w]. Since g* (w) = a'^w, [a'^w] = [w]. Let M = P'xC,W= U, xC with U,={z\\z\<e}, 5 = 0 x C , \ ^ = L^, x C and S = OxC. We denote a point on M by (z, [w]), and a point on Wby{z, [w]). Define a biholomorphic

§2.2. Compact Complex Manifolds map O of W-S

onto W-S

55 by

0 : ( z , [ w ] ) ^ ( z , [ w ] ) = (z,[zw]). Put W, = U,/2 X C and W, = U,/2 x C. Then since ^( W,-S)=W,obtain by surgery a complex manifold

M = iM-W)u

5, we

W.

M is a Hopf manifold given in Example 2.9. To be precise, letting G be the infinite cyclic group of automorphisms of C^ — {0} generated by g- (zi, Z2) -> (azi, az2), we have M =

(C^-{0})/G.

To see this, put iV = (C^-{0})/G. We want to construct a biholomorphic map of N onto M. We denote by [zj, Z2] the point of N corresponding to (zi, Z2)GC^ —{0}. Since Z1/Z2 is invariant under g,/(zi, Z2) = Z1/Z2 is a meromorphic function on TV, and / : [zu Z2]^z =f{zu zi) = Z1/Z2 is a holomorphic map of N onto P^ = Cuoo. Let Vi = 7V-/~^(0), and V2 =r\U,). Then N = Vi u V2. Since Zi ?^ 0 on Vi, and Z27^ 0 on V2, if we put >l>i:[zi,Z2]^(z,[w]) = (z,[zi]), ^2:[^i,Z2]-^(^,[w]) = (z,[z2]), ^ 1 maps Vi onto M - S, and ^2 maps V2 onto \ ^ biholomorphically. For 0 ( z i , . . . , z j . Then 0 maps M-S biholomorphically onto C"-{0} = M - 5 . Therefore M = (M - 5) u 5 as required.

§2.2. Compact Complex Manifolds

57

We have explained above how we construct M from M = C" by replacing O G M by S = P''~\ Let W, be the ^-neighbourhood of 0 in C" with e>0. Then the above procedure does not affect the complement of W^. Thus we have

M = (M-wju w;,

w, = ^-\w,),

and We is a complex manifold obtained from W^ by replacing OeW^ by

W^ = {W,-{0})uS.

(2.20)

Given an arbitrary complex manifold M " and any point ^ G M", we can construct a new complex manifold M " from M", replacing q e M" by P"~^ as follows. Let p^Zq(p) be local coordinates on M " with centre q, and ^ e ( ^ ) = {p \\^qip)\ < ^} where e is sufficiently small. Then the map/? -^ ^q(p) maps W^iq) biholomorphically onto the ^-neighbourhood W^ of 0 in C", and Zq{q) = 0. Thus identifying W^{q) with W^ via z^, and putting We{q) = We, we obtain from (2.20) WAq) = iWAq)-{q})uS, where S = P"~\ Thus letting M" = ( M " - W e ( ^ ) ) u W e ( ^ ) , we have M«=.(M"-{^})u5

with S = [

as required. Let O be the holomorphic map of We(^) = Wg onto W^{q)= W^ defined above. Extending by putting ^(p)=p for peM""— W^{q), we obtain a holomorphic map of M " onto M which maps S to the point ^ and M^'-J biholomorphically onto M " - g . Thus "HM") = M " and ~^(^) = *S. We denote 0~^ by Qq and call the quadratic transformation with centre q. For example, let M^ = P^, and let (WQ, WJ, W2) be its homogeneous coordinates and ^ = (1,0, 0). Put M^ = Qq(P^). We denote by P ^ the projective line defined by WQ = 0. Then P^ = C ^ u P i „

C^=

L/O = { W G P ^ | W O 7 ^ 0 } .

As in the case of P^ = C u {00}, we call Plo the line at infinity, and any points (0, Wi, W2) on PIO a point at infinity. Let (zi, Z2) = (wi/wo, W2/H'O) be the non-homogeneous coordinates on C^=Uo. Any line ^ = (^0,^1) on C^

58

2. Complex Manifolds

= (1,0,0)

wo = 0

Figure 4

QM)

(?,(p')

*|

ixl — I —

Figure 5

through q = (0, 0) extends to the projective Hne

which passes through (1, 0, 0) and (0, ^o, fi) on P l Since Q^(P^) = Q^(C^) u and Q^(C^) = U^ ^ x f, we have 009

§2.3. Complex Analytic Family

59

Thus Qq(P^) is a submanifold of P^ x P ^ The restriction ^ of the projection P ^ x p U p i to Qq(P^) maps Qg(P^) onto P \ and ^ " H f ) = f x ^ . Qg(g) is a line on Qq(P^) which does not intersect P | - | ^ R e w ^ i 1^1 = 1} is a fundamental

§2.3. Complex Analytic Family

69

domain of T and H^/T is biholomorphic to C. Consequently there is a F-invariant holomorphic function J((o) defined on H^ which induces a biholomorphic map of W^/T onto C. J((o) is called the elliptic modular function. C^ and C^> are biholomorphic if and only if J((o) = J(co'). Thus the complex structure of C^ varies "continuously" as co moves in H^. Example 2.15. By a Hopf surface we mean a Hopf manifold of dimension 2. Let W = C^ — {0}, and g, an automorphism of W given by gt' (Zi,z2)->(azi + rzi,az2), where 0 < |Q:| < 1, and teC. Let G^ = {g^ | ^ e 2} be an infinite cyclic group generated by g,. Then Gt acts on \y in a properly discontinuous manner without fixed point. Put M^ = W/ Gf MQ is a Hopf surface given in Example 2.9. M, for tT^O is also called a Hopf surface. { M J r G C} forms a complex analytic family. In fact, an automorphism of \ ^ X C given by g: (zi, Z2, 0 -> {azi + rz2, az2, t) generates an infinitely cyclic group G, which is properly discontinuous and fixed point free. Hence M= WxC/G is a complex manifold. Since the projection of T^ x C to C commutes with g, it induces a holomorphic map vj of M to C. Clearly the rank of the Jacobian matrix of m is equal to 1. Thus {M, C, m) is a complex analytic family with m~\t) = W/Gt = Mt. Apparently the complex structure of Mf seems to vary as t moves in C, but this is not true. Let U = C-{0}. Then the restriction (Mi^U, XJJJJ) of ( ^ , C, XJT) to L^ is proved to be trivial. This follows immediately from the equality

\0 J \ 0 a/\0

rV

\0 a}'

In fact, introduce new coordinates (wi, W2, 0 = (^1? ^^2? 0 on Wx U. Then in terms of these coordinates, g is represented as g : ( w i , W2, 0 - ^ ( « W i , a:W2+Wi, 0 -

Therefore

Mu= WxU/'^=

W/G,xU = M,xU,

hence (^u, U, xuu) = (Mi x U, U, TT). Thus M^ has the same complex structure as that of Mj for 19^ 0. But the complex structure of MQ is different from that of M, with ^ 7^ 0. To see this, consider holomorphic vector fields on M^. A holomorphic vector field on

70

2. Complex Manifolds

Mt is induced from a Grinvariant holomorphic vector field on W. In what follows we write z' = (zi,Z2) instead of (a"'zi + ma"^~Uz2, a'^Z2) for simplicity. In this notation we have g r : ^ = ( z i , z , ) ^ z ' = (z;,z^). Let v,iz)^+V2iz)^ dZi

(2.30)

dZ2

be a Grinvariant holomorphic vector field on W, where Vi{z) and V2iz) are holomorphic functions on W. Since — =a

—7,

—=ma

dZi

dz[

dZ2

t—-4-a —-, dz\

dZ2

the vector field (2.30) is transformed by gT into the vector field

dZi

dZ2

Since (2.30) is G^invariant, we have Viiz') = V2(z^) =

a'^v,{z)-hma'"''tV2{z),

(2.31)

a-V2(z).

According to the Corollary to Theorem 1.8 (Hartog's theorem), holomorphic functions Vi{zi, Z2) and ^2(^1, ^2) on W are extended to holomorphic functions on all of C^ Therefore we may assume that i^i(zi, Z2) and 1^2(^1, ^2) are holorhorphic on all of C^. By (2.31), we have V2(zi,Z2) =-;;;; V2(a"'z,-^ma'^-'tZ2,a'"z2). a Consequently, since 0 < | a | < 1, letting +00

^2(^1,2:2)=

Z

ChkzUi

h,k = 0

be the power series expansion of ^2(^1, ^2), we have t;2(zi,Z2)= lim A;;l Cn,(a'^z, +

ma'^-'tZ2)'{a-Z2)'

y {^00, ^->+oo \a

\ I

m^+00 a

h,k

,rnt a

§2.3. Complex Analytic Family

71

In order that this Hmit may exist for any Zi, Z2, CQO must be zero, and, if 17^ 0, Cio must also be zero. Thus we have

where Cio = 0 if r 7^ 0. Let +00

1^1(^1,^2)= I

bhkzU2'

h,k = 0

Then by (2.31) we have

1^1(^1,-^2)= lim (—^Vi{a'^z^

+ ma'^'Uz2,a"'z2)

1^2(^1,2:2))

= lim ( - ^ + ^ 1 0 ^ 1 + — ^10^2+^01^2 m^+00 \a = 1™ ( ~ ^ + h ' l O

a

(^io^i + CoiZ2))

a J [^10-Coi]^2+^01^2).

C10U1+

Hence we have 600 = 0, and, if r 7^ 0, we also have ^10 = CQI. Therefore 1^1(^1, 2-2) = 610^1+ ^01^2 holds where 610 = CQI if r T^ 0. To sum u p , if we put c, = 610, ^2 = ^oi, ^3 = ^lo, and c^= CQI, a h o l o m o r p h i c vector field on MQ is given by

CiZi

d

I-C2Z2

dZi

d

f-CsZi

dZi

d

f-C4Z2

dZ2

d

,

dZ2

while a holomorphic vector field on M^ with / 5^ 0 is given by f

d

Ci{ Zx

\

d\ hZ2

azi

^az2/

d +^2^2

•

azi

Consequently there are four linearly independent holomorphic vector fields on Mo, while on M, with t ^ 0, there are only two such ones. Hence M Q has a different complex structure from M^ with tr^O. Thus the complex structure of M^ " j u m p s " at ^ = 0. Example 2.16. Consider deformations of the surface M ^ given in Example 2.10. Here by a surface we mean a compact complex manifold of dimension 2. Write P^ - C u 00 as P^ = «7i u t/2 where (7, = C and t/2 = P^ - { 0 } . Let Zj

72

2. Complex Manifolds

and Z2 be the non-homogeneous coordinates on Ui and U2 respectively. Then on Uin U2, ZiZ2= I holds. In Example 2.10 we have defined the surface M^ as M^ = ( M - 5 ) u W, Since M-S=

W=

U,xP\

f/2 x P \ putting e = 00, we have

M^= U2XP'u

U,xP\

where (zi, ^i)e Ui xp^ and (z2, ^2)^ U2XP^ are the same point on M^ if ZiZ2=l

and

^i = Z2"^2.

(2.32)

Note that here we write C\, C2, and Zj = I/Z2 for 1/^, 1/^, and z respectively in the notation given in Example 2.10. We define a complex analytic family {M^ \teC} with MQ = M^ as follows. Fix a natural number k^ rn/2, and define M^ as M , = ^ i X P ^ u (72XP\

where (zj, fi) e t/j x P ' and (z2, ^2) e (72 xP*^ are the same point of M^ if ZiZ2=l

and

(2.33)

^i = z ^ ^ 2 + ^ ^ 2

Clearly M^ is a surface, and { M j r e C } forms a complex analytic family. For t = 0, MQ= M^ since in this case (2.33) becomes (2.32). For t9^0, Mt = Mm-2k' To see this we introduce new coordinates (z^, ^J) on Ui x P \ / = 1, 2, as follows.

Since the determinants made by the coefficients of the Unear transformations C2 tC\

^^2 ^2 + t

are given by Zi

-t

t

0

= r 7^ 0

and

1

0

rz2"-'

e

= r 7^ 0,

73

§2.3. Complex Analytic Family

respectively, (z„ ^J) actually define coordinates on (7, x P ^ By (2.33), we have

m-2k ^, b2 m-2k _^_^2__fc2___ fe2^^m-fc, . 2 ~ ^ 2 ^2 i^rny,.2k + f ^^2 rZ2 fe2 "I" * ^2

Thus in terms of these new coordinates, the relation (2.33) is given by ^[ =

and

ZiOZ2=l

zr^^Ci

hence, M^ = M^-ikThus for any natural number / c ^ m / 2 , M^ is a deformation o/M^_2fc. Hence by putting k = m/2 if m is even, and k = m/2-\ if m is odd, we see that Mrn is a deformation of Mo = P^ xP^ if m is even, and a deformation of Ml ifm is odd. Therefore by Theorem 2.3, M^ is diffeomorphic to P^ xP^ if m is even and to M^ if m is odd. We have already proved in Example 2.10 that Ml and P^ xP^ are not diffeomorphic. Thus Mrn and M„ are diffeomorphic if m = n (mod 2), but they are not biholomorphic, if m T^ n. Consequently in the family {Mj te C} described above, M^ = ^m-2k does not change its complex structure for all t9^0, and the complex structure of M, jumps to that of MQ = Mrn at t = 0. We show that Mrn is not biholomorphic to M„ if m T^ n, by computing the number of linearly independent holomorphic vector fields on them. First consider holomorphic vector fields on P^ = f/i u U2. A holomorphic vector field on P^ is represented as Vi{zi)(d/dzi) on Ui, V2{z2){d/dz2) on U2, where Vi(zi) are entire functions of z, on L/, for 1 = 1,2. On Uin U2, they must coincide: (2.34)

v,{z,)-^=V2{z2)-^. az2 azi Since Zj = I/Z2, we have d dz2

dzx

d

dz2 dzi

1

d _

Z2 dzx

dzi

Hence, substituting this into (2.34), we obtain Viizi) = -z^iV2i — ]. Therefore, putting Vi{zi)= X Ci„z"

and

1^2(^2)= I C2„Z2,

74

2. Complex Manifolds

we have ^lO"*" ^ 1 1 ^ 1 "^ ^ 1 2 ^ 1 "^ * * * ~

A • • • A dxA I

?p-

1

t " ' " 7 ^ ^ ^ A ^X" A t/x"> A • • • A dx"- =^0.

=- I Z I Thus we obtain

(3.10)

dt/(p = 0.

Next we define the integral of a differential form. For this purpose first we define the orientation of a differentiable manifold. We call a differentiable manifold S orientable if we can choose a system of local C°° coordinates {xi, X2,...} such that on each L^ n L4 T^ 0 , det - i

Aaxf)^./3 = l,...,m

- c i e t j f • ' " Vx>Q-

(3.11)

0, or r = 3. In what follows the indices a, jS, y represent numbers from 1 to m - 1. Define ^ = ^^1

^a^dx-Adx^

by

A • • • A t/z/p A dzf^ A • • • A J z f s

and ^^(P,^) ^ _ _ ^ ^ ^^^.^^ ^^^_^ ^_^ A dzj^ A • • • A t/z/p A t/zf • A • • • A J z f .

Then we have

where d(p^^''^^ is a (;7 4-1, ^)-form, and dcp^^'^^ is a (/?, ^4-l)-form. Hence the invariance of d under the coordinate transformation z^ -> z^ implies those of d and 5. As in (3.6), the coefficients of dcp^^'"^^ are given by (p,q)\

_ v^ /

1 \5 ^^J«o«i"-«s-i«s+r"«p^r

(^